Subgroups — #22574152

Order 8576.

1152 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864131	C₂	1	1	yes	maximal, characteristic, complemented
2	#16864099	C₂	1	1	yes	maximal, characteristic, complemented
3	#16864094	C₂	1	2	yes	maximal, complemented
4	#16864084	C₂	1	1	yes	maximal, characteristic, complemented
5	#16864073	C₂	1	2	yes	maximal, complemented
6	#16864073	C₂	1	2	yes	maximal, complemented
7	#16864049	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864041	C₂	1	1	yes	maximal, characteristic, complemented
9	#16864041	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864037	C₂	1	2	yes	maximal, complemented
11	D₄×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864020	C₂	1	1	yes	maximal, characteristic, complemented
13	#16864002	C₂	1	1	yes	maximal, characteristic, complemented
14	#16863995	C₂	1	2	yes	maximal, complemented
15	#16863986	C₂	1	2	yes	maximal, complemented
16	#16863933	C₂	1	1	yes	maximal, characteristic, complemented
17	#16863931	C₂	1	1	yes	maximal, characteristic, complemented
18	#16863909	C₂	1	2	yes	maximal, complemented
19	C₂×#14524028	C₂	1	2	yes	maximal, complemented
20	#16863875	C₂	1	1	yes	maximal, characteristic, complemented
21	#16863868	C₂	1	1	yes	maximal, characteristic, complemented
22	C₄×D₄×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
23	#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11761629	C₂²	1	2	yes	complemented
25	C₂×#11761627	C₂²	1	1	yes	characteristic
26	C₂²×D₂₆₈	C₂²	1	2	yes	—
27	C₂×D₄×D₆₇	C₂²	1	1	yes	characteristic
28	C₂×D₄×D₆₇	C₂²	1	1	yes	characteristic
29	C₂²×C₄×D₆₇	C₂²	1	2	yes	complemented
30	#14524040	C₂²	1	1	yes	characteristic
31	#14524038	C₂²	1	2	yes	complemented
32	#14524038	C₂²	1	1	yes	characteristic, complemented
33	#14524038	C₂²	1	2	yes	complemented
34	#14524038	C₂²	1	2	yes	complemented
35	#14524038	C₂²	1	1	yes	characteristic, complemented
36	#14524038	C₂²	1	2	yes	complemented
37	#14524034	C₂²	1	2	yes	complemented
38	#14524034	C₂²	1	1	yes	characteristic, complemented
39	#14524032	C₂²	1	2	yes	complemented
40	#14524032	C₂²	1	2	yes	complemented
41	#14524032	C₂²	1	1	yes	characteristic, complemented
42	#14524031	C₂²	1	1	yes	characteristic, complemented
43	#14524031	C₂²	1	1	yes	characteristic, complemented
44	#14524028	C₂²	1	2	yes	—
45	#14524028	C₂²	1	1	yes	characteristic
46	#14524028	C₂²	1	2	yes	complemented
47	#14524028	C₂²	1	2	yes	complemented
48	#14524028	C₂²	1	2	yes	complemented
49	#14524028	C₂²	1	2	yes	complemented
50	#14524028	C₂²	1	2	yes	complemented
51	#14524028	C₂²	1	2	yes	complemented
52	#14524028	C₂²	1	2	yes	complemented
53	#14524028	C₂²	1	2	yes	complemented
54	#14524028	C₂²	1	2	yes	complemented
55	#14524028	C₂²	1	1	yes	characteristic, complemented
56	#14524027	C₂²	1	2	yes	complemented
57	#14524027	C₂²	1	2	yes	complemented
58	#14524023	C₂²	1	1	yes	characteristic, complemented
59	#14524023	C₂²	1	1	yes	characteristic, complemented
60	#14524019	C₂²	1	2	yes	complemented
61	#14524019	C₂²	1	2	yes	complemented
62	#14524019	C₂²	1	1	yes	characteristic, complemented
63	#14524018	C₂²	1	1	yes	characteristic
64	#14524018	C₂²	1	1	yes	characteristic
65	#14524018	C₂²	1	1	yes	characteristic
66	#14524018	C₂²	1	2	yes	complemented
67	#14524018	C₂²	1	1	yes	characteristic, complemented
68	#14524018	C₂²	1	1	yes	characteristic, complemented
69	#14524018	C₂²	1	2	yes	complemented
70	#14524018	C₂²	1	2	yes	complemented
71	#14524017	C₂²	1	2	yes	—
72	#14524017	C₂²	1	2	yes	complemented
73	#14524017	C₂²	1	2	yes	complemented
74	#14524013	C₂²	1	2	yes	—
75	#14524013	C₂²	1	2	yes	—
76	#14524013	C₂²	1	2	yes	complemented
77	#14524013	C₂²	1	2	yes	complemented
78	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic, complemented
79	#14524008	C₂²	1	1	yes	characteristic, complemented
80	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
81	C₄×D₂₆₈	C₂²	1	1	yes	characteristic, complemented
82	C₄×D₂₆₈	C₂²	1	1	yes	characteristic, complemented
83	C₄×D₂₆₈	C₂²	1	1	yes	characteristic, complemented
84	C₄×D₂₆₈	C₂²	1	1	yes	characteristic, complemented
85	D₄×Dic₆₇	C₂²	1	1	yes	characteristic
86	#14524005	C₂²	1	1	yes	characteristic, complemented
87	#14524005	C₂²	1	1	yes	characteristic, complemented
88	#14524005	C₂²	1	2	yes	complemented
89	#14524005	C₂²	1	2	yes	complemented
90	#14524005	C₂²	1	2	yes	complemented
91	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	complemented
92	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	complemented
93	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	complemented
94	(C₂²⋊D₆₇)⋊C₄	C₂²	1	2	yes	complemented
95	(C₂²⋊D₆₇)⋊C₄	C₂²	1	2	yes	complemented
96	(C₂²⋊D₆₇)⋊C₄	C₂²	1	2	yes	complemented
97	#14524001	C₂²	1	2	yes	complemented
98	#14524001	C₂²	1	2	yes	complemented
99	#14524001	C₂²	1	1	yes	characteristic, complemented
100	#14524000	C₂²	1	1	yes	characteristic, complemented
101	#14524000	C₂²	1	1	yes	characteristic, complemented
102	#14523999	C₂²	1	2	yes	complemented
103	C₂×Dic₆₇⋊C₄	C₂²	1	2	yes	complemented
104	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic, complemented
105	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic, complemented
106	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	complemented
107	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	complemented
108	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	2	yes	complemented
109	C₂²⋊C₄×D₆₇	C₂²	1	2	yes	complemented
110	C₂²⋊C₄×D₆₇	C₂²	1	2	yes	complemented
111	C₄²×D₆₇	C₂²	1	1	yes	characteristic, complemented
112	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
113	#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
114	#125×C₆₇	C₂²	1	1	yes	characteristic, complemented
115	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
116	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
117	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
118	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
119	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
120	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
121	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
123	C₂³×D₆₇	C₂³	1	2	yes	—
124	C₂³×D₆₇	C₂³	1	2	yes	—
125	#11761629	—	4	4	no	—
126	#11761627	—	2	4	no	—
127	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
128	C₂×D₂₆₈	—	4	4	no	—
129	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
130	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
131	C₂×D₂₆₈	C₂³	1	2	yes	—
132	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
133	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
134	C₂×D₂₆₈	C₂³	1	2	yes	—
135	C₂×D₂₆₈	C₂³	1	2	yes	—
136	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
137	C₂×D₂₆₈	C₂³	1	2	yes	—
138	C₂×D₂₆₈	C₂³	1	2	yes	—
139	C₂×D₂₆₈	C₂³	1	2	yes	—
140	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
141	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
142	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
143	D₄×D₆₇	—	2	4	no	—
144	D₄×D₆₇	—	2	4	no	—
145	C₂²×Dic₆₇	C₂³	1	2	yes	—
146	C₂²×Dic₆₇	C₂³	1	2	yes	—
147	C₂×C₄×D₆₇	—	4	4	no	—
148	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
149	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄×D₆₇	C₂³	1	2	yes	—
151	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
152	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
153	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
154	C₂×C₄×D₆₇	C₂³	1	2	yes	—
155	C₂×C₄×D₆₇	C₂³	1	2	yes	—
156	C₂×C₄×D₆₇	C₂³	1	2	yes	—
157	C₂×C₄×D₆₇	C₂³	1	2	yes	—
158	C₂×C₄×D₆₇	C₂³	1	2	yes	—
159	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
160	C₄⋊Dic₆₇	C₂³	1	2	yes	—
161	C₄⋊Dic₆₇	C₂³	1	2	yes	—
162	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
163	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
164	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
165	Dic₆₇⋊C₄	C₂³	1	2	yes	—
166	Dic₆₇⋊C₄	C₂³	1	2	yes	—
167	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
168	Dic₆₇⋊C₄	C₂³	1	2	yes	—
169	Dic₆₇⋊C₄	C₂³	1	2	yes	—
170	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
171	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
172	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
173	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
174	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
175	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
176	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
177	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
178	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
179	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
180	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
181	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
182	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
183	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
184	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
185	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
186	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
187	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
188	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
189	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
190	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
191	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
192	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
193	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
194	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
195	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
196	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
197	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
198	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
199	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
200	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
201	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
202	C₄×Dic₆₇	C₂³	1	2	yes	—
203	C₄×Dic₆₇	C₂³	1	2	yes	—
204	C₄×Dic₆₇	C₂³	1	2	yes	—
205	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
206	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
207	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
208	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
209	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
210	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
211	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
212	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
213	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
214	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
215	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
216	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
217	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
218	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
219	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
220	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
221	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
222	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
223	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
224	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
225	C₄²×C₆₇	C₂³	1	1	yes	characteristic
226	C₄²×C₆₇	C₂³	1	1	yes	characteristic
227	C₄²×C₆₇	C₂³	1	1	yes	characteristic
228	C₂³×C₆₇	C₂⁴	1	2	yes	—
229	C₂²×D₆₇	—	2	4	no	—
230	C₂²×D₆₇	—	2	4	no	—
231	C₂²×D₆₇	—	2	4	no	—
232	C₂²×D₆₇	—	4	2	no	—
233	C₂²×D₆₇	—	4	2	no	—
234	C₂²×D₆₇	—	4	2	no	—
235	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
236	C₂²×D₆₇	C₂⁴	1	2	yes	—
237	C₂²×D₆₇	C₂⁴	1	2	yes	—
238	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
239	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
240	Q₈×C₆₇	—	4	1	no	—
241	Q₈×C₆₇	—	4	2	no	—
242	D₄×C₆₇	#41	1	4	yes	complemented
243	C₂×C₄×C₆₇	—	4	2	no	—
244	C₂×C₄×C₆₇	—	4	2	no	—
245	C₂×C₄×C₆₇	—	4	2	no	—
246	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
247	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
248	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
249	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
252	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
253	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
254	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
255	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
256	C₂²⋊D₆₇	—	2	4	no	—
257	C₂²⋊D₆₇	—	2	4	no	—
258	C₂²⋊D₆₇	—	2	4	no	—
259	C₂×Dic₆₇	—	2	4	no	—
260	C₂×Dic₆₇	—	4	2	no	—
261	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
262	C₂×Dic₆₇	C₂⁴	1	2	yes	—
263	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
264	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
265	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
266	C₂×Dic₆₇	C₂⁴	1	2	yes	—
267	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
268	D₂₆₈	—	2	2	no	—
269	D₂₆₈	—	2	4	no	—
270	D₂₆₈	—	2	2	no	—
271	D₂₆₈	—	2	2	no	—
272	D₂₆₈	—	2	2	no	—
273	D₂₆₈	—	4	2	no	—
274	D₂₆₈	—	4	2	no	—
275	D₂₆₈	—	4	2	no	—
276	D₂₆₈	—	4	1	no	—
277	D₂₆₈	—	4	2	no	—
278	D₂₆₈	—	4	2	no	—
279	C₄×D₆₇	—	2	2	no	—
280	C₄×D₆₇	—	2	4	no	—
281	C₄×D₆₇	—	2	2	no	—
282	C₄×D₆₇	—	2	2	no	—
283	C₄×D₆₇	—	2	2	no	—
284	C₄×D₆₇	—	4	2	no	—
285	C₄×D₆₇	—	4	2	no	—
286	C₄×D₆₇	—	4	2	no	—
287	C₄×D₆₇	—	4	1	no	—
288	C₄×D₆₇	—	4	2	no	—
289	C₄×D₆₇	—	4	2	no	—
290	Dic₁₃₄	—	2	2	no	—
291	C₂²×C₆₇	—	2	2	no	—
292	C₂²×C₆₇	—	2	2	no	—
293	C₂²×C₆₇	C₂×#41	1	4	yes	—
294	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
295	C₂×D₆₇	—	2	1	no	—
296	C₂×D₆₇	—	2	1	no	—
297	C₂×D₆₇	—	2	1	no	—
298	C₂×D₆₇	—	2	1	no	—
299	C₂×D₆₇	—	2	2	no	—
300	C₂×D₆₇	—	2	2	no	—
301	C₂×D₆₇	—	2	1	no	—
302	C₂×D₆₇	—	2	2	no	—
303	C₂×D₆₇	—	2	2	no	—
304	C₂×D₆₇	—	2	1	no	—
305	C₂×D₆₇	—	2	2	no	—
306	C₂×D₆₇	—	2	1	no	—
307	C₂×D₆₇	—	2	2	no	—
308	C₂×D₆₇	—	2	1	no	—
309	C₂×D₆₇	—	4	8	no	—
310	C₂×D₆₇	—	8	4	no	—
311	C₂×D₆₇	H(5,𝔽₂)	1	2	yes	—
312	C₄×C₆₇	—	2	1	no	—
313	C₄×C₆₇	—	2	2	no	—
314	C₄×C₆₇	—	2	1	no	—
315	C₄×C₆₇	—	2	2	no	—
316	C₄×C₆₇	—	2	1	no	—
317	C₄×C₆₇	—	2	2	no	—
318	C₄×C₆₇	—	2	1	no	—
319	C₄×C₆₇	—	2	1	no	—
320	C₄×C₆₇	—	2	1	no	—
321	C₄×C₆₇	C₂×#41	1	2	yes	—
322	Dic₆₇	—	2	2	no	—
323	Dic₆₇	—	2	1	no	—
324	Dic₆₇	—	2	1	no	—
325	Dic₆₇	—	2	1	no	—
326	Dic₆₇	—	2	2	no	—
327	Dic₆₇	—	2	1	no	—
328	Dic₆₇	—	2	1	no	—
329	C₂×C₆₇	—	2	4	no	—
330	C₂×C₆₇	C₂²×#41	1	1	yes	characteristic
331	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
332	C₂×C₆₇	#585	1	1	yes	characteristic
333	D₆₇	—	2	2	no	—
334	D₆₇	—	4	1	no	—
335	D₆₇	—	4	2	no	—
336	D₆₇	—	4	2	no	—
337	D₆₇	—	4	1	no	—
338	#3513	—	67	1	no	maximal, Sylow
339	C₆₇	#3513	1	1	yes	characteristic, complemented, Sylow
340	#559	—	67	1	no	—
341	#554	D₆₇	1	1	yes	characteristic, complemented
342	#553	—	67	1	no	—
343	#553	—	67	2	no	—
344	#552	—	67	1	no	—
345	#551	—	67	2	no	—
346	#551	—	67	2	no	—
347	#547	—	67	1	no	—
348	#546	—	67	2	no	—
349	#546	—	67	1	no	—
350	#546	—	67	1	no	—
351	D₄²	—	67	1	no	—
352	#542	—	67	1	no	—
353	#540	—	67	1	no	—
354	#539	—	67	2	no	—
355	#538	—	67	2	no	—
356	#529	—	67	1	no	—
357	#529	—	67	1	no	—
358	#525	—	67	2	no	—
359	C₂×#123	—	67	2	no	—
360	#520	—	67	1	no	—
361	#518	—	67	1	no	—
362	C₂×C₄×D₄	—	67	1	no	—
363	C₂×#41	—	67	2	no	—
364	C₂×#41	—	67	1	no	—
365	C₂²×D₄	—	67	1	no	—
366	C₂²×D₄	—	67	1	no	—
367	C₂²×D₄	—	67	2	no	—
368	C₂³×C₄	—	67	2	no	—
369	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
370	#127	—	67	1	no	—
371	#126	—	67	2	no	—
372	#126	—	67	1	no	—
373	#126	—	67	1	no	—
374	#126	—	67	2	no	—
375	#126	—	67	2	no	—
376	#126	—	67	2	no	—
377	#125	—	67	1	no	—
378	#125	—	67	2	no	—
379	#125	C₂×D₆₇	1	1	yes	characteristic, complemented
380	#125	C₂×D₆₇	1	1	yes	characteristic, complemented
381	#124	—	67	1	no	—
382	#124	—	67	2	no	—
383	#124	—	67	2	no	—
384	#124	—	67	1	no	—
385	#124	—	67	1	no	—
386	#123	—	67	1	no	—
387	#123	—	67	2	no	—
388	#123	—	67	2	no	—
389	#123	—	67	2	no	—
390	#123	—	67	2	no	—
391	#123	—	67	2	no	—
392	#123	—	67	2	no	—
393	#123	—	67	2	no	—
394	#123	—	67	1	no	—
395	#123	—	67	2	no	—
396	#123	—	67	2	no	—
397	#123	—	67	2	no	—
398	#123	—	67	2	no	—
399	#123	—	67	2	no	—
400	C₂²⋊Q₈	—	67	2	no	—
401	C₂²⋊Q₈	—	67	1	no	—
402	C₂²⋊Q₈	—	67	2	no	—
403	C₂²⋊Q₈	—	67	1	no	—
404	C₂²⋊Q₈	—	67	1	no	—
405	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	complemented
406	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	complemented
407	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	complemented
408	#121	—	67	2	no	—
409	#121	—	67	2	no	—
410	#121	—	67	2	no	—
411	#121	—	67	2	no	—
412	#121	—	67	1	no	—
413	#121	—	67	1	no	—
414	#121	—	67	2	no	—
415	#121	—	67	1	no	—
416	#121	—	67	2	no	—
417	#121	—	67	1	no	—
418	#121	—	67	1	no	—
419	#120	—	67	2	no	—
420	#120	—	67	2	no	—
421	#120	—	67	2	no	—
422	#120	—	67	2	no	—
423	C₄×Q₈	—	67	1	no	—
424	C₄×Q₈	—	67	1	no	—
425	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
426	C₄×D₄	—	67	1	no	—
427	C₄×D₄	—	67	2	no	—
428	C₄×D₄	—	67	2	no	—
429	C₄×D₄	—	67	2	no	—
430	C₄×D₄	—	67	1	no	—
431	C₄×D₄	—	67	1	no	—
432	C₄×D₄	—	67	2	no	—
433	C₄×D₄	—	67	2	no	—
434	C₄×D₄	—	67	2	no	—
435	C₄×D₄	—	67	2	no	—
436	C₄×D₄	—	67	2	no	—
437	C₄×D₄	—	67	2	no	—
438	C₄×D₄	—	67	1	no	—
439	C₄×D₄	—	67	1	no	—
440	C₄×D₄	—	67	1	no	—
441	C₄×D₄	—	67	1	no	—
442	C₄×D₄	—	67	1	no	—
443	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
444	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
445	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
446	#117	—	67	1	no	—
447	#117	—	67	2	no	—
448	#117	—	67	2	no	—
449	#117	—	67	2	no	—
450	#117	—	67	1	no	—
451	#117	—	67	1	no	—
452	C₂×C₄⋊C₄	—	67	2	no	—
453	C₂×C₄⋊C₄	—	67	1	no	—
454	C₂×C₄⋊C₄	—	67	1	no	—
455	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	complemented
456	C₂×C₂²⋊C₄	—	67	2	no	—
457	C₂×C₂²⋊C₄	—	67	2	no	—
458	C₂×C₂²⋊C₄	—	67	2	no	—
459	C₂×C₂²⋊C₄	—	67	2	no	—
460	C₂×C₂²⋊C₄	—	67	2	no	—
461	C₂×C₄²	—	67	1	no	—
462	C₂⁴	—	67	2	no	—
463	C₂⁴	—	67	2	no	—
464	#41	—	268	4	no	—
465	#41	—	134	4	no	—
466	C₂×Q₈	—	67	1	no	—
467	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
468	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
469	C₂×D₄	—	268	4	no	—
470	C₂×D₄	—	67	2	no	—
471	C₂×D₄	—	67	2	no	—
472	C₂×D₄	—	67	2	no	—
473	C₂×D₄	—	67	1	no	—
474	C₂×D₄	—	67	2	no	—
475	C₂×D₄	—	67	2	no	—
476	C₂×D₄	—	67	2	no	—
477	C₂×D₄	—	67	2	no	—
478	C₂×D₄	—	67	2	no	—
479	C₂×D₄	—	67	1	no	—
480	C₂×D₄	—	67	1	no	—
481	C₂×D₄	—	67	1	no	—
482	C₂×D₄	—	67	2	no	—
483	C₂×D₄	—	67	1	no	—
484	C₂×D₄	—	134	4	no	—
485	C₂×D₄	—	134	4	no	—
486	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
487	C₂²×C₄	—	268	4	no	—
488	C₂²×C₄	—	67	2	no	—
489	C₂²×C₄	—	67	1	no	—
490	C₂²×C₄	—	67	2	no	—
491	C₂²×C₄	—	67	2	no	—
492	C₂²×C₄	—	67	2	no	—
493	C₂²×C₄	—	67	2	no	—
494	C₂²×C₄	—	67	2	no	—
495	C₂²×C₄	—	67	2	no	—
496	C₂²×C₄	—	67	1	no	—
497	C₂²×C₄	—	67	1	no	—
498	C₂²×C₄	—	67	2	no	—
499	C₂²×C₄	—	67	1	no	—
500	C₂²×C₄	—	67	1	no	—
501	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
502	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
503	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
504	C₄⋊C₄	—	67	1	no	—
505	C₄⋊C₄	—	67	2	no	—
506	C₄⋊C₄	—	67	1	no	—
507	C₄⋊C₄	—	67	2	no	—
508	C₄⋊C₄	—	67	2	no	—
509	C₄⋊C₄	—	67	2	no	—
510	C₄⋊C₄	—	67	1	no	—
511	C₄⋊C₄	—	67	1	no	—
512	C₄⋊C₄	—	67	1	no	—
513	C₄⋊C₄	—	67	2	no	—
514	C₄⋊C₄	—	67	2	no	—
515	C₄⋊C₄	—	67	1	no	—
516	C₄⋊C₄	—	67	1	no	—
517	C₄⋊C₄	—	67	1	no	—
518	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
519	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
520	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
521	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
522	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
523	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
524	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
525	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
526	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
527	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
528	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
529	C₂²⋊C₄	—	67	2	no	—
530	C₂²⋊C₄	—	67	2	no	—
531	C₂²⋊C₄	—	67	1	no	—
532	C₂²⋊C₄	—	67	1	no	—
533	C₂²⋊C₄	—	67	1	no	—
534	C₂²⋊C₄	—	67	1	no	—
535	C₂²⋊C₄	—	67	2	no	—
536	C₂²⋊C₄	—	67	2	no	—
537	C₂²⋊C₄	—	67	2	no	—
538	C₂²⋊C₄	—	67	2	no	—
539	C₂²⋊C₄	—	67	2	no	—
540	C₂²⋊C₄	—	67	2	no	—
541	C₂²⋊C₄	—	67	2	no	—
542	C₂²⋊C₄	—	67	2	no	—
543	C₂²⋊C₄	—	67	2	no	—
544	C₂²⋊C₄	—	67	2	no	—
545	C₂²⋊C₄	—	67	1	no	—
546	C₂²⋊C₄	—	67	1	no	—
547	C₂²⋊C₄	—	67	1	no	—
548	C₂²⋊C₄	—	67	2	no	—
549	C₂²⋊C₄	—	67	1	no	—
550	C₂²⋊C₄	—	67	1	no	—
551	C₂²⋊C₄	—	67	2	no	—
552	C₂²⋊C₄	—	67	2	no	—
553	C₂²⋊C₄	—	67	2	no	—
554	C₂²⋊C₄	—	67	1	no	—
555	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
556	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
557	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
558	C₄²	—	67	1	no	—
559	C₄²	—	67	1	no	—
560	C₄²	—	67	1	no	—
561	C₄²	—	67	2	no	—
562	C₄²	—	67	2	no	—
563	C₄²	—	67	2	no	—
564	C₄²	C₂²×D₆₇	1	1	yes	characteristic
565	C₄²	C₂²×D₆₇	1	1	yes	characteristic
566	C₄²	C₂²×D₆₇	1	1	yes	characteristic
567	C₂³	—	268	2	no	—
568	C₂³	—	268	2	no	—
569	C₂³	—	268	2	no	—
570	C₂³	—	67	1	no	—
571	C₂³	—	67	1	no	—
572	C₂³	—	67	2	no	—
573	C₂³	—	67	2	no	—
574	C₂³	—	67	1	no	—
575	C₂³	—	134	4	no	—
576	C₂³	—	134	4	no	—
577	C₂³	—	134	4	no	—
578	C₂³	C₂³×D₆₇	1	2	yes	—
579	Q₈	—	4	2	no	—
580	Q₈	—	4	1	no	—
581	Q₈	—	134	2	no	—
582	D₄	—	268	2	no	—
583	D₄	—	268	2	no	—
584	D₄	—	268	2	no	—
585	D₄	—	268	2	no	—
586	D₄	—	268	1	no	—
587	D₄	—	268	2	no	—
588	D₄	—	134	2	no	—
589	D₄	—	134	4	no	—
590	D₄	—	134	2	no	—
591	D₄	—	134	4	no	—
592	D₄	—	134	4	no	—
593	D₄	—	134	4	no	—
594	D₄	—	134	2	no	—
595	D₄	—	134	2	no	—
596	D₄	#11761629	1	4	yes	complemented
597	C₂×C₄	—	4	2	no	—
598	C₂×C₄	—	4	2	no	—
599	C₂×C₄	—	4	2	no	—
600	C₂×C₄	—	268	2	no	—
601	C₂×C₄	—	268	2	no	—
602	C₂×C₄	—	268	2	no	—
603	C₂×C₄	—	268	2	no	—
604	C₂×C₄	—	268	2	no	—
605	C₂×C₄	—	268	1	no	—
606	C₂×C₄	—	268	2	no	—
607	C₂×C₄	—	67	1	no	—
608	C₂×C₄	—	67	2	no	—
609	C₂×C₄	—	67	1	no	—
610	C₂×C₄	—	67	1	no	—
611	C₂×C₄	—	67	1	no	—
612	C₂×C₄	—	67	2	no	—
613	C₂×C₄	—	67	1	no	—
614	C₂×C₄	—	134	2	no	—
615	C₂×C₄	—	134	4	no	—
616	C₂×C₄	—	134	4	no	—
617	C₂×C₄	—	134	2	no	—
618	C₂×C₄	—	134	2	no	—
619	C₂×C₄	—	134	2	no	—
620	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
621	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
622	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
623	C₂×C₄	C₂³×D₆₇	1	2	yes	—
624	C₂×C₄	C₂³×D₆₇	1	2	yes	—
625	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
626	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
627	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
628	C₂×C₄	C₂³×D₆₇	1	2	yes	—
629	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
630	C₂²	—	2	2	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	268	8	no	—
633	C₂²	—	536	4	no	—
634	C₂²	—	67	2	no	—
635	C₂²	—	134	1	no	—
636	C₂²	—	134	1	no	—
637	C₂²	—	134	2	no	—
638	C₂²	—	134	2	no	—
639	C₂²	—	134	1	no	—
640	C₂²	—	134	2	no	—
641	C₂²	—	134	2	no	—
642	C₂²	—	134	1	no	—
643	C₂²	—	134	2	no	—
644	C₂²	—	134	1	no	—
645	C₂²	—	134	2	no	—
646	C₂²	—	134	1	no	—
647	C₂²	—	134	1	no	—
648	C₂²	—	134	1	no	—
649	C₂²	C₂×#11761629	1	4	yes	—
650	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
651	C₄	—	2	1	no	—
652	C₄	—	2	1	no	—
653	C₄	—	2	2	no	—
654	C₄	—	2	2	no	—
655	C₄	—	2	1	no	—
656	C₄	—	2	1	no	—
657	C₄	—	2	2	no	—
658	C₄	—	2	1	no	—
659	C₄	—	2	1	no	—
660	C₄	—	134	1	no	—
661	C₄	—	134	1	no	—
662	C₄	—	134	1	no	—
663	C₄	—	134	2	no	—
664	C₄	—	134	1	no	—
665	C₄	—	134	2	no	—
666	C₄	—	134	1	no	—
667	C₄	C₂×#11761629	1	2	yes	—
668	C₂	—	2	4	no	—
669	C₂	—	268	2	no	—
670	C₂	—	268	1	no	—
671	C₂	—	268	2	no	—
672	C₂	—	268	1	no	—
673	C₂	—	134	2	no	—
674	C₂	C₂²×#11761629	1	1	yes	characteristic
675	C₂	H(5,𝔽₂)×D₆₇	1	1	yes	characteristic
676	C₂	#16864232	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#554×C₆₇	C₂
hypercenter	#554	D₆₇
hypocenter	C₆₇	#3513
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #554

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #16864131	C₂	semidirect
2. #16864099	C₂	semidirect
3. #16864094	C₂	semidirect
4. #16864084	C₂	semidirect
5. #16864073	C₂	semidirect
6. #16864073	C₂	semidirect
7. #16864049	C₂	semidirect
8. #16864041	C₂	semidirect
9. #16864041	C₂	semidirect
10. #16864037	C₂	semidirect
11. D₄×D₂₆₈	C₂	semidirect
12. #16864020	C₂	semidirect
13. #16864002	C₂	semidirect
14. #16863995	C₂	semidirect
15. #16863986	C₂	semidirect
16. #16863933	C₂	semidirect
17. #16863931	C₂	semidirect
18. #16863909	C₂	semidirect
19. C₂×#14524028	C₂	semidirect
20. #16863875	C₂	semidirect
21. #16863868	C₂	semidirect
22. C₄×D₄×D₆₇	C₂	semidirect
23. #554×C₆₇	C₂	semidirect
24. C₂×#11761629	C₂²	semidirect
29. C₂²×C₄×D₆₇	C₂²	semidirect
31. #14524038	C₂²	semidirect
32. #14524038	C₂²	semidirect
33. #14524038	C₂²	semidirect
34. #14524038	C₂²	semidirect
35. #14524038	C₂²	semidirect
36. #14524038	C₂²	semidirect
37. #14524034	C₂²	semidirect
38. #14524034	C₂²	semidirect
39. #14524032	C₂²	semidirect
40. #14524032	C₂²	semidirect
41. #14524032	C₂²	semidirect
42. #14524031	C₂²	semidirect
43. #14524031	C₂²	semidirect
46. #14524028	C₂²	semidirect
47. #14524028	C₂²	semidirect
48. #14524028	C₂²	semidirect
49. #14524028	C₂²	semidirect
50. #14524028	C₂²	semidirect
51. #14524028	C₂²	semidirect
52. #14524028	C₂²	semidirect
53. #14524028	C₂²	semidirect
54. #14524028	C₂²	semidirect
55. #14524028	C₂²	semidirect
56. #14524027	C₂²	semidirect
57. #14524027	C₂²	semidirect
58. #14524023	C₂²	semidirect
59. #14524023	C₂²	semidirect
60. #14524019	C₂²	semidirect
61. #14524019	C₂²	semidirect
62. #14524019	C₂²	semidirect
66. #14524018	C₂²	semidirect
67. #14524018	C₂²	semidirect
68. #14524018	C₂²	semidirect
69. #14524018	C₂²	semidirect
70. #14524018	C₂²	semidirect
72. #14524017	C₂²	semidirect
73. #14524017	C₂²	semidirect
76. #14524013	C₂²	semidirect
77. #14524013	C₂²	semidirect
78. C₄×Dic₁₃₄	C₂²	semidirect
79. #14524008	C₂²	semidirect
81. C₄×D₂₆₈	C₂²	semidirect
82. C₄×D₂₆₈	C₂²	semidirect
83. C₄×D₂₆₈	C₂²	semidirect
84. C₄×D₂₆₈	C₂²	semidirect
86. #14524005	C₂²	semidirect
87. #14524005	C₂²	semidirect
88. #14524005	C₂²	semidirect
89. #14524005	C₂²	semidirect
90. #14524005	C₂²	semidirect
91. C₄×C₂²⋊D₆₇	C₂²	semidirect
92. C₄×C₂²⋊D₆₇	C₂²	semidirect
93. C₄×C₂²⋊D₆₇	C₂²	semidirect
94. (C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
95. (C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
96. (C₂²⋊D₆₇)⋊C₄	C₂²	semidirect
97. #14524001	C₂²	semidirect
98. #14524001	C₂²	semidirect
99. #14524001	C₂²	semidirect
100. #14524000	C₂²	semidirect
101. #14524000	C₂²	semidirect
102. #14523999	C₂²	semidirect
103. C₂×Dic₆₇⋊C₄	C₂²	semidirect
104. C₄⋊C₄×D₆₇	C₂²	semidirect
105. C₄⋊C₄×D₆₇	C₂²	semidirect
106. C₂×(C₂×D₆₇)⋊C₄	C₂²	semidirect
107. C₂×(C₂×D₆₇)⋊C₄	C₂²	semidirect
108. C₂×(C₂×D₆₇)⋊C₄	C₂²	semidirect
109. C₂²⋊C₄×D₆₇	C₂²	semidirect
110. C₂²⋊C₄×D₆₇	C₂²	semidirect
111. C₄²×D₆₇	C₂²	semidirect
112. C₄⋊Q₈×C₆₇	C₂²	semidirect
113. #125×C₆₇	C₂²	semidirect
114. #125×C₆₇	C₂²	semidirect
115. C₂²⋊Q₈×C₆₇	C₂²	semidirect
116. C₂²⋊Q₈×C₆₇	C₂²	semidirect
117. C₂²⋊Q₈×C₆₇	C₂²	semidirect
118. C₄×Q₈×C₆₇	C₂²	semidirect
122. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
242. D₄×C₆₇	#41	semidirect
339. C₆₇	#3513	semidirect
341. #554	D₆₇	semidirect
369. C₄⋊Q₈	C₂×D₆₇	semidirect
379. #125	C₂×D₆₇	semidirect
380. #125	C₂×D₆₇	semidirect
405. C₂²⋊Q₈	C₂×D₆₇	semidirect
406. C₂²⋊Q₈	C₂×D₆₇	semidirect
407. C₂²⋊Q₈	C₂×D₆₇	semidirect
425. C₄×Q₈	C₂×D₆₇	semidirect
455. C₂×C₄⋊C₄	C₂×D₆₇	semidirect
596. D₄	#11761629	semidirect