Subgroups — #22574649

Order 8576.

1004 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864147	C₂	1	2	yes	maximal, complemented
2	#16864145	C₂	1	2	yes	maximal, complemented
3	#16864123	C₂	1	1	yes	maximal, characteristic, complemented
4	#16864114	C₂	1	1	yes	maximal, characteristic, complemented
5	#16864107	C₂	1	1	yes	maximal, characteristic, complemented
6	#16864101	C₂	1	1	yes	maximal, characteristic, complemented
7	#16864099	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864090	C₂	1	1	yes	maximal, characteristic, complemented
9	#16864061	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864061	C₂	1	2	yes	maximal, complemented
11	#16864056	C₂	1	2	yes	maximal, complemented
12	#16864047	C₂	1	1	yes	maximal, characteristic, complemented
13	#16864026	C₂	1	1	yes	maximal, characteristic, complemented
14	#16864020	C₂	1	1	yes	maximal, characteristic, complemented
15	#16864017	C₂	1	1	yes	maximal, characteristic, complemented
16	#16864005	C₂	1	2	yes	maximal, complemented
17	#16863994	C₂	1	2	yes	maximal, complemented
18	#16863992	C₂	1	2	yes	maximal, complemented
19	#16863969	C₂	1	2	yes	maximal, complemented
20	C₄⋊Q₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
21	#16863931	C₂	1	1	yes	maximal, characteristic, complemented
22	#16863924	C₂	1	1	yes	maximal, characteristic, complemented
23	#544×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
24	C₂×#11761628	C₂²	1	2	yes	—
25	C₂×Q₈×D₆₇	C₂²	1	2	yes	—
26	C₂×Q₈×D₆₇	C₂²	1	2	yes	—
27	#14524043	C₂²	1	1	yes	characteristic
28	#14524042	C₂²	1	2	yes	—
29	#14524042	C₂²	1	1	yes	characteristic
30	#14524042	C₂²	1	2	yes	—
31	#14524042	C₂²	1	1	yes	characteristic
32	#14524042	C₂²	1	1	yes	characteristic
33	#14524042	C₂²	1	2	yes	—
34	#14524041	C₂²	1	2	yes	—
35	#14524038	C₂²	1	2	yes	—
36	#14524038	C₂²	1	2	yes	—
37	#14524038	C₂²	1	2	yes	—
38	#14524038	C₂²	1	1	yes	characteristic
39	#14524038	C₂²	1	2	yes	—
40	#14524037	C₂²	1	2	yes	—
41	#14524037	C₂²	1	2	yes	—
42	#14524036	C₂²	1	2	yes	—
43	#14524036	C₂²	1	1	yes	characteristic
44	#14524034	C₂²	1	2	yes	—
45	#14524034	C₂²	1	1	yes	characteristic
46	#14524033	C₂²	1	1	yes	characteristic
47	#14524033	C₂²	1	2	yes	—
48	#14524033	C₂²	1	2	yes	—
49	#14524033	C₂²	1	2	yes	—
50	#14524033	C₂²	1	2	yes	—
51	#14524032	C₂²	1	2	yes	—
52	#14524031	C₂²	1	1	yes	characteristic
53	#14524029	C₂²	1	2	yes	—
54	#14524029	C₂²	1	2	yes	—
55	#14524029	C₂²	1	1	yes	characteristic
56	#14524028	C₂²	1	2	yes	—
57	#14524028	C₂²	1	2	yes	—
58	#14524028	C₂²	1	2	yes	—
59	#14524028	C₂²	1	1	yes	characteristic
60	#14524028	C₂²	1	2	yes	—
61	#14524026	C₂²	1	2	yes	—
62	#14524025	C₂²	1	2	yes	—
63	#14524025	C₂²	1	2	yes	—
64	#14524025	C₂²	1	1	yes	characteristic
65	#14524023	C₂²	1	1	yes	characteristic
66	#14524023	C₂²	1	2	yes	—
67	#14524023	C₂²	1	1	yes	characteristic
68	#14524022	C₂²	1	1	yes	characteristic
69	#14524022	C₂²	1	2	yes	—
70	#14524021	C₂²	1	2	yes	—
71	#14524021	C₂²	1	2	yes	—
72	#14524021	C₂²	1	2	yes	—
73	#14524021	C₂²	1	2	yes	—
74	#14524021	C₂²	1	1	yes	characteristic
75	#14524021	C₂²	1	2	yes	—
76	#14524021	C₂²	1	2	yes	—
77	#14524019	C₂²	1	2	yes	—
78	#14524019	C₂²	1	2	yes	—
79	#14524019	C₂²	1	2	yes	—
80	#14524019	C₂²	1	2	yes	—
81	#14524018	C₂²	1	1	yes	characteristic
82	#14524018	C₂²	1	2	yes	—
83	#14524018	C₂²	1	1	yes	characteristic
84	#14524017	C₂²	1	1	yes	characteristic
85	#14524016	C₂²	1	1	yes	characteristic
86	#14524016	C₂²	1	2	yes	—
87	C₄×Dic₁₃₄	C₂²	1	2	yes	—
88	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
89	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
90	#14524008	C₂²	1	2	yes	—
91	#14524008	C₂²	1	1	yes	characteristic
92	#14524008	C₂²	1	1	yes	characteristic
93	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
94	C₄×D₂₆₈	C₂²	1	2	yes	—
95	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
96	#14524005	C₂²	1	1	yes	characteristic
97	#14524005	C₂²	1	1	yes	characteristic
98	#14524005	C₂²	1	2	yes	—
99	#14524005	C₂²	1	2	yes	—
100	#14524005	C₂²	1	2	yes	—
101	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
102	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
103	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
104	#14524001	C₂²	1	2	yes	—
105	#14524001	C₂²	1	1	yes	characteristic
106	#14524000	C₂²	1	1	yes	characteristic
107	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic
108	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic
109	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic
110	C₄⋊C₄×D₆₇	C₂²	1	2	yes	—
111	C₄²×D₆₇	C₂²	1	1	yes	characteristic
112	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
113	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
114	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
115	#125×C₆₇	C₂²	1	2	yes	—
116	#125×C₆₇	C₂²	1	1	yes	characteristic
117	#125×C₆₇	C₂²	1	1	yes	characteristic
118	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
119	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
120	#117×C₆₇	C₂²	1	1	yes	characteristic
121	#117×C₆₇	C₂²	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
123	#11761628	—	4	4	no	—
124	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
125	C₂×Dic₁₃₄	C₂³	1	2	yes	—
126	C₂×Dic₁₃₄	C₂³	1	2	yes	—
127	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
128	C₂×Dic₁₃₄	C₂³	1	2	yes	—
129	C₂×Dic₁₃₄	C₂³	1	2	yes	—
130	Q₈×D₆₇	—	4	4	no	—
131	Q₈×D₆₇	—	4	4	no	—
132	C₂×D₂₆₈	C₂³	1	2	yes	—
133	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
134	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
135	C₂×D₂₆₈	C₂³	1	2	yes	—
136	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
137	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
138	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
139	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
140	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
141	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
142	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
143	C₂×C₄×D₆₇	C₂³	1	2	yes	—
144	C₂×C₄×D₆₇	C₂³	1	2	yes	—
145	C₂×C₄×D₆₇	C₂³	1	2	yes	—
146	C₂×C₄×D₆₇	C₂³	1	2	yes	—
147	C₂×C₄×D₆₇	C₂³	1	2	yes	—
148	C₂×C₄×D₆₇	C₂³	1	2	yes	—
149	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
150	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
151	C₄⋊Dic₆₇	C₂³	1	2	yes	—
152	C₄⋊Dic₆₇	C₂³	1	2	yes	—
153	C₄⋊Dic₆₇	C₂³	1	2	yes	—
154	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
155	C₄⋊Dic₆₇	C₂³	1	2	yes	—
156	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
157	Dic₆₇⋊C₄	C₂³	1	2	yes	—
158	Dic₆₇⋊C₄	C₂³	1	2	yes	—
159	Dic₆₇⋊C₄	C₂³	1	2	yes	—
160	Dic₆₇⋊C₄	C₂³	1	2	yes	—
161	Dic₆₇⋊C₄	C₂³	1	2	yes	—
162	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
163	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
164	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
165	Dic₆₇⋊C₄	C₂³	1	2	yes	—
166	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
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	2	yes	—
171	Dic₆₇⋊C₄	C₂³	1	2	yes	—
172	Dic₆₇⋊C₄	C₂³	1	2	yes	—
173	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
174	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
175	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
176	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
177	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
178	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
179	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
180	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
181	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
182	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
183	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
184	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
185	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
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	2	yes	—
190	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
191	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
192	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
193	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
194	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
195	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
196	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
197	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
198	C₄×Dic₆₇	C₂³	1	2	yes	—
199	C₄×Dic₆₇	C₂³	1	2	yes	—
200	C₄×Dic₆₇	C₂³	1	2	yes	—
201	C₄×Dic₆₇	C₂³	1	2	yes	—
202	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
203	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
204	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
205	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
206	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
207	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
208	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
209	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
210	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
211	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
212	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
213	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
214	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
215	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
216	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
217	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
218	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
219	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
220	C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
221	C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
222	C₄²×C₆₇	C₂³	1	1	yes	characteristic
223	C₄²×C₆₇	C₂³	1	1	yes	characteristic
224	C₄²×C₆₇	C₂³	1	2	yes	—
225	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
226	C₂²×D₆₇	C₂⁴	1	2	yes	—
227	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
228	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
229	Q₈×C₆₇	—	4	2	no	—
230	Q₈×C₆₇	—	4	2	no	—
231	C₂×C₄×C₆₇	—	2	2	no	—
232	C₂×C₄×C₆₇	—	4	2	no	—
233	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
234	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
235	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
236	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
238	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
239	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
240	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
241	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
242	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
243	C₂²⋊D₆₇	—	2	2	no	—
244	C₂²⋊D₆₇	—	2	2	no	—
245	C₂²⋊D₆₇	—	4	2	no	—
246	C₂×Dic₆₇	C₂⁴	1	2	yes	—
247	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
248	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
249	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
250	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
251	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
252	C₂×Dic₆₇	C₂⁴	1	2	yes	—
253	C₂×Dic₆₇	C₂⁴	1	2	yes	—
254	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
255	D₂₆₈	—	2	4	no	—
256	D₂₆₈	—	2	2	no	—
257	D₂₆₈	—	2	2	no	—
258	D₂₆₈	—	4	2	no	—
259	C₄×D₆₇	—	2	2	no	—
260	C₄×D₆₇	—	2	2	no	—
261	C₄×D₆₇	—	2	2	no	—
262	C₄×D₆₇	—	2	2	no	—
263	C₄×D₆₇	—	2	4	no	—
264	C₄×D₆₇	—	4	2	no	—
265	C₄×D₆₇	—	4	2	no	—
266	C₄×D₆₇	—	4	2	no	—
267	C₄×D₆₇	—	4	2	no	—
268	C₄×D₆₇	—	4	2	no	—
269	Dic₁₃₄	—	2	2	no	—
270	Dic₁₃₄	—	2	2	no	—
271	Dic₁₃₄	—	2	4	no	—
272	Dic₁₃₄	—	2	4	no	—
273	Dic₁₃₄	—	4	2	no	—
274	Dic₁₃₄	—	4	2	no	—
275	C₂²×C₆₇	—	2	1	no	—
276	C₂²×C₆₇	—	2	1	no	—
277	C₂²×C₆₇	—	2	1	no	—
278	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
279	C₂×D₆₇	—	2	1	no	—
280	C₂×D₆₇	—	2	1	no	—
281	C₂×D₆₇	—	2	2	no	—
282	C₂×D₆₇	—	2	1	no	—
283	C₂×D₆₇	—	2	1	no	—
284	C₂×D₆₇	—	2	2	no	—
285	C₂×D₆₇	—	2	1	no	—
286	C₂×D₆₇	—	2	2	no	—
287	C₂×D₆₇	2₋¹⁺⁴	1	2	yes	—
288	C₄×C₆₇	—	2	1	no	—
289	C₄×C₆₇	—	2	2	no	—
290	C₄×C₆₇	—	2	2	no	—
291	C₄×C₆₇	—	2	1	no	—
292	C₄×C₆₇	—	2	2	no	—
293	C₄×C₆₇	—	2	1	no	—
294	C₄×C₆₇	—	2	1	no	—
295	C₄×C₆₇	—	2	2	no	—
296	C₄×C₆₇	—	2	1	no	—
297	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
298	Dic₆₇	—	2	1	no	—
299	Dic₆₇	—	2	2	no	—
300	Dic₆₇	—	2	2	no	—
301	Dic₆₇	—	2	2	no	—
302	Dic₆₇	—	2	1	no	—
303	Dic₆₇	—	2	1	no	—
304	Dic₆₇	—	2	1	no	—
305	Dic₆₇	—	2	1	no	—
306	Dic₆₇	H(5,𝔽₂)	1	2	yes	—
307	C₂×C₆₇	—	4	1	no	—
308	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
309	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
310	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
311	D₆₇	—	2	2	no	—
312	D₆₇	—	4	2	no	—
313	D₆₇	—	4	1	no	—
314	#3552	—	67	1	no	maximal, Sylow
315	C₆₇	#3552	1	1	yes	characteristic, complemented, Sylow
316	#563	—	67	2	no	—
317	#563	—	67	2	no	—
318	#557	—	67	1	no	—
319	#556	—	67	1	no	—
320	#554	—	67	1	no	—
321	#554	—	67	1	no	—
322	#553	—	67	1	no	—
323	#553	—	67	1	no	—
324	D₄×Q₈	—	67	1	no	—
325	D₄×Q₈	—	67	2	no	—
326	#548	—	67	2	no	—
327	#547	—	67	1	no	—
328	#544	—	67	1	no	—
329	#544	D₆₇	1	1	yes	characteristic, complemented
330	#542	—	67	1	no	—
331	#542	—	67	1	no	—
332	#541	—	67	2	no	—
333	#539	—	67	2	no	—
334	#539	—	67	2	no	—
335	#536	—	67	2	no	—
336	C₂×C₄⋊Q₈	—	67	1	no	—
337	#529	—	67	1	no	—
338	#529	—	67	1	no	—
339	C₂×#41	—	67	2	no	—
340	C₂²×Q₈	—	67	2	no	—
341	C₂²×Q₈	—	67	2	no	—
342	C₄⋊Q₈	—	67	2	no	—
343	C₄⋊Q₈	—	67	2	no	—
344	C₄⋊Q₈	—	67	1	no	—
345	C₄⋊Q₈	—	67	1	no	—
346	C₄⋊Q₈	—	67	2	no	—
347	C₄⋊Q₈	—	67	1	no	—
348	C₄⋊Q₈	—	67	2	no	—
349	C₄⋊Q₈	—	67	1	no	—
350	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
351	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
352	C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
353	#126	—	67	1	no	—
354	#126	—	67	2	no	—
355	#126	—	67	2	no	—
356	#126	—	67	2	no	—
357	#126	—	67	2	no	—
358	#126	—	67	1	no	—
359	#126	—	67	2	no	—
360	#126	—	67	2	no	—
361	#126	—	67	2	no	—
362	#125	—	67	1	no	—
363	#125	—	67	2	no	—
364	#125	—	67	2	no	—
365	#125	—	67	2	no	—
366	#125	—	67	2	no	—
367	#125	—	67	2	no	—
368	#125	—	67	1	no	—
369	#125	C₂×D₆₇	1	1	yes	characteristic
370	#125	C₂×D₆₇	1	2	yes	—
371	#125	C₂×D₆₇	1	1	yes	characteristic
372	#124	—	67	2	no	—
373	#124	—	67	1	no	—
374	#124	—	67	2	no	—
375	#124	—	67	2	no	—
376	#124	—	67	1	no	—
377	#123	—	67	2	no	—
378	#123	—	67	1	no	—
379	#123	—	67	2	no	—
380	#123	—	67	2	no	—
381	#123	—	67	2	no	—
382	#123	—	67	2	no	—
383	#123	—	67	1	no	—
384	#123	—	67	2	no	—
385	#123	—	67	2	no	—
386	C₂²⋊Q₈	—	67	2	no	—
387	C₂²⋊Q₈	—	67	2	no	—
388	C₂²⋊Q₈	—	67	2	no	—
389	C₂²⋊Q₈	—	67	2	no	—
390	C₂²⋊Q₈	—	67	2	no	—
391	C₂²⋊Q₈	—	67	1	no	—
392	C₂²⋊Q₈	—	67	1	no	—
393	C₂²⋊Q₈	—	67	1	no	—
394	C₂²⋊Q₈	—	67	2	no	—
395	C₂²⋊Q₈	—	67	2	no	—
396	C₂²⋊Q₈	—	67	2	no	—
397	C₂²⋊Q₈	—	67	2	no	—
398	C₂²⋊Q₈	—	67	2	no	—
399	C₂²⋊Q₈	—	67	2	no	—
400	C₂²⋊Q₈	—	67	1	no	—
401	C₂²⋊Q₈	—	67	2	no	—
402	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
403	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	—
404	#121	—	67	1	no	—
405	#121	—	67	1	no	—
406	#121	—	67	2	no	—
407	#121	—	67	1	no	—
408	#121	—	67	1	no	—
409	#121	—	67	2	no	—
410	C₄×Q₈	—	67	1	no	—
411	C₄×Q₈	—	67	2	no	—
412	C₄×Q₈	—	67	1	no	—
413	C₄×Q₈	—	67	1	no	—
414	C₄×Q₈	—	67	2	no	—
415	C₄×Q₈	—	67	1	no	—
416	C₄×D₄	—	67	1	no	—
417	C₄×D₄	—	67	1	no	—
418	C₄×D₄	—	67	2	no	—
419	C₄×D₄	—	67	1	no	—
420	C₄×D₄	—	67	2	no	—
421	C₄×D₄	—	67	2	no	—
422	C₄×D₄	—	67	2	no	—
423	C₄×D₄	—	67	1	no	—
424	C₄×D₄	—	67	1	no	—
425	C₄×D₄	—	67	2	no	—
426	C₄×D₄	—	67	1	no	—
427	#117	—	67	2	no	—
428	#117	—	67	1	no	—
429	#117	—	67	1	no	—
430	#117	C₂×D₆₇	1	1	yes	characteristic
431	#117	C₂×D₆₇	1	1	yes	characteristic
432	C₂×C₄⋊C₄	—	67	2	no	—
433	C₂×C₄⋊C₄	—	67	1	no	—
434	C₂×C₄⋊C₄	—	67	1	no	—
435	C₂×C₄⋊C₄	—	67	1	no	—
436	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
437	C₂×C₄²	—	67	1	no	—
438	#41	—	268	4	no	—
439	C₂×Q₈	—	268	4	no	—
440	C₂×Q₈	—	268	4	no	—
441	C₂×Q₈	—	67	2	no	—
442	C₂×Q₈	—	67	2	no	—
443	C₂×Q₈	—	67	1	no	—
444	C₂×Q₈	—	67	1	no	—
445	C₂×Q₈	—	67	2	no	—
446	C₂×Q₈	—	67	2	no	—
447	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
448	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
449	C₂×D₄	—	67	2	no	—
450	C₂×D₄	—	67	1	no	—
451	C₂×D₄	—	67	1	no	—
452	C₂×D₄	—	67	2	no	—
453	C₂×D₄	—	67	2	no	—
454	C₂×D₄	—	67	1	no	—
455	C₂×D₄	—	67	1	no	—
456	C₂²×C₄	—	67	2	no	—
457	C₂²×C₄	—	67	2	no	—
458	C₂²×C₄	—	67	2	no	—
459	C₂²×C₄	—	67	2	no	—
460	C₂²×C₄	—	67	2	no	—
461	C₂²×C₄	—	67	1	no	—
462	C₂²×C₄	—	67	1	no	—
463	C₂²×C₄	—	67	1	no	—
464	C₂²×C₄	—	67	1	no	—
465	C₂²×C₄	—	67	2	no	—
466	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
467	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
468	C₄⋊C₄	—	67	1	no	—
469	C₄⋊C₄	—	67	1	no	—
470	C₄⋊C₄	—	67	2	no	—
471	C₄⋊C₄	—	67	2	no	—
472	C₄⋊C₄	—	67	2	no	—
473	C₄⋊C₄	—	67	2	no	—
474	C₄⋊C₄	—	67	2	no	—
475	C₄⋊C₄	—	67	2	no	—
476	C₄⋊C₄	—	67	2	no	—
477	C₄⋊C₄	—	67	1	no	—
478	C₄⋊C₄	—	67	1	no	—
479	C₄⋊C₄	—	67	1	no	—
480	C₄⋊C₄	—	67	2	no	—
481	C₄⋊C₄	—	67	2	no	—
482	C₄⋊C₄	—	67	1	no	—
483	C₄⋊C₄	—	67	1	no	—
484	C₄⋊C₄	—	67	1	no	—
485	C₄⋊C₄	—	67	2	no	—
486	C₄⋊C₄	—	67	2	no	—
487	C₄⋊C₄	—	67	2	no	—
488	C₄⋊C₄	—	67	2	no	—
489	C₄⋊C₄	—	67	2	no	—
490	C₄⋊C₄	—	67	1	no	—
491	C₄⋊C₄	—	67	1	no	—
492	C₄⋊C₄	—	67	2	no	—
493	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
494	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
495	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
496	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
497	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
498	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
499	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
500	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
501	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
502	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
503	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
504	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
505	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
506	C₂²⋊C₄	—	67	1	no	—
507	C₂²⋊C₄	—	67	2	no	—
508	C₂²⋊C₄	—	67	1	no	—
509	C₂²⋊C₄	—	67	2	no	—
510	C₂²⋊C₄	—	67	1	no	—
511	C₂²⋊C₄	—	67	2	no	—
512	C₂²⋊C₄	—	67	1	no	—
513	C₂²⋊C₄	—	67	2	no	—
514	C₂²⋊C₄	—	67	2	no	—
515	C₂²⋊C₄	—	67	1	no	—
516	C₂²⋊C₄	—	67	1	no	—
517	C₂²⋊C₄	—	67	2	no	—
518	C₂²⋊C₄	—	67	2	no	—
519	C₂²⋊C₄	—	67	2	no	—
520	C₂²⋊C₄	—	67	2	no	—
521	C₂²⋊C₄	—	67	2	no	—
522	C₂²⋊C₄	—	67	1	no	—
523	C₂²⋊C₄	—	67	2	no	—
524	C₂²⋊C₄	—	67	2	no	—
525	C₂²⋊C₄	—	67	1	no	—
526	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
527	C₂²⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
528	C₂²⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
529	C₄²	—	67	1	no	—
530	C₄²	—	67	1	no	—
531	C₄²	—	67	1	no	—
532	C₄²	—	67	1	no	—
533	C₄²	—	67	2	no	—
534	C₄²	—	67	2	no	—
535	C₄²	—	67	2	no	—
536	C₄²	—	67	2	no	—
537	C₄²	C₂²×D₆₇	1	1	yes	characteristic
538	C₄²	C₂²×D₆₇	1	1	yes	characteristic
539	C₄²	C₂²×D₆₇	1	2	yes	—
540	C₂³	—	67	1	no	—
541	C₂³	—	67	2	no	—
542	C₂³	—	67	1	no	—
543	C₂³	C₂³×D₆₇	1	1	yes	characteristic
544	Q₈	—	4	2	no	—
545	Q₈	—	4	2	no	—
546	Q₈	—	268	2	no	—
547	Q₈	—	268	2	no	—
548	Q₈	—	134	4	no	—
549	Q₈	—	134	2	no	—
550	Q₈	—	134	4	no	—
551	Q₈	—	134	2	no	—
552	D₄	—	268	2	no	—
553	D₄	—	268	2	no	—
554	D₄	—	134	4	no	—
555	D₄	—	134	2	no	—
556	D₄	—	134	2	no	—
557	D₄	—	134	2	no	—
558	D₄	—	134	2	no	—
559	C₂×C₄	—	2	2	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	268	2	no	—
562	C₂×C₄	—	268	2	no	—
563	C₂×C₄	—	268	2	no	—
564	C₂×C₄	—	268	2	no	—
565	C₂×C₄	—	268	2	no	—
566	C₂×C₄	—	67	1	no	—
567	C₂×C₄	—	67	2	no	—
568	C₂×C₄	—	67	1	no	—
569	C₂×C₄	—	67	1	no	—
570	C₂×C₄	—	67	1	no	—
571	C₂×C₄	—	67	1	no	—
572	C₂×C₄	—	67	1	no	—
573	C₂×C₄	—	67	2	no	—
574	C₂×C₄	—	67	2	no	—
575	C₂×C₄	—	134	2	no	—
576	C₂×C₄	—	134	4	no	—
577	C₂×C₄	—	134	2	no	—
578	C₂×C₄	—	134	2	no	—
579	C₂×C₄	—	134	2	no	—
580	C₂×C₄	C₂³×D₆₇	1	2	yes	—
581	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
582	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
583	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
584	C₂×C₄	C₂³×D₆₇	1	2	yes	—
585	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
586	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
587	C₂×C₄	C₂³×D₆₇	1	2	yes	—
588	C₂×C₄	C₂³×D₆₇	1	2	yes	—
589	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
590	C₂²	—	2	1	no	—
591	C₂²	—	2	1	no	—
592	C₂²	—	2	1	no	—
593	C₂²	—	67	2	no	—
594	C₂²	—	134	2	no	—
595	C₂²	—	134	1	no	—
596	C₂²	—	134	2	no	—
597	C₂²	—	134	1	no	—
598	C₂²	—	134	1	no	—
599	C₂²	—	134	2	no	—
600	C₂²	—	134	1	no	—
601	C₂²	—	134	1	no	—
602	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
603	C₄	—	2	2	no	—
604	C₄	—	2	2	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	1	no	—
607	C₄	—	2	1	no	—
608	C₄	—	2	1	no	—
609	C₄	—	2	2	no	—
610	C₄	—	2	1	no	—
611	C₄	—	2	1	no	—
612	C₄	—	67	2	no	—
613	C₄	—	134	2	no	—
614	C₄	—	134	2	no	—
615	C₄	—	134	1	no	—
616	C₄	—	134	1	no	—
617	C₄	—	134	1	no	—
618	C₄	—	134	1	no	—
619	C₄	—	134	1	no	—
620	C₄	—	134	2	no	—
621	C₄	#14524091	1	2	yes	—
622	C₂	—	4	1	no	—
623	C₂	—	268	2	no	—
624	C₂	—	268	1	no	—
625	C₂	—	134	2	no	—
626	C₂	H(5,𝔽₂)×D₆₇	1	1	yes	characteristic
627	C₂	2₋¹⁺⁴×D₆₇	1	1	yes	characteristic
628	C₂	C₂×#14524091	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#544×C₆₇	C₂
hypercenter	#544	D₆₇
hypocenter	C₆₇	#3552
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #544

Complements

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

N	H ≅ G/N	type
1. #16864147	C₂	semidirect
2. #16864145	C₂	semidirect
3. #16864123	C₂	semidirect
4. #16864114	C₂	semidirect
5. #16864107	C₂	semidirect
6. #16864101	C₂	semidirect
7. #16864099	C₂	semidirect
8. #16864090	C₂	semidirect
9. #16864061	C₂	semidirect
10. #16864061	C₂	semidirect
11. #16864056	C₂	semidirect
12. #16864047	C₂	semidirect
13. #16864026	C₂	semidirect
14. #16864020	C₂	semidirect
15. #16864017	C₂	semidirect
16. #16864005	C₂	semidirect
17. #16863994	C₂	semidirect
18. #16863992	C₂	semidirect
19. #16863969	C₂	semidirect
20. C₄⋊Q₈×D₆₇	C₂	semidirect
21. #16863931	C₂	semidirect
22. #16863924	C₂	semidirect
23. #544×C₆₇	C₂	semidirect
315. C₆₇	#3552	semidirect
329. #544	D₆₇	semidirect