Subgroups — #22574006

Order 8576.

1100 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16864107	C₂	1	1	yes	maximal, characteristic, complemented
2	#16864104	C₂	1	4	yes	maximal, complemented
3	#16864103	C₂	1	2	yes	maximal, complemented
4	#16864102	C₂	1	2	yes	maximal, complemented
5	#16864101	C₂	1	2	yes	maximal, complemented
6	Q₈×D₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
7	D₄×Dic₁₃₄	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864059	C₂	1	2	yes	maximal, complemented
9	#16864026	C₂	1	2	yes	maximal, complemented
10	#16864021	C₂	1	4	yes	maximal, complemented
11	#16863951	C₂	1	1	yes	maximal, characteristic, complemented
12	#16863950	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14524023	C₂	1	2	yes	maximal, complemented
14	C₂²⋊Q₈×D₆₇	C₂	1	2	yes	maximal, complemented
15	#16863868	C₂	1	2	yes	maximal
16	C₄×D₄×D₆₇	C₂	1	1	yes	maximal, characteristic
17	#554×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂²×Dic₁₃₄	C₂²	1	2	yes	complemented
19	C₂×Q₈×D₆₇	C₂²	1	2	yes	complemented
20	C₂×D₄×D₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×D₆₇	C₂²	1	2	yes	—
22	#14524043	C₂²	1	2	yes	complemented
23	#14524043	C₂²	1	1	yes	characteristic, complemented
24	#14524042	C₂²	1	4	yes	complemented
25	#14524042	C₂²	1	2	yes	complemented
26	#14524042	C₂²	1	1	yes	characteristic, complemented
27	#14524041	C₂²	1	1	yes	characteristic, complemented
28	#14524034	C₂²	1	2	yes	complemented
29	#14524034	C₂²	1	2	yes	complemented
30	#14524033	C₂²	1	4	yes	complemented
31	#14524033	C₂²	1	2	yes	complemented
32	#14524023	C₂²	1	4	yes	—
33	#14524023	C₂²	1	2	yes	—
34	#14524023	C₂²	1	1	yes	characteristic
35	#14524023	C₂²	1	4	yes	complemented
36	#14524023	C₂²	1	2	yes	complemented
37	#14524023	C₂²	1	2	yes	complemented
38	#14524023	C₂²	1	4	yes	complemented
39	#14524022	C₂²	1	4	yes	complemented
40	#14524022	C₂²	1	2	yes	complemented
41	#14524021	C₂²	1	2	yes	—
42	#14524021	C₂²	1	4	yes	—
43	#14524021	C₂²	1	4	yes	complemented
44	#14524021	C₂²	1	4	yes	complemented
45	#14524020	C₂²	1	4	yes	complemented
46	#14524020	C₂²	1	4	yes	complemented
47	#14524020	C₂²	1	4	yes	complemented
48	#14524019	C₂²	1	2	yes	—
49	#14524019	C₂²	1	1	yes	characteristic
50	#14524019	C₂²	1	4	yes	complemented
51	Q₈×Dic₆₇	C₂²	1	2	yes	complemented
52	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic, complemented
53	C₄×Dic₁₃₄	C₂²	1	2	yes	complemented
54	#14524008	C₂²	1	2	yes	complemented
55	C₄×D₂₆₈	C₂²	1	2	yes	—
56	C₄×D₂₆₈	C₂²	1	1	yes	characteristic
57	D₄×Dic₆₇	C₂²	1	2	yes	—
58	D₄×Dic₆₇	C₂²	1	1	yes	characteristic
59	#14524005	C₂²	1	1	yes	characteristic
60	#14524005	C₂²	1	2	yes	—
61	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
62	C₄×C₂²⋊D₆₇	C₂²	1	4	yes	—
63	(C₂²⋊D₆₇)⋊C₄	C₂²	1	4	yes	—
64	(C₂²⋊D₆₇)⋊C₄	C₂²	1	2	yes	—
65	#14524001	C₂²	1	2	yes	—
66	#14524000	C₂²	1	2	yes	—
67	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	complemented
68	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic
69	C₄⋊C₄×D₆₇	C₂²	1	2	yes	complemented
70	C₄⋊C₄×D₆₇	C₂²	1	4	yes	complemented
71	C₂×(C₂×D₆₇)⋊C₄	C₂²	1	4	yes	—
72	C₂²⋊C₄×D₆₇	C₂²	1	2	yes	—
73	C₂²⋊C₄×D₆₇	C₂²	1	4	yes	—
74	C₄²×D₆₇	C₂²	1	1	yes	characteristic
75	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
76	#125×C₆₇	C₂²	1	2	yes	complemented
77	C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	complemented
78	C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	complemented
79	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic, complemented
80	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
81	C₄×D₄×C₆₇	C₂²	1	2	yes	—
82	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	complemented
83	C₂³×D₆₇	C₂³	1	2	yes	—
84	C₂×Dic₁₃₄	—	4	4	no	—
85	C₂×Dic₁₃₄	C₂³	1	2	yes	—
86	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
87	C₂×Dic₁₃₄	C₂³	1	2	yes	—
88	C₂×Dic₁₃₄	C₂³	1	4	yes	—
89	C₂×Dic₁₃₄	C₂³	1	4	yes	—
90	Q₈×D₆₇	—	4	4	no	—
91	C₂×D₂₆₈	C₂³	1	1	yes	characteristic
92	C₂×C₂²⋊D₆₇	C₂³	1	2	yes	—
93	D₄×D₆₇	Q₈	1	8	yes	complemented
94	C₂²×Dic₆₇	C₂³	1	4	yes	—
95	C₂²×Dic₆₇	C₂³	1	2	yes	—
96	C₂×C₄×D₆₇	—	4	4	no	—
97	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
98	C₂×C₄×D₆₇	C₂³	1	2	yes	—
99	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
100	C₂×C₄×D₆₇	C₂³	1	2	yes	—
101	C₂×C₄×D₆₇	C₂³	1	4	yes	—
102	C₂×C₄×D₆₇	C₂³	1	4	yes	—
103	C₂×C₄×D₆₇	C₂³	1	2	yes	—
104	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
105	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
106	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
107	C₄⋊Dic₆₇	C₂³	1	4	yes	—
108	C₄⋊Dic₆₇	C₂³	1	2	yes	—
109	C₄⋊Dic₆₇	C₂³	1	4	yes	—
110	C₄⋊Dic₆₇	C₂³	1	4	yes	—
111	C₄⋊Dic₆₇	C₂³	1	2	yes	—
112	C₄⋊Dic₆₇	C₂³	1	2	yes	—
113	Dic₆₇⋊C₄	C₂³	1	2	yes	—
114	Dic₆₇⋊C₄	C₂³	1	2	yes	—
115	Dic₆₇⋊C₄	C₂³	1	4	yes	—
116	Dic₆₇⋊C₄	C₂³	1	4	yes	—
117	Dic₆₇⋊C₄	C₂³	1	2	yes	—
118	Dic₆₇⋊C₄	C₂³	1	2	yes	—
119	Dic₆₇⋊C₄	C₂³	1	4	yes	—
120	Dic₆₇⋊C₄	C₂³	1	2	yes	—
121	Dic₆₇⋊C₄	C₂³	1	2	yes	—
122	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
123	C₂²⋊Dic₆₇	C₂³	1	4	yes	—
124	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
125	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
126	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
127	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
128	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
129	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
130	(C₂×D₆₇)⋊C₄	C₂³	1	4	yes	—
131	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
132	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
133	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
134	C₄×Dic₆₇	C₂³	1	2	yes	—
135	C₄×Dic₆₇	C₂³	1	4	yes	—
136	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
137	C₂×Q₈×C₆₇	C₂³	1	2	yes	—
138	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
140	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
141	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
142	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
143	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
144	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
145	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
146	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
147	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
148	C₂²⋊C₄×C₆₇	C₂³	1	4	yes	—
149	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
150	C₄²×C₆₇	C₂³	1	2	yes	—
151	C₄²×C₆₇	C₂³	1	1	yes	characteristic
152	C₂³×C₆₇	C₂⁴	1	2	yes	—
153	C₂²×D₆₇	—	4	2	no	—
154	C₂²×D₆₇	—	4	2	no	—
155	C₂²×D₆₇	C₂×Q₈	1	8	yes	complemented
156	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
157	C₂²×D₆₇	C₂⁴	1	2	yes	—
158	Q₈×C₆₇	—	2	2	no	—
159	Q₈×C₆₇	—	4	2	no	—
160	D₄×C₆₇	C₂×Q₈	1	4	yes	complemented
161	C₂×C₄×C₆₇	—	2	4	no	—
162	C₂×C₄×C₆₇	—	4	4	no	—
163	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
164	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
165	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
166	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
167	C₂×C₄×C₆₇	C₂⁴	1	4	yes	—
168	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
169	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂²⋊D₆₇	C₂×Q₈	1	8	yes	complemented
171	C₂×Dic₆₇	—	4	2	no	—
172	C₂×Dic₆₇	—	4	4	no	—
173	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
174	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
175	C₂×Dic₆₇	C₂⁴	1	2	yes	—
176	C₂×Dic₆₇	C₂⁴	1	4	yes	—
177	C₂×Dic₆₇	C₂⁴	1	2	yes	—
178	C₂×Dic₆₇	C₂⁴	1	2	yes	—
179	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
180	D₂₆₈	C₂×Q₈	1	4	yes	complemented
181	C₄×D₆₇	—	2	2	no	—
182	C₄×D₆₇	—	2	4	no	—
183	C₄×D₆₇	—	4	2	no	—
184	C₄×D₆₇	—	4	2	no	—
185	C₄×D₆₇	—	4	1	no	—
186	C₄×D₆₇	—	4	4	no	—
187	C₄×D₆₇	—	4	4	no	—
188	C₄×D₆₇	C₂×Q₈	1	4	yes	complemented
189	Dic₁₃₄	—	2	4	no	—
190	Dic₁₃₄	—	2	2	no	—
191	Dic₁₃₄	—	4	4	no	—
192	Dic₁₃₄	—	4	4	no	—
193	Dic₁₃₄	—	4	2	no	—
194	C₂²×C₆₇	—	2	2	no	—
195	C₂²×C₆₇	—	2	2	no	—
196	C₂²×C₆₇	C₂²×Q₈	1	4	yes	—
197	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
198	C₂×D₆₇	—	2	2	no	—
199	C₂×D₆₇	—	2	1	no	—
200	C₂×D₆₇	—	2	2	no	—
201	C₂×D₆₇	—	2	1	no	—
202	C₂×D₆₇	—	4	8	no	—
203	C₂×D₆₇	C₂²×Q₈	1	2	yes	—
204	C₂×D₆₇	C₂²×Q₈	1	4	yes	—
205	C₄×C₆₇	—	2	2	no	—
206	C₄×C₆₇	—	2	2	no	—
207	C₄×C₆₇	—	2	2	no	—
208	C₄×C₆₇	—	2	4	no	—
209	C₄×C₆₇	—	2	1	no	—
210	C₄×C₆₇	C₂²×Q₈	1	2	yes	—
211	C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
212	Dic₆₇	—	2	4	no	—
213	Dic₆₇	—	2	2	no	—
214	Dic₆₇	—	2	2	no	—
215	Dic₆₇	—	2	1	no	—
216	Dic₆₇	—	2	1	no	—
217	Dic₆₇	—	2	2	no	—
218	Dic₆₇	C₂²×Q₈	1	2	yes	—
219	C₂×C₆₇	—	2	4	no	—
220	C₂×C₆₇	C₂³×Q₈	1	1	yes	characteristic
221	C₂×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
222	C₂×C₆₇	#585	1	1	yes	characteristic
223	D₆₇	—	2	2	no	—
224	D₆₇	—	2	4	no	—
225	#3501	—	67	1	no	maximal, Sylow
226	C₆₇	#3501	1	1	yes	characteristic, complemented, Sylow
227	#554	—	67	1	no	—
228	#554	—	67	2	no	—
229	#554	—	67	4	no	—
230	#554	—	67	2	no	—
231	#554	—	67	2	no	—
232	#554	D₆₇	1	1	yes	characteristic, complemented
233	D₄×Q₈	—	67	2	no	—
234	D₄×Q₈	—	67	1	no	—
235	D₄×Q₈	—	67	1	no	—
236	#544	—	67	2	no	—
237	#543	—	67	4	no	—
238	#533	—	67	1	no	—
239	#533	—	67	1	no	—
240	C₂×C₂²⋊Q₈	—	67	2	no	—
241	C₂×C₂²⋊Q₈	—	67	2	no	—
242	#518	—	67	2	no	—
243	C₂×C₄×D₄	—	67	1	no	—
244	C₂²×Q₈	—	67	2	no	—
245	C₂²×Q₈	—	67	2	no	—
246	C₂²×D₄	—	67	1	no	—
247	C₂³×C₄	—	67	2	no	—
248	C₄⋊Q₈	—	67	1	no	—
249	C₄⋊Q₈	—	67	1	no	—
250	C₄⋊Q₈	—	67	4	no	—
251	C₄⋊Q₈	—	67	2	no	—
252	C₄⋊Q₈	—	67	1	no	—
253	C₄⋊Q₈	—	67	2	no	—
254	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
255	#125	—	67	2	no	—
256	#125	—	67	2	no	—
257	#125	—	67	4	no	—
258	#125	—	67	2	no	—
259	#125	C₂×D₆₇	1	2	yes	complemented
260	C₂²⋊Q₈	—	67	1	no	—
261	C₂²⋊Q₈	—	67	2	no	—
262	C₂²⋊Q₈	—	67	4	no	—
263	C₂²⋊Q₈	—	67	2	no	—
264	C₂²⋊Q₈	—	67	4	no	—
265	C₂²⋊Q₈	—	67	2	no	—
266	C₂²⋊Q₈	—	67	1	no	—
267	C₂²⋊Q₈	—	67	4	no	—
268	C₂²⋊Q₈	—	67	4	no	—
269	C₂²⋊Q₈	—	67	2	no	—
270	C₂²⋊Q₈	—	67	2	no	—
271	C₂²⋊Q₈	—	67	4	no	—
272	C₂²⋊Q₈	—	67	2	no	—
273	C₂²⋊Q₈	—	67	4	no	—
274	C₂²⋊Q₈	—	67	4	no	—
275	C₂²⋊Q₈	—	67	4	no	—
276	C₂²⋊Q₈	—	67	4	no	—
277	C₂²⋊Q₈	—	67	4	no	—
278	C₂²⋊Q₈	—	67	4	no	—
279	C₂²⋊Q₈	C₂×D₆₇	1	2	yes	complemented
280	C₂²⋊Q₈	C₂×D₆₇	1	4	yes	complemented
281	C₄×Q₈	—	67	2	no	—
282	C₄×Q₈	—	67	2	no	—
283	C₄×Q₈	—	67	1	no	—
284	C₄×Q₈	—	67	2	no	—
285	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic, complemented
286	C₄×D₄	—	67	1	no	—
287	C₄×D₄	—	67	2	no	—
288	C₄×D₄	—	67	2	no	—
289	C₄×D₄	—	67	4	no	—
290	C₄×D₄	—	67	1	no	—
291	C₄×D₄	—	67	2	no	—
292	C₄×D₄	—	67	2	no	—
293	C₄×D₄	—	67	4	no	—
294	C₄×D₄	—	67	1	no	—
295	C₄×D₄	—	67	2	no	—
296	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
297	C₄×D₄	C₂×D₆₇	1	2	yes	—
298	#117	—	67	2	no	—
299	#117	—	67	2	no	—
300	C₂×C₄⋊C₄	—	67	4	no	—
301	C₂×C₄⋊C₄	—	67	1	no	—
302	C₂×C₄⋊C₄	—	67	4	no	—
303	C₂×C₄⋊C₄	—	67	2	no	—
304	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	complemented
305	C₂×C₂²⋊C₄	—	67	4	no	—
306	C₂×C₂²⋊C₄	—	67	2	no	—
307	C₂×C₂²⋊C₄	—	67	4	no	—
308	C₂×C₄²	—	67	1	no	—
309	C₂⁴	—	67	2	no	—
310	C₂×Q₈	—	268	4	no	—
311	C₂×Q₈	—	268	4	no	—
312	C₂×Q₈	—	67	2	no	—
313	C₂×Q₈	—	67	4	no	—
314	C₂×Q₈	—	67	4	no	—
315	C₂×Q₈	—	67	2	no	—
316	C₂×Q₈	—	67	1	no	—
317	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
318	C₂×Q₈	C₂²×D₆₇	1	2	yes	—
319	C₂×D₄	—	67	2	no	—
320	C₂×D₄	—	67	8	no	—
321	C₂×D₄	—	67	1	no	—
322	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
323	C₂²×C₄	—	268	4	no	—
324	C₂²×C₄	—	67	4	no	—
325	C₂²×C₄	—	67	2	no	—
326	C₂²×C₄	—	67	1	no	—
327	C₂²×C₄	—	67	4	no	—
328	C₂²×C₄	—	67	4	no	—
329	C₂²×C₄	—	67	2	no	—
330	C₂²×C₄	—	67	2	no	—
331	C₂²×C₄	—	67	1	no	—
332	C₂²×C₄	—	67	1	no	—
333	C₂²×C₄	—	67	2	no	—
334	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
335	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
336	C₄⋊C₄	—	67	2	no	—
337	C₄⋊C₄	—	67	4	no	—
338	C₄⋊C₄	—	67	2	no	—
339	C₄⋊C₄	—	67	1	no	—
340	C₄⋊C₄	—	67	2	no	—
341	C₄⋊C₄	—	67	1	no	—
342	C₄⋊C₄	—	67	4	no	—
343	C₄⋊C₄	—	67	4	no	—
344	C₄⋊C₄	—	67	4	no	—
345	C₄⋊C₄	—	67	4	no	—
346	C₄⋊C₄	—	67	2	no	—
347	C₄⋊C₄	—	67	2	no	—
348	C₄⋊C₄	—	67	2	no	—
349	C₄⋊C₄	—	67	4	no	—
350	C₄⋊C₄	—	67	2	no	—
351	C₄⋊C₄	—	67	2	no	—
352	C₄⋊C₄	—	67	2	no	—
353	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
354	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
355	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
356	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
357	C₄⋊C₄	C₂²×D₆₇	1	4	yes	—
358	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
359	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
360	C₂²⋊C₄	—	67	2	no	—
361	C₂²⋊C₄	—	67	4	no	—
362	C₂²⋊C₄	—	67	2	no	—
363	C₂²⋊C₄	—	67	4	no	—
364	C₂²⋊C₄	—	67	2	no	—
365	C₂²⋊C₄	—	67	2	no	—
366	C₂²⋊C₄	—	67	4	no	—
367	C₂²⋊C₄	—	67	4	no	—
368	C₂²⋊C₄	—	67	2	no	—
369	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
370	C₂²⋊C₄	C₂²×D₆₇	1	4	yes	—
371	C₄²	—	67	1	no	—
372	C₄²	—	67	1	no	—
373	C₄²	—	67	1	no	—
374	C₄²	—	67	4	no	—
375	C₄²	—	67	2	no	—
376	C₄²	C₂²×D₆₇	1	1	yes	characteristic
377	C₄²	C₂²×D₆₇	1	2	yes	—
378	C₂³	—	268	2	no	—
379	C₂³	—	268	2	no	—
380	C₂³	—	67	8	no	—
381	C₂³	—	67	1	no	—
382	C₂³	—	67	2	no	—
383	C₂³	C₂³×D₆₇	1	2	yes	—
384	Q₈	—	2	2	no	—
385	Q₈	—	4	2	no	—
386	Q₈	—	268	4	no	—
387	Q₈	—	268	4	no	—
388	Q₈	—	268	2	no	—
389	Q₈	—	134	2	no	—
390	Q₈	—	134	4	no	—
391	D₄	—	67	8	no	—
392	D₄	—	67	4	no	—
393	D₄	Q₈×D₆₇	1	4	yes	complemented
394	C₂×C₄	—	2	4	no	—
395	C₂×C₄	—	4	4	no	—
396	C₂×C₄	—	268	2	no	—
397	C₂×C₄	—	268	2	no	—
398	C₂×C₄	—	268	4	no	—
399	C₂×C₄	—	268	4	no	—
400	C₂×C₄	—	268	1	no	—
401	C₂×C₄	—	268	2	no	—
402	C₂×C₄	—	268	4	no	—
403	C₂×C₄	—	67	1	no	—
404	C₂×C₄	—	67	4	no	—
405	C₂×C₄	—	67	1	no	—
406	C₂×C₄	—	67	1	no	—
407	C₂×C₄	—	67	2	no	—
408	C₂×C₄	—	67	4	no	—
409	C₂×C₄	—	67	2	no	—
410	C₂×C₄	—	67	2	no	—
411	C₂×C₄	—	134	2	no	—
412	C₂×C₄	—	134	4	no	—
413	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
414	C₂×C₄	C₂³×D₆₇	1	4	yes	—
415	C₂×C₄	C₂³×D₆₇	1	2	yes	—
416	C₂×C₄	C₂³×D₆₇	1	2	yes	—
417	C₂×C₄	C₂³×D₆₇	1	2	yes	—
418	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
419	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
420	C₂²	—	2	2	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	268	8	no	—
423	C₂²	—	67	2	no	—
424	C₂²	—	67	4	no	—
425	C₂²	—	134	2	no	—
426	C₂²	—	134	1	no	—
427	C₂²	—	134	2	no	—
428	C₂²	—	134	1	no	—
429	C₂²	C₂×Q₈×D₆₇	1	4	yes	—
430	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
431	C₄	—	2	2	no	—
432	C₄	—	2	2	no	—
433	C₄	—	2	2	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	1	no	—
436	C₄	—	67	2	no	—
437	C₄	—	134	1	no	—
438	C₄	—	134	2	no	—
439	C₄	—	134	4	no	—
440	C₄	—	134	2	no	—
441	C₄	—	134	2	no	—
442	C₄	—	134	1	no	—
443	C₄	C₂×Q₈×D₆₇	1	2	yes	—
444	C₄	#14524089	1	2	yes	—
445	C₂	—	2	4	no	—
446	C₂	—	134	2	no	—
447	C₂	—	134	4	no	—
448	C₂	C₂²×Q₈×D₆₇	1	1	yes	characteristic
449	C₂	C₂×#14524089	1	1	yes	characteristic
450	C₂	#16864231	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	#3501
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. #16864107	C₂	semidirect
2. #16864104	C₂	semidirect
3. #16864103	C₂	semidirect
4. #16864102	C₂	semidirect
5. #16864101	C₂	semidirect
6. Q₈×D₂₆₈	C₂	semidirect
7. D₄×Dic₁₃₄	C₂	semidirect
8. #16864059	C₂	semidirect
9. #16864026	C₂	semidirect
10. #16864021	C₂	semidirect
11. #16863951	C₂	semidirect
12. #16863950	C₂	semidirect
13. C₂×#14524023	C₂	semidirect
14. C₂²⋊Q₈×D₆₇	C₂	semidirect
17. #554×C₆₇	C₂	semidirect
18. C₂²×Dic₁₃₄	C₂²	semidirect
19. C₂×Q₈×D₆₇	C₂²	semidirect
22. #14524043	C₂²	semidirect
23. #14524043	C₂²	semidirect
24. #14524042	C₂²	semidirect
25. #14524042	C₂²	semidirect
26. #14524042	C₂²	semidirect
27. #14524041	C₂²	semidirect
28. #14524034	C₂²	semidirect
29. #14524034	C₂²	semidirect
30. #14524033	C₂²	semidirect
31. #14524033	C₂²	semidirect
35. #14524023	C₂²	semidirect
36. #14524023	C₂²	semidirect
37. #14524023	C₂²	semidirect
38. #14524023	C₂²	semidirect
39. #14524022	C₂²	semidirect
40. #14524022	C₂²	semidirect
43. #14524021	C₂²	semidirect
44. #14524021	C₂²	semidirect
45. #14524020	C₂²	semidirect
46. #14524020	C₂²	semidirect
47. #14524020	C₂²	semidirect
50. #14524019	C₂²	semidirect
51. Q₈×Dic₆₇	C₂²	semidirect
52. C₄×Dic₁₃₄	C₂²	semidirect
53. C₄×Dic₁₃₄	C₂²	semidirect
54. #14524008	C₂²	semidirect
67. C₂×C₄⋊Dic₆₇	C₂²	semidirect
69. C₄⋊C₄×D₆₇	C₂²	semidirect
70. C₄⋊C₄×D₆₇	C₂²	semidirect
75. C₄⋊Q₈×C₆₇	C₂²	semidirect
76. #125×C₆₇	C₂²	semidirect
77. C₂²⋊Q₈×C₆₇	C₂²	semidirect
78. C₂²⋊Q₈×C₆₇	C₂²	semidirect
79. C₄×Q₈×C₆₇	C₂²	semidirect
82. C₂×C₄⋊C₄×C₆₇	C₂²	semidirect
93. D₄×D₆₇	Q₈	semidirect
155. C₂²×D₆₇	C₂×Q₈	semidirect
160. D₄×C₆₇	C₂×Q₈	semidirect
170. C₂²⋊D₆₇	C₂×Q₈	semidirect
180. D₂₆₈	C₂×Q₈	semidirect
188. C₄×D₆₇	C₂×Q₈	semidirect
226. C₆₇	#3501	semidirect
232. #554	D₆₇	semidirect
254. C₄⋊Q₈	C₂×D₆₇	semidirect
259. #125	C₂×D₆₇	semidirect
279. C₂²⋊Q₈	C₂×D₆₇	semidirect
280. C₂²⋊Q₈	C₂×D₆₇	semidirect
285. C₄×Q₈	C₂×D₆₇	semidirect
304. C₂×C₄⋊C₄	C₂×D₆₇	semidirect
393. D₄	Q₈×D₆₇	semidirect