Subgroups — #22574816

Order 8576.

912 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇⋊#564	C₂	1	2	yes	maximal, complemented
2	#16864149	C₂	1	2	yes	maximal, complemented
3	#16864123	C₂	1	1	yes	maximal, characteristic, complemented
4	#16864122	C₂	1	1	yes	maximal, characteristic, complemented
5	#16864104	C₂	1	1	yes	maximal, characteristic, complemented
6	#16864101	C₂	1	1	yes	maximal, characteristic, complemented
7	#16864086	C₂	1	1	yes	maximal, characteristic, complemented
8	#16864085	C₂	1	2	yes	maximal, complemented
9	#16864085	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864082	C₂	1	2	yes	maximal, complemented
11	#16864080	C₂	1	1	yes	maximal, characteristic, complemented
12	#16864027	C₂	1	1	yes	maximal, characteristic, complemented
13	#16864026	C₂	1	2	yes	maximal, complemented
14	#16864024	C₂	1	1	yes	maximal, characteristic, complemented
15	#16864008	C₂	1	2	yes	maximal, complemented
16	#16864006	C₂	1	1	yes	maximal, characteristic, complemented
17	#16864006	C₂	1	1	yes	maximal, characteristic, complemented
18	#16864003	C₂	1	2	yes	maximal, complemented
19	#16863989	C₂	1	2	yes	maximal
20	#16863950	C₂	1	1	yes	maximal, characteristic, complemented
21	#16863928	C₂	1	1	yes	maximal, characteristic
22	#125×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
23	#552×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
24	#14524043	C₂²	1	2	yes	—
25	#14524042	C₂²	1	2	yes	—
26	#14524042	C₂²	1	1	yes	characteristic
27	#14524042	C₂²	1	1	yes	characteristic
28	#14524042	C₂²	1	2	yes	—
29	#14524042	C₂²	1	2	yes	—
30	#14524042	C₂²	1	1	yes	characteristic
31	#14524041	C₂²	1	1	yes	characteristic
32	#14524036	C₂²	1	1	yes	characteristic
33	#14524036	C₂²	1	2	yes	—
34	#14524036	C₂²	1	2	yes	—
35	#14524036	C₂²	1	2	yes	—
36	#14524036	C₂²	1	2	yes	—
37	#14524036	C₂²	1	2	yes	—
38	#14524036	C₂²	1	1	yes	characteristic
39	#14524036	C₂²	1	2	yes	—
40	#14524036	C₂²	1	2	yes	—
41	#14524035	C₂²	1	2	yes	—
42	#14524035	C₂²	1	2	yes	—
43	#14524035	C₂²	1	2	yes	—
44	#14524035	C₂²	1	1	yes	characteristic
45	#14524034	C₂²	1	2	yes	—
46	#14524034	C₂²	1	1	yes	characteristic
47	#14524034	C₂²	1	1	yes	characteristic
48	#14524034	C₂²	1	2	yes	—
49	#14524034	C₂²	1	1	yes	characteristic
50	#14524034	C₂²	1	2	yes	—
51	#14524034	C₂²	1	1	yes	characteristic
52	#14524034	C₂²	1	2	yes	—
53	#14524034	C₂²	1	1	yes	characteristic
54	#14524033	C₂²	1	2	yes	—
55	#14524033	C₂²	1	2	yes	—
56	#14524033	C₂²	1	2	yes	—
57	#14524033	C₂²	1	2	yes	—
58	#14524033	C₂²	1	2	yes	—
59	#14524033	C₂²	1	2	yes	—
60	#14524033	C₂²	1	2	yes	—
61	#14524033	C₂²	1	2	yes	—
62	#14524033	C₂²	1	2	yes	—
63	#14524033	C₂²	1	2	yes	—
64	#14524033	C₂²	1	2	yes	—
65	#14524030	C₂²	1	1	yes	characteristic
66	#14524025	C₂²	1	2	yes	—
67	#14524025	C₂²	1	1	yes	characteristic
68	#14524025	C₂²	1	2	yes	—
69	#14524025	C₂²	1	1	yes	characteristic
70	#14524024	C₂²	1	2	yes	—
71	#14524023	C₂²	1	1	yes	characteristic
72	#14524023	C₂²	1	2	yes	—
73	#14524023	C₂²	1	1	yes	characteristic
74	#14524022	C₂²	1	1	yes	characteristic
75	#14524022	C₂²	1	1	yes	characteristic
76	#14524022	C₂²	1	2	yes	—
77	#14524022	C₂²	1	2	yes	—
78	#14524022	C₂²	1	2	yes	—
79	#14524021	C₂²	1	2	yes	—
80	#14524021	C₂²	1	2	yes	—
81	#14524020	C₂²	1	2	yes	—
82	#14524020	C₂²	1	2	yes	—
83	#14524020	C₂²	1	2	yes	—
84	#14524019	C₂²	1	1	yes	characteristic
85	#14524014	C₂²	1	1	yes	characteristic
86	Q₈×Dic₆₇	C₂²	1	1	yes	characteristic
87	Q₈×Dic₆₇	C₂²	1	1	yes	characteristic
88	C₄×Dic₁₃₄	C₂²	1	2	yes	—
89	C₄×Dic₁₃₄	C₂²	1	2	yes	—
90	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
91	C₄×Dic₁₃₄	C₂²	1	1	yes	characteristic
92	#14524008	C₂²	1	2	yes	—
93	#14524008	C₂²	1	1	yes	characteristic
94	#14524008	C₂²	1	1	yes	characteristic
95	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
96	C₄×C₂²⋊D₆₇	C₂²	1	2	yes	—
97	C₄×C₂²⋊D₆₇	C₂²	1	1	yes	characteristic
98	(C₂²⋊D₆₇)⋊C₄	C₂²	1	2	yes	—
99	#14524002	C₂²	1	1	yes	characteristic
100	#14524002	C₂²	1	1	yes	characteristic
101	#14524002	C₂²	1	1	yes	characteristic
102	#14524001	C₂²	1	2	yes	—
103	#14524000	C₂²	1	2	yes	—
104	#14524000	C₂²	1	2	yes	—
105	#14523999	C₂²	1	2	yes	—
106	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
107	C₄⋊C₄×D₆₇	C₂²	1	2	yes	—
108	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic
109	C₄⋊C₄×D₆₇	C₂²	1	2	yes	—
110	C₄⋊C₄×D₆₇	C₂²	1	1	yes	characteristic
111	C₄²×D₆₇	C₂²	1	1	yes	characteristic
112	#126×C₆₇	C₂²	1	2	yes	—
113	#125×C₆₇	C₂²	1	2	yes	—
114	#125×C₆₇	C₂²	1	1	yes	characteristic
115	#123×C₆₇	C₂²	1	2	yes	—
116	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
117	C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
118	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
119	C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
120	#117×C₆₇	C₂²	1	2	yes	—
121	#117×C₆₇	C₂²	1	1	yes	characteristic
122	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
123	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
124	C₂×Dic₁₃₄	C₂³	1	2	yes	—
125	C₂×Dic₁₃₄	C₂³	1	2	yes	—
126	C₂×Dic₁₃₄	C₂³	1	2	yes	—
127	C₂×Dic₁₃₄	C₂³	1	1	yes	characteristic
128	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
129	C₂×C₂²⋊D₆₇	C₂³	1	1	yes	characteristic
130	C₂²×Dic₆₇	C₂³	1	2	yes	—
131	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
132	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
133	C₂×C₄×D₆₇	C₂³	1	2	yes	—
134	C₂×C₄×D₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄×D₆₇	C₂³	1	2	yes	—
136	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
137	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
138	C₄⋊Dic₆₇	C₂³	1	2	yes	—
139	C₄⋊Dic₆₇	C₂³	1	2	yes	—
140	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
141	C₄⋊Dic₆₇	C₂³	1	2	yes	—
142	C₄⋊Dic₆₇	C₂³	1	2	yes	—
143	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
144	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
145	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
146	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
147	C₄⋊Dic₆₇	C₂³	1	1	yes	characteristic
148	C₄⋊Dic₆₇	C₂³	1	2	yes	—
149	C₄⋊Dic₆₇	C₂³	1	2	yes	—
150	C₄⋊Dic₆₇	C₂³	1	2	yes	—
151	C₄⋊Dic₆₇	C₂³	1	2	yes	—
152	Dic₆₇⋊C₄	C₂³	1	2	yes	—
153	Dic₆₇⋊C₄	C₂³	1	2	yes	—
154	Dic₆₇⋊C₄	C₂³	1	2	yes	—
155	Dic₆₇⋊C₄	C₂³	1	2	yes	—
156	Dic₆₇⋊C₄	C₂³	1	2	yes	—
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	2	yes	—
163	Dic₆₇⋊C₄	C₂³	1	2	yes	—
164	Dic₆₇⋊C₄	C₂³	1	2	yes	—
165	Dic₆₇⋊C₄	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
172	Dic₆₇⋊C₄	C₂³	1	2	yes	—
173	Dic₆₇⋊C₄	C₂³	1	2	yes	—
174	Dic₆₇⋊C₄	C₂³	1	2	yes	—
175	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
176	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
177	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
178	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
179	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
180	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
181	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
182	C₂²⋊Dic₆₇	C₂³	1	2	yes	—
183	C₂²⋊Dic₆₇	C₂³	1	1	yes	characteristic
184	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
185	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
186	(C₂×D₆₇)⋊C₄	C₂³	1	2	yes	—
187	(C₂×D₆₇)⋊C₄	C₂³	1	1	yes	characteristic
188	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
189	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
190	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
191	C₄×Dic₆₇	C₂³	1	2	yes	—
192	C₄×Dic₆₇	C₂³	1	2	yes	—
193	C₄×Dic₆₇	C₂³	1	2	yes	—
194	C₄×Dic₆₇	C₂³	1	2	yes	—
195	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
196	C₄×Dic₆₇	C₂³	1	1	yes	characteristic
197	C₄×Dic₆₇	C₂³	1	2	yes	—
198	C₄×Dic₆₇	C₂³	1	2	yes	—
199	C₂×Q₈×C₆₇	C₂³	1	1	yes	characteristic
200	C₂×D₄×C₆₇	C₂³	1	1	yes	characteristic
201	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
202	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
203	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
204	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
205	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
206	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
207	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
208	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
209	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
210	C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
211	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
212	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
213	C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
214	C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
215	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
216	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
217	C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
218	C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
219	C₄²×C₆₇	C₂³	1	2	yes	—
220	C₄²×C₆₇	C₂³	1	2	yes	—
221	C₄²×C₆₇	C₂³	1	1	yes	characteristic
222	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
223	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
224	C₂²×D₆₇	C₂⁴	1	1	yes	characteristic
225	Q₈×C₆₇	—	2	2	no	—
226	D₄×C₆₇	—	4	1	no	—
227	C₂×C₄×C₆₇	—	2	2	no	—
228	C₂×C₄×C₆₇	—	2	2	no	—
229	C₂×C₄×C₆₇	—	4	2	no	—
230	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
231	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
232	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
233	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
234	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
235	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
236	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
237	C₂×C₄×C₆₇	C₂⁴	1	2	yes	—
238	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
239	C₂²⋊D₆₇	—	2	2	no	—
240	C₂²⋊D₆₇	—	2	2	no	—
241	C₂×Dic₆₇	—	4	2	no	—
242	C₂×Dic₆₇	C₂⁴	1	2	yes	—
243	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
244	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
245	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
246	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
247	C₂×Dic₆₇	C₂⁴	1	2	yes	—
248	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
249	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
250	C₂×Dic₆₇	C₂⁴	1	2	yes	—
251	C₂×Dic₆₇	C₂⁴	1	1	yes	characteristic
252	C₂×Dic₆₇	C₂⁴	1	2	yes	—
253	C₄×D₆₇	—	2	2	no	—
254	C₄×D₆₇	—	2	4	no	—
255	C₄×D₆₇	—	2	2	no	—
256	C₄×D₆₇	—	4	1	no	—
257	C₄×D₆₇	—	4	2	no	—
258	Dic₁₃₄	—	2	2	no	—
259	Dic₁₃₄	—	4	1	no	—
260	Dic₁₃₄	—	4	2	no	—
261	Dic₁₃₄	—	4	2	no	—
262	Dic₁₃₄	—	4	2	no	—
263	C₂²×C₆₇	—	2	1	no	—
264	C₂²×C₆₇	—	2	1	no	—
265	C₂²×C₆₇	—	2	1	no	—
266	C₂²×C₆₇	—	2	1	no	—
267	C₂²×C₆₇	—	2	1	no	—
268	C₂²×C₆₇	—	2	1	no	—
269	C₂²×C₆₇	C₂⁵	1	1	yes	characteristic
270	C₂×D₆₇	—	2	1	no	—
271	C₂×D₆₇	—	2	1	no	—
272	C₂×D₆₇	2₋¹⁺⁴	1	2	yes	—
273	C₄×C₆₇	—	2	1	no	—
274	C₄×C₆₇	—	2	2	no	—
275	C₄×C₆₇	—	2	1	no	—
276	C₄×C₆₇	—	2	1	no	—
277	C₄×C₆₇	—	2	2	no	—
278	C₄×C₆₇	—	2	2	no	—
279	C₄×C₆₇	—	2	2	no	—
280	C₄×C₆₇	—	2	1	no	—
281	C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
282	Dic₆₇	—	2	2	no	—
283	Dic₆₇	—	2	1	no	—
284	Dic₆₇	—	2	2	no	—
285	Dic₆₇	—	2	1	no	—
286	Dic₆₇	—	2	1	no	—
287	Dic₆₇	—	2	1	no	—
288	Dic₆₇	—	2	1	no	—
289	Dic₆₇	—	2	1	no	—
290	Dic₆₇	—	2	1	no	—
291	Dic₆₇	—	2	2	no	—
292	Dic₆₇	—	2	2	no	—
293	C₂×C₆₇	—	4	1	no	—
294	C₂×C₆₇	—	4	1	no	—
295	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
296	C₂×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
297	C₂×C₆₇	#585	1	1	yes	characteristic
298	D₆₇	—	2	2	no	—
299	#3563	—	67	1	no	maximal, Sylow
300	C₆₇	#3563	1	1	yes	characteristic, complemented, Sylow
301	#564	—	67	2	no	—
302	#563	—	67	2	no	—
303	#557	—	67	1	no	—
304	#557	—	67	1	no	—
305	#554	—	67	1	no	—
306	#554	—	67	1	no	—
307	#552	—	67	2	no	—
308	#552	—	67	1	no	—
309	#552	—	67	1	no	—
310	#552	—	67	2	no	—
311	#552	—	67	1	no	—
312	#552	D₆₇	1	1	yes	characteristic, complemented
313	#544	—	67	2	no	—
314	#544	—	67	1	no	—
315	#544	—	67	1	no	—
316	#541	—	67	2	no	—
317	#541	—	67	1	no	—
318	#541	—	67	1	no	—
319	#541	—	67	2	no	—
320	#539	—	67	2	no	—
321	#533	—	67	1	no	—
322	#529	—	67	1	no	—
323	C₂×#125	—	67	1	no	—
324	C₄⋊Q₈	—	67	1	no	—
325	C₄⋊Q₈	—	67	1	no	—
326	C₄⋊Q₈	—	67	2	no	—
327	C₄⋊Q₈	—	67	1	no	—
328	C₄⋊Q₈	—	67	1	no	—
329	C₄⋊Q₈	—	67	2	no	—
330	C₄⋊Q₈	—	67	2	no	—
331	C₄⋊Q₈	—	67	2	no	—
332	#126	—	67	1	no	—
333	#126	—	67	1	no	—
334	#126	—	67	2	no	—
335	#126	—	67	2	no	—
336	#126	—	67	2	no	—
337	#126	—	67	2	no	—
338	#126	—	67	2	no	—
339	#126	—	67	2	no	—
340	#126	—	67	2	no	—
341	#126	C₂×D₆₇	1	2	yes	—
342	#125	—	67	2	no	—
343	#125	—	67	1	no	—
344	#125	—	67	1	no	—
345	#125	—	67	2	no	—
346	#125	—	67	1	no	—
347	#125	—	67	2	no	—
348	#125	—	67	2	no	—
349	#125	—	67	2	no	—
350	#125	—	67	2	no	—
351	#125	—	67	2	no	—
352	#125	—	67	2	no	—
353	#125	—	67	2	no	—
354	#125	—	67	2	no	—
355	#125	—	67	2	no	—
356	#125	—	67	2	no	—
357	#125	—	67	2	no	—
358	#125	—	67	1	no	—
359	#125	—	67	1	no	—
360	#125	—	67	2	no	—
361	#125	—	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	C₂×D₆₇	1	1	yes	characteristic
367	#125	C₂×D₆₇	1	2	yes	—
368	#124	—	67	1	no	—
369	#123	—	67	2	no	—
370	#123	—	67	1	no	—
371	#123	—	67	1	no	—
372	#123	—	67	2	no	—
373	#123	—	67	2	no	—
374	#123	C₂×D₆₇	1	2	yes	—
375	C₂²⋊Q₈	—	67	1	no	—
376	C₂²⋊Q₈	—	67	2	no	—
377	C₂²⋊Q₈	—	67	1	no	—
378	C₂²⋊Q₈	—	67	1	no	—
379	C₂²⋊Q₈	—	67	2	no	—
380	C₂²⋊Q₈	—	67	2	no	—
381	C₂²⋊Q₈	—	67	1	no	—
382	C₂²⋊Q₈	—	67	2	no	—
383	C₂²⋊Q₈	—	67	1	no	—
384	C₂²⋊Q₈	—	67	2	no	—
385	C₂²⋊Q₈	—	67	2	no	—
386	C₂²⋊Q₈	—	67	2	no	—
387	C₂²⋊Q₈	—	67	2	no	—
388	C₂²⋊Q₈	—	67	2	no	—
389	C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
390	C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
391	#121	—	67	1	no	—
392	C₄×Q₈	—	67	1	no	—
393	C₄×Q₈	—	67	1	no	—
394	C₄×Q₈	—	67	2	no	—
395	C₄×Q₈	—	67	1	no	—
396	C₄×Q₈	—	67	1	no	—
397	C₄×Q₈	—	67	1	no	—
398	C₄×Q₈	—	67	2	no	—
399	C₄×Q₈	—	67	1	no	—
400	C₄×Q₈	—	67	2	no	—
401	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
402	C₄×D₄	—	67	1	no	—
403	C₄×D₄	—	67	1	no	—
404	C₄×D₄	—	67	2	no	—
405	C₄×D₄	—	67	2	no	—
406	C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
407	#117	—	67	1	no	—
408	#117	—	67	1	no	—
409	#117	—	67	1	no	—
410	#117	—	67	2	no	—
411	#117	—	67	2	no	—
412	#117	—	67	2	no	—
413	#117	—	67	2	no	—
414	#117	C₂×D₆₇	1	2	yes	—
415	#117	C₂×D₆₇	1	1	yes	characteristic
416	C₂×C₄⋊C₄	—	67	1	no	—
417	C₂×C₄⋊C₄	—	67	2	no	—
418	C₂×C₄⋊C₄	—	67	1	no	—
419	C₂×C₄⋊C₄	—	67	2	no	—
420	C₂×C₄⋊C₄	—	67	1	no	—
421	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
422	C₂×C₄²	—	67	1	no	—
423	C₂×Q₈	—	67	2	no	—
424	C₂×Q₈	—	67	2	no	—
425	C₂×Q₈	—	67	2	no	—
426	C₂×Q₈	—	67	1	no	—
427	C₂×Q₈	—	67	1	no	—
428	C₂×Q₈	C₂²×D₆₇	1	1	yes	characteristic
429	C₂×D₄	—	67	1	no	—
430	C₂×D₄	—	67	1	no	—
431	C₂×D₄	C₂²×D₆₇	1	1	yes	characteristic
432	C₂²×C₄	—	67	2	no	—
433	C₂²×C₄	—	67	1	no	—
434	C₂²×C₄	—	67	2	no	—
435	C₂²×C₄	—	67	1	no	—
436	C₂²×C₄	—	67	1	no	—
437	C₂²×C₄	—	67	2	no	—
438	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
439	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
440	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
441	C₄⋊C₄	—	67	1	no	—
442	C₄⋊C₄	—	67	1	no	—
443	C₄⋊C₄	—	67	1	no	—
444	C₄⋊C₄	—	67	2	no	—
445	C₄⋊C₄	—	67	2	no	—
446	C₄⋊C₄	—	67	2	no	—
447	C₄⋊C₄	—	67	2	no	—
448	C₄⋊C₄	—	67	2	no	—
449	C₄⋊C₄	—	67	2	no	—
450	C₄⋊C₄	—	67	2	no	—
451	C₄⋊C₄	—	67	2	no	—
452	C₄⋊C₄	—	67	2	no	—
453	C₄⋊C₄	—	67	2	no	—
454	C₄⋊C₄	—	67	2	no	—
455	C₄⋊C₄	—	67	2	no	—
456	C₄⋊C₄	—	67	2	no	—
457	C₄⋊C₄	—	67	1	no	—
458	C₄⋊C₄	—	67	2	no	—
459	C₄⋊C₄	—	67	1	no	—
460	C₄⋊C₄	—	67	1	no	—
461	C₄⋊C₄	—	67	1	no	—
462	C₄⋊C₄	—	67	2	no	—
463	C₄⋊C₄	—	67	2	no	—
464	C₄⋊C₄	—	67	2	no	—
465	C₄⋊C₄	—	67	1	no	—
466	C₄⋊C₄	—	67	1	no	—
467	C₄⋊C₄	—	67	1	no	—
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	2	no	—
478	C₄⋊C₄	—	67	2	no	—
479	C₄⋊C₄	—	67	2	no	—
480	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
481	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
482	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
483	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
484	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
485	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
486	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
487	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
488	C₄⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
489	C₄⋊C₄	C₂²×D₆₇	1	2	yes	—
490	C₂²⋊C₄	—	67	1	no	—
491	C₂²⋊C₄	—	67	1	no	—
492	C₂²⋊C₄	—	67	2	no	—
493	C₂²⋊C₄	—	67	1	no	—
494	C₂²⋊C₄	—	67	1	no	—
495	C₂²⋊C₄	—	67	2	no	—
496	C₂²⋊C₄	—	67	2	no	—
497	C₂²⋊C₄	—	67	1	no	—
498	C₂²⋊C₄	—	67	1	no	—
499	C₂²⋊C₄	—	67	2	no	—
500	C₂²⋊C₄	—	67	1	no	—
501	C₂²⋊C₄	—	67	1	no	—
502	C₂²⋊C₄	—	67	2	no	—
503	C₂²⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
504	C₂²⋊C₄	C₂²×D₆₇	1	1	yes	characteristic
505	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
506	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
507	C₂²⋊C₄	C₂²×D₆₇	1	2	yes	—
508	C₄²	—	67	1	no	—
509	C₄²	—	67	1	no	—
510	C₄²	—	67	1	no	—
511	C₄²	—	67	2	no	—
512	C₄²	—	67	2	no	—
513	C₄²	—	67	1	no	—
514	C₄²	—	67	1	no	—
515	C₄²	—	67	2	no	—
516	C₄²	—	67	2	no	—
517	C₄²	—	67	2	no	—
518	C₄²	—	67	2	no	—
519	C₄²	C₂²×D₆₇	1	1	yes	characteristic
520	C₄²	C₂²×D₆₇	1	2	yes	—
521	C₄²	C₂²×D₆₇	1	2	yes	—
522	C₂³	—	67	1	no	—
523	C₂³	C₂³×D₆₇	1	1	yes	characteristic
524	C₂³	C₂³×D₆₇	1	1	yes	characteristic
525	Q₈	—	2	2	no	—
526	Q₈	—	268	2	no	—
527	Q₈	—	268	2	no	—
528	Q₈	—	268	2	no	—
529	Q₈	—	268	1	no	—
530	Q₈	—	134	2	no	—
531	D₄	—	4	1	no	—
532	D₄	—	134	2	no	—
533	D₄	—	134	2	no	—
534	C₂×C₄	—	2	2	no	—
535	C₂×C₄	—	2	2	no	—
536	C₂×C₄	—	4	2	no	—
537	C₂×C₄	—	268	2	no	—
538	C₂×C₄	—	268	1	no	—
539	C₂×C₄	—	268	2	no	—
540	C₂×C₄	—	67	2	no	—
541	C₂×C₄	—	67	1	no	—
542	C₂×C₄	—	67	1	no	—
543	C₂×C₄	—	67	1	no	—
544	C₂×C₄	—	67	1	no	—
545	C₂×C₄	—	67	1	no	—
546	C₂×C₄	—	67	2	no	—
547	C₂×C₄	—	67	1	no	—
548	C₂×C₄	—	67	1	no	—
549	C₂×C₄	—	67	2	no	—
550	C₂×C₄	—	67	2	no	—
551	C₂×C₄	—	134	2	no	—
552	C₂×C₄	—	134	4	no	—
553	C₂×C₄	—	134	2	no	—
554	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
555	C₂×C₄	C₂³×D₆₇	1	2	yes	—
556	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
557	C₂×C₄	C₂³×D₆₇	1	2	yes	—
558	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
559	C₂×C₄	C₂³×D₆₇	1	2	yes	—
560	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
561	C₂×C₄	C₂³×D₆₇	1	2	yes	—
562	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
563	C₂²	—	2	1	no	—
564	C₂²	—	2	1	no	—
565	C₂²	—	2	1	no	—
566	C₂²	—	2	1	no	—
567	C₂²	—	2	1	no	—
568	C₂²	—	2	1	no	—
569	C₂²	—	67	2	no	—
570	C₂²	—	134	1	no	—
571	C₂²	—	134	1	no	—
572	C₂²	C₂⁴×D₆₇	1	1	yes	characteristic
573	C₄	—	2	1	no	—
574	C₄	—	2	2	no	—
575	C₄	—	2	1	no	—
576	C₄	—	2	1	no	—
577	C₄	—	2	2	no	—
578	C₄	—	2	2	no	—
579	C₄	—	2	2	no	—
580	C₄	—	2	1	no	—
581	C₄	—	134	1	no	—
582	C₄	—	134	2	no	—
583	C₄	—	134	1	no	—
584	C₄	—	134	1	no	—
585	C₄	—	134	1	no	—
586	C₄	—	134	2	no	—
587	C₄	—	134	2	no	—
588	C₄	—	134	1	no	—
589	C₄	—	134	2	no	—
590	C₄	—	134	1	no	—
591	C₄	—	134	1	no	—
592	C₄	#14524090	1	2	yes	—
593	C₂	—	4	1	no	—
594	C₂	—	4	1	no	—
595	C₂	—	134	2	no	—
596	C₂	2₋¹⁺⁴×D₆₇	1	1	yes	characteristic
597	C₂	C₂×#14524090	1	1	yes	characteristic
598	C₂	#16864232	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₆₇
commutator, socle	C₂²×C₆₇	C₂⁵
Fitting	#552×C₆₇	C₂
hypercenter	#552	D₆₇
hypocenter	C₆₇	#3563
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ #552

Complements

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

N	H ≅ G/N	type
1. C₆₇⋊#564	C₂	semidirect
2. #16864149	C₂	semidirect
3. #16864123	C₂	semidirect
4. #16864122	C₂	semidirect
5. #16864104	C₂	semidirect
6. #16864101	C₂	semidirect
7. #16864086	C₂	semidirect
8. #16864085	C₂	semidirect
9. #16864085	C₂	semidirect
10. #16864082	C₂	semidirect
11. #16864080	C₂	semidirect
12. #16864027	C₂	semidirect
13. #16864026	C₂	semidirect
14. #16864024	C₂	semidirect
15. #16864008	C₂	semidirect
16. #16864006	C₂	semidirect
17. #16864006	C₂	semidirect
18. #16864003	C₂	semidirect
20. #16863950	C₂	semidirect
22. #125×D₆₇	C₂	semidirect
23. #552×C₆₇	C₂	semidirect
300. C₆₇	#3563	semidirect
312. #552	D₆₇	semidirect