Subgroups — #30825826

Order 17152.

700 conjugacy classes of subgroups.

The subgroup lattice has 480 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#22568921	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559088	C₂	1	2	yes	maximal, complemented
4	#22559074	C₂	1	2	yes	maximal, complemented
5	SD₁₆×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
6	D₈×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
7	#22558998	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558648	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558626	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558490	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#3106	C₂	1	1	yes	maximal, characteristic
12	C₂×#14524050	—	2	1	no	—
13	Q₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	Q₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	#16863474	—	2	1	no	—
16	#16863474	C₂²	1	2	yes	—
17	D₄×C₆₇⋊C₈	—	2	1	no	—
18	D₄×C₆₇⋊C₈	—	2	1	no	—
19	D₄×C₆₇⋊C₈	—	2	1	no	—
20	D₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
21	D₄×C₆₇⋊C₈	C₂²	1	2	yes	—
22	D₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	#16863449	—	2	1	no	—
24	#16863449	C₂²	1	1	yes	characteristic
25	#16863420	C₂²	1	1	yes	characteristic
26	#16863363	—	2	1	no	—
27	C₄×#11761612	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	#16863149	C₂²	1	2	yes	—
31	#16863147	C₂²	1	2	yes	—
32	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
33	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
34	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
35	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	#16863118	C₂²	1	2	yes	—
37	C₈×C₆₇⋊C₈	C₂²	1	2	yes	—
38	#532×C₆₇	C₂²	1	1	yes	characteristic
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	#482×C₆₇	C₄	1	2	yes	—
41	#477×C₆₇	C₄	1	1	yes	characteristic
42	#476×C₆₇	C₄	1	1	yes	characteristic
43	#460×C₆₇	C₄	1	1	yes	characteristic
44	#459×C₆₇	C₄	1	1	yes	characteristic
45	#449×C₆₇	C₄	1	2	yes	—
46	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
47	C₄×D₈×C₆₇	C₂²	1	2	yes	—
48	#431×C₆₇	C₂²	1	1	yes	characteristic
49	#14524050	—	4	4	no	—
50	C₂×#11761612	—	2	2	no	—
51	C₂×#11761612	—	2	1	no	—
52	C₂×#11761612	—	2	1	no	—
53	C₂²×C₆₇⋊C₈	—	2	1	no	—
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂²×C₆₇⋊C₈	—	2	2	no	—
56	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
57	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
58	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
59	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
60	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
61	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
62	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
63	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
64	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
65	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
66	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
67	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
68	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
69	#14523954	—	2	1	no	—
70	C₄×C₆₇⋊C₈	—	2	1	no	—
71	C₄×C₆₇⋊C₈	—	2	1	no	—
72	C₄×C₆₇⋊C₈	—	2	1	no	—
73	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
74	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
75	C₄×C₆₇⋊C₈	C₂³	1	2	yes	—
76	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×#41×C₆₇	—	2	1	no	—
79	C₂×#41×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
81	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
82	C₄⋊Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
83	#127×C₆₇	C₂×C₄	1	1	yes	characteristic
84	#124×C₆₇	—	2	1	no	—
85	#121×C₆₇	—	2	1	no	—
86	#121×C₆₇	C₂×C₄	1	2	yes	—
87	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
89	C₄×D₄×C₆₇	—	2	1	no	—
90	C₄×D₄×C₆₇	—	2	1	no	—
91	C₄×D₄×C₆₇	—	2	1	no	—
92	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
93	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×D₄×C₆₇	C₂³	1	2	yes	—
95	#117×C₆₇	—	2	1	no	—
96	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
97	C₂×C₄²×C₆₇	—	2	1	no	—
98	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
99	#107×C₆₇	C₂×C₄	1	2	yes	—
100	#106×C₆₇	C₂×C₄	1	2	yes	—
101	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₄⋊C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
103	#103×C₆₇	C₂×C₄	1	2	yes	—
104	#102×C₆₇	C₂×C₄	1	2	yes	—
105	#102×C₆₇	C₂×C₄	1	2	yes	—
106	#102×C₆₇	C₂×C₄	1	2	yes	—
107	C₂²⋊C₈×C₆₇	C₂×C₄	1	2	yes	—
108	C₄×C₈×C₆₇	C₂³	1	2	yes	—
109	#11761612	—	2	4	no	—
110	#11761612	—	4	4	no	—
111	#11761612	—	4	2	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	2	no	—
114	C₂×C₆₇⋊C₈	—	2	1	no	—
115	C₂×C₆₇⋊C₈	—	2	1	no	—
116	C₂×C₆₇⋊C₈	—	2	2	no	—
117	C₂×C₆₇⋊C₈	—	4	1	no	—
118	C₂×C₆₇⋊C₈	—	4	2	no	—
119	C₂×C₆₇⋊C₈	—	4	2	no	—
120	C₂×C₆₇⋊C₈	—	4	4	no	—
121	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
122	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
125	#41×C₆₇	—	2	2	no	—
126	#41×C₆₇	—	2	2	no	—
127	#41×C₆₇	—	4	2	no	—
128	#41×C₆₇	—	4	2	no	—
129	C₂×Q₈×C₆₇	—	2	1	no	—
130	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂×D₄×C₆₇	—	2	1	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	—	2	1	no	—
134	C₂×D₄×C₆₇	—	2	1	no	—
135	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
136	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	1	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
142	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
144	SD₁₆×C₆₇	—	2	4	no	—
145	D₈×C₆₇	—	2	4	no	—
146	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
147	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
148	C₄⋊C₄×C₆₇	—	2	1	no	—
149	C₄⋊C₄×C₆₇	—	2	1	no	—
150	C₄⋊C₄×C₆₇	—	2	1	no	—
151	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
153	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
154	C₂²⋊C₄×C₆₇	—	2	1	no	—
155	C₂²⋊C₄×C₆₇	—	2	1	no	—
156	C₂²⋊C₄×C₆₇	—	2	2	no	—
157	C₂²⋊C₄×C₆₇	—	2	1	no	—
158	C₂²⋊C₄×C₆₇	—	2	2	no	—
159	C₄²×C₆₇	—	2	1	no	—
160	C₄²×C₆₇	—	2	1	no	—
161	C₄²×C₆₇	—	2	1	no	—
162	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
164	C₄²×C₆₇	#41	1	1	yes	characteristic
165	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
166	C₂³×C₆₇	—	2	1	no	—
167	C₂³×C₆₇	—	2	2	no	—
168	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
169	Q₈×C₆₇	—	2	1	no	—
170	Q₈×C₆₇	—	4	2	no	—
171	Q₈×C₆₇	#131	1	2	yes	complemented
172	D₄×C₆₇	—	2	4	no	—
173	D₄×C₆₇	—	2	1	no	—
174	D₄×C₆₇	—	2	2	no	—
175	D₄×C₆₇	—	2	2	no	—
176	D₄×C₆₇	—	4	2	no	—
177	D₄×C₆₇	—	4	4	no	—
178	D₄×C₆₇	—	4	2	no	—
179	D₄×C₆₇	#131	1	2	yes	complemented
180	C₂×C₄×C₆₇	—	2	4	no	—
181	C₂×C₄×C₆₇	—	2	1	no	—
182	C₂×C₄×C₆₇	—	2	1	no	—
183	C₂×C₄×C₆₇	—	2	1	no	—
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	1	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	1	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
196	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
197	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
198	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
199	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
201	C₈×C₆₇	—	2	2	no	—
202	C₈×C₆₇	—	2	2	no	—
203	C₆₇⋊C₈	—	2	4	no	—
204	C₆₇⋊C₈	—	2	1	no	—
205	C₆₇⋊C₈	—	2	1	no	—
206	C₆₇⋊C₈	—	2	1	no	—
207	C₆₇⋊C₈	—	4	1	no	—
208	C₆₇⋊C₈	—	4	2	no	—
209	C₆₇⋊C₈	—	4	1	no	—
210	C₆₇⋊C₈	AGL(1,ℤ/8)	1	2	yes	—
211	C₂²×C₆₇	—	2	1	no	—
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	2	1	no	—
214	C₂²×C₆₇	—	2	1	no	—
215	C₂²×C₆₇	—	4	1	no	—
216	C₂²×C₆₇	—	4	1	no	—
217	C₂²×C₆₇	—	4	2	no	—
218	C₂²×C₆₇	—	4	2	no	—
219	C₂²×C₆₇	—	4	2	no	—
220	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
221	C₄×C₆₇	—	2	1	no	—
222	C₄×C₆₇	—	2	2	no	—
223	C₄×C₆₇	—	2	1	no	—
224	C₄×C₆₇	—	2	1	no	—
225	C₄×C₆₇	—	4	1	no	—
226	C₄×C₆₇	—	4	2	no	—
227	C₄×C₆₇	—	4	1	no	—
228	C₄×C₆₇	#443	1	2	yes	—
229	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
230	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
231	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
232	C₄×C₆₇	#575	1	1	yes	characteristic
233	#17912	—	67	1	no	maximal, Sylow
234	C₂×C₆₇	—	4	1	no	—
235	C₂×C₆₇	—	4	2	no	—
236	C₂×C₆₇	—	8	2	no	—
237	C₂×C₆₇	#2929	1	1	yes	characteristic
238	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₂×C₆₇	C₂×#443	1	1	yes	characteristic
240	#3106	D₆₇	1	1	yes	characteristic
241	C₈×#41	—	67	1	no	—
242	#2931	—	67	1	no	—
243	#1588	—	67	2	no	—
244	#1586	—	67	2	no	—
245	C₈×SD₁₆	—	67	2	no	—
246	C₈×D₈	—	67	2	no	—
247	#1574	—	67	1	no	—
248	#1489	—	67	1	no	—
249	#1484	—	67	1	no	—
250	C₈×M₁₆	—	67	1	no	—
251	C₆₇	#17912	1	1	yes	characteristic, complemented, Sylow
252	C₂×#131	—	134	1	no	—
253	#532	C₂×D₆₇	1	1	yes	characteristic
254	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
255	#482	Dic₆₇	1	2	yes	—
256	#477	Dic₆₇	1	1	yes	characteristic
257	#476	Dic₆₇	1	1	yes	characteristic
258	#460	Dic₆₇	1	1	yes	characteristic
259	#459	Dic₆₇	1	1	yes	characteristic
260	#449	Dic₆₇	1	2	yes	—
261	C₈×Q₈	—	67	1	no	—
262	C₈×Q₈	—	67	1	no	—
263	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
264	C₄×D₈	C₂×D₆₇	1	2	yes	—
265	#436	—	67	2	no	—
266	#436	—	134	1	no	—
267	C₈×D₄	—	67	1	no	—
268	C₈×D₄	—	67	2	no	—
269	C₈×D₄	—	67	1	no	—
270	C₈×D₄	—	134	1	no	—
271	C₈×D₄	—	134	1	no	—
272	C₈×D₄	—	134	1	no	—
273	#431	—	67	1	no	—
274	#431	—	134	1	no	—
275	#431	C₂×D₆₇	1	1	yes	characteristic
276	#423	—	67	1	no	—
277	#408	—	134	1	no	—
278	C₄×M₁₆	—	67	1	no	—
279	C₂×C₄×C₈	—	67	1	no	—
280	C₂×C₄×C₈	—	134	1	no	—
281	#335	—	67	2	no	—
282	#334	—	67	2	no	—
283	Q₈⋊C₈	—	67	2	no	—
284	D₄⋊C₈	—	67	2	no	—
285	D₄⋊C₈	—	67	2	no	—
286	D₄⋊C₈	—	67	2	no	—
287	#322	—	67	2	no	—
288	C₈²	—	67	2	no	—
289	C₂×#41	—	2	1	no	—
290	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
291	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
292	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
293	#131	—	268	4	no	—
294	C₂×M₁₆	—	134	2	no	—
295	C₂×M₁₆	—	134	1	no	—
296	C₂×M₁₆	—	134	1	no	—
297	C₂²×C₈	—	67	1	no	—
298	C₂²×C₈	—	67	1	no	—
299	C₂²×C₈	—	134	2	no	—
300	C₂²×C₈	—	134	1	no	—
301	C₂²×C₈	—	134	1	no	—
302	C₄⋊Q₈	C₂×Dic₆₇	1	1	yes	characteristic
303	#127	C₂×Dic₆₇	1	1	yes	characteristic
304	#124	—	2	1	no	—
305	#121	—	2	1	no	—
306	#121	C₂×Dic₆₇	1	2	yes	—
307	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
308	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
309	C₄×D₄	—	2	1	no	—
310	C₄×D₄	—	2	1	no	—
311	C₄×D₄	—	2	1	no	—
312	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
313	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
314	C₄×D₄	C₂²×D₆₇	1	2	yes	—
315	#117	—	2	1	no	—
316	#117	C₂×Dic₆₇	1	1	yes	characteristic
317	C₂×C₄²	—	2	1	no	—
318	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
319	#107	C₂×Dic₆₇	1	2	yes	—
320	#106	C₂×Dic₆₇	1	2	yes	—
321	C₄⋊C₈	—	67	1	no	—
322	C₄⋊C₈	—	67	1	no	—
323	C₄⋊C₈	—	67	2	no	—
324	C₄⋊C₈	—	134	1	no	—
325	C₄⋊C₈	—	134	1	no	—
326	C₄⋊C₈	—	134	1	no	—
327	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
328	C₄⋊C₈	C₂×Dic₆₇	1	1	yes	characteristic
329	#103	C₂×Dic₆₇	1	2	yes	—
330	#102	C₂×Dic₆₇	1	2	yes	—
331	#102	C₂×Dic₆₇	1	2	yes	—
332	#102	C₂×Dic₆₇	1	2	yes	—
333	C₂²⋊C₈	—	134	2	no	—
334	C₂²⋊C₈	—	134	2	no	—
335	C₂²⋊C₈	—	134	1	no	—
336	C₂²⋊C₈	—	134	1	no	—
337	C₂²⋊C₈	—	134	1	no	—
338	C₂²⋊C₈	C₂×Dic₆₇	1	2	yes	—
339	#97	—	134	1	no	—
340	C₄×C₈	—	67	1	no	—
341	C₄×C₈	—	67	1	no	—
342	C₄×C₈	—	67	1	no	—
343	C₄×C₈	—	67	1	no	—
344	C₄×C₈	—	67	2	no	—
345	C₄×C₈	—	134	1	no	—
346	C₄×C₈	—	134	1	no	—
347	C₄×C₈	—	134	1	no	—
348	C₄×C₈	C₂²×D₆₇	1	2	yes	—
349	#41	—	2	2	no	—
350	#41	—	2	2	no	—
351	#41	—	4	2	no	—
352	#41	—	4	2	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
355	C₂×D₄	—	2	1	no	—
356	C₂×D₄	—	2	2	no	—
357	C₂×D₄	—	2	1	no	—
358	C₂×D₄	—	2	1	no	—
359	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
360	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
361	C₂²×C₄	—	2	1	no	—
362	C₂²×C₄	—	2	1	no	—
363	C₂²×C₄	—	2	1	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
366	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
367	C₂²×C₄	#11761627	1	1	yes	characteristic
368	SD₁₆	—	2	4	no	—
369	D₈	—	2	4	no	—
370	M₁₆	—	268	2	no	—
371	M₁₆	—	268	4	no	—
372	M₁₆	—	134	4	no	—
373	C₂×C₈	—	268	4	no	—
374	C₂×C₈	—	268	2	no	—
375	C₂×C₈	—	268	2	no	—
376	C₂×C₈	—	268	1	no	—
377	C₂×C₈	—	67	1	no	—
378	C₂×C₈	—	67	1	no	—
379	C₂×C₈	—	67	1	no	—
380	C₂×C₈	—	67	1	no	—
381	C₂×C₈	—	134	1	no	—
382	C₂×C₈	—	134	1	no	—
383	C₂×C₈	—	134	2	no	—
384	C₂×C₈	—	134	2	no	—
385	C₂×C₈	—	134	2	no	—
386	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
387	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	1	no	—
390	C₄⋊C₄	—	2	1	no	—
391	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
392	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
393	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
394	C₂²⋊C₄	—	2	1	no	—
395	C₂²⋊C₄	—	2	1	no	—
396	C₂²⋊C₄	—	2	2	no	—
397	C₂²⋊C₄	—	2	1	no	—
398	C₂²⋊C₄	—	2	2	no	—
399	C₄²	—	2	1	no	—
400	C₄²	—	2	1	no	—
401	C₄²	—	2	1	no	—
402	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
403	C₄²	D₄×D₆₇	1	1	yes	characteristic
404	C₄²	#11761627	1	1	yes	characteristic
405	C₄²	C₂³×D₆₇	1	1	yes	characteristic
406	C₂³	—	2	1	no	—
407	C₂³	—	2	2	no	—
408	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
409	Q₈	—	2	1	no	—
410	Q₈	—	4	2	no	—
411	Q₈	#14524050	1	2	yes	complemented
412	D₄	—	2	4	no	—
413	D₄	—	2	1	no	—
414	D₄	—	2	2	no	—
415	D₄	—	2	2	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	4	no	—
418	D₄	—	4	2	no	—
419	D₄	#14524050	1	2	yes	complemented
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	1	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
436	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
437	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
438	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
439	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
440	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
441	C₈	—	2	2	no	—
442	C₈	—	2	2	no	—
443	C₈	—	268	2	no	—
444	C₈	—	268	1	no	—
445	C₈	—	268	1	no	—
446	C₈	—	67	2	no	—
447	C₈	—	134	4	no	—
448	C₈	—	134	1	no	—
449	C₈	—	134	1	no	—
450	C₈	—	134	1	no	—
451	C₂²	—	2	1	no	—
452	C₂²	—	2	1	no	—
453	C₂²	—	2	1	no	—
454	C₂²	—	2	2	no	—
455	C₂²	—	4	1	no	—
456	C₂²	—	4	1	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	2	no	—
460	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
461	C₄	—	2	1	no	—
462	C₄	—	2	1	no	—
463	C₄	—	2	1	no	—
464	C₄	—	2	2	no	—
465	C₄	—	4	2	no	—
466	C₄	—	4	1	no	—
467	C₄	—	4	1	no	—
468	C₄	#16863513	1	2	yes	—
469	C₄	C₂×#14524050	1	1	yes	characteristic
470	C₄	C₂×#14524050	1	1	yes	characteristic
471	C₄	AGL(1,ℤ/8)×D₆₇	1	1	yes	characteristic
472	C₄	#16864192	1	1	yes	characteristic
473	C₂	—	4	1	no	—
474	C₂	—	4	2	no	—
475	C₂	—	8	2	no	—
476	C₂	#22568904	1	1	yes	characteristic
477	C₂	AGL(1,ℤ/8)×Dic₆₇	1	1	yes	characteristic
478	C₂	C₂×#16863513	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting, prime radical	C₆₇×#3106	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#3106	D₆₇
hypocenter	C₆₇	#17912
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3106

Complements

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

N	H ≅ G/N	type
1. #41×C₆₇⋊C₈	C₂	semidirect
2. #22568921	C₂	semidirect
3. #22559088	C₂	semidirect
4. #22559074	C₂	semidirect
5. SD₁₆×C₆₇⋊C₈	C₂	semidirect
6. D₈×C₆₇⋊C₈	C₂	semidirect
7. #22558998	C₂	semidirect
8. #22558648	C₂	semidirect
9. #22558626	C₂	semidirect
10. #22558490	C₂	semidirect
171. Q₈×C₆₇	#131	semidirect
179. D₄×C₆₇	#131	semidirect
251. C₆₇	#17912	semidirect
411. Q₈	#14524050	semidirect
419. D₄	#14524050	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#41×C₆₇⋊C₈	2
#22568921	2
#22559088	2
#22559074	2
SD₁₆×C₆₇⋊C₈	2
D₈×C₆₇⋊C₈	2
#22558998	2
#22558648	2
#22558626	2
#22558490	2
C₆₇×#3106	2
#17912	67