Subgroups — #30825439

Order 17152.

738 conjugacy classes of subgroups.

The subgroup lattice has 494 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	C₂×#16863474	C₂	1	1	yes	maximal, characteristic, complemented
3	#22559088	C₂	1	2	yes	maximal, complemented
4	#22559074	C₂	1	2	yes	maximal, complemented
5	#22559053	C₂	1	2	yes	maximal, complemented
6	#22559039	C₂	1	2	yes	maximal, complemented
7	#22558961	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558600	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558596	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558496	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×AGL(1,ℤ/8)×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×#11761612	—	2	1	no	—
13	Q₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
14	Q₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
15	#16863474	—	2	1	no	—
16	#16863474	—	2	1	no	—
17	#16863474	C₂²	1	2	yes	—
18	#16863474	C₂²	1	2	yes	—
19	D₄×C₆₇⋊C₈	—	2	1	no	—
20	D₄×C₆₇⋊C₈	—	2	1	no	—
21	D₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
22	D₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic, complemented
23	#16863449	—	2	1	no	—
24	#16863449	C₂²	1	1	yes	characteristic
25	#16863449	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
28	C₂×C₄×C₆₇⋊C₈	—	2	1	no	—
29	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
30	#16863149	C₂²	1	2	yes	complemented
31	#16863147	C₂²	1	2	yes	complemented
32	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	complemented
33	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
34	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
35	D₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	complemented
36	#16863118	C₂²	1	2	yes	complemented
37	#16863118	C₂²	1	2	yes	complemented
38	C₂×AGL(1,ℤ/8)×C₆₇	C₄	1	1	yes	characteristic
39	C₄×#41×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
41	#442×C₆₇	C₄	1	2	yes	—
42	#440×C₆₇	C₄	1	2	yes	—
43	C₄×SD₁₆×C₆₇	C₂²	1	2	yes	—
44	C₄×D₈×C₆₇	C₂²	1	2	yes	—
45	#428×C₆₇	C₄	1	1	yes	characteristic
46	#418×C₆₇	C₄	1	1	yes	characteristic
47	#417×C₆₇	C₄	1	1	yes	characteristic
48	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
49	C₂×#11761612	—	2	2	no	—
50	C₂×#11761612	—	2	2	no	—
51	C₂×#11761612	—	4	4	no	—
52	C₂²×C₆₇⋊C₈	—	2	1	no	—
53	C₂²×C₆₇⋊C₈	—	2	1	no	—
54	C₂²×C₆₇⋊C₈	—	2	1	no	—
55	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
56	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
57	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
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	C₂²⋊(C₆₇⋊C₈)	—	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	C₂×C₄	1	1	yes	characteristic
79	C₂²×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
80	C₂³×C₄×C₆₇	—	2	1	no	—
81	AGL(1,ℤ/8)×C₆₇	—	2	4	no	—
82	C₂×SD₁₆×C₆₇	C₂×C₄	1	2	yes	—
83	C₂×D₈×C₆₇	C₂×C₄	1	2	yes	—
84	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×M₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
88	C₄×D₄×C₆₇	—	2	1	no	—
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	2	yes	—
93	C₄×D₄×C₆₇	C₂×C₄	1	1	yes	characteristic
94	C₄×D₄×C₆₇	C₂³	1	2	yes	—
95	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
96	#117×C₆₇	—	2	1	no	—
97	#117×C₆₇	C₂×C₄	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
99	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
100	C₂×C₄²×C₆₇	—	2	1	no	—
101	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
102	#107×C₆₇	C₂×C₄	1	2	yes	—
103	#106×C₆₇	C₂×C₄	1	2	yes	—
104	#103×C₆₇	C₂×C₄	1	2	yes	—
105	#102×C₆₇	C₂×C₄	1	2	yes	—
106	#102×C₆₇	C₂×C₄	1	2	yes	—
107	#102×C₆₇	C₂×C₄	1	2	yes	—
108	#97×C₆₇	C₂×C₄	1	2	yes	—
109	C₄×C₈×C₆₇	C₂³	1	2	yes	—
110	#11761612	—	4	4	no	—
111	#11761612	—	4	4	no	—
112	C₂×C₆₇⋊C₈	—	2	2	no	—
113	C₂×C₆₇⋊C₈	—	2	1	no	—
114	C₂×C₆₇⋊C₈	—	2	1	no	—
115	C₂×C₆₇⋊C₈	—	2	2	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	2	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	C₂⁴×C₆₇	—	2	1	no	—
126	#41×C₆₇	—	2	2	no	—
127	#41×C₆₇	M₁₆	1	4	yes	complemented
128	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×D₄×C₆₇	—	2	1	no	—
130	C₂×D₄×C₆₇	—	2	1	no	—
131	C₂×D₄×C₆₇	—	2	2	no	—
132	C₂×D₄×C₆₇	—	2	2	no	—
133	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
135	C₂²×C₄×C₆₇	—	2	1	no	—
136	C₂²×C₄×C₆₇	—	2	1	no	—
137	C₂²×C₄×C₆₇	—	2	1	no	—
138	C₂²×C₄×C₆₇	—	2	2	no	—
139	C₂²×C₄×C₆₇	—	2	1	no	—
140	C₂²×C₄×C₆₇	—	2	2	no	—
141	C₂²×C₄×C₆₇	—	4	2	no	—
142	C₂²×C₄×C₆₇	—	4	2	no	—
143	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
144	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
146	SD₁₆×C₆₇	—	2	4	no	—
147	D₈×C₆₇	—	2	4	no	—
148	M₁₆×C₆₇	—	2	2	no	—
149	M₁₆×C₆₇	—	2	2	no	—
150	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
151	C₂×C₈×C₆₇	C₂²×C₄	1	2	yes	—
152	C₄⋊C₄×C₆₇	—	2	1	no	—
153	C₄⋊C₄×C₆₇	—	2	1	no	—
154	C₄⋊C₄×C₆₇	—	2	1	no	—
155	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
156	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
158	C₂²⋊C₄×C₆₇	—	2	1	no	—
159	C₂²⋊C₄×C₆₇	—	2	2	no	—
160	C₂²⋊C₄×C₆₇	—	2	1	no	—
161	C₂²⋊C₄×C₆₇	—	2	2	no	—
162	C₂²⋊C₄×C₆₇	—	2	1	no	—
163	C₄²×C₆₇	—	2	1	no	—
164	C₄²×C₆₇	—	2	1	no	—
165	C₄²×C₆₇	—	2	1	no	—
166	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
168	C₄²×C₆₇	#41	1	1	yes	characteristic
169	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
170	C₂³×C₆₇	—	2	1	no	—
171	C₂³×C₆₇	—	2	2	no	—
172	C₂³×C₆₇	—	4	2	no	—
173	C₂³×C₆₇	—	4	2	no	—
174	C₂³×C₆₇	—	4	2	no	—
175	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
176	Q₈×C₆₇	—	2	1	no	—
177	Q₈×C₆₇	C₂×M₁₆	1	2	yes	complemented
178	D₄×C₆₇	—	2	4	no	—
179	D₄×C₆₇	—	2	2	no	—
180	D₄×C₆₇	—	2	1	no	—
181	D₄×C₆₇	—	2	2	no	—
182	D₄×C₆₇	—	4	2	no	—
183	D₄×C₆₇	C₂×M₁₆	1	2	yes	complemented
184	C₂×C₄×C₆₇	—	2	1	no	—
185	C₂×C₄×C₆₇	—	2	1	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	4	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	1	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	—	4	1	no	—
192	C₂×C₄×C₆₇	—	4	2	no	—
193	C₂×C₄×C₆₇	—	4	4	no	—
194	C₂×C₄×C₆₇	—	4	2	no	—
195	C₂×C₄×C₆₇	—	4	2	no	—
196	C₂×C₄×C₆₇	—	4	1	no	—
197	C₂×C₄×C₆₇	—	4	2	no	—
198	C₂×C₄×C₆₇	—	4	1	no	—
199	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
200	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
203	C₂×C₄×C₆₇	#131	1	2	yes	—
204	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
206	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
207	C₈×C₆₇	—	2	2	no	—
208	C₈×C₆₇	—	2	2	no	—
209	C₆₇⋊C₈	—	2	1	no	—
210	C₆₇⋊C₈	—	2	1	no	—
211	C₆₇⋊C₈	—	2	1	no	—
212	C₆₇⋊C₈	—	4	2	no	—
213	C₆₇⋊C₈	—	4	1	no	—
214	C₆₇⋊C₈	—	4	2	no	—
215	C₆₇⋊C₈	—	4	1	no	—
216	C₆₇⋊C₈	AGL(1,ℤ/8)	1	2	yes	complemented
217	C₂²×C₆₇	—	2	1	no	—
218	C₂²×C₆₇	—	2	1	no	—
219	C₂²×C₆₇	—	2	2	no	—
220	C₂²×C₆₇	—	4	1	no	—
221	C₂²×C₆₇	—	4	1	no	—
222	C₂²×C₆₇	—	4	2	no	—
223	C₂²×C₆₇	—	4	2	no	—
224	C₂²×C₆₇	—	4	2	no	—
225	C₂²×C₆₇	—	8	4	no	—
226	C₂²×C₆₇	#435	1	2	yes	—
227	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	1	no	—
230	C₄×C₆₇	—	2	1	no	—
231	C₄×C₆₇	—	2	1	no	—
232	C₄×C₆₇	—	4	1	no	—
233	C₄×C₆₇	—	4	2	no	—
234	C₄×C₆₇	—	4	1	no	—
235	C₄×C₆₇	#435	1	2	yes	—
236	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
237	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
238	C₄×C₆₇	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
239	C₄×C₆₇	#575	1	1	yes	characteristic
240	#17871	—	67	1	no	maximal, Sylow
241	C₂×C₆₇	—	2	2	no	—
242	C₂×C₆₇	—	4	2	no	—
243	C₂×C₆₇	—	8	2	no	—
244	C₂×C₆₇	C₂×#435	1	1	yes	characteristic
245	C₂×C₆₇	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic
246	C₂×C₆₇	#2955	1	1	yes	characteristic
247	C₈×#41	—	67	1	no	—
248	C₄×AGL(1,ℤ/8)	D₆₇	1	1	yes	characteristic
249	C₂×#436	—	67	1	no	—
250	#1588	—	67	2	no	—
251	#1586	—	67	2	no	—
252	#1583	—	67	2	no	—
253	#1581	—	67	2	no	—
254	#1565	—	67	1	no	—
255	#1478	—	67	1	no	—
256	#1477	—	67	1	no	—
257	#1451	—	67	1	no	—
258	C₆₇	#17871	1	1	yes	characteristic, complemented, Sylow
259	C₂×AGL(1,ℤ/8)	Dic₆₇	1	1	yes	characteristic
260	C₂²×M₁₆	—	134	1	no	—
261	C₄×#41	C₂×D₆₇	1	1	yes	characteristic
262	C₂×C₄×D₄	C₂×D₆₇	1	1	yes	characteristic
263	C₈×Q₈	—	67	1	no	—
264	C₈×Q₈	—	67	1	no	—
265	#442	Dic₆₇	1	2	yes	—
266	#440	Dic₆₇	1	2	yes	—
267	C₄×SD₁₆	C₂×D₆₇	1	2	yes	—
268	C₄×D₈	C₂×D₆₇	1	2	yes	—
269	#436	—	67	2	no	—
270	#436	—	67	2	no	—
271	#436	—	134	1	no	—
272	#436	—	134	1	no	—
273	C₈×D₄	—	67	1	no	—
274	C₈×D₄	—	67	1	no	—
275	C₈×D₄	—	134	1	no	—
276	C₈×D₄	—	134	1	no	—
277	#431	—	67	1	no	—
278	#431	—	67	1	no	—
279	#431	—	134	1	no	—
280	#428	Dic₆₇	1	1	yes	characteristic
281	C₂×C₄⋊C₈	—	67	1	no	—
282	#418	Dic₆₇	1	1	yes	characteristic
283	#417	Dic₆₇	1	1	yes	characteristic
284	C₂×C₂²⋊C₈	—	134	1	no	—
285	C₄×M₁₆	C₂×D₆₇	1	1	yes	characteristic
286	C₂×C₄×C₈	—	67	1	no	—
287	C₂×C₄×C₈	—	134	1	no	—
288	#335	—	67	2	no	—
289	#334	—	67	2	no	—
290	Q₈⋊C₈	—	67	2	no	—
291	D₄⋊C₈	—	67	2	no	—
292	D₄⋊C₈	—	67	2	no	—
293	D₄⋊C₈	—	67	2	no	—
294	#322	—	67	2	no	—
295	#322	—	67	2	no	—
296	C₂×#41	C₂×Dic₆₇	1	1	yes	characteristic
297	C₂²×D₄	C₂×Dic₆₇	1	1	yes	characteristic
298	C₂³×C₄	—	2	1	no	—
299	AGL(1,ℤ/8)	—	2	4	no	—
300	C₂×SD₁₆	C₂×Dic₆₇	1	2	yes	—
301	C₂×D₈	C₂×Dic₆₇	1	2	yes	—
302	C₂×M₁₆	—	268	4	no	—
303	C₂×M₁₆	—	134	2	no	—
304	C₂×M₁₆	—	134	2	no	—
305	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
306	C₂×M₁₆	C₂×Dic₆₇	1	1	yes	characteristic
307	C₂²×C₈	—	67	1	no	—
308	C₂²×C₈	—	67	1	no	—
309	C₂²×C₈	—	134	1	no	—
310	C₂²×C₈	—	134	1	no	—
311	C₂²×C₈	—	134	1	no	—
312	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
313	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
314	C₄×D₄	—	2	1	no	—
315	C₄×D₄	—	2	1	no	—
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	C₂×Dic₆₇	1	2	yes	—
319	C₄×D₄	C₂×Dic₆₇	1	1	yes	characteristic
320	C₄×D₄	C₂²×D₆₇	1	2	yes	—
321	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	C₂×Dic₆₇	1	1	yes	characteristic
324	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
325	C₂×C₂²⋊C₄	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
328	#107	C₂×Dic₆₇	1	2	yes	—
329	#106	C₂×Dic₆₇	1	2	yes	—
330	C₄⋊C₈	—	67	1	no	—
331	C₄⋊C₈	—	67	1	no	—
332	C₄⋊C₈	—	67	2	no	—
333	C₄⋊C₈	—	134	1	no	—
334	C₄⋊C₈	—	134	1	no	—
335	C₄⋊C₈	—	134	1	no	—
336	C₄⋊C₈	—	134	1	no	—
337	#103	C₂×Dic₆₇	1	2	yes	—
338	#102	C₂×Dic₆₇	1	2	yes	—
339	#102	C₂×Dic₆₇	1	2	yes	—
340	#102	C₂×Dic₆₇	1	2	yes	—
341	C₂²⋊C₈	—	134	2	no	—
342	C₂²⋊C₈	—	134	2	no	—
343	C₂²⋊C₈	—	134	1	no	—
344	C₂²⋊C₈	—	134	1	no	—
345	C₂²⋊C₈	—	134	1	no	—
346	C₂²⋊C₈	—	134	1	no	—
347	#97	C₂×Dic₆₇	1	2	yes	—
348	C₄×C₈	—	67	1	no	—
349	C₄×C₈	—	67	1	no	—
350	C₄×C₈	—	67	1	no	—
351	C₄×C₈	—	67	1	no	—
352	C₄×C₈	—	67	2	no	—
353	C₄×C₈	—	134	1	no	—
354	C₄×C₈	—	134	1	no	—
355	C₄×C₈	—	134	1	no	—
356	C₄×C₈	C₂²×D₆₇	1	2	yes	—
357	C₂⁴	—	2	1	no	—
358	#41	—	2	2	no	—
359	#41	#11761612	1	4	yes	complemented
360	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
361	C₂×D₄	—	2	1	no	—
362	C₂×D₄	—	2	1	no	—
363	C₂×D₄	—	2	2	no	—
364	C₂×D₄	—	2	2	no	—
365	C₂×D₄	C₂²×Dic₆₇	1	2	yes	—
366	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
367	C₂²×C₄	—	2	1	no	—
368	C₂²×C₄	—	2	1	no	—
369	C₂²×C₄	—	2	1	no	—
370	C₂²×C₄	—	2	2	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	4	2	no	—
374	C₂²×C₄	—	4	2	no	—
375	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
376	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
377	C₂²×C₄	#11761627	1	1	yes	characteristic
378	SD₁₆	—	2	4	no	—
379	D₈	—	2	4	no	—
380	M₁₆	—	2	2	no	—
381	M₁₆	—	2	2	no	—
382	M₁₆	—	268	4	no	—
383	M₁₆	—	268	4	no	—
384	C₂×C₈	—	268	2	no	—
385	C₂×C₈	—	268	2	no	—
386	C₂×C₈	—	268	2	no	—
387	C₂×C₈	—	268	1	no	—
388	C₂×C₈	—	67	1	no	—
389	C₂×C₈	—	67	1	no	—
390	C₂×C₈	—	67	1	no	—
391	C₂×C₈	—	67	1	no	—
392	C₂×C₈	—	134	1	no	—
393	C₂×C₈	—	134	1	no	—
394	C₂×C₈	—	134	2	no	—
395	C₂×C₈	—	134	2	no	—
396	C₂×C₈	—	134	2	no	—
397	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
398	C₂×C₈	C₂²×Dic₆₇	1	2	yes	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	1	no	—
402	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
403	C₄⋊C₄	C₂²×Dic₆₇	1	1	yes	characteristic
404	C₄⋊C₄	C₂²×Dic₆₇	1	2	yes	—
405	C₂²⋊C₄	—	2	1	no	—
406	C₂²⋊C₄	—	2	2	no	—
407	C₂²⋊C₄	—	2	1	no	—
408	C₂²⋊C₄	—	2	2	no	—
409	C₂²⋊C₄	—	2	1	no	—
410	C₄²	—	2	1	no	—
411	C₄²	—	2	1	no	—
412	C₄²	—	2	1	no	—
413	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
414	C₄²	D₄×D₆₇	1	1	yes	characteristic
415	C₄²	#11761627	1	1	yes	characteristic
416	C₄²	C₂³×D₆₇	1	1	yes	characteristic
417	C₂³	—	2	1	no	—
418	C₂³	—	2	2	no	—
419	C₂³	—	4	2	no	—
420	C₂³	—	4	2	no	—
421	C₂³	—	4	2	no	—
422	C₂³	D₄×Dic₆₇	1	1	yes	characteristic
423	Q₈	—	2	1	no	—
424	Q₈	C₂×#11761612	1	2	yes	complemented
425	D₄	—	2	4	no	—
426	D₄	—	2	2	no	—
427	D₄	—	2	1	no	—
428	D₄	—	2	2	no	—
429	D₄	—	4	2	no	—
430	D₄	C₂×#11761612	1	2	yes	complemented
431	C₂×C₄	—	2	1	no	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	4	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	4	no	—
440	C₂×C₄	—	4	2	no	—
441	C₂×C₄	—	4	2	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
447	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
448	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂×#11761612	1	2	yes	—
450	C₂×C₄	#14524050	1	2	yes	—
451	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
453	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	268	2	no	—
457	C₈	—	268	2	no	—
458	C₈	—	268	1	no	—
459	C₈	—	268	1	no	—
460	C₈	—	67	2	no	—
461	C₈	—	134	1	no	—
462	C₈	—	134	1	no	—
463	C₈	—	134	1	no	—
464	C₂²	—	2	1	no	—
465	C₂²	—	2	1	no	—
466	C₂²	—	2	2	no	—
467	C₂²	—	4	1	no	—
468	C₂²	—	4	1	no	—
469	C₂²	—	4	2	no	—
470	C₂²	—	4	2	no	—
471	C₂²	—	4	2	no	—
472	C₂²	—	8	4	no	—
473	C₂²	#16863468	1	2	yes	—
474	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
475	C₄	—	2	1	no	—
476	C₄	—	2	1	no	—
477	C₄	—	2	1	no	—
478	C₄	—	2	2	no	—
479	C₄	—	4	2	no	—
480	C₄	—	4	1	no	—
481	C₄	—	4	1	no	—
482	C₄	#16863468	1	2	yes	—
483	C₄	C₂²×#11761612	1	1	yes	characteristic
484	C₄	C₂×#14524050	1	1	yes	characteristic
485	C₄	AGL(1,ℤ/8)×D₆₇	1	1	yes	characteristic
486	C₄	#16864192	1	1	yes	characteristic
487	C₂	—	2	2	no	—
488	C₂	—	4	2	no	—
489	C₂	—	8	2	no	—
490	C₂	C₂×#16863468	1	1	yes	characteristic
491	C₂	AGL(1,ℤ/8)×Dic₆₇	1	1	yes	characteristic
492	C₂	#22569112	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₄×AGL(1,ℤ/8)×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×AGL(1,ℤ/8)×C₆₇	C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₄×AGL(1,ℤ/8)	D₆₇
hypocenter	C₆₇	#17871
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #41×C₆₇⋊C₈	C₂	semidirect
2. C₂×#16863474	C₂	semidirect
3. #22559088	C₂	semidirect
4. #22559074	C₂	semidirect
5. #22559053	C₂	semidirect
6. #22559039	C₂	semidirect
7. #22558961	C₂	semidirect
8. #22558600	C₂	semidirect
9. #22558596	C₂	semidirect
10. #22558496	C₂	semidirect
13. Q₈×C₆₇⋊C₈	C₂²	semidirect
14. Q₈×C₆₇⋊C₈	C₂²	semidirect
21. D₄×C₆₇⋊C₈	C₂²	semidirect
22. D₄×C₆₇⋊C₈	C₂²	semidirect
30. #16863149	C₂²	semidirect
31. #16863147	C₂²	semidirect
32. Q₈⋊(C₆₇⋊C₈)	C₂²	semidirect
35. D₄⋊(C₆₇⋊C₈)	C₂²	semidirect
36. #16863118	C₂²	semidirect
37. #16863118	C₂²	semidirect
127. #41×C₆₇	M₁₆	semidirect
177. Q₈×C₆₇	C₂×M₁₆	semidirect
183. D₄×C₆₇	C₂×M₁₆	semidirect
216. C₆₇⋊C₈	AGL(1,ℤ/8)	semidirect
258. C₆₇	#17871	semidirect
359. #41	#11761612	semidirect
424. Q₈	C₂×#11761612	semidirect
430. D₄	C₂×#11761612	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
C₂×#16863474	2
#22559088	2
#22559074	2
#22559053	2
#22559039	2
#22558961	2
#22558600	2
#22558596	2
#22558496	2
C₄×AGL(1,ℤ/8)×C₆₇	2
#17871	67