Subgroups — #30825259

Order 17152.

714 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11761627	C₂	1	1	yes	maximal, characteristic
2	C₈×Q₈×D₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#14524072	C₂	1	1	yes	maximal, characteristic
4	#22559029	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559026	C₂	1	1	yes	maximal, characteristic, complemented
6	#22559021	C₂	1	1	yes	maximal, characteristic, complemented
7	#22559015	C₂	1	1	yes	maximal, characteristic, complemented
8	#22559013	C₂	1	1	yes	maximal, characteristic, complemented
9	#22559012	C₂	1	1	yes	maximal, characteristic, complemented
10	SD₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
11	#22558960	C₂	1	1	yes	maximal, characteristic, complemented
12	#22558602	C₂	1	1	yes	maximal, characteristic, complemented
13	#22558593	C₂	1	1	yes	maximal, characteristic
14	#22558488	C₂	1	1	yes	maximal, characteristic, complemented
15	C₆₇×SD₁₆⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14524072	C₄	1	1	yes	characteristic
17	C₄×#11761627	C₂²	1	1	yes	characteristic
18	C₄×Q₈×D₆₇	C₂²	1	1	yes	characteristic
19	Q₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₃₄	—	2	1	no	—
21	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
22	C₈×Dic₁₃₄	C₂²	1	1	yes	characteristic
23	#16863520	—	2	1	no	—
24	#16863520	C₂²	1	1	yes	characteristic
25	#16863520	C₂²	1	1	yes	characteristic
26	#16863502	C₄	1	1	yes	characteristic
27	#16863500	C₄	1	1	yes	characteristic
28	#16863497	C₄	1	1	yes	characteristic
29	#16863493	C₄	1	1	yes	characteristic
30	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
31	C₄×#11761618	C₂²	1	1	yes	characteristic
32	C₄×#11761616	C₂²	1	1	yes	characteristic
33	D₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
34	C₈×C₂²⋊D₆₇	—	2	1	no	—
35	(C₂²⋊D₆₇)⋊C₈	—	2	1	no	—
36	#16863448	C₂²	1	1	yes	characteristic
37	#16863447	C₂²	1	1	yes	characteristic
38	#16863446	—	2	1	no	—
39	#16863435	C₄	1	1	yes	characteristic
40	C₄⋊C₈×D₆₇	—	2	1	no	—
41	C₄⋊C₈×D₆₇	C₂²	1	1	yes	characteristic
42	#16863403	C₄	1	1	yes	characteristic
43	#16863394	C₄	1	1	yes	characteristic
44	C₄×#11761611	C₂²	1	1	yes	characteristic
45	C₂×C₈×Dic₆₇	—	2	1	no	—
46	C₄×C₈×D₆₇	—	2	1	no	—
47	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
48	#16863148	C₂²	1	1	yes	characteristic
49	#16863147	C₂²	1	1	yes	characteristic
50	#16863146	C₂²	1	1	yes	characteristic
51	#16863128	C₂²	1	1	yes	characteristic
52	#16863128	C₂²	1	1	yes	characteristic
53	Q₈⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
54	#16863126	C₂²	1	1	yes	characteristic
55	#16863126	C₂²	1	1	yes	characteristic
56	D₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
57	#16863118	C₂²	1	1	yes	characteristic
58	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
59	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
60	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
61	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
62	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
63	#335×C₆₇	C₂²	1	1	yes	characteristic
64	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
65	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
66	#322×C₆₇	C₂²	1	1	yes	characteristic
67	C₂×#11761627	C₂×C₄	1	1	yes	characteristic
68	C₂×Q₈×D₆₇	C₂×C₄	1	1	yes	characteristic
69	#14524072	C₈	1	8	yes	complemented
70	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
71	C₂×#11761618	C₂×C₄	1	1	yes	characteristic
72	C₂×#11761616	C₂×C₄	1	1	yes	characteristic
73	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
74	C₂×#11761611	C₂×C₄	1	1	yes	characteristic
75	C₂²×C₆₇⋊C₈	—	2	1	no	—
76	C₂×C₈×D₆₇	—	2	1	no	—
77	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
78	C₂×C₈×D₆₇	D₄	1	1	yes	characteristic
79	Q₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₁₃₄	—	2	1	no	—
81	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
82	C₄×Dic₁₃₄	C₂³	1	1	yes	characteristic
83	#14524008	—	2	1	no	—
84	#14524008	C₂×C₄	1	1	yes	characteristic
85	#14524008	C₂×C₄	1	1	yes	characteristic
86	D₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
87	C₄×C₂²⋊D₆₇	—	2	1	no	—
88	(C₂²⋊D₆₇)⋊C₄	—	2	1	no	—
89	#14524001	C₂×C₄	1	1	yes	characteristic
90	#14523999	—	2	1	no	—
91	C₄⋊C₄×D₆₇	—	2	1	no	—
92	C₄⋊C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
93	C₂×C₄×Dic₆₇	—	2	1	no	—
94	C₄²×D₆₇	—	2	1	no	—
95	C₄²×D₆₇	C₂³	1	1	yes	characteristic
96	#14523977	C₂×C₄	1	1	yes	characteristic
97	#14523976	C₂×C₄	1	1	yes	characteristic
98	#14523974	C₂×C₄	1	1	yes	characteristic
99	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
100	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
101	C₄⋊(C₆₇⋊C₈)	C₂³	1	1	yes	characteristic
102	Dic₆₇⋊C₈	—	2	1	no	—
103	Dic₆₇⋊C₈	—	2	1	no	—
104	Dic₆₇⋊C₈	—	2	1	no	—
105	Dic₆₇⋊C₈	—	2	1	no	—
106	Dic₆₇⋊C₈	—	2	1	no	—
107	Dic₆₇⋊C₈	—	2	1	no	—
108	#14523968	C₂×C₄	1	1	yes	characteristic
109	#14523968	C₂×C₄	1	1	yes	characteristic
110	#14523967	C₂×C₄	1	1	yes	characteristic
111	#14523966	C₂×C₄	1	1	yes	characteristic
112	#14523966	C₂×C₄	1	1	yes	characteristic
113	#14523964	C₂×C₄	1	1	yes	characteristic
114	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
115	(C₂×D₆₇)⋊C₈	—	2	1	no	—
116	(C₂×D₆₇)⋊C₈	—	2	1	no	—
117	#14523953	C₂×C₄	1	1	yes	characteristic
118	#14523953	C₂×C₄	1	1	yes	characteristic
119	C₄×C₆₇⋊C₈	—	2	1	no	—
120	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
121	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
122	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
123	C₈×Dic₆₇	—	2	1	no	—
124	C₈×Dic₆₇	—	2	1	no	—
125	C₈×Dic₆₇	—	2	1	no	—
126	C₈×Dic₆₇	—	2	1	no	—
127	C₈×Dic₆₇	—	2	1	no	—
128	C₈×Dic₆₇	D₄	1	1	yes	characteristic
129	C₈×Dic₆₇	D₄	1	1	yes	characteristic
130	C₂×SD₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
131	C₂²×C₈×C₆₇	—	2	1	no	—
132	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
133	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
134	#106×C₆₇	C₂×C₄	1	1	yes	characteristic
135	C₄⋊C₈×C₆₇	—	2	1	no	—
136	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
137	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
138	#103×C₆₇	C₂×C₄	1	1	yes	characteristic
139	#102×C₆₇	C₂×C₄	1	1	yes	characteristic
140	C₂²⋊C₈×C₆₇	—	2	1	no	—
141	C₄×C₈×C₆₇	—	2	1	no	—
142	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
143	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
144	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
145	#11761627	—	2	2	no	—
146	#11761627	C₂×C₈	1	4	yes	—
147	C₂×Dic₁₃₄	—	2	1	no	—
148	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
149	C₂×Dic₁₃₄	C₂²×C₄	1	1	yes	characteristic
150	Q₈×D₆₇	—	2	2	no	—
151	Q₈×D₆₇	C₂×C₈	1	4	yes	complemented
152	C₂×C₂²⋊D₆₇	—	2	1	no	—
153	C₂²×Dic₆₇	—	2	1	no	—
154	C₂²×Dic₆₇	—	2	1	no	—
155	C₂×C₄×D₆₇	—	2	1	no	—
156	C₂×C₄×D₆₇	—	2	1	no	—
157	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
158	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
159	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
160	Dic₂₆₈	C₂×C₈	1	4	yes	complemented
161	#11761618	C₂×C₈	1	4	yes	complemented
162	#11761616	C₂×C₈	1	4	yes	complemented
163	#11761611	—	2	2	no	—
164	#11761611	C₂×C₈	1	4	yes	complemented
165	C₂×C₆₇⋊C₈	—	2	1	no	—
166	C₂×C₆₇⋊C₈	—	2	1	no	—
167	C₂×C₆₇⋊C₈	—	2	1	no	—
168	C₂×C₆₇⋊C₈	—	4	2	no	—
169	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
170	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
171	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
172	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
173	C₈×D₆₇	—	2	2	no	—
174	C₈×D₆₇	—	4	1	no	—
175	C₈×D₆₇	—	4	2	no	—
176	C₄⋊Dic₆₇	—	2	1	no	—
177	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
178	C₄⋊Dic₆₇	C₂²×C₄	1	1	yes	characteristic
179	Dic₆₇⋊C₄	—	2	1	no	—
180	Dic₆₇⋊C₄	—	2	1	no	—
181	Dic₆₇⋊C₄	—	2	1	no	—
182	Dic₆₇⋊C₄	—	2	1	no	—
183	Dic₆₇⋊C₄	—	2	1	no	—
184	C₂²⋊Dic₆₇	—	2	1	no	—
185	(C₂×D₆₇)⋊C₄	—	2	1	no	—
186	C₄×Dic₆₇	—	2	1	no	—
187	C₄×Dic₆₇	—	2	1	no	—
188	C₄×Dic₆₇	—	2	1	no	—
189	C₄×Dic₆₇	—	2	1	no	—
190	C₄×Dic₆₇	—	2	1	no	—
191	C₄×Dic₆₇	—	2	1	no	—
192	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
193	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
194	C₄×Dic₆₇	#41	1	1	yes	characteristic
195	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
196	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
197	C₂²×C₄×C₆₇	—	2	1	no	—
198	SD₁₆×C₆₇	C₂×C₈	1	4	yes	complemented
199	C₂×C₈×C₆₇	—	2	1	no	—
200	C₂×C₈×C₆₇	—	2	1	no	—
201	C₂×C₈×C₆₇	—	2	1	no	—
202	C₂×C₈×C₆₇	—	4	2	no	—
203	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
204	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
205	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
206	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
207	C₄⋊C₄×C₆₇	—	2	1	no	—
208	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
209	C₄⋊C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₆₇	—	2	1	no	—
211	C₄²×C₆₇	—	2	1	no	—
212	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
213	C₂³×C₆₇	—	2	1	no	—
214	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
215	Q₈×C₆₇	—	2	1	no	—
216	Q₈×C₆₇	C₂²×C₈	1	2	yes	—
217	D₄×C₆₇	—	2	1	no	—
218	D₄×C₆₇	C₂²×C₈	1	2	yes	—
219	C₂×C₄×C₆₇	—	2	1	no	—
220	C₂×C₄×C₆₇	—	2	1	no	—
221	C₂×C₄×C₆₇	—	2	1	no	—
222	C₂×C₄×C₆₇	—	4	1	no	—
223	C₂×C₄×C₆₇	—	4	1	no	—
224	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
226	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
227	C₂²⋊D₆₇	—	2	4	no	—
228	C₂×Dic₆₇	—	2	1	no	—
229	C₂×Dic₆₇	—	2	1	no	—
230	C₂×Dic₆₇	—	2	4	no	—
231	C₂×Dic₆₇	—	2	1	no	—
232	C₂×Dic₆₇	—	2	1	no	—
233	C₂×Dic₆₇	—	4	2	no	—
234	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
235	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
236	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
237	C₄×D₆₇	—	2	4	no	—
238	C₄×D₆₇	—	2	1	no	—
239	C₄×D₆₇	—	2	1	no	—
240	C₄×D₆₇	—	4	1	no	—
241	C₄×D₆₇	—	4	2	no	—
242	C₄×D₆₇	C₂²×C₈	1	2	yes	—
243	C₄×D₆₇	#131	1	2	yes	—
244	Dic₁₃₄	—	2	1	no	—
245	Dic₁₃₄	—	2	1	no	—
246	Dic₁₃₄	—	2	4	no	—
247	Dic₁₃₄	C₂²×C₈	1	2	yes	—
248	Dic₁₃₄	C₂²×C₈	1	2	yes	—
249	C₈×C₆₇	—	2	1	no	—
250	C₈×C₆₇	—	2	1	no	—
251	C₈×C₆₇	—	2	1	no	—
252	C₈×C₆₇	—	4	1	no	—
253	C₈×C₆₇	—	4	1	no	—
254	C₈×C₆₇	—	4	1	no	—
255	C₈×C₆₇	C₂²×C₈	1	2	yes	—
256	C₆₇⋊C₈	—	2	1	no	—
257	C₆₇⋊C₈	—	2	1	no	—
258	C₆₇⋊C₈	—	4	1	no	—
259	C₆₇⋊C₈	—	4	1	no	—
260	C₆₇⋊C₈	—	4	1	no	—
261	C₆₇⋊C₈	C₂²×C₈	1	2	yes	—
262	C₆₇⋊C₈	#137	1	2	yes	complemented
263	C₂²×C₆₇	—	2	2	no	—
264	C₂²×C₆₇	—	4	1	no	—
265	C₂²×C₆₇	—	4	1	no	—
266	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
267	C₂×D₆₇	—	2	1	no	—
268	C₂×D₆₇	—	2	1	no	—
269	C₂×D₆₇	C₈×D₄	1	2	yes	—
270	C₄×C₆₇	—	2	2	no	—
271	C₄×C₆₇	—	2	1	no	—
272	C₄×C₆₇	—	4	1	no	—
273	C₄×C₆₇	—	4	1	no	—
274	C₄×C₆₇	C₂³×C₈	1	1	yes	characteristic
275	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
276	C₄×C₆₇	C₂×#137	1	1	yes	characteristic
277	C₄×C₆₇	#575	1	1	yes	characteristic
278	Dic₆₇	—	2	2	no	—
279	Dic₆₇	—	2	2	no	—
280	Dic₆₇	—	2	1	no	—
281	Dic₆₇	—	2	1	no	—
282	Dic₆₇	—	4	1	no	—
283	Dic₆₇	—	4	1	no	—
284	Dic₆₇	C₈×D₄	1	2	yes	—
285	C₈×#137	—	67	1	no	maximal, Sylow
286	C₂×C₆₇	—	4	2	no	—
287	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
288	C₂×C₆₇	C₄×#137	1	1	yes	characteristic
289	C₂×C₆₇	#2955	1	1	yes	characteristic
290	D₆₇	—	2	2	no	—
291	C₈×#41	—	67	1	no	—
292	C₂×C₈×Q₈	—	67	1	no	—
293	C₄×#137	—	67	1	no	—
294	#1579	—	67	1	no	—
295	#1579	—	67	1	no	—
296	SD₁₆⋊C₈	—	67	1	no	—
297	SD₁₆⋊C₈	D₆₇	1	1	yes	characteristic, complemented
298	C₈×Q₁₆	—	67	1	no	—
299	C₈×Q₁₆	—	67	1	no	—
300	C₈×SD₁₆	—	67	1	no	—
301	C₈×SD₁₆	—	67	1	no	—
302	#1565	—	67	1	no	—
303	#1479	—	67	1	no	—
304	#1477	—	67	1	no	—
305	C₈×M₁₆	—	67	1	no	—
306	C₆₇	C₈×#137	1	1	yes	characteristic, complemented, Sylow
307	C₂×#137	—	67	1	no	—
308	C₄×#41	—	67	1	no	—
309	C₂×C₄×Q₈	—	67	1	no	—
310	C₈×Q₈	—	67	1	no	—
311	C₈×Q₈	—	67	1	no	—
312	C₈×Q₈	—	67	1	no	—
313	C₈×Q₈	—	67	1	no	—
314	C₈×Q₈	—	67	1	no	—
315	C₈×Q₈	—	134	1	no	—
316	C₈×Q₈	—	134	1	no	—
317	C₈×Q₈	C₂×D₆₇	1	1	yes	characteristic
318	#441	—	67	1	no	—
319	#441	—	67	1	no	—
320	#440	—	67	1	no	—
321	#440	—	67	1	no	—
322	C₄×Q₁₆	—	67	1	no	—
323	C₄×Q₁₆	—	67	1	no	—
324	C₄×SD₁₆	—	67	1	no	—
325	C₄×SD₁₆	C₂×D₆₇	1	1	yes	characteristic
326	C₈×D₄	—	67	1	no	—
327	C₈×D₄	—	134	1	no	—
328	C₈×D₄	—	134	1	no	—
329	C₈×D₄	C₂×D₆₇	1	1	yes	characteristic
330	#431	—	67	1	no	—
331	#431	—	67	1	no	—
332	#431	—	134	1	no	—
333	#428	—	67	1	no	—
334	C₂×C₄⋊C₈	—	67	1	no	—
335	C₂×C₄⋊C₈	—	134	1	no	—
336	#419	—	67	1	no	—
337	#417	—	67	1	no	—
338	C₄×M₁₆	—	67	1	no	—
339	C₂×C₄×C₈	—	67	1	no	—
340	C₂×C₄×C₈	—	134	1	no	—
341	C₂×C₄×C₈	—	134	1	no	—
342	#335	—	67	1	no	—
343	#335	C₂×D₆₇	1	1	yes	characteristic
344	#334	—	67	1	no	—
345	#334	—	67	1	no	—
346	Q₈⋊C₈	—	67	1	no	—
347	Q₈⋊C₈	—	67	1	no	—
348	Q₈⋊C₈	—	67	1	no	—
349	Q₈⋊C₈	—	67	1	no	—
350	Q₈⋊C₈	—	67	1	no	—
351	Q₈⋊C₈	C₂×D₆₇	1	1	yes	characteristic
352	D₄⋊C₈	—	67	1	no	—
353	D₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
354	#322	—	67	1	no	—
355	#322	C₂×D₆₇	1	1	yes	characteristic
356	C₈²	—	67	1	no	—
357	C₈²	—	67	1	no	—
358	C₂×#41	—	67	1	no	—
359	C₂²×Q₈	—	67	1	no	—
360	#137	—	67	8	no	—
361	C₂×Q₁₆	—	67	1	no	—
362	C₂×Q₁₆	—	67	1	no	—
363	C₂×SD₁₆	—	67	1	no	—
364	C₂×SD₁₆	C₄×D₆₇	1	1	yes	characteristic
365	C₂×M₁₆	—	67	1	no	—
366	C₂×M₁₆	—	67	1	no	—
367	C₂²×C₈	—	2	1	no	—
368	C₂²×C₈	—	67	1	no	—
369	C₂²×C₈	—	67	1	no	—
370	C₂²×C₈	—	134	1	no	—
371	C₂²×C₈	—	134	1	no	—
372	C₄×Q₈	—	67	1	no	—
373	C₄×Q₈	—	67	1	no	—
374	C₄×Q₈	—	67	1	no	—
375	C₄×Q₈	—	67	1	no	—
376	C₄×Q₈	—	67	1	no	—
377	C₄×Q₈	—	134	1	no	—
378	C₄×Q₈	—	134	1	no	—
379	C₄×Q₈	C₂²×D₆₇	1	1	yes	characteristic
380	C₄×D₄	—	67	1	no	—
381	C₄×D₄	—	134	1	no	—
382	C₄×D₄	—	134	1	no	—
383	C₄×D₄	C₂²×D₆₇	1	1	yes	characteristic
384	#117	—	67	1	no	—
385	#117	—	134	1	no	—
386	C₂×C₄⋊C₄	—	67	1	no	—
387	C₂×C₄⋊C₄	—	134	1	no	—
388	C₂×C₄²	—	67	1	no	—
389	C₂×C₄²	—	134	1	no	—
390	C₂×C₄²	—	134	1	no	—
391	#107	—	67	1	no	—
392	#107	—	67	1	no	—
393	#106	—	67	1	no	—
394	#106	C₄×D₆₇	1	1	yes	characteristic
395	C₄⋊C₈	—	2	1	no	—
396	C₄⋊C₈	—	67	1	no	—
397	C₄⋊C₈	—	67	1	no	—
398	C₄⋊C₈	—	134	1	no	—
399	C₄⋊C₈	—	134	1	no	—
400	C₄⋊C₈	—	134	1	no	—
401	C₄⋊C₈	—	134	1	no	—
402	C₄⋊C₈	—	134	1	no	—
403	C₄⋊C₈	—	134	1	no	—
404	C₄⋊C₈	—	134	1	no	—
405	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
406	C₄⋊C₈	C₂²×D₆₇	1	1	yes	characteristic
407	#103	—	67	1	no	—
408	#103	—	67	1	no	—
409	#103	—	67	1	no	—
410	#103	—	67	1	no	—
411	#103	—	67	1	no	—
412	#103	C₄×D₆₇	1	1	yes	characteristic
413	#102	—	67	1	no	—
414	#102	C₄×D₆₇	1	1	yes	characteristic
415	C₂²⋊C₈	—	2	1	no	—
416	C₂²⋊C₈	—	134	1	no	—
417	C₂²⋊C₈	—	134	1	no	—
418	C₂²⋊C₈	—	134	1	no	—
419	#97	—	67	1	no	—
420	#97	—	67	1	no	—
421	C₄×C₈	—	2	1	no	—
422	C₄×C₈	—	67	1	no	—
423	C₄×C₈	—	67	1	no	—
424	C₄×C₈	—	67	1	no	—
425	C₄×C₈	—	67	1	no	—
426	C₄×C₈	—	67	1	no	—
427	C₄×C₈	—	134	1	no	—
428	C₄×C₈	—	134	1	no	—
429	C₄×C₈	—	134	1	no	—
430	C₄×C₈	—	134	1	no	—
431	C₄×C₈	—	134	1	no	—
432	C₄×C₈	—	134	1	no	—
433	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
434	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
435	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
436	#41	—	67	4	no	—
437	#41	—	134	2	no	—
438	C₂×Q₈	—	67	4	no	—
439	C₂×Q₈	—	67	1	no	—
440	C₂×Q₈	—	67	1	no	—
441	C₂×Q₈	—	134	1	no	—
442	C₂×Q₈	—	134	2	no	—
443	C₂×Q₈	C₂×C₄×D₆₇	1	1	yes	characteristic
444	C₂×D₄	—	134	1	no	—
445	C₂×D₄	C₂×C₄×D₆₇	1	1	yes	characteristic
446	C₂²×C₄	—	2	1	no	—
447	C₂²×C₄	—	67	1	no	—
448	C₂²×C₄	—	67	1	no	—
449	C₂²×C₄	—	67	1	no	—
450	C₂²×C₄	—	134	1	no	—
451	C₂²×C₄	—	134	1	no	—
452	C₂²×C₄	—	134	1	no	—
453	C₂²×C₄	—	134	1	no	—
454	Q₁₆	—	67	4	no	—
455	Q₁₆	—	67	4	no	—
456	SD₁₆	—	67	4	no	—
457	SD₁₆	C₈×D₆₇	1	4	yes	complemented
458	M₁₆	—	67	4	no	—
459	M₁₆	—	134	2	no	—
460	C₂×C₈	—	2	1	no	—
461	C₂×C₈	—	2	1	no	—
462	C₂×C₈	—	2	1	no	—
463	C₂×C₈	—	4	2	no	—
464	C₂×C₈	—	268	1	no	—
465	C₂×C₈	—	268	2	no	—
466	C₂×C₈	—	268	2	no	—
467	C₂×C₈	—	67	1	no	—
468	C₂×C₈	—	67	1	no	—
469	C₂×C₈	—	67	1	no	—
470	C₂×C₈	—	67	1	no	—
471	C₂×C₈	—	134	2	no	—
472	C₂×C₈	—	134	1	no	—
473	C₂×C₈	—	134	1	no	—
474	C₂×C₈	—	134	1	no	—
475	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
476	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
477	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
478	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
479	C₄⋊C₄	—	2	1	no	—
480	C₄⋊C₄	—	67	1	no	—
481	C₄⋊C₄	—	67	1	no	—
482	C₄⋊C₄	—	134	1	no	—
483	C₄⋊C₄	—	134	1	no	—
484	C₄⋊C₄	—	134	1	no	—
485	C₄⋊C₄	—	134	1	no	—
486	C₄⋊C₄	—	134	1	no	—
487	C₄⋊C₄	—	134	1	no	—
488	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
489	C₄⋊C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
490	C₂²⋊C₄	—	2	1	no	—
491	C₂²⋊C₄	—	134	1	no	—
492	C₂²⋊C₄	—	134	1	no	—
493	C₄²	—	2	1	no	—
494	C₄²	—	67	1	no	—
495	C₄²	—	67	1	no	—
496	C₄²	—	67	1	no	—
497	C₄²	—	134	1	no	—
498	C₄²	—	134	1	no	—
499	C₄²	—	134	1	no	—
500	C₄²	—	134	1	no	—
501	C₄²	—	134	1	no	—
502	C₄²	—	134	1	no	—
503	C₄²	C₂³×D₆₇	1	1	yes	characteristic
504	C₂³	—	2	1	no	—
505	C₂³	—	67	1	no	—
506	Q₈	—	2	1	no	—
507	Q₈	—	67	2	no	—
508	Q₈	—	67	2	no	—
509	Q₈	—	134	4	no	—
510	Q₈	—	134	1	no	—
511	Q₈	—	134	1	no	—
512	Q₈	C₂×C₈×D₆₇	1	2	yes	—
513	D₄	—	2	1	no	—
514	D₄	—	134	4	no	—
515	D₄	C₂×C₈×D₆₇	1	2	yes	—
516	C₂×C₄	—	2	1	no	—
517	C₂×C₄	—	2	1	no	—
518	C₂×C₄	—	2	1	no	—
519	C₂×C₄	—	4	1	no	—
520	C₂×C₄	—	4	1	no	—
521	C₂×C₄	—	268	1	no	—
522	C₂×C₄	—	268	2	no	—
523	C₂×C₄	—	268	2	no	—
524	C₂×C₄	—	67	1	no	—
525	C₂×C₄	—	67	2	no	—
526	C₂×C₄	—	67	2	no	—
527	C₂×C₄	—	67	1	no	—
528	C₂×C₄	—	67	1	no	—
529	C₂×C₄	—	134	4	no	—
530	C₂×C₄	—	134	1	no	—
531	C₂×C₄	—	134	1	no	—
532	C₂×C₄	—	134	4	no	—
533	C₂×C₄	—	134	1	no	—
534	C₂×C₄	—	134	1	no	—
535	C₂×C₄	—	134	1	no	—
536	C₂×C₄	—	134	1	no	—
537	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
538	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
539	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
540	C₈	—	2	1	no	—
541	C₈	—	2	1	no	—
542	C₈	—	2	1	no	—
543	C₈	—	4	1	no	—
544	C₈	—	4	1	no	—
545	C₈	—	4	1	no	—
546	C₈	—	268	1	no	—
547	C₈	—	268	1	no	—
548	C₈	—	268	1	no	—
549	C₈	—	67	2	no	—
550	C₈	—	67	2	no	—
551	C₈	—	134	1	no	—
552	C₈	—	134	1	no	—
553	C₈	C₂×C₈×D₆₇	1	2	yes	—
554	C₂²	—	2	2	no	—
555	C₂²	—	4	1	no	—
556	C₂²	—	4	1	no	—
557	C₂²	—	67	2	no	—
558	C₂²	—	134	1	no	—
559	C₂²	—	134	1	no	—
560	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
561	C₄	—	2	1	no	—
562	C₄	—	2	2	no	—
563	C₄	—	4	1	no	—
564	C₄	—	4	1	no	—
565	C₄	—	268	1	no	—
566	C₄	—	268	1	no	—
567	C₄	—	67	2	no	—
568	C₄	—	134	1	no	—
569	C₄	—	134	1	no	—
570	C₄	—	134	2	no	—
571	C₄	—	134	2	no	—
572	C₄	C₂²×C₈×D₆₇	1	1	yes	characteristic
573	C₄	#131×D₆₇	1	1	yes	characteristic
574	C₄	#137×D₆₇	1	1	yes	characteristic
575	C₄	#16864192	1	1	yes	characteristic
576	C₂	—	4	2	no	—
577	C₂	—	134	2	no	—
578	C₂	C₈×D₄×D₆₇	1	1	yes	characteristic
579	C₂	#22569031	1	1	yes	characteristic
580	C₂	#22569109	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₄×C₆₇	C₂³×C₈
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×SD₁₆⋊C₈	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂×#11761627	C₂×C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	SD₁₆⋊C₈	D₆₇
hypocenter	C₆₇	C₈×#137
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ SD₁₆⋊C₈

Complements

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

N	H ≅ G/N	type
2. C₈×Q₈×D₆₇	C₂	semidirect
4. #22559029	C₂	semidirect
5. #22559026	C₂	semidirect
6. #22559021	C₂	semidirect
7. #22559015	C₂	semidirect
8. #22559013	C₂	semidirect
9. #22559012	C₂	semidirect
10. SD₁₆×C₆₇⋊C₈	C₂	semidirect
11. #22558960	C₂	semidirect
12. #22558602	C₂	semidirect
14. #22558488	C₂	semidirect
15. C₆₇×SD₁₆⋊C₈	C₂	semidirect
69. #14524072	C₈	semidirect
151. Q₈×D₆₇	C₂×C₈	semidirect
160. Dic₂₆₈	C₂×C₈	semidirect
161. #11761618	C₂×C₈	semidirect
162. #11761616	C₂×C₈	semidirect
164. #11761611	C₂×C₈	semidirect
198. SD₁₆×C₆₇	C₂×C₈	semidirect
262. C₆₇⋊C₈	#137	semidirect
297. SD₁₆⋊C₈	D₆₇	semidirect
306. C₆₇	C₈×#137	semidirect
457. SD₁₆	C₈×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11761627	2
C₈×Q₈×D₆₇	2
C₄×#14524072	2
#22559029	2
#22559026	2
#22559021	2
#22559015	2
#22559013	2
#22559012	2
SD₁₆×C₆₇⋊C₈	2
#22558960	2
#22558602	2
#22558593	2
#22558488	2
C₆₇×SD₁₆⋊C₈	2
C₈×#137	67