Subgroups — #30748559

Order 17152.

658 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560189	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863133	C₂	1	2	yes	maximal
3	#22558439	C₂	1	1	yes	maximal, characteristic
4	#22558087	C₂	1	2	yes	maximal
5	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄⋊Dic₆₇	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
8	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
9	C₂×C₆₇⋊#95	—	2	1	no	—
10	C₂×C₆₇⋊#95	C₂²	1	2	yes	—
11	#16863151	—	2	1	no	—
12	#16863133	—	2	4	no	—
13	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
15	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
16	#391×C₆₇	C₄	1	1	yes	characteristic, complemented
17	#389×C₆₇	C₄	1	1	yes	characteristic, complemented
18	#387×C₆₇	—	2	1	no	—
19	#385×C₆₇	—	2	1	no	—
20	#380×C₆₇	C₂²	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
22	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic, complemented
23	C₄×C₂²⋊C₄×C₆₇	—	2	1	no	—
24	C₂³×Dic₆₇	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₂×C₄⋊Dic₆₇	—	2	4	no	—
27	C₂×C₄⋊Dic₆₇	—	4	2	no	—
28	C₂×C₄⋊Dic₆₇	—	4	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
30	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
31	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₆₇⋊#95	—	2	2	no	—
33	C₆₇⋊#95	—	2	2	no	—
34	C₆₇⋊#95	—	4	2	no	—
35	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
37	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
38	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
39	C₂²⋊Q₈×C₆₇	—	2	1	no	—
40	C₂²⋊Q₈×C₆₇	—	2	1	no	—
41	C₂²⋊Q₈×C₆₇	D₄	1	2	yes	—
42	C₂²⋊Q₈×C₆₇	Q₈	1	2	yes	—
43	C₄×Q₈×C₆₇	—	4	2	no	—
44	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
45	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
46	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
51	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
52	C₂×C₄²×C₆₇	—	2	1	no	—
53	C₂×C₄²×C₆₇	—	2	1	no	—
54	C₂×C₄²×C₆₇	—	4	2	no	—
55	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
59	#95×C₆₇	—	2	1	no	—
60	#95×C₆₇	—	2	1	no	—
61	#95×C₆₇	—	2	1	no	—
62	#95×C₆₇	—	2	1	no	—
63	#95×C₆₇	—	2	1	no	—
64	C₂²×Dic₆₇	—	2	4	no	—
65	C₂²×Dic₆₇	—	4	2	no	—
66	C₂²×Dic₆₇	—	4	2	no	—
67	C₂²×Dic₆₇	—	4	2	no	—
68	C₂²×Dic₆₇	—	4	2	no	—
69	C₂²×Dic₆₇	—	4	2	no	—
70	C₂²×Dic₆₇	—	4	2	no	—
71	C₂×C₆₇⋊C₈	—	4	2	no	—
72	C₂×C₆₇⋊C₈	—	4	2	no	—
73	C₂×C₆₇⋊C₈	—	4	2	no	—
74	C₄⋊Dic₆₇	—	4	8	no	—
75	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×Q₈×C₆₇	—	2	1	no	—
77	C₂×Q₈×C₆₇	—	4	1	no	—
78	C₂×Q₈×C₆₇	—	4	1	no	—
79	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	complemented
80	C₂²×C₄×C₆₇	—	2	2	no	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	1	no	—
83	C₂²×C₄×C₆₇	—	2	1	no	—
84	C₂²×C₄×C₆₇	—	2	1	no	—
85	C₂²×C₄×C₆₇	—	2	1	no	—
86	C₂²×C₄×C₆₇	—	2	1	no	—
87	C₂²×C₄×C₆₇	—	2	1	no	—
88	C₂²×C₄×C₆₇	—	4	1	no	—
89	C₂²×C₄×C₆₇	—	4	1	no	—
90	C₂²×C₄×C₆₇	—	4	1	no	—
91	C₂²×C₄×C₆₇	—	4	1	no	—
92	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
94	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
95	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
97	C₂²×C₄×C₆₇	Q₁₆	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
102	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
104	C₄⋊C₄×C₆₇	—	2	1	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	—	4	1	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₄⋊C₄×C₆₇	—	4	2	no	—
110	C₄⋊C₄×C₆₇	—	4	2	no	—
111	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	complemented
112	C₂²⋊C₄×C₆₇	—	2	2	no	—
113	C₂²⋊C₄×C₆₇	—	2	2	no	—
114	C₂²⋊C₄×C₆₇	—	4	1	no	—
115	C₂²⋊C₄×C₆₇	—	4	1	no	—
116	C₄²×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	2	2	no	—
118	C₄²×C₆₇	—	4	1	no	—
119	C₄²×C₆₇	—	4	1	no	—
120	C₄²×C₆₇	—	4	2	no	—
121	C₄²×C₆₇	—	4	2	no	—
122	C₂³×C₆₇	—	2	1	no	—
123	C₂³×C₆₇	—	2	1	no	—
124	C₂³×C₆₇	—	2	1	no	—
125	C₂³×C₆₇	—	2	1	no	—
126	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂³×C₆₇	#121	1	1	yes	characteristic
128	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
130	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
131	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂³×C₆₇	#137	1	1	yes	characteristic
133	Q₈×C₆₇	—	4	2	no	—
134	Q₈×C₆₇	—	4	2	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	2	1	no	—
141	C₂×C₄×C₆₇	—	2	4	no	—
142	C₂×C₄×C₆₇	—	2	4	no	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	1	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	1	no	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	4	1	no	—
152	C₂×C₄×C₆₇	—	4	1	no	—
153	C₂×C₄×C₆₇	—	4	1	no	—
154	C₂×C₄×C₆₇	—	4	1	no	—
155	C₂×C₄×C₆₇	—	4	2	no	—
156	C₂×C₄×C₆₇	—	4	2	no	—
157	C₂×C₄×C₆₇	—	4	1	no	—
158	C₂×C₄×C₆₇	—	4	1	no	—
159	C₂×C₄×C₆₇	—	4	1	no	—
160	C₂×C₄×C₆₇	—	4	1	no	—
161	C₂×C₄×C₆₇	—	4	1	no	—
162	C₂×C₄×C₆₇	—	4	1	no	—
163	C₂×C₄×C₆₇	—	4	1	no	—
164	C₂×C₄×C₆₇	—	4	1	no	—
165	C₂×C₄×C₆₇	—	4	1	no	—
166	C₂×C₄×C₆₇	—	4	1	no	—
167	C₂×C₄×C₆₇	#99	1	2	yes	—
168	C₂×C₄×C₆₇	#103	1	2	yes	—
169	C₂×C₄×C₆₇	#103	1	2	yes	—
170	C₂×C₄×C₆₇	#104	1	2	yes	—
171	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
173	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
174	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
175	C₂×Dic₆₇	—	4	4	no	—
176	C₂×Dic₆₇	—	4	4	no	—
177	C₂×Dic₆₇	—	4	4	no	—
178	C₂×Dic₆₇	—	8	2	no	—
179	C₂×Dic₆₇	—	8	2	no	—
180	C₂×Dic₆₇	—	8	2	no	—
181	C₂×Dic₆₇	—	8	2	no	—
182	C₆₇⋊C₈	—	8	2	no	—
183	C₂²×C₆₇	—	2	1	no	—
184	C₂²×C₆₇	—	2	1	no	—
185	C₂²×C₆₇	—	2	1	no	—
186	C₂²×C₆₇	—	2	1	no	—
187	C₂²×C₆₇	—	2	1	no	—
188	C₂²×C₆₇	—	2	1	no	—
189	C₂²×C₆₇	—	2	1	no	—
190	C₂²×C₆₇	—	2	1	no	—
191	C₂²×C₆₇	—	2	1	no	—
192	C₂²×C₆₇	—	2	1	no	—
193	C₂²×C₆₇	—	2	1	no	—
194	C₂²×C₆₇	—	2	1	no	—
195	C₂²×C₆₇	#380	1	1	yes	characteristic
196	C₂²×C₆₇	C₂×#99	1	1	yes	characteristic
197	C₂²×C₆₇	#410	1	1	yes	characteristic
198	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
199	C₂²×C₆₇	#417	1	1	yes	characteristic
200	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
201	C₂²×C₆₇	#421	1	1	yes	characteristic
202	C₂²×C₆₇	#460	1	1	yes	characteristic
203	C₂²×C₆₇	#462	1	1	yes	characteristic
204	C₂²×C₆₇	#475	1	1	yes	characteristic
205	C₂²×C₆₇	#477	1	1	yes	characteristic
206	C₄×C₆₇	—	2	1	no	—
207	C₄×C₆₇	—	2	1	no	—
208	C₄×C₆₇	—	2	1	no	—
209	C₄×C₆₇	—	2	1	no	—
210	C₄×C₆₇	—	2	2	no	—
211	C₄×C₆₇	—	4	2	no	—
212	C₄×C₆₇	—	4	2	no	—
213	C₄×C₆₇	—	4	1	no	—
214	C₄×C₆₇	—	4	1	no	—
215	C₄×C₆₇	—	8	1	no	—
216	C₄×C₆₇	—	8	1	no	—
217	C₄×C₆₇	#328	1	4	yes	complemented
218	Dic₆₇	—	8	4	no	—
219	#8572	—	67	1	no	maximal, Sylow
220	C₂×C₆₇	—	2	1	no	—
221	C₂×C₆₇	—	2	1	no	—
222	C₂×C₆₇	—	2	1	no	—
223	C₂×C₆₇	—	2	1	no	—
224	C₂×C₆₇	C₂×#328	1	1	yes	characteristic
225	C₂×C₆₇	#1509	1	1	yes	characteristic
226	C₂×C₆₇	#1513	1	1	yes	characteristic
227	C₂×C₆₇	#1518	1	1	yes	characteristic
228	C₂×C₆₇	#1799	1	1	yes	characteristic
229	C₂×C₆₇	#1803	1	1	yes	characteristic
230	C₂×C₆₇	#1806	1	1	yes	characteristic
231	C₄×C₂²⋊Q₈	D₆₇	1	1	yes	characteristic
232	#1797	—	67	1	no	—
233	C₂×#328	—	67	2	no	—
234	C₄⋊#95	—	67	1	no	—
235	#1294	—	67	2	no	—
236	C₆₇	#8572	1	1	yes	characteristic, complemented, Sylow
237	C₂×C₂²⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
238	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic, complemented
239	C₂²×C₄⋊C₄	—	134	1	no	—
240	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
241	C₂×C₄⋊C₈	—	134	1	no	—
242	C₂×C₂²⋊C₈	—	67	2	no	—
243	#391	Dic₆₇	1	1	yes	characteristic, complemented
244	#389	Dic₆₇	1	1	yes	characteristic, complemented
245	#387	—	2	1	no	—
246	#385	—	2	1	no	—
247	#380	C₂×D₆₇	1	1	yes	characteristic
248	C₄×C₄⋊C₄	—	2	1	no	—
249	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic, complemented
250	C₄×C₂²⋊C₄	—	2	1	no	—
251	C₂×#95	—	67	2	no	—
252	C₂×#95	—	134	1	no	—
253	#336	—	134	1	no	—
254	#328	—	134	4	no	—
255	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
256	C₂³×C₄	—	134	2	no	—
257	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
258	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
259	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
260	C₂²×C₈	—	134	2	no	—
261	C₂²⋊Q₈	—	2	1	no	—
262	C₂²⋊Q₈	—	2	1	no	—
263	C₂²⋊Q₈	Dic₁₃₄	1	2	yes	—
264	C₂²⋊Q₈	D₂₆₈	1	2	yes	—
265	C₄×Q₈	—	4	2	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	268	2	no	—
270	C₂×C₄⋊C₄	—	268	2	no	—
271	C₂×C₄⋊C₄	—	134	4	no	—
272	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
273	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
274	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
275	C₂×C₂²⋊C₄	—	2	1	no	—
276	C₂×C₂²⋊C₄	—	2	1	no	—
277	C₂×C₄²	—	2	1	no	—
278	C₂×C₄²	—	2	1	no	—
279	C₂×C₄²	—	4	2	no	—
280	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
281	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
282	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
283	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
284	C₄⋊C₈	—	268	2	no	—
285	C₂²⋊C₈	—	134	2	no	—
286	C₂²⋊C₈	—	134	2	no	—
287	#95	—	2	1	no	—
288	#95	—	2	1	no	—
289	#95	—	2	1	no	—
290	#95	—	2	1	no	—
291	#95	—	2	1	no	—
292	#95	—	268	2	no	—
293	#95	—	134	2	no	—
294	#95	—	134	2	no	—
295	C₂⁴	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
296	C₂×Q₈	—	2	1	no	—
297	C₂×Q₈	—	4	1	no	—
298	C₂×Q₈	—	4	1	no	—
299	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	complemented
300	C₂²×C₄	—	2	2	no	—
301	C₂²×C₄	—	2	2	no	—
302	C₂²×C₄	—	2	1	no	—
303	C₂²×C₄	—	2	1	no	—
304	C₂²×C₄	—	2	1	no	—
305	C₂²×C₄	—	2	1	no	—
306	C₂²×C₄	—	2	1	no	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	4	1	no	—
309	C₂²×C₄	—	4	1	no	—
310	C₂²×C₄	—	4	1	no	—
311	C₂²×C₄	—	4	1	no	—
312	C₂²×C₄	—	268	2	no	—
313	C₂²×C₄	—	268	2	no	—
314	C₂²×C₄	—	268	2	no	—
315	C₂²×C₄	—	268	2	no	—
316	C₂²×C₄	—	268	2	no	—
317	C₂²×C₄	—	268	2	no	—
318	C₂²×C₄	—	134	4	no	—
319	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
320	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
321	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
322	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
323	C₂²×C₄	#11761615	1	2	yes	—
324	C₂²×C₄	#11761618	1	2	yes	—
325	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
326	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
327	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
328	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
329	C₂²×C₄	#11761628	1	1	yes	characteristic
330	C₂²×C₄	#11761628	1	1	yes	characteristic
331	C₂×C₈	—	268	2	no	—
332	C₂×C₈	—	268	2	no	—
333	C₂×C₈	—	268	2	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	2	no	—
337	C₄⋊C₄	—	4	1	no	—
338	C₄⋊C₄	—	4	1	no	—
339	C₄⋊C₄	—	4	2	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	—	268	8	no	—
342	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	complemented
343	C₂²⋊C₄	—	2	2	no	—
344	C₂²⋊C₄	—	2	2	no	—
345	C₂²⋊C₄	—	4	1	no	—
346	C₂²⋊C₄	—	4	1	no	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	4	1	no	—
350	C₄²	—	4	1	no	—
351	C₄²	—	4	2	no	—
352	C₄²	—	4	2	no	—
353	C₂³	—	2	1	no	—
354	C₂³	—	2	1	no	—
355	C₂³	—	2	1	no	—
356	C₂³	—	2	1	no	—
357	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
358	C₂³	#14524017	1	1	yes	characteristic
359	C₂³	#14524022	1	1	yes	characteristic
360	C₂³	C₂×#11761615	1	1	yes	characteristic
361	C₂³	C₂×#11761618	1	1	yes	characteristic
362	C₂³	#14524068	1	1	yes	characteristic
363	C₂³	#14524073	1	1	yes	characteristic
364	Q₈	—	4	2	no	—
365	Q₈	—	4	2	no	—
366	C₂×C₄	—	2	1	no	—
367	C₂×C₄	—	2	1	no	—
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	4	no	—
374	C₂×C₄	—	2	2	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	1	no	—
377	C₂×C₄	—	2	1	no	—
378	C₂×C₄	—	2	1	no	—
379	C₂×C₄	—	2	1	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	1	no	—
386	C₂×C₄	—	4	2	no	—
387	C₂×C₄	—	4	2	no	—
388	C₂×C₄	—	4	1	no	—
389	C₂×C₄	—	4	1	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	—	4	1	no	—
392	C₂×C₄	—	4	1	no	—
393	C₂×C₄	—	4	1	no	—
394	C₂×C₄	—	4	1	no	—
395	C₂×C₄	—	4	1	no	—
396	C₂×C₄	—	4	1	no	—
397	C₂×C₄	—	4	1	no	—
398	C₂×C₄	—	268	4	no	—
399	C₂×C₄	—	268	4	no	—
400	C₂×C₄	—	268	4	no	—
401	C₂×C₄	—	536	2	no	—
402	C₂×C₄	—	536	2	no	—
403	C₂×C₄	—	536	2	no	—
404	C₂×C₄	—	536	2	no	—
405	C₂×C₄	#14523958	1	2	yes	—
406	C₂×C₄	#14523967	1	2	yes	—
407	C₂×C₄	#14523967	1	2	yes	—
408	C₂×C₄	#14523970	1	2	yes	—
409	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
410	C₂×C₄	#14524002	1	1	yes	characteristic
411	C₂×C₄	#14524017	1	1	yes	characteristic
412	C₂×C₄	#14524022	1	1	yes	characteristic
413	C₈	—	536	2	no	—
414	C₂²	—	2	1	no	—
415	C₂²	—	2	1	no	—
416	C₂²	—	2	1	no	—
417	C₂²	—	2	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	2	1	no	—
422	C₂²	—	2	1	no	—
423	C₂²	—	2	1	no	—
424	C₂²	—	2	1	no	—
425	C₂²	—	2	1	no	—
426	C₂²	#16863256	1	1	yes	characteristic
427	C₂²	C₂×#14523958	1	1	yes	characteristic
428	C₂²	#16863370	1	1	yes	characteristic
429	C₂²	C₂×#14523967	1	1	yes	characteristic
430	C₂²	#16863397	1	1	yes	characteristic
431	C₂²	C₂×#14523970	1	1	yes	characteristic
432	C₂²	#16863413	1	1	yes	characteristic
433	C₂²	#16863596	1	1	yes	characteristic
434	C₂²	#16863606	1	1	yes	characteristic
435	C₂²	#16863681	1	1	yes	characteristic
436	C₂²	#16863691	1	1	yes	characteristic
437	C₄	—	2	1	no	—
438	C₄	—	2	1	no	—
439	C₄	—	2	1	no	—
440	C₄	—	2	1	no	—
441	C₄	—	2	2	no	—
442	C₄	—	4	1	no	—
443	C₄	—	4	1	no	—
444	C₄	—	4	2	no	—
445	C₄	—	4	2	no	—
446	C₄	—	8	1	no	—
447	C₄	—	8	1	no	—
448	C₄	—	536	4	no	—
449	C₄	#16863133	1	4	yes	complemented
450	C₂	—	2	1	no	—
451	C₂	—	2	1	no	—
452	C₂	—	2	1	no	—
453	C₂	—	2	1	no	—
454	C₂	C₂×#16863133	1	1	yes	characteristic
455	C₂	#22558734	1	1	yes	characteristic
456	C₂	#22558752	1	1	yes	characteristic
457	C₂	#22558773	1	1	yes	characteristic
458	C₂	#22560197	1	1	yes	characteristic
459	C₂	#22560214	1	1	yes	characteristic
460	C₂	#22560229	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊Dic₆₇
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₆₇
Fitting	C₄×C₂²⋊Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₆₇	C₂×D₄
cosocle	C₂³×C₄×C₆₇	C₂³
hypercenter	C₄×C₂²⋊Q₈	D₆₇
hypocenter	C₆₇	#8572
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₄×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
14. C₂×C₄×Q₈×C₆₇	C₄	semidirect
16. #391×C₆₇	C₄	semidirect
17. #389×C₆₇	C₄	semidirect
22. C₄×C₄⋊C₄×C₆₇	C₄	semidirect
79. C₂×Q₈×C₆₇	C₄⋊C₄	semidirect
111. C₄⋊C₄×C₆₇	C₄⋊C₄	semidirect
217. C₄×C₆₇	#328	semidirect
236. C₆₇	#8572	semidirect
238. C₂×C₄×Q₈	Dic₆₇	semidirect
243. #391	Dic₆₇	semidirect
244. #389	Dic₆₇	semidirect
249. C₄×C₄⋊C₄	Dic₆₇	semidirect
299. C₂×Q₈	C₄⋊Dic₆₇	semidirect
342. C₄⋊C₄	C₄⋊Dic₆₇	semidirect
449. C₄	#16863133	semidirect

Maximal subgroups

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

subgroup	index
#22560189	2
C₂×#16863133	2
#22558439	2
#22558087	2
C₄×C₂²⋊Q₈×C₆₇	2
#8572	67