Subgroups — #30824901

Order 17152.

668 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22568837	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863447	C₂	1	1	yes	maximal, characteristic
3	#431×D₆₇	C₂	1	1	yes	maximal, characteristic
4	#22558988	C₂	1	2	yes	maximal, complemented
5	#22558986	C₂	1	2	yes	maximal, complemented
6	#22558982	C₂	1	2	yes	maximal, complemented
7	#22558978	C₂	1	2	yes	maximal, complemented
8	#22558957	C₂	1	1	yes	maximal, characteristic, complemented
9	#22558955	C₂	1	1	yes	maximal, characteristic, complemented
10	#22558955	C₂	1	1	yes	maximal, characteristic, complemented
11	C₆₇×#1564	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
13	#16863452	C₂²	1	2	yes	—
14	#16863450	—	2	1	no	—
15	#16863449	C₂²	1	1	yes	characteristic
16	#16863449	C₂²	1	1	yes	characteristic
17	#16863448	C₂²	1	2	yes	—
18	#16863447	—	2	1	no	—
19	#16863447	—	2	1	no	—
20	#16863447	C₄	1	1	yes	characteristic
21	#16863447	C₄	1	1	yes	characteristic
22	#16863447	C₄	1	1	yes	characteristic
23	#16863447	C₄	1	1	yes	characteristic
24	#16863447	C₂²	1	1	yes	characteristic
25	#16863447	C₂²	1	1	yes	characteristic
26	#16863447	C₂²	1	1	yes	characteristic
27	#16863447	C₂²	1	1	yes	characteristic
28	#16863446	—	2	1	no	—
29	#16863418	C₂²	1	2	yes	—
30	C₂×Dic₆₇⋊C₈	C₄	1	1	yes	characteristic
31	C₂×Dic₆₇⋊C₈	C₄	1	1	yes	characteristic
32	C₄⋊C₈×D₆₇	C₂²	1	2	yes	—
33	#16863358	—	2	1	no	—
34	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
35	C₂×(C₂×D₆₇)⋊C₈	C₄	1	1	yes	characteristic
36	C₂²⋊C₈×D₆₇	—	2	1	no	—
37	C₄×#11761611	—	2	1	no	—
38	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
39	C₄×C₈×D₆₇	—	2	1	no	—
40	#16863149	C₂²	1	2	yes	complemented
41	#16863148	C₂²	1	2	yes	complemented
42	#16863148	C₂²	1	2	yes	complemented
43	#16863147	C₂²	1	2	yes	complemented
44	#16863146	C₂²	1	2	yes	complemented
45	#16863146	C₂²	1	2	yes	complemented
46	#431×C₆₇	C₂²	1	1	yes	characteristic
47	#431×C₆₇	C₂²	1	1	yes	characteristic
48	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
49	#335×C₆₇	C₂²	1	2	yes	complemented
50	#334×C₆₇	C₂²	1	2	yes	complemented
51	C₂²×C₄×D₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×D₆₇	D₄	1	1	yes	characteristic
53	C₂²×C₄×D₆₇	Q₈	1	1	yes	characteristic
54	C₂×#11761611	—	2	2	no	—
55	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₈×D₆₇	—	2	2	no	—
58	C₂×C₄×Dic₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
60	C₂×C₄×Dic₆₇	Q₈	1	1	yes	characteristic
61	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
62	C₄²×D₆₇	C₂×C₄	1	1	yes	characteristic
63	C₄²×D₆₇	D₄	1	1	yes	characteristic
64	C₄²×D₆₇	D₄	1	1	yes	characteristic
65	C₄²×D₆₇	Q₈	1	1	yes	characteristic
66	C₄²×D₆₇	Q₈	1	1	yes	characteristic
67	C₄²×D₆₇	C₂³	1	1	yes	characteristic
68	C₄²×D₆₇	C₂³	1	1	yes	characteristic
69	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
70	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
71	Dic₆₇⋊C₈	—	2	2	no	—
72	Dic₆₇⋊C₈	—	2	2	no	—
73	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
74	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
75	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
76	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
77	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
78	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
79	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
80	Dic₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
81	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
82	C₂²⋊(C₆₇⋊C₈)	—	2	1	no	—
83	(C₂×D₆₇)⋊C₈	—	2	2	no	—
84	(C₂×D₆₇)⋊C₈	—	2	2	no	—
85	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
86	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
87	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
88	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
89	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
90	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
91	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
92	(C₂×D₆₇)⋊C₈	C₂×C₄	1	1	yes	characteristic
93	#14523953	—	2	2	no	—
94	C₄×C₆₇⋊C₈	—	2	1	no	—
95	C₄×C₆₇⋊C₈	—	2	1	no	—
96	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
97	C₄×C₆₇⋊C₈	C₂×C₄	1	1	yes	characteristic
98	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
99	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
100	C₈×Dic₆₇	—	2	2	no	—
101	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
103	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
104	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
105	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
106	C₂²⋊C₈×C₆₇	—	2	1	no	—
107	C₂²⋊C₈×C₆₇	—	2	1	no	—
108	C₄×C₈×C₆₇	—	2	1	no	—
109	C₄×C₈×C₆₇	—	2	1	no	—
110	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
111	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
112	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
113	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
114	C₂³×D₆₇	C₄⋊C₄	1	1	yes	characteristic
115	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
116	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
117	C₂²×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×D₆₇	—	2	2	no	—
119	C₂×C₄×D₆₇	—	2	2	no	—
120	C₂×C₄×D₆₇	—	2	4	no	—
121	C₂×C₄×D₆₇	—	4	2	no	—
122	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
123	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×D₆₇	C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
130	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
131	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
133	C₂×C₄×D₆₇	C₂×Q₈	1	1	yes	characteristic
134	#11761611	—	4	4	no	—
135	C₂×C₆₇⋊C₈	—	2	2	no	—
136	C₂×C₆₇⋊C₈	—	2	2	no	—
137	C₂×C₆₇⋊C₈	—	2	2	no	—
138	C₂×C₆₇⋊C₈	—	2	2	no	—
139	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
140	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
141	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
142	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
143	C₈×D₆₇	—	4	4	no	—
144	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
145	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
146	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
147	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
148	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
149	C₄×Dic₆₇	C₄⋊C₄	1	1	yes	characteristic
150	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
151	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
152	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
153	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
154	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
155	C₄×Dic₆₇	C₂×Q₈	1	1	yes	characteristic
156	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
157	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
159	C₂×C₈×C₆₇	—	2	2	no	—
160	C₂×C₈×C₆₇	—	2	2	no	—
161	C₂×C₈×C₆₇	—	2	2	no	—
162	C₂×C₈×C₆₇	—	2	2	no	—
163	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
164	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
165	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
166	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
167	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
170	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
171	C₂³×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂²×D₆₇	—	2	2	no	—
173	C₂²×D₆₇	—	2	2	no	—
174	C₂²×D₆₇	—	4	1	no	—
175	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂²×D₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	2	no	—
179	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
181	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₄×C₆₇	#131	1	2	yes	—
183	C₂×C₄×C₆₇	#131	1	2	yes	—
184	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
185	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
186	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
187	C₂×Dic₆₇	—	2	2	no	—
188	C₂×Dic₆₇	—	2	2	no	—
189	C₂×Dic₆₇	—	4	1	no	—
190	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
191	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
192	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
193	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
194	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
195	C₂×Dic₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
196	C₄×D₆₇	—	2	2	no	—
197	C₄×D₆₇	—	2	1	no	—
198	C₄×D₆₇	—	2	1	no	—
199	C₄×D₆₇	—	2	1	no	—
200	C₄×D₆₇	—	2	1	no	—
201	C₄×D₆₇	—	2	2	no	—
202	C₄×D₆₇	—	2	2	no	—
203	C₄×D₆₇	—	2	2	no	—
204	C₄×D₆₇	—	2	2	no	—
205	C₄×D₆₇	—	2	2	no	—
206	C₄×D₆₇	—	2	2	no	—
207	C₄×D₆₇	—	4	1	no	—
208	C₄×D₆₇	—	4	1	no	—
209	C₄×D₆₇	#135	1	4	yes	—
210	C₈×C₆₇	—	2	1	no	—
211	C₈×C₆₇	—	2	1	no	—
212	C₈×C₆₇	—	2	1	no	—
213	C₈×C₆₇	—	2	1	no	—
214	C₈×C₆₇	—	4	2	no	—
215	C₈×C₆₇	—	4	2	no	—
216	C₆₇⋊C₈	—	2	1	no	—
217	C₆₇⋊C₈	—	2	1	no	—
218	C₆₇⋊C₈	—	2	1	no	—
219	C₆₇⋊C₈	—	2	1	no	—
220	C₆₇⋊C₈	—	4	2	no	—
221	C₆₇⋊C₈	—	4	2	no	—
222	C₂²×C₆₇	—	2	1	no	—
223	C₂²×C₆₇	—	2	1	no	—
224	C₂²×C₆₇	#424	1	2	yes	—
225	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
226	C₂×D₆₇	—	2	1	no	—
227	C₂×D₆₇	—	2	1	no	—
228	C₂×D₆₇	—	2	1	no	—
229	C₂×D₆₇	—	2	1	no	—
230	C₂×D₆₇	—	2	1	no	—
231	C₂×D₆₇	—	4	4	no	—
232	C₂×D₆₇	#427	1	2	yes	—
233	C₄×C₆₇	—	2	1	no	—
234	C₄×C₆₇	—	2	1	no	—
235	C₄×C₆₇	#424	1	2	yes	—
236	C₄×C₆₇	#430	1	2	yes	—
237	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
238	C₄×C₆₇	C₂×#131	1	1	yes	characteristic
239	C₄×C₆₇	C₂×#135	1	2	yes	—
240	Dic₆₇	—	2	1	no	—
241	Dic₆₇	—	2	1	no	—
242	Dic₆₇	—	2	1	no	—
243	Dic₆₇	—	2	1	no	—
244	Dic₆₇	—	2	1	no	—
245	Dic₆₇	#427	1	2	yes	—
246	#17803	—	67	1	no	maximal, Sylow
247	C₂×C₆₇	—	2	2	no	—
248	C₂×C₆₇	C₂×#424	1	1	yes	characteristic
249	C₂×C₆₇	C₂×#427	1	1	yes	characteristic
250	C₂×C₆₇	C₂×#430	1	1	yes	characteristic
251	D₆₇	—	2	2	no	—
252	D₆₇	—	4	1	no	—
253	#2920	—	67	1	no	—
254	C₂×#431	—	67	1	no	—
255	C₂×#431	—	67	1	no	—
256	#1572	—	67	2	no	—
257	#1572	—	67	2	no	—
258	#1571	—	67	2	no	—
259	#1570	—	67	2	no	—
260	#1564	—	67	1	no	—
261	#1564	—	67	1	no	—
262	#1564	—	67	1	no	—
263	#1564	D₆₇	1	1	yes	characteristic, complemented
264	C₆₇	#17803	1	1	yes	characteristic, complemented, Sylow
265	C₂²×C₄²	—	67	1	no	—
266	#432	—	67	2	no	—
267	#432	—	134	1	no	—
268	#431	—	67	1	no	—
269	#431	—	67	1	no	—
270	#431	—	67	1	no	—
271	#431	—	67	1	no	—
272	#431	—	67	1	no	—
273	#431	—	67	1	no	—
274	#431	—	67	1	no	—
275	#431	—	67	1	no	—
276	#431	—	67	1	no	—
277	#431	—	67	1	no	—
278	#431	—	67	2	no	—
279	#431	—	134	1	no	—
280	#431	—	134	1	no	—
281	#431	—	134	1	no	—
282	#431	C₂×D₆₇	1	1	yes	characteristic
283	#431	C₂×D₆₇	1	1	yes	characteristic
284	#423	—	67	2	no	—
285	C₂×C₄⋊C₈	—	67	1	no	—
286	C₂×C₄⋊C₈	—	67	1	no	—
287	C₂×C₄⋊C₈	—	67	2	no	—
288	#407	—	134	1	no	—
289	C₂×C₂²⋊C₈	—	67	1	no	—
290	C₂×C₂²⋊C₈	—	67	1	no	—
291	C₂×C₂²⋊C₈	—	134	1	no	—
292	C₄×M₁₆	—	134	1	no	—
293	C₂×C₄×C₈	—	67	1	no	—
294	C₂×C₄×C₈	—	134	1	no	—
295	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
296	#335	—	67	2	no	—
297	#335	—	67	2	no	—
298	#335	—	67	2	no	—
299	#335	C₂×D₆₇	1	2	yes	complemented
300	#334	—	67	2	no	—
301	#334	—	67	2	no	—
302	#334	—	67	2	no	—
303	#334	C₂×D₆₇	1	2	yes	complemented
304	C₂³×C₄	—	67	1	no	—
305	C₂³×C₄	—	67	1	no	—
306	C₂³×C₄	—	67	1	no	—
307	C₂×M₁₆	—	134	2	no	—
308	C₂²×C₈	—	67	1	no	—
309	C₂²×C₈	—	67	1	no	—
310	C₂²×C₈	—	134	2	no	—
311	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
312	C₂²×C₈	C₄×D₆₇	1	1	yes	characteristic
313	C₂×C₄²	—	67	1	no	—
314	C₂×C₄²	—	67	1	no	—
315	C₂×C₄²	—	67	1	no	—
316	C₂×C₄²	—	67	1	no	—
317	C₂×C₄²	—	67	1	no	—
318	C₂×C₄²	—	67	1	no	—
319	C₂×C₄²	—	67	1	no	—
320	C₂×C₄²	—	67	1	no	—
321	C₂×C₄²	—	67	1	no	—
322	C₂×C₄²	—	67	1	no	—
323	C₂×C₄²	—	67	1	no	—
324	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
325	C₄⋊C₈	—	67	1	no	—
326	C₄⋊C₈	—	67	1	no	—
327	C₄⋊C₈	—	67	1	no	—
328	C₄⋊C₈	—	67	1	no	—
329	C₄⋊C₈	—	67	1	no	—
330	C₄⋊C₈	—	67	1	no	—
331	C₄⋊C₈	—	67	1	no	—
332	C₄⋊C₈	—	67	1	no	—
333	C₄⋊C₈	—	67	2	no	—
334	C₄⋊C₈	—	67	2	no	—
335	C₄⋊C₈	—	134	2	no	—
336	C₄⋊C₈	—	134	2	no	—
337	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
338	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
339	C₂²⋊C₈	—	2	1	no	—
340	C₂²⋊C₈	—	2	1	no	—
341	C₂²⋊C₈	—	67	1	no	—
342	C₂²⋊C₈	—	67	1	no	—
343	C₂²⋊C₈	—	67	1	no	—
344	C₂²⋊C₈	—	67	1	no	—
345	C₂²⋊C₈	—	67	1	no	—
346	C₂²⋊C₈	—	67	1	no	—
347	C₂²⋊C₈	—	67	1	no	—
348	C₂²⋊C₈	—	67	1	no	—
349	C₂²⋊C₈	—	134	2	no	—
350	C₂²⋊C₈	—	134	2	no	—
351	C₂²⋊C₈	—	134	1	no	—
352	C₂²⋊C₈	—	134	1	no	—
353	#97	—	134	2	no	—
354	C₄×C₈	—	2	1	no	—
355	C₄×C₈	—	2	1	no	—
356	C₄×C₈	—	67	1	no	—
357	C₄×C₈	—	67	1	no	—
358	C₄×C₈	—	67	1	no	—
359	C₄×C₈	—	67	1	no	—
360	C₄×C₈	—	134	2	no	—
361	C₄×C₈	—	134	1	no	—
362	C₄×C₈	—	134	1	no	—
363	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
364	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
365	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
366	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
367	C₂⁴	—	67	1	no	—
368	C₂²×C₄	—	268	2	no	—
369	C₂²×C₄	—	67	1	no	—
370	C₂²×C₄	—	67	1	no	—
371	C₂²×C₄	—	67	1	no	—
372	C₂²×C₄	—	67	1	no	—
373	C₂²×C₄	—	67	1	no	—
374	C₂²×C₄	—	67	1	no	—
375	C₂²×C₄	—	67	1	no	—
376	C₂²×C₄	—	67	1	no	—
377	C₂²×C₄	—	67	1	no	—
378	C₂²×C₄	—	67	1	no	—
379	C₂²×C₄	—	67	1	no	—
380	C₂²×C₄	—	67	1	no	—
381	C₂²×C₄	—	67	1	no	—
382	C₂²×C₄	—	67	1	no	—
383	C₂²×C₄	—	67	1	no	—
384	C₂²×C₄	—	134	4	no	—
385	C₂²×C₄	—	134	2	no	—
386	C₂²×C₄	—	134	2	no	—
387	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
388	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
389	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
390	M₁₆	—	268	4	no	—
391	C₂×C₈	—	2	2	no	—
392	C₂×C₈	—	2	2	no	—
393	C₂×C₈	—	2	2	no	—
394	C₂×C₈	—	2	2	no	—
395	C₂×C₈	—	268	4	no	—
396	C₂×C₈	—	67	1	no	—
397	C₂×C₈	—	67	1	no	—
398	C₂×C₈	—	67	1	no	—
399	C₂×C₈	—	67	1	no	—
400	C₂×C₈	—	134	2	no	—
401	C₂×C₈	—	134	2	no	—
402	C₂×C₈	—	134	2	no	—
403	C₂×C₈	—	134	2	no	—
404	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
405	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
406	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
407	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
408	C₄²	—	67	1	no	—
409	C₄²	—	67	1	no	—
410	C₄²	—	67	1	no	—
411	C₄²	—	67	1	no	—
412	C₄²	—	67	1	no	—
413	C₄²	—	67	1	no	—
414	C₄²	—	67	1	no	—
415	C₄²	—	67	1	no	—
416	C₄²	—	67	1	no	—
417	C₄²	—	67	1	no	—
418	C₄²	—	67	1	no	—
419	C₄²	—	67	1	no	—
420	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
421	C₄²	D₄×D₆₇	1	1	yes	characteristic
422	C₄²	Q₈×D₆₇	1	1	yes	characteristic
423	C₄²	C₂³×D₆₇	1	1	yes	characteristic
424	C₂³	—	268	1	no	—
425	C₂³	—	67	1	no	—
426	C₂³	—	67	1	no	—
427	C₂³	—	134	2	no	—
428	C₂³	—	134	2	no	—
429	C₂³	C₄⋊C₄×D₆₇	1	1	yes	characteristic
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	268	1	no	—
433	C₂×C₄	—	268	1	no	—
434	C₂×C₄	—	268	1	no	—
435	C₂×C₄	—	67	1	no	—
436	C₂×C₄	—	67	4	no	—
437	C₂×C₄	—	67	1	no	—
438	C₂×C₄	—	67	1	no	—
439	C₂×C₄	—	67	1	no	—
440	C₂×C₄	—	67	1	no	—
441	C₂×C₄	—	67	1	no	—
442	C₂×C₄	—	134	2	no	—
443	C₂×C₄	—	134	2	no	—
444	C₂×C₄	—	134	2	no	—
445	C₂×C₄	—	134	2	no	—
446	C₂×C₄	—	134	1	no	—
447	C₂×C₄	—	134	1	no	—
448	C₂×C₄	—	134	1	no	—
449	C₂×C₄	—	134	1	no	—
450	C₂×C₄	—	134	2	no	—
451	C₂×C₄	—	134	2	no	—
452	C₂×C₄	—	134	2	no	—
453	C₂×C₄	—	134	2	no	—
454	C₂×C₄	—	134	2	no	—
455	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
456	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
457	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
458	C₂×C₄	#14524049	1	2	yes	—
459	C₂×C₄	#14524051	1	2	yes	—
460	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
461	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
462	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
463	C₈	—	2	1	no	—
464	C₈	—	2	1	no	—
465	C₈	—	2	1	no	—
466	C₈	—	2	1	no	—
467	C₈	—	4	2	no	—
468	C₈	—	4	2	no	—
469	C₈	—	268	2	no	—
470	C₈	—	268	2	no	—
471	C₈	—	134	1	no	—
472	C₈	—	134	1	no	—
473	C₈	—	134	1	no	—
474	C₈	—	134	1	no	—
475	C₂²	—	2	1	no	—
476	C₂²	—	2	1	no	—
477	C₂²	—	268	4	no	—
478	C₂²	—	67	2	no	—
479	C₂²	—	134	1	no	—
480	C₂²	—	134	1	no	—
481	C₂²	—	134	1	no	—
482	C₂²	—	134	1	no	—
483	C₂²	—	134	1	no	—
484	C₂²	#16863421	1	2	yes	—
485	C₂²	C₂×C₄⋊C₄×D₆₇	1	1	yes	characteristic
486	C₄	—	2	1	no	—
487	C₄	—	2	1	no	—
488	C₄	—	67	2	no	—
489	C₄	—	134	1	no	—
490	C₄	—	134	1	no	—
491	C₄	—	134	1	no	—
492	C₄	—	134	1	no	—
493	C₄	—	134	1	no	—
494	C₄	#16863421	1	2	yes	—
495	C₄	#16863443	1	2	yes	—
496	C₄	C₂×#14524049	1	1	yes	characteristic
497	C₄	C₂×#14524051	1	1	yes	characteristic
498	C₄	#135×D₆₇	1	2	yes	—
499	C₂	—	2	2	no	—
500	C₂	—	268	1	no	—
501	C₂	—	134	2	no	—
502	C₂	C₂×#16863421	1	1	yes	characteristic
503	C₂	#427×D₆₇	1	1	yes	characteristic
504	C₂	C₂×#16863443	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₆₇×#1564	C₂
socle	C₂²×C₆₇	C₂²×C₄⋊C₄
prime radical	C₂³×D₆₇	C₄⋊C₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	#1564	D₆₇
hypocenter	C₆₇	#17803
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
4. #22558988	C₂	semidirect
5. #22558986	C₂	semidirect
6. #22558982	C₂	semidirect
7. #22558978	C₂	semidirect
8. #22558957	C₂	semidirect
9. #22558955	C₂	semidirect
10. #22558955	C₂	semidirect
11. C₆₇×#1564	C₂	semidirect
40. #16863149	C₂²	semidirect
41. #16863148	C₂²	semidirect
42. #16863148	C₂²	semidirect
43. #16863147	C₂²	semidirect
44. #16863146	C₂²	semidirect
45. #16863146	C₂²	semidirect
49. #335×C₆₇	C₂²	semidirect
50. #334×C₆₇	C₂²	semidirect
263. #1564	D₆₇	semidirect
264. C₆₇	#17803	semidirect
299. #335	C₂×D₆₇	semidirect
303. #334	C₂×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22568837	2
C₂×#16863447	2
#431×D₆₇	2
#22558988	2
#22558986	2
#22558982	2
#22558978	2
#22558957	2
#22558955	2
C₆₇×#1564	2
#17803	67