Subgroups — C₆₇×#22519 #30701770

Order 17152.

1086 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₆₇×#2355	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#1991	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1990	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#1989	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1804	C₂	1	2	yes	maximal
9	C₆₇×#1802	C₂	1	1	yes	maximal, characteristic
10	C₄×#102×C₆₇	C₂	1	1	yes	maximal, characteristic
11	C₄×#102×C₆₇	C₂	1	2	yes	maximal
12	#554×C₆₇	—	2	4	no	—
13	#554×C₆₇	C₄	1	8	yes	complemented
14	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#125×C₆₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊Q₈×C₆₇	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₆₇	—	2	2	no	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₆₇	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
22	C₈×Q₈×C₆₇	—	2	4	no	—
23	C₂×C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
25	C₂×#102×C₆₇	C₂²	1	2	yes	—
26	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#102×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
30	#391×C₆₇	—	2	1	no	—
31	#387×C₆₇	—	2	2	no	—
32	#384×C₆₇	C₂²	1	1	yes	characteristic
33	#382×C₆₇	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
37	#340×C₆₇	C₂²	1	2	yes	—
38	#340×C₆₇	C₂²	1	2	yes	—
39	#340×C₆₇	C₂²	1	1	yes	characteristic
40	#340×C₆₇	C₂²	1	2	yes	—
41	#340×C₆₇	C₂²	1	2	yes	—
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₆₇	—	2	1	no	—
45	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
46	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
47	C₂³×C₄×C₆₇	—	2	1	no	—
48	C₂³×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
50	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₆₇	—	2	2	no	—
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
54	#125×C₆₇	—	2	4	no	—
55	#125×C₆₇	C₂×C₄	1	8	yes	—
56	C₂²⋊Q₈×C₆₇	—	2	8	no	—
57	C₂²⋊Q₈×C₆₇	—	2	16	no	—
58	C₄×Q₈×C₆₇	C₂×C₄	1	4	yes	—
59	C₄×Q₈×C₆₇	D₄	1	4	yes	—
60	C₄×D₄×C₆₇	—	2	4	no	—
61	C₄×D₄×C₆₇	—	2	2	no	—
62	C₄×D₄×C₆₇	C₂×C₄	1	8	yes	—
63	C₄×D₄×C₆₇	C₂×C₄	1	4	yes	—
64	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
65	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
70	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
71	C₂×C₄⋊C₄×C₆₇	D₄	1	2	yes	—
72	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
73	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
78	C₂×C₂²⋊C₄×C₆₇	—	2	2	no	—
79	C₂×C₄²×C₆₇	—	2	1	no	—
80	C₂×C₄²×C₆₇	—	2	2	no	—
81	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
82	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
83	C₄⋊C₈×C₆₇	—	2	4	no	—
84	C₄⋊C₈×C₆₇	—	2	2	no	—
85	#102×C₆₇	—	2	4	no	—
86	#102×C₆₇	—	2	2	no	—
87	#102×C₆₇	Q₈	1	4	yes	complemented
88	C₄×C₈×C₆₇	—	2	4	no	—
89	C₄×C₈×C₆₇	—	2	2	no	—
90	#95×C₆₇	—	2	2	no	—
91	#95×C₆₇	—	2	1	no	—
92	#95×C₆₇	—	2	2	no	—
93	#95×C₆₇	—	2	2	no	—
94	C₂⁴×C₆₇	—	2	1	no	—
95	C₂×Q₈×C₆₇	—	2	4	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	2	2	no	—
98	C₂×Q₈×C₆₇	—	8	1	no	—
99	C₂×Q₈×C₆₇	—	8	1	no	—
100	C₂×Q₈×C₆₇	C₂²⋊C₄	1	4	yes	—
101	C₂×D₄×C₆₇	—	2	1	no	—
102	C₂×D₄×C₆₇	—	2	1	no	—
103	C₂×D₄×C₆₇	—	2	1	no	—
104	C₂×D₄×C₆₇	C₂²×C₄	1	2	yes	—
105	C₂×D₄×C₆₇	C₂×Q₈	1	2	yes	—
106	C₂×D₄×C₆₇	#41	1	2	yes	—
107	C₂²×C₄×C₆₇	—	2	1	no	—
108	C₂²×C₄×C₆₇	—	2	1	no	—
109	C₂²×C₄×C₆₇	—	2	2	no	—
110	C₂²×C₄×C₆₇	—	2	4	no	—
111	C₂²×C₄×C₆₇	—	2	8	no	—
112	C₂²×C₄×C₆₇	—	2	2	no	—
113	C₂²×C₄×C₆₇	—	2	1	no	—
114	C₂²×C₄×C₆₇	—	4	2	no	—
115	C₂²×C₄×C₆₇	—	4	2	no	—
116	C₂²×C₄×C₆₇	—	8	2	no	—
117	C₂²×C₄×C₆₇	—	8	2	no	—
118	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
121	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
122	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
123	C₂×C₈×C₆₇	—	2	1	no	—
124	C₂×C₈×C₆₇	—	2	1	no	—
125	C₂×C₈×C₆₇	—	2	1	no	—
126	C₂×C₈×C₆₇	—	2	2	no	—
127	C₂×C₈×C₆₇	—	2	2	no	—
128	C₂×C₈×C₆₇	—	2	1	no	—
129	C₂×C₈×C₆₇	—	2	2	no	—
130	C₂×C₈×C₆₇	—	2	1	no	—
131	C₂×C₈×C₆₇	C₂×Q₈	1	2	yes	—
132	C₄⋊C₄×C₆₇	—	2	8	no	—
133	C₄⋊C₄×C₆₇	—	2	4	no	—
134	C₄⋊C₄×C₆₇	—	2	8	no	—
135	C₄⋊C₄×C₆₇	—	2	1	no	—
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	—	2	1	no	—
138	C₄⋊C₄×C₆₇	—	2	1	no	—
139	C₄⋊C₄×C₆₇	—	2	2	no	—
140	C₄⋊C₄×C₆₇	—	2	1	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	4	2	no	—
143	C₄⋊C₄×C₆₇	—	8	1	no	—
144	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	4	yes	—
145	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	8	yes	—
146	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
147	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
148	C₄⋊C₄×C₆₇	C₂×Q₈	1	2	yes	—
149	C₂²⋊C₄×C₆₇	—	2	4	no	—
150	C₂²⋊C₄×C₆₇	—	2	8	no	—
151	C₄²×C₆₇	—	4	2	no	—
152	C₄²×C₆₇	—	8	2	no	—
153	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
154	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
155	C₄²×C₆₇	C₂×D₄	1	4	yes	—
156	C₄²×C₆₇	C₂×D₄	1	2	yes	—
157	C₂³×C₆₇	—	2	2	no	—
158	C₂³×C₆₇	—	2	2	no	—
159	C₂³×C₆₇	—	2	2	no	—
160	C₂³×C₆₇	—	4	1	no	—
161	C₂³×C₆₇	—	4	1	no	—
162	C₂³×C₆₇	—	4	1	no	—
163	C₂³×C₆₇	—	4	1	no	—
164	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
165	Q₈×C₆₇	—	2	8	no	—
166	Q₈×C₆₇	—	8	4	no	—
167	D₄×C₆₇	—	2	2	no	—
168	D₄×C₆₇	—	2	2	no	—
169	D₄×C₆₇	—	2	2	no	—
170	D₄×C₆₇	C₄×Q₈	1	4	yes	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	2	no	—
174	C₂×C₄×C₆₇	—	2	1	no	—
175	C₂×C₄×C₆₇	—	2	2	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	2	no	—
178	C₂×C₄×C₆₇	—	2	4	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	4	1	no	—
181	C₂×C₄×C₆₇	—	4	1	no	—
182	C₂×C₄×C₆₇	—	4	1	no	—
183	C₂×C₄×C₆₇	—	4	1	no	—
184	C₂×C₄×C₆₇	—	4	2	no	—
185	C₂×C₄×C₆₇	—	4	2	no	—
186	C₂×C₄×C₆₇	—	4	8	no	—
187	C₂×C₄×C₆₇	—	4	1	no	—
188	C₂×C₄×C₆₇	—	4	2	no	—
189	C₂×C₄×C₆₇	—	4	1	no	—
190	C₂×C₄×C₆₇	—	4	2	no	—
191	C₂×C₄×C₆₇	—	4	2	no	—
192	C₂×C₄×C₆₇	—	4	1	no	—
193	C₂×C₄×C₆₇	—	8	8	no	—
194	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
195	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
196	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	4	yes	—
197	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	2	yes	—
198	C₂×C₄×C₆₇	#135	1	2	yes	—
199	C₂×C₄×C₆₇	#135	1	2	yes	—
200	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
205	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
206	C₈×C₆₇	—	2	2	no	—
207	C₈×C₆₇	—	4	1	no	—
208	C₈×C₆₇	—	4	2	no	—
209	C₂²×C₆₇	—	2	4	no	—
210	C₂²×C₆₇	—	4	2	no	—
211	C₂²×C₆₇	—	4	2	no	—
212	C₂²×C₆₇	—	4	2	no	—
213	C₂²×C₆₇	—	4	2	no	—
214	C₂²×C₆₇	—	4	2	no	—
215	C₂²×C₆₇	—	4	2	no	—
216	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
217	C₂²×C₆₇	#548	1	1	yes	characteristic
218	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
219	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
221	C₂²×C₆₇	#576	1	1	yes	characteristic
222	C₂²×C₆₇	#577	1	1	yes	characteristic
223	C₄×C₆₇	—	2	4	no	—
224	C₄×C₆₇	—	2	4	no	—
225	C₄×C₆₇	—	2	2	no	—
226	C₄×C₆₇	—	4	2	no	—
227	C₄×C₆₇	—	4	2	no	—
228	C₄×C₆₇	—	4	4	no	—
229	C₄×C₆₇	—	8	1	no	—
230	C₄×C₆₇	—	8	2	no	—
231	C₄×C₆₇	#416	1	4	yes	—
232	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
233	C₄×C₆₇	#519	1	1	yes	characteristic
234	C₄×C₆₇	#548	1	1	yes	characteristic
235	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
236	#22519	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
237	C₂×C₆₇	—	4	4	no	—
238	C₂×C₆₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
239	C₂×C₆₇	C₂×#416	1	1	yes	characteristic
240	C₂×C₆₇	#2897	1	1	yes	characteristic
241	C₂×C₆₇	#3360	1	1	yes	characteristic
242	C₂×C₆₇	#3361	1	1	yes	characteristic
243	C₂×C₆₇	#3380	1	1	yes	characteristic
244	C₂×C₆₇	#3381	1	1	yes	characteristic
245	C₂×#554	C₂×C₆₇	1	1	yes	characteristic
246	C₂×C₈×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
247	#2355	C₂×C₆₇	1	1	yes	characteristic, complemented
248	#1991	C₂×C₆₇	1	2	yes	complemented
249	#1990	C₂×C₆₇	1	1	yes	characteristic, complemented
250	#1989	C₂×C₆₇	1	2	yes	complemented
251	#1987	C₂×C₆₇	1	1	yes	characteristic, complemented
252	#1804	C₂×C₆₇	1	2	yes	—
253	#1802	C₂×C₆₇	1	1	yes	characteristic
254	C₄×#102	C₂×C₆₇	1	1	yes	characteristic
255	C₄×#102	C₂×C₆₇	1	2	yes	—
256	C₆₇	#22519	1	1	yes	characteristic, direct product factor, Sylow
257	#554	—	2	4	no	—
258	#554	C₄×C₆₇	1	8	yes	complemented
259	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
260	C₂×#125	C₂²×C₆₇	1	2	yes	—
261	C₂×C₂²⋊Q₈	—	2	1	no	—
262	C₂×C₂²⋊Q₈	—	2	2	no	—
263	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
264	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
265	C₂×C₄×D₄	C₂²×C₆₇	1	2	yes	—
266	C₂²×C₄⋊C₄	—	2	1	no	—
267	C₈×Q₈	—	2	4	no	—
268	C₂×C₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
269	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
270	C₂×#102	C₂²×C₆₇	1	2	yes	—
271	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
272	C₂×#102	C₂²×C₆₇	1	1	yes	characteristic
273	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
274	C₂×C₄×C₈	C₂²×C₆₇	1	2	yes	—
275	#391	—	2	1	no	—
276	#387	—	2	2	no	—
277	#384	C₂²×C₆₇	1	1	yes	characteristic
278	#382	C₂²×C₆₇	1	2	yes	—
279	C₄×C₄⋊C₄	—	2	1	no	—
280	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
281	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
282	#340	C₂²×C₆₇	1	2	yes	—
283	#340	C₂²×C₆₇	1	2	yes	—
284	#340	C₂²×C₆₇	1	2	yes	—
285	#340	C₂²×C₆₇	1	2	yes	—
286	#340	C₂²×C₆₇	1	1	yes	characteristic
287	#340	C₂²×C₆₇	1	2	yes	—
288	#340	C₂²×C₆₇	1	1	yes	characteristic
289	C₂²×Q₈	—	2	1	no	—
290	C₂²×Q₈	D₄×C₆₇	1	1	yes	characteristic
291	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
292	C₂³×C₄	—	2	1	no	—
293	C₂³×C₄	—	2	2	no	—
294	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
295	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
296	C₂²×C₈	C₂³×C₆₇	1	1	yes	characteristic
297	C₄⋊Q₈	—	2	2	no	—
298	C₄⋊Q₈	C₂×C₄×C₆₇	1	4	yes	—
299	#125	—	2	4	no	—
300	#125	C₂×C₄×C₆₇	1	8	yes	—
301	C₂²⋊Q₈	—	2	8	no	—
302	C₂²⋊Q₈	—	2	16	no	—
303	C₄×Q₈	C₂×C₄×C₆₇	1	4	yes	—
304	C₄×Q₈	D₄×C₆₇	1	4	yes	—
305	C₄×D₄	—	2	4	no	—
306	C₄×D₄	—	2	2	no	—
307	C₄×D₄	C₂×C₄×C₆₇	1	8	yes	—
308	C₄×D₄	C₂×C₄×C₆₇	1	4	yes	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	2	2	no	—
312	C₂×C₄⋊C₄	—	2	8	no	—
313	C₂×C₄⋊C₄	—	2	1	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	D₄×C₆₇	1	1	yes	characteristic
316	C₂×C₄⋊C₄	D₄×C₆₇	1	2	yes	—
317	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
318	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
319	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
320	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
321	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
322	C₂×C₂²⋊C₄	—	2	1	no	—
323	C₂×C₂²⋊C₄	—	2	2	no	—
324	C₂×C₄²	—	2	1	no	—
325	C₂×C₄²	—	2	2	no	—
326	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
327	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
328	C₄⋊C₈	—	2	4	no	—
329	C₄⋊C₈	—	2	2	no	—
330	#102	—	2	4	no	—
331	#102	—	2	2	no	—
332	#102	Q₈×C₆₇	1	4	yes	complemented
333	C₄×C₈	—	2	4	no	—
334	C₄×C₈	—	2	2	no	—
335	#95	—	2	2	no	—
336	#95	—	2	1	no	—
337	#95	—	2	2	no	—
338	#95	—	2	2	no	—
339	C₂⁴	—	2	1	no	—
340	C₂×Q₈	—	2	4	no	—
341	C₂×Q₈	—	2	2	no	—
342	C₂×Q₈	—	2	2	no	—
343	C₂×Q₈	—	8	1	no	—
344	C₂×Q₈	—	8	1	no	—
345	C₂×Q₈	C₂²⋊C₄×C₆₇	1	4	yes	—
346	C₂×D₄	—	2	1	no	—
347	C₂×D₄	—	2	1	no	—
348	C₂×D₄	—	2	1	no	—
349	C₂×D₄	C₂²×C₄×C₆₇	1	2	yes	—
350	C₂×D₄	C₂×Q₈×C₆₇	1	2	yes	—
351	C₂×D₄	#41×C₆₇	1	2	yes	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	2	no	—
355	C₂²×C₄	—	2	4	no	—
356	C₂²×C₄	—	2	8	no	—
357	C₂²×C₄	—	2	1	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	4	2	no	—
360	C₂²×C₄	—	4	2	no	—
361	C₂²×C₄	—	8	2	no	—
362	C₂²×C₄	—	8	2	no	—
363	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₆₇	1	2	yes	—
365	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
367	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
368	C₂×C₈	—	2	1	no	—
369	C₂×C₈	—	2	1	no	—
370	C₂×C₈	—	2	2	no	—
371	C₂×C₈	—	2	1	no	—
372	C₂×C₈	—	2	2	no	—
373	C₂×C₈	—	2	1	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	1	no	—
376	C₂×C₈	C₂×Q₈×C₆₇	1	2	yes	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	4	no	—
379	C₄⋊C₄	—	2	8	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	2	no	—
382	C₄⋊C₄	—	2	1	no	—
383	C₄⋊C₄	—	2	1	no	—
384	C₄⋊C₄	—	2	2	no	—
385	C₄⋊C₄	—	2	1	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	—	4	2	no	—
388	C₄⋊C₄	—	8	1	no	—
389	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	4	yes	—
390	C₄⋊C₄	C₂²⋊C₄×C₆₇	1	8	yes	—
391	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
392	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
393	C₄⋊C₄	C₂×Q₈×C₆₇	1	2	yes	—
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	—	2	8	no	—
396	C₄²	—	4	2	no	—
397	C₄²	—	8	2	no	—
398	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
399	C₄²	C₂²×C₄×C₆₇	1	2	yes	—
400	C₄²	C₂×D₄×C₆₇	1	4	yes	—
401	C₄²	C₂×D₄×C₆₇	1	2	yes	—
402	C₂³	—	2	2	no	—
403	C₂³	—	2	2	no	—
404	C₂³	—	2	2	no	—
405	C₂³	—	4	1	no	—
406	C₂³	—	4	1	no	—
407	C₂³	—	4	1	no	—
408	C₂³	—	4	1	no	—
409	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
410	Q₈	—	2	8	no	—
411	Q₈	—	8	4	no	—
412	D₄	—	2	2	no	—
413	D₄	—	2	2	no	—
414	D₄	—	2	2	no	—
415	D₄	C₄×Q₈×C₆₇	1	4	yes	—
416	C₂×C₄	—	2	2	no	—
417	C₂×C₄	—	2	2	no	—
418	C₂×C₄	—	2	2	no	—
419	C₂×C₄	—	2	1	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	4	1	no	—
426	C₂×C₄	—	4	1	no	—
427	C₂×C₄	—	4	1	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	2	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	8	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	1	no	—
435	C₂×C₄	—	4	2	no	—
436	C₂×C₄	—	4	2	no	—
437	C₂×C₄	—	4	1	no	—
438	C₂×C₄	—	8	8	no	—
439	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
440	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
441	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	4	yes	—
442	C₂×C₄	C₂×C₂²⋊C₄×C₆₇	1	2	yes	—
443	C₂×C₄	#135×C₆₇	1	2	yes	—
444	C₂×C₄	#135×C₆₇	1	2	yes	—
445	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
447	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
448	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
449	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
450	C₂×C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
451	C₈	—	2	2	no	—
452	C₈	—	4	2	no	—
453	C₈	—	4	1	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	4	2	no	—
456	C₂²	—	4	2	no	—
457	C₂²	—	4	2	no	—
458	C₂²	—	4	2	no	—
459	C₂²	—	4	2	no	—
460	C₂²	—	4	2	no	—
461	C₂²	C₂²×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
462	C₂²	#548×C₆₇	1	1	yes	characteristic
463	C₂²	D₄×Q₈×C₆₇	1	1	yes	characteristic
464	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
465	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
466	C₂²	#576×C₆₇	1	1	yes	characteristic
467	C₂²	#577×C₆₇	1	1	yes	characteristic
468	C₄	—	2	4	no	—
469	C₄	—	2	4	no	—
470	C₄	—	2	2	no	—
471	C₄	—	4	2	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	4	no	—
474	C₄	—	8	1	no	—
475	C₄	—	8	2	no	—
476	C₄	#416×C₆₇	1	4	yes	—
477	C₄	C₂×C₄×Q₈×C₆₇	1	1	yes	characteristic
478	C₄	#519×C₆₇	1	1	yes	characteristic
479	C₄	#548×C₆₇	1	1	yes	characteristic
480	C₄	D₄×Q₈×C₆₇	1	1	yes	characteristic
481	C₂	—	4	4	no	—
482	C₂	Q₈×C₂²⋊C₄×C₆₇	1	1	yes	characteristic
483	C₂	C₂×#416×C₆₇	1	1	yes	characteristic
484	C₂	C₆₇×#2897	1	1	yes	characteristic
485	C₂	C₆₇×#3360	1	1	yes	characteristic
486	C₂	C₆₇×#3361	1	1	yes	characteristic
487	C₂	C₆₇×#3380	1	1	yes	characteristic
488	C₂	C₆₇×#3381	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂×C₄×Q₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₆₇	C₂	semidirect
3. C₆₇×#2355	C₂	semidirect
4. C₆₇×#1991	C₂	semidirect
5. C₆₇×#1990	C₂	semidirect
6. C₆₇×#1989	C₂	semidirect
7. C₆₇×#1987	C₂	semidirect
13. #554×C₆₇	C₄	semidirect
87. #102×C₆₇	Q₈	semidirect
236. #22519	C₆₇	direct (N × H)
246. C₂×C₈×Q₈	C₂×C₆₇	semidirect
247. #2355	C₂×C₆₇	semidirect
248. #1991	C₂×C₆₇	semidirect
249. #1990	C₂×C₆₇	semidirect
250. #1989	C₂×C₆₇	semidirect
251. #1987	C₂×C₆₇	semidirect
256. C₆₇	#22519	direct (N × H)
258. #554	C₄×C₆₇	semidirect
332. #102	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#554×C₆₇	2
C₂×C₈×Q₈×C₆₇	2
C₆₇×#2355	2
C₆₇×#1991	2
C₆₇×#1990	2
C₆₇×#1989	2
C₆₇×#1987	2
C₆₇×#1804	2
C₆₇×#1802	2
C₄×#102×C₆₇	2
#22519	67

Subgroups — C67×#22519 #30701770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#22519 #30701770