Subgroups — C₄×#123×C₆₇ #22556732

Order 8576.

688 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#123×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×D₄×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₄²×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#388×C₆₇	C₂	1	2	yes	maximal, complemented
5	#387×C₆₇	C₂	1	2	yes	maximal, complemented
6	#386×C₆₇	C₂	1	1	yes	maximal, characteristic
7	#385×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
8	#381×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₄⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
10	C₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal, complemented
11	C₄×C₂²⋊C₄×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₆₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
17	#123×C₆₇	C₄	1	8	yes	direct product factor
18	C₄×D₄×C₆₇	—	2	4	no	—
19	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
21	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
22	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₂×C₄²×C₆₇	—	2	4	no	—
29	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
30	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
31	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
33	#95×C₆₇	C₂²	1	2	yes	—
34	#95×C₆₇	C₂²	1	2	yes	—
35	#95×C₆₇	C₂²	1	2	yes	—
36	#95×C₆₇	C₂²	1	2	yes	—
37	#95×C₆₇	C₂²	1	2	yes	—
38	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
39	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₆₇	—	2	2	no	—
41	C₂×D₄×C₆₇	—	2	2	no	—
42	C₂×D₄×C₆₇	C₂×C₄	1	4	yes	—
43	C₂²×C₄×C₆₇	—	2	2	no	—
44	C₂²×C₄×C₆₇	—	2	2	no	—
45	C₂²×C₄×C₆₇	—	2	2	no	—
46	C₂²×C₄×C₆₇	—	2	4	no	—
47	C₂²×C₄×C₆₇	—	2	4	no	—
48	C₂²×C₄×C₆₇	—	2	2	no	—
49	C₂²×C₄×C₆₇	—	2	4	no	—
50	C₂²×C₄×C₆₇	C₂×C₄	1	4	yes	complemented
51	C₂²×C₄×C₆₇	D₄	1	4	yes	—
52	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
56	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
61	C₄⋊C₄×C₆₇	—	2	2	no	—
62	C₄⋊C₄×C₆₇	—	2	4	no	—
63	C₄⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
64	C₂²⋊C₄×C₆₇	—	2	2	no	—
65	C₂²⋊C₄×C₆₇	—	2	2	no	—
66	C₂²⋊C₄×C₆₇	—	2	2	no	—
67	C₂²⋊C₄×C₆₇	—	2	4	no	—
68	C₂²⋊C₄×C₆₇	C₂×C₄	1	4	yes	complemented
69	C₂²⋊C₄×C₆₇	C₂×C₄	1	8	yes	complemented
70	C₄²×C₆₇	—	2	4	no	—
71	C₄²×C₆₇	—	2	4	no	—
72	C₄²×C₆₇	—	2	2	no	—
73	C₄²×C₆₇	—	2	4	no	—
74	C₂³×C₆₇	—	2	2	no	—
75	C₂³×C₆₇	—	2	2	no	—
76	C₂³×C₆₇	—	2	1	no	—
77	C₂³×C₆₇	—	2	2	no	—
78	C₂³×C₆₇	—	2	2	no	—
79	C₂³×C₆₇	—	2	1	no	—
80	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂³×C₆₇	C₂²×C₄	1	2	yes	—
82	C₂³×C₆₇	C₂×D₄	1	2	yes	—
83	C₂³×C₆₇	#41	1	2	yes	—
84	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
85	D₄×C₆₇	—	2	8	no	—
86	C₂×C₄×C₆₇	—	2	8	no	—
87	C₂×C₄×C₆₇	—	2	8	no	—
88	C₂×C₄×C₆₇	—	2	2	no	—
89	C₂×C₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	1	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	2	2	no	—
93	C₂×C₄×C₆₇	—	2	2	no	—
94	C₂×C₄×C₆₇	—	2	2	no	—
95	C₂×C₄×C₆₇	—	2	2	no	—
96	C₂×C₄×C₆₇	—	2	1	no	—
97	C₂×C₄×C₆₇	—	2	8	no	—
98	C₂×C₄×C₆₇	—	2	1	no	—
99	C₂×C₄×C₆₇	—	2	1	no	—
100	C₂×C₄×C₆₇	—	2	2	no	—
101	C₂×C₄×C₆₇	—	2	2	no	—
102	C₂×C₄×C₆₇	—	2	2	no	—
103	C₂×C₄×C₆₇	—	2	2	no	—
104	C₂×C₄×C₆₇	—	2	2	no	—
105	C₂×C₄×C₆₇	—	4	4	no	—
106	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
107	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
108	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
110	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
111	C₂×C₄×C₆₇	#41	1	4	yes	—
112	C₂×C₄×C₆₇	#41	1	4	yes	—
113	C₂²×C₆₇	—	2	4	no	—
114	C₂²×C₆₇	—	2	2	no	—
115	C₂²×C₆₇	—	2	2	no	—
116	C₂²×C₆₇	—	2	4	no	—
117	C₂²×C₆₇	—	2	2	no	—
118	C₂²×C₆₇	—	2	4	no	—
119	C₂²×C₆₇	—	4	1	no	—
120	C₂²×C₆₇	—	4	1	no	—
121	C₂²×C₆₇	—	4	2	no	—
122	C₂²×C₆₇	C₄×D₄	1	4	yes	—
123	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
124	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
125	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
126	C₂²×C₆₇	C₂×#41	1	2	yes	—
127	C₂²×C₆₇	C₂×#41	1	2	yes	—
128	C₄×C₆₇	—	2	4	no	—
129	C₄×C₆₇	—	2	2	no	—
130	C₄×C₆₇	—	2	4	no	—
131	C₄×C₆₇	—	2	2	no	—
132	C₄×C₆₇	—	2	4	no	—
133	C₄×C₆₇	—	4	1	no	—
134	C₄×C₆₇	—	4	1	no	—
135	C₄×C₆₇	—	4	2	no	—
136	C₄×C₆₇	#123	1	4	yes	direct product factor
137	C₂×C₆₇	—	2	4	no	—
138	C₂×C₆₇	—	4	2	no	—
139	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
140	C₂×C₆₇	C₄×#41	1	2	yes	—
141	C₂×C₆₇	C₂×#123	1	1	yes	characteristic
142	C₂×C₆₇	#525	1	1	yes	characteristic
143	C₂×C₆₇	#529	1	2	yes	—
144	C₄×#123	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
145	C₆₇	C₄×#123	1	1	yes	characteristic, direct product factor, Sylow
146	C₂×#123	C₂×C₆₇	1	1	yes	characteristic
147	C₂×C₄×D₄	C₂×C₆₇	1	1	yes	characteristic
148	C₂²×C₄²	C₂×C₆₇	1	1	yes	characteristic, complemented
149	#388	C₂×C₆₇	1	2	yes	complemented
150	#387	C₂×C₆₇	1	2	yes	complemented
151	#386	C₂×C₆₇	1	1	yes	characteristic
152	#385	C₂×C₆₇	1	1	yes	characteristic, complemented
153	#381	C₂×C₆₇	1	1	yes	characteristic, complemented
154	C₄×C₄⋊C₄	C₂×C₆₇	1	2	yes	complemented
155	C₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	complemented
156	C₄×C₂²⋊C₄	C₂×C₆₇	1	1	yes	characteristic, complemented
157	C₂²×D₄	C₂²×C₆₇	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
159	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
160	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
161	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
162	#123	C₄×C₆₇	1	8	yes	direct product factor
163	C₄×D₄	—	2	4	no	—
164	C₂×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
165	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
166	C₂×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
170	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
171	C₂×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
172	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
173	C₂×C₄²	—	2	4	no	—
174	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
175	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
176	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
177	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
178	#95	C₂²×C₆₇	1	2	yes	—
179	#95	C₂²×C₆₇	1	2	yes	—
180	#95	C₂²×C₆₇	1	2	yes	—
181	#95	C₂²×C₆₇	1	2	yes	—
182	#95	C₂²×C₆₇	1	2	yes	—
183	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
184	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	C₂×C₄×C₆₇	1	4	yes	—
188	C₂²×C₄	—	2	2	no	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	4	no	—
192	C₂²×C₄	—	2	4	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂×C₄×C₆₇	1	4	yes	complemented
196	C₂²×C₄	D₄×C₆₇	1	4	yes	—
197	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
198	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
199	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
200	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
201	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
202	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
203	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
204	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
205	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
206	C₄⋊C₄	—	2	2	no	—
207	C₄⋊C₄	—	2	4	no	—
208	C₄⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
209	C₂²⋊C₄	—	2	2	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	C₂×C₄×C₆₇	1	4	yes	complemented
214	C₂²⋊C₄	C₂×C₄×C₆₇	1	8	yes	complemented
215	C₄²	—	2	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	4	no	—
219	C₂³	—	2	2	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	1	no	—
222	C₂³	—	2	2	no	—
223	C₂³	—	2	2	no	—
224	C₂³	—	2	1	no	—
225	C₂³	C₂²×C₄×C₆₇	1	2	yes	—
226	C₂³	C₂²×C₄×C₆₇	1	2	yes	—
227	C₂³	C₂×D₄×C₆₇	1	2	yes	—
228	C₂³	#41×C₆₇	1	2	yes	—
229	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
230	D₄	—	2	8	no	—
231	C₂×C₄	—	2	8	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	8	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	2	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	4	4	no	—
251	C₂×C₄	C₂²×C₄×C₆₇	1	4	yes	—
252	C₂×C₄	C₂²×C₄×C₆₇	1	4	yes	—
253	C₂×C₄	C₂²×C₄×C₆₇	1	2	yes	—
254	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
255	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
256	C₂×C₄	#41×C₆₇	1	4	yes	—
257	C₂×C₄	#41×C₆₇	1	4	yes	—
258	C₂²	—	2	4	no	—
259	C₂²	—	2	2	no	—
260	C₂²	—	2	2	no	—
261	C₂²	—	2	4	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	4	no	—
264	C₂²	—	4	1	no	—
265	C₂²	—	4	1	no	—
266	C₂²	—	4	2	no	—
267	C₂²	C₄×D₄×C₆₇	1	4	yes	—
268	C₂²	C₂³×C₄×C₆₇	1	1	yes	characteristic
269	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
270	C₂²	C₂×#41×C₆₇	1	1	yes	characteristic
271	C₂²	C₂×#41×C₆₇	1	2	yes	—
272	C₂²	C₂×#41×C₆₇	1	2	yes	—
273	C₄	—	2	4	no	—
274	C₄	—	2	2	no	—
275	C₄	—	2	4	no	—
276	C₄	—	2	2	no	—
277	C₄	—	2	4	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	—	4	2	no	—
281	C₄	#123×C₆₇	1	4	yes	direct product factor
282	C₂	—	2	4	no	—
283	C₂	—	4	2	no	—
284	C₂	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
285	C₂	C₄×#41×C₆₇	1	2	yes	—
286	C₂	C₂×#123×C₆₇	1	1	yes	characteristic
287	C₂	#525×C₆₇	1	1	yes	characteristic
288	C₂	#529×C₆₇	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
3. C₂²×C₄²×C₆₇	C₂	semidirect
4. #388×C₆₇	C₂	semidirect
5. #387×C₆₇	C₂	semidirect
7. #385×C₆₇	C₂	semidirect
8. #381×C₆₇	C₂	semidirect
9. C₄×C₄⋊C₄×C₆₇	C₂	semidirect
10. C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
11. C₄×C₂²⋊C₄×C₆₇	C₂	semidirect
17. #123×C₆₇	C₄	direct (N × H)
50. C₂²×C₄×C₆₇	C₂×C₄	semidirect
63. C₄⋊C₄×C₆₇	C₂×C₄	semidirect
68. C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
69. C₂²⋊C₄×C₆₇	C₂×C₄	semidirect
136. C₄×C₆₇	#123	direct (N × H)
144. C₄×#123	C₆₇	direct (N × H)
145. C₆₇	C₄×#123	direct (N × H)
148. C₂²×C₄²	C₂×C₆₇	semidirect
149. #388	C₂×C₆₇	semidirect
150. #387	C₂×C₆₇	semidirect
152. #385	C₂×C₆₇	semidirect
153. #381	C₂×C₆₇	semidirect
154. C₄×C₄⋊C₄	C₂×C₆₇	semidirect
155. C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
156. C₄×C₂²⋊C₄	C₂×C₆₇	semidirect
162. #123	C₄×C₆₇	direct (N × H)
195. C₂²×C₄	C₂×C₄×C₆₇	semidirect
208. C₄⋊C₄	C₂×C₄×C₆₇	semidirect
213. C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
214. C₂²⋊C₄	C₂×C₄×C₆₇	semidirect
281. C₄	#123×C₆₇	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#123×C₆₇	2
C₂×C₄×D₄×C₆₇	2
C₂²×C₄²×C₆₇	2
#388×C₆₇	2
#387×C₆₇	2
#386×C₆₇	2
#385×C₆₇	2
#381×C₆₇	2
C₄×C₄⋊C₄×C₆₇	2
C₄×C₂²⋊C₄×C₆₇	2
C₄×#123	67

Subgroups — C4×#123×C67 #22556732

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#123×C₆₇ #22556732