Subgroups — #30745203

Order 17152.

486 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22559954	C₂	1	3	yes	maximal
2	C₂×#16863118	C₂	1	3	yes	maximal
3	C₂×C₈²×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
5	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
6	#16863150	C₄	1	12	yes	—
7	#16863118	C₄	1	12	yes	complemented
8	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
9	C₈²×C₆₇	C₄	1	4	yes	complemented
10	C₂²×C₆₇⋊C₈	C₂×C₄	1	6	yes	—
11	C₂×C₄×Dic₆₇	C₂×C₄	1	6	yes	—
12	C₄×C₆₇⋊C₈	C₂×C₄	1	12	yes	—
13	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
14	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
15	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
16	C₄×C₈×C₆₇	C₂×C₄	1	6	yes	—
17	C₂²×Dic₆₇	C₄²	1	4	yes	—
18	C₂×C₆₇⋊C₈	—	2	12	no	—
19	C₂×C₆₇⋊C₈	C₄²	1	12	yes	—
20	C₄×Dic₆₇	—	2	12	no	—
21	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
22	C₂×C₈×C₆₇	—	2	12	no	—
23	C₂×C₈×C₆₇	C₄²	1	12	yes	—
24	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
25	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
26	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
27	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
28	C₂×C₄×C₆₇	C₂×C₄²	1	3	yes	—
29	C₂×C₄×C₆₇	#131	1	12	yes	—
30	C₂×Dic₆₇	—	2	12	no	—
31	C₈×C₆₇	—	2	12	no	—
32	C₆₇⋊C₈	—	2	12	no	—
33	C₂²×C₆₇	C₄³	1	1	yes	characteristic
34	C₂²×C₆₇	#405	1	6	yes	—
35	C₄×C₆₇	#405	1	6	yes	—
36	C₄×C₆₇	#405	1	6	yes	—
37	Dic₆₇	—	4	4	no	—
38	#7845	—	67	1	no	maximal, Sylow
39	C₂×C₆₇	#1454	1	4	yes	—
40	C₂×C₆₇	#1726	1	3	yes	—
41	#1744	—	67	3	no	—
42	C₂×#322	—	67	3	no	—
43	C₂×C₈²	D₆₇	1	1	yes	characteristic
44	C₆₇	#7845	1	1	yes	characteristic, complemented, Sylow
45	C₂×C₄×C₈	—	67	3	no	—
46	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
47	C₄³	—	67	1	no	—
48	#336	—	67	12	no	—
49	#322	—	67	12	no	—
50	C₈²	Dic₆₇	1	4	yes	complemented
51	C₂²×C₈	—	67	6	no	—
52	C₂²×C₈	C₄×D₆₇	1	6	yes	—
53	C₂×C₄²	—	67	6	no	—
54	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
55	C₄×C₈	—	67	12	no	—
56	C₄×C₈	C₄×D₆₇	1	6	yes	—
57	C₄×C₈	C₂×Dic₆₇	1	6	yes	—
58	C₂²×C₄	—	67	4	no	—
59	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
60	C₂×C₈	—	2	12	no	—
61	C₂×C₈	—	67	12	no	—
62	C₂×C₈	—	134	12	no	—
63	C₂×C₈	C₄×Dic₆₇	1	12	yes	—
64	C₄²	—	134	12	no	—
65	C₄²	C₂×C₄×D₆₇	1	3	yes	—
66	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
67	C₂³	C₄²×D₆₇	1	1	yes	characteristic
68	C₂×C₄	—	134	12	no	—
69	C₂×C₄	C₄²×D₆₇	1	3	yes	—
70	C₂×C₄	C₂×C₄×Dic₆₇	1	3	yes	—
71	C₂×C₄	#14524049	1	12	yes	—
72	C₈	—	2	12	no	—
73	C₈	—	134	12	no	—
74	C₂²	C₄²×Dic₆₇	1	1	yes	characteristic
75	C₂²	#16863350	1	6	yes	—
76	C₄	—	268	4	no	—
77	C₄	#16863350	1	6	yes	—
78	C₄	#16863351	1	6	yes	—
79	C₂	#22558508	1	4	yes	—
80	C₂	#22559895	1	3	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator	C₂²×C₆₇	C₄³
Fitting	C₂×C₈²×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×C₈²	D₆₇
hypocenter	C₆₇	#7845
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
7. #16863118	C₄	semidirect
9. C₈²×C₆₇	C₄	semidirect
44. C₆₇	#7845	semidirect
50. C₈²	Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
#22559954	2
C₂×#16863118	2
C₂×C₈²×C₆₇	2
#7845	67