Subgroups — C₆₇×#33301 #30712552

Order 17152.

2046 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#493×C₆₇	C₂	1	12	yes	maximal, complemented
2	C₆₇×#2867	C₂	1	2	yes	maximal, complemented
3	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₈×C₆₇	C₂²	1	16	yes	—
5	C₂×#127×C₆₇	—	2	8	no	—
6	C₂×#127×C₆₇	C₂²	1	12	yes	—
7	#493×C₆₇	—	2	48	no	—
8	C₂×C₄×C₈×C₆₇	C₂²	1	6	yes	—
9	C₄³×C₆₇	C₂²	1	1	yes	characteristic
10	C₂²×D₄×C₆₇	—	2	24	no	—
11	C₂²×D₄×C₆₇	C₂³	1	8	yes	—
12	C₂×D₈×C₆₇	—	2	96	no	—
13	C₂²×C₈×C₆₇	C₂³	1	4	yes	—
14	#127×C₆₇	—	2	48	no	—
15	#127×C₆₇	—	4	32	no	—
16	C₂×C₄²×C₆₇	D₄	1	4	yes	—
17	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
18	C₄×C₈×C₆₇	D₄	1	24	yes	complemented
19	C₂⁴×C₆₇	—	2	8	no	—
20	C₂×D₄×C₆₇	—	2	48	no	—
21	C₂×D₄×C₆₇	—	4	48	no	—
22	C₂×D₄×C₆₇	—	4	96	no	—
23	C₂²×C₄×C₆₇	C₂×D₄	1	6	yes	—
24	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
25	D₈×C₆₇	—	4	64	no	—
26	C₂×C₈×C₆₇	C₂×D₄	1	24	yes	—
27	C₄²×C₆₇	D₈	1	16	yes	complemented
28	C₄²×C₆₇	C₂×D₄	1	12	yes	—
29	C₂³×C₆₇	—	4	24	no	—
30	C₂³×C₆₇	—	4	32	no	—
31	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
32	D₄×C₆₇	—	4	32	no	—
33	D₄×C₆₇	—	8	96	no	—
34	C₂×C₄×C₆₇	#127	1	12	yes	—
35	C₂×C₄×C₆₇	C₂×D₈	1	24	yes	—
36	C₂×C₄×C₆₇	C₂²×D₄	1	6	yes	—
37	C₈×C₆₇	#127	1	16	yes	complemented
38	C₂²×C₆₇	—	8	8	no	—
39	C₂²×C₆₇	—	8	48	no	—
40	C₂²×C₆₇	C₂×#127	1	3	yes	—
41	C₂²×C₆₇	C₂²×D₈	1	4	yes	—
42	C₄×C₆₇	#493	1	24	yes	complemented
43	C₄×C₆₇	C₂×#127	1	4	yes	—
44	#33301	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂×C₆₇	—	16	8	no	—
46	C₂×C₆₇	#2867	1	1	yes	characteristic
47	C₂×C₆₇	C₂×#493	1	6	yes	—
48	C₂×#493	C₂×C₆₇	1	12	yes	complemented
49	#2867	C₂×C₆₇	1	2	yes	complemented
50	C₄²×C₈	C₂×C₆₇	1	1	yes	characteristic, complemented
51	C₆₇	#33301	1	1	yes	characteristic, direct product factor, Sylow
52	C₂²×D₈	C₂²×C₆₇	1	16	yes	—
53	C₂×#127	—	2	8	no	—
54	C₂×#127	C₂²×C₆₇	1	12	yes	—
55	#493	—	2	48	no	—
56	C₂×C₄×C₈	C₂²×C₆₇	1	6	yes	—
57	C₄³	C₂²×C₆₇	1	1	yes	characteristic
58	C₂²×D₄	—	2	24	no	—
59	C₂²×D₄	C₂³×C₆₇	1	8	yes	—
60	C₂×D₈	—	2	96	no	—
61	C₂²×C₈	C₂³×C₆₇	1	4	yes	—
62	#127	—	2	48	no	—
63	#127	—	4	32	no	—
64	C₂×C₄²	D₄×C₆₇	1	4	yes	—
65	C₂×C₄²	C₂³×C₆₇	1	3	yes	—
66	C₄×C₈	D₄×C₆₇	1	24	yes	complemented
67	C₂⁴	—	2	8	no	—
68	C₂×D₄	—	2	48	no	—
69	C₂×D₄	—	4	48	no	—
70	C₂×D₄	—	4	96	no	—
71	C₂²×C₄	C₂×D₄×C₆₇	1	6	yes	—
72	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
73	D₈	—	4	64	no	—
74	C₂×C₈	C₂×D₄×C₆₇	1	24	yes	—
75	C₄²	D₈×C₆₇	1	16	yes	complemented
76	C₄²	C₂×D₄×C₆₇	1	12	yes	—
77	C₂³	—	4	24	no	—
78	C₂³	—	4	32	no	—
79	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
80	D₄	—	4	32	no	—
81	D₄	—	8	96	no	—
82	C₂×C₄	#127×C₆₇	1	12	yes	—
83	C₂×C₄	C₂×D₈×C₆₇	1	24	yes	—
84	C₂×C₄	C₂²×D₄×C₆₇	1	6	yes	—
85	C₈	#127×C₆₇	1	16	yes	complemented
86	C₂²	—	8	8	no	—
87	C₂²	—	8	48	no	—
88	C₂²	C₂×#127×C₆₇	1	3	yes	—
89	C₂²	C₂²×D₈×C₆₇	1	4	yes	—
90	C₄	#493×C₆₇	1	24	yes	complemented
91	C₄	C₂×#127×C₆₇	1	4	yes	—
92	C₂	—	16	8	no	—
93	C₂	C₆₇×#2867	1	1	yes	characteristic
94	C₂	C₂×#493×C₆₇	1	6	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₄³×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#493×C₆₇	C₂	semidirect
2. C₆₇×#2867	C₂	semidirect
3. C₄²×C₈×C₆₇	C₂	semidirect
18. C₄×C₈×C₆₇	D₄	semidirect
27. C₄²×C₆₇	D₈	semidirect
37. C₈×C₆₇	#127	semidirect
42. C₄×C₆₇	#493	semidirect
44. #33301	C₆₇	direct (N × H)
48. C₂×#493	C₂×C₆₇	semidirect
49. #2867	C₂×C₆₇	semidirect
50. C₄²×C₈	C₂×C₆₇	semidirect
51. C₆₇	#33301	direct (N × H)
66. C₄×C₈	D₄×C₆₇	semidirect
75. C₄²	D₈×C₆₇	semidirect
85. C₈	#127×C₆₇	semidirect
90. C₄	#493×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#493×C₆₇	2
C₆₇×#2867	2
C₄²×C₈×C₆₇	2
#33301	67

Subgroups — C67×#33301 #30712552

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#33301 #30712552