Subgroups — C₆₇×#1408 #22555839

Order 8576.

120 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#493×C₆₇	C₂	1	1	yes	maximal, characteristic
2	#353×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₂×D₈×C₆₇	—	2	2	no	—
4	#127×C₆₇	C₄	1	1	yes	characteristic, complemented
5	#127×C₆₇	C₂²	1	1	yes	characteristic
6	#99×C₆₇	—	2	2	no	—
7	C₄×C₈×C₆₇	C₄	1	1	yes	characteristic, complemented
8	C₂×D₄×C₆₇	—	2	1	no	—
9	C₂×D₄×C₆₇	—	4	1	no	—
10	C₂×D₄×C₆₇	—	4	1	no	—
11	C₂×D₄×C₆₇	D₄	1	2	yes	—
12	D₈×C₆₇	—	4	4	no	—
13	C₂×C₈×C₆₇	—	2	1	no	—
14	C₂²⋊C₄×C₆₇	—	4	2	no	—
15	C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂³×C₆₇	—	2	2	no	—
17	C₂³×C₆₇	—	4	1	no	—
18	D₄×C₆₇	—	4	2	no	—
19	D₄×C₆₇	—	4	2	no	—
20	D₄×C₆₇	—	8	2	no	—
21	D₄×C₆₇	—	8	2	no	—
22	C₂×C₄×C₆₇	—	2	1	no	—
23	C₂×C₄×C₆₇	—	8	2	no	—
24	C₂×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₈×C₆₇	—	2	2	no	—
26	C₂²×C₆₇	—	4	2	no	—
27	C₂²×C₆₇	—	8	2	no	—
28	C₂²×C₆₇	—	8	2	no	—
29	C₂²×C₆₇	—	8	1	no	—
30	C₂²×C₆₇	#99	1	1	yes	characteristic
31	C₄×C₆₇	—	2	2	no	—
32	C₄×C₆₇	—	2	1	no	—
33	C₄×C₆₇	—	16	2	no	—
34	C₂×C₆₇	—	2	1	no	—
35	C₂×C₆₇	—	8	2	no	—
36	C₂×C₆₇	—	16	1	no	—
37	C₂×C₆₇	#353	1	1	yes	characteristic
38	#1408	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₆₇	#1408	1	1	yes	characteristic, direct product factor, Sylow
40	#493	C₂×C₆₇	1	1	yes	characteristic
41	#353	C₂×C₆₇	1	2	yes	complemented
42	C₂×D₈	—	2	2	no	—
43	#127	C₄×C₆₇	1	1	yes	characteristic, complemented
44	#127	C₂²×C₆₇	1	1	yes	characteristic
45	#99	—	2	2	no	—
46	C₄×C₈	C₄×C₆₇	1	1	yes	characteristic, complemented
47	C₂×D₄	—	2	1	no	—
48	C₂×D₄	—	4	1	no	—
49	C₂×D₄	—	4	1	no	—
50	C₂×D₄	D₄×C₆₇	1	2	yes	—
51	D₈	—	4	4	no	—
52	C₂×C₈	—	2	1	no	—
53	C₂²⋊C₄	—	4	2	no	—
54	C₄²	C₂×C₄×C₆₇	1	1	yes	characteristic
55	C₂³	—	2	2	no	—
56	C₂³	—	4	1	no	—
57	D₄	—	4	2	no	—
58	D₄	—	4	2	no	—
59	D₄	—	8	2	no	—
60	D₄	—	8	2	no	—
61	C₂×C₄	—	2	1	no	—
62	C₂×C₄	—	8	2	no	—
63	C₂×C₄	C₂²⋊C₄×C₆₇	1	1	yes	characteristic
64	C₈	—	2	2	no	—
65	C₂²	—	4	2	no	—
66	C₂²	—	8	2	no	—
67	C₂²	—	8	2	no	—
68	C₂²	—	8	1	no	—
69	C₂²	#99×C₆₇	1	1	yes	characteristic
70	C₄	—	2	2	no	—
71	C₄	—	2	1	no	—
72	C₄	—	16	2	no	—
73	C₂	—	2	1	no	—
74	C₂	—	8	2	no	—
75	C₂	—	16	1	no	—
76	C₂	#353×C₆₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₆₇	#353
commutator	C₄²	C₂×C₄×C₆₇
Frattini, cosocle	#127	C₂²×C₆₇
prime radical	#493×C₆₇	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₆₇ ◁ C₂²×C₆₇ ◁ C₂×C₄×C₆₇ ◁ #127×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #353×C₆₇	C₂	semidirect
4. #127×C₆₇	C₄	semidirect
7. C₄×C₈×C₆₇	C₄	semidirect
38. #1408	C₆₇	direct (N × H)
39. C₆₇	#1408	direct (N × H)
41. #353	C₂×C₆₇	semidirect
43. #127	C₄×C₆₇	semidirect
46. C₄×C₈	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
#493×C₆₇	2
#353×C₆₇	2
#1408	67

Subgroups — C67×#1408 #22555839

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#1408 #22555839