Subgroups — C₆₇×#2158 #22556589

Order 8576.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
2	#427×C₆₇	C₂	1	2	yes	maximal
3	#367×C₆₇	C₂	1	2	yes	maximal, complemented
4	#366×C₆₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
6	#117×C₆₇	—	2	2	no	—
7	C₂×C₁₆×C₆₇	C₄	1	4	yes	complemented
8	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
10	#107×C₆₇	C₂²	1	4	yes	—
11	#106×C₆₇	—	2	2	no	—
12	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
13	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
14	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
15	C₂×C₈×C₆₇	Q₈	1	2	yes	—
16	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
17	C₄⋊C₄×C₆₇	—	2	4	no	—
18	C₂²⋊C₄×C₆₇	—	4	2	no	—
19	C₄²×C₆₇	—	4	2	no	—
20	C₁₆×C₆₇	C₂×C₄	1	2	yes	—
21	C₁₆×C₆₇	C₂×C₄	1	2	yes	—
22	C₂³×C₆₇	D₈	1	1	yes	characteristic
23	C₂×C₄×C₆₇	—	4	4	no	—
24	C₂×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
25	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
26	C₂×C₄×C₆₇	Q₁₆	1	2	yes	—
27	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
28	C₈×C₆₇	C₄⋊C₄	1	2	yes	—
29	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
31	C₂²×C₆₇	—	2	2	no	—
32	C₂²×C₆₇	#107	1	2	yes	—
33	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
34	C₄×C₆₇	—	8	4	no	—
35	C₄×C₆₇	#107	1	2	yes	—
36	C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
37	C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
38	C₂×C₆₇	—	2	2	no	—
39	C₂×C₆₇	C₂×#107	1	1	yes	characteristic
40	C₂×C₆₇	#508	1	2	yes	—
41	#2158	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₆₇	#2158	1	1	yes	characteristic, direct product factor, Sylow
43	C₂²×C₁₆	C₂×C₆₇	1	1	yes	characteristic
44	#427	C₂×C₆₇	1	2	yes	—
45	#367	C₂×C₆₇	1	2	yes	complemented
46	#366	C₂×C₆₇	1	2	yes	complemented
47	C₂²×C₈	C₂²×C₆₇	1	1	yes	characteristic
48	#117	—	2	2	no	—
49	C₂×C₁₆	C₄×C₆₇	1	4	yes	complemented
50	C₂×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
51	C₂×C₁₆	C₂²×C₆₇	1	1	yes	characteristic
52	#107	C₂²×C₆₇	1	4	yes	—
53	#106	—	2	2	no	—
54	C₂²×C₄	D₄×C₆₇	1	1	yes	characteristic
55	C₂×C₈	C₂×C₄×C₆₇	1	2	yes	—
56	C₂×C₈	D₄×C₆₇	1	1	yes	characteristic
57	C₂×C₈	Q₈×C₆₇	1	2	yes	—
58	C₂×C₈	C₂³×C₆₇	1	1	yes	characteristic
59	C₄⋊C₄	—	2	4	no	—
60	C₂²⋊C₄	—	4	2	no	—
61	C₄²	—	4	2	no	—
62	C₁₆	C₂×C₄×C₆₇	1	2	yes	—
63	C₁₆	C₂×C₄×C₆₇	1	2	yes	—
64	C₂³	D₈×C₆₇	1	1	yes	characteristic
65	C₂×C₄	—	4	4	no	—
66	C₂×C₄	C₄⋊C₄×C₆₇	1	2	yes	—
67	C₂×C₄	D₈×C₆₇	1	1	yes	characteristic
68	C₂×C₄	Q₁₆×C₆₇	1	2	yes	—
69	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
70	C₈	C₄⋊C₄×C₆₇	1	2	yes	—
71	C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
72	C₈	C₂×Q₈×C₆₇	1	1	yes	characteristic
73	C₂²	—	2	2	no	—
74	C₂²	#107×C₆₇	1	2	yes	—
75	C₂²	C₂×D₈×C₆₇	1	1	yes	characteristic
76	C₄	—	8	4	no	—
77	C₄	#107×C₆₇	1	2	yes	—
78	C₄	C₂×C₄⋊C₄×C₆₇	1	1	yes	characteristic
79	C₄	C₂×Q₁₆×C₆₇	1	1	yes	characteristic
80	C₂	—	2	2	no	—
81	C₂	C₂×#107×C₆₇	1	1	yes	characteristic
82	C₂	#508×C₆₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₆₇	D₈
commutator	C₈	C₂²×C₄×C₆₇
Frattini, cosocle	C₂×C₈	C₂³×C₆₇
socle	C₂²×C₆₇	C₂×D₈
prime radical	C₂³×C₆₇	D₈
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₂²×C₄×C₆₇ ◁ C₂²×C₈×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. #367×C₆₇	C₂	semidirect
4. #366×C₆₇	C₂	semidirect
7. C₂×C₁₆×C₆₇	C₄	semidirect
41. #2158	C₆₇	direct (N × H)
42. C₆₇	#2158	direct (N × H)
45. #367	C₂×C₆₇	semidirect
46. #366	C₂×C₆₇	semidirect
49. C₂×C₁₆	C₄×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₆×C₆₇	2
#427×C₆₇	2
#367×C₆₇	2
#366×C₆₇	2
#2158	67

Subgroups — C67×#2158 #22556589

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2158 #22556589