Subgroups — #36326433

Order 21376.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923685	C₂	1	1	yes	maximal, characteristic
2	#24923102	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₆₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₂×D₁₆₇	C₂	1	2	yes	maximal, complemented
5	M₆₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128838	C₄	1	1	yes	characteristic
7	#19128772	C₄	1	1	yes	characteristic
8	#19128771	C₄	1	2	yes	—
9	C₁₆×D₁₆₇	C₂²	1	2	yes	—
10	C₁₆₇⋊C₃₂	C₂²	1	1	yes	characteristic
11	C₁₆₇⋊C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₃₂×C₁₆₇	C₂²	1	2	yes	—
14	#14581950	C₈	1	1	yes	characteristic
15	#14581934	C₈	1	1	yes	characteristic
16	#14581933	C₈	1	2	yes	—
17	C₈×D₁₆₇	C₂×C₄	1	2	yes	—
18	C₁₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₁₆₇	C₂×C₄	1	1	yes	characteristic
20	C₁₆×C₁₆₇	C₂×C₄	1	1	yes	characteristic
21	C₁₆×C₁₆₇	C₂³	1	1	yes	characteristic
22	Dic₃₃₄	C₁₆	1	1	yes	characteristic
23	D₆₆₈	C₁₆	1	1	yes	characteristic
24	C₂²⋊D₁₆₇	C₁₆	1	2	yes	—
25	C₄×D₁₆₇	C₂×C₈	1	2	yes	—
26	C₁₆₇⋊C₈	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₁₆₇	C₂×C₈	1	1	yes	characteristic
28	C₈×C₁₆₇	C₂×C₈	1	1	yes	characteristic
29	C₈×C₁₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₂²×C₁₆₇	C₂×C₁₆	1	1	yes	characteristic
31	C₂×D₁₆₇	C₂×C₁₆	1	2	yes	—
32	C₄×C₁₆₇	C₂×C₁₆	1	1	yes	characteristic
33	C₄×C₁₆₇	C₂²×C₈	1	1	yes	characteristic
34	Dic₁₆₇	C₂×C₁₆	1	2	yes	—
35	C₂×C₁₆₇	—	2	1	no	—
36	C₂×C₁₆₇	C₂²×C₁₆	1	1	yes	characteristic
37	D₁₆₇	—	2	2	no	—
38	C₁₆₇	#2258	1	1	yes	characteristic, complemented, Sylow
39	#2258	—	167	1	no	maximal, Sylow
40	#504	—	167	1	no	—
41	M₆₄	—	167	2	no	—
42	M₆₄	D₁₆₇	1	1	yes	characteristic, complemented
43	C₂×C₃₂	—	167	1	no	—
44	C₂×C₃₂	—	167	2	no	—
45	#131	—	167	1	no	—
46	M₃₂	—	167	1	no	—
47	M₃₂	—	167	2	no	—
48	C₂×C₁₆	—	167	2	no	—
49	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
50	C₃₂	—	167	1	no	—
51	C₃₂	—	167	1	no	—
52	C₃₂	C₂×D₁₆₇	1	2	yes	—
53	#41	—	167	1	no	—
54	M₁₆	—	167	1	no	—
55	M₁₆	—	167	2	no	—
56	C₂×C₈	—	167	2	no	—
57	C₂×C₈	C₄×D₁₆₇	1	1	yes	characteristic
58	C₁₆	—	167	2	no	—
59	C₁₆	C₄×D₁₆₇	1	1	yes	characteristic
60	C₁₆	C₂²×D₁₆₇	1	1	yes	characteristic
61	Q₈	—	167	1	no	—
62	D₄	—	167	2	no	—
63	D₄	—	167	1	no	—
64	C₂×C₄	—	167	2	no	—
65	C₂×C₄	C₈×D₁₆₇	1	1	yes	characteristic
66	C₈	—	167	2	no	—
67	C₈	C₈×D₁₆₇	1	1	yes	characteristic
68	C₈	C₂×C₄×D₁₆₇	1	1	yes	characteristic
69	C₂²	—	167	2	no	—
70	C₂²	C₁₆×D₁₆₇	1	1	yes	characteristic
71	C₄	—	167	2	no	—
72	C₄	C₁₆×D₁₆₇	1	1	yes	characteristic
73	C₄	C₂×C₈×D₁₆₇	1	1	yes	characteristic
74	C₂	—	2	1	no	—
75	C₂	—	334	2	no	—
76	C₂	C₂×C₁₆×D₁₆₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₁₆	C₂²×D₁₆₇
commutator, socle	C₂×C₁₆₇	C₂²×C₁₆
Fitting	M₆₄×C₁₆₇	C₂
prime radical	#14581950	C₈
cosocle	C₁₆×C₁₆₇	C₂³
hypercenter	M₆₄	D₁₆₇
hypocenter	C₁₆₇	#2258
LCS	G ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #24923102	C₂	semidirect
3. C₂×C₁₆₇⋊C₃₂	C₂	semidirect
4. C₃₂×D₁₆₇	C₂	semidirect
5. M₆₄×C₁₆₇	C₂	semidirect
38. C₁₆₇	#2258	semidirect
42. M₆₄	D₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923685	2
#24923102	2
C₂×C₁₆₇⋊C₃₂	2
C₃₂×D₁₆₇	2
M₆₄×C₁₆₇	2
#2258	167