Subgroups — #36322326

Order 21376.

74 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923701	C₂	1	1	yes	maximal, characteristic
2	#24923097	C₂	1	1	yes	maximal, characteristic, complemented
3	M₆₄×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128854	—	2	1	no	—
5	Dic₁₃₃₆	C₄	1	1	yes	characteristic, complemented
6	#19128777	—	2	1	no	—
7	D₂₆₇₂	C₄	1	1	yes	characteristic, complemented
8	#19128766	—	2	1	no	—
9	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
10	C₃₂×C₁₆₇	—	2	1	no	—
11	#14581950	—	4	1	no	—
12	Dic₆₆₈	—	2	1	no	—
13	#14581939	—	4	1	no	—
14	D₁₃₃₆	—	2	1	no	—
15	C₄⋊Dic₁₆₇	—	4	1	no	—
16	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₆₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₆₇	D₄	1	1	yes	characteristic
19	Dic₃₃₄	—	4	1	no	—
20	D₆₆₈	—	4	1	no	—
21	C₂²⋊D₁₆₇	—	8	1	no	—
22	C₂×Dic₁₆₇	—	8	1	no	—
23	C₄×D₁₆₇	—	8	1	no	—
24	C₂×C₄×C₁₆₇	D₈	1	1	yes	characteristic
25	C₈×C₁₆₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₆₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₆₇	D₁₆	1	1	yes	characteristic
28	C₂×D₁₆₇	—	8	1	no	—
29	C₄×C₁₆₇	#102	1	1	yes	characteristic
30	C₄×C₁₆₇	SD₃₂	1	1	yes	characteristic
31	Dic₁₆₇	—	8	1	no	—
32	Dic₁₆₇	—	16	1	no	—
33	C₂×C₁₆₇	—	2	1	no	—
34	C₂×C₁₆₇	#357	1	1	yes	characteristic
35	D₁₆₇	—	16	1	no	—
36	C₁₆₇	#1418	1	1	yes	characteristic, complemented, Sylow
37	#1418	—	167	1	no	maximal, Sylow
38	#508	—	167	1	no	—
39	M₆₄	D₁₆₇	1	1	yes	characteristic, complemented
40	#367	—	167	1	no	—
41	#135	—	334	1	no	—
42	Q₃₂	—	167	1	no	—
43	SD₃₂	—	334	1	no	—
44	D₁₆	—	167	1	no	—
45	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
46	#107	—	334	1	no	—
47	C₃₂	—	2	1	no	—
48	#41	—	668	1	no	—
49	Q₁₆	—	334	1	no	—
50	SD₁₆	—	668	1	no	—
51	D₈	—	334	1	no	—
52	C₂×C₈	D₆₆₈	1	1	yes	characteristic
53	C₄⋊C₄	—	668	1	no	—
54	C₁₆	C₄×D₁₆₇	1	1	yes	characteristic
55	C₁₆	C₂²⋊D₁₆₇	1	1	yes	characteristic
56	Q₈	—	668	1	no	—
57	D₄	—	1336	1	no	—
58	D₄	—	668	1	no	—
59	C₂×C₄	—	1336	1	no	—
60	C₂×C₄	—	1336	1	no	—
61	C₂×C₄	D₁₃₃₆	1	1	yes	characteristic
62	C₈	(C₂×D₁₆₇)⋊C₄	1	1	yes	characteristic
63	C₈	#14581939	1	1	yes	characteristic
64	C₂²	—	1336	1	no	—
65	C₂²	D₂₆₇₂	1	1	yes	characteristic
66	C₄	—	1336	1	no	—
67	C₄	—	2672	1	no	—
68	C₄	#19128754	1	1	yes	characteristic
69	C₄	#19128777	1	1	yes	characteristic
70	C₂	—	2	1	no	—
71	C₂	—	2672	1	no	—
72	C₂	#24923071	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#24923071
commutator	C₁₆×C₁₆₇	C₂×C₄
Frattini	C₂×C₁₆	C₂×D₁₆₇
Fitting	M₆₄×C₁₆₇	C₂
socle	C₂×C₁₆₇	#357
prime radical	#24923701	C₂
cosocle	C₂×C₁₆×C₁₆₇	C₂²
hypercenter	M₆₄	D₁₆₇
hypocenter	C₁₆₇	#1418
LCS	G ▷ C₁₆×C₁₆₇ ▷ C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ M₆₄

Complements

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

N	H ≅ G/N	type
2. #24923097	C₂	semidirect
3. M₆₄×C₁₆₇	C₂	semidirect
5. Dic₁₃₃₆	C₄	semidirect
7. D₂₆₇₂	C₄	semidirect
36. C₁₆₇	#1418	semidirect
39. M₆₄	D₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923701	2
#24923097	2
M₆₄×C₁₆₇	2
#1418	167