Subgroups — #36326380

Order 21376.

152 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923671	C₂	1	1	yes	maximal, characteristic, complemented
2	#24923646	C₂	1	1	yes	maximal, characteristic
3	#24923071	C₂	1	4	yes	maximal, complemented
4	#346×C₁₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₃₃₆	—	2	1	no	—
6	C₂×D₁₃₃₆	C₂²	1	2	yes	—
7	#19128829	—	2	1	no	—
8	#19128824	—	2	1	no	—
9	#19128766	C₂²	1	2	yes	—
10	#19128764	—	2	1	no	—
11	C₂×C₁₆×C₁₆₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
13	C₂×D₆₆₈	—	2	2	no	—
14	C₂×D₆₆₈	—	4	2	no	—
15	D₁₃₃₆	—	2	4	no	—
16	D₁₃₃₆	—	4	2	no	—
17	C₄⋊Dic₁₆₇	—	2	2	no	—
18	Dic₁₆₇⋊C₄	—	4	2	no	—
19	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
20	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
22	C₁₆×C₁₆₇	—	2	2	no	—
23	C₂²×D₁₆₇	—	4	2	no	—
24	D₆₆₈	—	4	2	no	—
25	D₆₆₈	—	4	2	no	—
26	D₆₆₈	—	8	4	no	—
27	C₂×Dic₁₆₇	—	4	2	no	—
28	C₂×C₄×C₁₆₇	D₈	1	2	yes	—
29	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
30	C₈×C₁₆₇	SD₁₆	1	2	yes	—
31	C₈×C₁₆₇	#41	1	2	yes	—
32	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
33	C₂×D₁₆₇	—	8	2	no	—
34	C₂×D₁₆₇	—	8	4	no	—
35	C₄×C₁₆₇	—	2	2	no	—
36	C₄×C₁₆₇	#124	1	1	yes	characteristic
37	C₄×C₁₆₇	C₂×SD₁₆	1	1	yes	characteristic
38	Dic₁₆₇	—	8	2	no	—
39	C₂×C₁₆₇	#486	1	1	yes	characteristic
40	C₂×C₁₆₇	#509	1	2	yes	—
41	D₁₆₇	—	16	2	no	—
42	C₁₆₇	#2244	1	1	yes	characteristic, complemented, Sylow
43	#2244	—	167	1	no	maximal, Sylow
44	#499	—	167	1	no	—
45	#493	—	167	1	no	—
46	#357	—	167	4	no	—
47	#346	D₁₆₇	1	1	yes	characteristic, complemented
48	C₂×D₈	—	334	1	no	—
49	C₂×D₈	—	167	2	no	—
50	#127	—	334	1	no	—
51	#125	—	334	1	no	—
52	C₂×C₁₆	C₂×D₁₆₇	1	2	yes	—
53	#107	—	167	2	no	—
54	#106	—	334	1	no	—
55	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
56	C₂×D₄	—	334	2	no	—
57	C₂×D₄	—	668	2	no	—
58	D₈	—	334	4	no	—
59	D₈	—	668	2	no	—
60	C₂×C₈	D₆₆₈	1	1	yes	characteristic
61	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
62	C₄⋊C₄	—	334	2	no	—
63	C₄⋊C₄	—	668	2	no	—
64	C₄²	D₆₆₈	1	1	yes	characteristic
65	C₁₆	—	2	2	no	—
66	C₂³	—	668	2	no	—
67	D₄	—	1336	4	no	—
68	D₄	—	668	2	no	—
69	D₄	—	668	2	no	—
70	C₂×C₄	—	668	2	no	—
71	C₂×C₄	D₁₃₃₆	1	2	yes	—
72	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
73	C₈	#14581939	1	2	yes	—
74	C₈	#14581950	1	2	yes	—
75	C₂²	—	1336	4	no	—
76	C₂²	—	1336	2	no	—
77	C₂²	C₂×D₁₃₃₆	1	1	yes	characteristic
78	C₄	—	2	2	no	—
79	C₄	—	1336	2	no	—
80	C₄	#19128820	1	1	yes	characteristic
81	C₄	C₂×#14581939	1	1	yes	characteristic
82	C₂	—	2672	2	no	—
83	C₂	#24923616	1	1	yes	characteristic
84	C₂	#24923707	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₃₆
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	#346×C₁₆₇	C₂
socle	C₂²×C₁₆₇	C₂×D₈
prime radical	#24923646	C₂
hypercenter	#346	D₁₆₇
hypocenter	C₁₆₇	#2244
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #346

Complements

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

N	H ≅ G/N	type
1. #24923671	C₂	semidirect
3. #24923071	C₂	semidirect
4. #346×C₁₆₇	C₂	semidirect
42. C₁₆₇	#2244	semidirect
47. #346	D₁₆₇	semidirect

Maximal subgroups

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

subgroup	index
#24923671	2
#24923646	2
#24923071	2
#346×C₁₆₇	2
#2244	167