Subgroups — #36326417

Order 21376.

116 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24923671	C₂	1	2	yes	maximal
2	#24923097	C₂	1	2	yes	maximal
3	#24923095	C₂	1	2	yes	maximal
4	C₄×C₁₆×C₁₆₇	C₂	1	1	yes	maximal, characteristic
5	#19128824	—	2	2	no	—
6	#19128766	C₂²	1	4	yes	—
7	#19128764	—	2	2	no	—
8	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₁₆×C₁₆₇	C₂²	1	1	yes	characteristic
10	C₄×C₈×C₁₆₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₁₆₇	—	2	4	no	—
12	Dic₁₆₇⋊C₄	—	4	4	no	—
13	C₂×C₈×C₁₆₇	D₄	1	1	yes	characteristic
14	C₂×C₈×C₁₆₇	C₂³	1	1	yes	characteristic
15	C₄²×C₁₆₇	D₄	1	1	yes	characteristic
16	C₁₆×C₁₆₇	Q₈	1	2	yes	—
17	C₁₆×C₁₆₇	Q₈	1	2	yes	—
18	C₂×Dic₁₆₇	—	4	4	no	—
19	C₂×C₄×C₁₆₇	D₈	1	2	yes	—
20	C₂×C₄×C₁₆₇	C₂×D₄	1	1	yes	characteristic
21	C₈×C₁₆₇	Q₁₆	1	2	yes	—
22	C₈×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
23	C₈×C₁₆₇	C₂×Q₈	1	1	yes	characteristic
24	C₂²×C₁₆₇	C₂×D₈	1	1	yes	characteristic
25	C₄×C₁₆₇	—	2	2	no	—
26	C₄×C₁₆₇	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₁₆₇	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₁₆₇	—	8	4	no	—
29	C₂×C₁₆₇	#500	1	1	yes	characteristic
30	C₂×C₁₆₇	#508	1	2	yes	—
31	C₁₆₇	#2253	1	1	yes	characteristic, complemented, Sylow
32	#2253	—	167	1	no	maximal, Sylow
33	#499	—	167	2	no	—
34	#367	—	167	2	no	—
35	#366	—	167	2	no	—
36	C₄×C₁₆	D₁₆₇	1	1	yes	characteristic
37	#125	—	334	2	no	—
38	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
39	C₂×C₁₆	C₂×D₁₆₇	1	1	yes	characteristic
40	#107	—	167	4	no	—
41	#106	—	334	2	no	—
42	C₄×C₈	C₂×D₁₆₇	1	1	yes	characteristic
43	C₂×C₈	D₆₆₈	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₁₆₇	1	1	yes	characteristic
45	C₄⋊C₄	—	334	4	no	—
46	C₄⋊C₄	—	668	4	no	—
47	C₄²	D₆₆₈	1	1	yes	characteristic
48	C₁₆	Dic₃₃₄	1	2	yes	—
49	C₁₆	Dic₃₃₄	1	2	yes	—
50	C₂×C₄	—	668	4	no	—
51	C₂×C₄	D₁₃₃₆	1	2	yes	—
52	C₂×C₄	C₂×D₆₆₈	1	1	yes	characteristic
53	C₈	Dic₆₆₈	1	2	yes	—
54	C₈	C₂×Dic₃₃₄	1	1	yes	characteristic
55	C₈	C₂×Dic₃₃₄	1	1	yes	characteristic
56	C₂²	C₂×D₁₃₃₆	1	1	yes	characteristic
57	C₄	—	2	2	no	—
58	C₄	—	1336	4	no	—
59	C₄	#19128832	1	1	yes	characteristic
60	C₄	C₂×Dic₆₆₈	1	1	yes	characteristic
61	C₂	#24923674	1	1	yes	characteristic
62	C₂	#24923701	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₃₃₆
commutator, cosocle	C₂×C₈×C₁₆₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₆₇
Fitting	C₄×C₁₆×C₁₆₇	C₂
socle, prime radical	C₂²×C₁₆₇	C₂×D₈
hypercenter	C₄×C₁₆	D₁₆₇
hypocenter	C₁₆₇	#2253
LCS	G ▷ C₂×C₈×C₁₆₇ ▷ C₄×C₁₆₇ ▷ C₂×C₁₆₇ ▷ C₁₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
31. C₁₆₇	#2253	semidirect

Maximal subgroups

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

subgroup	index
#24923671	2
#24923097	2
#24923095	2
C₄×C₁₆×C₁₆₇	2
#2253	167