Subgroups — #34098208

Order 20096.

70 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757026	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₅₇	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11929434	—	2	2	no	—
4	C₂×C₁₆×C₁₅₇	C₄	1	1	yes	characteristic
5	C₂×C₁₆×C₁₅₇	C₄	1	1	yes	characteristic
6	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
7	C₂×#11929434	—	4	2	no	—
8	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
9	C₂×C₈×C₁₅₇	Q₈	1	1	yes	characteristic
10	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
11	C₁₆×C₁₅₇	—	2	1	no	—
12	C₁₆×C₁₅₇	C₈	1	2	yes	complemented
13	#11929434	—	8	2	no	—
14	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
15	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
16	C₂×C₄×C₁₅₇	Q₁₆	1	1	yes	characteristic
17	C₈×C₁₅₇	—	2	1	no	—
18	C₈×C₁₅₇	C₂×C₈	1	1	yes	characteristic
19	C₈×C₁₅₇	M₁₆	1	1	yes	characteristic
20	C₂²×C₁₅₇	#107	1	1	yes	characteristic
21	C₄×C₁₅₇	—	2	1	no	—
22	C₄×C₁₅₇	C₄⋊C₈	1	1	yes	characteristic
23	C₄×C₁₅₇	#108	1	1	yes	characteristic
24	C₄×C₁₅₇	D₁₆	1	1	yes	characteristic
25	C₄×C₁₅₇	Q₃₂	1	1	yes	characteristic
26	C₂×C₁₅₇	#335	1	1	yes	characteristic
27	C₂×C₁₅₇	#366	1	1	yes	characteristic
28	C₂×C₁₅₇	#368	1	1	yes	characteristic
29	C₁₅₇	#1371	1	1	yes	characteristic, complemented, Sylow
30	#1371	—	157	1	no	maximal, Sylow
31	C₄×C₁₆	D₁₅₇	1	1	yes	characteristic
32	#335	—	157	2	no	—
33	C₂×C₁₆	Dic₁₅₇	1	1	yes	characteristic
34	C₂×C₁₆	Dic₁₅₇	1	1	yes	characteristic
35	C₄⋊C₈	—	314	2	no	—
36	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
37	C₂×C₈	—	628	2	no	—
38	C₂×C₈	C₂×Dic₁₅₇	1	1	yes	characteristic
39	C₂×C₈	Dic₃₁₄	1	1	yes	characteristic
40	C₄²	D₆₂₈	1	1	yes	characteristic
41	C₁₆	—	2	1	no	—
42	C₁₆	#11929434	1	2	yes	complemented
43	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
44	C₂×C₄	D₁₂₅₆	1	1	yes	characteristic
45	C₂×C₄	Dic₆₂₈	1	1	yes	characteristic
46	C₈	—	2	1	no	—
47	C₈	—	1256	2	no	—
48	C₈	C₂×#11929434	1	1	yes	characteristic
49	C₈	#14578938	1	1	yes	characteristic
50	C₂²	#19097705	1	1	yes	characteristic
51	C₄	—	2	1	no	—
52	C₄	C₄⋊#11929434	1	1	yes	characteristic
53	C₄	#19097707	1	1	yes	characteristic
54	C₄	D₂₅₁₂	1	1	yes	characteristic
55	C₄	Dic₁₂₅₆	1	1	yes	characteristic
56	C₂	#23757026	1	1	yes	characteristic
57	C₂	#23757099	1	1	yes	characteristic
58	C₂	#23757103	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₁₂₅₆
commutator	C₈×C₁₅₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₅₇
Fitting	C₄×C₁₆×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#107
cosocle	C₄×C₈×C₁₅₇	C₂²
hypercenter	C₄×C₁₆	D₁₅₇
hypocenter	C₁₅₇	#1371
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
12. C₁₆×C₁₅₇	C₈	semidirect
29. C₁₅₇	#1371	semidirect
42. C₁₆	#11929434	semidirect

Maximal subgroups

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

subgroup	index
#23757026	2
C₄×C₁₆×C₁₅₇	2
#1371	157