Subgroups — #34099743

Order 20096.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757650	C₂	1	1	yes	maximal, characteristic
2	#23757621	C₂	1	4	yes	maximal, complemented
3	C₈×#11929434	C₂	1	1	yes	maximal, characteristic, complemented
4	#494×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×D₁₂₅₆	—	2	2	no	—
6	#19097768	C₂²	1	2	yes	—
7	#19097691	—	2	8	no	—
8	C₄×#11929434	C₂²	1	2	yes	—
9	C₂×Q₁₆×C₁₅₇	—	2	2	no	—
10	C₄⋊Q₈×C₁₅₇	C₂²	1	2	yes	—
11	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
12	C₂×D₆₂₈	—	2	4	no	—
13	C₂×D₆₂₈	—	2	2	no	—
14	D₁₂₅₆	—	4	4	no	—
15	C₂×#11929434	D₄	1	4	yes	—
16	C₂×Q₈×C₁₅₇	—	2	2	no	—
17	Q₁₆×C₁₅₇	—	4	4	no	—
18	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
19	C₄⋊C₄×C₁₅₇	—	2	4	no	—
20	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
21	C₂²×D₁₅₇	—	4	2	no	—
22	D₆₂₈	—	4	4	no	—
23	D₆₂₈	—	4	8	no	—
24	#11929434	—	2	4	no	—
25	Q₈×C₁₅₇	—	4	4	no	—
26	C₂×C₄×C₁₅₇	—	4	2	no	—
27	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
28	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
29	C₈×C₁₅₇	D₈	1	4	yes	—
30	C₂²×C₁₅₇	#127	1	1	yes	characteristic
31	C₂×D₁₅₇	—	8	2	no	—
32	C₂×D₁₅₇	—	8	4	no	—
33	C₄×C₁₅₇	—	8	2	no	—
34	C₄×C₁₅₇	C₂×D₈	1	2	yes	—
35	C₄×C₁₅₇	#135	1	4	yes	—
36	C₂×C₁₅₇	#493	1	1	yes	characteristic
37	C₂×C₁₅₇	#495	1	2	yes	—
38	D₁₅₇	—	16	2	no	—
39	C₁₅₇	#1709	1	1	yes	characteristic, complemented, Sylow
40	#1709	—	157	1	no	maximal, Sylow
41	#494	D₁₅₇	1	1	yes	characteristic, complemented
42	#493	—	157	1	no	—
43	#486	—	157	4	no	—
44	C₈²	—	157	1	no	—
45	C₂×Q₁₆	—	2	2	no	—
46	C₂×D₈	—	314	2	no	—
47	C₄⋊Q₈	C₂×D₁₅₇	1	2	yes	—
48	#127	—	157	2	no	—
49	#102	—	314	8	no	—
50	C₄×C₈	—	157	2	no	—
51	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	314	4	no	—
54	C₂×D₄	—	314	2	no	—
55	Q₁₆	—	4	4	no	—
56	D₈	—	628	4	no	—
57	C₂×C₈	—	157	4	no	—
58	C₂×C₈	C₂²⋊D₁₅₇	1	2	yes	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
61	C₂³	—	628	2	no	—
62	Q₈	—	4	4	no	—
63	D₄	—	628	4	no	—
64	D₄	—	628	8	no	—
65	C₂×C₄	—	4	2	no	—
66	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
67	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
68	C₈	—	314	4	no	—
69	C₈	#14578939	1	4	yes	—
70	C₂²	—	1256	2	no	—
71	C₂²	—	1256	4	no	—
72	C₂²	#19097767	1	1	yes	characteristic
73	C₄	—	8	2	no	—
74	C₄	C₂×#14578939	1	2	yes	—
75	C₄	#19097794	1	4	yes	—
76	C₂	—	2512	2	no	—
77	C₂	#23757649	1	1	yes	characteristic
78	C₂	#23757658	1	2	yes	—

	subgroup	G/H
center	C₂²	#19097767
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#494×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#127
prime radical	#23757650	C₂
hypercenter	#494	D₁₅₇
hypocenter	C₁₅₇	#1709
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

Complements

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

N	H ≅ G/N	type
2. #23757621	C₂	semidirect
3. C₈×#11929434	C₂	semidirect
4. #494×C₁₅₇	C₂	semidirect
39. C₁₅₇	#1709	semidirect
41. #494	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757650	2
#23757621	2
C₈×#11929434	2
#494×C₁₅₇	2
#1709	157