Subgroups — #34103317

Order 20096.

424 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757180	C₂	1	4	yes	maximal
2	#23757159	C₂	1	8	yes	maximal
3	#23757141	C₂	1	2	yes	maximal
4	C₂×C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₁₅₇	C₂²	1	4	yes	—
6	C₂×C₄×Dic₁₅₇	C₂²	1	4	yes	—
7	#19097678	C₂²	1	8	yes	—
8	#19097678	C₂²	1	4	yes	—
9	#19097678	C₂²	1	8	yes	—
10	C₂²×Q₈×C₁₅₇	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₁₅₇	C₄	1	8	yes	complemented
12	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	4	yes	—
13	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₇	C₂³	1	8	yes	—
15	C₄⋊Dic₁₅₇	—	4	4	no	—
16	C₄×Dic₁₅₇	—	4	4	no	—
17	C₂×Q₈×C₁₅₇	—	4	2	no	—
18	C₂×Q₈×C₁₅₇	—	4	2	no	—
19	C₂×Q₈×C₁₅₇	C₂×C₄	1	8	yes	—
20	C₂²×C₄×C₁₅₇	C₂³	1	2	yes	—
21	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
22	C₂²×C₄×C₁₅₇	C₂³	1	4	yes	—
23	C₄⋊C₄×C₁₅₇	C₂×C₄	1	16	yes	—
24	C₄²×C₁₅₇	C₂×C₄	1	4	yes	—
25	C₂×Dic₁₅₇	—	2	8	no	—
26	C₂×Dic₁₅₇	—	2	16	no	—
27	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
28	Q₈×C₁₅₇	—	4	8	no	—
29	C₂×C₄×C₁₅₇	—	2	2	no	—
30	C₂×C₄×C₁₅₇	—	2	4	no	—
31	C₂×C₄×C₁₅₇	—	2	4	no	—
32	C₂×C₄×C₁₅₇	—	2	4	no	—
33	C₂×C₄×C₁₅₇	C₂²×C₄	1	4	yes	—
34	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
35	C₂×C₄×C₁₅₇	C₂²×C₄	1	8	yes	—
36	C₂²×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
37	C₂²×C₁₅₇	H(5,𝔽₂)	1	2	yes	—
38	C₂²×C₁₅₇	2₋¹⁺⁴	1	4	yes	—
39	C₄×C₁₅₇	—	2	8	no	—
40	C₄×C₁₅₇	—	2	2	no	—
41	C₄×C₁₅₇	—	2	4	no	—
42	Dic₁₅₇	—	4	8	no	—
43	C₂×C₁₅₇	#518	1	1	yes	characteristic
44	C₂×C₁₅₇	#519	1	2	yes	—
45	C₂×C₁₅₇	#563	1	4	yes	—
46	C₁₅₇	#2381	1	1	yes	characteristic, complemented, Sylow
47	#2381	—	157	1	no	maximal, Sylow
48	C₂×C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic
49	#391	—	157	4	no	—
50	#387	—	157	8	no	—
51	#383	—	157	2	no	—
52	C₂²×Q₈	C₂×D₁₅₇	1	2	yes	—
53	C₄⋊Q₈	Dic₁₅₇	1	8	yes	complemented
54	C₂×C₄⋊C₄	—	157	4	no	—
55	C₂×C₄⋊C₄	C₂×D₁₅₇	1	4	yes	—
56	C₂×C₄²	—	157	4	no	—
57	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
58	#95	—	157	8	no	—
59	#95	—	157	4	no	—
60	#95	—	157	8	no	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	4	2	no	—
63	C₂×Q₈	C₂×Dic₁₅₇	1	8	yes	—
64	C₂²×C₄	—	157	8	no	—
65	C₂²×C₄	C₂²×D₁₅₇	1	2	yes	—
66	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
67	C₂²×C₄	C₂²×D₁₅₇	1	4	yes	—
68	C₄⋊C₄	—	628	4	no	—
69	C₄⋊C₄	C₂×Dic₁₅₇	1	16	yes	—
70	C₄²	—	628	4	no	—
71	C₄²	C₂×Dic₁₅₇	1	4	yes	—
72	C₂³	C₂³×D₁₅₇	1	1	yes	characteristic
73	Q₈	—	4	8	no	—
74	C₂×C₄	—	2	4	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	314	8	no	—
79	C₂×C₄	—	314	16	no	—
80	C₂×C₄	C₂²×Dic₁₅₇	1	4	yes	—
81	C₂×C₄	C₂²×Dic₁₅₇	1	2	yes	—
82	C₂×C₄	C₂²×Dic₁₅₇	1	8	yes	—
83	C₂²	C₂³×Dic₁₅₇	1	1	yes	characteristic
84	C₂²	#19097816	1	2	yes	—
85	C₂²	#19097819	1	4	yes	—
86	C₄	—	2	2	no	—
87	C₄	—	2	8	no	—
88	C₄	—	2	4	no	—
89	C₄	—	628	8	no	—
90	C₂	#23757748	1	1	yes	characteristic
91	C₂	#23757751	1	2	yes	—
92	C₂	#23758022	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₇
commutator	C₂²×C₁₅₇	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₁₅₇	C₂
socle, prime radical, cosocle	C₂³×C₁₅₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#2381
LCS	G ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
11. C₄⋊Q₈×C₁₅₇	C₄	semidirect
46. C₁₅₇	#2381	semidirect
53. C₄⋊Q₈	Dic₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757180	2
#23757159	2
#23757141	2
C₂×C₄⋊Q₈×C₁₅₇	2
#2381	157