Subgroups — #34100276

Order 20096.

180 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757297	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
3	#23757037	C₂	1	4	yes	maximal
4	#423×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14578938	C₂²	1	2	yes	—
6	C₂²×#11929434	C₂²	1	2	yes	—
7	C₄⋊#11929434	—	2	2	no	—
8	#19097682	C₄	1	4	yes	—
9	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₅₇	—	2	2	no	—
12	#14578938	—	2	4	no	—
13	C₂×#11929434	—	2	2	no	—
14	C₂×#11929434	—	2	2	no	—
15	C₂×#11929434	—	2	2	no	—
16	C₂×#11929434	C₂×C₄	1	4	yes	—
17	C₂²×C₄×C₁₅₇	Q₈	1	2	yes	—
18	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
19	M₁₆×C₁₅₇	—	2	4	no	—
20	C₂×C₈×C₁₅₇	—	2	2	no	—
21	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
22	C₄²×C₁₅₇	D₄	1	2	yes	—
23	#11929434	—	2	4	no	—
24	#11929434	—	4	2	no	—
25	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₄×C₁₅₇	—	4	2	no	—
27	C₂×C₄×C₁₅₇	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	#41	1	2	yes	—
31	C₂×C₄×C₁₅₇	#41	1	2	yes	—
32	C₈×C₁₅₇	—	4	2	no	—
33	C₂²×C₁₅₇	—	2	2	no	—
34	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂²×C₁₅₇	#125	1	2	yes	—
36	C₄×C₁₅₇	—	2	4	no	—
37	C₄×C₁₅₇	#117	1	2	yes	—
38	C₄×C₁₅₇	#124	1	1	yes	characteristic
39	C₄×C₁₅₇	#124	1	1	yes	characteristic
40	C₂×C₁₅₇	—	2	2	no	—
41	C₂×C₁₅₇	#382	1	1	yes	characteristic
42	C₂×C₁₅₇	#430	1	2	yes	—
43	C₁₅₇	#1838	1	1	yes	characteristic, complemented, Sylow
44	#1838	—	157	1	no	maximal, Sylow
45	#423	—	157	1	no	—
46	#423	D₁₅₇	1	1	yes	characteristic
47	C₂×#97	—	157	1	no	—
48	#341	—	157	4	no	—
49	C₂×M₁₆	—	157	2	no	—
50	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
51	C₂²×C₈	—	157	2	no	—
52	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	314	2	no	—
55	#97	—	157	4	no	—
56	C₂²×C₄	Dic₃₁₄	1	2	yes	—
57	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
58	M₁₆	—	2	4	no	—
59	M₁₆	—	314	4	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	314	2	no	—
62	C₂×C₈	—	314	2	no	—
63	C₂×C₈	—	314	2	no	—
64	C₂×C₈	—	157	4	no	—
65	C₄²	C₄×D₁₅₇	1	2	yes	—
66	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
67	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
68	C₂×C₄	—	4	2	no	—
69	C₂×C₄	#14578933	1	4	yes	—
70	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
71	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
72	C₂×C₄	#14578953	1	2	yes	—
73	C₂×C₄	#14578955	1	2	yes	—
74	C₈	—	4	2	no	—
75	C₈	—	314	4	no	—
76	C₈	—	628	2	no	—
77	C₂²	—	2	2	no	—
78	C₂²	C₂×#14578933	1	1	yes	characteristic
79	C₂²	#19097762	1	2	yes	—
80	C₄	—	2	4	no	—
81	C₄	#19097729	1	2	yes	—
82	C₄	#19097758	1	1	yes	characteristic
83	C₄	#19097760	1	1	yes	characteristic
84	C₂	—	2	2	no	—
85	C₂	#23757138	1	1	yes	characteristic
86	C₂	#23757321	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#423×C₁₅₇	C₂
socle	C₂²×C₁₅₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₅₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#423	D₁₅₇
hypocenter	C₁₅₇	#1838
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

Complements

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

N	H ≅ G/N	type
43. C₁₅₇	#1838	semidirect

Maximal subgroups

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

subgroup	index
#23757297	2
C₂×#19097682	2
#23757037	2
#423×C₁₅₇	2
#1838	157