Subgroups — #34100310

Order 20096.

212 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19097706	C₂	1	4	yes	maximal
2	#23757297	C₂	1	1	yes	maximal, characteristic
3	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
4	#423×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#14578938	C₂²	1	2	yes	—
6	C₂²×#11929434	C₂²	1	2	yes	—
7	#19097706	—	2	8	no	—
8	C₄⋊#11929434	—	2	2	no	—
9	#19097682	C₄	1	4	yes	—
10	C₂×M₁₆×C₁₅₇	C₂²	1	2	yes	—
11	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₁₅₇	—	2	2	no	—
13	#14578938	—	2	4	no	—
14	C₂×#11929434	—	2	2	no	—
15	C₂×#11929434	C₂×C₄	1	4	yes	—
16	C₂×#11929434	D₄	1	4	yes	—
17	C₂×#11929434	Q₈	1	4	yes	—
18	C₂²×C₄×C₁₅₇	Q₈	1	2	yes	—
19	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
20	M₁₆×C₁₅₇	—	2	4	no	—
21	C₂×C₈×C₁₅₇	—	2	2	no	—
22	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
23	C₄²×C₁₅₇	D₄	1	2	yes	—
24	#11929434	—	4	2	no	—
25	#11929434	C₄⋊C₄	1	8	yes	—
26	C₂³×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×C₄×C₁₅₇	—	4	2	no	—
28	C₂×C₄×C₁₅₇	C₄⋊C₄	1	4	yes	—
29	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
31	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
33	C₈×C₁₅₇	—	4	2	no	—
34	C₂²×C₁₅₇	—	2	2	no	—
35	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	C₂²×C₁₅₇	C₄⋊Q₈	1	2	yes	—
37	C₄×C₁₅₇	—	2	4	no	—
38	C₄×C₁₅₇	C₂×C₄⋊C₄	1	2	yes	—
39	C₄×C₁₅₇	#127	1	1	yes	characteristic
40	C₄×C₁₅₇	C₄⋊Q₈	1	1	yes	characteristic
41	C₂×C₁₅₇	—	2	2	no	—
42	C₂×C₁₅₇	#384	1	1	yes	characteristic
43	C₂×C₁₅₇	#430	1	2	yes	—
44	C₁₅₇	#1849	1	1	yes	characteristic, complemented, Sylow
45	#1849	—	157	1	no	maximal, Sylow
46	C₂×#108	—	157	4	no	—
47	#423	—	157	1	no	—
48	#423	D₁₅₇	1	1	yes	characteristic
49	C₂×#97	—	157	1	no	—
50	C₂×M₁₆	—	157	2	no	—
51	C₂×M₁₆	C₂×D₁₅₇	1	2	yes	—
52	C₂²×C₈	—	157	2	no	—
53	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
54	#108	—	314	8	no	—
55	C₄⋊C₈	—	2	2	no	—
56	C₄⋊C₈	—	314	2	no	—
57	#97	—	157	4	no	—
58	C₂²×C₄	Dic₃₁₄	1	2	yes	—
59	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
60	M₁₆	—	2	4	no	—
61	M₁₆	—	314	4	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	314	2	no	—
64	C₂×C₈	—	157	4	no	—
65	C₂×C₈	—	157	4	no	—
66	C₂×C₈	—	157	4	no	—
67	C₄²	C₄×D₁₅₇	1	2	yes	—
68	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
69	C₂³	C₂×Dic₃₁₄	1	1	yes	characteristic
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	#14578933	1	4	yes	—
72	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
73	C₂×C₄	D₄×D₁₅₇	1	2	yes	—
74	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
75	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
76	C₈	—	4	2	no	—
77	C₈	—	628	2	no	—
78	C₈	—	157	8	no	—
79	C₂²	—	2	2	no	—
80	C₂²	C₂×#14578933	1	1	yes	characteristic
81	C₂²	#19097770	1	2	yes	—
82	C₄	—	2	4	no	—
83	C₄	C₄⋊C₄×D₁₅₇	1	2	yes	—
84	C₄	#19097767	1	1	yes	characteristic
85	C₄	#19097769	1	1	yes	characteristic
86	C₂	—	2	2	no	—
87	C₂	#23757142	1	1	yes	characteristic
88	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	#1849
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
44. C₁₅₇	#1849	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19097706	2
#23757297	2
C₂×#19097682	2
#423×C₁₅₇	2
#1849	157