Subgroups — #34100207

Order 20096.

292 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₂²⋊#11929434	C₂	1	2	yes	maximal
2	#23757028	C₂	1	4	yes	maximal, complemented
3	C₂×#117×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929434	C₂²	1	4	yes	—
5	C₂²⋊#11929434	—	2	4	no	—
6	C₂³×C₄×C₁₅₇	C₂²	1	1	yes	characteristic
7	#117×C₁₅₇	—	2	4	no	—
8	C₂×C₄⋊C₄×C₁₅₇	C₄	1	2	yes	—
9	C₂×C₂²⋊C₄×C₁₅₇	C₄	1	2	yes	—
10	C₂×C₄²×C₁₅₇	C₂²	1	2	yes	—
11	C₂×#11929434	—	2	4	no	—
12	C₂×#11929434	—	2	8	no	—
13	C₂⁴×C₁₅₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₅₇	—	2	4	no	—
15	C₂²×C₄×C₁₅₇	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₅₇	D₄	1	4	yes	—
18	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₁₅₇	—	2	4	no	—
20	C₂²⋊C₄×C₁₅₇	—	2	4	no	—
21	C₄²×C₁₅₇	—	2	4	no	—
22	#11929434	—	4	4	no	—
23	C₂³×C₁₅₇	—	2	2	no	—
24	C₂³×C₁₅₇	—	2	4	no	—
25	C₂³×C₁₅₇	C₂²⋊C₄	1	2	yes	—
26	C₂³×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₅₇	—	2	2	no	—
28	C₂×C₄×C₁₅₇	—	2	8	no	—
29	C₂×C₄×C₁₅₇	—	2	4	no	—
30	C₂×C₄×C₁₅₇	—	2	8	no	—
31	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	2	yes	—
32	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₁₅₇	#41	1	4	yes	—
34	C₂²×C₁₅₇	—	2	2	no	—
35	C₂²×C₁₅₇	—	2	4	no	—
36	C₂²×C₁₅₇	—	4	4	no	—
37	C₂²×C₁₅₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₁₅₇	#117	1	2	yes	—
39	C₂²×C₁₅₇	#131	1	4	yes	—
40	C₄×C₁₅₇	—	2	2	no	—
41	C₄×C₁₅₇	—	4	4	no	—
42	C₄×C₁₅₇	#123	1	4	yes	—
43	C₂×C₁₅₇	—	4	2	no	—
44	C₂×C₁₅₇	#381	1	1	yes	characteristic
45	C₂×C₁₅₇	#408	1	2	yes	—
46	C₂×C₁₅₇	#433	1	4	yes	—
47	C₁₅₇	#1818	1	1	yes	characteristic, complemented, Sylow
48	#1818	—	157	1	no	maximal, Sylow
49	C₂×#117	D₁₅₇	1	1	yes	characteristic
50	C₂×C₂²⋊C₈	—	157	2	no	—
51	#336	—	157	4	no	—
52	C₂³×C₄	C₂×D₁₅₇	1	1	yes	characteristic
53	C₂²×C₈	—	157	4	no	—
54	#117	—	2	4	no	—
55	C₂×C₄⋊C₄	Dic₁₅₇	1	2	yes	—
56	C₂×C₂²⋊C₄	Dic₁₅₇	1	2	yes	—
57	C₂×C₄²	C₂×D₁₅₇	1	2	yes	—
58	C₂²⋊C₈	—	314	4	no	—
59	C₂⁴	C₂×Dic₁₅₇	1	1	yes	characteristic
60	C₂²×C₄	—	2	4	no	—
61	C₂²×C₄	C₂×Dic₁₅₇	1	4	yes	—
62	C₂²×C₄	C₂×Dic₁₅₇	1	1	yes	characteristic
63	C₂²×C₄	C₂²⋊D₁₅₇	1	4	yes	—
64	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
65	C₂×C₈	—	314	4	no	—
66	C₂×C₈	—	314	8	no	—
67	C₄⋊C₄	—	2	4	no	—
68	C₂²⋊C₄	—	2	4	no	—
69	C₄²	—	2	4	no	—
70	C₂³	—	2	2	no	—
71	C₂³	—	2	4	no	—
72	C₂³	C₂²⋊Dic₁₅₇	1	2	yes	—
73	C₂³	C₂²×Dic₁₅₇	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	8	no	—
76	C₂×C₄	—	2	4	no	—
77	C₂×C₄	—	2	8	no	—
78	C₂×C₄	C₂²⋊Dic₁₅₇	1	2	yes	—
79	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
80	C₂×C₄	#14578954	1	4	yes	—
81	C₈	—	628	4	no	—
82	C₂²	—	2	2	no	—
83	C₂²	—	2	4	no	—
84	C₂²	—	4	4	no	—
85	C₂²	C₂×C₂²⋊Dic₁₅₇	1	1	yes	characteristic
86	C₂²	#19097730	1	2	yes	—
87	C₂²	#19097778	1	4	yes	—
88	C₄	—	2	2	no	—
89	C₄	—	4	4	no	—
90	C₄	#19097754	1	4	yes	—
91	C₂	—	4	2	no	—
92	C₂	#23757136	1	1	yes	characteristic
93	C₂	#23757240	1	2	yes	—
94	C₂	#23757334	1	4	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₇
commutator, socle	C₂³×C₁₅₇	C₂²×C₄
Fitting	C₂×#117×C₁₅₇	C₂
prime radical	C₂⁴×C₁₅₇	C₂×C₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	C₂×#117	D₁₅₇
hypocenter	C₁₅₇	#1818
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

Complements

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

N	H ≅ G/N	type
2. #23757028	C₂	semidirect
47. C₁₅₇	#1818	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₂²⋊#11929434	2
#23757028	2
C₂×#117×C₁₅₇	2
#1818	157