Subgroups — #34102365

Order 20096.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757653	C₂	1	1	yes	maximal, characteristic
2	#23757076	C₂	1	4	yes	maximal
3	C₄×#14578929	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₆₂₈	—	2	1	no	—
6	C₂×Dic₆₂₈	C₂²	1	2	yes	—
7	#19097771	—	2	1	no	—
8	C₂×#14578929	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₅₇	—	2	1	no	—
10	#107×C₁₅₇	—	2	1	no	—
11	#107×C₁₅₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₃₁₄	—	2	2	no	—
14	Dic₆₂₈	—	2	4	no	—
15	Dic₆₂₈	—	4	2	no	—
16	C₄⋊Dic₁₅₇	—	4	1	no	—
17	C₄⋊Dic₁₅₇	—	4	1	no	—
18	#14578929	—	2	2	no	—
19	C₂×Q₈×C₁₅₇	—	4	1	no	—
20	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₅₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₅₇	—	2	2	no	—
23	C₄⋊C₄×C₁₅₇	—	4	1	no	—
24	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
25	Dic₃₁₄	—	4	2	no	—
26	Dic₃₁₄	—	4	2	no	—
27	C₂×Dic₁₅₇	—	4	2	no	—
28	Q₈×C₁₅₇	—	8	2	no	—
29	C₂×C₄×C₁₅₇	—	4	2	no	—
30	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
31	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₅₇	SD₁₆	1	2	yes	—
34	C₈×C₁₅₇	#41	1	2	yes	—
35	C₂²×C₁₅₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₅₇	—	8	2	no	—
37	C₄×C₁₅₇	SD₃₂	1	2	yes	—
38	C₄×C₁₅₇	Q₃₂	1	2	yes	—
39	C₄×C₁₅₇	#124	1	1	yes	characteristic
40	C₄×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₁₅₇	—	8	2	no	—
42	C₂×C₁₅₇	#486	1	1	yes	characteristic
43	C₂×C₁₅₇	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₁₅₇	C₂×Q₃₂	1	1	yes	characteristic
45	C₁₅₇	#2241	1	1	yes	characteristic, complemented, Sylow
46	#2241	—	157	1	no	maximal, Sylow
47	#500	D₁₅₇	1	1	yes	characteristic
48	#494	—	157	1	no	—
49	#358	—	157	4	no	—
50	C₄×C₁₆	—	157	1	no	—
51	C₂×Q₁₆	—	314	1	no	—
52	C₂×Q₁₆	—	157	2	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	314	1	no	—
55	C₂×C₁₆	—	157	2	no	—
56	#107	—	2	1	no	—
57	#107	C₂×D₁₅₇	1	2	yes	—
58	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	314	2	no	—
61	Q₁₆	—	314	4	no	—
62	Q₁₆	—	628	2	no	—
63	C₂×C₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₅₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	628	1	no	—
68	C₄⋊C₄	—	628	1	no	—
69	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
70	C₁₆	—	314	2	no	—
71	Q₈	—	8	2	no	—
72	Q₈	—	628	2	no	—
73	Q₈	—	628	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	628	2	no	—
76	C₂×C₄	#14578939	1	1	yes	characteristic
77	C₂×C₄	#14578939	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
79	C₈	#14578941	1	2	yes	—
80	C₈	#14578955	1	2	yes	—
81	C₂²	C₂×#14578939	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1256	2	no	—
84	C₄	#19097715	1	2	yes	—
85	C₄	#19097717	1	2	yes	—
86	C₄	#19097760	1	1	yes	characteristic
87	C₄	C₂×#14578941	1	1	yes	characteristic
88	C₂	#23757621	1	1	yes	characteristic
89	C₂	C₂×#19097715	1	1	yes	characteristic
90	C₂	C₂×#19097717	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14578939
commutator, cosocle	C₂×C₈×C₁₅₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₅₇
Fitting	#500×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	C₂×D₈
hypercenter	#500	D₁₅₇
hypocenter	C₁₅₇	#2241
LCS	G ▷ C₂×C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

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

N	H ≅ G/N	type
45. C₁₅₇	#2241	semidirect

Maximal subgroups

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

subgroup	index
#23757653	2
#23757076	2
C₄×#14578929	2
#500×C₁₅₇	2
#2241	157