Subgroups — #34098296

Order 20096.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757070	C₂	1	1	yes	maximal, characteristic
2	#23757068	C₂	1	1	yes	maximal, characteristic
3	#497×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	#19097690	—	2	1	no	—
5	#19097686	—	2	1	no	—
6	C₂×Q₁₆×C₁₅₇	—	2	1	no	—
7	C₂×SD₁₆×C₁₅₇	—	2	1	no	—
8	C₄⋊Q₈×C₁₅₇	C₄	1	1	yes	characteristic, complemented
9	#124×C₁₅₇	C₂²	1	1	yes	characteristic
10	#97×C₁₅₇	C₄	1	1	yes	characteristic, complemented
11	#14578938	—	4	1	no	—
12	C₂²⋊Dic₁₅₇	—	4	1	no	—
13	C₂×Q₈×C₁₅₇	—	2	1	no	—
14	C₂×Q₈×C₁₅₇	D₄	1	1	yes	characteristic
15	C₂×D₄×C₁₅₇	D₄	1	1	yes	characteristic
16	Q₁₆×C₁₅₇	—	4	2	no	—
17	SD₁₆×C₁₅₇	—	4	2	no	—
18	C₂×C₈×C₁₅₇	—	2	1	no	—
19	C₄⋊C₄×C₁₅₇	—	4	1	no	—
20	C₂²⋊C₄×C₁₅₇	—	4	1	no	—
21	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₅₇	—	8	1	no	—
23	#11929434	—	8	1	no	—
24	C₂³×C₁₅₇	—	2	1	no	—
25	Q₈×C₁₅₇	—	4	1	no	—
26	Q₈×C₁₅₇	—	4	1	no	—
27	Q₈×C₁₅₇	—	4	1	no	—
28	D₄×C₁₅₇	—	4	1	no	—
29	C₂×C₄×C₁₅₇	—	2	1	no	—
30	C₂×C₄×C₁₅₇	—	2	1	no	—
31	C₂×C₄×C₁₅₇	—	4	1	no	—
32	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₈×C₁₅₇	—	2	2	no	—
34	C₂²×C₁₅₇	—	4	1	no	—
35	C₂²×C₁₅₇	—	8	1	no	—
36	C₂²×C₁₅₇	#99	1	1	yes	characteristic
37	C₄×C₁₅₇	—	2	1	no	—
38	C₄×C₁₅₇	—	4	1	no	—
39	C₄×C₁₅₇	—	4	1	no	—
40	C₄×C₁₅₇	—	8	1	no	—
41	Dic₁₅₇	—	16	1	no	—
42	C₂×C₁₅₇	—	2	1	no	—
43	C₂×C₁₅₇	—	8	1	no	—
44	C₂×C₁₅₇	#353	1	1	yes	characteristic
45	C₁₅₇	#1407	1	1	yes	characteristic, complemented, Sylow
46	#1407	—	157	1	no	maximal, Sylow
47	#497	D₁₅₇	1	1	yes	characteristic
48	#355	—	157	1	no	—
49	#354	—	157	1	no	—
50	C₂×Q₁₆	—	2	1	no	—
51	C₂×SD₁₆	—	2	1	no	—
52	C₄⋊Q₈	Dic₁₅₇	1	1	yes	characteristic, complemented
53	#124	C₂×D₁₅₇	1	1	yes	characteristic
54	#101	—	314	1	no	—
55	#99	—	314	1	no	—
56	#97	Dic₁₅₇	1	1	yes	characteristic, complemented
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	C₂²⋊D₁₅₇	1	1	yes	characteristic
59	C₂×D₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
60	Q₁₆	—	4	2	no	—
61	SD₁₆	—	4	2	no	—
62	M₁₆	—	628	1	no	—
63	C₂×C₈	—	2	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₂²⋊C₄	—	4	1	no	—
66	C₂²⋊C₄	—	628	1	no	—
67	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
68	C₂³	—	2	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	4	1	no	—
72	D₄	—	4	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	1256	1	no	—
77	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
78	C₈	—	2	2	no	—
79	C₈	—	1256	1	no	—
80	C₂²	—	4	1	no	—
81	C₂²	—	8	1	no	—
82	C₂²	#19097686	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	8	1	no	—
87	C₄	—	2512	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	8	1	no	—
90	C₂	#23757066	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23757066
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	#124	C₂×D₁₅₇
Fitting	#497×C₁₅₇	C₂
socle	C₂×C₁₅₇	#353
prime radical	C₂×D₄×C₁₅₇	D₄
cosocle	#124×C₁₅₇	C₂²
hypercenter	#497	D₁₅₇
hypocenter	C₁₅₇	#1407
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #124 ◁ #497

Complements

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

N	H ≅ G/N	type
8. C₄⋊Q₈×C₁₅₇	C₄	semidirect
10. #97×C₁₅₇	C₄	semidirect
45. C₁₅₇	#1407	semidirect
52. C₄⋊Q₈	Dic₁₅₇	semidirect
56. #97	Dic₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757070	2
#23757068	2
#497×C₁₅₇	2
#1407	157