Subgroups — #34099807

Order 20096.

198 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757678	C₂	1	2	yes	maximal, complemented
2	#23757653	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757646	C₂	1	1	yes	maximal, characteristic
4	#23757620	C₂	1	2	yes	maximal
5	#322×C₁₅₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₂₈	—	2	2	no	—
7	C₂×#14578943	—	2	2	no	—
8	#19097771	C₂²	1	2	yes	—
9	#19097771	C₂²	1	1	yes	characteristic
10	#19097768	C₂²	1	1	yes	characteristic
11	#19097705	—	2	4	no	—
12	#19097693	—	2	4	no	—
13	C₄×C₈×C₁₅₇	C₂²	1	2	yes	—
14	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₄	—	2	1	no	—
16	C₂×Dic₃₁₄	—	2	2	no	—
17	C₂×D₆₂₈	—	2	2	no	—
18	C₂×D₆₂₈	—	2	1	no	—
19	Dic₆₂₈	—	4	4	no	—
20	#14578943	—	4	4	no	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₄⋊Dic₁₅₇	—	2	2	no	—
23	C₄⋊Dic₁₅₇	—	2	2	no	—
24	C₂×C₈×C₁₅₇	D₄	1	4	yes	—
25	C₂×C₈×C₁₅₇	D₄	1	2	yes	—
26	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
27	C₂²×D₁₅₇	—	4	1	no	—
28	Dic₃₁₄	—	4	4	no	—
29	Dic₃₁₄	—	4	2	no	—
30	D₆₂₈	—	4	2	no	—
31	D₆₂₈	—	4	4	no	—
32	C₂×Dic₁₅₇	—	4	1	no	—
33	C₂×Dic₁₅₇	—	4	2	no	—
34	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
36	C₈×C₁₅₇	—	2	4	no	—
37	C₈×C₁₅₇	D₈	1	4	yes	—
38	C₂²×C₁₅₇	#127	1	1	yes	characteristic
39	C₂×D₁₅₇	—	8	1	no	—
40	C₂×D₁₅₇	—	8	2	no	—
41	C₄×C₁₅₇	C₂×D₈	1	2	yes	—
42	C₄×C₁₅₇	#137	1	4	yes	—
43	Dic₁₅₇	—	8	2	no	—
44	Dic₁₅₇	—	8	1	no	—
45	C₂×C₁₅₇	#493	1	1	yes	characteristic
46	C₂×C₁₅₇	#497	1	2	yes	—
47	D₁₅₇	—	16	1	no	—
48	C₁₅₇	#1722	1	1	yes	characteristic, complemented, Sylow
49	#1722	—	157	1	no	maximal, Sylow
50	#500	—	157	2	no	—
51	#494	—	157	1	no	—
52	#492	—	157	1	no	—
53	#486	—	157	2	no	—
54	#322	D₁₅₇	1	1	yes	characteristic, complemented
55	C₂×Q₁₆	—	314	2	no	—
56	C₂×SD₁₆	—	314	2	no	—
57	C₄⋊Q₈	—	157	2	no	—
58	C₄⋊Q₈	—	157	1	no	—
59	#127	—	157	1	no	—
60	#107	—	314	4	no	—
61	#102	—	314	4	no	—
62	C₄×C₈	C₂×D₁₅₇	1	2	yes	—
63	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
64	C₂×Q₈	—	314	1	no	—
65	C₂×Q₈	—	314	2	no	—
66	C₂×D₄	—	314	2	no	—
67	C₂×D₄	—	314	1	no	—
68	Q₁₆	—	628	4	no	—
69	SD₁₆	—	628	4	no	—
70	C₂×C₈	D₆₂₈	1	4	yes	—
71	C₂×C₈	D₆₂₈	1	2	yes	—
72	C₄⋊C₄	—	314	2	no	—
73	C₄⋊C₄	—	314	2	no	—
74	C₄⋊C₄	—	314	2	no	—
75	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
76	C₂³	—	628	1	no	—
77	Q₈	—	628	2	no	—
78	Q₈	—	628	4	no	—
79	D₄	—	628	2	no	—
80	D₄	—	628	4	no	—
81	C₂×C₄	—	628	2	no	—
82	C₂×C₄	—	628	1	no	—
83	C₂×C₄	C₂×D₆₂₈	1	1	yes	characteristic
84	C₂×C₄	C₂×D₆₂₈	1	2	yes	—
85	C₈	—	2	4	no	—
86	C₈	D₁₂₅₆	1	4	yes	—
87	C₂²	—	1256	1	no	—
88	C₂²	—	1256	2	no	—
89	C₂²	#19097768	1	1	yes	characteristic
90	C₄	—	1256	1	no	—
91	C₄	—	1256	2	no	—
92	C₄	C₂×D₁₂₅₆	1	2	yes	—
93	C₄	#19097804	1	4	yes	—
94	C₂	—	2512	1	no	—
95	C₂	#23757650	1	1	yes	characteristic
96	C₂	#23757667	1	2	yes	—

	subgroup	G/H
center	C₂²	#19097768
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting	#322×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#127
prime radical	#19097768	C₂²
hypercenter	#322	D₁₅₇
hypocenter	C₁₅₇	#1722
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

Complements

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

N	H ≅ G/N	type
1. #23757678	C₂	semidirect
2. #23757653	C₂	semidirect
5. #322×C₁₅₇	C₂	semidirect
48. C₁₅₇	#1722	semidirect
54. #322	D₁₅₇	semidirect

Maximal subgroups

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

subgroup	index
#23757678	2
#23757653	2
#23757646	2
#23757620	2
#322×C₁₅₇	2
#1722	157