Subgroups — #34098099

Order 20096.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×Dic₆₂₈	C₂	1	1	yes	maximal, characteristic
2	C₄×#14578929	C₂	1	1	yes	maximal, characteristic
3	#335×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₆₂₈	C₄	1	1	yes	characteristic
5	C₄×Dic₃₁₄	—	2	1	no	—
6	C₂×#14578929	—	2	1	no	—
7	#19097705	C₄	1	1	yes	characteristic
8	#19097696	—	2	1	no	—
9	C₄⋊C₈×C₁₅₇	—	2	1	no	—
10	C₄×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₂×Dic₃₁₄	—	2	1	no	—
12	Dic₆₂₈	—	2	1	no	—
13	Dic₆₂₈	C₈	1	2	yes	complemented
14	C₄⋊Dic₁₅₇	—	2	1	no	—
15	#14578933	—	4	1	no	—
16	C₄×Dic₁₅₇	—	4	1	no	—
17	#14578929	—	2	2	no	—
18	C₂×C₈×C₁₅₇	—	4	1	no	—
19	C₂×C₈×C₁₅₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₁₅₇	D₄	1	1	yes	characteristic
21	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
22	Dic₃₁₄	—	2	2	no	—
23	Dic₃₁₄	—	4	1	no	—
24	C₂×Dic₁₅₇	—	4	1	no	—
25	C₂×Dic₁₅₇	—	4	1	no	—
26	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×C₄×C₁₅₇	D₈	1	1	yes	characteristic
28	C₂×C₄×C₁₅₇	SD₁₆	1	1	yes	characteristic
29	C₈×C₁₅₇	—	2	1	no	—
30	C₈×C₁₅₇	—	8	1	no	—
31	C₈×C₁₅₇	C₂×C₈	1	1	yes	characteristic
32	C₈×C₁₅₇	M₁₆	1	1	yes	characteristic
33	C₂²×C₁₅₇	#102	1	1	yes	characteristic
34	C₄×C₁₅₇	—	2	1	no	—
35	C₄×C₁₅₇	C₂²⋊C₈	1	1	yes	characteristic
36	C₄×C₁₅₇	#104	1	1	yes	characteristic
37	C₄×C₁₅₇	SD₃₂	1	1	yes	characteristic
38	C₄×C₁₅₇	Q₃₂	1	1	yes	characteristic
39	Dic₁₅₇	—	4	2	no	—
40	Dic₁₅₇	—	8	1	no	—
41	C₂×C₁₅₇	D₄⋊C₈	1	1	yes	characteristic
42	C₂×C₁₅₇	#358	1	1	yes	characteristic
43	C₂×C₁₅₇	#359	1	1	yes	characteristic
44	C₁₅₇	#1332	1	1	yes	characteristic, complemented, Sylow
45	#1332	—	157	1	no	maximal, Sylow
46	C₄×Q₁₆	—	157	1	no	—
47	C₄×C₁₆	—	157	1	no	—
48	#335	D₁₅₇	1	1	yes	characteristic
49	C₂×Q₁₆	—	157	1	no	—
50	C₄×Q₈	—	314	1	no	—
51	C₂×C₁₆	—	314	1	no	—
52	#107	—	157	1	no	—
53	C₄⋊C₈	—	2	1	no	—
54	#103	—	314	1	no	—
55	C₄×C₈	C₂×D₁₅₇	1	1	yes	characteristic
56	C₂×Q₈	—	314	1	no	—
57	Q₁₆	—	314	1	no	—
58	Q₁₆	—	157	2	no	—
59	C₂×C₈	—	4	1	no	—
60	C₂×C₈	C₄×D₁₅₇	1	1	yes	characteristic
61	C₂×C₈	D₆₂₈	1	1	yes	characteristic
62	C₄⋊C₄	—	314	1	no	—
63	C₄⋊C₄	—	628	1	no	—
64	C₄²	—	628	1	no	—
65	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
66	C₁₆	—	314	2	no	—
67	Q₈	—	314	2	no	—
68	Q₈	—	628	1	no	—
69	C₂×C₄	—	628	1	no	—
70	C₂×C₄	—	628	1	no	—
71	C₂×C₄	#14578931	1	1	yes	characteristic
72	C₂×C₄	#14578939	1	1	yes	characteristic
73	C₂×C₄	#14578941	1	1	yes	characteristic
74	C₈	—	2	1	no	—
75	C₈	—	8	1	no	—
76	C₈	C₈×D₁₅₇	1	1	yes	characteristic
77	C₈	#14578937	1	1	yes	characteristic
78	C₂²	#19097691	1	1	yes	characteristic
79	C₄	—	2	1	no	—
80	C₄	—	628	2	no	—
81	C₄	—	1256	1	no	—
82	C₄	#19097683	1	1	yes	characteristic
83	C₄	#19097697	1	1	yes	characteristic
84	C₄	#19097715	1	1	yes	characteristic
85	C₄	#19097717	1	1	yes	characteristic
86	C₂	#23757000	1	1	yes	characteristic
87	C₂	#23757076	1	1	yes	characteristic
88	C₂	#23757079	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14578939
commutator	C₈×C₁₅₇	C₂×C₈
Frattini	C₄×C₈	C₂×D₁₅₇
Fitting	#335×C₁₅₇	C₂
socle, prime radical	C₂²×C₁₅₇	#102
cosocle	C₄×C₈×C₁₅₇	C₂²
hypercenter	#335	D₁₅₇
hypocenter	C₁₅₇	#1332
LCS	G ▷ C₈×C₁₅₇ ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×C₈ ◁ #335

Complements

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

N	H ≅ G/N	type
13. Dic₆₂₈	C₈	semidirect
44. C₁₅₇	#1332	semidirect

Maximal subgroups

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

subgroup	index
C₄×Dic₆₂₈	2
C₄×#14578929	2
#335×C₁₅₇	2
#1332	157