Subgroups — #34115103

Order 20096.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#14578957	C₂	1	1	yes	maximal, characteristic
2	C₄×#14578957	C₂	1	1	yes	maximal, characteristic
3	#23757397	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578957	—	2	1	no	—
5	C₂×#14578957	—	2	1	no	—
6	C₄×Dic₃₁₄	C₄	1	1	yes	characteristic
7	#19097700	—	2	1	no	—
8	C₄×#11929434	C₂²	1	1	yes	characteristic
9	C₈×Dic₁₅₇	—	2	1	no	—
10	C₄⋊C₈×C₁₅₇	C₄	1	1	yes	characteristic
11	#14578957	—	2	2	no	—
12	#14578957	—	4	1	no	—
13	C₂×Dic₃₁₄	C₈	1	1	yes	characteristic
14	C₂×#11929434	D₄	1	1	yes	characteristic
15	C₂×#11929434	D₄	1	1	yes	characteristic
16	C₄⋊Dic₁₅₇	C₈	1	1	yes	characteristic
17	#14578933	—	2	1	no	—
18	C₄×Dic₁₅₇	—	2	1	no	—
19	C₂×C₈×C₁₅₇	—	2	1	no	—
20	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
21	Dic₃₁₄	—	2	1	no	—
22	Dic₃₁₄	C₁₆	1	2	yes	complemented
23	C₂×Dic₁₅₇	—	2	1	no	—
24	C₂×Dic₁₅₇	—	2	1	no	—
25	#11929434	—	2	1	no	—
26	#11929434	SD₁₆	1	1	yes	characteristic
27	#11929434	Q₁₆	1	1	yes	characteristic
28	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₁₅₇	M₁₆	1	1	yes	characteristic
31	C₈×C₁₅₇	—	4	1	no	—
32	C₂²×C₁₅₇	C₂²⋊C₈	1	1	yes	characteristic
33	C₄×C₁₅₇	—	2	1	no	—
34	C₄×C₁₅₇	#103	1	1	yes	characteristic
35	C₄×C₁₅₇	#104	1	1	yes	characteristic
36	C₄×C₁₅₇	C₂×C₁₆	1	1	yes	characteristic
37	C₄×C₁₅₇	M₃₂	1	1	yes	characteristic
38	Dic₁₅₇	—	2	2	no	—
39	Dic₁₅₇	—	4	1	no	—
40	C₂×C₁₅₇	Q₈⋊C₈	1	1	yes	characteristic
41	C₂×C₁₅₇	C₂²⋊C₁₆	1	1	yes	characteristic
42	C₂×C₁₅₇	#350	1	1	yes	characteristic
43	C₁₅₇	Q₈⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
44	Q₈⋊C₁₆	—	157	1	no	maximal, Sylow
45	C₈×Q₈	—	157	1	no	—
46	C₄⋊C₁₆	—	157	1	no	—
47	C₄×C₁₆	—	157	1	no	—
48	C₄×Q₈	—	157	1	no	—
49	C₂×C₁₆	—	314	1	no	—
50	C₂×C₁₆	—	314	1	no	—
51	C₄⋊C₈	—	314	1	no	—
52	C₄⋊C₈	#10597845	1	1	yes	characteristic
53	C₄×C₈	—	314	1	no	—
54	C₄×C₈	—	157	1	no	—
55	C₂×Q₈	—	157	1	no	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	—	157	1	no	—
58	C₂×C₈	—	157	1	no	—
59	C₄⋊C₄	—	314	1	no	—
60	C₄⋊C₄	—	157	1	no	—
61	C₄²	—	314	1	no	—
62	C₄²	C₂×#10597845	1	1	yes	characteristic
63	C₁₆	—	314	2	no	—
64	C₁₆	—	628	1	no	—
65	Q₈	—	314	1	no	—
66	Q₈	—	157	2	no	—
67	C₂×C₄	—	314	1	no	—
68	C₂×C₄	—	314	1	no	—
69	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
70	C₂×C₄	D₁₅₇⋊C₈	1	1	yes	characteristic
71	C₂×C₄	#14578964	1	1	yes	characteristic
72	C₈	—	4	1	no	—
73	C₈	—	314	1	no	—
74	C₈	—	157	1	no	—
75	C₈	—	157	1	no	—
76	C₂²	#19097828	1	1	yes	characteristic
77	C₄	—	2	1	no	—
78	C₄	—	314	2	no	—
79	C₄	—	628	1	no	—
80	C₄	#19097834	1	1	yes	characteristic
81	C₄	#19097835	1	1	yes	characteristic
82	C₄	D₁₅₇⋊C₁₆	1	1	yes	characteristic
83	C₄	#19097846	1	1	yes	characteristic
84	C₂	#23758123	1	1	yes	characteristic
85	C₂	#23758168	1	1	yes	characteristic
86	C₂	#23758172	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10597845
commutator	C₄×C₁₅₇	C₂×C₁₆
Frattini	C₄²	C₂×#10597845
Fitting	C₄⋊C₈×C₁₅₇	C₄
socle, prime radical	C₂²×C₁₅₇	C₂²⋊C₈
cosocle	C₄×#11929434	C₂²
hypercenter	C₄⋊C₈	#10597845
hypocenter	C₁₅₇	Q₈⋊C₁₆
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
22. Dic₃₁₄	C₁₆	semidirect
43. C₁₅₇	Q₈⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#14578957	2
C₄×#14578957	2
#23757397	2
Q₈⋊C₁₆	157