Subgroups — #34115265

Order 20096.

98 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23758182	C₂	1	1	yes	maximal, characteristic
2	#23758180	C₂	1	1	yes	maximal, characteristic
3	#23757665	C₂	1	1	yes	maximal, characteristic
4	#19097832	—	2	1	no	—
5	#19097830	—	2	1	no	—
6	C₂×#14578944	—	2	1	no	—
7	C₂×#14578941	—	2	1	no	—
8	#19097759	C₂²	1	1	yes	characteristic
9	#19097682	C₄	1	1	yes	characteristic, complemented
10	C₄⋊Q₈×C₁₅₇	C₄	1	1	yes	characteristic, complemented
11	#14578963	—	4	1	no	—
12	C₂²⋊#10597845	—	4	1	no	—
13	C₂×Dic₃₁₄	D₄	1	1	yes	characteristic
14	C₂×D₆₂₈	D₄	1	1	yes	characteristic
15	#14578944	—	4	2	no	—
16	#14578941	—	4	2	no	—
17	C₂×#11929434	—	2	1	no	—
18	#14578931	—	4	1	no	—
19	C₂×Q₈×C₁₅₇	—	2	1	no	—
20	C₄⋊C₄×C₁₅₇	—	4	1	no	—
21	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
22	C₂×#10597845	—	8	1	no	—
23	#11929441	—	8	1	no	—
24	C₂²×D₁₅₇	—	2	1	no	—
25	Dic₃₁₄	—	4	1	no	—
26	D₆₂₈	—	4	1	no	—
27	C₂×Dic₁₅₇	—	2	1	no	—
28	#11929434	—	2	2	no	—
29	Q₈×C₁₅₇	—	4	1	no	—
30	Q₈×C₁₅₇	—	4	1	no	—
31	C₂×C₄×C₁₅₇	—	2	1	no	—
32	C₂×C₄×C₁₅₇	—	4	1	no	—
33	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂²×C₁₅₇	#99	1	1	yes	characteristic
35	C₂×D₁₅₇	—	4	1	no	—
36	C₂×D₁₅₇	—	8	1	no	—
37	#10597845	—	16	1	no	—
38	C₄×C₁₅₇	—	2	1	no	—
39	C₄×C₁₅₇	—	4	1	no	—
40	C₄×C₁₅₇	—	8	1	no	—
41	Dic₁₅₇	—	4	1	no	—
42	C₂×C₁₅₇	—	2	1	no	—
43	C₂×C₁₅₇	#353	1	1	yes	characteristic
44	D₁₅₇	—	8	1	no	—
45	C₁₅₇	#1407	1	1	yes	characteristic, complemented, Sylow
46	#1407	—	157	1	no	maximal, Sylow
47	#497	—	157	1	no	—
48	#355	—	157	1	no	—
49	#354	—	157	1	no	—
50	C₂×Q₁₆	—	314	1	no	—
51	C₂×SD₁₆	—	314	1	no	—
52	C₄⋊Q₈	#10597845	1	1	yes	characteristic, complemented
53	#124	—	157	1	no	—
54	#101	—	314	1	no	—
55	#99	—	314	1	no	—
56	#97	—	157	1	no	—
57	C₂×Q₈	—	2	1	no	—
58	C₂×Q₈	—	157	1	no	—
59	C₂×D₄	—	157	1	no	—
60	Q₁₆	—	628	2	no	—
61	SD₁₆	—	628	2	no	—
62	M₁₆	—	628	1	no	—
63	C₂×C₈	—	314	1	no	—
64	C₄⋊C₄	—	4	1	no	—
65	C₂²⋊C₄	—	628	1	no	—
66	C₂²⋊C₄	—	628	1	no	—
67	C₄²	C₂×#10597845	1	1	yes	characteristic
68	C₂³	—	314	1	no	—
69	Q₈	—	4	1	no	—
70	Q₈	—	4	1	no	—
71	Q₈	—	628	1	no	—
72	D₄	—	628	1	no	—
73	C₂×C₄	—	2	1	no	—
74	C₂×C₄	—	4	1	no	—
75	C₂×C₄	—	314	1	no	—
76	C₂×C₄	—	1256	1	no	—
77	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
78	C₈	—	314	2	no	—
79	C₈	—	1256	1	no	—
80	C₂²	—	628	1	no	—
81	C₂²	—	1256	1	no	—
82	C₂²	#19097829	1	1	yes	characteristic
83	C₄	—	2	1	no	—
84	C₄	—	4	1	no	—
85	C₄	—	8	1	no	—
86	C₄	—	628	1	no	—
87	C₄	—	2512	1	no	—
88	C₂	—	2	1	no	—
89	C₂	—	1256	1	no	—
90	C₂	#23758177	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23758177
commutator	C₄²×C₁₅₇	C₂×C₄
Frattini	C₄²	C₂×#10597845
Fitting	C₄⋊Q₈×C₁₅₇	C₄
socle	C₂×C₁₅₇	#353
prime radical	C₂×D₆₂₈	D₄
cosocle	#19097759	C₂²
hypercenter	C₄⋊Q₈	#10597845
hypocenter	C₁₅₇	#1407
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
9. #19097682	C₄	semidirect
10. C₄⋊Q₈×C₁₅₇	C₄	semidirect
45. C₁₅₇	#1407	semidirect
52. C₄⋊Q₈	#10597845	semidirect

Maximal subgroups

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

subgroup	index
#23758182	2
#23758180	2
#23757665	2
#1407	157