Subgroups — #34115088

Order 20096.

100 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, complemented
2	#23758161	C₂	1	1	yes	maximal, characteristic, complemented
3	#23757351	C₂	1	1	yes	maximal, characteristic
4	C₂×#14578957	—	2	1	no	—
5	C₂×#14578957	—	2	1	no	—
6	C₂×#14578938	—	2	1	no	—
7	C₄⋊#11929434	C₄	1	1	yes	characteristic
8	C₂²⋊#11929434	—	2	1	no	—
9	C₄×#11929434	C₂²	1	1	yes	characteristic
10	C₄×D₄×C₁₅₇	C₄	1	1	yes	characteristic
11	#14578957	—	2	2	no	—
12	#14578957	—	4	1	no	—
13	#14578938	—	4	1	no	—
14	#14578938	—	4	1	no	—
15	C₂×#11929434	—	2	1	no	—
16	C₂×#11929434	D₄	1	1	yes	characteristic
17	C₂×#11929434	D₄	1	1	yes	characteristic
18	C₂×D₄×C₁₅₇	C₈	1	1	yes	characteristic
19	C₂²×C₄×C₁₅₇	—	2	1	no	—
20	C₄⋊C₄×C₁₅₇	C₈	1	1	yes	characteristic
21	C₂²⋊C₄×C₁₅₇	—	2	1	no	—
22	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
23	#11929434	—	2	1	no	—
24	#11929434	—	4	1	no	—
25	#11929434	D₈	1	1	yes	characteristic
26	#11929434	SD₁₆	1	1	yes	characteristic
27	C₂³×C₁₅₇	—	2	1	no	—
28	D₄×C₁₅₇	—	4	1	no	—
29	C₂×C₄×C₁₅₇	—	2	1	no	—
30	C₂×C₄×C₁₅₇	—	4	1	no	—
31	C₂×C₄×C₁₅₇	—	4	1	no	—
32	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂×C₈	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	M₁₆	1	1	yes	characteristic
35	C₂²×C₁₅₇	—	4	1	no	—
36	C₂²×C₁₅₇	—	4	1	no	—
37	C₂²×C₁₅₇	—	4	1	no	—
38	C₂²×C₁₅₇	C₂²⋊C₈	1	1	yes	characteristic
39	C₄×C₁₅₇	—	2	1	no	—
40	C₄×C₁₅₇	—	2	1	no	—
41	C₄×C₁₅₇	—	4	1	no	—
42	C₄×C₁₅₇	#102	1	1	yes	characteristic
43	C₄×C₁₅₇	#104	1	1	yes	characteristic
44	C₂×C₁₅₇	—	8	1	no	—
45	C₂×C₁₅₇	D₄⋊C₈	1	1	yes	characteristic
46	C₂×C₁₅₇	#349	1	1	yes	characteristic
47	C₂×C₁₅₇	#350	1	1	yes	characteristic
48	C₁₅₇	#1330	1	1	yes	characteristic, complemented, Sylow
49	#1330	—	157	1	no	maximal, Sylow
50	#436	—	157	1	no	—
51	C₄⋊C₁₆	—	157	1	no	—
52	#346	—	157	1	no	—
53	C₂×M₁₆	—	314	1	no	—
54	C₄×D₄	#10597845	1	1	yes	characteristic
55	C₂×C₁₆	—	314	1	no	—
56	C₂×C₁₆	—	314	1	no	—
57	C₄⋊C₈	—	157	1	no	—
58	C₂²⋊C₈	—	314	1	no	—
59	C₄×C₈	—	157	1	no	—
60	C₂×D₄	#11929441	1	1	yes	characteristic
61	C₂²×C₄	—	2	1	no	—
62	M₁₆	—	628	1	no	—
63	M₁₆	—	628	1	no	—
64	C₂×C₈	—	314	1	no	—
65	C₂×C₈	—	157	1	no	—
66	C₂×C₈	—	157	1	no	—
67	C₄⋊C₄	#11929441	1	1	yes	characteristic
68	C₂²⋊C₄	—	2	1	no	—
69	C₄²	C₂×#10597845	1	1	yes	characteristic
70	C₁₆	—	314	2	no	—
71	C₁₆	—	628	1	no	—
72	C₂³	—	2	1	no	—
73	D₄	—	4	1	no	—
74	C₂×C₄	—	2	1	no	—
75	C₂×C₄	—	4	1	no	—
76	C₂×C₄	—	4	1	no	—
77	C₂×C₄	C₂²⋊#10597845	1	1	yes	characteristic
78	C₂×C₄	C₂×#11929441	1	1	yes	characteristic
79	C₂×C₄	#14578963	1	1	yes	characteristic
80	C₈	—	314	1	no	—
81	C₈	—	628	1	no	—
82	C₈	—	157	1	no	—
83	C₈	—	157	1	no	—
84	C₂²	—	4	1	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	C₂²⋊#11929441	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	4	1	no	—
91	C₄	#19097833	1	1	yes	characteristic
92	C₄	#19097835	1	1	yes	characteristic
93	C₂	—	8	1	no	—
94	C₂	D₄⋊#11929441	1	1	yes	characteristic
95	C₂	#23758169	1	1	yes	characteristic
96	C₂	#23758171	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#10597845
commutator	C₂×C₄×C₁₅₇	C₂×C₈
Frattini	C₄²	C₂×#10597845
Fitting	C₄×D₄×C₁₅₇	C₄
socle	C₂²×C₁₅₇	C₂²⋊C₈
prime radical	C₂×D₄×C₁₅₇	C₈
cosocle	C₄×#11929434	C₂²
hypercenter	C₄×D₄	#10597845
hypocenter	C₁₅₇	#1330
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#14578957	C₂	semidirect
2. #23758161	C₂	semidirect
48. C₁₅₇	#1330	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
#23758161	2
#23757351	2
#1330	157