Group #34097966 — subgroups

Subgroups — #34097966

Order 20096.

152 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 77 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23757326	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097682	C₂	1	1	yes	maximal, characteristic
3	#431×C₁₅₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929434	C₄	1	2	yes	—
5	C₄⋊#11929434	—	2	1	no	—
6	C₂²⋊#11929434	—	2	1	no	—
7	#19097682	—	2	1	no	—
8	#19097682	—	2	1	no	—
9	C₄×#11929434	C₄	1	2	yes	—
10	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₁₅₇	—	2	1	no	—
12	C₂²⋊C₈×C₁₅₇	—	2	1	no	—
13	C₄×C₈×C₁₅₇	C₄	1	2	yes	—
14	C₂×#11929434	—	2	2	no	—
15	C₂×#11929434	—	2	2	no	—
16	C₂×#11929434	—	2	2	no	—
17	C₂×#11929434	C₈	1	4	yes	complemented
18	C₂²×C₄×C₁₅₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₇	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₁₅₇	—	2	2	no	—
22	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₁₅₇	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
25	C₄²×C₁₅₇	D₄	1	1	yes	characteristic
26	#11929434	—	2	4	no	—
27	#11929434	—	4	2	no	—
28	C₂³×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₅₇	—	4	1	no	—
30	C₂×C₄×C₁₅₇	—	4	1	no	—
31	C₂×C₄×C₁₅₇	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₅₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₇	C₄⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₇	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₁₅₇	M₁₆	1	2	yes	—
39	C₈×C₁₅₇	—	4	2	no	—
40	C₂²×C₁₅₇	—	2	1	no	—
41	C₂²×C₁₅₇	—	2	1	no	—
42	C₂²×C₁₅₇	#95	1	1	yes	characteristic
43	C₂²×C₁₅₇	C₄⋊C₈	1	2	yes	—
44	C₄×C₁₅₇	—	2	1	no	—
45	C₄×C₁₅₇	—	2	1	no	—
46	C₄×C₁₅₇	—	2	1	no	—
47	C₄×C₁₅₇	—	2	1	no	—
48	C₄×C₁₅₇	C₄×C₈	1	1	yes	characteristic
49	C₄×C₁₅₇	#97	1	1	yes	characteristic
50	C₄×C₁₅₇	C₂²⋊C₈	1	2	yes	—
51	C₂×C₁₅₇	—	2	2	no	—
52	C₂×C₁₅₇	#336	1	1	yes	characteristic
53	C₂×C₁₅₇	H(3,ℤ/4)	1	1	yes	characteristic
54	C₂×C₁₅₇	#338	1	1	yes	characteristic
55	C₁₅₇	#1275	1	1	yes	characteristic, complemented, Sylow
56	#1275	—	157	1	no	maximal, Sylow
57	#431	—	157	1	no	—
58	#431	D₁₅₇	1	1	yes	characteristic
59	C₂×#97	—	157	1	no	—
60	C₂²×C₈	—	157	2	no	—
61	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
62	C₄⋊C₈	—	2	1	no	—
63	C₄⋊C₈	—	314	1	no	—
64	C₂²⋊C₈	—	2	1	no	—
65	C₂²⋊C₈	—	314	1	no	—
66	#97	—	314	1	no	—
67	#97	—	314	1	no	—
68	C₄×C₈	—	157	2	no	—
69	C₄×C₈	Dic₁₅₇	1	2	yes	—
70	C₂²×C₄	C₄×D₁₅₇	1	1	yes	characteristic
71	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
72	C₂²×C₄	Dic₃₁₄	1	1	yes	characteristic
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	314	2	no	—
75	C₂×C₈	—	314	2	no	—
76	C₂×C₈	—	314	2	no	—
77	C₂×C₈	—	157	4	no	—
78	C₄²	C₄×D₁₅₇	1	1	yes	characteristic
79	C₄²	C₂×Dic₁₅₇	1	1	yes	characteristic
80	C₄²	C₂²⋊D₁₅₇	1	1	yes	characteristic
81	C₄²	D₆₂₈	1	1	yes	characteristic
82	C₂³	#14578933	1	1	yes	characteristic
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	C₄×Dic₁₅₇	1	1	yes	characteristic
86	C₂×C₄	#14578931	1	1	yes	characteristic
87	C₂×C₄	#14578931	1	1	yes	characteristic
88	C₂×C₄	C₂²⋊Dic₁₅₇	1	1	yes	characteristic
89	C₂×C₄	#14578933	1	1	yes	characteristic
90	C₂×C₄	C₄⋊Dic₁₅₇	1	1	yes	characteristic
91	C₂×C₄	C₈×D₁₅₇	1	2	yes	—
92	C₂×C₄	#14578937	1	2	yes	—
93	C₈	—	4	2	no	—
94	C₈	—	314	4	no	—
95	C₈	—	628	2	no	—
96	C₂²	—	2	1	no	—
97	C₂²	—	2	1	no	—
98	C₂²	#19097678	1	1	yes	characteristic
99	C₂²	#19097700	1	2	yes	—
100	C₄	—	2	1	no	—
101	C₄	—	2	1	no	—
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	C₈×Dic₁₅₇	1	1	yes	characteristic
105	C₄	#19097681	1	1	yes	characteristic
106	C₄	#19097683	1	2	yes	—
107	C₂	—	2	2	no	—
108	C₂	#23757027	1	1	yes	characteristic
109	C₂	#23757029	1	1	yes	characteristic
110	C₂	#23757031	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757326

C₂

yes

maximal, characteristic

C₂×#19097682

C₂

yes

maximal, characteristic

#431×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₄

yes

—

—

—

—

—

C₄

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₇

—

C₂²⋊C₈×C₁₅₇

—

C₄×C₈×C₁₅₇

C₄

yes

—

—

—

—

C₈

yes

complemented

C₂²×C₄×C₁₅₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

Q₈

yes

characteristic

C₂×C₈×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

C₂×C₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

C₄²×C₁₅₇

D₄

yes

characteristic

#11929434

—

#11929434

—

C₂³×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄²

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×C₈

yes

—

C₂×C₄×C₁₅₇

M₁₆

yes

—

C₈×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#95

yes

characteristic

C₂²×C₁₅₇

C₄⋊C₈

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×C₈

yes

characteristic

C₄×C₁₅₇

#97

yes

characteristic

C₄×C₁₅₇

C₂²⋊C₈

yes

—

C₂×C₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1275

—

157

maximal, Sylow

#431

—

157

—

#431

D₁₅₇

yes

characteristic

C₂×#97

—

157

—

C₂²×C₈

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

314

—

C₂²⋊C₈

—

C₂²⋊C₈

—

314

—

#97

—

314

—

#97

—

314

—

C₄×C₈

—

157

—

C₄×C₈

Dic₁₅₇

yes

—

C₂²×C₄

C₄×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

characteristic

C₂²×C₄

Dic₃₁₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₄²

C₄×D₁₅₇

yes

characteristic

C₄²

C₂×Dic₁₅₇

yes

characteristic

C₄²

C₂²⋊D₁₅₇

yes

characteristic

C₄²

D₆₂₈

yes

characteristic

C₂³

#14578933

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₅₇

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Dic₁₅₇

yes

characteristic

C₂×C₄

#14578933

yes

characteristic

C₂×C₄

C₄⋊Dic₁₅₇

yes

characteristic

C₂×C₄

C₈×D₁₅₇

yes

—

C₂×C₄

#14578937

yes

—

C₈

—

C₈

—

314

—

C₈

—

628

—

C₂²

—

C₂²

—

C₂²

#19097678

yes

characteristic

C₂²

#19097700

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₈×Dic₁₅₇

yes

characteristic

105

C₄

#19097681

yes

characteristic

106

C₄

#19097683

yes

—

107

C₂

—

108

C₂

#23757027

yes

characteristic

109

C₂

#23757029

yes

characteristic

110

C₂

#23757031

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₄×Dic₁₅₇
commutator	C₄×C₁₅₇	C₄×C₈
Frattini	C₂×C₄²	C₂×D₁₅₇
Fitting	#431×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#95
prime radical	C₂³×C₁₅₇	C₄⋊C₄
cosocle	C₂×C₄²×C₁₅₇	C₂²
hypercenter	#431	D₁₅₇
hypocenter	C₁₅₇	#1275
LCS	G ▷ C₄×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center

C₂×C₄

C₄×Dic₁₅₇

commutator

C₄×C₁₅₇

C₄×C₈

Frattini

C₂×C₄²

C₂×D₁₅₇

Fitting

#431×C₁₅₇

C₂

socle

C₂²×C₁₅₇

#95

prime radical

C₂³×C₁₅₇

C₄⋊C₄

cosocle

C₂×C₄²×C₁₅₇

C₂²

D₁₅₇

C₁₅₇

G ▷ C₄×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
17. C₂×#11929434	C₈	semidirect
55. C₁₅₇	#1275	semidirect

H ≅ G/N

type

17. C₂×#11929434

C₈

semidirect

55. C₁₅₇

#1275

semidirect

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

subgroup	index
#23757326	2
C₂×#19097682	2
#431×C₁₅₇	2
#1275	157

subgroup

index

#23757326

C₂×#19097682

#431×C₁₅₇

#1275

157

Subgroups — #34097966

Conjugacy classes of subgroups

Complements

Maximal subgroups