Group #34099281 — subgroups

Subgroups — #34099281

Order 20096.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#23757652	C₂	1	1	yes	maximal, characteristic, complemented
2	#23757563	C₂	1	2	yes	maximal, complemented
3	#23757022	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊#11929434	C₂	1	2	yes	maximal, complemented
5	#555×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#14578944	—	2	2	no	—
7	#19097771	C₂²	1	1	yes	characteristic
8	#19097702	—	2	2	no	—
9	C₄⋊#11929434	C₂²	1	2	yes	—
10	#19097692	—	2	2	no	—
11	C₄×#11929434	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₅₇	C₂²	1	1	yes	characteristic
13	#124×C₁₅₇	—	2	2	no	—
14	#121×C₁₅₇	—	2	2	no	—
15	C₄×D₄×C₁₅₇	C₂²	1	2	yes	—
16	#117×C₁₅₇	—	2	2	no	—
17	C₂×Dic₃₁₄	—	2	1	no	—
18	#14578944	—	4	4	no	—
19	C₂×#11929434	—	2	1	no	—
20	C₂×#11929434	—	2	2	no	—
21	C₄⋊Dic₁₅₇	—	2	2	no	—
22	C₂×Q₈×C₁₅₇	D₄	1	2	yes	—
23	C₂×D₄×C₁₅₇	—	2	2	no	—
24	C₂×D₄×C₁₅₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₅₇	—	2	2	no	—
26	C₄⋊C₄×C₁₅₇	—	2	2	no	—
27	C₄⋊C₄×C₁₅₇	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₅₇	—	2	4	no	—
29	C₂²⋊C₄×C₁₅₇	—	2	2	no	—
30	C₄²×C₁₅₇	—	2	2	no	—
31	C₄²×C₁₅₇	C₂³	1	1	yes	characteristic
32	Dic₃₁₄	—	4	2	no	—
33	C₂×Dic₁₅₇	—	4	1	no	—
34	#11929434	—	2	2	no	—
35	#11929434	—	4	2	no	—
36	C₂³×C₁₅₇	—	2	2	no	—
37	Q₈×C₁₅₇	—	4	2	no	—
38	D₄×C₁₅₇	—	2	2	no	—
39	D₄×C₁₅₇	—	2	2	no	—
40	D₄×C₁₅₇	—	8	2	no	—
41	C₂×C₄×C₁₅₇	—	2	2	no	—
42	C₂×C₄×C₁₅₇	—	2	2	no	—
43	C₂×C₄×C₁₅₇	—	4	2	no	—
44	C₂×C₄×C₁₅₇	—	4	2	no	—
45	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
47	C₂²×C₁₅₇	—	4	2	no	—
48	C₂²×C₁₅₇	—	4	2	no	—
49	C₂²×C₁₅₇	—	4	2	no	—
50	C₂²×C₁₅₇	#120	1	1	yes	characteristic
51	C₄×C₁₅₇	—	2	1	no	—
52	C₄×C₁₅₇	—	4	2	no	—
53	C₄×C₁₅₇	—	4	2	no	—
54	C₄×C₁₅₇	#135	1	2	yes	—
55	C₄×C₁₅₇	#137	1	2	yes	—
56	Dic₁₅₇	—	8	1	no	—
57	C₂×C₁₅₇	—	8	2	no	—
58	C₂×C₁₅₇	#452	1	2	yes	—
59	C₂×C₁₅₇	#453	1	1	yes	characteristic
60	C₁₅₇	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	157	1	no	maximal, Sylow
62	#555	D₁₅₇	1	1	yes	characteristic
63	#494	—	157	1	no	—
64	#476	—	157	2	no	—
65	#333	—	157	1	no	—
66	D₄⋊C₈	—	157	2	no	—
67	C₂×Q₁₆	—	314	2	no	—
68	C₄⋊Q₈	—	157	1	no	—
69	C₄⋊Q₈	C₂×D₁₅₇	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂×D₁₅₇	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	314	2	no	—
75	C₄⋊C₈	—	157	2	no	—
76	#102	—	314	2	no	—
77	C₄×C₈	—	157	1	no	—
78	C₂×Q₈	—	314	1	no	—
79	C₂×Q₈	C₂²⋊D₁₅₇	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂²⋊D₁₅₇	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	628	4	no	—
84	C₂×C₈	—	314	1	no	—
85	C₂×C₈	—	314	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	314	2	no	—
88	C₄⋊C₄	C₂²⋊D₁₅₇	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂²×D₁₅₇	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	628	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	628	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
106	C₈	—	314	2	no	—
107	C₈	—	628	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#19097740	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	1256	1	no	—
116	C₄	#19097791	1	2	yes	—
117	C₄	#19097801	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	#23757435	1	2	yes	—
120	C₂	#23757441	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23757652

C₂

yes

maximal, characteristic, complemented

#23757563

C₂

yes

maximal, complemented

#23757022

C₂

yes

maximal, characteristic, complemented

D₄⋊#11929434

C₂

yes

maximal, complemented

#555×C₁₅₇

C₂

yes

maximal, characteristic

C₂×#14578944

—

#19097771

C₂²

yes

characteristic

#19097702

—

C₄⋊#11929434

C₂²

yes

—

#19097692

—

C₄×#11929434

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₇

C₂²

yes

characteristic

#124×C₁₅₇

—

#121×C₁₅₇

—

C₄×D₄×C₁₅₇

C₂²

yes

—

#117×C₁₅₇

—

C₂×Dic₃₁₄

—

#14578944

—

C₂×#11929434

—

C₂×#11929434

—

C₄⋊Dic₁₅₇

—

C₂×Q₈×C₁₅₇

D₄

yes

—

C₂×D₄×C₁₅₇

—

C₂×D₄×C₁₅₇

D₄

yes

—

C₂²×C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

D₄

yes

—

C₂²⋊C₄×C₁₅₇

—

C₂²⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

—

C₄²×C₁₅₇

C₂³

yes

characteristic

Dic₃₁₄

—

C₂×Dic₁₅₇

—

#11929434

—

#11929434

—

C₂³×C₁₅₇

—

Q₈×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

D₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

—

C₂²×C₁₅₇

#120

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#135

yes

—

C₄×C₁₅₇

#137

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

—

C₂×C₁₅₇

#452

yes

—

C₂×C₁₅₇

#453

yes

characteristic

C₁₅₇

#1634

yes

characteristic, complemented, Sylow

#1634

—

157

maximal, Sylow

#555

D₁₅₇

yes

characteristic

#494

—

157

—

#476

—

157

—

#333

—

157

—

D₄⋊C₈

—

157

—

C₂×Q₁₆

—

314

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

C₂×D₁₅₇

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂×D₁₅₇

yes

—

#117

—

#106

—

314

—

C₄⋊C₈

—

157

—

#102

—

314

—

C₄×C₈

—

157

—

C₂×Q₈

—

314

—

C₂×Q₈

C₂²⋊D₁₅₇

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₅₇

yes

—

C₂²×C₄

—

Q₁₆

—

628

—

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

C₂²⋊D₁₅₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₅₇

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

628

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

628

—

104

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

106

C₈

—

314

—

107

C₈

—

628

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#19097740

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

1256

—

116

C₄

#19097791

yes

—

117

C₄

#19097801

yes

—

118

C₂

—

119

C₂

#23757435

yes

—

120

C₂

#23757441

yes

characteristic

	subgroup	G/H
center	C₂²	#19097740
commutator, cosocle	C₄²×C₁₅₇	C₂³
Frattini	C₄²	C₂²×D₁₅₇
Fitting, prime radical	#555×C₁₅₇	C₂
socle	C₂²×C₁₅₇	#120
hypercenter	#555	D₁₅₇
hypocenter	C₁₅₇	#1634
LCS	G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#19097740

commutator, cosocle

C₄²×C₁₅₇

C₂³

Frattini

C₄²

C₂²×D₁₅₇

Fitting, prime radical

#555×C₁₅₇

C₂

C₂²×C₁₅₇

D₁₅₇

C₁₅₇

G ▷ C₄²×C₁₅₇ ▷ C₂²×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #23757652	C₂	semidirect
2. #23757563	C₂	semidirect
3. #23757022	C₂	semidirect
4. D₄⋊#11929434	C₂	semidirect
60. C₁₅₇	#1634	semidirect

H ≅ G/N

type

1. #23757652

C₂

semidirect

2. #23757563

C₂

semidirect

3. #23757022

C₂

semidirect

4. D₄⋊#11929434

C₂

semidirect

60. C₁₅₇

#1634

semidirect

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

subgroup	index
#23757652	2
#23757563	2
#23757022	2
D₄⋊#11929434	2
#555×C₁₅₇	2
#1634	157

subgroup

index

#555×C₁₅₇

157

Subgroups — #34099281

Conjugacy classes of subgroups

Complements

Maximal subgroups