Group #34100263 — subgroups

Subgroups — #34100263

Order 20096.

248 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 133 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	C₂×C₄×#11929434	C₂	1	1	yes	maximal, characteristic
2	#23757143	C₂	1	1	yes	maximal, characteristic
3	#23757035	C₂	1	4	yes	maximal
4	#384×C₁₅₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11929434	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₅₇	—	2	1	no	—
7	C₂×C₄⋊Dic₁₅₇	—	2	1	no	—
8	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
9	C₄×#11929434	C₄	1	4	yes	complemented
10	C₂×C₄⋊C₄×C₁₅₇	—	2	1	no	—
11	C₂×C₄⋊C₄×C₁₅₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₅₇	—	2	2	no	—
15	C₂×#11929434	—	2	2	no	—
16	C₂×#11929434	—	2	2	no	—
17	C₂×#11929434	C₂×C₄	1	4	yes	—
18	C₄⋊Dic₁₅₇	—	2	4	no	—
19	C₄⋊Dic₁₅₇	—	4	2	no	—
20	C₄⋊Dic₁₅₇	—	4	2	no	—
21	C₂²×C₄×C₁₅₇	—	2	2	no	—
22	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₁₅₇	—	2	4	no	—
26	C₄⋊C₄×C₁₅₇	—	4	2	no	—
27	C₄⋊C₄×C₁₅₇	—	4	2	no	—
28	C₄²×C₁₅₇	C₂×C₄	1	2	yes	—
29	C₄²×C₁₅₇	Q₈	1	2	yes	—
30	C₂×Dic₁₅₇	—	4	2	no	—
31	C₂×Dic₁₅₇	—	4	2	no	—
32	C₂×Dic₁₅₇	—	4	2	no	—
33	#11929434	—	2	4	no	—
34	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₇	—	4	2	no	—
36	C₂×C₄×C₁₅₇	—	4	2	no	—
37	C₂×C₄×C₁₅₇	—	4	2	no	—
38	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₅₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₅₇	D₈	1	2	yes	—
41	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
42	C₂×C₄×C₁₅₇	SD₁₆	1	2	yes	—
43	C₂×C₄×C₁₅₇	Q₁₆	1	2	yes	—
44	C₂×C₄×C₁₅₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₅₇	#41	1	2	yes	—
47	C₂×C₄×C₁₅₇	#41	1	2	yes	—
48	C₂²×C₁₅₇	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₅₇	#124	1	1	yes	characteristic
50	C₂²×C₁₅₇	#124	1	1	yes	characteristic
51	C₂²×C₁₅₇	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
53	C₂²×C₁₅₇	C₂×SD₁₆	1	1	yes	characteristic
54	C₂²×C₁₅₇	C₂×Q₁₆	1	1	yes	characteristic
55	C₄×C₁₅₇	—	8	2	no	—
56	C₄×C₁₅₇	#106	1	4	yes	—
57	C₄×C₁₅₇	#107	1	4	yes	—
58	C₄×C₁₅₇	#117	1	2	yes	—
59	C₄×C₁₅₇	#125	1	2	yes	—
60	Dic₁₅₇	—	8	2	no	—
61	C₂×C₁₅₇	#382	1	1	yes	characteristic
62	C₂×C₁₅₇	C₂×#106	1	1	yes	characteristic
63	C₂×C₁₅₇	C₂×#107	1	1	yes	characteristic
64	C₂×C₁₅₇	#486	1	1	yes	characteristic
65	C₂×C₁₅₇	#486	1	1	yes	characteristic
66	C₂×C₁₅₇	#487	1	1	yes	characteristic
67	C₂×C₁₅₇	#487	1	1	yes	characteristic
68	C₁₅₇	#1834	1	1	yes	characteristic, complemented, Sylow
69	#1834	—	157	1	no	maximal, Sylow
70	C₂×C₄×C₈	—	157	1	no	—
71	#384	—	157	1	no	—
72	#384	D₁₅₇	1	1	yes	characteristic
73	#340	—	157	4	no	—
74	C₂²×C₈	—	157	2	no	—
75	C₂×C₄⋊C₄	—	2	1	no	—
76	C₂×C₄⋊C₄	—	2	1	no	—
77	C₂×C₄⋊C₄	—	314	1	no	—
78	C₂×C₄⋊C₄	—	314	1	no	—
79	C₂×C₄⋊C₄	—	157	2	no	—
80	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
81	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
82	C₄×C₈	—	157	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	314	2	no	—
85	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
88	C₂×C₈	—	314	2	no	—
89	C₂×C₈	—	314	2	no	—
90	C₂×C₈	—	157	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	314	4	no	—
95	C₄⋊C₄	—	628	2	no	—
96	C₄⋊C₄	—	628	2	no	—
97	C₄²	C₄×D₁₅₇	1	2	yes	—
98	C₄²	Dic₃₁₄	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	628	2	no	—
104	C₂×C₄	—	628	2	no	—
105	C₂×C₄	—	628	2	no	—
106	C₂×C₄	#14578933	1	2	yes	—
107	C₂×C₄	#14578933	1	2	yes	—
108	C₂×C₄	#14578939	1	2	yes	—
109	C₂×C₄	#14578941	1	2	yes	—
110	C₂×C₄	#14578942	1	2	yes	—
111	C₂×C₄	#14578944	1	2	yes	—
112	C₂×C₄	C₂×C₄×D₁₅₇	1	1	yes	characteristic
113	C₂×C₄	C₂×Dic₃₁₄	1	1	yes	characteristic
114	C₂×C₄	#14578953	1	2	yes	—
115	C₂×C₄	#14578955	1	2	yes	—
116	C₈	—	314	4	no	—
117	C₂²	C₂×#14578933	1	1	yes	characteristic
118	C₂²	#19097758	1	1	yes	characteristic
119	C₂²	#19097760	1	1	yes	characteristic
120	C₂²	C₂×#14578939	1	1	yes	characteristic
121	C₂²	C₂×#14578941	1	1	yes	characteristic
122	C₂²	C₂×#14578942	1	1	yes	characteristic
123	C₂²	C₂×#14578944	1	1	yes	characteristic
124	C₄	—	8	2	no	—
125	C₄	—	1256	2	no	—
126	C₄	#19097702	1	4	yes	—
127	C₄	#19097704	1	4	yes	—
128	C₄	#19097729	1	2	yes	—
129	C₄	#19097762	1	2	yes	—
130	C₂	#23757138	1	1	yes	characteristic
131	C₂	C₂×#19097702	1	1	yes	characteristic
132	C₂	C₂×#19097704	1	1	yes	characteristic
133	C₂	#23757619	1	1	yes	characteristic
134	C₂	#23757621	1	1	yes	characteristic
135	C₂	#23757623	1	1	yes	characteristic
136	C₂	#23757625	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×#11929434

C₂

yes

maximal, characteristic

#23757143

C₂

yes

maximal, characteristic

#23757035

C₂

yes

maximal

#384×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₄×#11929434

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂²×Dic₁₅₇

—

C₂×#11929434

—

C₂×#11929434

—

C₂×#11929434

C₂×C₄

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

D₄

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

C₂×C₄

yes

—

C₄²×C₁₅₇

Q₈

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₇

D₈

yes

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

SD₁₆

yes

—

C₂×C₄×C₁₅₇

Q₁₆

yes

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂×C₄⋊C₄

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

yes

C₂²×C₁₅₇

C₂×D₈

yes

characteristic

C₂²×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₅₇

C₂×SD₁₆

yes

characteristic

C₂²×C₁₅₇

C₂×Q₁₆

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

#106

yes

—

C₄×C₁₅₇

#107

yes

—

C₄×C₁₅₇

#117

yes

—

C₄×C₁₅₇

#125

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₁₅₇

yes

characteristic, complemented, Sylow

#1834

—

157

maximal, Sylow

C₂×C₄×C₈

—

157

—

#384

—

157

—

#384

D₁₅₇

yes

characteristic

#340

—

157

—

C₂²×C₈

—

157

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

C₄×C₈

—

157

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

C₂²⋊D₁₅₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

314

—

C₂×C₈

—

314

—

C₂×C₈

—

157

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

628

—

C₄⋊C₄

—

628

—

C₄²

C₄×D₁₅₇

yes

—

C₄²

Dic₃₁₄

yes

—

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

628

—

104

C₂×C₄

—

628

—

105

C₂×C₄

—

628

—

106

C₂×C₄

#14578933

yes

—

107

C₂×C₄

#14578933

yes

—

108

C₂×C₄

#14578939

yes

—

109

C₂×C₄

#14578941

yes

—

110

C₂×C₄

#14578942

yes

—

111

C₂×C₄

#14578944

yes

—

112

C₂×C₄

C₂×C₄×D₁₅₇

yes

characteristic

113

C₂×C₄

C₂×Dic₃₁₄

yes

characteristic

114

C₂×C₄

#14578953

yes

—

115

C₂×C₄

#14578955

yes

—

116

C₈

—

314

—

117

C₂²

C₂×#14578933

yes

characteristic

118

C₂²

#19097758

yes

characteristic

119

C₂²

#19097760

yes

characteristic

120

C₂²

C₂×#14578939

yes

characteristic

121

C₂²

C₂×#14578941

yes

characteristic

122

C₂²

C₂×#14578942

yes

characteristic

123

C₂²

C₂×#14578944

yes

characteristic

124

C₄

—

125

C₄

—

1256

—

126

C₄

#19097702

yes

—

127

C₄

#19097704

yes

—

128

C₄

#19097729

yes

—

129

C₄

#19097762

yes

—

130

C₂

#23757138

yes

characteristic

131

C₂

C₂×#19097702

yes

characteristic

132

C₂

C₂×#19097704

yes

characteristic

133

C₂

#23757619

yes

characteristic

134

C₂

#23757621

yes

characteristic

135

C₂

#23757623

yes

characteristic

136

C₂

#23757625

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator	C₂×C₄×C₁₅₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₇
Fitting	#384×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂²×C₄×C₁₅₇	C₂³
hypercenter	#384	D₁₅₇
hypocenter	C₁₅₇	#1834
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator

C₂×C₄×C₁₅₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₇

Fitting

#384×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂²×C₄×C₁₅₇

C₂³

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
9. C₄×#11929434	C₄	semidirect
68. C₁₅₇	#1834	semidirect

H ≅ G/N

type

9. C₄×#11929434

C₄

semidirect

68. C₁₅₇

#1834

semidirect

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

subgroup	index
C₂×C₄×#11929434	2
#23757143	2
#23757035	2
#384×C₁₅₇	2
#1834	157

subgroup

index

#384×C₁₅₇

157

Subgroups — #34100263

Conjugacy classes of subgroups

Complements

Maximal subgroups