Group #34108513 — subgroups

Subgroups — #34108513

Order 20096.

504 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 197 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×#19097771	C₂	1	1	yes	maximal, characteristic
2	#23757180	C₂	1	4	yes	maximal
3	#23757178	C₂	1	4	yes	maximal
4	#23757138	C₂	1	4	yes	maximal
5	C₄²×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₄	C₂²	1	2	yes	—
8	#19097771	—	2	4	no	—
9	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
11	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
12	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
13	C₂×C₄×Dic₁₅₇	C₂²	1	2	yes	—
14	#19097678	C₂²	1	16	yes	—
15	C₂²×Q₈×C₁₅₇	C₂²	1	2	yes	—
16	C₄⋊Q₈×C₁₅₇	—	2	4	no	—
17	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
18	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₅₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₁₄	—	2	4	no	—
21	C₂×Dic₃₁₄	—	2	8	no	—
22	C₂²×Dic₁₅₇	C₂³	1	4	yes	—
23	C₂²×Dic₁₅₇	C₂³	1	4	yes	—
24	C₄⋊Dic₁₅₇	—	2	4	no	—
25	C₄⋊Dic₁₅₇	—	2	4	no	—
26	C₄×Dic₁₅₇	—	2	4	no	—
27	C₄×Dic₁₅₇	—	2	4	no	—
28	C₄×Dic₁₅₇	Q₈	1	8	yes	complemented
29	C₂×Q₈×C₁₅₇	—	2	4	no	—
30	C₂×Q₈×C₁₅₇	—	2	8	no	—
31	C₂²×C₄×C₁₅₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
33	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₅₇	—	2	4	no	—
36	C₄⋊C₄×C₁₅₇	—	2	4	no	—
37	C₄²×C₁₅₇	D₄	1	4	yes	—
38	Dic₃₁₄	—	4	8	no	—
39	C₂×Dic₁₅₇	—	2	4	no	—
40	C₂×Dic₁₅₇	—	2	4	no	—
41	C₂×Dic₁₅₇	—	2	4	no	—
42	C₂×Dic₁₅₇	—	2	8	no	—
43	C₂×Dic₁₅₇	C₂×Q₈	1	8	yes	—
44	C₂³×C₁₅₇	C₂⁴	1	1	yes	characteristic
45	Q₈×C₁₅₇	—	4	8	no	—
46	C₂×C₄×C₁₅₇	—	2	4	no	—
47	C₂×C₄×C₁₅₇	—	2	8	no	—
48	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
52	C₂×C₄×C₁₅₇	#41	1	4	yes	—
53	C₂×C₄×C₁₅₇	#41	1	4	yes	—
54	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
55	C₂²×C₁₅₇	C₂²×Q₈	1	2	yes	—
56	C₂²×C₁₅₇	C₂×#41	1	2	yes	—
57	C₂²×C₁₅₇	C₂×#41	1	2	yes	—
58	C₄×C₁₅₇	—	4	4	no	—
59	C₄×C₁₅₇	#124	1	4	yes	—
60	C₄×C₁₅₇	#124	1	4	yes	—
61	C₄×C₁₅₇	C₄⋊Q₈	1	4	yes	—
62	Dic₁₅₇	—	2	8	no	—
63	Dic₁₅₇	—	4	4	no	—
64	C₂×C₁₅₇	C₂×#124	1	1	yes	characteristic
65	C₂×C₁₅₇	C₂×#124	1	1	yes	characteristic
66	C₂×C₁₅₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
67	C₂×C₁₅₇	#533	1	4	yes	—
68	C₁₅₇	#2862	1	1	yes	characteristic, complemented, Sylow
69	#2862	—	157	1	no	maximal, Sylow
70	C₂×C₄⋊Q₈	—	157	1	no	—
71	C₂×C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic
72	#391	—	157	4	no	—
73	#391	—	157	4	no	—
74	#382	—	157	4	no	—
75	C₄³	—	157	1	no	—
76	C₂²×Q₈	—	157	2	no	—
77	C₂²×Q₈	C₂×D₁₅₇	1	2	yes	—
78	C₄⋊Q₈	—	2	4	no	—
79	C₄⋊Q₈	—	314	4	no	—
80	C₂×C₄⋊C₄	—	157	2	no	—
81	C₂×C₄⋊C₄	—	157	2	no	—
82	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
83	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
84	C₂×C₄²	—	157	2	no	—
85	C₂×C₄²	—	157	2	no	—
86	C₂×C₄²	—	157	2	no	—
87	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
88	#95	—	157	16	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	2	8	no	—
91	C₂×Q₈	—	314	4	no	—
92	C₂×Q₈	—	314	8	no	—
93	C₂²×C₄	—	157	4	no	—
94	C₂²×C₄	—	157	4	no	—
95	C₂²×C₄	C₂²×D₁₅₇	1	4	yes	—
96	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
98	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
99	C₄⋊C₄	—	2	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	314	4	no	—
102	C₄⋊C₄	—	314	4	no	—
103	C₄²	—	314	4	no	—
104	C₄²	—	314	4	no	—
105	C₄²	—	157	8	no	—
106	C₄²	C₂²⋊D₁₅₇	1	4	yes	—
107	C₂³	C₂³×D₁₅₇	1	1	yes	characteristic
108	Q₈	—	4	8	no	—
109	Q₈	—	628	8	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	314	4	no	—
113	C₂×C₄	—	314	4	no	—
114	C₂×C₄	—	314	4	no	—
115	C₂×C₄	—	314	8	no	—
116	C₂×C₄	—	157	8	no	—
117	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
118	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	2	yes	—
120	C₂×C₄	Q₈×D₁₅₇	1	4	yes	—
121	C₂×C₄	#14578953	1	4	yes	—
122	C₂×C₄	#14578955	1	4	yes	—
123	C₂²	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
124	C₂²	C₂×Q₈×D₁₅₇	1	2	yes	—
125	C₂²	C₂×#14578953	1	2	yes	—
126	C₂²	C₂×#14578955	1	2	yes	—
127	C₄	—	4	4	no	—
128	C₄	—	314	8	no	—
129	C₄	—	628	4	no	—
130	C₄	#19097758	1	4	yes	—
131	C₄	#19097760	1	4	yes	—
132	C₄	#19097769	1	4	yes	—
133	C₂	C₂×#19097758	1	1	yes	characteristic
134	C₂	C₂×#19097760	1	1	yes	characteristic
135	C₂	C₂×#19097769	1	1	yes	characteristic
136	C₂	#23757828	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097771

C₂

yes

maximal, characteristic

#23757180

C₂

yes

maximal

#23757178

C₂

yes

maximal

#23757138

C₂

yes

maximal

C₄²×Dic₁₅₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₂²

yes

—

#19097771

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

C₂×C₄×Dic₁₅₇

C₂²

yes

—

#19097678

C₂²

yes

—

C₂²×Q₈×C₁₅₇

C₂²

yes

—

C₄⋊Q₈×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

C₂³

yes

—

C₂²×Dic₁₅₇

C₂³

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

Q₈

yes

complemented

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

—

C₄²×C₁₅₇

D₄

yes

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×Q₈

yes

—

C₂³×C₁₅₇

C₂⁴

yes

characteristic

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

#41

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₂²×C₁₅₇

C₂²×Q₈

yes

—

C₂²×C₁₅₇

C₂×#41

yes

—

C₂²×C₁₅₇

C₂×#41

yes

—

C₄×C₁₅₇

—

C₄×C₁₅₇

#124

yes

—

C₄×C₁₅₇

#124

yes

—

C₄×C₁₅₇

C₄⋊Q₈

yes

—

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

yes

C₂×C₁₅₇

yes

C₂×C₁₅₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₇

#533

yes

—

C₁₅₇

#2862

yes

characteristic, complemented, Sylow

#2862

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

C₂×C₄⋊Q₈

D₁₅₇

yes

characteristic

#391

—

157

—

#391

—

157

—

#382

—

157

—

C₄³

—

157

—

C₂²×Q₈

—

157

—

C₂²×Q₈

C₂×D₁₅₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

—

157

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#95

—

157

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

314

—

102

C₄⋊C₄

—

314

—

103

C₄²

—

314

—

104

C₄²

—

314

—

105

C₄²

—

157

—

106

C₄²

C₂²⋊D₁₅₇

yes

—

107

C₂³

C₂³×D₁₅₇

yes

characteristic

108

Q₈

—

109

Q₈

—

628

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

314

—

113

C₂×C₄

—

314

—

114

C₂×C₄

—

314

—

115

C₂×C₄

—

314

—

116

C₂×C₄

—

157

—

117

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

118

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

120

C₂×C₄

Q₈×D₁₅₇

yes

—

121

C₂×C₄

#14578953

yes

—

122

C₂×C₄

#14578955

yes

—

123

C₂²

C₂²×C₂²⋊D₁₅₇

yes

characteristic

124

C₂²

C₂×Q₈×D₁₅₇

yes

—

125

C₂²

C₂×#14578953

yes

—

126

C₂²

C₂×#14578955

yes

—

127

C₄

—

128

C₄

—

314

—

129

C₄

—

628

—

130

C₄

#19097758

yes

—

131

C₄

#19097760

yes

—

132

C₄

#19097769

yes

—

133

C₂

C₂×#19097758

yes

characteristic

134

C₂

C₂×#19097760

yes

characteristic

135

C₂

C₂×#19097769

yes

characteristic

136

C₂

#23757828

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₁₅₇
commutator, socle, prime radical, cosocle	C₂³×C₁₅₇	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₁₅₇	C₂
hypercenter	C₂×C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	#2862
LCS	G ▷ C₂³×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₁₅₇

commutator, socle, prime radical, cosocle

C₂³×C₁₅₇

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₁₅₇

C₂

hypercenter

C₂×C₄⋊Q₈

D₁₅₇

C₁₅₇

G ▷ C₂³×C₁₅₇ ▷ C₁₅₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
28. C₄×Dic₁₅₇	Q₈	semidirect
68. C₁₅₇	#2862	semidirect

H ≅ G/N

type

28. C₄×Dic₁₅₇

Q₈

semidirect

68. C₁₅₇

#2862

semidirect

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

subgroup	index
C₂×#19097771	2
#23757180	2
#23757178	2
#23757138	2
C₄²×Dic₁₅₇	2
C₂×C₄⋊Q₈×C₁₅₇	2
#2862	157

subgroup

index

C₄²×Dic₁₅₇

C₂×C₄⋊Q₈×C₁₅₇

#2862

157

Subgroups — #34108513

Conjugacy classes of subgroups

Complements

Maximal subgroups