Group #34110642 — subgroups

Subgroups — C₂×#23757676 #34110642

Order 20096.

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

normal subgroups only

This group has 424 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	#23757676	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19097704	C₂	1	4	yes	maximal
4	C₂×C₄×#11929434	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₁₄	—	2	1	no	—
7	C₂²×#11929434	C₂²	1	2	yes	—
8	#19097771	—	2	2	no	—
9	#19097771	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₅₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
12	#19097704	C₂²	1	16	yes	—
13	C₄×#11929434	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₅₇	—	2	1	no	—
15	C₄⋊Q₈×C₁₅₇	—	2	2	no	—
16	C₄⋊Q₈×C₁₅₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₅₇	—	2	1	no	—
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	—	4	4	no	—
22	C₂²×Dic₁₅₇	—	2	2	no	—
23	C₂×#11929434	Q₈	1	8	yes	complemented
24	C₂×#11929434	C₂³	1	4	yes	—
25	C₄⋊Dic₁₅₇	—	2	4	no	—
26	C₄⋊Dic₁₅₇	—	2	2	no	—
27	C₄⋊Dic₁₅₇	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₅₇	—	2	4	no	—
29	C₂×Q₈×C₁₅₇	—	4	4	no	—
30	C₂²×C₄×C₁₅₇	—	2	2	no	—
31	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₅₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₅₇	—	2	4	no	—
35	C₄⋊C₄×C₁₅₇	—	2	2	no	—
36	C₄⋊C₄×C₁₅₇	C₂³	1	4	yes	—
37	C₄²×C₁₅₇	D₄	1	2	yes	—
38	C₄²×C₁₅₇	C₂³	1	2	yes	—
39	Dic₃₁₄	—	4	8	no	—
40	C₂×Dic₁₅₇	—	2	4	no	—
41	C₂×Dic₁₅₇	—	4	4	no	—
42	#11929434	C₂×Q₈	1	8	yes	complemented
43	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₅₇	—	4	8	no	—
45	C₂×C₄×C₁₅₇	—	2	4	no	—
46	C₂×C₄×C₁₅₇	—	4	4	no	—
47	C₂×C₄×C₁₅₇	D₈	1	4	yes	—
48	C₂×C₄×C₁₅₇	Q₁₆	1	4	yes	—
49	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₅₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₅₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
54	C₂²×C₁₅₇	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₅₇	C₂×D₈	1	2	yes	—
56	C₂²×C₁₅₇	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₅₇	—	4	4	no	—
59	C₄×C₁₅₇	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₅₇	C₂×D₈	1	4	yes	—
61	C₄×C₁₅₇	C₂×Q₁₆	1	4	yes	—
62	C₄×C₁₅₇	C₂²×Q₈	1	2	yes	—
63	Dic₁₅₇	—	4	4	no	—
64	C₂×C₁₅₇	#500	1	4	yes	complemented
65	C₂×C₁₅₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₁₅₇	C₂²×D₈	1	1	yes	characteristic
67	C₂×C₁₅₇	C₂²×Q₁₆	1	1	yes	characteristic
68	C₁₅₇	C₂×#500	1	1	yes	characteristic, complemented, Sylow
69	C₂×#500	—	157	1	no	maximal, Sylow
70	C₂×C₄⋊Q₈	—	157	1	no	—
71	C₂×C₄⋊Q₈	D₁₅₇	1	1	yes	characteristic
72	#500	—	157	8	no	—
73	C₂×#107	—	157	4	no	—
74	C₂×C₄×C₈	—	157	1	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	314	1	no	—
77	C₂²×C₈	—	157	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	314	2	no	—
80	C₄⋊Q₈	—	157	4	no	—
81	C₄⋊Q₈	C₂×D₁₅₇	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	314	1	no	—
84	C₂×C₄⋊C₄	—	157	2	no	—
85	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
86	C₂×C₄²	C₂×D₁₅₇	1	1	yes	characteristic
87	#107	—	157	16	no	—
88	C₄×C₈	—	157	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	314	4	no	—
92	C₂×Q₈	—	628	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	314	2	no	—
95	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₅₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
98	C₂×C₈	—	157	8	no	—
99	C₂×C₈	—	157	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	314	4	no	—
103	C₄⋊C₄	—	314	2	no	—
104	C₄⋊C₄	—	157	4	no	—
105	C₄⋊C₄	C₂²×D₁₅₇	1	4	yes	—
106	C₄²	C₂²⋊D₁₅₇	1	2	yes	—
107	C₄²	C₂²×D₁₅₇	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	628	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	314	4	no	—
114	C₂×C₄	—	628	4	no	—
115	C₂×C₄	#14578939	1	4	yes	—
116	C₂×C₄	#14578944	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₅₇	1	1	yes	characteristic
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	C₂³×D₁₅₇	1	1	yes	characteristic
122	C₈	—	157	8	no	—
123	C₂²	#19097769	1	2	yes	—
124	C₂²	C₂×#14578939	1	2	yes	—
125	C₂²	C₂×#14578944	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₁₅₇	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	628	4	no	—
129	C₄	#19097769	1	2	yes	—
130	C₄	C₂×#14578939	1	4	yes	—
131	C₄	C₂×#14578944	1	4	yes	—
132	C₄	C₂×Q₈×D₁₅₇	1	2	yes	—
133	C₂	#23757676	1	4	yes	direct product factor
134	C₂	C₂×#19097769	1	1	yes	characteristic
135	C₂	C₂²×#14578939	1	1	yes	characteristic
136	C₂	C₂²×#14578944	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097771

C₂

yes

maximal, characteristic

#23757676

C₂

yes

maximal, direct product factor

C₂×#19097704

C₂

yes

maximal

C₂×C₄×#11929434

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

—

C₂²×#11929434

C₂²

yes

—

#19097771

—

#19097771

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

#19097704

C₂²

yes

—

C₄×#11929434

C₂²

yes

—

C₂²×Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

—

C₄⋊Q₈×C₁₅₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₅₇

—

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

C₂×C₄²×C₁₅₇

C₂²

yes

characteristic

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂²×Dic₁₅₇

—

C₂×#11929434

Q₈

yes

complemented

C₂×#11929434

C₂³

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂³

yes

—

C₂×Q₈×C₁₅₇

—

C₂×Q₈×C₁₅₇

—

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

C₂³

yes

—

C₄²×C₁₅₇

D₄

yes

—

C₄²×C₁₅₇

C₂³

yes

—

Dic₃₁₄

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

#11929434

C₂×Q₈

yes

complemented

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

Q₈×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

D₈

yes

—

C₂×C₄×C₁₅₇

Q₁₆

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₂²×C₁₅₇

C₄⋊Q₈

yes

—

C₂²×C₁₅₇

C₂×D₈

yes

—

C₂²×C₁₅₇

C₂×Q₁₆

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄⋊Q₈

yes

—

C₄×C₁₅₇

C₂×D₈

yes

—

C₄×C₁₅₇

C₂×Q₁₆

yes

—

C₄×C₁₅₇

C₂²×Q₈

yes

—

Dic₁₅₇

—

C₂×C₁₅₇

#500

yes

complemented

C₂×C₁₅₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₇

C₂²×D₈

yes

characteristic

C₂×C₁₅₇

C₂²×Q₁₆

yes

characteristic

C₁₅₇

C₂×#500

yes

characteristic, complemented, Sylow

C₂×#500

—

157

maximal, Sylow

C₂×C₄⋊Q₈

—

157

—

C₂×C₄⋊Q₈

D₁₅₇

yes

characteristic

#500

—

157

—

C₂×#107

—

157

—

C₂×C₄×C₈

—

157

—

C₂²×Q₈

—

C₂²×Q₈

—

314

—

C₂²×C₈

—

157

—

C₄⋊Q₈

—

C₄⋊Q₈

—

314

—

C₄⋊Q₈

—

157

—

C₄⋊Q₈

C₂×D₁₅₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

C₂×D₁₅₇

yes

characteristic

#107

—

157

—

C₄×C₈

—

157

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

314

—

C₂×Q₈

—

628

—

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

—

157

—

C₂×C₈

—

157

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

314

—

103

C₄⋊C₄

—

314

—

104

C₄⋊C₄

—

157

—

105

C₄⋊C₄

C₂²×D₁₅₇

yes

—

106

C₄²

C₂²⋊D₁₅₇

yes

—

107

C₄²

C₂²×D₁₅₇

yes

—

108

C₂³

C₂×C₂²⋊D₁₅₇

yes

characteristic

109

Q₈

—

110

Q₈

—

628

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

314

—

114

C₂×C₄

—

628

—

115

C₂×C₄

#14578939

yes

—

116

C₂×C₄

#14578944

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₅₇

yes

—

120

C₂×C₄

Q₈×D₁₅₇

yes

—

121

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

122

C₈

—

157

—

123

C₂²

#19097769

yes

—

124

C₂²

C₂×#14578939

yes

—

125

C₂²

C₂×#14578944

yes

—

126

C₂²

C₂²×C₂²⋊D₁₅₇

yes

characteristic

127

C₄

—

128

C₄

—

628

—

129

C₄

#19097769

yes

—

130

C₄

C₂×#14578939

yes

—

131

C₄

C₂×#14578944

yes

—

132

C₄

C₂×Q₈×D₁₅₇

yes

—

133

C₂

#23757676

yes

direct product factor

134

C₂

C₂×#19097769

yes

characteristic

135

C₂

C₂²×#14578939

yes

characteristic

136

C₂

C₂²×#14578944

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×C₄⋊Q₈×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₅₇
hypocenter	C₁₅₇	C₂×#500
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

C₂×C₄⋊Q₈×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₅₇

hypocenter

C₁₅₇

C₂×#500

LCS

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #23757676	C₂	direct (N × H)
23. C₂×#11929434	Q₈	semidirect
42. #11929434	C₂×Q₈	semidirect
64. C₂×C₁₅₇	#500	semidirect
68. C₁₅₇	C₂×#500	semidirect
133. C₂	#23757676	direct (N × H)

H ≅ G/N

type

2. #23757676

C₂

direct (N × H)

23. C₂×#11929434

Q₈

semidirect

42. #11929434

C₂×Q₈

semidirect

64. C₂×C₁₅₇

#500

semidirect

68. C₁₅₇

C₂×#500

semidirect

133. C₂

#23757676

direct (N × H)

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

subgroup	index
C₂×#19097771	2
#23757676	2
C₂×#19097704	2
C₂×C₄×#11929434	2
C₂×C₄⋊Q₈×C₁₅₇	2
C₂×#500	157

subgroup

index

C₂×C₄⋊Q₈×C₁₅₇

157

Subgroups — C₂×#23757676 #34110642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23757676 #34110642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23757676 #34110642