Group #34108922 — subgroups

Subgroups — C₂×#23757379 #34108922

Order 20096.

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

normal subgroups only

This group has 480 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	C₂²×Dic₆₂₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#19097736	C₂	1	2	yes	maximal
3	#23757379	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19097694	C₂	1	2	yes	maximal
5	C₂×#19097681	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₃₁₄	C₂²	1	2	yes	—
8	C₂×Dic₆₂₈	C₄	1	8	yes	complemented
9	C₂×Dic₆₂₈	C₂²	1	4	yes	—
10	C₂²×#11929434	C₂²	1	1	yes	characteristic
11	#19097736	—	2	4	no	—
12	#19097736	C₂²	1	8	yes	—
13	C₂×#14578933	—	2	2	no	—
14	C₂×C₄×Dic₁₅₇	—	2	2	no	—
15	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
16	#19097694	C₂²	1	8	yes	—
17	#19097681	C₂²	1	4	yes	—
18	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
20	#106×C₁₅₇	C₂²	1	4	yes	—
21	C₂×Dic₃₁₄	—	2	2	no	—
22	C₂×Dic₃₁₄	—	2	4	no	—
23	C₂×Dic₃₁₄	C₂×C₄	1	8	yes	—
24	C₂×Dic₃₁₄	C₂³	1	4	yes	—
25	C₂²×Dic₁₅₇	—	2	2	no	—
26	C₂²×Dic₁₅₇	D₄	1	2	yes	—
27	Dic₆₂₈	C₂×C₄	1	16	yes	complemented
28	C₂×#11929434	—	2	2	no	—
29	C₂×#11929434	C₂³	1	2	yes	—
30	#14578933	—	2	8	no	—
31	C₄×Dic₁₅₇	—	2	8	no	—
32	C₄×Dic₁₅₇	D₄	1	2	yes	—
33	C₄×Dic₁₅₇	C₂³	1	2	yes	—
34	C₂²×C₄×C₁₅₇	—	2	2	no	—
35	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₁₅₇	C₂×C₄	1	4	yes	—
37	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
38	C₄⋊C₄×C₁₅₇	—	2	2	no	—
39	C₄⋊C₄×C₁₅₇	C₂³	1	4	yes	—
40	Dic₃₁₄	—	2	8	no	—
41	Dic₃₁₄	—	2	4	no	—
42	Dic₃₁₄	C₂²×C₄	1	8	yes	—
43	C₂×Dic₁₅₇	—	2	8	no	—
44	C₂×Dic₁₅₇	—	2	4	no	—
45	C₂×Dic₁₅₇	—	2	4	no	—
46	C₂×Dic₁₅₇	C₂×D₄	1	4	yes	—
47	#11929434	—	2	2	no	—
48	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₁₅₇	—	2	4	no	—
50	C₂×C₄×C₁₅₇	—	4	4	no	—
51	C₂×C₄×C₁₅₇	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₅₇	#41	1	2	yes	—
54	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
55	C₈×C₁₅₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₁₅₇	C₄×D₄	1	2	yes	—
57	C₂²×C₁₅₇	#137	1	4	yes	—
58	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₁₅₇	—	4	4	no	—
60	C₄×C₁₅₇	C₄×D₄	1	2	yes	—
61	C₄×C₁₅₇	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₁₅₇	C₂×#41	1	1	yes	characteristic
63	Dic₁₅₇	—	2	8	no	—
64	Dic₁₅₇	—	2	4	no	—
65	C₂×C₁₅₇	#441	1	4	yes	complemented
66	C₂×C₁₅₇	C₂×C₄×D₄	1	1	yes	characteristic
67	C₂×C₁₅₇	C₂×#137	1	2	yes	—
68	C₁₅₇	C₂×#441	1	1	yes	characteristic, complemented, Sylow
69	C₂×#441	—	157	1	no	maximal, Sylow
70	C₂²×Q₁₆	—	157	1	no	—
71	C₂×C₄×Q₈	—	157	2	no	—
72	#441	—	157	8	no	—
73	C₂×#106	D₁₅₇	1	1	yes	characteristic
74	C₂×#103	—	157	2	no	—
75	C₂×#97	—	157	1	no	—
76	C₂²×Q₈	—	157	2	no	—
77	C₂×Q₁₆	—	157	8	no	—
78	C₂×Q₁₆	—	157	4	no	—
79	C₂²×C₈	—	157	1	no	—
80	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
81	C₄×Q₈	—	314	4	no	—
82	C₄×Q₈	—	157	8	no	—
83	C₂×C₄⋊C₄	—	314	2	no	—
84	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
85	C₂×C₄²	—	314	2	no	—
86	C₂×C₄²	—	157	1	no	—
87	#106	C₂×D₁₅₇	1	4	yes	—
88	#103	—	157	8	no	—
89	#97	—	157	4	no	—
90	C₂×Q₈	—	314	2	no	—
91	C₂×Q₈	—	314	4	no	—
92	C₂×Q₈	—	157	8	no	—
93	C₂×Q₈	—	157	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	314	2	no	—
96	C₂²×C₄	—	157	2	no	—
97	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
98	Q₁₆	—	157	16	no	—
99	C₂×C₈	—	314	2	no	—
100	C₂×C₈	—	157	2	no	—
101	C₂×C₈	C₄×D₁₅₇	1	4	yes	—
102	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	314	8	no	—
105	C₄⋊C₄	C₂²×D₁₅₇	1	4	yes	—
106	C₄²	—	314	8	no	—
107	C₄²	—	157	2	no	—
108	C₄²	—	157	2	no	—
109	C₂³	D₄×D₁₅₇	1	1	yes	characteristic
110	Q₈	—	314	8	no	—
111	Q₈	—	314	4	no	—
112	Q₈	—	157	8	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	314	8	no	—
116	C₂×C₄	—	314	4	no	—
117	C₂×C₄	—	314	4	no	—
118	C₂×C₄	—	157	4	no	—
119	C₂×C₄	C₂×C₄×D₁₅₇	1	2	yes	—
120	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
121	C₂×C₄	#14578955	1	2	yes	—
122	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
123	C₈	—	314	2	no	—
124	C₈	C₂×C₄×D₁₅₇	1	4	yes	—
125	C₂²	#19097733	1	2	yes	—
126	C₂²	#19097800	1	4	yes	—
127	C₂²	C₂×D₄×D₁₅₇	1	1	yes	characteristic
128	C₄	—	4	4	no	—
129	C₄	—	314	8	no	—
130	C₄	—	314	4	no	—
131	C₄	#19097733	1	2	yes	—
132	C₄	C₂²×C₄×D₁₅₇	1	1	yes	characteristic
133	C₄	C₂×#14578955	1	1	yes	characteristic
134	C₂	#23757379	1	4	yes	direct product factor
135	C₂	C₂×#19097733	1	1	yes	characteristic
136	C₂	C₂×#19097800	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₆₂₈

C₂

yes

maximal, characteristic

C₂×#19097736

C₂

yes

maximal

#23757379

C₂

yes

maximal, direct product factor

C₂×#19097694

C₂

yes

maximal

C₂×#19097681

C₂

yes

maximal, characteristic

C₂×#106×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₁₄

C₂²

yes

—

C₂×Dic₆₂₈

C₄

yes

complemented

C₂×Dic₆₂₈

C₂²

yes

—

C₂²×#11929434

C₂²

yes

characteristic

#19097736

—

#19097736

C₂²

yes

—

C₂×#14578933

—

C₂×C₄×Dic₁₅₇

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097694

C₂²

yes

—

#19097681

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

#106×C₁₅₇

C₂²

yes

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

—

C₂×Dic₃₁₄

C₂×C₄

yes

—

C₂×Dic₃₁₄

C₂³

yes

—

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

D₄

yes

—

Dic₆₂₈

C₂×C₄

yes

complemented

C₂×#11929434

—

C₂×#11929434

C₂³

yes

—

#14578933

—

C₄×Dic₁₅₇

—

C₄×Dic₁₅₇

D₄

yes

—

C₄×Dic₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

C₂×C₄

yes

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

C₂³

yes

—

Dic₃₁₄

—

Dic₃₁₄

—

Dic₃₁₄

C₂²×C₄

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×D₄

yes

—

#11929434

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂²×C₄

yes

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

#41

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

C₂²×C₄

yes

—

C₂²×C₁₅₇

C₄×D₄

yes

—

C₂²×C₁₅₇

#137

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄×D₄

yes

—

C₄×C₁₅₇

C₂³×C₄

yes

characteristic

C₄×C₁₅₇

C₂×#41

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#441

yes

complemented

C₂×C₁₅₇

C₂×C₄×D₄

yes

characteristic

C₂×C₁₅₇

C₂×#137

yes

—

C₁₅₇

C₂×#441

yes

characteristic, complemented, Sylow

C₂×#441

—

157

maximal, Sylow

C₂²×Q₁₆

—

157

—

C₂×C₄×Q₈

—

157

—

#441

—

157

—

C₂×#106

D₁₅₇

yes

characteristic

C₂×#103

—

157

—

C₂×#97

—

157

—

C₂²×Q₈

—

157

—

C₂×Q₁₆

—

157

—

C₂×Q₁₆

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

C₄×Q₈

—

314

—

C₄×Q₈

—

157

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

—

314

—

C₂×C₄²

—

157

—

#106

C₂×D₁₅₇

yes

—

#103

—

157

—

#97

—

157

—

C₂×Q₈

—

314

—

C₂×Q₈

—

314

—

C₂×Q₈

—

157

—

C₂×Q₈

—

157

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

Q₁₆

—

157

—

C₂×C₈

—

314

—

100

C₂×C₈

—

157

—

101

C₂×C₈

C₄×D₁₅₇

yes

—

102

C₂×C₈

C₂²×D₁₅₇

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

314

—

105

C₄⋊C₄

C₂²×D₁₅₇

yes

—

106

C₄²

—

314

—

107

C₄²

—

157

—

108

C₄²

—

157

—

109

C₂³

D₄×D₁₅₇

yes

characteristic

110

Q₈

—

314

—

111

Q₈

—

314

—

112

Q₈

—

157

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

314

—

116

C₂×C₄

—

314

—

117

C₂×C₄

—

314

—

118

C₂×C₄

—

157

—

119

C₂×C₄

C₂×C₄×D₁₅₇

yes

—

120

C₂×C₄

D₄×D₁₅₇

yes

characteristic

121

C₂×C₄

#14578955

yes

—

122

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

123

C₈

—

314

—

124

C₈

C₂×C₄×D₁₅₇

yes

—

125

C₂²

#19097733

yes

—

126

C₂²

#19097800

yes

—

127

C₂²

C₂×D₄×D₁₅₇

yes

characteristic

128

C₄

—

129

C₄

—

314

—

130

C₄

—

314

—

131

C₄

#19097733

yes

—

132

C₄

C₂²×C₄×D₁₅₇

yes

characteristic

133

C₄

C₂×#14578955

yes

characteristic

134

C₂

#23757379

yes

direct product factor

135

C₂

C₂×#19097733

yes

characteristic

136

C₂

C₂×#19097800

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₅₇
commutator	C₄×C₁₅₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×#106×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
cosocle	C₂×C₄×C₁₅₇	C₂⁴
hypercenter	C₂×#106	D₁₅₇
hypocenter	C₁₅₇	C₂×#441
LCS	G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₅₇

commutator

C₄×C₁₅₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

C₂×#106×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

cosocle

C₂×C₄×C₁₅₇

C₂⁴

D₁₅₇

C₁₅₇

G ▷ C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #23757379	C₂	direct (N × H)
8. C₂×Dic₆₂₈	C₄	semidirect
27. Dic₆₂₈	C₂×C₄	semidirect
65. C₂×C₁₅₇	#441	semidirect
68. C₁₅₇	C₂×#441	semidirect
134. C₂	#23757379	direct (N × H)

H ≅ G/N

type

3. #23757379

C₂

direct (N × H)

8. C₂×Dic₆₂₈

C₄

semidirect

27. Dic₆₂₈

C₂×C₄

semidirect

65. C₂×C₁₅₇

#441

semidirect

68. C₁₅₇

C₂×#441

semidirect

134. C₂

#23757379

direct (N × H)

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

subgroup	index
C₂²×Dic₆₂₈	2
C₂×#19097736	2
#23757379	2
C₂×#19097694	2
C₂×#19097681	2
C₂×#106×C₁₅₇	2
C₂×#441	157

subgroup

index

C₂²×Dic₆₂₈

C₂×#106×C₁₅₇

157

Subgroups — C₂×#23757379 #34108922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23757379 #34108922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23757379 #34108922