Group #34110638 — subgroups

Subgroups — C₂×#23757673 #34110638

Order 20096.

360 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 165 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×#19097762	C₂	1	2	yes	maximal
2	#23757673	C₂	1	8	yes	maximal, direct product factor
3	C₂×#19097704	C₂	1	2	yes	maximal
4	C₂×#19097703	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₁₅₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₁₅₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11929434	C₂²	1	1	yes	characteristic
8	#19097762	—	2	4	no	—
9	#19097762	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₁₅₇	—	2	2	no	—
11	C₂×C₄⋊Dic₁₅₇	C₂²	1	2	yes	—
12	C₂×#14578933	—	2	2	no	—
13	C₂×C₄×Dic₁₅₇	C₂²	1	1	yes	characteristic
14	#19097704	C₂²	1	8	yes	—
15	#19097703	C₂²	1	4	yes	—
16	C₈×Dic₁₅₇	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₅₇	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₅₇	C₂²	1	2	yes	—
19	#106×C₁₅₇	C₂²	1	4	yes	—
20	C₂²×Dic₁₅₇	—	2	2	no	—
21	C₂²×Dic₁₅₇	D₄	1	2	yes	—
22	C₂×#11929434	Q₈	1	4	yes	—
23	C₂×#11929434	C₂³	1	2	yes	—
24	C₄⋊Dic₁₅₇	—	2	8	no	—
25	C₄⋊Dic₁₅₇	—	2	2	no	—
26	C₄⋊Dic₁₅₇	C₂³	1	4	yes	—
27	#14578933	—	2	8	no	—
28	C₄×Dic₁₅₇	D₄	1	2	yes	—
29	C₄×Dic₁₅₇	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₅₇	—	2	2	no	—
31	C₂²×C₄×C₁₅₇	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₅₇	Q₈	1	4	yes	—
33	C₂×C₈×C₁₅₇	C₂³	1	2	yes	—
34	C₄⋊C₄×C₁₅₇	—	2	2	no	—
35	C₄⋊C₄×C₁₅₇	C₂³	1	4	yes	—
36	C₂×Dic₁₅₇	—	2	4	no	—
37	C₂×Dic₁₅₇	—	2	4	no	—
38	C₂×Dic₁₅₇	—	4	4	no	—
39	C₂×Dic₁₅₇	C₂×D₄	1	4	yes	—
40	#11929434	C₂×Q₈	1	4	yes	—
41	C₂³×C₁₅₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₇	—	2	4	no	—
43	C₂×C₄×C₁₅₇	—	4	4	no	—
44	C₂×C₄×C₁₅₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₅₇	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₅₇	C₂⁴	1	1	yes	characteristic
48	C₈×C₁₅₇	C₂×Q₈	1	4	yes	—
49	C₂²×C₁₅₇	C₄⋊Q₈	1	2	yes	—
50	C₂²×C₁₅₇	#135	1	4	yes	—
51	C₂²×C₁₅₇	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₁₅₇	—	4	4	no	—
53	C₄×C₁₅₇	C₄⋊Q₈	1	2	yes	—
54	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
55	C₄×C₁₅₇	C₂²×Q₈	1	1	yes	characteristic
56	Dic₁₅₇	—	2	4	no	—
57	Dic₁₅₇	—	4	4	no	—
58	C₂×C₁₅₇	#499	1	4	yes	complemented
59	C₂×C₁₅₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₁₅₇	C₂×#135	1	2	yes	—
61	C₁₅₇	C₂×#499	1	1	yes	characteristic, complemented, Sylow
62	C₂×#499	—	157	1	no	maximal, Sylow
63	C₂×#125	—	157	2	no	—
64	#499	—	157	8	no	—
65	C₂×#107	—	157	2	no	—
66	C₂×#106	—	157	1	no	—
67	C₂×#106	D₁₅₇	1	1	yes	characteristic
68	C₂×C₄×C₈	—	157	1	no	—
69	C₂²×C₈	—	157	1	no	—
70	C₂²×C₈	C₂×D₁₅₇	1	1	yes	characteristic
71	#125	—	314	4	no	—
72	#125	—	157	8	no	—
73	C₂×C₄⋊C₄	—	314	2	no	—
74	C₂×C₄⋊C₄	—	314	2	no	—
75	C₂×C₄⋊C₄	—	157	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₅₇	1	2	yes	—
77	C₂×C₄²	—	157	1	no	—
78	#107	—	157	8	no	—
79	#106	—	157	4	no	—
80	#106	C₂×D₁₅₇	1	4	yes	—
81	C₄×C₈	—	157	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	314	2	no	—
84	C₂²×C₄	—	157	2	no	—
85	C₂²×C₄	C₂²×D₁₅₇	1	1	yes	characteristic
86	C₂×C₈	—	157	4	no	—
87	C₂×C₈	—	157	2	no	—
88	C₂×C₈	Dic₃₁₄	1	4	yes	—
89	C₂×C₈	C₂²×D₁₅₇	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	314	8	no	—
92	C₄⋊C₄	—	314	8	no	—
93	C₄⋊C₄	—	314	2	no	—
94	C₄⋊C₄	—	157	4	no	—
95	C₄⋊C₄	C₂²×D₁₅₇	1	4	yes	—
96	C₄²	—	157	2	no	—
97	C₄²	—	157	2	no	—
98	C₂³	D₄×D₁₅₇	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	314	4	no	—
102	C₂×C₄	—	314	4	no	—
103	C₂×C₄	—	628	4	no	—
104	C₂×C₄	—	157	4	no	—
105	C₂×C₄	D₄×D₁₅₇	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₅₇	1	2	yes	—
107	C₂×C₄	C₂×Dic₃₁₄	1	2	yes	—
108	C₂×C₄	C₂³×D₁₅₇	1	1	yes	characteristic
109	C₈	—	157	4	no	—
110	C₈	C₂×Dic₃₁₄	1	4	yes	—
111	C₂²	#19097770	1	2	yes	—
112	C₂²	#19097792	1	4	yes	—
113	C₂²	C₂×D₄×D₁₅₇	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	314	4	no	—
116	C₄	—	628	4	no	—
117	C₄	#19097770	1	2	yes	—
118	C₄	C₂×Q₈×D₁₅₇	1	1	yes	characteristic
119	C₄	C₂²×Dic₃₁₄	1	1	yes	characteristic
120	C₂	#23757673	1	4	yes	direct product factor
121	C₂	C₂×#19097770	1	1	yes	characteristic
122	C₂	C₂×#19097792	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#19097762

C₂

yes

maximal

#23757673

C₂

yes

maximal, direct product factor

C₂×#19097704

C₂

yes

maximal

C₂×#19097703

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₅₇

C₂

yes

maximal, characteristic

C₂×#106×C₁₅₇

C₂

yes

maximal, characteristic

C₂²×#11929434

C₂²

yes

characteristic

#19097762

—

#19097762

C₂²

yes

—

C₂×C₄⋊Dic₁₅₇

—

C₂×C₄⋊Dic₁₅₇

C₂²

yes

—

C₂×#14578933

—

C₂×C₄×Dic₁₅₇

C₂²

yes

characteristic

#19097704

C₂²

yes

—

#19097703

C₂²

yes

—

C₈×Dic₁₅₇

C₂²

yes

—

C₂²×C₈×C₁₅₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₇

C₂²

yes

—

#106×C₁₅₇

C₂²

yes

—

C₂²×Dic₁₅₇

—

C₂²×Dic₁₅₇

D₄

yes

—

C₂×#11929434

Q₈

yes

—

C₂×#11929434

C₂³

yes

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

—

C₄⋊Dic₁₅₇

C₂³

yes

—

#14578933

—

C₄×Dic₁₅₇

D₄

yes

—

C₄×Dic₁₅₇

C₂³

yes

—

C₂²×C₄×C₁₅₇

—

C₂²×C₄×C₁₅₇

C₂³

yes

characteristic

C₂×C₈×C₁₅₇

Q₈

yes

—

C₂×C₈×C₁₅₇

C₂³

yes

—

C₄⋊C₄×C₁₅₇

—

C₄⋊C₄×C₁₅₇

C₂³

yes

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

—

C₂×Dic₁₅₇

C₂×D₄

yes

—

#11929434

C₂×Q₈

yes

—

C₂³×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

—

C₂×C₄×C₁₅₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂×Q₈

yes

—

C₂×C₄×C₁₅₇

C₂⁴

yes

characteristic

C₈×C₁₅₇

C₂×Q₈

yes

—

C₂²×C₁₅₇

C₄⋊Q₈

yes

—

C₂²×C₁₅₇

#135

yes

—

C₂²×C₁₅₇

C₂²×D₄

yes

characteristic

C₄×C₁₅₇

—

C₄×C₁₅₇

C₄⋊Q₈

yes

—

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

C₄×C₁₅₇

C₂²×Q₈

yes

characteristic

Dic₁₅₇

—

Dic₁₅₇

—

C₂×C₁₅₇

#499

yes

complemented

C₂×C₁₅₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₇

C₂×#135

yes

—

C₁₅₇

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

157

maximal, Sylow

C₂×#125

—

157

—

#499

—

157

—

C₂×#107

—

157

—

C₂×#106

—

157

—

C₂×#106

D₁₅₇

yes

characteristic

C₂×C₄×C₈

—

157

—

C₂²×C₈

—

157

—

C₂²×C₈

C₂×D₁₅₇

yes

characteristic

#125

—

314

—

#125

—

157

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

314

—

C₂×C₄⋊C₄

—

157

—

C₂×C₄⋊C₄

C₂×D₁₅₇

yes

—

C₂×C₄²

—

157

—

#107

—

157

—

#106

—

157

—

#106

C₂×D₁₅₇

yes

—

C₄×C₈

—

157

—

C₂²×C₄

—

C₂²×C₄

—

314

—

C₂²×C₄

—

157

—

C₂²×C₄

C₂²×D₁₅₇

yes

characteristic

C₂×C₈

—

157

—

C₂×C₈

—

157

—

C₂×C₈

Dic₃₁₄

yes

—

C₂×C₈

C₂²×D₁₅₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

314

—

C₄⋊C₄

—

157

—

C₄⋊C₄

C₂²×D₁₅₇

yes

—

C₄²

—

157

—

C₄²

—

157

—

C₂³

D₄×D₁₅₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

314

—

102

C₂×C₄

—

314

—

103

C₂×C₄

—

628

—

104

C₂×C₄

—

157

—

105

C₂×C₄

D₄×D₁₅₇

yes

characteristic

106

C₂×C₄

Q₈×D₁₅₇

yes

—

107

C₂×C₄

C₂×Dic₃₁₄

yes

—

108

C₂×C₄

C₂³×D₁₅₇

yes

characteristic

109

C₈

—

157

—

110

C₈

C₂×Dic₃₁₄

yes

—

111

C₂²

#19097770

yes

—

112

C₂²

#19097792

yes

—

113

C₂²

C₂×D₄×D₁₅₇

yes

characteristic

114

C₄

—

115

C₄

—

314

—

116

C₄

—

628

—

117

C₄

#19097770

yes

—

118

C₄

C₂×Q₈×D₁₅₇

yes

characteristic

119

C₄

C₂²×Dic₃₁₄

yes

characteristic

120

C₂

#23757673

yes

direct product factor

121

C₂

C₂×#19097770

yes

characteristic

122

C₂

C₂×#19097792

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₅₇
commutator, cosocle	C₂×C₄×C₁₅₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₇
Fitting	C₂×#106×C₁₅₇	C₂
socle, prime radical	C₂³×C₁₅₇	C₂×D₄
hypercenter	C₂×#106	D₁₅₇
hypocenter	C₁₅₇	C₂×#499
LCS	G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₅₇

commutator, cosocle

C₂×C₄×C₁₅₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₇

Fitting

C₂×#106×C₁₅₇

C₂

socle, prime radical

C₂³×C₁₅₇

C₂×D₄

D₁₅₇

C₁₅₇

G ▷ C₂×C₄×C₁₅₇ ▷ C₂×C₁₅₇ ▷ C₁₅₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #23757673	C₂	direct (N × H)
58. C₂×C₁₅₇	#499	semidirect
61. C₁₅₇	C₂×#499	semidirect
120. C₂	#23757673	direct (N × H)

H ≅ G/N

type

2. #23757673

C₂

direct (N × H)

58. C₂×C₁₅₇

#499

semidirect

61. C₁₅₇

C₂×#499

semidirect

120. C₂

#23757673

direct (N × H)

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

subgroup	index
C₂×#19097762	2
#23757673	2
C₂×#19097704	2
C₂×#19097703	2
C₂×C₈×Dic₁₅₇	2
C₂×#106×C₁₅₇	2
C₂×#499	157

subgroup

index

C₂×C₈×Dic₁₅₇

C₂×#106×C₁₅₇

C₂×#499

157

Subgroups — C₂×#23757673 #34110638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23757673 #34110638

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23757673 #34110638