Group #34079612 — subgroups

Subgroups — C₂×#23752612 #34079612

Order 19328.

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×#18012730	C₂	1	2	yes	maximal
2	#23752612	C₂	1	8	yes	maximal, direct product factor
3	C₂×#18012672	C₂	1	2	yes	maximal
4	C₂×#18012671	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#106×C₁₅₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
8	#18012730	—	2	4	no	—
9	#18012730	C₂²	1	8	yes	—
10	C₂×C₄⋊Dic₁₅₁	—	2	2	no	—
11	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
12	C₂×Dic₁₅₁⋊C₄	—	2	2	no	—
13	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
14	#18012672	C₂²	1	8	yes	—
15	#18012671	C₂²	1	4	yes	—
16	C₈×Dic₁₅₁	C₂²	1	4	yes	—
17	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
19	#106×C₁₅₁	C₂²	1	4	yes	—
20	C₂²×Dic₁₅₁	—	2	2	no	—
21	C₂²×Dic₁₅₁	D₄	1	2	yes	—
22	C₂×C₁₅₁⋊C₈	Q₈	1	4	yes	—
23	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
24	C₄⋊Dic₁₅₁	—	2	8	no	—
25	C₄⋊Dic₁₅₁	—	2	2	no	—
26	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
27	Dic₁₅₁⋊C₄	—	2	8	no	—
28	C₄×Dic₁₅₁	D₄	1	2	yes	—
29	C₄×Dic₁₅₁	C₂³	1	2	yes	—
30	C₂²×C₄×C₁₅₁	—	2	2	no	—
31	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
32	C₂×C₈×C₁₅₁	Q₈	1	4	yes	—
33	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
34	C₄⋊C₄×C₁₅₁	—	2	2	no	—
35	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
36	C₂×Dic₁₅₁	—	2	4	no	—
37	C₂×Dic₁₅₁	—	2	4	no	—
38	C₂×Dic₁₅₁	—	4	4	no	—
39	C₂×Dic₁₅₁	C₂×D₄	1	4	yes	—
40	C₁₅₁⋊C₈	C₂×Q₈	1	4	yes	—
41	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	—	2	4	no	—
43	C₂×C₄×C₁₅₁	—	4	4	no	—
44	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
46	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
48	C₈×C₁₅₁	C₂×Q₈	1	4	yes	—
49	C₂²×C₁₅₁	C₄⋊Q₈	1	2	yes	—
50	C₂²×C₁₅₁	#135	1	4	yes	—
51	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
52	C₄×C₁₅₁	—	4	4	no	—
53	C₄×C₁₅₁	C₄⋊Q₈	1	2	yes	—
54	C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
55	C₄×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
56	Dic₁₅₁	—	2	4	no	—
57	Dic₁₅₁	—	4	4	no	—
58	C₂×C₁₅₁	#499	1	4	yes	complemented
59	C₂×C₁₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
60	C₂×C₁₅₁	C₂×#135	1	2	yes	—
61	C₁₅₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
62	C₂×#499	—	151	1	no	maximal, Sylow
63	C₂×#125	—	151	2	no	—
64	#499	—	151	8	no	—
65	C₂×#107	—	151	2	no	—
66	C₂×#106	—	151	1	no	—
67	C₂×#106	D₁₅₁	1	1	yes	characteristic
68	C₂×C₄×C₈	—	151	1	no	—
69	C₂²×C₈	—	151	1	no	—
70	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
71	#125	—	302	4	no	—
72	#125	—	151	8	no	—
73	C₂×C₄⋊C₄	—	302	2	no	—
74	C₂×C₄⋊C₄	—	302	2	no	—
75	C₂×C₄⋊C₄	—	151	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
77	C₂×C₄²	—	151	1	no	—
78	#107	—	151	8	no	—
79	#106	—	151	4	no	—
80	#106	C₂×D₁₅₁	1	4	yes	—
81	C₄×C₈	—	151	4	no	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	302	2	no	—
84	C₂²×C₄	—	151	2	no	—
85	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
86	C₂×C₈	—	151	4	no	—
87	C₂×C₈	—	151	2	no	—
88	C₂×C₈	Dic₃₀₂	1	4	yes	—
89	C₂×C₈	C₂²×D₁₅₁	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	302	8	no	—
92	C₄⋊C₄	—	302	8	no	—
93	C₄⋊C₄	—	302	2	no	—
94	C₄⋊C₄	—	151	4	no	—
95	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
96	C₄²	—	151	2	no	—
97	C₄²	—	151	2	no	—
98	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	—	302	4	no	—
102	C₂×C₄	—	302	4	no	—
103	C₂×C₄	—	604	4	no	—
104	C₂×C₄	—	151	4	no	—
105	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
107	C₂×C₄	C₂×Dic₃₀₂	1	2	yes	—
108	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
109	C₈	—	151	4	no	—
110	C₈	C₂×Dic₃₀₂	1	4	yes	—
111	C₂²	#18012738	1	2	yes	—
112	C₂²	#18012760	1	4	yes	—
113	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
114	C₄	—	4	4	no	—
115	C₄	—	302	4	no	—
116	C₄	—	604	4	no	—
117	C₄	#18012738	1	2	yes	—
118	C₄	C₂×Q₈×D₁₅₁	1	1	yes	characteristic
119	C₄	C₂²×Dic₃₀₂	1	1	yes	characteristic
120	C₂	#23752612	1	4	yes	direct product factor
121	C₂	C₂×#18012738	1	1	yes	characteristic
122	C₂	C₂×#18012760	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012730

C₂

yes

maximal

#23752612

C₂

yes

maximal, direct product factor

C₂×#18012672

C₂

yes

maximal

C₂×#18012671

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₅₁

C₂

yes

maximal, characteristic

C₂×#106×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

#18012730

—

#18012730

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×Dic₁₅₁⋊C₄

—

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

#18012672

C₂²

yes

—

#18012671

C₂²

yes

—

C₈×Dic₁₅₁

C₂²

yes

—

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

#106×C₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

—

C₂²×Dic₁₅₁

D₄

yes

—

C₂×C₁₅₁⋊C₈

Q₈

yes

—

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂³

yes

—

Dic₁₅₁⋊C₄

—

C₄×Dic₁₅₁

D₄

yes

—

C₄×Dic₁₅₁

C₂³

yes

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

Q₈

yes

—

C₂×C₈×C₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×D₄

yes

—

C₁₅₁⋊C₈

C₂×Q₈

yes

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₈×C₁₅₁

C₂×Q₈

yes

—

C₂²×C₁₅₁

C₄⋊Q₈

yes

—

C₂²×C₁₅₁

#135

yes

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

C₄⋊Q₈

yes

—

C₄×C₁₅₁

C₂²×Q₈

yes

characteristic

C₄×C₁₅₁

C₂²×Q₈

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#499

yes

complemented

C₂×C₁₅₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₁

C₂×#135

yes

—

C₁₅₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

151

maximal, Sylow

C₂×#125

—

151

—

#499

—

151

—

C₂×#107

—

151

—

C₂×#106

—

151

—

C₂×#106

D₁₅₁

yes

characteristic

C₂×C₄×C₈

—

151

—

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

#125

—

302

—

#125

—

151

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

#107

—

151

—

#106

—

151

—

#106

C₂×D₁₅₁

yes

—

C₄×C₈

—

151

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

Dic₃₀₂

yes

—

C₂×C₈

C₂²×D₁₅₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

151

—

C₄⋊C₄

C₂²×D₁₅₁

yes

—

C₄²

—

151

—

C₄²

—

151

—

C₂³

D₄×D₁₅₁

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

302

—

102

C₂×C₄

—

302

—

103

C₂×C₄

—

604

—

104

C₂×C₄

—

151

—

105

C₂×C₄

D₄×D₁₅₁

yes

characteristic

106

C₂×C₄

Q₈×D₁₅₁

yes

—

107

C₂×C₄

C₂×Dic₃₀₂

yes

—

108

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

109

C₈

—

151

—

110

C₈

C₂×Dic₃₀₂

yes

—

111

C₂²

#18012738

yes

—

112

C₂²

#18012760

yes

—

113

C₂²

C₂×D₄×D₁₅₁

yes

characteristic

114

C₄

—

115

C₄

—

302

—

116

C₄

—

604

—

117

C₄

#18012738

yes

—

118

C₄

C₂×Q₈×D₁₅₁

yes

characteristic

119

C₄

C₂²×Dic₃₀₂

yes

characteristic

120

C₂

#23752612

yes

direct product factor

121

C₂

C₂×#18012738

yes

characteristic

122

C₂

C₂×#18012760

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	C₂×#106×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
hypercenter	C₂×#106	D₁₅₁
hypocenter	C₁₅₁	C₂×#499
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

C₂×#106×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
2. #23752612	C₂	direct (N × H)
58. C₂×C₁₅₁	#499	semidirect
61. C₁₅₁	C₂×#499	semidirect
120. C₂	#23752612	direct (N × H)

H ≅ G/N

type

2. #23752612

C₂

direct (N × H)

58. C₂×C₁₅₁

#499

semidirect

61. C₁₅₁

C₂×#499

semidirect

120. C₂

#23752612

direct (N × H)

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

subgroup	index
C₂×#18012730	2
#23752612	2
C₂×#18012672	2
C₂×#18012671	2
C₂×C₈×Dic₁₅₁	2
C₂×#106×C₁₅₁	2
C₂×#499	151

subgroup

index

C₂×C₈×Dic₁₅₁

C₂×#106×C₁₅₁

C₂×#499

151

Subgroups — C₂×#23752612 #34079612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#23752612 #34079612

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23752612 #34079612