Group #34079619 — subgroups

Subgroups — #34079619

Order 19328.

356 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 145 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	#23752767	C₂	1	2	yes	maximal
2	#23752615	C₂	1	2	yes	maximal, complemented
3	#23752612	C₂	1	4	yes	maximal, complemented
4	#23752610	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752247	C₂	1	4	yes	maximal
6	C₄×C₈×D₁₅₁	C₂	1	1	yes	maximal, characteristic
7	#498×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₁₅₁	C₂²	1	2	yes	—
9	#18012739	C₂²	1	2	yes	—
10	#18012730	C₂²	1	4	yes	—
11	#18012719	—	2	2	no	—
12	#18012715	—	2	2	no	—
13	Q₈×Dic₁₅₁	—	2	2	no	—
14	#18012704	—	2	2	no	—
15	#18012697	C₂²	1	4	yes	—
16	C₄²×D₁₅₁	C₂²	1	1	yes	characteristic
17	#18012672	C₂²	1	8	yes	—
18	#18012671	C₂²	1	4	yes	—
19	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₁₅₁	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₁₅₁	C₂²	1	2	yes	—
22	#106×C₁₅₁	C₂²	1	4	yes	—
23	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
24	C₂×Dic₃₀₂	—	2	2	no	—
25	C₂×C₄×D₁₅₁	D₄	1	2	yes	—
26	C₂×C₄×D₁₅₁	C₂³	1	1	yes	characteristic
27	C₂×C₁₅₁⋊C₈	C₂³	1	2	yes	—
28	C₈×D₁₅₁	Q₈	1	8	yes	complemented
29	C₄⋊Dic₁₅₁	—	2	4	no	—
30	C₄⋊Dic₁₅₁	—	2	2	no	—
31	C₄⋊Dic₁₅₁	C₂³	1	4	yes	—
32	Dic₁₅₁⋊C₄	—	2	4	no	—
33	(C₂×D₁₅₁)⋊C₄	—	2	4	no	—
34	C₄×Dic₁₅₁	—	2	4	no	—
35	C₄×Dic₁₅₁	D₄	1	2	yes	—
36	C₄×Dic₁₅₁	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₅₁	—	2	2	no	—
38	C₂×C₈×C₁₅₁	C₂³	1	2	yes	—
39	C₄⋊C₄×C₁₅₁	—	2	2	no	—
40	C₄⋊C₄×C₁₅₁	C₂³	1	4	yes	—
41	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
42	C₂²×D₁₅₁	C₂×D₄	1	1	yes	characteristic
43	Dic₃₀₂	—	4	4	no	—
44	C₂×Dic₁₅₁	—	2	4	no	—
45	C₂×Dic₁₅₁	C₂×D₄	1	2	yes	—
46	C₂×Dic₁₅₁	C₂×D₄	1	1	yes	characteristic
47	C₄×D₁₅₁	—	2	4	no	—
48	C₄×D₁₅₁	C₂×Q₈	1	4	yes	—
49	C₁₅₁⋊C₈	C₂×Q₈	1	4	yes	—
50	Q₈×C₁₅₁	—	4	4	no	—
51	C₂×C₄×C₁₅₁	—	2	4	no	—
52	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
54	C₈×C₁₅₁	C₂×Q₈	1	4	yes	—
55	C₂²×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
56	C₂×D₁₅₁	—	2	2	no	—
57	C₂×D₁₅₁	C₄⋊Q₈	1	2	yes	—
58	C₄×C₁₅₁	—	4	4	no	—
59	C₄×C₁₅₁	#135	1	4	yes	—
60	C₄×C₁₅₁	C₂²×Q₈	1	2	yes	—
61	Dic₁₅₁	—	2	2	no	—
62	Dic₁₅₁	—	4	4	no	—
63	Dic₁₅₁	C₄⋊Q₈	1	2	yes	—
64	C₂×C₁₅₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₁₅₁	C₂×#135	1	2	yes	—
66	D₁₅₁	—	2	2	no	—
67	C₁₅₁	#3160	1	1	yes	characteristic, complemented, Sylow
68	#3160	—	151	1	no	maximal, Sylow
69	#533	—	151	2	no	—
70	#500	—	151	2	no	—
71	#499	—	151	4	no	—
72	#498	—	151	1	no	—
73	#498	D₁₅₁	1	1	yes	characteristic, complemented
74	#427	—	151	4	no	—
75	C₂×C₄×C₈	—	151	1	no	—
76	C₂²×C₈	—	151	2	no	—
77	C₄⋊Q₈	—	151	2	no	—
78	C₄⋊Q₈	C₂×D₁₅₁	1	2	yes	—
79	#125	—	151	4	no	—
80	C₂²⋊Q₈	—	302	2	no	—
81	C₂²⋊Q₈	—	302	2	no	—
82	C₄×Q₈	—	302	2	no	—
83	C₄×Q₈	—	302	2	no	—
84	#117	—	151	4	no	—
85	C₂×C₄²	—	151	1	no	—
86	#107	—	151	8	no	—
87	#106	—	151	4	no	—
88	#106	C₂×D₁₅₁	1	4	yes	—
89	C₄×C₈	—	151	2	no	—
90	C₄×C₈	—	151	1	no	—
91	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	302	2	no	—
94	C₂²×C₄	—	151	2	no	—
95	C₂²×C₄	—	151	1	no	—
96	C₂×C₈	—	151	2	no	—
97	C₂×C₈	—	151	8	no	—
98	C₂×C₈	C₂²×D₁₅₁	1	2	yes	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	302	4	no	—
101	C₄⋊C₄	—	302	4	no	—
102	C₄⋊C₄	—	302	2	no	—
103	C₄⋊C₄	—	151	4	no	—
104	C₄⋊C₄	C₂²×D₁₅₁	1	4	yes	—
105	C₂²⋊C₄	—	302	4	no	—
106	C₄²	—	302	4	no	—
107	C₄²	—	151	2	no	—
108	C₄²	—	151	1	no	—
109	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
110	C₂³	—	151	1	no	—
111	Q₈	—	4	4	no	—
112	Q₈	—	604	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	302	4	no	—
115	C₂×C₄	—	302	4	no	—
116	C₂×C₄	—	151	2	no	—
117	C₂×C₄	—	151	1	no	—
118	C₂×C₄	—	151	4	no	—
119	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
120	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
121	C₈	—	151	4	no	—
122	C₈	Q₈×D₁₅₁	1	4	yes	—
123	C₂²	—	302	2	no	—
124	C₂²	—	151	2	no	—
125	C₂²	C₂×D₄×D₁₅₁	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	302	2	no	—
128	C₄	—	604	4	no	—
129	C₄	—	151	2	no	—
130	C₄	#18012760	1	4	yes	—
131	C₄	C₂×Q₈×D₁₅₁	1	2	yes	—
132	C₂	—	302	2	no	—
133	C₂	C₄⋊Q₈×D₁₅₁	1	1	yes	characteristic
134	C₂	C₂×#18012760	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752767

C₂

yes

maximal

#23752615

C₂

yes

maximal, complemented

#23752612

C₂

yes

maximal, complemented

#23752610

C₂

yes

maximal, characteristic, complemented

#23752247

C₂

yes

maximal

C₄×C₈×D₁₅₁

C₂

yes

maximal, characteristic

#498×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₁₅₁

C₂²

yes

—

#18012739

C₂²

yes

—

#18012730

C₂²

yes

—

#18012719

—

#18012715

—

Q₈×Dic₁₅₁

—

#18012704

—

#18012697

C₂²

yes

—

C₄²×D₁₅₁

C₂²

yes

characteristic

#18012672

C₂²

yes

—

#18012671

C₂²

yes

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₈×Dic₁₅₁

C₂²

yes

—

C₄⋊Q₈×C₁₅₁

C₂²

yes

—

#106×C₁₅₁

C₂²

yes

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×C₄×D₁₅₁

D₄

yes

—

C₂×C₄×D₁₅₁

C₂³

yes

characteristic

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₈×D₁₅₁

Q₈

yes

complemented

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

C₂³

yes

—

Dic₁₅₁⋊C₄

—

(C₂×D₁₅₁)⋊C₄

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

D₄

yes

—

C₄×Dic₁₅₁

C₂³

yes

characteristic

C₂×Q₈×C₁₅₁

—

C₂×C₈×C₁₅₁

C₂³

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

C₂³

yes

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

C₂×D₄

yes

characteristic

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂×D₄

yes

—

C₂×Dic₁₅₁

C₂×D₄

yes

characteristic

C₄×D₁₅₁

—

C₄×D₁₅₁

C₂×Q₈

yes

—

C₁₅₁⋊C₈

C₂×Q₈

yes

—

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₈×C₁₅₁

C₂×Q₈

yes

—

C₂²×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

C₄⋊Q₈

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#135

yes

—

C₄×C₁₅₁

C₂²×Q₈

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

C₄⋊Q₈

yes

—

C₂×C₁₅₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₅₁

C₂×#135

yes

—

D₁₅₁

—

C₁₅₁

#3160

yes

characteristic, complemented, Sylow

#3160

—

151

maximal, Sylow

#533

—

151

—

#500

—

151

—

#499

—

151

—

#498

—

151

—

#498

D₁₅₁

yes

characteristic, complemented

#427

—

151

—

C₂×C₄×C₈

—

151

—

C₂²×C₈

—

151

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

—

#125

—

151

—

C₂²⋊Q₈

—

302

—

C₂²⋊Q₈

—

302

—

C₄×Q₈

—

302

—

C₄×Q₈

—

302

—

#117

—

151

—

C₂×C₄²

—

151

—

#107

—

151

—

#106

—

151

—

#106

C₂×D₁₅₁

yes

—

C₄×C₈

—

151

—

C₄×C₈

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

C₂²×D₁₅₁

yes

—

C₄⋊C₄

—

100

C₄⋊C₄

—

302

—

101

C₄⋊C₄

—

302

—

102

C₄⋊C₄

—

302

—

103

C₄⋊C₄

—

151

—

104

C₄⋊C₄

C₂²×D₁₅₁

yes

—

105

C₂²⋊C₄

—

302

—

106

C₄²

—

302

—

107

C₄²

—

151

—

108

C₄²

—

151

—

109

C₄²

C₂²×D₁₅₁

yes

characteristic

110

C₂³

—

151

—

111

Q₈

—

112

Q₈

—

604

—

113

C₂×C₄

—

114

C₂×C₄

—

302

—

115

C₂×C₄

—

302

—

116

C₂×C₄

—

151

—

117

C₂×C₄

—

151

—

118

C₂×C₄

—

151

—

119

C₂×C₄

D₄×D₁₅₁

yes

—

120

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

121

C₈

—

151

—

122

C₈

Q₈×D₁₅₁

yes

—

123

C₂²

—

302

—

124

C₂²

—

151

—

125

C₂²

C₂×D₄×D₁₅₁

yes

characteristic

126

C₄

—

127

C₄

—

302

—

128

C₄

—

604

—

129

C₄

—

151

—

130

C₄

#18012760

yes

—

131

C₄

C₂×Q₈×D₁₅₁

yes

—

132

C₂

—

302

—

133

C₂

C₄⋊Q₈×D₁₅₁

yes

characteristic

134

C₂

C₂×#18012760

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₁₅₁
commutator, cosocle	C₂×C₄×C₁₅₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	#498×C₁₅₁	C₂
socle	C₂²×C₁₅₁	C₂²×D₄
prime radical	C₂²×D₁₅₁	C₂×D₄
hypercenter	#498	D₁₅₁
hypocenter	C₁₅₁	#3160
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

C₂×D₄×D₁₅₁

commutator, cosocle

C₂×C₄×C₁₅₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

#498×C₁₅₁

C₂

socle

C₂²×C₁₅₁

C₂²×D₄

prime radical

C₂²×D₁₅₁

C₂×D₄

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #23752615	C₂	semidirect
3. #23752612	C₂	semidirect
4. #23752610	C₂	semidirect
7. #498×C₁₅₁	C₂	semidirect
28. C₈×D₁₅₁	Q₈	semidirect
67. C₁₅₁	#3160	semidirect
73. #498	D₁₅₁	semidirect

H ≅ G/N

type

2. #23752615

C₂

semidirect

3. #23752612

C₂

semidirect

4. #23752610

C₂

semidirect

7. #498×C₁₅₁

C₂

semidirect

28. C₈×D₁₅₁

Q₈

semidirect

67. C₁₅₁

#3160

semidirect

73. #498

D₁₅₁

semidirect

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

subgroup	index
#23752767	2
#23752615	2
#23752612	2
#23752610	2
#23752247	2
C₄×C₈×D₁₅₁	2
#498×C₁₅₁	2
#3160	151

subgroup

index

C₄×C₈×D₁₅₁

#498×C₁₅₁

#3160

151

Subgroups — #34079619

Conjugacy classes of subgroups

Complements

Maximal subgroups