Group #34069912 — subgroups

Subgroups — #34069912

Order 19328.

332 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 141 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×#18012739	C₂	1	1	yes	maximal, characteristic
2	C₂×#18012664	C₂	1	4	yes	maximal
3	#23752082	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₀₂	C₂²	1	2	yes	—
6	#18012739	—	2	2	no	—
7	#18012739	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₁₅₁	—	2	2	no	—
9	C₂×C₄⋊Dic₁₅₁	—	2	2	no	—
10	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
11	#18012664	—	2	8	no	—
12	C₂²×C₈×C₁₅₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₁₅₁	—	2	2	no	—
15	C₂×Dic₃₀₂	—	2	2	no	—
16	C₂×Dic₃₀₂	—	2	4	no	—
17	C₂×Dic₃₀₂	—	2	4	no	—
18	C₂×Dic₃₀₂	C₂×C₄	1	4	yes	—
19	C₂²×Dic₁₅₁	—	2	2	no	—
20	C₂²×Dic₁₅₁	—	2	2	no	—
21	C₄⋊Dic₁₅₁	—	2	8	no	—
22	C₄⋊Dic₁₅₁	—	2	4	no	—
23	C₄⋊Dic₁₅₁	—	4	4	no	—
24	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
25	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₁₅₁	—	2	2	no	—
27	C₂×C₈×C₁₅₁	—	2	2	no	—
28	C₂×C₈×C₁₅₁	D₄	1	4	yes	—
29	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
30	C₄²×C₁₅₁	D₄	1	2	yes	—
31	Dic₃₀₂	—	2	8	no	—
32	Dic₃₀₂	—	4	4	no	—
33	C₂×Dic₁₅₁	—	2	4	no	—
34	C₂×Dic₁₅₁	—	4	2	no	—
35	C₂×Dic₁₅₁	—	4	2	no	—
36	C₂×Dic₁₅₁	—	4	2	no	—
37	C₂×Dic₁₅₁	—	4	4	no	—
38	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₁₅₁	SD₁₆	1	4	yes	—
41	C₂×C₄×C₁₅₁	Q₁₆	1	4	yes	—
42	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₅₁	#41	1	2	yes	—
46	C₈×C₁₅₁	—	2	4	no	—
47	C₂²×C₁₅₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₅₁	#124	1	1	yes	characteristic
49	C₂²×C₁₅₁	#127	1	1	yes	characteristic
50	C₂²×C₁₅₁	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₁₅₁	C₂×Q₁₆	1	2	yes	—
52	C₄×C₁₅₁	#103	1	8	yes	complemented
53	C₄×C₁₅₁	C₄×D₄	1	2	yes	—
54	C₄×C₁₅₁	#121	1	2	yes	—
55	Dic₁₅₁	—	4	4	no	—
56	Dic₁₅₁	—	8	2	no	—
57	C₂×C₁₅₁	#390	1	1	yes	characteristic
58	C₂×C₁₅₁	C₂×#103	1	2	yes	—
59	C₂×C₁₅₁	#487	1	2	yes	—
60	C₂×C₁₅₁	#492	1	1	yes	characteristic
61	C₂×C₁₅₁	#494	1	1	yes	characteristic
62	C₁₅₁	#1956	1	1	yes	characteristic, complemented, Sylow
63	#1956	—	151	1	no	maximal, Sylow
64	C₂×C₄⋊Q₈	—	151	1	no	—
65	C₂×#103	—	151	4	no	—
66	C₂×C₄×C₈	D₁₅₁	1	1	yes	characteristic
67	#384	—	151	1	no	—
68	C₂²×Q₈	—	151	2	no	—
69	C₂²×C₈	C₂×D₁₅₁	1	2	yes	—
70	C₄⋊Q₈	—	302	2	no	—
71	C₄⋊Q₈	—	151	4	no	—
72	C₂×C₄⋊C₄	—	302	2	no	—
73	C₂×C₄⋊C₄	—	302	2	no	—
74	C₂×C₄⋊C₄	—	151	2	no	—
75	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
76	#103	—	302	8	no	—
77	C₄×C₈	—	2	2	no	—
78	C₂×Q₈	—	302	2	no	—
79	C₂×Q₈	—	302	4	no	—
80	C₂×Q₈	—	302	4	no	—
81	C₂×Q₈	—	151	4	no	—
82	C₂²×C₄	—	302	2	no	—
83	C₂²×C₄	—	302	2	no	—
84	C₂²×C₄	D₆₀₄	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	D₆₀₄	1	4	yes	—
89	C₄⋊C₄	—	302	8	no	—
90	C₄⋊C₄	—	302	4	no	—
91	C₄⋊C₄	—	604	4	no	—
92	C₄²	C₄×D₁₅₁	1	2	yes	—
93	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
94	C₂³	C₂×D₆₀₄	1	1	yes	characteristic
95	Q₈	—	302	8	no	—
96	Q₈	—	604	4	no	—
97	C₂×C₄	—	302	4	no	—
98	C₂×C₄	—	604	2	no	—
99	C₂×C₄	—	604	2	no	—
100	C₂×C₄	—	604	2	no	—
101	C₂×C₄	—	604	4	no	—
102	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	4	yes	—
103	C₂×C₄	#14558993	1	4	yes	—
104	C₂×C₄	Dic₆₀₄	1	4	yes	—
105	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
107	C₂×C₄	C₂×D₆₀₄	1	2	yes	—
108	C₂×C₄	#14559004	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₂²	C₂×(C₂×D₁₅₁)⋊C₄	1	1	yes	characteristic
111	C₂²	#18012727	1	1	yes	characteristic
112	C₂²	#18012736	1	1	yes	characteristic
113	C₂²	C₂×#14558993	1	2	yes	—
114	C₂²	C₂×Dic₆₀₄	1	2	yes	—
115	C₄	—	604	4	no	—
116	C₄	—	1208	2	no	—
117	C₄	#18012664	1	8	yes	complemented
118	C₄	C₄×D₆₀₄	1	2	yes	—
119	C₄	#18012713	1	2	yes	—
120	C₂	#23752114	1	1	yes	characteristic
121	C₂	C₂×#18012664	1	2	yes	—
122	C₂	#23752563	1	2	yes	—
123	C₂	#23752585	1	1	yes	characteristic
124	C₂	#23752592	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012739

C₂

yes

maximal, characteristic

C₂×#18012664

C₂

yes

maximal

#23752082

C₂

yes

maximal, characteristic

C₂×C₄×C₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

C₂²

yes

—

#18012739

—

#18012739

C₄

yes

complemented

C₂×C₄⋊Dic₁₅₁

—

C₂×C₄⋊Dic₁₅₁

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

#18012664

—

C₂²×C₈×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₄×C₈×C₁₅₁

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

C₂×C₄

yes

—

C₂²×Dic₁₅₁

—

C₂²×Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

D₄

yes

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

SD₁₆

yes

—

C₂×C₄×C₁₅₁

Q₁₆

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₂²⋊C₄

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

C₂×SD₁₆

yes

—

C₂²×C₁₅₁

C₂×Q₁₆

yes

—

C₄×C₁₅₁

#103

yes

complemented

C₄×C₁₅₁

C₄×D₄

yes

—

C₄×C₁₅₁

#121

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#390

yes

characteristic

C₂×C₁₅₁

C₂×#103

yes

—

C₂×C₁₅₁

#487

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1956

—

151

maximal, Sylow

C₂×C₄⋊Q₈

—

151

—

C₂×#103

—

151

—

C₂×C₄×C₈

D₁₅₁

yes

characteristic

#384

—

151

—

C₂²×Q₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

—

C₄⋊Q₈

—

302

—

C₄⋊Q₈

—

151

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#103

—

302

—

C₄×C₈

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

151

—

C₂²×C₄

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

D₆₀₄

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₆₀₄

yes

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

604

—

C₄²

C₄×D₁₅₁

yes

—

C₄²

C₂²⋊D₁₅₁

yes

—

C₂³

C₂×D₆₀₄

yes

characteristic

Q₈

—

302

—

Q₈

—

604

—

C₂×C₄

—

302

—

C₂×C₄

—

604

—

C₂×C₄

—

604

—

100

C₂×C₄

—

604

—

101

C₂×C₄

—

604

—

102

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

103

C₂×C₄

#14558993

yes

—

104

C₂×C₄

Dic₆₀₄

yes

—

105

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

107

C₂×C₄

C₂×D₆₀₄

yes

—

108

C₂×C₄

#14559004

yes

—

109

C₈

—

110

C₂²

C₂×(C₂×D₁₅₁)⋊C₄

yes

characteristic

111

C₂²

#18012727

yes

characteristic

112

C₂²

#18012736

yes

characteristic

113

C₂²

C₂×#14558993

yes

—

114

C₂²

C₂×Dic₆₀₄

yes

—

115

C₄

—

604

—

116

C₄

—

1208

—

117

C₄

#18012664

yes

complemented

118

C₄

C₄×D₆₀₄

yes

—

119

C₄

#18012713

yes

—

120

C₂

#23752114

yes

characteristic

121

C₂

C₂×#18012664

yes

—

122

C₂

#23752563

yes

—

123

C₂

#23752585

yes

characteristic

124

C₂

#23752592

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₀₄
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×C₄×C₈×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×C₄×C₈	D₁₅₁
hypocenter	C₁₅₁	#1956
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₆₀₄

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×C₄×C₈×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

hypercenter

C₂×C₄×C₈

D₁₅₁

hypocenter

C₁₅₁

#1956

LCS

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
7. #18012739	C₄	semidirect
52. C₄×C₁₅₁	#103	semidirect
62. C₁₅₁	#1956	semidirect
117. C₄	#18012664	semidirect

H ≅ G/N

type

7. #18012739

C₄

semidirect

52. C₄×C₁₅₁

#103

semidirect

62. C₁₅₁

#1956

semidirect

117. C₄

#18012664

semidirect

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

subgroup	index
C₂×#18012739	2
C₂×#18012664	2
#23752082	2
C₂×C₄×C₈×C₁₅₁	2
#1956	151

subgroup

index

C₂×C₄×C₈×C₁₅₁

151

Subgroups — #34069912

Conjugacy classes of subgroups

Complements

Maximal subgroups