Group #34069271 — subgroups

Subgroups — #34069271

Order 19328.

248 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 117 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×#18012672	C₂	1	2	yes	maximal
2	C₂×#18012671	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
4	#23752077	C₂	1	2	yes	maximal
5	C₂×#106×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
8	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
9	#18012672	—	2	4	no	—
10	#18012671	—	2	2	no	—
11	C₈×Dic₁₅₁	C₄	1	4	yes	complemented
12	C₁₅₁⋊#95	—	2	4	no	—
13	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
15	#106×C₁₅₁	—	2	2	no	—
16	C₂²×Dic₁₅₁	—	2	2	no	—
17	C₂²×Dic₁₅₁	D₄	1	2	yes	—
18	C₂×C₁₅₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
20	C₂×C₁₅₁⋊C₈	Q₈	1	2	yes	—
21	C₄⋊Dic₁₅₁	—	2	2	no	—
22	C₄⋊Dic₁₅₁	—	2	2	no	—
23	C₄×Dic₁₅₁	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₅₁	Q₈	1	2	yes	—
25	C₂²×C₄×C₁₅₁	—	2	2	no	—
26	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
27	C₂×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
29	C₂×C₈×C₁₅₁	Q₈	1	2	yes	—
30	C₄⋊C₄×C₁₅₁	—	2	2	no	—
31	C₄⋊C₄×C₁₅₁	—	2	2	no	—
32	C₂×Dic₁₅₁	—	2	4	no	—
33	C₂×Dic₁₅₁	—	4	2	no	—
34	C₂×Dic₁₅₁	—	4	4	no	—
35	C₂×Dic₁₅₁	C₄⋊C₄	1	4	yes	—
36	C₁₅₁⋊C₈	C₄⋊C₄	1	4	yes	—
37	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₁₅₁	—	4	2	no	—
39	C₂×C₄×C₁₅₁	—	4	4	no	—
40	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
46	C₈×C₁₅₁	C₄⋊C₄	1	4	yes	—
47	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₅₁	#127	1	1	yes	characteristic
49	C₂²×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₅₁	#135	1	4	yes	—
51	C₄×C₁₅₁	—	8	2	no	—
52	C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
55	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₁₅₁	—	2	4	no	—
57	Dic₁₅₁	—	8	2	no	—
58	C₂×C₁₅₁	#384	1	1	yes	characteristic
59	C₂×C₁₅₁	#427	1	2	yes	—
60	C₂×C₁₅₁	#495	1	2	yes	—
61	C₂×C₁₅₁	#499	1	2	yes	—
62	C₁₅₁	#1846	1	1	yes	characteristic, complemented, Sylow
63	#1846	—	151	1	no	maximal, Sylow
64	C₂×#107	—	151	2	no	—
65	C₂×#106	—	151	1	no	—
66	C₂×#106	D₁₅₁	1	1	yes	characteristic
67	C₂×C₄×C₈	—	151	1	no	—
68	#382	—	151	2	no	—
69	C₂²×C₈	—	151	1	no	—
70	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
71	C₂×C₄⋊C₄	—	151	2	no	—
72	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
73	C₂×C₄²	—	151	1	no	—
74	#107	—	302	4	no	—
75	#106	—	2	2	no	—
76	#106	—	302	2	no	—
77	C₄×C₈	—	151	4	no	—
78	#95	—	302	4	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	302	2	no	—
81	C₂²×C₄	—	151	2	no	—
82	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
83	C₂×C₈	—	151	2	no	—
84	C₂×C₈	—	151	2	no	—
85	C₂×C₈	—	151	2	no	—
86	C₂×C₈	C₄×D₁₅₁	1	2	yes	—
87	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
88	C₂×C₈	Dic₃₀₂	1	2	yes	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	302	2	no	—
92	C₄⋊C₄	—	302	2	no	—
93	C₄²	—	151	2	no	—
94	C₄²	—	151	2	no	—
95	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	302	4	no	—
99	C₂×C₄	—	604	2	no	—
100	C₂×C₄	—	604	4	no	—
101	C₂×C₄	—	151	4	no	—
102	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
103	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
105	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₃₀₂	1	1	yes	characteristic
108	C₈	—	151	4	no	—
109	C₈	Dic₁₅₁⋊C₄	1	4	yes	—
110	C₂²	C₄⋊C₄×D₁₅₁	1	1	yes	characteristic
111	C₂²	#18012735	1	1	yes	characteristic
112	C₂²	#18012738	1	1	yes	characteristic
113	C₂²	#18012760	1	4	yes	—
114	C₄	—	8	2	no	—
115	C₄	—	302	4	no	—
116	C₄	—	1208	2	no	—
117	C₄	C₄⋊C₄×D₁₅₁	1	1	yes	characteristic
118	C₄	C₂×Dic₁₅₁⋊C₄	1	1	yes	characteristic
119	C₄	#18012737	1	1	yes	characteristic
120	C₄	#18012738	1	1	yes	characteristic
121	C₂	#23752081	1	1	yes	characteristic
122	C₂	#23752247	1	2	yes	—
123	C₂	#23752593	1	2	yes	—
124	C₂	#23752612	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012672

C₂

yes

maximal

C₂×#18012671

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₅₁

C₂

yes

maximal, characteristic

#23752077

C₂

yes

maximal

C₂×#106×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

#18012672

—

#18012671

—

C₈×Dic₁₅₁

C₄

yes

complemented

C₁₅₁⋊#95

—

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

#106×C₁₅₁

—

C₂²×Dic₁₅₁

—

C₂²×Dic₁₅₁

D₄

yes

—

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×C₁₅₁⋊C₈

Q₈

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄×Dic₁₅₁

C₂×C₄

yes

—

C₄×Dic₁₅₁

Q₈

yes

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

C₂×C₄

yes

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₂×C₈×C₁₅₁

Q₈

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₄⋊C₄

yes

—

C₁₅₁⋊C₈

C₄⋊C₄

yes

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₈×C₁₅₁

C₄⋊C₄

yes

—

C₂²×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₁

#127

yes

characteristic

C₂²×C₁₅₁

C₄⋊Q₈

yes

characteristic

C₂²×C₁₅₁

#135

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#384

yes

characteristic

C₂×C₁₅₁

#427

yes

—

C₂×C₁₅₁

#495

yes

—

C₂×C₁₅₁

#499

yes

—

C₁₅₁

#1846

yes

characteristic, complemented, Sylow

#1846

—

151

maximal, Sylow

C₂×#107

—

151

—

C₂×#106

—

151

—

C₂×#106

D₁₅₁

yes

characteristic

C₂×C₄×C₈

—

151

—

#382

—

151

—

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

#107

—

302

—

#106

—

#106

—

302

—

C₄×C₈

—

151

—

#95

—

302

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

C₄×D₁₅₁

yes

—

C₂×C₈

C₂²⋊D₁₅₁

yes

—

C₂×C₈

Dic₃₀₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄²

—

151

—

C₄²

—

151

—

C₂³

D₄×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

604

—

100

C₂×C₄

—

604

—

101

C₂×C₄

—

151

—

102

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

103

C₂×C₄

D₄×D₁₅₁

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

105

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

106

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

107

C₂×C₄

C₂×Dic₃₀₂

yes

characteristic

108

C₈

—

151

—

109

C₈

Dic₁₅₁⋊C₄

yes

—

110

C₂²

C₄⋊C₄×D₁₅₁

yes

characteristic

111

C₂²

#18012735

yes

characteristic

112

C₂²

#18012738

yes

characteristic

113

C₂²

#18012760

yes

—

114

C₄

—

115

C₄

—

302

—

116

C₄

—

1208

—

117

C₄

C₄⋊C₄×D₁₅₁

yes

characteristic

118

C₄

C₂×Dic₁₅₁⋊C₄

yes

characteristic

119

C₄

#18012737

yes

characteristic

120

C₄

#18012738

yes

characteristic

121

C₂

#23752081

yes

characteristic

122

C₂

#23752247

yes

—

123

C₂

#23752593

yes

—

124

C₂

#23752612

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×#106×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×#106	D₁₅₁
hypocenter	C₁₅₁	#1846
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂³

D₄×D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×#106×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
11. C₈×Dic₁₅₁	C₄	semidirect
62. C₁₅₁	#1846	semidirect

H ≅ G/N

type

11. C₈×Dic₁₅₁

C₄

semidirect

62. C₁₅₁

#1846

semidirect

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

subgroup	index
C₂×#18012672	2
C₂×#18012671	2
C₂×C₈×Dic₁₅₁	2
#23752077	2
C₂×#106×C₁₅₁	2
#1846	151

subgroup

index

C₂×C₈×Dic₁₅₁

C₂×#106×C₁₅₁

151

Subgroups — #34069271

Conjugacy classes of subgroups

Complements

Maximal subgroups