Group #34069239 — subgroups

Subgroups — #34069239

Order 19328.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×C₈×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#23752077	C₂	1	2	yes	maximal
3	#23751975	C₂	1	1	yes	maximal, characteristic
4	#23751974	C₂	1	2	yes	maximal
5	#340×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	C₈×Dic₁₅₁	C₄	1	4	yes	complemented
10	C₁₅₁⋊#95	—	2	2	no	—
11	C₁₅₁⋊#95	—	2	2	no	—
12	C₂²×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
14	C₂²×Dic₁₅₁	—	2	2	no	—
15	C₂²×Dic₁₅₁	D₄	1	1	yes	characteristic
16	C₂²×Dic₁₅₁	D₄	1	1	yes	characteristic
17	C₂×C₁₅₁⋊C₈	—	2	2	no	—
18	C₂×C₁₅₁⋊C₈	C₂×C₄	1	2	yes	—
19	C₄⋊Dic₁₅₁	—	2	2	no	—
20	C₄⋊Dic₁₅₁	—	2	2	no	—
21	C₄×Dic₁₅₁	C₂×C₄	1	2	yes	—
22	C₄×Dic₁₅₁	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₅₁	—	2	2	no	—
24	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₅₁	—	2	2	no	—
26	C₂×C₈×C₁₅₁	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₁₅₁	—	2	2	no	—
28	C₄⋊C₄×C₁₅₁	—	2	2	no	—
29	C₂×Dic₁₅₁	—	2	1	no	—
30	C₂×Dic₁₅₁	—	2	2	no	—
31	C₂×Dic₁₅₁	—	2	1	no	—
32	C₂×Dic₁₅₁	—	4	2	no	—
33	C₂×Dic₁₅₁	—	4	2	no	—
34	C₂×Dic₁₅₁	—	4	2	no	—
35	C₂×Dic₁₅₁	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₁₅₁	C₄⋊C₄	1	2	yes	—
37	C₁₅₁⋊C₈	—	2	2	no	—
38	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₅₁	—	4	2	no	—
40	C₂×C₄×C₁₅₁	—	4	2	no	—
41	C₂×C₄×C₁₅₁	—	4	2	no	—
42	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₅₁	#41	1	2	yes	—
45	C₂×C₄×C₁₅₁	#41	1	2	yes	—
46	C₈×C₁₅₁	—	2	2	no	—
47	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂²×C₁₅₁	#124	1	2	yes	—
49	C₂²×C₁₅₁	#135	1	1	yes	characteristic
50	C₂²×C₁₅₁	#135	1	2	yes	—
51	C₂²×C₁₅₁	#135	1	1	yes	characteristic
52	C₄×C₁₅₁	—	8	2	no	—
53	C₄×C₁₅₁	#117	1	1	yes	characteristic
54	C₄×C₁₅₁	#117	1	1	yes	characteristic
55	C₄×C₁₅₁	#125	1	2	yes	—
56	Dic₁₅₁	—	2	2	no	—
57	Dic₁₅₁	—	2	2	no	—
58	Dic₁₅₁	—	8	2	no	—
59	C₂×C₁₅₁	#382	1	1	yes	characteristic
60	C₂×C₁₅₁	#427	1	1	yes	characteristic
61	C₂×C₁₅₁	#427	1	1	yes	characteristic
62	C₂×C₁₅₁	#488	1	2	yes	—
63	C₂×C₁₅₁	#488	1	2	yes	—
64	C₁₅₁	#1835	1	1	yes	characteristic, complemented, Sylow
65	#1835	—	151	1	no	maximal, Sylow
66	C₂×C₄×C₈	—	151	1	no	—
67	#382	—	151	2	no	—
68	#340	—	151	1	no	—
69	#340	—	151	2	no	—
70	#340	D₁₅₁	1	1	yes	characteristic
71	C₂²×C₈	—	151	1	no	—
72	C₂²×C₈	C₂×D₁₅₁	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	151	2	no	—
74	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
75	C₂×C₄²	—	151	1	no	—
76	C₄×C₈	—	151	4	no	—
77	#95	—	302	2	no	—
78	#95	—	302	2	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	302	2	no	—
81	C₂²×C₄	—	151	1	no	—
82	C₂²×C₄	—	151	1	no	—
83	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	302	2	no	—
86	C₂×C₈	—	151	2	no	—
87	C₂×C₈	C₄×D₁₅₁	1	2	yes	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	302	2	no	—
91	C₄⋊C₄	—	302	2	no	—
92	C₄²	—	151	2	no	—
93	C₄²	—	151	2	no	—
94	C₂³	D₄×D₁₅₁	1	1	yes	characteristic
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	302	1	no	—
99	C₂×C₄	—	302	2	no	—
100	C₂×C₄	—	302	1	no	—
101	C₂×C₄	—	604	2	no	—
102	C₂×C₄	—	604	2	no	—
103	C₂×C₄	—	604	2	no	—
104	C₂×C₄	—	151	2	no	—
105	C₂×C₄	—	151	2	no	—
106	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
107	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
108	C₂×C₄	#14559003	1	2	yes	—
109	C₂×C₄	#14559004	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	302	2	no	—
112	C₂²	C₄⋊C₄×D₁₅₁	1	1	yes	characteristic
113	C₂²	#18012725	1	2	yes	—
114	C₂²	#18012758	1	1	yes	characteristic
115	C₂²	#18012760	1	2	yes	—
116	C₂²	#18012762	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	—	302	2	no	—
119	C₄	—	302	2	no	—
120	C₄	—	1208	2	no	—
121	C₄	#18012695	1	1	yes	characteristic
122	C₄	#18012696	1	1	yes	characteristic
123	C₄	#18012729	1	2	yes	—
124	C₂	#23752076	1	1	yes	characteristic
125	C₂	#23752247	1	1	yes	characteristic
126	C₂	#23752249	1	1	yes	characteristic
127	C₂	#23752565	1	2	yes	—
128	C₂	#23752568	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×Dic₁₅₁

C₂

yes

maximal, characteristic

#23752077

C₂

yes

maximal

#23751975

C₂

yes

maximal, characteristic

#23751974

C₂

yes

maximal

#340×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

C₈×Dic₁₅₁

C₄

yes

complemented

C₁₅₁⋊#95

—

C₁₅₁⋊#95

—

C₂²×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

—

C₂²×Dic₁₅₁

D₄

yes

characteristic

C₂²×Dic₁₅₁

D₄

yes

characteristic

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

C₂×C₄

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×C₈×C₁₅₁

C₂×C₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₄⋊C₄

yes

—

C₂×Dic₁₅₁

C₄⋊C₄

yes

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₁

#124

yes

—

C₂²×C₁₅₁

#135

yes

characteristic

C₂²×C₁₅₁

#135

yes

—

C₂²×C₁₅₁

#135

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₂×C₁₅₁

#488

yes

—

C₁₅₁

#1835

yes

characteristic, complemented, Sylow

#1835

—

151

maximal, Sylow

C₂×C₄×C₈

—

151

—

#382

—

151

—

#340

—

151

—

#340

—

151

—

#340

D₁₅₁

yes

characteristic

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

characteristic

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

C₄×C₈

—

151

—

#95

—

302

—

#95

—

302

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

302

—

C₂×C₈

—

151

—

C₂×C₈

C₄×D₁₅₁

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

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

100

C₂×C₄

—

302

—

101

C₂×C₄

—

604

—

102

C₂×C₄

—

604

—

103

C₂×C₄

—

604

—

104

C₂×C₄

—

151

—

105

C₂×C₄

—

151

—

106

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

107

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

108

C₂×C₄

#14559003

yes

—

109

C₂×C₄

#14559004

yes

—

110

C₈

—

111

C₈

—

302

—

112

C₂²

C₄⋊C₄×D₁₅₁

yes

characteristic

113

C₂²

#18012725

yes

—

114

C₂²

#18012758

yes

characteristic

115

C₂²

#18012760

yes

—

116

C₂²

#18012762

yes

characteristic

117

C₄

—

118

C₄

—

302

—

119

C₄

—

302

—

120

C₄

—

1208

—

121

C₄

#18012695

yes

characteristic

122

C₄

#18012696

yes

characteristic

123

C₄

#18012729

yes

—

124

C₂

#23752076

yes

characteristic

125

C₂

#23752247

yes

characteristic

126

C₂

#23752249

yes

characteristic

127

C₂

#23752565

yes

—

128

C₂

#23752568

yes

—

	subgroup	G/H
center	C₂³	D₄×D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#340×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#340	D₁₅₁
hypocenter	C₁₅₁	#1835
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#340×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 ◁ #340

N	H ≅ G/N	type
9. C₈×Dic₁₅₁	C₄	semidirect
64. C₁₅₁	#1835	semidirect

H ≅ G/N

type

9. C₈×Dic₁₅₁

C₄

semidirect

64. C₁₅₁

#1835

semidirect

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

subgroup	index
C₂×C₈×Dic₁₅₁	2
#23752077	2
#23751975	2
#23751974	2
#340×C₁₅₁	2
#1835	151

subgroup

index

C₂×C₈×Dic₁₅₁

#340×C₁₅₁

151

Subgroups — #34069239

Conjugacy classes of subgroups

Complements

Maximal subgroups