Group #34069254 — subgroups

Subgroups — #34069254

Order 19328.

196 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 85 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	#23752058	C₂	1	1	yes	maximal, characteristic
2	#23751967	C₂	1	1	yes	maximal, characteristic
3	#23751966	C₂	1	2	yes	maximal
4	#23751966	C₂	1	2	yes	maximal
5	#336×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
9	C₁₅₁⋊#95	C₄	1	2	yes	—
10	C₁₅₁⋊#95	C₄	1	2	yes	—
11	C₂²×C₈×C₁₅₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₅₁	C₂×C₄	1	2	yes	—
16	C₂×C₁₅₁⋊C₈	—	2	2	no	—
17	C₂×C₁₅₁⋊C₈	—	2	2	no	—
18	C₂×C₁₅₁⋊C₈	—	2	2	no	—
19	C₄×Dic₁₅₁	—	2	1	no	—
20	C₄×Dic₁₅₁	—	2	2	no	—
21	C₄×Dic₁₅₁	—	2	1	no	—
22	C₂²×C₄×C₁₅₁	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
24	C₂×C₈×C₁₅₁	—	2	2	no	—
25	C₂×C₈×C₁₅₁	—	2	2	no	—
26	C₂×C₈×C₁₅₁	—	2	2	no	—
27	C₄²×C₁₅₁	—	2	2	no	—
28	C₂×Dic₁₅₁	—	2	1	no	—
29	C₂×Dic₁₅₁	—	2	1	no	—
30	C₂×Dic₁₅₁	—	2	2	no	—
31	C₂×Dic₁₅₁	—	2	2	no	—
32	C₂×Dic₁₅₁	—	2	2	no	—
33	C₂×Dic₁₅₁	—	2	2	no	—
34	C₂×Dic₁₅₁	—	2	2	no	—
35	C₁₅₁⋊C₈	—	4	2	no	—
36	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₁	—	2	2	no	—
38	C₂×C₄×C₁₅₁	—	2	2	no	—
39	C₂×C₄×C₁₅₁	—	2	2	no	—
40	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
41	C₂×C₄×C₁₅₁	#41	1	2	yes	—
42	C₂×C₄×C₁₅₁	#41	1	2	yes	—
43	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
44	C₈×C₁₅₁	—	4	2	no	—
45	C₂²×C₁₅₁	#117	1	1	yes	characteristic
46	C₂²×C₁₅₁	#117	1	1	yes	characteristic
47	C₂²×C₁₅₁	#117	1	1	yes	characteristic
48	C₂²×C₁₅₁	#131	1	2	yes	—
49	C₂²×C₁₅₁	#131	1	2	yes	—
50	C₄×C₁₅₁	—	4	2	no	—
51	C₄×C₁₅₁	#126	1	2	yes	—
52	C₄×C₁₅₁	#126	1	2	yes	—
53	Dic₁₅₁	—	4	1	no	—
54	Dic₁₅₁	—	4	2	no	—
55	Dic₁₅₁	—	4	1	no	—
56	C₂×C₁₅₁	#383	1	1	yes	characteristic
57	C₂×C₁₅₁	#433	1	2	yes	—
58	C₂×C₁₅₁	#433	1	1	yes	characteristic
59	C₂×C₁₅₁	#433	1	2	yes	—
60	C₂×C₁₅₁	#433	1	1	yes	characteristic
61	C₁₅₁	#1840	1	1	yes	characteristic, complemented, Sylow
62	#1840	—	151	1	no	maximal, Sylow
63	#376	—	151	1	no	—
64	#336	—	151	2	no	—
65	#336	—	151	2	no	—
66	#336	—	151	1	no	—
67	#336	D₁₅₁	1	1	yes	characteristic
68	C₂²×C₈	—	151	2	no	—
69	C₂²×C₈	C₂×D₁₅₁	1	2	yes	—
70	C₂×C₄²	—	151	1	no	—
71	C₂×C₄²	—	151	1	no	—
72	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
73	#95	—	151	2	no	—
74	#95	—	151	2	no	—
75	C₂²×C₄	—	151	1	no	—
76	C₂²×C₄	—	151	1	no	—
77	C₂²×C₄	—	151	2	no	—
78	C₂²×C₄	C₄×D₁₅₁	1	2	yes	—
79	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	302	2	no	—
84	C₂×C₈	—	302	2	no	—
85	C₂×C₈	—	302	2	no	—
86	C₄²	—	2	2	no	—
87	C₄²	—	302	1	no	—
88	C₄²	—	302	2	no	—
89	C₄²	—	302	1	no	—
90	C₂³	C₂×C₄×D₁₅₁	1	1	yes	characteristic
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	302	1	no	—
95	C₂×C₄	—	302	1	no	—
96	C₂×C₄	—	302	2	no	—
97	C₂×C₄	—	302	2	no	—
98	C₂×C₄	—	302	2	no	—
99	C₂×C₄	—	302	2	no	—
100	C₂×C₄	—	302	2	no	—
101	C₂×C₄	#14559003	1	1	yes	characteristic
102	C₂×C₄	#14559003	1	2	yes	—
103	C₂×C₄	#14559004	1	2	yes	—
104	C₂×C₄	#14559005	1	1	yes	characteristic
105	C₈	—	4	2	no	—
106	C₈	—	604	2	no	—
107	C₂²	#18012695	1	1	yes	characteristic
108	C₂²	#18012696	1	1	yes	characteristic
109	C₂²	#18012697	1	1	yes	characteristic
110	C₂²	#18012745	1	2	yes	—
111	C₂²	#18012747	1	2	yes	—
112	C₄	—	4	2	no	—
113	C₄	—	604	1	no	—
114	C₄	—	604	2	no	—
115	C₄	—	604	1	no	—
116	C₄	#18012732	1	2	yes	—
117	C₄	#18012734	1	2	yes	—
118	C₂	#23752079	1	1	yes	characteristic
119	C₂	#23752270	1	2	yes	—
120	C₂	#23752271	1	1	yes	characteristic
121	C₂	#23752272	1	2	yes	—
122	C₂	#23752274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752058

C₂

yes

maximal, characteristic

#23751967

C₂

yes

maximal, characteristic

#23751966

C₂

yes

maximal

#23751966

C₂

yes

maximal

#336×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

C₁₅₁⋊#95

C₄

yes

—

C₁₅₁⋊#95

C₄

yes

—

C₂²×C₈×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₅₁

C₂×C₄

yes

characteristic

C₂²×Dic₁₅₁

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₄²×C₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

#41

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

characteristic

C₈×C₁₅₁

—

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

—

C₂²×C₁₅₁

#131

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#126

yes

—

C₄×C₁₅₁

#126

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#383

yes

characteristic

C₂×C₁₅₁

#433

yes

—

C₂×C₁₅₁

#433

yes

characteristic

C₂×C₁₅₁

#433

yes

—

C₂×C₁₅₁

#433

yes

characteristic

C₁₅₁

#1840

yes

characteristic, complemented, Sylow

#1840

—

151

maximal, Sylow

#376

—

151

—

#336

—

151

—

#336

—

151

—

#336

—

151

—

#336

D₁₅₁

yes

characteristic

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

C₂×C₄²

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#95

—

151

—

#95

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₄×D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄²

—

C₄²

—

302

—

C₄²

—

302

—

C₄²

—

302

—

C₂³

C₂×C₄×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

100

C₂×C₄

—

302

—

101

C₂×C₄

#14559003

yes

characteristic

102

C₂×C₄

#14559003

yes

—

103

C₂×C₄

#14559004

yes

—

104

C₂×C₄

#14559005

yes

characteristic

105

C₈

—

106

C₈

—

604

—

107

C₂²

#18012695

yes

characteristic

108

C₂²

#18012696

yes

characteristic

109

C₂²

#18012697

yes

characteristic

110

C₂²

#18012745

yes

—

111

C₂²

#18012747

yes

—

112

C₄

—

113

C₄

—

604

—

114

C₄

—

604

—

115

C₄

—

604

—

116

C₄

#18012732

yes

—

117

C₄

#18012734

yes

—

118

C₂

#23752079

yes

characteristic

119

C₂

#23752270

yes

—

120

C₂

#23752271

yes

characteristic

121

C₂

#23752272

yes

—

122

C₂

#23752274

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₁
commutator, socle, prime radical	C₂³×C₁₅₁	C₂²×C₄
Fitting	#336×C₁₅₁	C₂
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#336	D₁₅₁
hypocenter	C₁₅₁	#1840
LCS	G ▷ C₂³×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₁

commutator, socle, prime radical

C₂³×C₁₅₁

C₂²×C₄

Fitting

#336×C₁₅₁

C₂

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
61. C₁₅₁	#1840	semidirect

H ≅ G/N

type

61. C₁₅₁

#1840

semidirect

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

subgroup	index
#23752058	2
#23751967	2
#23751966	2
#336×C₁₅₁	2
#1840	151

subgroup

index

#336×C₁₅₁

151

Subgroups — #34069254

Conjugacy classes of subgroups

Complements

Maximal subgroups