Group #34070258 — subgroups

Subgroups — #34070258

Order 19328.

258 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 87 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×#18012662	C₂	1	2	yes	maximal
2	#23752118	C₂	1	1	yes	maximal, characteristic
3	#23751974	C₂	1	2	yes	maximal
4	#23751967	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₀₂	C₂²	1	1	yes	characteristic
7	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₁₅₁	C₂²	1	1	yes	characteristic
9	#18012662	—	2	4	no	—
10	C₁₅₁⋊#95	—	2	2	no	—
11	#125×C₁₅₁	—	2	4	no	—
12	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₀₂	—	2	2	no	—
16	C₂×Dic₃₀₂	—	2	1	no	—
17	C₂×Dic₃₀₂	—	2	2	no	—
18	C₂×Dic₃₀₂	Q₈	1	2	yes	—
19	C₂²×Dic₁₅₁	—	2	1	no	—
20	C₂²×Dic₁₅₁	—	2	1	no	—
21	C₂×C₁₅₁⋊C₈	—	2	2	no	—
22	C₂×C₁₅₁⋊C₈	—	2	2	no	—
23	C₂×C₁₅₁⋊C₈	—	2	2	no	—
24	C₄⋊Dic₁₅₁	—	2	2	no	—
25	C₂²×C₄×C₁₅₁	—	2	2	no	—
26	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
27	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₁₅₁	—	2	8	no	—
29	C₄⋊C₄×C₁₅₁	—	2	2	no	—
30	C₄⋊C₄×C₁₅₁	D₄	1	4	yes	—
31	C₄²×C₁₅₁	—	2	2	no	—
32	Dic₃₀₂	—	2	4	no	—
33	Dic₃₀₂	—	4	2	no	—
34	C₂×Dic₁₅₁	—	2	2	no	—
35	C₂×Dic₁₅₁	—	4	1	no	—
36	C₂×Dic₁₅₁	—	4	1	no	—
37	C₂×Dic₁₅₁	—	4	1	no	—
38	C₂×Dic₁₅₁	—	4	2	no	—
39	C₁₅₁⋊C₈	—	4	2	no	—
40	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₅₁	—	2	2	no	—
42	C₂×C₄×C₁₅₁	—	2	4	no	—
43	C₂×C₄×C₁₅₁	—	2	2	no	—
44	C₂×C₄×C₁₅₁	—	2	2	no	—
45	C₂×C₄×C₁₅₁	—	4	2	no	—
46	C₂×C₄×C₁₅₁	—	4	2	no	—
47	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₄×C₁₅₁	#41	1	2	yes	—
50	C₂×C₄×C₁₅₁	#41	1	1	yes	characteristic
51	C₂²×C₁₅₁	#120	1	1	yes	characteristic
52	C₂²×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
53	C₂²×C₁₅₁	#124	1	1	yes	characteristic
54	C₂²×C₁₅₁	#135	1	2	yes	—
55	C₂²×C₁₅₁	#137	1	2	yes	—
56	C₄×C₁₅₁	—	4	2	no	—
57	C₄×C₁₅₁	—	4	4	no	—
58	C₄×C₁₅₁	C₂²⋊Q₈	1	2	yes	—
59	C₄×C₁₅₁	#124	1	2	yes	—
60	Dic₁₅₁	—	4	2	no	—
61	Dic₁₅₁	—	8	1	no	—
62	C₂×C₁₅₁	#393	1	1	yes	characteristic
63	C₂×C₁₅₁	#452	1	2	yes	—
64	C₂×C₁₅₁	#479	1	2	yes	—
65	C₂×C₁₅₁	#488	1	1	yes	characteristic
66	C₂×C₁₅₁	#491	1	1	yes	characteristic
67	C₁₅₁	#2026	1	1	yes	characteristic, complemented, Sylow
68	#2026	—	151	1	no	maximal, Sylow
69	C₂×#125	D₁₅₁	1	1	yes	characteristic
70	C₂×#103	—	151	2	no	—
71	#391	—	151	1	no	—
72	#340	—	151	2	no	—
73	#336	—	151	1	no	—
74	C₂²×Q₈	—	151	1	no	—
75	C₂²×C₈	—	151	2	no	—
76	#125	—	2	4	no	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	151	1	no	—
79	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
80	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
81	#103	—	302	4	no	—
82	#95	—	302	2	no	—
83	C₂×Q₈	—	302	1	no	—
84	C₂×Q₈	—	302	2	no	—
85	C₂×Q₈	—	302	2	no	—
86	C₂×Q₈	—	151	2	no	—
87	C₂²×C₄	—	2	2	no	—
88	C₂²×C₄	—	302	1	no	—
89	C₂²×C₄	—	302	1	no	—
90	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
91	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
92	C₂×C₈	—	302	2	no	—
93	C₂×C₈	—	302	2	no	—
94	C₂×C₈	—	302	2	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	302	2	no	—
98	C₄⋊C₄	C₂²⋊D₁₅₁	1	4	yes	—
99	C₄²	—	2	2	no	—
100	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
101	Q₈	—	302	4	no	—
102	Q₈	—	604	2	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	302	2	no	—
110	C₂×C₄	—	604	1	no	—
111	C₂×C₄	—	604	1	no	—
112	C₂×C₄	—	604	1	no	—
113	C₂×C₄	—	604	2	no	—
114	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
115	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
116	C₂×C₄	#14559005	1	2	yes	—
117	C₂×C₄	#14559005	1	1	yes	characteristic
118	C₈	—	604	2	no	—
119	C₂²	#18012708	1	1	yes	characteristic
120	C₂²	#18012715	1	1	yes	characteristic
121	C₂²	#18012728	1	1	yes	characteristic
122	C₂²	#18012759	1	2	yes	—
123	C₂²	#18012769	1	2	yes	—
124	C₄	—	4	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	604	2	no	—
127	C₄	—	1208	1	no	—
128	C₄	#18012715	1	2	yes	—
129	C₄	#18012728	1	2	yes	—
130	C₂	#23752123	1	1	yes	characteristic
131	C₂	#23752374	1	2	yes	—
132	C₂	#23752516	1	2	yes	—
133	C₂	#23752569	1	1	yes	characteristic
134	C₂	#23752582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012662

C₂

yes

maximal

#23752118

C₂

yes

maximal, characteristic

#23751974

C₂

yes

maximal

#23751967

C₂

yes

maximal, characteristic

C₂×#125×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₃₀₂

C₂²

yes

characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

characteristic

#18012662

—

C₁₅₁⋊#95

—

#125×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

—

C₂×Dic₃₀₂

Q₈

yes

—

C₂²×Dic₁₅₁

—

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

—

Dic₃₀₂

—

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

C₂²⋊Q₈

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

—

C₂²×C₁₅₁

#137

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂²⋊Q₈

yes

—

C₄×C₁₅₁

#124

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#393

yes

characteristic

C₂×C₁₅₁

#452

yes

—

C₂×C₁₅₁

#479

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#2026

—

151

maximal, Sylow

C₂×#125

D₁₅₁

yes

characteristic

C₂×#103

—

151

—

#391

—

151

—

#340

—

151

—

#336

—

151

—

C₂²×Q₈

—

151

—

C₂²×C₈

—

151

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#103

—

302

—

#95

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

302

—

C₂×Q₈

—

151

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

C₄²

—

100

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

101

Q₈

—

302

—

102

Q₈

—

604

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

302

—

110

C₂×C₄

—

604

—

111

C₂×C₄

—

604

—

112

C₂×C₄

—

604

—

113

C₂×C₄

—

604

—

114

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

115

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

116

C₂×C₄

#14559005

yes

—

117

C₂×C₄

#14559005

yes

characteristic

118

C₈

—

604

—

119

C₂²

#18012708

yes

characteristic

120

C₂²

#18012715

yes

characteristic

121

C₂²

#18012728

yes

characteristic

122

C₂²

#18012759

yes

—

123

C₂²

#18012769

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

604

—

127

C₄

—

1208

—

128

C₄

#18012715

yes

—

129

C₄

#18012728

yes

—

130

C₂

#23752123

yes

characteristic

131

C₂

#23752374

yes

—

132

C₂

#23752516

yes

—

133

C₂

#23752569

yes

characteristic

134

C₂

#23752582

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator, cosocle	C₂²×C₄×C₁₅₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	C₂×#125×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
hypercenter	C₂×#125	D₁₅₁
hypocenter	C₁₅₁	#2026
LCS	G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator, cosocle

C₂²×C₄×C₁₅₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

C₂×#125×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

D₁₅₁

C₁₅₁

G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
67. C₁₅₁	#2026	semidirect

H ≅ G/N

type

67. C₁₅₁

#2026

semidirect

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

subgroup	index
C₂×#18012662	2
#23752118	2
#23751974	2
#23751967	2
C₂×#125×C₁₅₁	2
#2026	151

subgroup

index

C₂×#125×C₁₅₁

151

Subgroups — #34070258

Conjugacy classes of subgroups

Complements

Maximal subgroups