Group #34068740 — subgroups

Subgroups — #34068740

Order 19328.

220 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 61 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	#23752608	C₂	1	1	yes	maximal, characteristic, complemented
2	#23752588	C₂	1	1	yes	maximal, characteristic
3	#23752586	C₂	1	1	yes	maximal, characteristic, complemented
4	#23752585	C₂	1	1	yes	maximal, characteristic, complemented
5	#23752558	C₂	1	1	yes	maximal, characteristic, complemented
6	#23751934	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14558993	—	2	2	no	—
9	C₂×#14558991	—	2	2	no	—
10	C₂×#14558989	—	2	2	no	—
11	#18012739	C₂²	1	1	yes	characteristic
12	#18012736	C₂²	1	1	yes	characteristic
13	#18012670	—	2	2	no	—
14	#18012660	—	2	2	no	—
15	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₁₅₁	C₂²	1	1	yes	characteristic
18	#127×C₁₅₁	C₂²	1	1	yes	characteristic
19	#102×C₁₅₁	—	2	2	no	—
20	C₄×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₂×Dic₃₀₂	—	2	1	no	—
22	C₂×D₆₀₄	—	2	1	no	—
23	C₂×D₆₀₄	—	2	1	no	—
24	C₂×D₆₀₄	—	2	1	no	—
25	#14558993	—	4	4	no	—
26	#14558991	—	4	4	no	—
27	#14558989	—	4	4	no	—
28	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
29	C₂×C₁₅₁⋊C₈	D₄	1	2	yes	—
30	C₄⋊Dic₁₅₁	—	2	1	no	—
31	C₄⋊Dic₁₅₁	—	2	1	no	—
32	C₂×Q₈×C₁₅₁	—	2	1	no	—
33	C₂×D₄×C₁₅₁	—	2	1	no	—
34	C₂×D₄×C₁₅₁	—	2	1	no	—
35	C₂×D₄×C₁₅₁	—	2	1	no	—
36	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
37	C₄⋊C₄×C₁₅₁	—	2	1	no	—
38	C₄⋊C₄×C₁₅₁	—	2	1	no	—
39	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
40	C₂²×D₁₅₁	—	4	1	no	—
41	Dic₃₀₂	—	4	2	no	—
42	D₆₀₄	—	4	2	no	—
43	D₆₀₄	—	4	2	no	—
44	D₆₀₄	—	4	2	no	—
45	C₂×Dic₁₅₁	—	4	1	no	—
46	C₁₅₁⋊C₈	—	2	2	no	—
47	C₁₅₁⋊C₈	SD₁₆	1	4	yes	complemented
48	C₂³×C₁₅₁	—	4	1	no	—
49	Q₈×C₁₅₁	—	4	2	no	—
50	D₄×C₁₅₁	—	4	2	no	—
51	D₄×C₁₅₁	—	4	2	no	—
52	D₄×C₁₅₁	—	4	2	no	—
53	C₂×C₄×C₁₅₁	—	4	1	no	—
54	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
57	C₈×C₁₅₁	—	2	2	no	—
58	C₂²×C₁₅₁	—	8	1	no	—
59	C₂²×C₁₅₁	—	8	1	no	—
60	C₂²×C₁₅₁	—	8	1	no	—
61	C₂²×C₁₅₁	#127	1	1	yes	characteristic
62	C₂×D₁₅₁	—	8	1	no	—
63	C₂×D₁₅₁	—	8	1	no	—
64	C₂×D₁₅₁	—	8	1	no	—
65	C₄×C₁₅₁	—	8	1	no	—
66	C₄×C₁₅₁	C₂×SD₁₆	1	2	yes	—
67	C₄×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
69	Dic₁₅₁	—	8	1	no	—
70	C₂×C₁₅₁	—	16	1	no	—
71	C₂×C₁₅₁	#492	1	1	yes	characteristic
72	C₂×C₁₅₁	#496	1	1	yes	characteristic
73	C₂×C₁₅₁	#496	1	1	yes	characteristic
74	D₁₅₁	—	16	1	no	—
75	C₁₅₁	#1715	1	1	yes	characteristic, complemented, Sylow
76	#1715	—	151	1	no	maximal, Sylow
77	#498	—	151	1	no	—
78	#493	—	151	1	no	—
79	#492	—	151	1	no	—
80	#492	—	151	1	no	—
81	#486	—	151	1	no	—
82	#486	D₁₅₁	1	1	yes	characteristic, complemented
83	#322	—	151	1	no	—
84	C₂×SD₁₆	—	302	2	no	—
85	C₂×SD₁₆	—	302	2	no	—
86	C₂×D₈	—	302	2	no	—
87	C₄⋊Q₈	—	151	1	no	—
88	C₄⋊Q₈	C₂×D₁₅₁	1	1	yes	characteristic
89	#127	—	151	1	no	—
90	#127	C₂×D₁₅₁	1	1	yes	characteristic
91	#106	—	302	2	no	—
92	#102	—	2	2	no	—
93	#102	—	302	2	no	—
94	C₄×C₈	—	151	1	no	—
95	C₄×C₈	—	151	1	no	—
96	C₄×C₈	C₂×D₁₅₁	1	1	yes	characteristic
97	C₂×Q₈	—	2	1	no	—
98	C₂×Q₈	—	302	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	302	1	no	—
103	C₂×D₄	—	302	1	no	—
104	C₂×D₄	—	302	1	no	—
105	SD₁₆	—	604	4	no	—
106	SD₁₆	—	604	4	no	—
107	D₈	—	604	4	no	—
108	C₂×C₈	—	151	2	no	—
109	C₂×C₈	—	151	2	no	—
110	C₂×C₈	C₂²⋊D₁₅₁	1	2	yes	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄⋊C₄	—	302	1	no	—
114	C₄⋊C₄	—	302	1	no	—
115	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
116	C₂³	—	4	1	no	—
117	C₂³	—	604	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	604	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	604	2	no	—
124	D₄	—	604	2	no	—
125	D₄	—	604	2	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	604	1	no	—
128	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
129	C₂×C₄	D₄×D₁₅₁	1	1	yes	characteristic
130	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
131	C₈	—	2	2	no	—
132	C₈	—	302	2	no	—
133	C₈	—	151	4	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	1208	1	no	—
138	C₂²	—	1208	1	no	—
139	C₂²	—	1208	1	no	—
140	C₂²	#18012735	1	1	yes	characteristic
141	C₄	—	8	1	no	—
142	C₄	—	1208	1	no	—
143	C₄	SD₁₆×D₁₅₁	1	2	yes	—
144	C₄	#18012763	1	2	yes	—
145	C₄	#18012764	1	2	yes	—
146	C₂	—	16	1	no	—
147	C₂	—	2416	1	no	—
148	C₂	#23752583	1	1	yes	characteristic
149	C₂	#23752600	1	1	yes	characteristic
150	C₂	#23752602	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752608

C₂

yes

maximal, characteristic, complemented

#23752588

C₂

yes

maximal, characteristic

#23752586

C₂

yes

maximal, characteristic, complemented

#23752585

C₂

yes

maximal, characteristic, complemented

#23752558

C₂

yes

maximal, characteristic, complemented

#23751934

C₂

yes

maximal, characteristic, complemented

#486×C₁₅₁

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

characteristic

#18012736

C₂²

yes

characteristic

#18012670

—

#18012660

—

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₅₁

C₂²

yes

characteristic

#127×C₁₅₁

C₂²

yes

characteristic

#102×C₁₅₁

—

C₄×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×Dic₃₀₂

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

C₂×D₆₀₄

—

#14558993

—

#14558991

—

#14558989

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₂×C₁₅₁⋊C₈

D₄

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂×Q₈×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×D₄×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

C₂²×D₁₅₁

—

Dic₃₀₂

—

D₆₀₄

—

D₆₀₄

—

D₆₀₄

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

SD₁₆

yes

complemented

C₂³×C₁₅₁

—

Q₈×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

D₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

#127

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×SD₁₆

yes

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

C₄×C₁₅₁

AGL(1,ℤ/8)

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

D₁₅₁

—

C₁₅₁

#1715

yes

characteristic, complemented, Sylow

#1715

—

151

maximal, Sylow

#498

—

151

—

#493

—

151

—

#492

—

151

—

#492

—

151

—

#486

—

151

—

#486

D₁₅₁

yes

characteristic, complemented

#322

—

151

—

C₂×SD₁₆

—

302

—

C₂×SD₁₆

—

302

—

C₂×D₈

—

302

—

C₄⋊Q₈

—

151

—

C₄⋊Q₈

C₂×D₁₅₁

yes

characteristic

#127

—

151

—

#127

C₂×D₁₅₁

yes

characteristic

#106

—

302

—

#102

—

#102

—

302

—

C₄×C₈

—

151

—

C₄×C₈

—

151

—

C₄×C₈

C₂×D₁₅₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

302

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

302

—

103

C₂×D₄

—

302

—

104

C₂×D₄

—

302

—

105

SD₁₆

—

604

—

106

SD₁₆

—

604

—

107

D₈

—

604

—

108

C₂×C₈

—

151

—

109

C₂×C₈

—

151

—

110

C₂×C₈

C₂²⋊D₁₅₁

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

302

—

114

C₄⋊C₄

—

302

—

115

C₄²

C₂²×D₁₅₁

yes

characteristic

116

C₂³

—

117

C₂³

—

604

—

118

Q₈

—

119

Q₈

—

604

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

604

—

124

D₄

—

604

—

125

D₄

—

604

—

126

C₂×C₄

—

127

C₂×C₄

—

604

—

128

C₂×C₄

D₄×D₁₅₁

yes

characteristic

129

C₂×C₄

D₄×D₁₅₁

yes

characteristic

130

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

131

C₈

—

132

C₈

—

302

—

133

C₈

—

151

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

1208

—

138

C₂²

—

1208

—

139

C₂²

—

1208

—

140

C₂²

#18012735

yes

characteristic

141

C₄

—

142

C₄

—

1208

—

143

C₄

SD₁₆×D₁₅₁

yes

—

144

C₄

#18012763

yes

—

145

C₄

#18012764

yes

—

146

C₂

—

147

C₂

—

2416

—

148

C₂

#23752583

yes

characteristic

149

C₂

#23752600

yes

characteristic

150

C₂

#23752602

yes

characteristic

	subgroup	G/H
center	C₂²	#18012735
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#486×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#127
prime radical	#23752588	C₂
hypercenter	#486	D₁₅₁
hypocenter	C₁₅₁	#1715
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#18012735

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#486×C₁₅₁

C₂

C₂²×C₁₅₁

C₂

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #23752608	C₂	semidirect
3. #23752586	C₂	semidirect
4. #23752585	C₂	semidirect
5. #23752558	C₂	semidirect
6. #23751934	C₂	semidirect
7. #486×C₁₅₁	C₂	semidirect
47. C₁₅₁⋊C₈	SD₁₆	semidirect
75. C₁₅₁	#1715	semidirect
82. #486	D₁₅₁	semidirect

H ≅ G/N

type

1. #23752608

C₂

semidirect

3. #23752586

C₂

semidirect

4. #23752585

C₂

semidirect

5. #23752558

C₂

semidirect

6. #23751934

C₂

semidirect

7. #486×C₁₅₁

C₂

semidirect

47. C₁₅₁⋊C₈

SD₁₆

semidirect

75. C₁₅₁

#1715

semidirect

82. #486

D₁₅₁

semidirect

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

subgroup	index
#23752608	2
#23752588	2
#23752586	2
#23752585	2
#23752558	2
#23751934	2
#486×C₁₅₁	2
#1715	151

subgroup

index

#486×C₁₅₁

151

Subgroups — #34068740

Conjugacy classes of subgroups

Complements

Maximal subgroups