Group #30675044 — subgroups

Subgroups — #30675044

Order 16768.

356 conjugacy classes of subgroups; only the 177 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 177 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×C₄×Dic₂₆₂	C₂	1	1	yes	maximal, characteristic
2	#22553442	C₂	1	4	yes	maximal, complemented
3	#22553439	C₂	1	4	yes	maximal, complemented
4	#22553366	C₂	1	4	yes	maximal
5	#22553365	C₂	1	1	yes	maximal, characteristic
6	#431×C₁₃₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₂₆₂	C₄	1	1	yes	characteristic
8	C₄×Dic₂₆₂	C₄	1	2	yes	—
9	C₄×Dic₂₆₂	C₄	1	2	yes	—
10	C₄×Dic₂₆₂	C₂²	1	2	yes	—
11	C₄×Dic₂₆₂	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₁₃₁	C₄	1	1	yes	characteristic
13	C₂×Dic₁₃₁⋊C₄	C₄	1	2	yes	—
14	C₂×C₄×Dic₁₃₁	C₂²	1	2	yes	—
15	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
16	Dic₁₃₁⋊C₈	C₂²	1	8	yes	—
17	C₂²⋊(C₁₃₁⋊C₈)	C₂²	1	2	yes	—
18	#16862156	C₂²	1	8	yes	—
19	C₄×C₁₃₁⋊C₈	C₂²	1	2	yes	—
20	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₁	C₂²	1	2	yes	—
22	C₂²⋊C₈×C₁₃₁	C₂²	1	2	yes	—
23	C₄×C₈×C₁₃₁	C₂²	1	2	yes	—
24	C₂×Dic₂₆₂	—	2	4	no	—
25	C₂×Dic₂₆₂	C₂×C₄	1	2	yes	—
26	C₂×Dic₂₆₂	C₂×C₄	1	2	yes	—
27	C₂²×Dic₁₃₁	C₂×C₄	1	2	yes	—
28	C₂²×Dic₁₃₁	C₂×C₄	1	2	yes	—
29	C₂×C₁₃₁⋊C₈	C₂³	1	4	yes	—
30	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
31	C₄⋊Dic₁₃₁	C₂×C₄	1	2	yes	—
32	Dic₁₃₁⋊C₄	C₂×C₄	1	4	yes	—
33	Dic₁₃₁⋊C₄	C₂×C₄	1	4	yes	—
34	C₄×Dic₁₃₁	C₂×C₄	1	4	yes	—
35	C₄×Dic₁₃₁	C₂³	1	4	yes	—
36	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₄×C₁₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
39	C₂×C₈×C₁₃₁	C₂³	1	4	yes	—
40	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
41	C₄²×C₁₃₁	C₂×C₄	1	1	yes	characteristic
42	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
43	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
44	Dic₂₆₂	—	2	4	no	—
45	Dic₂₆₂	—	2	4	no	—
46	C₂×Dic₁₃₁	—	2	4	no	—
47	C₂×Dic₁₃₁	—	4	2	no	—
48	C₂×Dic₁₃₁	C₂²×C₄	1	4	yes	—
49	C₂×Dic₁₃₁	C₂²×C₄	1	4	yes	—
50	C₁₃₁⋊C₈	—	2	4	no	—
51	C₂³×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₁	—	2	2	no	—
53	C₂×C₄×C₁₃₁	—	4	1	no	—
54	C₂×C₄×C₁₃₁	M₁₆	1	4	yes	—
55	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₄×C₁₃₁	C₂²×C₄	1	1	yes	characteristic
60	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
61	C₈×C₁₃₁	—	2	4	no	—
62	C₂²×C₁₃₁	—	2	1	no	—
63	C₂²×C₁₃₁	—	2	1	no	—
64	C₂²×C₁₃₁	C₂×M₁₆	1	2	yes	—
65	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₁₃₁	—	2	1	no	—
67	C₄×C₁₃₁	—	2	1	no	—
68	C₄×C₁₃₁	—	2	1	no	—
69	C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
70	C₄×C₁₃₁	C₂×M₁₆	1	2	yes	—
71	C₄×C₁₃₁	2₋¹⁺⁴	1	2	yes	—
72	Dic₁₃₁	—	2	4	no	—
73	Dic₁₃₁	—	2	4	no	—
74	C₂×C₁₃₁	—	2	2	no	—
75	C₂×C₁₃₁	#519	1	1	yes	characteristic
76	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
77	C₂×C₁₃₁	#568	1	1	yes	characteristic
78	C₁₃₁	#2983	1	1	yes	characteristic, complemented, Sylow
79	#2983	—	131	1	no	maximal, Sylow
80	C₂×C₄×Q₈	—	131	1	no	—
81	#446	—	131	4	no	—
82	#446	—	131	4	no	—
83	#432	—	131	4	no	—
84	#431	—	131	1	no	—
85	#431	D₁₃₁	1	1	yes	characteristic
86	C₂²×Q₈	—	131	1	no	—
87	C₄×Q₈	—	131	2	no	—
88	C₄×Q₈	—	131	2	no	—
89	C₄×Q₈	—	131	2	no	—
90	C₄×Q₈	—	131	2	no	—
91	C₂×C₄⋊C₄	—	131	2	no	—
92	C₂×C₄⋊C₄	—	131	1	no	—
93	C₂×C₄²	—	131	2	no	—
94	C₂×C₄²	C₂×D₁₃₁	1	1	yes	characteristic
95	C₄⋊C₈	—	131	8	no	—
96	C₄⋊C₈	—	131	2	no	—
97	C₄⋊C₈	C₂×D₁₃₁	1	2	yes	—
98	C₂²⋊C₈	—	131	2	no	—
99	C₂²⋊C₈	C₂×D₁₃₁	1	2	yes	—
100	#97	—	131	8	no	—
101	C₄×C₈	—	131	2	no	—
102	C₄×C₈	C₂×D₁₃₁	1	2	yes	—
103	C₂×Q₈	—	262	4	no	—
104	C₂×Q₈	—	131	2	no	—
105	C₂×Q₈	—	131	2	no	—
106	C₂²×C₄	—	131	2	no	—
107	C₂²×C₄	—	131	2	no	—
108	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
109	C₂²×C₄	C₄×D₁₃₁	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₁₃₁	1	1	yes	characteristic
111	C₂×C₈	—	131	4	no	—
112	C₂×C₈	C₂²×D₁₃₁	1	4	yes	—
113	C₄⋊C₄	—	131	4	no	—
114	C₄⋊C₄	—	131	4	no	—
115	C₄⋊C₄	—	131	2	no	—
116	C₄⋊C₄	—	131	2	no	—
117	C₄²	—	131	4	no	—
118	C₄²	—	131	4	no	—
119	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
120	C₄²	C₄×D₁₃₁	1	1	yes	characteristic
121	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
122	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
123	C₂³	C₂×C₄×D₁₃₁	1	1	yes	characteristic
124	Q₈	—	262	4	no	—
125	Q₈	—	262	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	262	4	no	—
129	C₂×C₄	—	524	2	no	—
130	C₂×C₄	—	131	4	no	—
131	C₂×C₄	—	131	4	no	—
132	C₂×C₄	#14523496	1	4	yes	—
133	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
135	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
136	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×D₁₃₁	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	262	4	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	C₂×#14523496	1	2	yes	—
144	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
145	C₄	—	2	1	no	—
146	C₄	—	2	1	no	—
147	C₄	—	2	1	no	—
148	C₄	—	262	4	no	—
149	C₄	—	262	4	no	—
150	C₄	C₂×#14523496	1	2	yes	—
151	C₄	C₂×#14523496	1	2	yes	—
152	C₄	#16862293	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#22553789	1	1	yes	characteristic
155	C₂	C₂²×#14523496	1	1	yes	characteristic
156	C₂	#22554078	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₂₆₂

C₂

yes

maximal, characteristic

#22553442

C₂

yes

maximal, complemented

#22553439

C₂

yes

maximal, complemented

#22553366

C₂

yes

maximal

#22553365

C₂

yes

maximal, characteristic

#431×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₂

C₄

yes

characteristic

C₄×Dic₂₆₂

C₄

yes

—

C₄×Dic₂₆₂

C₄

yes

—

C₄×Dic₂₆₂

C₂²

yes

—

C₄×Dic₂₆₂

C₂²

yes

—

C₂×C₄⋊Dic₁₃₁

C₄

yes

characteristic

C₂×Dic₁₃₁⋊C₄

C₄

yes

—

C₂×C₄×Dic₁₃₁

C₂²

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

Dic₁₃₁⋊C₈

C₂²

yes

—

C₂²⋊(C₁₃₁⋊C₈)

C₂²

yes

—

#16862156

C₂²

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

—

C₂²⋊C₈×C₁₃₁

C₂²

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

—

C₂×Dic₂₆₂

—

C₂×Dic₂₆₂

C₂×C₄

yes

—

C₂×Dic₂₆₂

C₂×C₄

yes

—

C₂²×Dic₁₃₁

C₂×C₄

yes

—

C₂²×Dic₁₃₁

C₂×C₄

yes

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

Dic₁₃₁⋊C₄

C₂×C₄

yes

—

Dic₁₃₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₈×C₁₃₁

C₂³

yes

—

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

C₂×C₄

yes

characteristic

C₄²×C₁₃₁

C₂³

yes

characteristic

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

C₂²×C₄

yes

—

C₂×Dic₁₃₁

C₂²×C₄

yes

—

C₁₃₁⋊C₈

—

C₂³×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

M₁₆

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

—

C₂²×C₁₃₁

C₂×M₁₆

yes

—

C₂²×C₁₃₁

C₂³×C₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×M₁₆

yes

—

C₄×C₁₃₁

C₂×M₁₆

yes

—

C₄×C₁₃₁

2₋¹⁺⁴

yes

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

—

C₂×C₁₃₁

#519

yes

characteristic

C₂×C₁₃₁

C₂²×M₁₆

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#2983

—

131

maximal, Sylow

C₂×C₄×Q₈

—

131

—

#446

—

131

—

#446

—

131

—

#432

—

131

—

#431

—

131

—

#431

D₁₃₁

yes

characteristic

C₂²×Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

—

131

—

C₂×C₄⋊C₄

—

131

—

C₂×C₄⋊C₄

—

131

—

C₂×C₄²

—

131

—

C₂×C₄²

C₂×D₁₃₁

yes

characteristic

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

C₂×D₁₃₁

yes

—

C₂²⋊C₈

—

131

—

C₂²⋊C₈

C₂×D₁₃₁

yes

—

100

#97

—

131

—

101

C₄×C₈

—

131

—

102

C₄×C₈

C₂×D₁₃₁

yes

—

103

C₂×Q₈

—

262

—

104

C₂×Q₈

—

131

—

105

C₂×Q₈

—

131

—

106

C₂²×C₄

—

131

—

107

C₂²×C₄

—

131

—

108

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

109

C₂²×C₄

C₄×D₁₃₁

yes

characteristic

110

C₂²×C₄

C₂²×D₁₃₁

yes

characteristic

111

C₂×C₈

—

131

—

112

C₂×C₈

C₂²×D₁₃₁

yes

—

113

C₄⋊C₄

—

131

—

114

C₄⋊C₄

—

131

—

115

C₄⋊C₄

—

131

—

116

C₄⋊C₄

—

131

—

117

C₄²

—

131

—

118

C₄²

—

131

—

119

C₄²

C₄×D₁₃₁

yes

characteristic

120

C₄²

C₄×D₁₃₁

yes

characteristic

121

C₄²

C₂²×D₁₃₁

yes

characteristic

122

C₄²

C₂²×D₁₃₁

yes

characteristic

123

C₂³

C₂×C₄×D₁₃₁

yes

characteristic

124

Q₈

—

262

—

125

Q₈

—

262

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

262

—

129

C₂×C₄

—

524

—

130

C₂×C₄

—

131

—

131

C₂×C₄

—

131

—

132

C₂×C₄

#14523496

yes

—

133

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

134

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

135

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

136

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

137

C₂×C₄

C₂×C₄×D₁₃₁

yes

characteristic

138

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

139

C₈

—

140

C₈

—

262

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×#14523496

yes

—

144

C₂²

C₂²×C₄×D₁₃₁

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

262

—

149

C₄

—

262

—

150

C₄

C₂×#14523496

yes

—

151

C₄

C₂×#14523496

yes

—

152

C₄

#16862293

yes

—

153

C₂

—

154

C₂

#22553789

yes

characteristic

155

C₂

C₂²×#14523496

yes

characteristic

156

C₂

#22554078

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	#431×C₁₃₁	C₂
prime radical	C₂³×C₁₃₁	C₂²×C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	#431	D₁₃₁
hypocenter	C₁₃₁	#2983
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ #431

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂³×C₄

Fitting

#431×C₁₃₁

C₂

prime radical

C₂³×C₁₃₁

C₂²×C₄

cosocle

C₂×C₄×C₁₃₁

C₂⁴

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂×C₄ ◁ #431

N	H ≅ G/N	type
2. #22553442	C₂	semidirect
3. #22553439	C₂	semidirect
78. C₁₃₁	#2983	semidirect

H ≅ G/N

type

2. #22553442

C₂

semidirect

3. #22553439

C₂

semidirect

78. C₁₃₁

#2983

semidirect

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

subgroup	index
C₂×C₄×Dic₂₆₂	2
#22553442	2
#22553439	2
#22553366	2
#22553365	2
#431×C₁₃₁	2
#2983	131

subgroup

index

C₂×C₄×Dic₂₆₂

#431×C₁₃₁

131

Subgroups — #30675044

Conjugacy classes of subgroups

Complements

Maximal subgroups