Group #30675092 — subgroups

Subgroups — #30675092

Order 16768.

394 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₄×#14523514	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#22553391	C₂	1	3	yes	maximal, complemented
4	#22553389	C₂	1	3	yes	maximal, complemented
5	#22553363	C₂	1	3	yes	maximal, complemented
6	#22553334	C₂	1	3	yes	maximal, complemented
7	C₈×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14523514	C₄	1	1	yes	characteristic
9	C₂×#14523496	C₂²	1	6	yes	—
10	Q₈×Dic₁₃₁	C₄	1	1	yes	characteristic
11	C₄×D₅₂₄	C₂²	1	3	yes	—
12	#16862208	C₄	1	3	yes	—
13	#16862204	C₄	1	3	yes	—
14	C₄²×D₁₃₁	C₂²	1	3	yes	—
15	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	3	yes	—
16	Dic₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
17	Dic₁₃₁⋊C₈	C₂²	1	3	yes	—
18	(C₂×D₁₃₁)⋊C₈	C₂²	1	3	yes	—
19	(C₂×D₁₃₁)⋊C₈	C₂²	1	3	yes	—
20	C₄×C₁₃₁⋊C₈	C₂²	1	3	yes	—
21	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	C₄⋊C₈×C₁₃₁	C₂²	1	3	yes	—
23	C₄×C₈×C₁₃₁	C₂²	1	3	yes	—
24	#14523514	—	2	4	no	—
25	C₂×D₅₂₄	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₁₃₁	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₁₃₁	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₁₃₁	C₂³	1	3	yes	—
29	#14523496	—	4	6	no	—
30	C₂×C₁₃₁⋊C₈	C₂³	1	3	yes	—
31	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₁₃₁	C₂×C₄	1	3	yes	—
33	(C₂×D₁₃₁)⋊C₄	C₂×C₄	1	6	yes	—
34	C₄×Dic₁₃₁	C₂×C₄	1	3	yes	—
35	C₄×Dic₁₃₁	C₂×C₄	1	3	yes	—
36	C₄×Dic₁₃₁	C₂³	1	1	yes	characteristic
37	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₈×C₁₃₁	C₂³	1	3	yes	—
39	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
40	C₄⋊C₄×C₁₃₁	C₂×C₄	1	3	yes	—
41	C₄²×C₁₃₁	C₂³	1	3	yes	—
42	C₂²×D₁₃₁	C₂²×C₄	1	3	yes	—
43	D₅₂₄	—	2	6	no	—
44	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₁₃₁	C₂²×C₄	1	3	yes	—
46	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
47	C₄×D₁₃₁	—	2	6	no	—
48	C₄×D₁₃₁	—	2	3	no	—
49	C₄×D₁₃₁	—	2	3	no	—
50	C₄×D₁₃₁	—	2	6	no	—
51	C₁₃₁⋊C₈	—	2	3	no	—
52	C₁₃₁⋊C₈	—	2	1	no	—
53	Q₈×C₁₃₁	M₁₆	1	4	yes	complemented
54	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
55	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
56	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
57	C₈×C₁₃₁	—	2	3	no	—
58	C₈×C₁₃₁	—	2	1	no	—
59	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
60	C₂×D₁₃₁	—	2	3	no	—
61	C₂×D₁₃₁	—	2	3	no	—
62	C₂×D₁₃₁	—	2	3	no	—
63	C₄×C₁₃₁	—	2	3	no	—
64	C₄×C₁₃₁	C₂×M₁₆	1	6	yes	—
65	C₄×C₁₃₁	H(5,𝔽₂)	1	1	yes	characteristic
66	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
67	Dic₁₃₁	—	2	3	no	—
68	Dic₁₃₁	—	2	1	no	—
69	Dic₁₃₁	#131	1	2	yes	—
70	C₂×C₁₃₁	#520	1	1	yes	characteristic
71	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
72	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
73	D₁₃₁	—	4	3	no	—
74	C₁₃₁	#2991	1	1	yes	characteristic, complemented, Sylow
75	#2991	—	131	1	no	maximal, Sylow
76	C₄×#41	—	131	1	no	—
77	C₈×Q₈	—	131	1	no	—
78	C₈×Q₈	D₁₃₁	1	1	yes	characteristic, complemented
79	#436	—	131	3	no	—
80	#436	—	131	3	no	—
81	#431	—	131	3	no	—
82	#423	—	131	3	no	—
83	C₂×#41	—	131	1	no	—
84	C₂×M₁₆	—	131	6	no	—
85	C₄×Q₈	—	131	1	no	—
86	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
87	C₄×D₄	—	131	3	no	—
88	C₄×D₄	—	131	3	no	—
89	#117	—	131	3	no	—
90	C₂×C₄²	—	131	3	no	—
91	C₄⋊C₈	—	131	3	no	—
92	C₄⋊C₈	—	131	1	no	—
93	C₄⋊C₈	—	131	3	no	—
94	C₄⋊C₈	C₂×D₁₃₁	1	3	yes	—
95	C₂²⋊C₈	—	131	3	no	—
96	C₂²⋊C₈	—	131	3	no	—
97	C₄×C₈	—	131	3	no	—
98	C₄×C₈	C₂×D₁₃₁	1	3	yes	—
99	#41	—	262	4	no	—
100	C₂×Q₈	C₄×D₁₃₁	1	1	yes	characteristic
101	C₂×D₄	—	131	3	no	—
102	C₂²×C₄	—	131	3	no	—
103	C₂²×C₄	—	131	3	no	—
104	C₂²×C₄	—	131	3	no	—
105	M₁₆	—	524	6	no	—
106	C₂×C₈	—	131	1	no	—
107	C₂×C₈	—	131	3	no	—
108	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₁₃₁	1	3	yes	—
110	C₄⋊C₄	—	131	3	no	—
111	C₄⋊C₄	C₄×D₁₃₁	1	3	yes	—
112	C₂²⋊C₄	—	131	6	no	—
113	C₄²	—	131	3	no	—
114	C₄²	—	131	3	no	—
115	C₄²	—	131	1	no	—
116	C₄²	C₂²×D₁₃₁	1	3	yes	—
117	C₂³	—	131	3	no	—
118	Q₈	#14523496	1	4	yes	complemented
119	D₄	—	262	6	no	—
120	C₂×C₄	—	262	6	no	—
121	C₂×C₄	—	262	6	no	—
122	C₂×C₄	—	262	3	no	—
123	C₂×C₄	—	262	3	no	—
124	C₂×C₄	—	131	1	no	—
125	C₂×C₄	—	131	3	no	—
126	C₂×C₄	—	131	1	no	—
127	C₂×C₄	C₂×C₄×D₁₃₁	1	3	yes	—
128	C₂×C₄	C₂×C₄×D₁₃₁	1	3	yes	—
129	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
130	C₈	—	2	3	no	—
131	C₈	—	2	1	no	—
132	C₈	—	262	3	no	—
133	C₈	—	262	1	no	—
134	C₂²	—	262	3	no	—
135	C₂²	—	262	3	no	—
136	C₂²	—	262	3	no	—
137	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	262	3	no	—
140	C₄	—	262	1	no	—
141	C₄	—	131	2	no	—
142	C₄	C₂×#14523496	1	6	yes	—
143	C₄	#16862292	1	1	yes	characteristic
144	C₄	#16862294	1	1	yes	characteristic
145	C₂	—	524	3	no	—
146	C₂	#22553794	1	1	yes	characteristic
147	C₂	C₂²×#14523496	1	1	yes	characteristic
148	C₂	#131×D₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#14523514

C₂

yes

maximal, characteristic

Q₈×C₁₃₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#22553391

C₂

yes

maximal, complemented

#22553389

C₂

yes

maximal, complemented

#22553363

C₂

yes

maximal, complemented

#22553334

C₂

yes

maximal, complemented

C₈×Q₈×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×#14523514

C₄

yes

characteristic

C₂×#14523496

C₂²

yes

—

Q₈×Dic₁₃₁

C₄

yes

characteristic

C₄×D₅₂₄

C₂²

yes

—

#16862208

C₄

yes

—

#16862204

C₄

yes

—

C₄²×D₁₃₁

C₂²

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

Dic₁₃₁⋊C₈

C₂²

yes

characteristic

Dic₁₃₁⋊C₈

C₂²

yes

—

(C₂×D₁₃₁)⋊C₈

C₂²

yes

—

(C₂×D₁₃₁)⋊C₈

C₂²

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

—

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

—

#14523514

—

C₂×D₅₂₄

C₂×C₄

yes

—

C₂×C₄×D₁₃₁

C₂×C₄

yes

—

C₂×C₄×D₁₃₁

C₂×C₄

yes

—

C₂×C₄×D₁₃₁

C₂³

yes

—

#14523496

—

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₂×C₁₃₁⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

(C₂×D₁₃₁)⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂³

yes

characteristic

C₂×Q₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

C₂³

yes

—

C₂×C₈×C₁₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₁

C₂×C₄

yes

—

C₄²×C₁₃₁

C₂³

yes

—

C₂²×D₁₃₁

C₂²×C₄

yes

—

D₅₂₄

—

C₂×Dic₁₃₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₃₁

C₂²×C₄

yes

—

C₂×Dic₁₃₁

C₂²×C₄

yes

characteristic

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

M₁₆

yes

complemented

C₂×C₄×C₁₃₁

C₂²×C₄

yes

—

C₂×C₄×C₁₃₁

C₂²×C₄

yes

—

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₈×C₁₃₁

—

C₈×C₁₃₁

—

C₂²×C₁₃₁

C₂³×C₄

yes

characteristic

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₂×D₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×M₁₆

yes

—

C₄×C₁₃₁

H(5,𝔽₂)

yes

characteristic

C₄×C₁₃₁

2₋¹⁺⁴

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

#131

yes

—

C₂×C₁₃₁

#520

yes

characteristic

C₂×C₁₃₁

C₂²×M₁₆

yes

characteristic

C₂×C₁₃₁

C₂×#131

yes

characteristic

D₁₃₁

—

C₁₃₁

#2991

yes

characteristic, complemented, Sylow

#2991

—

131

maximal, Sylow

C₄×#41

—

131

—

C₈×Q₈

—

131

—

C₈×Q₈

D₁₃₁

yes

characteristic, complemented

#436

—

131

—

#436

—

131

—

#431

—

131

—

#423

—

131

—

C₂×#41

—

131

—

C₂×M₁₆

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

C₄×D₄

—

131

—

C₄×D₄

—

131

—

#117

—

131

—

C₂×C₄²

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

C₂×D₁₃₁

yes

—

C₂²⋊C₈

—

131

—

C₂²⋊C₈

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

—

#41

—

262

—

100

C₂×Q₈

C₄×D₁₃₁

yes

characteristic

101

C₂×D₄

—

131

—

102

C₂²×C₄

—

131

—

103

C₂²×C₄

—

131

—

104

C₂²×C₄

—

131

—

105

M₁₆

—

524

—

106

C₂×C₈

—

131

—

107

C₂×C₈

—

131

—

108

C₂×C₈

C₂²×D₁₃₁

yes

characteristic

109

C₂×C₈

C₂²×D₁₃₁

yes

—

110

C₄⋊C₄

—

131

—

111

C₄⋊C₄

C₄×D₁₃₁

yes

—

112

C₂²⋊C₄

—

131

—

113

C₄²

—

131

—

114

C₄²

—

131

—

115

C₄²

—

131

—

116

C₄²

C₂²×D₁₃₁

yes

—

117

C₂³

—

131

—

118

Q₈

#14523496

yes

complemented

119

D₄

—

262

—

120

C₂×C₄

—

262

—

121

C₂×C₄

—

262

—

122

C₂×C₄

—

262

—

123

C₂×C₄

—

262

—

124

C₂×C₄

—

131

—

125

C₂×C₄

—

131

—

126

C₂×C₄

—

131

—

127

C₂×C₄

C₂×C₄×D₁₃₁

yes

—

128

C₂×C₄

C₂×C₄×D₁₃₁

yes

—

129

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

130

C₈

—

131

C₈

—

132

C₈

—

262

—

133

C₈

—

262

—

134

C₂²

—

262

—

135

C₂²

—

262

—

136

C₂²

—

262

—

137

C₂²

C₂²×C₄×D₁₃₁

yes

characteristic

138

C₄

—

139

C₄

—

262

—

140

C₄

—

262

—

141

C₄

—

131

—

142

C₄

C₂×#14523496

yes

—

143

C₄

#16862292

yes

characteristic

144

C₄

#16862294

yes

characteristic

145

C₂

—

524

—

146

C₂

#22553794

yes

characteristic

147

C₂

C₂²×#14523496

yes

characteristic

148

C₂

#131×D₁₃₁

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₃₁
commutator, socle	C₂²×C₁₃₁	C₂³×C₄
Fitting	C₈×Q₈×C₁₃₁	C₂
prime radical	C₂×#14523514	C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	C₈×Q₈	D₁₃₁
hypocenter	C₁₃₁	#2991
LCS	G ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×Q₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₁₃₁

commutator, socle

C₂²×C₁₃₁

C₂³×C₄

Fitting

C₈×Q₈×C₁₃₁

C₂

prime radical

C₂×#14523514

C₄

cosocle

C₂×C₄×C₁₃₁

C₂⁴

hypercenter

C₈×Q₈

D₁₃₁

C₁₃₁

G ▷ C₂²×C₁₃₁ ▷ C₁₃₁

C₁ ◁ C₂×C₄ ◁ C₈×Q₈

N	H ≅ G/N	type
2. Q₈×C₁₃₁⋊C₈	C₂	semidirect
3. #22553391	C₂	semidirect
4. #22553389	C₂	semidirect
5. #22553363	C₂	semidirect
6. #22553334	C₂	semidirect
7. C₈×Q₈×C₁₃₁	C₂	semidirect
53. Q₈×C₁₃₁	M₁₆	semidirect
74. C₁₃₁	#2991	semidirect
78. C₈×Q₈	D₁₃₁	semidirect
118. Q₈	#14523496	semidirect

H ≅ G/N

type

2. Q₈×C₁₃₁⋊C₈

C₂

semidirect

3. #22553391

C₂

semidirect

4. #22553389

C₂

semidirect

5. #22553363

C₂

semidirect

6. #22553334

C₂

semidirect

7. C₈×Q₈×C₁₃₁

C₂

semidirect

53. Q₈×C₁₃₁

M₁₆

semidirect

74. C₁₃₁

#2991

semidirect

78. C₈×Q₈

D₁₃₁

semidirect

118. Q₈

#14523496

semidirect

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

subgroup	index
C₄×#14523514	2
Q₈×C₁₃₁⋊C₈	2
#22553391	2
#22553389	2
#22553363	2
#22553334	2
C₈×Q₈×C₁₃₁	2
#2991	131

subgroup

index

Q₈×C₁₃₁⋊C₈

C₈×Q₈×C₁₃₁

131

Subgroups — #30675092

Conjugacy classes of subgroups

Complements

Maximal subgroups