Group #30675049 — subgroups

Subgroups — #30675049

Order 16768.

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

normal subgroups only

This group has 364 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×Q₈×D₁₃₁	C₂	1	1	yes	maximal, characteristic
2	#22553442	C₂	1	3	yes	maximal, complemented
3	#22553439	C₂	1	3	yes	maximal, complemented
4	Q₈×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	#22553368	C₂	1	3	yes	maximal
6	#22553363	C₂	1	3	yes	maximal
7	C₈×Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₁₃₁	C₄	1	1	yes	characteristic
9	Q₈×Dic₁₃₁	C₄	1	1	yes	characteristic
10	C₄×Dic₂₆₂	C₂²	1	3	yes	—
11	#16862211	C₄	1	3	yes	—
12	C₄⋊C₄×D₁₃₁	C₄	1	3	yes	—
13	C₄²×D₁₃₁	C₂²	1	3	yes	—
14	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	3	yes	—
15	Dic₁₃₁⋊C₈	C₂²	1	3	yes	—
16	Dic₁₃₁⋊C₈	C₂²	1	3	yes	—
17	(C₂×D₁₃₁)⋊C₈	C₂²	1	1	yes	characteristic
18	(C₂×D₁₃₁)⋊C₈	C₂²	1	3	yes	—
19	#16862156	C₂²	1	6	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	C₂×Dic₂₆₂	C₂×C₄	1	3	yes	—
25	Q₈×D₁₃₁	—	2	4	no	—
26	C₂×C₄×D₁₃₁	C₂×C₄	1	3	yes	—
27	C₂×C₄×D₁₃₁	C₂×C₄	1	3	yes	—
28	C₂×C₄×D₁₃₁	C₂³	1	1	yes	characteristic
29	C₂×C₁₃₁⋊C₈	C₂³	1	1	yes	characteristic
30	C₂×C₁₃₁⋊C₈	C₂³	1	3	yes	—
31	C₄⋊Dic₁₃₁	C₂×C₄	1	3	yes	—
32	Dic₁₃₁⋊C₄	C₂×C₄	1	6	yes	—
33	C₄×Dic₁₃₁	C₂×C₄	1	3	yes	—
34	C₄×Dic₁₃₁	C₂×C₄	1	3	yes	—
35	C₄×Dic₁₃₁	C₂³	1	3	yes	—
36	C₂×Q₈×C₁₃₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₈×C₁₃₁	C₂³	1	3	yes	—
39	C₄⋊C₄×C₁₃₁	C₂×C₄	1	3	yes	—
40	C₄²×C₁₃₁	C₂³	1	3	yes	—
41	C₂²×D₁₃₁	C₂²×C₄	1	1	yes	characteristic
42	Dic₂₆₂	—	2	6	no	—
43	C₂×Dic₁₃₁	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₁₃₁	C₂²×C₄	1	3	yes	—
45	C₂×Dic₁₃₁	C₂²×C₄	1	3	yes	—
46	C₄×D₁₃₁	—	2	6	no	—
47	C₄×D₁₃₁	—	2	6	no	—
48	C₄×D₁₃₁	—	2	1	no	—
49	C₄×D₁₃₁	—	2	1	no	—
50	C₁₃₁⋊C₈	—	2	1	no	—
51	C₁₃₁⋊C₈	—	2	3	no	—
52	Q₈×C₁₃₁	M₁₆	1	4	yes	—
53	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
54	C₂×C₄×C₁₃₁	C₂²×C₄	1	3	yes	—
55	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
56	C₈×C₁₃₁	—	2	3	no	—
57	C₈×C₁₃₁	—	2	1	no	—
58	C₂²×C₁₃₁	C₂³×C₄	1	1	yes	characteristic
59	C₂×D₁₃₁	—	2	1	no	—
60	C₂×D₁₃₁	—	2	1	no	—
61	C₂×D₁₃₁	#131	1	2	yes	—
62	C₄×C₁₃₁	—	2	3	no	—
63	C₄×C₁₃₁	C₂×M₁₆	1	6	yes	—
64	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
65	C₄×C₁₃₁	2₋¹⁺⁴	1	1	yes	characteristic
66	Dic₁₃₁	—	2	1	no	—
67	Dic₁₃₁	—	2	3	no	—
68	Dic₁₃₁	—	2	3	no	—
69	C₂×C₁₃₁	#519	1	1	yes	characteristic
70	C₂×C₁₃₁	C₂²×M₁₆	1	1	yes	characteristic
71	C₂×C₁₃₁	C₂×#131	1	1	yes	characteristic
72	D₁₃₁	—	2	2	no	—
73	C₁₃₁	#2984	1	1	yes	characteristic, complemented, Sylow
74	#2984	—	131	1	no	maximal, Sylow
75	C₂×C₄×Q₈	—	131	1	no	—
76	#446	—	131	3	no	—
77	#446	—	131	3	no	—
78	C₈×Q₈	—	131	1	no	—
79	C₈×Q₈	D₁₃₁	1	1	yes	characteristic, complemented
80	#432	—	131	3	no	—
81	#431	—	131	3	no	—
82	C₂²×Q₈	—	131	1	no	—
83	C₄×Q₈	—	131	1	no	—
84	C₄×Q₈	—	131	3	no	—
85	C₄×Q₈	—	131	3	no	—
86	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
87	C₂×C₄⋊C₄	—	131	3	no	—
88	C₂×C₄²	—	131	3	no	—
89	C₄⋊C₈	—	131	3	no	—
90	C₄⋊C₈	—	131	3	no	—
91	C₄⋊C₈	—	131	3	no	—
92	C₄⋊C₈	C₂×D₁₃₁	1	3	yes	—
93	C₂²⋊C₈	—	131	1	no	—
94	C₂²⋊C₈	—	131	3	no	—
95	#97	—	131	6	no	—
96	C₄×C₈	—	131	3	no	—
97	C₄×C₈	C₂×D₁₃₁	1	3	yes	—
98	C₂×Q₈	—	262	4	no	—
99	C₂×Q₈	—	131	3	no	—
100	C₂×Q₈	C₄×D₁₃₁	1	1	yes	characteristic
101	C₂²×C₄	—	131	3	no	—
102	C₂²×C₄	—	131	3	no	—
103	C₂²×C₄	—	131	1	no	—
104	C₂×C₈	—	131	1	no	—
105	C₂×C₈	—	131	3	no	—
106	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₁₃₁	1	3	yes	—
108	C₄⋊C₄	—	131	6	no	—
109	C₄⋊C₄	—	131	3	no	—
110	C₄⋊C₄	C₄×D₁₃₁	1	3	yes	—
111	C₄²	—	131	3	no	—
112	C₄²	—	131	3	no	—
113	C₄²	—	131	3	no	—
114	C₄²	C₂²×D₁₃₁	1	3	yes	—
115	C₂³	—	131	1	no	—
116	Q₈	—	262	6	no	—
117	Q₈	#14523496	1	4	yes	—
118	C₂×C₄	—	262	6	no	—
119	C₂×C₄	—	262	6	no	—
120	C₂×C₄	—	262	1	no	—
121	C₂×C₄	—	262	1	no	—
122	C₂×C₄	—	131	1	no	—
123	C₂×C₄	—	131	3	no	—
124	C₂×C₄	—	131	3	no	—
125	C₂×C₄	C₂×C₄×D₁₃₁	1	3	yes	—
126	C₂×C₄	C₂×C₄×D₁₃₁	1	3	yes	—
127	C₂×C₄	C₂³×D₁₃₁	1	1	yes	characteristic
128	C₈	—	2	3	no	—
129	C₈	—	2	1	no	—
130	C₈	—	262	1	no	—
131	C₈	—	262	3	no	—
132	C₂²	—	262	1	no	—
133	C₂²	—	262	1	no	—
134	C₂²	—	131	2	no	—
135	C₂²	C₂²×C₄×D₁₃₁	1	1	yes	characteristic
136	C₄	—	2	3	no	—
137	C₄	—	262	1	no	—
138	C₄	—	262	3	no	—
139	C₄	—	262	3	no	—
140	C₄	C₂×#14523496	1	6	yes	—
141	C₄	#16862293	1	1	yes	characteristic
142	C₄	#16862294	1	1	yes	characteristic
143	C₂	—	262	2	no	—
144	C₂	#22553788	1	1	yes	characteristic
145	C₂	C₂²×#14523496	1	1	yes	characteristic
146	C₂	#131×D₁₃₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Q₈×D₁₃₁

C₂

yes

maximal, characteristic

#22553442

C₂

yes

maximal, complemented

#22553439

C₂

yes

maximal, complemented

Q₈×C₁₃₁⋊C₈

C₂

yes

maximal, characteristic, complemented

#22553368

C₂

yes

maximal

#22553363

C₂

yes

maximal

C₈×Q₈×C₁₃₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₁₃₁

C₄

yes

characteristic

Q₈×Dic₁₃₁

C₄

yes

characteristic

C₄×Dic₂₆₂

C₂²

yes

—

#16862211

C₄

yes

—

C₄⋊C₄×D₁₃₁

C₄

yes

—

C₄²×D₁₃₁

C₂²

yes

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

—

Dic₁₃₁⋊C₈

C₂²

yes

—

Dic₁₃₁⋊C₈

C₂²

yes

—

(C₂×D₁₃₁)⋊C₈

C₂²

yes

characteristic

(C₂×D₁₃₁)⋊C₈

C₂²

yes

—

#16862156

C₂²

yes

—

C₄×C₁₃₁⋊C₈

C₂²

yes

—

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

—

C₄×C₈×C₁₃₁

C₂²

yes

—

C₂×Dic₂₆₂

C₂×C₄

yes

—

Q₈×D₁₃₁

—

C₂×C₄×D₁₃₁

C₂×C₄

yes

—

C₂×C₄×D₁₃₁

C₂×C₄

yes

—

C₂×C₄×D₁₃₁

C₂³

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂³

yes

characteristic

C₂×C₁₃₁⋊C₈

C₂³

yes

—

C₄⋊Dic₁₃₁

C₂×C₄

yes

—

Dic₁₃₁⋊C₄

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂×C₄

yes

—

C₄×Dic₁₃₁

C₂³

yes

—

C₂×Q₈×C₁₃₁

C₂×C₄

yes

characteristic

C₂×C₈×C₁₃₁

C₂³

yes

characteristic

C₂×C₈×C₁₃₁

C₂³

yes

—

C₄⋊C₄×C₁₃₁

C₂×C₄

yes

—

C₄²×C₁₃₁

C₂³

yes

—

C₂²×D₁₃₁

C₂²×C₄

yes

characteristic

Dic₂₆₂

—

C₂×Dic₁₃₁

C₂²×C₄

yes

characteristic

C₂×Dic₁₃₁

C₂²×C₄

yes

—

C₂×Dic₁₃₁

C₂²×C₄

yes

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₄×D₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

Q₈×C₁₃₁

M₁₆

yes

—

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

#131

yes

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×M₁₆

yes

—

C₄×C₁₃₁

2₋¹⁺⁴

yes

characteristic

C₄×C₁₃₁

2₋¹⁺⁴

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

#519

yes

characteristic

C₂×C₁₃₁

C₂²×M₁₆

yes

characteristic

C₂×C₁₃₁

C₂×#131

yes

characteristic

D₁₃₁

—

C₁₃₁

#2984

yes

characteristic, complemented, Sylow

#2984

—

131

maximal, Sylow

C₂×C₄×Q₈

—

131

—

#446

—

131

—

#446

—

131

—

C₈×Q₈

—

131

—

C₈×Q₈

D₁₃₁

yes

characteristic, complemented

#432

—

131

—

#431

—

131

—

C₂²×Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

—

131

—

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

C₂×C₄⋊C₄

—

131

—

C₂×C₄²

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

C₂×D₁₃₁

yes

—

C₂²⋊C₈

—

131

—

C₂²⋊C₈

—

131

—

#97

—

131

—

C₄×C₈

—

131

—

C₄×C₈

C₂×D₁₃₁

yes

—

C₂×Q₈

—

262

—

C₂×Q₈

—

131

—

100

C₂×Q₈

C₄×D₁₃₁

yes

characteristic

101

C₂²×C₄

—

131

—

102

C₂²×C₄

—

131

—

103

C₂²×C₄

—

131

—

104

C₂×C₈

—

131

—

105

C₂×C₈

—

131

—

106

C₂×C₈

C₂²×D₁₃₁

yes

characteristic

107

C₂×C₈

C₂²×D₁₃₁

yes

—

108

C₄⋊C₄

—

131

—

109

C₄⋊C₄

—

131

—

110

C₄⋊C₄

C₄×D₁₃₁

yes

—

111

C₄²

—

131

—

112

C₄²

—

131

—

113

C₄²

—

131

—

114

C₄²

C₂²×D₁₃₁

yes

—

115

C₂³

—

131

—

116

Q₈

—

262

—

117

Q₈

#14523496

yes

—

118

C₂×C₄

—

262

—

119

C₂×C₄

—

262

—

120

C₂×C₄

—

262

—

121

C₂×C₄

—

262

—

122

C₂×C₄

—

131

—

123

C₂×C₄

—

131

—

124

C₂×C₄

—

131

—

125

C₂×C₄

C₂×C₄×D₁₃₁

yes

—

126

C₂×C₄

C₂×C₄×D₁₃₁

yes

—

127

C₂×C₄

C₂³×D₁₃₁

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

262

—

131

C₈

—

262

—

132

C₂²

—

262

—

133

C₂²

—

262

—

134

C₂²

—

131

—

135

C₂²

C₂²×C₄×D₁₃₁

yes

characteristic

136

C₄

—

137

C₄

—

262

—

138

C₄

—

262

—

139

C₄

—

262

—

140

C₄

C₂×#14523496

yes

—

141

C₄

#16862293

yes

characteristic

142

C₄

#16862294

yes

characteristic

143

C₂

—

262

—

144

C₂

#22553788

yes

characteristic

145

C₂

C₂²×#14523496

yes

characteristic

146

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	C₂²×D₁₃₁	C₂²×C₄
cosocle	C₂×C₄×C₁₃₁	C₂⁴
hypercenter	C₈×Q₈	D₁₃₁
hypocenter	C₁₃₁	#2984
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

C₂²×D₁₃₁

C₂²×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. #22553442	C₂	semidirect
3. #22553439	C₂	semidirect
4. Q₈×C₁₃₁⋊C₈	C₂	semidirect
7. C₈×Q₈×C₁₃₁	C₂	semidirect
73. C₁₃₁	#2984	semidirect
79. C₈×Q₈	D₁₃₁	semidirect

H ≅ G/N

type

2. #22553442

C₂

semidirect

3. #22553439

C₂

semidirect

4. Q₈×C₁₃₁⋊C₈

C₂

semidirect

7. C₈×Q₈×C₁₃₁

C₂

semidirect

73. C₁₃₁

#2984

semidirect

79. C₈×Q₈

D₁₃₁

semidirect

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

subgroup	index
C₄×Q₈×D₁₃₁	2
#22553442	2
#22553439	2
Q₈×C₁₃₁⋊C₈	2
#22553368	2
#22553363	2
C₈×Q₈×C₁₃₁	2
#2984	131

subgroup

index

C₄×Q₈×D₁₃₁

Q₈×C₁₃₁⋊C₈

C₈×Q₈×C₁₃₁

131

Subgroups — #30675049

Conjugacy classes of subgroups

Complements

Maximal subgroups