Group #30665341 — subgroups

Subgroups — #30665341

Order 16768.

150 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 150 conjugacy classes of subgroups — too many to draw. Only its 57 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	#22553708	C₂	1	1	yes	maximal, characteristic
2	#22553607	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₁₃₁⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#22553062	C₂	1	1	yes	maximal, characteristic
5	#22553060	C₂	1	1	yes	maximal, characteristic
6	#22553058	C₂	1	1	yes	maximal, characteristic
7	#462×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	#16862246	C₂²	1	1	yes	characteristic
9	#16862179	—	2	1	no	—
10	#16862177	—	2	1	no	—
11	#16862177	—	2	1	no	—
12	C₄⋊(C₁₃₁⋊C₈)	—	2	1	no	—
13	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₁₃₁⋊C₈)	C₂²	1	1	yes	characteristic
15	#16862170	—	2	1	no	—
16	C₄×C₁₃₁⋊C₈	—	2	1	no	—
17	C₄×C₁₃₁⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₃₁	—	2	1	no	—
19	C₄⋊Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₃₁	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₃₁	C₂²	1	1	yes	characteristic
22	#103×C₁₃₁	—	2	1	no	—
23	C₂×Dic₂₆₂	—	2	1	no	—
24	C₂×C₁₃₁⋊C₈	—	2	1	no	—
25	C₂×C₁₃₁⋊C₈	—	2	1	no	—
26	C₂×C₁₃₁⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₁₃₁⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₃₁	—	2	1	no	—
29	C₄⋊Dic₁₃₁	—	2	1	no	—
30	C₂×Q₈×C₁₃₁	—	2	1	no	—
31	C₂×Q₈×C₁₃₁	D₄	1	1	yes	characteristic
32	Q₁₆×C₁₃₁	—	4	2	no	—
33	C₂×C₈×C₁₃₁	—	2	1	no	—
34	C₄⋊C₄×C₁₃₁	—	2	1	no	—
35	C₄⋊C₄×C₁₃₁	—	2	1	no	—
36	C₄⋊C₄×C₁₃₁	—	2	1	no	—
37	C₄⋊C₄×C₁₃₁	D₄	1	1	yes	characteristic
38	C₄²×C₁₃₁	—	2	1	no	—
39	C₄²×C₁₃₁	C₂³	1	1	yes	characteristic
40	Dic₂₆₂	—	8	1	no	—
41	C₂×Dic₁₃₁	—	4	1	no	—
42	C₁₃₁⋊C₈	—	2	1	no	—
43	C₁₃₁⋊C₈	—	4	1	no	—
44	C₁₃₁⋊C₈	—	4	1	no	—
45	C₁₃₁⋊C₈	Q₁₆	1	2	yes	complemented
46	Q₈×C₁₃₁	—	2	1	no	—
47	Q₈×C₁₃₁	—	4	1	no	—
48	Q₈×C₁₃₁	—	4	1	no	—
49	Q₈×C₁₃₁	SD₁₆	1	2	yes	—
50	C₂×C₄×C₁₃₁	—	2	1	no	—
51	C₂×C₄×C₁₃₁	—	2	1	no	—
52	C₂×C₄×C₁₃₁	—	4	1	no	—
53	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₃₁	#41	1	1	yes	characteristic
56	C₈×C₁₃₁	—	4	1	no	—
57	C₂²×C₁₃₁	#121	1	1	yes	characteristic
58	C₄×C₁₃₁	—	2	2	no	—
59	C₄×C₁₃₁	—	2	1	no	—
60	C₄×C₁₃₁	—	4	1	no	—
61	C₄×C₁₃₁	—	8	1	no	—
62	C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₁₃₁	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₁₃₁	#135	1	1	yes	characteristic
65	C₄×C₁₃₁	#137	1	1	yes	characteristic
66	Dic₁₃₁	—	8	1	no	—
67	C₂×C₁₃₁	#462	1	1	yes	characteristic
68	C₂×C₁₃₁	#465	1	1	yes	characteristic
69	C₂×C₁₃₁	#471	1	1	yes	characteristic
70	C₁₃₁	#1672	1	1	yes	characteristic, complemented, Sylow
71	#1672	—	131	1	no	maximal, Sylow
72	#498	—	131	1	no	—
73	#477	—	131	1	no	—
74	#462	D₁₃₁	1	1	yes	characteristic
75	C₈×Q₈	—	131	1	no	—
76	#334	—	131	1	no	—
77	#333	—	131	1	no	—
78	#332	—	131	1	no	—
79	C₂×Q₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	131	1	no	—
81	C₄⋊Q₈	C₂×D₁₃₁	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₁₃₁	1	1	yes	characteristic
83	#107	—	262	1	no	—
84	#106	—	262	1	no	—
85	#106	—	262	1	no	—
86	C₄⋊C₈	—	262	1	no	—
87	C₄⋊C₈	—	131	1	no	—
88	C₄⋊C₈	—	131	1	no	—
89	C₄⋊C₈	C₂×D₁₃₁	1	1	yes	characteristic
90	#103	—	2	1	no	—
91	#103	—	262	1	no	—
92	C₄×C₈	—	262	1	no	—
93	C₄×C₈	—	131	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	262	1	no	—
96	C₂×Q₈	C₂²⋊D₁₃₁	1	1	yes	characteristic
97	Q₁₆	—	4	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	262	1	no	—
100	C₂×C₈	—	262	1	no	—
101	C₂×C₈	—	131	1	no	—
102	C₂×C₈	—	131	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	262	1	no	—
107	C₄⋊C₄	—	262	1	no	—
108	C₄⋊C₄	C₂²⋊D₁₃₁	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₁₃₁	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	1048	1	no	—
115	Q₈	#14523501	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	524	1	no	—
120	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₃₁	1	1	yes	characteristic
122	C₂×C₄	#14523512	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	262	1	no	—
125	C₈	—	524	1	no	—
126	C₈	—	524	1	no	—
127	C₈	—	131	2	no	—
128	C₂²	#16862218	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	1048	1	no	—
134	C₄	C₂×#14523501	1	1	yes	characteristic
135	C₄	Q₁₆×D₁₃₁	1	1	yes	characteristic
136	C₄	#16862266	1	1	yes	characteristic
137	C₄	#16862278	1	1	yes	characteristic
138	C₂	#22553520	1	1	yes	characteristic
139	C₂	#22553539	1	1	yes	characteristic
140	C₂	#22553577	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553708

C₂

yes

maximal, characteristic

#22553607

C₂

yes

maximal, characteristic

Q₈×C₁₃₁⋊C₈

C₂

yes

maximal, characteristic

#22553062

C₂

yes

maximal, characteristic

#22553060

C₂

yes

maximal, characteristic

#22553058

C₂

yes

maximal, characteristic

#462×C₁₃₁

C₂

yes

maximal, characteristic

#16862246

C₂²

yes

—

—

—

C₄⋊(C₁₃₁⋊C₈)

—

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₃₁⋊C₈)

C₂²

yes

characteristic

#16862170

—

C₄×C₁₃₁⋊C₈

—

C₄×C₁₃₁⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₁

—

C₄⋊Q₈×C₁₃₁

C₂²

yes

characteristic

C₄×Q₈×C₁₃₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₃₁

C₂²

yes

characteristic

#103×C₁₃₁

—

C₂×Dic₂₆₂

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

—

C₂×C₁₃₁⋊C₈

D₄

yes

characteristic

C₂×C₁₃₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

D₄

yes

characteristic

Q₁₆×C₁₃₁

—

C₂×C₈×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

D₄

yes

characteristic

C₄²×C₁₃₁

—

C₄²×C₁₃₁

C₂³

yes

characteristic

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

—

C₁₃₁⋊C₈

Q₁₆

yes

complemented

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

SD₁₆

yes

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

#41

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

#121

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

C₂×SD₁₆

yes

characteristic

C₄×C₁₃₁

C₂×Q₁₆

yes

C₄×C₁₃₁

yes

C₄×C₁₃₁

yes

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#1672

—

131

maximal, Sylow

#498

—

131

—

#477

—

131

—

#462

D₁₃₁

yes

characteristic

C₈×Q₈

—

131

—

#334

—

131

—

#333

—

131

—

#332

—

131

—

C₂×Q₁₆

—

C₄⋊Q₈

—

131

—

C₄⋊Q₈

C₂×D₁₃₁

yes

characteristic

C₄×Q₈

C₂×D₁₃₁

yes

characteristic

#107

—

262

—

#106

—

262

—

#106

—

262

—

C₄⋊C₈

—

262

—

C₄⋊C₈

—

131

—

C₄⋊C₈

—

131

—

C₄⋊C₈

C₂×D₁₃₁

yes

characteristic

#103

—

#103

—

262

—

C₄×C₈

—

262

—

C₄×C₈

—

131

—

C₂×Q₈

—

C₂×Q₈

—

262

—

C₂×Q₈

C₂²⋊D₁₃₁

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

262

—

100

C₂×C₈

—

262

—

101

C₂×C₈

—

131

—

102

C₂×C₈

—

131

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

262

—

107

C₄⋊C₄

—

262

—

108

C₄⋊C₄

C₂²⋊D₁₃₁

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₁₃₁

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

1048

—

115

Q₈

#14523501

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

524

—

120

C₂×C₄

D₄×D₁₃₁

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₃₁

yes

characteristic

122

C₂×C₄

#14523512

yes

characteristic

123

C₈

—

124

C₈

—

262

—

125

C₈

—

524

—

126

C₈

—

524

—

127

C₈

—

131

—

128

C₂²

#16862218

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

1048

—

134

C₄

C₂×#14523501

yes

characteristic

135

C₄

Q₁₆×D₁₃₁

yes

characteristic

136

C₄

#16862266

yes

characteristic

137

C₄

#16862278

yes

characteristic

138

C₂

#22553520

yes

characteristic

139

C₂

#22553539

yes

characteristic

140

C₂

#22553577

yes

characteristic

	subgroup	G/H
center	C₂²	#16862218
commutator, cosocle	C₄²×C₁₃₁	C₂³
Frattini	C₄²	C₂²×D₁₃₁
Fitting	#462×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	#121
hypercenter	#462	D₁₃₁
hypocenter	C₁₃₁	#1672
LCS	G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#16862218

commutator, cosocle

C₄²×C₁₃₁

C₂³

Frattini

C₄²

C₂²×D₁₃₁

Fitting

#462×C₁₃₁

C₂

C₂²×C₁₃₁

D₁₃₁

C₁₃₁

G ▷ C₄²×C₁₃₁ ▷ C₂²×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
45. C₁₃₁⋊C₈	Q₁₆	semidirect
70. C₁₃₁	#1672	semidirect

H ≅ G/N

type

45. C₁₃₁⋊C₈

Q₁₆

semidirect

70. C₁₃₁

#1672

semidirect

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

subgroup	index
#22553708	2
#22553607	2
Q₈×C₁₃₁⋊C₈	2
#22553062	2
#22553060	2
#22553058	2
#462×C₁₃₁	2
#1672	131

subgroup

index

Q₈×C₁₃₁⋊C₈

#462×C₁₃₁

131

Subgroups — #30665341

Conjugacy classes of subgroups

Complements

Maximal subgroups