Group #30668205 — subgroups

Subgroups — #30668205

Order 16768.

132 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 132 conjugacy classes of subgroups — too many to draw. Only its 47 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	#22553713	C₂	1	1	yes	maximal, characteristic
2	#22553690	C₂	1	1	yes	maximal, characteristic
3	#22553117	C₂	1	1	yes	maximal, characteristic
4	#22553116	C₂	1	1	yes	maximal, characteristic
5	#22553115	C₂	1	1	yes	maximal, characteristic
6	#22553087	C₂	1	1	yes	maximal, characteristic
7	#358×C₁₃₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Dic₅₂₄	C₂²	1	1	yes	characteristic
9	C₂×#14523503	—	2	1	no	—
10	#16862244	—	2	1	no	—
11	#16862237	—	2	1	no	—
12	C₂×C₁₃₁⋊C₁₆	C₂²	1	1	yes	characteristic
13	#16862180	C₂²	1	1	yes	characteristic
14	#16862177	—	2	1	no	—
15	C₈×Dic₁₃₁	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₁₃₁	C₂²	1	1	yes	characteristic
18	#107×C₁₃₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₂₆₂	—	2	1	no	—
20	Dic₅₂₄	—	2	1	no	—
21	Dic₅₂₄	—	2	1	no	—
22	#14523503	—	4	2	no	—
23	C₂×C₁₃₁⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Dic₁₃₁	—	2	1	no	—
25	C₄⋊Dic₁₃₁	—	4	1	no	—
26	Dic₁₃₁⋊C₄	—	4	1	no	—
27	Dic₁₃₁⋊C₄	—	4	1	no	—
28	Dic₁₃₁⋊C₄	—	4	1	no	—
29	C₄×Dic₁₃₁	D₄	1	1	yes	characteristic
30	C₁₃₁⋊C₁₆	—	2	1	no	—
31	C₂×Q₈×C₁₃₁	—	2	1	no	—
32	Q₁₆×C₁₃₁	—	2	1	no	—
33	Q₁₆×C₁₃₁	—	2	1	no	—
34	C₂×C₈×C₁₃₁	C₂³	1	1	yes	characteristic
35	C₄⋊C₄×C₁₃₁	—	2	1	no	—
36	C₁₆×C₁₃₁	—	2	1	no	—
37	Dic₂₆₂	—	4	1	no	—
38	Dic₂₆₂	—	4	1	no	—
39	C₂×Dic₁₃₁	—	4	1	no	—
40	C₂×Dic₁₃₁	—	4	1	no	—
41	C₂×Dic₁₃₁	D₈	1	1	yes	characteristic
42	C₂×Dic₁₃₁	D₈	1	1	yes	characteristic
43	C₁₃₁⋊C₈	SD₁₆	1	2	yes	—
44	Q₈×C₁₃₁	—	4	1	no	—
45	Q₈×C₁₃₁	—	4	1	no	—
46	C₂×C₄×C₁₃₁	—	4	1	no	—
47	C₂×C₄×C₁₃₁	—	4	1	no	—
48	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₃₁	#41	1	1	yes	characteristic
50	C₈×C₁₃₁	#41	1	1	yes	characteristic
51	C₂²×C₁₃₁	C₂×D₈	1	1	yes	characteristic
52	C₄×C₁₃₁	—	8	1	no	—
53	C₄×C₁₃₁	—	8	1	no	—
54	C₄×C₁₃₁	#124	1	1	yes	characteristic
55	C₄×C₁₃₁	C₂×SD₁₆	1	1	yes	characteristic
56	Dic₁₃₁	—	2	1	no	—
57	Dic₁₃₁	—	2	1	no	—
58	Dic₁₃₁	—	8	1	no	—
59	Dic₁₃₁	—	8	1	no	—
60	C₂×C₁₃₁	#486	1	1	yes	characteristic
61	C₂×C₁₃₁	#510	1	1	yes	characteristic
62	C₂×C₁₃₁	#510	1	1	yes	characteristic
63	C₁₃₁	#2245	1	1	yes	characteristic, complemented, Sylow
64	#2245	—	131	1	no	maximal, Sylow
65	#499	—	131	1	no	—
66	#494	—	131	1	no	—
67	#358	—	131	1	no	—
68	#358	—	131	1	no	—
69	#358	—	131	1	no	—
70	#358	D₁₃₁	1	1	yes	characteristic
71	#346	—	131	1	no	—
72	C₂×Q₁₆	—	262	1	no	—
73	C₂×Q₁₆	—	131	1	no	—
74	C₂×Q₁₆	C₂×D₁₃₁	1	1	yes	characteristic
75	C₄⋊Q₈	—	262	1	no	—
76	#125	—	262	1	no	—
77	C₂×C₁₆	—	131	1	no	—
78	C₂×C₁₆	C₂×D₁₃₁	1	1	yes	characteristic
79	#107	—	131	1	no	—
80	#107	C₂×D₁₃₁	1	1	yes	characteristic
81	#106	—	262	1	no	—
82	C₄×C₈	—	131	1	no	—
83	C₂×Q₈	—	2	1	no	—
84	C₂×Q₈	—	262	1	no	—
85	Q₁₆	—	2	1	no	—
86	Q₁₆	—	2	1	no	—
87	Q₁₆	—	262	1	no	—
88	Q₁₆	—	262	1	no	—
89	Q₁₆	—	524	2	no	—
90	C₂×C₈	—	131	1	no	—
91	C₂×C₈	C₂²×D₁₃₁	1	1	yes	characteristic
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	262	1	no	—
94	C₄⋊C₄	—	524	1	no	—
95	C₄⋊C₄	—	524	1	no	—
96	C₄⋊C₄	—	524	1	no	—
97	C₄⋊C₄	—	524	1	no	—
98	C₄²	—	131	1	no	—
99	C₁₆	—	2	1	no	—
100	C₁₆	—	262	1	no	—
101	Q₈	—	4	1	no	—
102	Q₈	—	4	1	no	—
103	Q₈	—	524	1	no	—
104	Q₈	—	524	1	no	—
105	C₂×C₄	—	4	1	no	—
106	C₂×C₄	—	4	1	no	—
107	C₂×C₄	—	524	1	no	—
108	C₂×C₄	—	524	1	no	—
109	C₂×C₄	—	131	1	no	—
110	C₂×C₄	—	131	1	no	—
111	C₂×C₄	D₄×D₁₃₁	1	1	yes	characteristic
112	C₈	—	131	2	no	—
113	C₈	#14523512	1	1	yes	characteristic
114	C₈	#14523513	1	1	yes	characteristic
115	C₂²	D₈×D₁₃₁	1	1	yes	characteristic
116	C₄	—	262	1	no	—
117	C₄	—	262	1	no	—
118	C₄	—	8	1	no	—
119	C₄	—	8	1	no	—
120	C₄	—	1048	1	no	—
121	C₄	—	1048	1	no	—
122	C₄	#16862232	1	1	yes	characteristic
123	C₄	SD₁₆×D₁₃₁	1	1	yes	characteristic
124	C₂	#22553657	1	1	yes	characteristic
125	C₂	#22553751	1	1	yes	characteristic
126	C₂	#22553754	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22553713

C₂

yes

maximal, characteristic

#22553690

C₂

yes

maximal, characteristic

#22553117

C₂

yes

maximal, characteristic

#22553116

C₂

yes

maximal, characteristic

#22553115

C₂

yes

maximal, characteristic

#22553087

C₂

yes

maximal, characteristic

#358×C₁₃₁

C₂

yes

maximal, characteristic

C₂×Dic₅₂₄

C₂²

yes

—

—

—

C₂×C₁₃₁⋊C₁₆

C₂²

yes

characteristic

#16862180

C₂²

yes

characteristic

#16862177

—

C₈×Dic₁₃₁

C₂²

yes

characteristic

C₂×Q₁₆×C₁₃₁

C₂²

yes

characteristic

C₂×C₁₆×C₁₃₁

C₂²

yes

characteristic

#107×C₁₃₁

C₂²

yes

characteristic

C₂×Dic₂₆₂

—

Dic₅₂₄

—

Dic₅₂₄

—

#14523503

—

C₂×C₁₃₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₃₁

—

C₄⋊Dic₁₃₁

—

Dic₁₃₁⋊C₄

—

Dic₁₃₁⋊C₄

—

Dic₁₃₁⋊C₄

—

C₄×Dic₁₃₁

D₄

yes

characteristic

C₁₃₁⋊C₁₆

—

C₂×Q₈×C₁₃₁

—

Q₁₆×C₁₃₁

—

Q₁₆×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₃₁

—

C₁₆×C₁₃₁

—

Dic₂₆₂

—

Dic₂₆₂

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

—

C₂×Dic₁₃₁

D₈

yes

characteristic

C₂×Dic₁₃₁

D₈

yes

characteristic

C₁₃₁⋊C₈

SD₁₆

yes

—

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

C₈×C₁₃₁

yes

C₈×C₁₃₁

yes

C₂²×C₁₃₁

C₂×D₈

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#124

yes

characteristic

C₄×C₁₃₁

C₂×SD₁₆

yes

characteristic

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

Dic₁₃₁

—

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₂×C₁₃₁

yes

C₁₃₁

yes

characteristic, complemented, Sylow

#2245

—

131

maximal, Sylow

#499

—

131

—

#494

—

131

—

#358

—

131

—

#358

—

131

—

#358

—

131

—

#358

D₁₃₁

yes

characteristic

#346

—

131

—

C₂×Q₁₆

—

262

—

C₂×Q₁₆

—

131

—

C₂×Q₁₆

C₂×D₁₃₁

yes

characteristic

C₄⋊Q₈

—

262

—

#125

—

262

—

C₂×C₁₆

—

131

—

C₂×C₁₆

C₂×D₁₃₁

yes

characteristic

#107

—

131

—

#107

C₂×D₁₃₁

yes

characteristic

#106

—

262

—

C₄×C₈

—

131

—

C₂×Q₈

—

C₂×Q₈

—

262

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

262

—

Q₁₆

—

262

—

Q₁₆

—

524

—

C₂×C₈

—

131

—

C₂×C₈

C₂²×D₁₃₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

262

—

C₄⋊C₄

—

524

—

C₄⋊C₄

—

524

—

C₄⋊C₄

—

524

—

C₄⋊C₄

—

524

—

C₄²

—

131

—

C₁₆

—

100

C₁₆

—

262

—

101

Q₈

—

102

Q₈

—

103

Q₈

—

524

—

104

Q₈

—

524

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

524

—

108

C₂×C₄

—

524

—

109

C₂×C₄

—

131

—

110

C₂×C₄

—

131

—

111

C₂×C₄

D₄×D₁₃₁

yes

characteristic

112

C₈

—

131

—

113

C₈

#14523512

yes

characteristic

114

C₈

#14523513

yes

characteristic

115

C₂²

D₈×D₁₃₁

yes

characteristic

116

C₄

—

262

—

117

C₄

—

262

—

118

C₄

—

119

C₄

—

120

C₄

—

1048

—

121

C₄

—

1048

—

122

C₄

#16862232

yes

characteristic

123

C₄

SD₁₆×D₁₃₁

yes

characteristic

124

C₂

#22553657

yes

characteristic

125

C₂

#22553751

yes

characteristic

126

C₂

#22553754

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₁₃₁
commutator, cosocle	C₂×C₈×C₁₃₁	C₂³
Frattini	C₂×C₈	C₂²×D₁₃₁
Fitting	#358×C₁₃₁	C₂
socle, prime radical	C₂²×C₁₃₁	C₂×D₈
hypercenter	#358	D₁₃₁
hypocenter	C₁₃₁	#2245
LCS	G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₁₃₁

commutator, cosocle

C₂×C₈×C₁₃₁

C₂³

Frattini

C₂×C₈

C₂²×D₁₃₁

Fitting

#358×C₁₃₁

C₂

socle, prime radical

C₂²×C₁₃₁

C₂×D₈

D₁₃₁

C₁₃₁

G ▷ C₂×C₈×C₁₃₁ ▷ C₄×C₁₃₁ ▷ C₂×C₁₃₁ ▷ C₁₃₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

N	H ≅ G/N	type
63. C₁₃₁	#2245	semidirect

H ≅ G/N

type

63. C₁₃₁

#2245

semidirect

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

subgroup	index
#22553713	2
#22553690	2
#22553117	2
#22553116	2
#22553115	2
#22553087	2
#358×C₁₃₁	2
#2245	131

subgroup

index

#358×C₁₃₁

131

Subgroups — #30668205

Conjugacy classes of subgroups

Complements

Maximal subgroups