Group #26103148 — subgroups

Subgroups — #26103148

Order 12032.

242 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 151 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×#15688876	C₂	1	2	yes	maximal
2	C₂×C₄×C₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
4	#15688876	—	2	2	no	—
5	#15688876	C₄	1	4	yes	—
6	C₂²×C₁₆×C₄₇	C₄	1	2	yes	—
7	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
8	C₄×C₁₆×C₄₇	—	2	2	no	—
9	C₂²×C₄₇⋊C₈	—	4	2	no	—
10	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
11	C₄⋊(C₄₇⋊C₈)	—	4	2	no	—
12	C₂²×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₈×C₄₇	Q₈	1	1	yes	characteristic
14	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₄₇	—	2	2	no	—
16	C₂×C₁₆×C₄₇	—	2	2	no	—
17	C₂×C₁₆×C₄₇	C₈	1	4	yes	complemented
18	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₄₇	D₄	1	2	yes	—
20	C₂²×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₈	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
23	C₂×C₈×C₄₇	—	2	2	no	—
24	C₂×C₈×C₄₇	C₄²	1	1	yes	characteristic
25	C₂×C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₄₇	C₂×C₈	1	2	yes	—
28	C₂×C₈×C₄₇	M₁₆	1	2	yes	—
29	C₄²×C₄₇	C₂²⋊C₄	1	2	yes	—
30	C₄²×C₄₇	SD₁₆	1	2	yes	—
31	C₂×C₄₇⋊C₈	—	4	4	no	—
32	C₂×C₄₇⋊C₈	—	8	2	no	—
33	C₂×C₄₇⋊C₈	—	8	2	no	—
34	C₁₆×C₄₇	—	2	4	no	—
35	C₂³×C₄₇	#107	1	1	yes	characteristic
36	C₂×C₄×C₄₇	—	2	2	no	—
37	C₂×C₄×C₄₇	#95	1	1	yes	characteristic
38	C₂×C₄×C₄₇	#102	1	2	yes	—
39	C₂×C₄×C₄₇	#103	1	2	yes	—
40	C₂×C₄×C₄₇	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₄₇	#106	1	1	yes	characteristic
42	C₂×C₄×C₄₇	#108	1	2	yes	—
43	C₂×C₄×C₄₇	D₁₆	1	1	yes	characteristic
44	C₂×C₄×C₄₇	SD₃₂	1	2	yes	—
45	C₂×C₄×C₄₇	Q₃₂	1	1	yes	characteristic
46	C₈×C₄₇	—	2	2	no	—
47	C₈×C₄₇	C₄×C₈	1	1	yes	characteristic
48	C₈×C₄₇	#97	1	1	yes	characteristic
49	C₈×C₄₇	C₂²⋊C₈	1	2	yes	—
50	C₄₇⋊C₈	—	8	4	no	—
51	#7154	—	47	1	no	maximal, Sylow
52	C₂²×C₄₇	#335	1	2	yes	—
53	C₂²×C₄₇	#340	1	1	yes	characteristic
54	C₂²×C₄₇	#366	1	1	yes	characteristic
55	C₂²×C₄₇	#367	1	1	yes	characteristic
56	C₂²×C₄₇	#368	1	2	yes	—
57	C₄×C₄₇	—	2	2	no	—
58	C₄×C₄₇	#334	1	2	yes	—
59	C₄×C₄₇	#336	1	1	yes	characteristic
60	C₄×C₄₇	#341	1	1	yes	characteristic
61	C₄×C₄₇	#357	1	2	yes	—
62	C₄×C₄₇	#358	1	2	yes	—
63	C₂×C₄×C₁₆	D₄₇	1	1	yes	characteristic
64	C₂×#335	—	47	2	no	—
65	C₂×C₄₇	#1277	1	1	yes	characteristic
66	C₂×C₄₇	#1371	1	2	yes	—
67	C₂×C₄₇	#1372	1	2	yes	—
68	C₂×C₄₇	#1380	1	1	yes	characteristic
69	C₂×C₄₇	#1381	1	1	yes	characteristic
70	C₂²×C₁₆	Dic₄₇	1	2	yes	—
71	C₂×C₄⋊C₈	—	94	2	no	—
72	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
73	C₄×C₁₆	—	2	2	no	—
74	#335	—	47	4	no	—
75	#335	—	94	2	no	—
76	C₄₇	#7154	1	1	yes	characteristic, complemented, Sylow
77	C₂²×C₈	—	188	2	no	—
78	C₂²×C₈	Dic₉₄	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₄₇	1	1	yes	characteristic
80	C₂×C₄²	D₁₈₈	1	1	yes	characteristic
81	C₂×C₁₆	—	2	2	no	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	C₄₇⋊C₈	1	4	yes	complemented
84	C₄⋊C₈	—	94	4	no	—
85	C₄⋊C₈	—	188	2	no	—
86	C₄×C₈	C₄×D₄₇	1	2	yes	—
87	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
88	C₂²×C₄	D₃₇₆	1	1	yes	characteristic
89	C₂²×C₄	Dic₁₈₈	1	1	yes	characteristic
90	C₂²×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
91	C₂×C₈	—	2	2	no	—
92	C₂×C₈	—	376	2	no	—
93	C₂×C₈	—	376	2	no	—
94	C₂×C₈	—	188	4	no	—
95	C₂×C₈	C₂×C₄₇⋊C₈	1	2	yes	—
96	C₂×C₈	#10625329	1	2	yes	—
97	C₂×C₈	C₄×Dic₄₇	1	1	yes	characteristic
98	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
99	C₂×C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
100	C₄²	#10625325	1	2	yes	—
101	C₄²	(C₂×D₄₇)⋊C₄	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	#13093958	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	C₄₇⋊#95	1	1	yes	characteristic
106	C₂×C₄	#13093946	1	2	yes	—
107	C₂×C₄	#13093949	1	2	yes	—
108	C₂×C₄	C₄⋊(C₄₇⋊C₈)	1	2	yes	—
109	C₂×C₄	#13093956	1	1	yes	characteristic
110	C₂×C₄	#13093960	1	2	yes	—
111	C₂×C₄	D₇₅₂	1	1	yes	characteristic
112	C₂×C₄	#13093969	1	2	yes	—
113	C₂×C₄	Dic₃₇₆	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	376	4	no	—
116	C₈	C₄×C₄₇⋊C₈	1	1	yes	characteristic
117	C₈	#13093935	1	1	yes	characteristic
118	C₈	C₂²⋊(C₄₇⋊C₈)	1	2	yes	—
119	C₂²	#15688876	1	2	yes	—
120	C₂²	#15688886	1	1	yes	characteristic
121	C₂²	#15688949	1	1	yes	characteristic
122	C₂²	#15688951	1	1	yes	characteristic
123	C₂²	#15688953	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	#15688874	1	2	yes	—
126	C₄	#15688878	1	1	yes	characteristic
127	C₄	#15688888	1	1	yes	characteristic
128	C₄	#15688925	1	2	yes	—
129	C₄	#15688928	1	2	yes	—
130	C₂	#20241223	1	1	yes	characteristic
131	C₂	#20241460	1	2	yes	—
132	C₂	#20241462	1	2	yes	—
133	C₂	#20241480	1	1	yes	characteristic
134	C₂	#20241482	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688876

C₂

yes

maximal

C₂×C₄×C₁₆×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₄₇⋊C₈)

—

#15688876

—

#15688876

C₄

yes

—

C₂²×C₁₆×C₄₇

C₄

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄×C₁₆×C₄₇

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₄₇

Q₈

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₈

yes

complemented

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

D₈

yes

characteristic

C₂²×C₄×C₄₇

Q₁₆

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄²

yes

characteristic

C₂×C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

C₂×C₈

yes

—

C₂×C₈×C₄₇

M₁₆

yes

—

C₄²×C₄₇

C₂²⋊C₄

yes

—

C₄²×C₄₇

SD₁₆

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₁₆×C₄₇

—

C₂³×C₄₇

#107

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#95

yes

characteristic

C₂×C₄×C₄₇

#102

yes

—

C₂×C₄×C₄₇

#103

yes

—

C₂×C₄×C₄₇

C₄⋊C₈

yes

—

C₂×C₄×C₄₇

#106

yes

characteristic

C₂×C₄×C₄₇

#108

yes

—

C₂×C₄×C₄₇

D₁₆

yes

characteristic

C₂×C₄×C₄₇

SD₃₂

yes

—

C₂×C₄×C₄₇

Q₃₂

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₄×C₈

yes

characteristic

C₈×C₄₇

#97

yes

characteristic

C₈×C₄₇

C₂²⋊C₈

yes

—

C₄₇⋊C₈

—

#7154

—

maximal, Sylow

C₂²×C₄₇

#335

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

#334

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

#358

yes

—

C₂×C₄×C₁₆

D₄₇

yes

characteristic

C₂×#335

—

C₂×C₄₇

#1277

yes

characteristic

C₂×C₄₇

#1371

yes

—

C₂×C₄₇

#1372

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂²×C₁₆

Dic₄₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

C₄×C₁₆

—

#335

—

#335

—

C₄₇

#7154

yes

characteristic, complemented, Sylow

C₂²×C₈

—

188

—

C₂²×C₈

Dic₉₄

yes

characteristic

C₂²×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₄²

D₁₈₈

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄₇⋊C₈

yes

complemented

C₄⋊C₈

—

C₄⋊C₈

—

188

—

C₄×C₈

C₄×D₄₇

yes

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₂²×C₄

D₃₇₆

yes

characteristic

C₂²×C₄

Dic₁₈₈

yes

characteristic

C₂²×C₄

C₄⋊Dic₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

376

—

C₂×C₈

—

376

—

C₂×C₈

—

188

—

C₂×C₈

C₂×C₄₇⋊C₈

yes

—

C₂×C₈

#10625329

yes

—

C₂×C₈

C₄×Dic₄₇

yes

characteristic

C₂×C₈

Dic₄₇⋊C₄

yes

—

C₂×C₈

C₂²⋊Dic₄₇

yes

characteristic

100

C₄²

#10625325

yes

—

101

C₄²

(C₂×D₄₇)⋊C₄

yes

—

102

C₁₆

—

103

C₂³

#13093958

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

C₄₇⋊#95

yes

characteristic

106

C₂×C₄

#13093946

yes

—

107

C₂×C₄

#13093949

yes

—

108

C₂×C₄

C₄⋊(C₄₇⋊C₈)

yes

—

109

C₂×C₄

#13093956

yes

characteristic

110

C₂×C₄

#13093960

yes

—

111

C₂×C₄

D₇₅₂

yes

characteristic

112

C₂×C₄

#13093969

yes

—

113

C₂×C₄

Dic₃₇₆

yes

characteristic

114

C₈

—

115

C₈

—

376

—

116

C₈

C₄×C₄₇⋊C₈

yes

characteristic

117

C₈

#13093935

yes

characteristic

118

C₈

C₂²⋊(C₄₇⋊C₈)

yes

—

119

C₂²

#15688876

yes

—

120

C₂²

#15688886

yes

characteristic

121

C₂²

#15688949

yes

characteristic

122

C₂²

#15688951

yes

characteristic

123

C₂²

#15688953

yes

—

124

C₄

—

125

C₄

#15688874

yes

—

126

C₄

#15688878

yes

characteristic

127

C₄

#15688888

yes

characteristic

128

C₄

#15688925

yes

—

129

C₄

#15688928

yes

—

130

C₂

#20241223

yes

characteristic

131

C₂

#20241460

yes

—

132

C₂

#20241462

yes

—

133

C₂

#20241480

yes

characteristic

134

C₂

#20241482

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	D₃₇₆
commutator	C₈×C₄₇	C₄×C₈
Frattini	C₂×C₄×C₈	C₂×D₄₇
Fitting	C₂×C₄×C₁₆×C₄₇	C₂
socle, prime radical	C₂³×C₄₇	#107
cosocle	C₂×C₄×C₈×C₄₇	C₂²
hypercenter	C₂×C₄×C₁₆	D₄₇
hypocenter	C₄₇	#7154
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

subgroup

G/H

center

C₂²×C₄

D₃₇₆

commutator

C₈×C₄₇

C₄×C₈

Frattini

C₂×C₄×C₈

C₂×D₄₇

Fitting

C₂×C₄×C₁₆×C₄₇

C₂

socle, prime radical

C₂³×C₄₇

#107

cosocle

C₂×C₄×C₈×C₄₇

C₂²

hypercenter

C₂×C₄×C₁₆

D₄₇

hypocenter

C₄₇

#7154

LCS

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

N	H ≅ G/N	type
17. C₂×C₁₆×C₄₇	C₈	semidirect
76. C₄₇	#7154	semidirect
83. C₂×C₁₆	C₄₇⋊C₈	semidirect

H ≅ G/N

type

17. C₂×C₁₆×C₄₇

C₈

semidirect

76. C₄₇

#7154

semidirect

83. C₂×C₁₆

C₄₇⋊C₈

semidirect

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

subgroup	index
C₂×#15688876	2
C₂×C₄×C₁₆×C₄₇	2
#7154	47

subgroup

index

C₂×#15688876

C₂×C₄×C₁₆×C₄₇

#7154

Subgroups — #26103148

Conjugacy classes of subgroups

Complements

Maximal subgroups