Group #26131045 — subgroups

Subgroups — #26131045

Order 12032.

274 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20245640	C₂	1	2	yes	maximal
2	#20245617	C₂	1	2	yes	maximal
3	#20243006	C₂	1	1	yes	maximal, characteristic
4	#20241306	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1707	C₂	1	1	yes	maximal, characteristic
6	C₂×#13093970	—	2	4	no	—
7	#15689528	—	2	1	no	—
8	#15689528	—	2	1	no	—
9	#15689503	C₂²	1	2	yes	—
10	#15688926	—	2	4	no	—
11	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
12	C₄₇×#500	C₂²	1	1	yes	characteristic
13	C₄₇×#498	—	2	1	no	—
14	C₄₇×#494	C₂²	1	1	yes	characteristic
15	C₄₇×#487	—	2	1	no	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	2	2	no	—
18	C₂×Dic₁₈₈	—	2	2	no	—
19	#13094024	—	2	2	no	—
20	#13093970	—	4	8	no	—
21	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
22	#13093958	—	4	2	no	—
23	#13093958	—	4	2	no	—
24	C₂×Q₁₆×C₄₇	—	2	2	no	—
25	C₄⋊Q₈×C₄₇	—	2	1	no	—
26	C₄⋊Q₈×C₄₇	—	2	1	no	—
27	#107×C₄₇	—	2	2	no	—
28	#106×C₄₇	—	4	2	no	—
29	#103×C₄₇	—	4	2	no	—
30	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₄₇	D₄	1	1	yes	characteristic
32	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₄₇	—	4	1	no	—
34	C₂×Q₈×C₄₇	—	4	1	no	—
35	C₂×Dic₉₄	—	4	2	no	—
36	Q₁₆×C₄₇	—	4	4	no	—
37	C₂×C₈×C₄₇	D₈	1	2	yes	—
38	C₂×C₈×C₄₇	D₈	1	2	yes	—
39	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
40	C₄⋊C₄×C₄₇	—	4	1	no	—
41	C₄⋊C₄×C₄₇	—	4	1	no	—
42	C₄⋊C₄×C₄₇	—	4	1	no	—
43	C₄⋊C₄×C₄₇	—	4	1	no	—
44	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₄⋊Dic₄₇	—	4	2	no	—
46	C₄⋊Dic₄₇	—	4	2	no	—
47	Dic₁₈₈	—	4	4	no	—
48	Dic₁₈₈	—	4	4	no	—
49	C₄₇⋊C₁₆	—	2	4	no	—
50	Q₈×C₄₇	—	8	2	no	—
51	Q₈×C₄₇	—	8	2	no	—
52	C₂×C₄×C₄₇	—	8	1	no	—
53	C₂×C₄×C₄₇	—	8	1	no	—
54	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
57	C₂×Dic₄₇	—	8	2	no	—
58	Dic₉₄	—	8	2	no	—
59	Dic₉₄	—	8	2	no	—
60	C₈×C₄₇	—	2	2	no	—
61	C₈×C₄₇	Q₃₂	1	4	yes	complemented
62	C₈×C₄₇	AGL(1,ℤ/8)	1	4	yes	—
63	#12528	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	#493	1	1	yes	characteristic
65	C₄×C₄₇	—	16	1	no	—
66	C₄×C₄₇	—	16	1	no	—
67	C₄×C₄₇	#496	1	2	yes	—
68	C₄×C₄₇	C₂×Q₃₂	1	2	yes	—
69	C₄×C₄₇	#510	1	2	yes	—
70	Dic₄₇	—	16	2	no	—
71	#2247	—	47	2	no	—
72	#2241	—	47	2	no	—
73	#1713	—	47	1	no	—
74	#1707	D₄₇	1	1	yes	characteristic
75	#1312	—	47	1	no	—
76	C₂×C₄₇	#1721	1	1	yes	characteristic
77	C₂×C₄₇	#2247	1	1	yes	characteristic
78	C₂×C₄₇	#2251	1	1	yes	characteristic
79	C₂×Q₃₂	—	94	4	no	—
80	#500	—	94	1	no	—
81	#500	—	94	1	no	—
82	#500	C₂×D₄₇	1	1	yes	characteristic
83	#498	—	2	1	no	—
84	#494	—	47	2	no	—
85	#494	C₂×D₄₇	1	1	yes	characteristic
86	#487	—	2	1	no	—
87	#358	—	94	4	no	—
88	C₄×C₁₆	—	47	2	no	—
89	C₈²	C₂×D₄₇	1	1	yes	characteristic
90	C₄₇	#12528	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	94	2	no	—
93	C₂×Q₁₆	—	94	2	no	—
94	C₄⋊Q₈	—	2	1	no	—
95	C₄⋊Q₈	—	2	1	no	—
96	C₄⋊Q₈	—	94	2	no	—
97	Q₃₂	—	188	8	no	—
98	C₂×C₁₆	—	47	4	no	—
99	#107	—	2	2	no	—
100	#107	—	188	2	no	—
101	#107	—	188	2	no	—
102	#106	—	4	2	no	—
103	#103	—	4	2	no	—
104	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₄×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
107	C₂×Q₈	—	4	1	no	—
108	C₂×Q₈	—	4	1	no	—
109	C₂×Q₈	—	188	2	no	—
110	Q₁₆	—	4	4	no	—
111	Q₁₆	—	188	4	no	—
112	Q₁₆	—	188	4	no	—
113	C₂×C₈	#10625334	1	2	yes	—
114	C₂×C₈	#10625334	1	2	yes	—
115	C₂×C₈	D₄×D₄₇	1	2	yes	—
116	C₄⋊C₄	—	4	1	no	—
117	C₄⋊C₄	—	4	1	no	—
118	C₄⋊C₄	—	4	1	no	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	188	2	no	—
121	C₄⋊C₄	—	188	2	no	—
122	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
123	C₁₆	—	94	4	no	—
124	Q₈	—	8	2	no	—
125	Q₈	—	8	2	no	—
126	Q₈	—	376	2	no	—
127	Q₈	—	376	2	no	—
128	C₂×C₄	—	8	1	no	—
129	C₂×C₄	—	8	1	no	—
130	C₂×C₄	—	376	2	no	—
131	C₂×C₄	#13094020	1	1	yes	characteristic
132	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
133	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	#13093970	1	4	yes	complemented
136	C₈	#13094051	1	4	yes	—
137	C₂²	#15689499	1	1	yes	characteristic
138	C₄	—	16	1	no	—
139	C₄	—	16	1	no	—
140	C₄	—	752	2	no	—
141	C₄	#15689509	1	2	yes	—
142	C₄	C₂×#13093970	1	2	yes	—
143	C₄	#15689563	1	2	yes	—
144	C₂	#20243052	1	1	yes	characteristic
145	C₂	#20245640	1	1	yes	characteristic
146	C₂	#20245660	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245640

C₂

yes

maximal

#20245617

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic

#20241306

C₂

yes

maximal, characteristic

C₄₇×#1707

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#15688926

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#500

C₂²

yes

characteristic

C₄₇×#498

—

C₄₇×#494

C₂²

yes

characteristic

C₄₇×#487

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

#13093970

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093958

—

C₂×Q₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

—

C₄⋊Q₈×C₄₇

—

#107×C₄₇

—

#106×C₄₇

—

#103×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

Q₃₂

yes

complemented

C₈×C₄₇

AGL(1,ℤ/8)

yes

—

—

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#496

yes

—

C₄×C₄₇

C₂×Q₃₂

yes

—

C₄×C₄₇

#510

yes

—

Dic₄₇

—

—

—

—

D₄₇

yes

characteristic

#1312

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

—

#500

—

#500

—

#500

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

characteristic

#487

—

#358

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12528

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

Q₃₂

—

188

—

C₂×C₁₆

—

#107

—

100

#107

—

188

—

101

#107

—

188

—

102

#106

—

103

#103

—

104

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

105

C₄×C₈

C₂²⋊D₄₇

yes

characteristic

106

C₄×C₈

C₂²×D₄₇

yes

characteristic

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

188

—

110

Q₁₆

—

111

Q₁₆

—

188

—

112

Q₁₆

—

188

—

113

C₂×C₈

#10625334

yes

—

114

C₂×C₈

#10625334

yes

—

115

C₂×C₈

D₄×D₄₇

yes

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

188

—

121

C₄⋊C₄

—

188

—

122

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

123

C₁₆

—

124

Q₈

—

125

Q₈

—

126

Q₈

—

376

—

127

Q₈

—

376

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

376

—

131

C₂×C₄

#13094020

yes

characteristic

132

C₂×C₄

C₂×#10625334

yes

characteristic

133

C₂×C₄

C₂×#10625334

yes

characteristic

134

C₈

—

135

C₈

#13093970

yes

complemented

136

C₈

#13094051

yes

—

137

C₂²

#15689499

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

752

—

141

C₄

#15689509

yes

—

142

C₄

C₂×#13093970

yes

—

143

C₄

#15689563

yes

—

144

C₂

#20243052

yes

characteristic

145

C₂

#20245640

yes

characteristic

146

C₂

#20245660

yes

characteristic

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1707	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#1707	D₄₇
hypocenter	C₄₇	#12528
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

subgroup

G/H

center

C₂²

#15689499

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1707

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1707

N	H ≅ G/N	type
61. C₈×C₄₇	Q₃₂	semidirect
90. C₄₇	#12528	semidirect
135. C₈	#13093970	semidirect

H ≅ G/N

type

61. C₈×C₄₇

Q₃₂

semidirect

90. C₄₇

#12528

semidirect

135. C₈

#13093970

semidirect

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

subgroup	index
#20245640	2
#20245617	2
#20243006	2
#20241306	2
C₄₇×#1707	2
#12528	47

subgroup

index

C₄₇×#1707

Subgroups — #26131045

Conjugacy classes of subgroups

Complements

Maximal subgroups