Group #26131047 — subgroups

Subgroups — #26131047

Order 12032.

274 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 91 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	#20245641	C₂	1	2	yes	maximal
2	#20245616	C₂	1	2	yes	maximal
3	#20243006	C₂	1	1	yes	maximal, characteristic
4	#20242985	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1312	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₃₇₆	—	2	4	no	—
7	#15689528	—	2	1	no	—
8	#15689528	—	2	1	no	—
9	#15689528	C₂²	1	1	yes	characteristic
10	#15689521	—	2	1	no	—
11	#15689503	C₂²	1	1	yes	characteristic
12	#15689503	C₂²	1	2	yes	—
13	#15689474	—	2	1	no	—
14	#15688928	—	2	4	no	—
15	C₄×C₁₆×C₄₇	C₂²	1	2	yes	—
16	C₈²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₈₈	—	2	2	no	—
18	C₂×Dic₁₈₈	—	2	2	no	—
19	C₂×Dic₁₈₈	—	2	2	no	—
20	#13094024	—	2	1	no	—
21	#13094024	—	2	1	no	—
22	#13094024	—	2	2	no	—
23	Dic₃₇₆	—	4	8	no	—
24	#13093958	—	2	2	no	—
25	#13093958	—	4	2	no	—
26	#13093958	—	4	2	no	—
27	#13093956	—	4	2	no	—
28	#13093949	—	4	2	no	—
29	C₂×C₁₆×C₄₇	D₄	1	4	yes	—
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×Dic₉₄	—	4	1	no	—
34	C₂×Dic₉₄	—	4	2	no	—
35	C₂×Dic₉₄	—	4	1	no	—
36	C₂×C₈×C₄₇	D₈	1	2	yes	—
37	C₂×C₈×C₄₇	D₈	1	2	yes	—
38	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
39	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊Dic₄₇	—	4	1	no	—
41	C₄⋊Dic₄₇	—	4	1	no	—
42	C₄⋊Dic₄₇	—	4	1	no	—
43	C₄⋊Dic₄₇	—	4	1	no	—
44	C₄⋊Dic₄₇	—	4	2	no	—
45	C₄⋊Dic₄₇	—	4	2	no	—
46	Dic₁₈₈	—	4	4	no	—
47	Dic₁₈₈	—	4	4	no	—
48	Dic₁₈₈	—	4	4	no	—
49	C₁₆×C₄₇	—	2	4	no	—
50	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₄₇	—	8	1	no	—
54	C₂×Dic₄₇	—	8	2	no	—
55	C₂×Dic₄₇	—	8	1	no	—
56	Dic₉₄	—	8	2	no	—
57	Dic₉₄	—	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	#496	1	2	yes	—
66	C₄×C₄₇	C₂×Q₃₂	1	2	yes	—
67	C₄×C₄₇	#510	1	2	yes	—
68	Dic₄₇	—	16	1	no	—
69	Dic₄₇	—	16	2	no	—
70	Dic₄₇	—	16	1	no	—
71	#2247	—	47	2	no	—
72	#2241	—	47	2	no	—
73	#1713	—	47	1	no	—
74	#1707	—	47	1	no	—
75	#1312	D₄₇	1	1	yes	characteristic
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	—	47	1	no	—
81	#500	—	94	1	no	—
82	#500	—	94	1	no	—
83	#498	—	94	1	no	—
84	#494	—	47	1	no	—
85	#494	—	47	2	no	—
86	#487	—	94	1	no	—
87	#358	—	94	4	no	—
88	C₄×C₁₆	C₂×D₄₇	1	2	yes	—
89	C₈²	C₂×D₄₇	1	1	yes	characteristic
90	C₄₇	#12528	1	1	yes	characteristic, complemented, Sylow
91	C₂×Q₁₆	—	94	2	no	—
92	C₂×Q₁₆	—	94	2	no	—
93	C₂×Q₁₆	—	94	2	no	—
94	C₄⋊Q₈	—	94	1	no	—
95	C₄⋊Q₈	—	94	1	no	—
96	C₄⋊Q₈	—	94	2	no	—
97	Q₃₂	—	188	8	no	—
98	C₂×C₁₆	D₁₈₈	1	4	yes	—
99	#107	—	94	2	no	—
100	#107	—	188	2	no	—
101	#107	—	188	2	no	—
102	#106	—	188	2	no	—
103	#103	—	188	2	no	—
104	C₄×C₈	D₁₈₈	1	1	yes	characteristic
105	C₄×C₈	D₁₈₈	1	1	yes	characteristic
106	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
107	C₂×Q₈	—	188	1	no	—
108	C₂×Q₈	—	188	2	no	—
109	C₂×Q₈	—	188	1	no	—
110	Q₁₆	—	188	4	no	—
111	Q₁₆	—	188	4	no	—
112	Q₁₆	—	188	4	no	—
113	C₂×C₈	D₃₇₆	1	2	yes	—
114	C₂×C₈	D₃₇₆	1	2	yes	—
115	C₂×C₈	C₂×D₁₈₈	1	2	yes	—
116	C₄⋊C₄	—	188	1	no	—
117	C₄⋊C₄	—	188	1	no	—
118	C₄⋊C₄	—	188	1	no	—
119	C₄⋊C₄	—	188	1	no	—
120	C₄⋊C₄	—	188	2	no	—
121	C₄⋊C₄	—	188	2	no	—
122	C₄²	C₂×D₁₈₈	1	1	yes	characteristic
123	C₁₆	—	2	4	no	—
124	Q₈	—	376	2	no	—
125	Q₈	—	376	2	no	—
126	Q₈	—	376	2	no	—
127	Q₈	—	376	2	no	—
128	C₂×C₄	—	376	1	no	—
129	C₂×C₄	—	376	2	no	—
130	C₂×C₄	—	376	1	no	—
131	C₂×C₄	#13094021	1	1	yes	characteristic
132	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
133	C₂×C₄	C₂×D₃₇₆	1	1	yes	characteristic
134	C₈	—	2	2	no	—
135	C₈	Dic₃₇₆	1	4	yes	complemented
136	C₈	#13094052	1	4	yes	—
137	C₂²	#15689500	1	1	yes	characteristic
138	C₄	—	752	1	no	—
139	C₄	—	752	2	no	—
140	C₄	—	752	1	no	—
141	C₄	#15689512	1	2	yes	—
142	C₄	C₂×Dic₃₇₆	1	2	yes	—
143	C₄	#15689566	1	2	yes	—
144	C₂	#20243054	1	1	yes	characteristic
145	C₂	#20245641	1	1	yes	characteristic
146	C₂	#20245659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245641

C₂

yes

maximal

#20245616

C₂

yes

maximal

#20243006

C₂

yes

maximal, characteristic

#20242985

C₂

yes

maximal, characteristic

C₄₇×#1312

C₂

yes

maximal, characteristic

C₂×Dic₃₇₆

—

#15689528

—

#15689528

—

#15689528

C₂²

yes

characteristic

#15689521

—

#15689503

C₂²

yes

characteristic

#15689503

C₂²

yes

—

#15689474

—

#15688928

—

C₄×C₁₆×C₄₇

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

C₂×Dic₁₈₈

—

#13094024

—

#13094024

—

#13094024

—

Dic₃₇₆

—

—

—

—

—

—

C₂×C₁₆×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

D₄

yes

characteristic

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

Dic₁₈₈

—

Dic₁₈₈

—

Dic₁₈₈

—

C₁₆×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

Q₃₂

yes

complemented

C₈×C₄₇

AGL(1,ℤ/8)

yes

—

—

C₂²×C₄₇

yes

C₄×C₄₇

yes

—

C₄×C₄₇

C₂×Q₃₂

yes

—

C₄×C₄₇

#510

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×Q₃₂

—

—

—

—

—

—

—

—

—

C₄×C₁₆

C₂×D₄₇

yes

—

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

D₁₈₈

yes

—

#107

—

100

#107

—

188

—

101

#107

—

188

—

102

#106

—

188

—

103

#103

—

188

—

104

C₄×C₈

D₁₈₈

yes

characteristic

105

C₄×C₈

D₁₈₈

yes

characteristic

106

C₄×C₈

C₂²×D₄₇

yes

characteristic

107

C₂×Q₈

—

188

—

108

C₂×Q₈

—

188

—

109

C₂×Q₈

—

188

—

110

Q₁₆

—

188

—

111

Q₁₆

—

188

—

112

Q₁₆

—

188

—

113

C₂×C₈

D₃₇₆

yes

—

114

C₂×C₈

D₃₇₆

yes

—

115

C₂×C₈

C₂×D₁₈₈

yes

—

116

C₄⋊C₄

—

188

—

117

C₄⋊C₄

—

188

—

118

C₄⋊C₄

—

188

—

119

C₄⋊C₄

—

188

—

120

C₄⋊C₄

—

188

—

121

C₄⋊C₄

—

188

—

122

C₄²

C₂×D₁₈₈

yes

characteristic

123

C₁₆

—

124

Q₈

—

376

—

125

Q₈

—

376

—

126

Q₈

—

376

—

127

Q₈

—

376

—

128

C₂×C₄

—

376

—

129

C₂×C₄

—

376

—

130

C₂×C₄

—

376

—

131

C₂×C₄

#13094021

yes

characteristic

132

C₂×C₄

C₂×D₃₇₆

yes

characteristic

133

C₂×C₄

C₂×D₃₇₆

yes

characteristic

134

C₈

—

135

C₈

Dic₃₇₆

yes

complemented

136

C₈

#13094052

yes

—

137

C₂²

#15689500

yes

characteristic

138

C₄

—

752

—

139

C₄

—

752

—

140

C₄

—

752

—

141

C₄

#15689512

yes

—

142

C₄

C₂×Dic₃₇₆

yes

—

143

C₄

#15689566

yes

—

144

C₂

#20243054

yes

characteristic

145

C₂

#20245641

yes

characteristic

146

C₂

#20245659

yes

characteristic

	subgroup	G/H
center	C₂²	#15689500
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting	C₄₇×#1312	C₂
socle, prime radical	C₂²×C₄₇	#493
hypercenter	#1312	D₄₇
hypocenter	C₄₇	#12528
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#15689500

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting

C₄₇×#1312

C₂

C₂²×C₄₇

D₄₇

C₄₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
61. C₈×C₄₇	Q₃₂	semidirect
90. C₄₇	#12528	semidirect
135. C₈	Dic₃₇₆	semidirect

H ≅ G/N

type

61. C₈×C₄₇

Q₃₂

semidirect

90. C₄₇

#12528

semidirect

135. C₈

Dic₃₇₆

semidirect

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

subgroup	index
#20245641	2
#20245616	2
#20243006	2
#20242985	2
C₄₇×#1312	2
#12528	47

subgroup

index

C₄₇×#1312

Subgroups — #26131047

Conjugacy classes of subgroups

Complements

Maximal subgroups