Group #26110054 — subgroups

Subgroups — #26110054

Order 12032.

422 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 203 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	#20243548	C₂	1	1	yes	maximal, characteristic
2	#20243515	C₂	1	4	yes	maximal
3	C₂×C₈×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₄₇×#1845	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093958	—	2	2	no	—
6	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
7	#15688993	C₂²	1	2	yes	—
8	#15688885	—	2	8	no	—
9	C₈×C₄₇⋊C₈	C₄	1	4	yes	complemented
10	C₂×#107×C₄₇	—	2	2	no	—
11	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#384	C₂²	1	2	yes	—
13	C₂²×C₄₇⋊C₈	D₄	1	4	yes	—
14	C₂×C₄⋊Dic₄₇	—	2	4	no	—
15	C₂×C₄⋊Dic₄₇	—	2	2	no	—
16	#13093958	—	4	4	no	—
17	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
18	C₄×C₄₇⋊C₈	Q₈	1	4	yes	—
19	C₂²×C₈×C₄₇	D₄	1	2	yes	—
20	C₂×C₄⋊C₄×C₄₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
22	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
23	#107×C₄₇	—	4	4	no	—
24	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₄₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₄₇	—	4	2	no	—
27	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
29	C₂²×Dic₄₇	—	4	2	no	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₄₇	D₈	1	4	yes	—
32	C₂×C₈×C₄₇	Q₁₆	1	4	yes	—
33	C₄⋊C₄×C₄₇	—	4	4	no	—
34	C₄⋊C₄×C₄₇	—	4	8	no	—
35	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×Q₈	1	2	yes	—
38	C₄⋊Dic₄₇	—	4	4	no	—
39	C₄⋊Dic₄₇	—	4	8	no	—
40	C₂×C₄₇⋊C₈	—	2	4	no	—
41	C₂×C₄₇⋊C₈	—	2	4	no	—
42	C₂×C₄₇⋊C₈	C₄⋊C₄	1	8	yes	—
43	C₂³×C₄₇	#127	1	1	yes	characteristic
44	C₂×C₄×C₄₇	—	8	2	no	—
45	C₂×C₄×C₄₇	—	8	4	no	—
46	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
52	C₂×C₄×C₄₇	#135	1	4	yes	—
53	C₂×C₄×C₄₇	#135	1	4	yes	—
54	C₂×Dic₄₇	—	8	2	no	—
55	C₂×Dic₄₇	—	8	4	no	—
56	C₈×C₄₇	#107	1	8	yes	—
57	C₄₇⋊C₈	—	2	8	no	—
58	#8816	—	47	1	no	maximal, Sylow
59	C₂²×C₄₇	#384	1	1	yes	characteristic
60	C₂²×C₄₇	#493	1	1	yes	characteristic
61	C₂²×C₄₇	#494	1	1	yes	characteristic
62	C₂²×C₄₇	#495	1	2	yes	—
63	C₂²×C₄₇	#495	1	2	yes	—
64	C₄×C₄₇	—	16	2	no	—
65	C₄×C₄₇	C₂×#107	1	2	yes	—
66	C₄×C₄₇	#427	1	4	yes	—
67	C₄×C₄₇	#499	1	4	yes	—
68	C₄×C₄₇	#500	1	2	yes	—
69	Dic₄₇	—	16	2	no	—
70	#1845	—	47	1	no	—
71	#1845	D₄₇	1	1	yes	characteristic
72	#1834	—	47	4	no	—
73	C₂×C₈²	—	47	1	no	—
74	C₂×C₄₇	#1709	1	1	yes	characteristic
75	C₂×C₄₇	#1709	1	1	yes	characteristic
76	C₂×C₄₇	#1710	1	2	yes	—
77	C₂×C₄₇	#1845	1	1	yes	characteristic
78	C₂×C₄₇	#1846	1	2	yes	—
79	C₂×#107	—	2	2	no	—
80	C₂×#107	—	94	2	no	—
81	C₂×C₄×C₈	—	47	2	no	—
82	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
83	#384	—	47	2	no	—
84	#384	C₂×D₄₇	1	2	yes	—
85	#340	—	94	8	no	—
86	C₈²	—	47	4	no	—
87	C₄₇	#8816	1	1	yes	characteristic, complemented, Sylow
88	C₂²×C₈	—	47	4	no	—
89	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
90	C₂×C₄⋊C₄	—	2	4	no	—
91	C₂×C₄⋊C₄	—	2	2	no	—
92	C₂×C₄⋊C₄	—	94	4	no	—
93	C₂×C₄⋊C₄	—	94	2	no	—
94	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
95	#107	—	4	4	no	—
96	#107	—	188	4	no	—
97	C₄×C₈	—	47	4	no	—
98	C₄×C₈	—	47	4	no	—
99	C₄×C₈	Dic₉₄	1	2	yes	—
100	C₄×C₈	C₄×D₄₇	1	2	yes	—
101	C₂²×C₄	—	4	2	no	—
102	C₂²×C₄	—	188	2	no	—
103	C₂²×C₄	D₄×D₄₇	1	2	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂×C₈	—	47	8	no	—
106	C₂×C₈	—	94	4	no	—
107	C₂×C₈	—	94	4	no	—
108	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
109	C₂×C₈	#10625334	1	4	yes	—
110	C₂×C₈	#10625337	1	4	yes	—
111	C₄⋊C₄	—	4	4	no	—
112	C₄⋊C₄	—	4	8	no	—
113	C₄⋊C₄	—	188	4	no	—
114	C₄⋊C₄	—	188	8	no	—
115	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
116	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
117	C₄²	Q₈×D₄₇	1	2	yes	—
118	C₂³	#13094020	1	1	yes	characteristic
119	C₂×C₄	—	8	2	no	—
120	C₂×C₄	—	8	4	no	—
121	C₂×C₄	—	376	2	no	—
122	C₂×C₄	—	376	4	no	—
123	C₂×C₄	C₄⋊C₄×D₄₇	1	2	yes	—
124	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
125	C₂×C₄	#13094022	1	1	yes	characteristic
126	C₂×C₄	#13094023	1	2	yes	—
127	C₂×C₄	C₂×#10625334	1	2	yes	—
128	C₂×C₄	C₂×#10625337	1	2	yes	—
129	C₂×C₄	#13094043	1	4	yes	—
130	C₂×C₄	#13094047	1	4	yes	—
131	C₈	—	94	8	no	—
132	C₈	#13093957	1	8	yes	—
133	C₂²	#15688992	1	1	yes	characteristic
134	C₂²	#15689499	1	1	yes	characteristic
135	C₂²	#15689502	1	1	yes	characteristic
136	C₂²	#15689506	1	2	yes	—
137	C₂²	#15689508	1	2	yes	—
138	C₄	—	16	2	no	—
139	C₄	—	752	2	no	—
140	C₄	C₂×#13093957	1	2	yes	—
141	C₄	#15689160	1	4	yes	—
142	C₄	#15689524	1	4	yes	—
143	C₄	#15689526	1	2	yes	—
144	C₂	#20242993	1	1	yes	characteristic
145	C₂	#20242995	1	1	yes	characteristic
146	C₂	#20242997	1	2	yes	—
147	C₂	#20243547	1	1	yes	characteristic
148	C₂	#20243551	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243548

C₂

yes

maximal, characteristic

#20243515

C₂

yes

maximal

C₂×C₈×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₄₇×#1845

C₂

yes

maximal, characteristic

C₂×#13093958

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

#15688993

C₂²

yes

—

#15688885

—

C₈×C₄₇⋊C₈

C₄

yes

complemented

C₂×#107×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#384

C₂²

yes

—

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093958

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

Q₁₆

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×C₄×C₄₇

C₂×Q₁₆

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

#107

yes

—

C₄₇⋊C₈

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#495

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×#107

yes

—

C₄×C₄₇

#427

yes

—

C₄×C₄₇

#499

yes

—

C₄×C₄₇

#500

yes

—

Dic₄₇

—

#1845

—

#1845

D₄₇

yes

characteristic

#1834

—

C₂×C₈²

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#1845

yes

characteristic

C₂×C₄₇

#1846

yes

—

C₂×#107

—

C₂×#107

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#384

—

#384

C₂×D₄₇

yes

—

#340

—

C₈²

—

C₄₇

#8816

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#107

—

#107

—

188

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₉₄

yes

—

100

C₄×C₈

C₄×D₄₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

—

188

—

103

C₂²×C₄

D₄×D₄₇

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

Dic₄₇⋊C₄

yes

—

109

C₂×C₈

#10625334

yes

—

110

C₂×C₈

#10625337

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

188

—

114

C₄⋊C₄

—

188

—

115

C₄²

C₂×Dic₉₄

yes

characteristic

116

C₄²

C₂×C₄×D₄₇

yes

characteristic

117

C₄²

Q₈×D₄₇

yes

—

118

C₂³

#13094020

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

376

—

122

C₂×C₄

—

376

—

123

C₂×C₄

C₄⋊C₄×D₄₇

yes

—

124

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

125

C₂×C₄

#13094022

yes

characteristic

126

C₂×C₄

#13094023

yes

—

127

C₂×C₄

C₂×#10625334

yes

—

128

C₂×C₄

C₂×#10625337

yes

—

129

C₂×C₄

#13094043

yes

—

130

C₂×C₄

#13094047

yes

—

131

C₈

—

132

C₈

#13093957

yes

—

133

C₂²

#15688992

yes

characteristic

134

C₂²

#15689499

yes

characteristic

135

C₂²

#15689502

yes

characteristic

136

C₂²

#15689506

yes

—

137

C₂²

#15689508

yes

—

138

C₄

—

139

C₄

—

752

—

140

C₄

C₂×#13093957

yes

—

141

C₄

#15689160

yes

—

142

C₄

#15689524

yes

—

143

C₄

#15689526

yes

—

144

C₂

#20242993

yes

characteristic

145

C₂

#20242995

yes

characteristic

146

C₂

#20242997

yes

—

147

C₂

#20243547

yes

characteristic

148

C₂

#20243551

yes

—

	subgroup	G/H
center	C₂³	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1845	C₂
socle, prime radical	C₂³×C₄₇	#127
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1845	D₄₇
hypocenter	C₄₇	#8816
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

subgroup

G/H

center

C₂³

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1845

C₂

socle, prime radical

C₂³×C₄₇

#127

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

N	H ≅ G/N	type
9. C₈×C₄₇⋊C₈	C₄	semidirect
87. C₄₇	#8816	semidirect

H ≅ G/N

type

9. C₈×C₄₇⋊C₈

C₄

semidirect

87. C₄₇

#8816

semidirect

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

subgroup	index
#20243548	2
#20243515	2
C₂×C₈×C₄₇⋊C₈	2
C₄₇×#1845	2
#8816	47

subgroup

index

#20243548

#20243515

C₂×C₈×C₄₇⋊C₈

C₄₇×#1845

#8816

Subgroups — #26110054

Conjugacy classes of subgroups

Complements

Maximal subgroups