Group #26131693 — subgroups

Subgroups — #26131693

Order 12032.

306 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 135 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×#15688952	C₂	1	2	yes	maximal
2	C₂×#15688898	C₂	1	1	yes	maximal, characteristic
3	#108×Dic₄₇	C₂	1	2	yes	maximal
4	#20241482	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#13093960	C₂²	1	2	yes	—
8	M₁₆×Dic₄₇	—	2	2	no	—
9	C₂×C₈×Dic₄₇	C₂²	1	1	yes	characteristic
10	#15688952	C₄	1	8	yes	complemented
11	#15688898	C₄	1	4	yes	—
12	#15688887	—	2	2	no	—
13	C₂²×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₄₇	C₂²	1	2	yes	—
15	C₂×#10625329	—	2	2	no	—
16	C₂²×C₄₇⋊C₈	D₄	1	1	yes	characteristic
17	C₂×C₄×Dic₄₇	D₄	1	1	yes	characteristic
18	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
20	#13093960	—	2	2	no	—
21	#13093960	C₂×C₄	1	4	yes	—
22	#13093934	—	4	2	no	—
23	C₈×Dic₄₇	—	4	2	no	—
24	C₈×Dic₄₇	C₂×C₄	1	2	yes	—
25	C₈×Dic₄₇	Q₈	1	2	yes	—
26	C₂×M₁₆×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₁₆×C₄₇	—	2	2	no	—
29	C₂×C₁₆×C₄₇	C₂×C₄	1	2	yes	—
30	#108×C₄₇	—	2	2	no	—
31	#108×C₄₇	C₂×C₄	1	4	yes	—
32	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×Dic₄₇	Q₁₆	1	2	yes	—
34	M₁₆×C₄₇	—	2	4	no	—
35	M₁₆×C₄₇	—	4	2	no	—
36	C₂×C₈×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
39	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₄₇	#41	1	2	yes	—
41	C₄×Dic₄₇	C₄⋊C₄	1	2	yes	—
42	C₄×Dic₄₇	D₈	1	2	yes	—
43	#10625329	—	2	4	no	—
44	#10625329	—	4	2	no	—
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂×C₄₇⋊C₈	—	4	2	no	—
47	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
48	C₁₆×C₄₇	—	2	2	no	—
49	C₄₇⋊C₁₆	C₄²	1	4	yes	—
50	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#135	1	2	yes	—
55	C₂×Dic₄₇	—	4	2	no	—
56	C₂×Dic₄₇	#107	1	4	yes	—
57	C₈×C₄₇	—	4	4	no	—
58	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
59	C₈×C₄₇	#117	1	1	yes	characteristic
60	C₈×C₄₇	C₄×Q₈	1	2	yes	—
61	C₄₇⋊C₈	—	2	2	no	—
62	C₄₇⋊C₈	—	4	4	no	—
63	#12676	—	47	1	no	maximal, Sylow
64	C₂²×C₄₇	—	2	2	no	—
65	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
66	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
67	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
68	C₄×C₄₇	#427	1	1	yes	characteristic
69	C₄×C₄₇	C₄×D₈	1	2	yes	—
70	Dic₄₇	—	2	4	no	—
71	C₂×#368	—	47	2	no	—
72	C₂×#346	—	47	1	no	—
73	C₄×#108	—	47	2	no	—
74	#1381	—	47	1	no	—
75	#1381	D₄₇	1	1	yes	characteristic
76	C₂×C₄₇	—	2	2	no	—
77	C₂×C₄₇	C₄×#107	1	1	yes	characteristic
78	C₂×C₄₇	#2161	1	2	yes	—
79	C₂²×C₁₆	—	47	1	no	—
80	C₂²×C₁₆	C₂×D₄₇	1	1	yes	characteristic
81	C₂×#108	—	47	2	no	—
82	C₂×#108	C₂×D₄₇	1	2	yes	—
83	C₄×M₁₆	—	94	2	no	—
84	C₂×C₄×C₈	—	47	1	no	—
85	#368	—	47	8	no	—
86	#346	—	47	4	no	—
87	#341	—	94	2	no	—
88	C₄₇	#12676	1	1	yes	characteristic, complemented, Sylow
89	C₂×M₁₆	—	2	2	no	—
90	C₂×M₁₆	—	94	2	no	—
91	C₂²×C₈	—	47	1	no	—
92	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
93	C₂×C₄²	—	47	1	no	—
94	C₂×C₁₆	—	2	2	no	—
95	C₂×C₁₆	—	47	2	no	—
96	C₂×C₁₆	—	47	4	no	—
97	C₂×C₁₆	C₄×D₄₇	1	2	yes	—
98	#108	—	2	2	no	—
99	#108	—	47	4	no	—
100	#108	—	94	2	no	—
101	#108	C₄×D₄₇	1	4	yes	—
102	#97	—	188	2	no	—
103	C₄×C₈	—	47	2	no	—
104	C₄×C₈	—	47	2	no	—
105	C₄×C₈	—	188	2	no	—
106	C₂²×C₄	—	47	2	no	—
107	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
108	M₁₆	—	2	4	no	—
109	M₁₆	—	4	2	no	—
110	M₁₆	—	94	4	no	—
111	M₁₆	—	188	2	no	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	47	2	no	—
114	C₂×C₈	—	94	2	no	—
115	C₂×C₈	—	188	2	no	—
116	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
117	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
118	C₂×C₈	#10625352	1	2	yes	—
119	C₂×C₈	Q₈×D₄₇	1	1	yes	characteristic
120	C₄²	—	47	2	no	—
121	C₄²	—	47	2	no	—
122	C₁₆	—	2	2	no	—
123	C₁₆	—	47	4	no	—
124	C₂³	Q₁₆×D₄₇	1	1	yes	characteristic
125	C₂×C₄	—	47	4	no	—
126	C₂×C₄	—	188	2	no	—
127	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
128	C₂×C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
129	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
130	C₂×C₄	#13094045	1	2	yes	—
131	C₈	—	4	4	no	—
132	C₈	—	94	2	no	—
133	C₈	—	188	4	no	—
134	C₈	C₄²×D₄₇	1	1	yes	characteristic
135	C₈	#13093980	1	1	yes	characteristic
136	C₈	#13093989	1	2	yes	—
137	C₂²	—	2	2	no	—
138	C₂²	#107×D₄₇	1	1	yes	characteristic
139	C₂²	#15689215	1	2	yes	—
140	C₄	—	94	4	no	—
141	C₄	#15688974	1	1	yes	characteristic
142	C₄	#15689158	1	1	yes	characteristic
143	C₄	#15689203	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#20243280	1	1	yes	characteristic
146	C₂	#20245181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688952

C₂

yes

maximal

C₂×#15688898

C₂

yes

maximal, characteristic

#108×Dic₄₇

C₂

yes

maximal

#20241482

C₂

yes

maximal, characteristic

C₄₇×#1381

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

M₁₆×Dic₄₇

—

C₂×C₈×Dic₄₇

C₂²

yes

characteristic

#15688952

C₄

yes

complemented

#15688898

C₄

yes

—

#15688887

—

C₂²×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×#108×C₄₇

C₂²

yes

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

#13093960

—

#13093960

C₂×C₄

yes

—

#13093934

—

C₈×Dic₄₇

—

C₈×Dic₄₇

C₂×C₄

yes

—

C₈×Dic₄₇

Q₈

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂×C₄

yes

—

#108×C₄₇

—

#108×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×Dic₄₇

Q₁₆

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂²×C₄

yes

—

C₂×C₈×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₄₇

#41

yes

—

C₄×Dic₄₇

C₄⋊C₄

yes

—

C₄×Dic₄₇

D₈

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

C₄²

yes

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

#107

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₄²

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₄×Q₈

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#12676

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

C₄×Q₁₆

yes

—

C₄×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₄×C₄₇

#427

yes

characteristic

C₄×C₄₇

C₄×D₈

yes

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

C₄×#107

yes

characteristic

C₂×C₄₇

#2161

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₄₇

yes

characteristic

C₂×#108

—

C₂×#108

C₂×D₄₇

yes

—

C₄×M₁₆

—

C₂×C₄×C₈

—

#368

—

#346

—

#341

—

C₄₇

#12676

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₄₇

yes

—

#108

—

#108

—

100

#108

—

101

#108

C₄×D₄₇

yes

—

102

#97

—

188

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

188

—

106

C₂²×C₄

—

107

C₂²×C₄

D₄×D₄₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

M₁₆

—

188

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

188

—

116

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

117

C₂×C₈

C₂×C₄×D₄₇

yes

—

118

C₂×C₈

#10625352

yes

—

119

C₂×C₈

Q₈×D₄₇

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₁₆

—

123

C₁₆

—

124

C₂³

Q₁₆×D₄₇

yes

characteristic

125

C₂×C₄

—

126

C₂×C₄

—

188

—

127

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

128

C₂×C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

129

C₂×C₄

D₈×D₄₇

yes

characteristic

130

C₂×C₄

#13094045

yes

—

131

C₈

—

132

C₈

—

133

C₈

—

188

—

134

C₈

C₄²×D₄₇

yes

characteristic

135

C₈

#13093980

yes

characteristic

136

C₈

#13093989

yes

—

137

C₂²

—

138

C₂²

#107×D₄₇

yes

characteristic

139

C₂²

#15689215

yes

—

140

C₄

—

141

C₄

#15688974

yes

characteristic

142

C₄

#15689158

yes

characteristic

143

C₄

#15689203

yes

—

144

C₂

—

145

C₂

#20243280

yes

characteristic

146

C₂

#20245181

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄₇×#1381	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	#1381	D₄₇
hypocenter	C₄₇	#12676
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₄₇

commutator

C₈×C₄₇

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄₇×#1381

C₂

socle

C₂²×C₄₇

C₂×#107

prime radical

C₂³×C₄₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
10. #15688952	C₄	semidirect
88. C₄₇	#12676	semidirect

H ≅ G/N

type

10. #15688952

C₄

semidirect

88. C₄₇

#12676

semidirect

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

subgroup	index
C₂×#15688952	2
C₂×#15688898	2
#108×Dic₄₇	2
#20241482	2
C₄₇×#1381	2
#12676	47

subgroup

index

#108×Dic₄₇

C₄₇×#1381

Subgroups — #26131693

Conjugacy classes of subgroups

Complements

Maximal subgroups