Group #26131691 — subgroups

Subgroups — C₄×#15688952 #26131691

Order 12032.

314 conjugacy classes of subgroups; only the 159 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 159 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×C₄×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	C₄×#13093960	C₂	1	1	yes	maximal, characteristic
4	#20241481	C₂	1	2	yes	maximal
5	C₄×#108×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
8	C₂×#13093960	C₂²	1	2	yes	—
9	C₄×#10625329	—	2	1	no	—
10	#15688952	C₄	1	8	yes	direct product factor
11	C₄×C₄₇⋊C₁₆	C₄	1	4	yes	—
12	#15688888	—	2	1	no	—
13	C₂×#108×C₄₇	C₂²	1	2	yes	—
14	C₄×M₁₆×C₄₇	—	2	1	no	—
15	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄₇×#341	—	2	1	no	—
17	C₂×#10625329	—	2	2	no	—
18	C₂×C₄₇⋊C₁₆	—	2	2	no	—
19	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
20	C₂×C₄₇⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₄₇⋊C₁₆	C₂×C₄	1	4	yes	—
22	#13093960	—	2	2	no	—
23	#13093960	C₂×C₄	1	4	yes	—
24	#13093935	—	4	1	no	—
25	C₄×C₄₇⋊C₈	—	4	1	no	—
26	C₂×M₁₆×C₄₇	—	2	2	no	—
27	C₂²×C₈×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
30	#108×C₄₇	—	2	2	no	—
31	#108×C₄₇	C₂×C₄	1	4	yes	—
32	#97×C₄₇	—	4	1	no	—
33	C₄×C₈×C₄₇	—	4	1	no	—
34	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
35	C₄×C₈×C₄₇	Q₈	1	2	yes	—
36	C₂²×C₄×C₄₇	Q₁₆	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
38	M₁₆×C₄₇	—	2	4	no	—
39	M₁₆×C₄₇	—	4	2	no	—
40	C₂×C₈×C₄₇	—	2	2	no	—
41	C₂×C₈×C₄₇	—	4	2	no	—
42	C₂×C₈×C₄₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₄₇	C₂²×C₄	1	2	yes	—
45	C₂×C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₄₇	#41	1	2	yes	—
47	C₄²×C₄₇	C₄⋊C₄	1	2	yes	—
48	C₄²×C₄₇	D₈	1	1	yes	characteristic
49	C₄²×C₄₇	D₈	1	1	yes	characteristic
50	#10625329	—	2	4	no	—
51	#10625329	—	4	2	no	—
52	C₂×C₄₇⋊C₈	—	4	2	no	—
53	C₄₇⋊C₁₆	—	2	2	no	—
54	C₄₇⋊C₁₆	C₄²	1	4	yes	—
55	C₂³×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	#107	1	2	yes	—
58	C₂×C₄×C₄₇	#107	1	2	yes	—
59	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
61	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₄₇	#135	1	2	yes	—
63	C₈×C₄₇	—	2	2	no	—
64	C₈×C₄₇	—	4	4	no	—
65	C₈×C₄₇	C₂×C₄²	1	1	yes	characteristic
66	C₈×C₄₇	#117	1	1	yes	characteristic
67	C₈×C₄₇	C₄×Q₈	1	2	yes	—
68	C₄₇⋊C₈	—	4	4	no	—
69	C₄×#368	—	47	1	no	maximal, Sylow
70	C₂²×C₄₇	—	2	2	no	—
71	C₂²×C₄₇	C₂×#107	1	1	yes	characteristic
72	C₂²×C₄₇	C₄×Q₁₆	1	2	yes	—
73	C₄×C₄₇	—	2	2	no	—
74	C₄×C₄₇	#368	1	4	yes	complemented
75	C₄×C₄₇	C₄×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₄₇	#427	1	1	yes	characteristic
77	C₄×C₄₇	C₄×D₈	1	2	yes	—
78	C₂×#368	—	47	2	no	—
79	C₂×C₄×C₁₆	—	47	1	no	—
80	C₄×#108	—	47	1	no	—
81	C₄×#108	D₄₇	1	1	yes	characteristic
82	#1381	—	47	2	no	—
83	C₂×C₄₇	—	2	2	no	—
84	C₂×C₄₇	C₄×#107	1	1	yes	characteristic
85	C₂×C₄₇	C₂×#368	1	2	yes	—
86	C₂²×C₁₆	—	47	1	no	—
87	C₂²×C₁₆	—	47	1	no	—
88	C₂×#108	—	47	2	no	—
89	C₂×#108	C₂×D₄₇	1	2	yes	—
90	C₄×M₁₆	—	2	1	no	—
91	C₄×M₁₆	—	94	1	no	—
92	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
93	#368	—	47	8	no	—
94	C₄×C₁₆	—	47	4	no	—
95	#341	—	2	1	no	—
96	#341	—	94	1	no	—
97	C₄₇	C₄×#368	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	94	2	no	—
100	C₂²×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
101	C₂²×C₈	C₂²×D₄₇	1	1	yes	characteristic
102	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
103	C₂×C₁₆	—	47	2	no	—
104	C₂×C₁₆	—	47	4	no	—
105	C₂×C₁₆	—	47	2	no	—
106	C₂×C₁₆	—	94	2	no	—
107	#108	—	2	2	no	—
108	#108	—	47	4	no	—
109	#108	—	94	2	no	—
110	#108	C₄×D₄₇	1	4	yes	—
111	#97	—	4	1	no	—
112	#97	—	188	1	no	—
113	C₄×C₈	—	4	1	no	—
114	C₄×C₈	—	188	1	no	—
115	C₄×C₈	Dic₉₄	1	2	yes	—
116	C₄×C₈	C₄×D₄₇	1	2	yes	—
117	C₂²×C₄	#10625337	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
119	M₁₆	—	2	4	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	94	4	no	—
122	M₁₆	—	188	2	no	—
123	C₂×C₈	—	2	2	no	—
124	C₂×C₈	—	4	2	no	—
125	C₂×C₈	—	188	2	no	—
126	C₂×C₈	Dic₄₇⋊C₄	1	2	yes	—
127	C₂×C₈	C₂×Dic₉₄	1	1	yes	characteristic
128	C₂×C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
129	C₂×C₈	C₂×C₄×D₄₇	1	2	yes	—
130	C₂×C₈	#10625350	1	2	yes	—
131	C₄²	Dic₄₇⋊C₄	1	2	yes	—
132	C₄²	#10625334	1	1	yes	characteristic
133	C₄²	#10625334	1	1	yes	characteristic
134	C₁₆	—	47	4	no	—
135	C₁₆	—	94	2	no	—
136	C₂³	C₂×#10625337	1	1	yes	characteristic
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	#13093957	1	2	yes	—
139	C₂×C₄	#13093957	1	2	yes	—
140	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
141	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
142	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
143	C₂×C₄	#13094044	1	2	yes	—
144	C₈	—	2	2	no	—
145	C₈	—	4	4	no	—
146	C₈	—	188	4	no	—
147	C₈	C₄²×D₄₇	1	1	yes	characteristic
148	C₈	#13093981	1	1	yes	characteristic
149	C₈	C₄×Dic₉₄	1	2	yes	—
150	C₂²	—	2	2	no	—
151	C₂²	C₂×#13093957	1	1	yes	characteristic
152	C₂²	C₄×#10625337	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	#15688952	1	4	yes	direct product factor
155	C₄	C₄×Dic₄₇⋊C₄	1	1	yes	characteristic
156	C₄	#15689159	1	1	yes	characteristic
157	C₄	C₄×#10625334	1	2	yes	—
158	C₂	—	2	2	no	—
159	C₂	C₄×#13093957	1	1	yes	characteristic
160	C₂	C₂×#15688952	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15688952

C₂

yes

maximal

C₂×C₄×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic

C₄×#13093960

C₂

yes

maximal, characteristic

#20241481

C₂

yes

maximal

C₄×#108×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂²×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×#13093960

C₂²

yes

—

C₄×#10625329

—

#15688952

C₄

yes

direct product factor

C₄×C₄₇⋊C₁₆

C₄

yes

—

#15688888

—

C₂×#108×C₄₇

C₂²

yes

—

C₄×M₁₆×C₄₇

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#341

—

C₂×#10625329

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

C₂×C₄₇⋊C₁₆

C₂×C₄

yes

—

#13093960

—

#13093960

C₂×C₄

yes

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

characteristic

C₂²×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄²×C₄₇

D₄

yes

characteristic

#108×C₄₇

—

#108×C₄₇

C₂×C₄

yes

—

#97×C₄₇

—

C₄×C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

Q₁₆

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

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

—

C₄²×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

D₈

yes

characteristic

C₄²×C₄₇

D₈

yes

characteristic

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

C₄²

yes

—

C₂³×C₄₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#107

yes

—

C₂×C₄×C₄₇

#107

yes

—

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

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×C₄²

yes

characteristic

C₈×C₄₇

#117

yes

characteristic

C₈×C₄₇

C₄×Q₈

yes

—

C₄₇⋊C₈

—

C₄×#368

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

C₂×#107

yes

characteristic

C₂²×C₄₇

C₄×Q₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

#368

yes

complemented

C₄×C₄₇

C₄×C₄⋊C₄

yes

characteristic

C₄×C₄₇

#427

yes

characteristic

C₄×C₄₇

C₄×D₈

yes

—

C₂×#368

—

C₂×C₄×C₁₆

—

C₄×#108

—

C₄×#108

D₄₇

yes

characteristic

#1381

—

C₂×C₄₇

—

C₂×C₄₇

C₄×#107

yes

characteristic

C₂×C₄₇

C₂×#368

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

—

C₂×#108

—

C₂×#108

C₂×D₄₇

yes

—

C₄×M₁₆

—

C₄×M₁₆

—

C₂×C₄×C₈

C₂×D₄₇

yes

characteristic

#368

—

C₄×C₁₆

—

#341

—

#341

—

C₄₇

C₄×#368

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

C₂²⋊D₄₇

yes

characteristic

101

C₂²×C₈

C₂²×D₄₇

yes

characteristic

102

C₂×C₄²

C₂²⋊D₄₇

yes

characteristic

103

C₂×C₁₆

—

104

C₂×C₁₆

—

105

C₂×C₁₆

—

106

C₂×C₁₆

—

107

#108

—

108

#108

—

109

#108

—

110

#108

C₄×D₄₇

yes

—

111

#97

—

112

#97

—

188

—

113

C₄×C₈

—

114

C₄×C₈

—

188

—

115

C₄×C₈

Dic₉₄

yes

—

116

C₄×C₈

C₄×D₄₇

yes

—

117

C₂²×C₄

#10625337

yes

—

118

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

119

M₁₆

—

120

M₁₆

—

121

M₁₆

—

122

M₁₆

—

188

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

188

—

126

C₂×C₈

Dic₄₇⋊C₄

yes

—

127

C₂×C₈

C₂×Dic₉₄

yes

characteristic

128

C₂×C₈

C₂×C₄×D₄₇

yes

characteristic

129

C₂×C₈

C₂×C₄×D₄₇

yes

—

130

C₂×C₈

#10625350

yes

—

131

C₄²

Dic₄₇⋊C₄

yes

—

132

C₄²

#10625334

yes

characteristic

133

C₄²

#10625334

yes

characteristic

134

C₁₆

—

135

C₁₆

—

136

C₂³

C₂×#10625337

yes

characteristic

137

C₂×C₄

—

138

C₂×C₄

#13093957

yes

—

139

C₂×C₄

#13093957

yes

—

140

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

141

C₂×C₄

C₄×C₂²⋊D₄₇

yes

—

142

C₂×C₄

C₂×#10625334

yes

characteristic

143

C₂×C₄

#13094044

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

188

—

147

C₈

C₄²×D₄₇

yes

characteristic

148

C₈

#13093981

yes

characteristic

149

C₈

C₄×Dic₉₄

yes

—

150

C₂²

—

151

C₂²

C₂×#13093957

yes

characteristic

152

C₂²

C₄×#10625337

yes

—

153

C₄

—

154

C₄

#15688952

yes

direct product factor

155

C₄

C₄×Dic₄₇⋊C₄

yes

characteristic

156

C₄

#15689159

yes

characteristic

157

C₄

C₄×#10625334

yes

—

158

C₂

—

159

C₂

C₄×#13093957

yes

characteristic

160

C₂

C₂×#15688952

yes

—

	subgroup	G/H
center	C₄²	#10625334
commutator	C₈×C₄₇	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₄₇
Fitting	C₄×#108×C₄₇	C₂
socle	C₂²×C₄₇	C₂×#107
prime radical	C₂³×C₄₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₄₇	C₂³
hypercenter	C₄×#108	D₄₇
hypocenter	C₄₇	C₄×#368
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

subgroup

G/H

center

C₄²

#10625334

commutator

C₈×C₄₇

C₂×C₄²

Frattini

C₂²×C₈

C₂²×D₄₇

Fitting

C₄×#108×C₄₇

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 ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₄×#108

N	H ≅ G/N	type
10. #15688952	C₄	direct (N × H)
74. C₄×C₄₇	#368	semidirect
97. C₄₇	C₄×#368	semidirect
154. C₄	#15688952	direct (N × H)

H ≅ G/N

type

10. #15688952

C₄

direct (N × H)

74. C₄×C₄₇

#368

semidirect

97. C₄₇

C₄×#368

semidirect

154. C₄

#15688952

direct (N × H)

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

subgroup	index
C₂×#15688952	2
C₂×C₄×C₄₇⋊C₁₆	2
C₄×#13093960	2
#20241481	2
C₄×#108×C₄₇	2
C₄×#368	47

subgroup

index

C₂×C₄×C₄₇⋊C₁₆

C₄×#108×C₄₇

Subgroups — C₄×#15688952 #26131691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C4×#15688952 #26131691

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#15688952 #26131691