Group #26110095 — subgroups

Subgroups — #26110095

Order 12032.

306 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20243525	C₂	1	2	yes	maximal
2	#20242147	C₂	1	1	yes	maximal, characteristic
3	#20242143	C₂	1	2	yes	maximal
4	C₂×#15688845	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1567	C₂	1	1	yes	maximal, characteristic
6	#15689147	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	#15688887	—	2	4	no	—
10	#15688876	—	2	2	no	—
11	#15688873	—	2	4	no	—
12	#15688845	C₄	1	4	yes	—
13	C₄₇×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#335	—	2	2	no	—
16	C₂×#10625329	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	D₄	1	2	yes	—
18	C₂²×C₄₇⋊C₈	Q₈	1	2	yes	—
19	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
20	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
22	C₄×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
23	C₄×C₄₇⋊C₈	D₄	1	2	yes	—
24	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	Q₈	1	2	yes	—
27	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₈×C₄₇	—	2	2	no	—
29	C₄⋊C₈×C₄₇	—	2	2	no	—
30	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
31	C₄×C₈×C₄₇	D₄	1	2	yes	—
32	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
34	C₂²×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
35	M₁₆×C₄₇	—	4	4	no	—
36	C₂×C₈×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	—	4	2	no	—
38	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
39	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
43	#10625329	—	4	4	no	—
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	2	2	no	—
46	C₂×C₄₇⋊C₈	—	4	2	no	—
47	C₂×C₄₇⋊C₈	—	4	2	no	—
48	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
49	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
50	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#135	1	2	yes	—
60	C₂×C₄×C₄₇	#135	1	2	yes	—
61	C₈×C₄₇	—	2	4	no	—
62	C₈×C₄₇	—	8	2	no	—
63	C₄₇⋊C₈	—	2	4	no	—
64	C₄₇⋊C₈	—	2	4	no	—
65	C₄₇⋊C₈	—	8	2	no	—
66	#8827	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	—	2	1	no	—
68	C₂²×C₄₇	—	2	1	no	—
69	C₂²×C₄₇	#384	1	1	yes	characteristic
70	C₂²×C₄₇	#499	1	2	yes	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	#427	1	2	yes	—
74	C₄×C₄₇	#430	1	2	yes	—
75	C₄×C₄₇	#430	1	2	yes	—
76	C₄×C₄₇	#495	1	1	yes	characteristic
77	C₄×C₄₇	#495	1	1	yes	characteristic
78	#1837	—	47	2	no	—
79	#1567	—	47	1	no	—
80	#1567	D₄₇	1	1	yes	characteristic
81	#1566	—	47	2	no	—
82	C₂×#322	—	47	1	no	—
83	C₂×C₄₇	—	2	2	no	—
84	C₂×C₄₇	#1846	1	1	yes	characteristic
85	C₂×C₄₇	#1849	1	1	yes	characteristic
86	C₂×C₄₇	#1849	1	1	yes	characteristic
87	#423	—	47	2	no	—
88	#423	C₂×D₄₇	1	2	yes	—
89	C₂×C₄×C₈	—	47	1	no	—
90	C₂×C₄×C₈	—	47	1	no	—
91	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
92	#341	—	94	4	no	—
93	#335	—	2	2	no	—
94	#335	—	94	2	no	—
95	#334	—	94	4	no	—
96	#322	—	47	4	no	—
97	C₄₇	#8827	1	1	yes	characteristic, complemented, Sylow
98	C₂×M₁₆	—	2	2	no	—
99	C₂×M₁₆	—	94	2	no	—
100	C₂²×C₈	—	47	2	no	—
101	C₂²×C₈	—	47	2	no	—
102	C₂²×C₈	Dic₉₄	1	2	yes	—
103	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	2	2	no	—
106	C₄⋊C₈	—	94	2	no	—
107	C₄⋊C₈	—	94	2	no	—
108	C₄×C₈	—	47	2	no	—
109	C₄×C₈	—	47	2	no	—
110	C₄×C₈	—	47	2	no	—
111	C₄×C₈	—	47	2	no	—
112	C₄×C₈	C₄×D₄₇	1	2	yes	—
113	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
114	C₂²×C₄	C₂×Dic₉₄	1	1	yes	characteristic
115	C₂²×C₄	D₄×D₄₇	1	1	yes	characteristic
116	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
117	M₁₆	—	4	4	no	—
118	M₁₆	—	188	4	no	—
119	C₂×C₈	—	4	2	no	—
120	C₂×C₈	—	4	2	no	—
121	C₂×C₈	—	47	4	no	—
122	C₂×C₈	—	47	4	no	—
123	C₂×C₈	—	94	2	no	—
124	C₂×C₈	—	94	2	no	—
125	C₂×C₈	—	188	2	no	—
126	C₂×C₈	—	188	2	no	—
127	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
128	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
129	C₄²	D₄×D₄₇	1	1	yes	characteristic
130	C₄²	Q₈×D₄₇	1	1	yes	characteristic
131	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
132	C₂³	#13094023	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
136	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
137	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
138	C₂×C₄	#13094020	1	1	yes	characteristic
139	C₂×C₄	#13094022	1	1	yes	characteristic
140	C₂×C₄	#13094023	1	1	yes	characteristic
141	C₂×C₄	#13094043	1	2	yes	—
142	C₂×C₄	#13094047	1	2	yes	—
143	C₈	—	2	4	no	—
144	C₈	—	8	2	no	—
145	C₈	—	94	4	no	—
146	C₈	—	94	4	no	—
147	C₈	—	376	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	#15688992	1	1	yes	characteristic
151	C₂²	#15689524	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	2	1	no	—
154	C₄	#15689160	1	2	yes	—
155	C₄	#15689170	1	2	yes	—
156	C₄	#15689171	1	2	yes	—
157	C₄	#15689506	1	1	yes	characteristic
158	C₄	#15689508	1	1	yes	characteristic
159	C₂	—	2	2	no	—
160	C₂	#20243551	1	1	yes	characteristic
161	C₂	#20243561	1	1	yes	characteristic
162	C₂	#20243562	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243525

C₂

yes

maximal

#20242147

C₂

yes

maximal, characteristic

#20242143

C₂

yes

maximal

C₂×#15688845

C₂

yes

maximal, characteristic

C₄₇×#1567

C₂

yes

maximal, characteristic

#15689147

C₂²

yes

—

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

—

—

—

C₄

yes

—

C₄₇×#423

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#335

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

D₄

yes

—

C₂²×C₄₇⋊C₈

Q₈

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

C₂×C₄

yes

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

Q₈

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×Q₈

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₄₇⋊C₈

C₄⋊C₄

yes

—

C₂³×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8827

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#384

yes

characteristic

C₂²×C₄₇

#499

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#427

yes

—

C₄×C₄₇

#430

yes

—

C₄×C₄₇

#430

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

—

D₄₇

yes

characteristic

#1566

—

C₂×#322

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×D₄₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

—

—

—

C₄₇

#8827

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

—

100

C₂²×C₈

—

101

C₂²×C₈

—

102

C₂²×C₈

Dic₉₄

yes

—

103

C₂×C₄²

C₂²×D₄₇

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₄×C₈

—

112

C₄×C₈

C₄×D₄₇

yes

—

113

C₄×C₈

C₂²⋊D₄₇

yes

—

114

C₂²×C₄

C₂×Dic₉₄

yes

characteristic

115

C₂²×C₄

D₄×D₄₇

yes

characteristic

116

C₂²×C₄

Q₈×D₄₇

yes

characteristic

117

M₁₆

—

118

M₁₆

—

188

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

—

123

C₂×C₈

—

124

C₂×C₈

—

125

C₂×C₈

—

188

—

126

C₂×C₈

—

188

—

127

C₂×C₈

Dic₄₇⋊C₄

yes

—

128

C₄²

C₂×C₄×D₄₇

yes

characteristic

129

C₄²

D₄×D₄₇

yes

characteristic

130

C₄²

Q₈×D₄₇

yes

characteristic

131

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

132

C₂³

#13094023

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

136

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

137

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

138

C₂×C₄

#13094020

yes

characteristic

139

C₂×C₄

#13094022

yes

characteristic

140

C₂×C₄

#13094023

yes

characteristic

141

C₂×C₄

#13094043

yes

—

142

C₂×C₄

#13094047

yes

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

376

—

148

C₂²

—

149

C₂²

—

150

C₂²

#15688992

yes

characteristic

151

C₂²

#15689524

yes

—

152

C₄

—

153

C₄

—

154

C₄

#15689160

yes

—

155

C₄

#15689170

yes

—

156

C₄

#15689171

yes

—

157

C₄

#15689506

yes

characteristic

158

C₄

#15689508

yes

characteristic

159

C₂

—

160

C₂

#20243551

yes

characteristic

161

C₂

#20243561

yes

characteristic

162

C₂

#20243562

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1567	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1567	D₄₇
hypocenter	C₄₇	#8827
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

subgroup

G/H

center

C₂×C₄

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1567

C₂

socle

C₂²×C₄₇

#384

prime radical

C₂³×C₄₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

N	H ≅ G/N	type
97. C₄₇	#8827	semidirect

H ≅ G/N

type

97. C₄₇

#8827

semidirect

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

subgroup	index
#20243525	2
#20242147	2
#20242143	2
C₂×#15688845	2
C₄₇×#1567	2
#8827	47

subgroup

index

C₄₇×#1567

Subgroups — #26110095

Conjugacy classes of subgroups

Complements

Maximal subgroups