Group #26110093 — subgroups

Subgroups — #26110093

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	#20243526	C₂	1	1	yes	maximal, characteristic
2	#20242146	C₂	1	2	yes	maximal
3	#20242143	C₂	1	2	yes	maximal
4	C₂×#15688844	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1837	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	#15688888	—	2	2	no	—
10	#15688875	—	2	4	no	—
11	#15688873	—	2	4	no	—
12	#15688844	C₄	1	4	yes	—
13	C₄₇×#423	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₄₇×#341	—	2	2	no	—
16	C₂×#10625329	—	2	2	no	—
17	C₂²×C₄₇⋊C₈	Q₈	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	D₄	1	2	yes	—
25	C₂×M₁₆×C₄₇	—	2	2	no	—
26	C₂²×C₈×C₄₇	D₄	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	Q₈	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	—	2	4	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	—	4	2	no	—
45	C₂×C₄₇⋊C₈	—	4	2	no	—
46	C₂×C₄₇⋊C₈	—	4	2	no	—
47	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
48	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
49	C₂³×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
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	#127	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#135	1	4	yes	—
59	C₈×C₄₇	—	2	4	no	—
60	C₈×C₄₇	—	8	2	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₄₇⋊C₈	—	2	4	no	—
63	C₄₇⋊C₈	—	8	2	no	—
64	#8827	—	47	1	no	maximal, Sylow
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	2	1	no	—
67	C₂²×C₄₇	#384	1	1	yes	characteristic
68	C₂²×C₄₇	#499	1	2	yes	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	#427	1	2	yes	—
72	C₄×C₄₇	#430	1	2	yes	—
73	C₄×C₄₇	#430	1	2	yes	—
74	C₄×C₄₇	#495	1	2	yes	—
75	#1837	—	47	1	no	—
76	#1837	D₄₇	1	1	yes	characteristic
77	#1567	—	47	2	no	—
78	#1566	—	47	2	no	—
79	C₂×#322	—	47	1	no	—
80	C₂×C₄₇	—	2	2	no	—
81	C₂×C₄₇	#1846	1	1	yes	characteristic
82	C₂×C₄₇	#1849	1	1	yes	characteristic
83	C₂×C₄₇	#1849	1	1	yes	characteristic
84	#423	—	47	2	no	—
85	#423	C₂×D₄₇	1	2	yes	—
86	C₂×C₄×C₈	—	47	1	no	—
87	C₂×C₄×C₈	—	47	1	no	—
88	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
89	#341	—	2	2	no	—
90	#341	—	94	2	no	—
91	#335	—	94	4	no	—
92	#334	—	94	4	no	—
93	#322	—	47	4	no	—
94	C₄₇	#8827	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	2	2	no	—
96	C₂×M₁₆	—	94	2	no	—
97	C₂²×C₈	—	47	2	no	—
98	C₂²×C₈	—	47	2	no	—
99	C₂²×C₈	C₂²⋊D₄₇	1	2	yes	—
100	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
101	C₄⋊C₈	—	2	2	no	—
102	C₄⋊C₈	—	2	2	no	—
103	C₄⋊C₈	—	94	2	no	—
104	C₄⋊C₈	—	94	2	no	—
105	C₄×C₈	—	47	2	no	—
106	C₄×C₈	—	47	2	no	—
107	C₄×C₈	—	47	2	no	—
108	C₄×C₈	—	47	2	no	—
109	C₄×C₈	Dic₉₄	1	2	yes	—
110	C₄×C₈	C₄×D₄₇	1	2	yes	—
111	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
112	C₂²×C₄	Q₈×D₄₇	1	1	yes	characteristic
113	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
114	M₁₆	—	4	4	no	—
115	M₁₆	—	188	4	no	—
116	C₂×C₈	—	2	4	no	—
117	C₂×C₈	—	4	2	no	—
118	C₂×C₈	—	47	4	no	—
119	C₂×C₈	—	47	4	no	—
120	C₂×C₈	—	188	2	no	—
121	C₂×C₈	—	188	2	no	—
122	C₂×C₈	—	188	2	no	—
123	C₂×C₈	Dic₄₇⋊C₄	1	4	yes	—
124	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
125	C₄²	C₂×C₄×D₄₇	1	1	yes	characteristic
126	C₄²	D₄×D₄₇	1	1	yes	characteristic
127	C₄²	D₄×D₄₇	1	1	yes	characteristic
128	C₂³	#13094022	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
132	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
133	C₂×C₄	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
134	C₂×C₄	#13094020	1	1	yes	characteristic
135	C₂×C₄	#13094023	1	1	yes	characteristic
136	C₂×C₄	#13094023	1	1	yes	characteristic
137	C₂×C₄	#13094044	1	4	yes	—
138	C₈	—	2	4	no	—
139	C₈	—	8	2	no	—
140	C₈	—	94	4	no	—
141	C₈	—	94	4	no	—
142	C₈	—	376	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	#15688992	1	1	yes	characteristic
146	C₂²	#15689522	1	2	yes	—
147	C₄	—	2	1	no	—
148	C₄	—	2	1	no	—
149	C₄	#15689159	1	2	yes	—
150	C₄	#15689170	1	2	yes	—
151	C₄	#15689170	1	2	yes	—
152	C₄	#15689505	1	2	yes	—
153	C₂	—	2	2	no	—
154	C₂	#20243550	1	1	yes	characteristic
155	C₂	#20243561	1	1	yes	characteristic
156	C₂	#20243561	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243526

C₂

yes

maximal, characteristic

#20242146

C₂

yes

maximal

#20242143

C₂

yes

maximal

C₂×#15688844

C₂

yes

maximal, characteristic

C₄₇×#1837

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×#341

—

C₂×#10625329

—

C₂²×C₄₇⋊C₈

Q₈

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

D₄

yes

—

C₂×M₁₆×C₄₇

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₄⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

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

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

—

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

#495

yes

—

#1837

—

#1837

D₄₇

yes

—

—

—

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

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₄₇

yes

—

100

C₂×C₄²

C₂²×D₄₇

yes

characteristic

101

C₄⋊C₈

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

Dic₉₄

yes

—

110

C₄×C₈

C₄×D₄₇

yes

—

111

C₂²×C₄

Q₈×D₄₇

yes

characteristic

112

C₂²×C₄

Q₈×D₄₇

yes

characteristic

113

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

114

M₁₆

—

115

M₁₆

—

188

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

188

—

121

C₂×C₈

—

188

—

122

C₂×C₈

—

188

—

123

C₂×C₈

Dic₄₇⋊C₄

yes

—

124

C₄²

C₂×Dic₉₄

yes

characteristic

125

C₄²

C₂×C₄×D₄₇

yes

characteristic

126

C₄²

D₄×D₄₇

yes

characteristic

127

C₄²

D₄×D₄₇

yes

characteristic

128

C₂³

#13094022

yes

characteristic

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

132

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

133

C₂×C₄

C₂×Dic₄₇⋊C₄

yes

characteristic

134

C₂×C₄

#13094020

yes

characteristic

135

C₂×C₄

#13094023

yes

characteristic

136

C₂×C₄

#13094023

yes

characteristic

137

C₂×C₄

#13094044

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

376

—

143

C₂²

—

144

C₂²

—

145

C₂²

#15688992

yes

characteristic

146

C₂²

#15689522

yes

—

147

C₄

—

148

C₄

—

149

C₄

#15689159

yes

—

150

C₄

#15689170

yes

—

151

C₄

#15689170

yes

—

152

C₄

#15689505

yes

—

153

C₂

—

154

C₂

#20243550

yes

characteristic

155

C₂

#20243561

yes

characteristic

156

C₂

#20243561

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13094020
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1837	C₂
socle	C₂²×C₄₇	#384
prime radical	C₂³×C₄₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1837	D₄₇
hypocenter	C₄₇	#8827
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1837

subgroup

G/H

center

C₂×C₄

#13094020

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1837

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 ◁ #1837

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

H ≅ G/N

type

94. C₄₇

#8827

semidirect

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

subgroup	index
#20243526	2
#20242146	2
#20242143	2
C₂×#15688844	2
C₄₇×#1837	2
#8827	47

subgroup

index

C₄₇×#1837

Subgroups — #26110093

Conjugacy classes of subgroups

Complements

Maximal subgroups