Group #26109930 — subgroups

Subgroups — #26109930

Order 12032.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20243140	C₂	1	1	yes	maximal, characteristic
2	#20242134	C₂	1	2	yes	maximal
3	#20241232	C₂	1	2	yes	maximal
4	#20241217	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1274	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×#13093935	C₂²	1	1	yes	characteristic
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	#15688878	C₄	1	2	yes	—
10	#15688878	C₄	1	2	yes	—
11	#15688875	—	2	2	no	—
12	#15688873	—	2	2	no	—
13	C₄₇×#431	C₂²	1	2	yes	—
14	C₄₇×#376	C₂²	1	1	yes	characteristic
15	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
20	#13093935	—	2	2	no	—
21	C₄×C₄₇⋊C₈	—	2	2	no	—
22	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₄₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
25	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
26	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₄₇	—	2	2	no	—
28	C₂²⋊C₈×C₄₇	—	2	2	no	—
29	C₄×C₈×C₄₇	—	2	2	no	—
30	#95×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₄₇	#41	1	2	yes	—
34	C₂×C₈×C₄₇	—	4	2	no	—
35	C₄²×C₄₇	—	2	2	no	—
36	C₄²×C₄₇	—	2	2	no	—
37	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
38	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
39	C₄²×C₄₇	#41	1	2	yes	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂×C₄₇⋊C₈	—	2	1	no	—
43	C₂×C₄₇⋊C₈	—	2	1	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	C₄⋊C₄	1	2	yes	—
48	C₂×C₄₇⋊C₈	C₄⋊C₄	1	2	yes	—
49	C₂³×C₄₇	#117	1	1	yes	characteristic
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	—	4	4	no	—
52	C₂×C₄×C₄₇	—	4	2	no	—
53	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#124	1	2	yes	—
56	C₂×C₄×C₄₇	#125	1	2	yes	—
57	C₂×C₄×C₄₇	#131	1	2	yes	—
58	C₂×C₄×C₄₇	#131	1	2	yes	—
59	C₈×C₄₇	—	8	2	no	—
60	C₄₇⋊C₈	—	2	2	no	—
61	C₄₇⋊C₈	—	2	2	no	—
62	C₄₇⋊C₈	—	4	1	no	—
63	C₄₇⋊C₈	—	4	1	no	—
64	C₄₇⋊C₈	—	8	2	no	—
65	#8782	—	47	1	no	maximal, Sylow
66	C₂²×C₄₇	—	2	2	no	—
67	C₂²×C₄₇	#382	1	1	yes	characteristic
68	C₂²×C₄₇	#433	1	2	yes	—
69	C₄×C₄₇	—	2	2	no	—
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	—	8	2	no	—
72	C₄×C₄₇	#424	1	2	yes	—
73	C₄×C₄₇	#433	1	1	yes	characteristic
74	C₄×C₄₇	#433	1	1	yes	characteristic
75	#1746	—	47	1	no	—
76	#1564	—	47	2	no	—
77	#1281	—	47	2	no	—
78	#1274	—	47	1	no	—
79	#1274	D₄₇	1	1	yes	characteristic
80	C₂×C₄₇	—	2	2	no	—
81	C₂×C₄₇	#1832	1	1	yes	characteristic
82	C₂×C₄₇	#1833	1	2	yes	—
83	#431	—	47	2	no	—
84	#431	C₂×D₄₇	1	2	yes	—
85	C₂×#97	—	47	1	no	—
86	C₂×C₄×C₈	—	47	1	no	—
87	#376	C₂×D₄₇	1	1	yes	characteristic
88	#336	—	47	2	no	—
89	#336	—	47	2	no	—
90	#335	—	94	2	no	—
91	#334	—	94	2	no	—
92	C₄₇	#8782	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₈	—	47	2	no	—
94	C₂²×C₈	—	47	1	no	—
95	C₂²×C₈	—	47	1	no	—
96	C₂×C₄²	C₄×D₄₇	1	2	yes	—
97	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
98	C₄⋊C₈	—	2	2	no	—
99	C₄⋊C₈	—	94	2	no	—
100	C₂²⋊C₈	—	2	2	no	—
101	C₂²⋊C₈	—	94	2	no	—
102	#97	—	94	2	no	—
103	C₄×C₈	—	2	2	no	—
104	C₄×C₈	—	47	1	no	—
105	C₄×C₈	—	47	1	no	—
106	C₄×C₈	—	47	2	no	—
107	C₄×C₈	—	94	2	no	—
108	#95	—	2	2	no	—
109	C₂²×C₄	—	2	2	no	—
110	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
111	C₂²×C₄	#10625352	1	2	yes	—
112	C₂×C₈	—	4	2	no	—
113	C₂×C₈	—	47	2	no	—
114	C₂×C₈	—	47	2	no	—
115	C₂×C₈	—	94	2	no	—
116	C₂×C₈	—	94	2	no	—
117	C₂×C₈	—	94	1	no	—
118	C₂×C₈	—	94	1	no	—
119	C₂×C₈	—	94	2	no	—
120	C₂×C₈	—	94	2	no	—
121	C₂×C₈	—	188	2	no	—
122	C₄²	—	2	2	no	—
123	C₄²	—	2	2	no	—
124	C₄²	#10625350	1	2	yes	—
125	C₄²	D₄×D₄₇	1	1	yes	characteristic
126	C₄²	Q₈×D₄₇	1	1	yes	characteristic
127	C₂³	#13093980	1	1	yes	characteristic
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
132	C₂×C₄	#13093981	1	1	yes	characteristic
133	C₂×C₄	#13094010	1	2	yes	—
134	C₂×C₄	#13094014	1	2	yes	—
135	C₂×C₄	#13094030	1	2	yes	—
136	C₂×C₄	#13094032	1	2	yes	—
137	C₈	—	8	2	no	—
138	C₈	—	94	2	no	—
139	C₈	—	94	2	no	—
140	C₈	—	376	2	no	—
141	C₈	—	188	1	no	—
142	C₈	—	188	1	no	—
143	C₂²	—	2	2	no	—
144	C₂²	#15688987	1	1	yes	characteristic
145	C₂²	#15689181	1	2	yes	—
146	C₄	—	2	2	no	—
147	C₄	—	2	2	no	—
148	C₄	—	8	2	no	—
149	C₄	#15689148	1	2	yes	—
150	C₄	#15689182	1	1	yes	characteristic
151	C₄	#15689185	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	#20243506	1	1	yes	characteristic
154	C₂	#20243510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243140

C₂

yes

maximal, characteristic

#20242134

C₂

yes

maximal

#20241232

C₂

yes

maximal

#20241217

C₂

yes

maximal, characteristic

C₄₇×#1274

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×#13093935

C₂²

yes

characteristic

C₂×C₄×C₄₇⋊C₈

C₂²

yes

characteristic

#15688878

C₄

yes

—

#15688878

C₄

yes

—

#15688875

—

#15688873

—

C₄₇×#431

C₂²

yes

—

C₄₇×#376

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂²×C₄₇⋊C₈

C₂×C₄

yes

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

D₄

yes

characteristic

C₄×C₄₇⋊C₈

Q₈

yes

—

C₂×C₄²×C₄₇

C₂×C₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

—

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

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

#117

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

—

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₂×C₄×C₄₇

#131

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#8782

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#382

yes

characteristic

C₂²×C₄₇

#433

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#424

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

—

—

—

—

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

#1832

yes

characteristic

C₂×C₄₇

#1833

yes

—

#431

—

#431

C₂×D₄₇

yes

—

C₂×#97

—

C₂×C₄×C₈

—

#376

C₂×D₄₇

yes

—

—

—

—

C₄₇

#8782

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

C₄×D₄₇

yes

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

C₂²⋊C₈

—

102

#97

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

#95

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

111

C₂²×C₄

#10625352

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

188

—

122

C₄²

—

123

C₄²

—

124

C₄²

#10625350

yes

—

125

C₄²

D₄×D₄₇

yes

characteristic

126

C₄²

Q₈×D₄₇

yes

characteristic

127

C₂³

#13093980

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

132

C₂×C₄

#13093981

yes

characteristic

133

C₂×C₄

#13094010

yes

—

134

C₂×C₄

#13094014

yes

—

135

C₂×C₄

#13094030

yes

—

136

C₂×C₄

#13094032

yes

—

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

376

—

141

C₈

—

188

—

142

C₈

—

188

—

143

C₂²

—

144

C₂²

#15688987

yes

characteristic

145

C₂²

#15689181

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#15689148

yes

—

150

C₄

#15689182

yes

characteristic

151

C₄

#15689185

yes

characteristic

152

C₂

—

153

C₂

#20243506

yes

characteristic

154

C₂

#20243510

yes

—

	subgroup	G/H
center	C₂×C₄	#13093981
commutator	C₂²×C₄×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1274	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#117
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1274	D₄₇
hypocenter	C₄₇	#8782
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

subgroup

G/H

center

C₂×C₄

#13093981

commutator

C₂²×C₄×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1274

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ #376 ◁ #1274

N	H ≅ G/N	type
92. C₄₇	#8782	semidirect

H ≅ G/N

type

92. C₄₇

#8782

semidirect

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

subgroup	index
#20243140	2
#20242134	2
#20241232	2
#20241217	2
C₄₇×#1274	2
#8782	47

subgroup

index

C₄₇×#1274

Subgroups — #26109930

Conjugacy classes of subgroups

Complements

Maximal subgroups