Group #26109934 — subgroups

Subgroups — #26109934

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	2	yes	—
23	C₄×C₄₇⋊C₈	Q₈	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
25	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	C₄⋊C₈×C₄₇	—	2	2	no	—
27	C₂²⋊C₈×C₄₇	—	2	2	no	—
28	C₄×C₈×C₄₇	—	2	2	no	—
29	#95×C₄₇	—	2	2	no	—
30	C₂²×C₄×C₄₇	—	2	2	no	—
31	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₄₇	#41	1	2	yes	—
33	C₂×C₈×C₄₇	—	4	2	no	—
34	C₄²×C₄₇	—	2	2	no	—
35	C₄²×C₄₇	—	2	2	no	—
36	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
38	C₄²×C₄₇	#41	1	2	yes	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂×C₄₇⋊C₈	—	2	1	no	—
42	C₂×C₄₇⋊C₈	—	2	1	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂×C₄₇⋊C₈	—	2	2	no	—
45	C₂×C₄₇⋊C₈	—	4	2	no	—
46	C₂×C₄₇⋊C₈	C₄⋊C₄	1	4	yes	—
47	C₂³×C₄₇	#117	1	1	yes	characteristic
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	—	4	4	no	—
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#124	1	2	yes	—
54	C₂×C₄×C₄₇	#125	1	2	yes	—
55	C₂×C₄×C₄₇	#131	1	2	yes	—
56	C₂×C₄×C₄₇	#131	1	2	yes	—
57	C₈×C₄₇	—	8	2	no	—
58	C₄₇⋊C₈	—	2	4	no	—
59	C₄₇⋊C₈	—	4	1	no	—
60	C₄₇⋊C₈	—	4	1	no	—
61	C₄₇⋊C₈	—	8	2	no	—
62	#8783	—	47	1	no	maximal, Sylow
63	C₂²×C₄₇	—	2	2	no	—
64	C₂²×C₄₇	#382	1	1	yes	characteristic
65	C₂²×C₄₇	#433	1	2	yes	—
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	8	2	no	—
69	C₄×C₄₇	#424	1	2	yes	—
70	C₄×C₄₇	#433	1	1	yes	characteristic
71	C₄×C₄₇	#433	1	1	yes	characteristic
72	#1746	—	47	1	no	—
73	#1564	—	47	2	no	—
74	#1281	—	47	2	no	—
75	#1274	—	47	1	no	—
76	#1274	D₄₇	1	1	yes	characteristic
77	C₂×C₄₇	—	2	2	no	—
78	C₂×C₄₇	#1832	1	1	yes	characteristic
79	C₂×C₄₇	#1833	1	2	yes	—
80	#431	—	47	2	no	—
81	#431	C₂×D₄₇	1	2	yes	—
82	C₂×#97	—	47	1	no	—
83	C₂×C₄×C₈	—	47	1	no	—
84	#376	C₂×D₄₇	1	1	yes	characteristic
85	#336	—	47	2	no	—
86	#336	—	47	2	no	—
87	#335	—	94	2	no	—
88	#334	—	94	2	no	—
89	C₄₇	#8783	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₈	—	47	2	no	—
91	C₂²×C₈	—	47	1	no	—
92	C₂²×C₈	—	47	1	no	—
93	C₂×C₄²	C₄×D₄₇	1	2	yes	—
94	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
95	C₄⋊C₈	—	2	2	no	—
96	C₄⋊C₈	—	94	2	no	—
97	C₂²⋊C₈	—	2	2	no	—
98	C₂²⋊C₈	—	94	2	no	—
99	#97	—	94	2	no	—
100	C₄×C₈	—	2	2	no	—
101	C₄×C₈	—	47	2	no	—
102	C₄×C₈	—	47	2	no	—
103	C₄×C₈	—	94	2	no	—
104	#95	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
107	C₂²×C₄	#10625352	1	2	yes	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	—	47	4	no	—
110	C₂×C₈	—	94	2	no	—
111	C₂×C₈	—	94	2	no	—
112	C₂×C₈	—	94	1	no	—
113	C₂×C₈	—	94	1	no	—
114	C₂×C₈	—	94	2	no	—
115	C₂×C₈	—	94	2	no	—
116	C₂×C₈	—	188	2	no	—
117	C₄²	—	2	2	no	—
118	C₄²	—	2	2	no	—
119	C₄²	#10625350	1	2	yes	—
120	C₄²	D₄×D₄₇	1	1	yes	characteristic
121	C₄²	Q₈×D₄₇	1	1	yes	characteristic
122	C₂³	#13093980	1	1	yes	characteristic
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	4	no	—
125	C₂×C₄	—	4	2	no	—
126	C₂×C₄	C₄⋊C₄×D₄₇	1	1	yes	characteristic
127	C₂×C₄	#13093981	1	1	yes	characteristic
128	C₂×C₄	#13094010	1	2	yes	—
129	C₂×C₄	#13094014	1	2	yes	—
130	C₂×C₄	#13094030	1	2	yes	—
131	C₂×C₄	#13094032	1	2	yes	—
132	C₈	—	8	2	no	—
133	C₈	—	94	4	no	—
134	C₈	—	376	2	no	—
135	C₈	—	188	1	no	—
136	C₈	—	188	1	no	—
137	C₂²	—	2	2	no	—
138	C₂²	#15688987	1	1	yes	characteristic
139	C₂²	#15689181	1	2	yes	—
140	C₄	—	2	2	no	—
141	C₄	—	2	2	no	—
142	C₄	—	8	2	no	—
143	C₄	#15689148	1	2	yes	—
144	C₄	#15689182	1	1	yes	characteristic
145	C₄	#15689185	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	#20243506	1	1	yes	characteristic
148	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

—

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

—

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

—

#8783

—

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

#8783

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

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

—

104

#95

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×C₄×D₄₇

yes

characteristic

107

C₂²×C₄

#10625352

yes

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

188

—

117

C₄²

—

118

C₄²

—

119

C₄²

#10625350

yes

—

120

C₄²

D₄×D₄₇

yes

characteristic

121

C₄²

Q₈×D₄₇

yes

characteristic

122

C₂³

#13093980

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₄⋊C₄×D₄₇

yes

characteristic

127

C₂×C₄

#13093981

yes

characteristic

128

C₂×C₄

#13094010

yes

—

129

C₂×C₄

#13094014

yes

—

130

C₂×C₄

#13094030

yes

—

131

C₂×C₄

#13094032

yes

—

132

C₈

—

133

C₈

—

134

C₈

—

376

—

135

C₈

—

188

—

136

C₈

—

188

—

137

C₂²

—

138

C₂²

#15688987

yes

characteristic

139

C₂²

#15689181

yes

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

#15689148

yes

—

144

C₄

#15689182

yes

characteristic

145

C₄

#15689185

yes

characteristic

146

C₂

—

147

C₂

#20243506

yes

characteristic

148

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	#8783
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
89. C₄₇	#8783	semidirect

H ≅ G/N

type

89. C₄₇

#8783

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
#8783	47

subgroup

index

C₄₇×#1274

Subgroups — #26109934

Conjugacy classes of subgroups

Complements

Maximal subgroups