Group #26121313 — subgroups

Subgroups — #26121313

Order 12032.

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

normal subgroups only

This group has 316 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	#20243542	C₂	1	1	yes	maximal, characteristic
2	#20242160	C₂	1	2	yes	maximal
3	#20241791	C₂	1	2	yes	maximal
4	#20241217	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1995	C₂	1	1	yes	maximal, characteristic
6	#15689176	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
8	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
9	#15688873	—	2	4	no	—
10	Q₈⋊(C₄₇⋊C₈)	—	2	4	no	—
11	C₄₇×#533	C₂²	1	1	yes	characteristic
12	C₄₇×#423	C₂²	1	1	yes	characteristic
13	C₄₇×#376	C₂²	1	1	yes	characteristic
14	C₄₇×#339	—	2	2	no	—
15	C₂²×C₄₇⋊C₈	—	2	2	no	—
16	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
17	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
19	C₄⋊(C₄₇⋊C₈)	—	4	4	no	—
20	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
21	C₄×C₄₇⋊C₈	D₄	1	4	yes	—
22	C₂×M₁₆×C₄₇	—	2	1	no	—
23	C₄⋊Q₈×C₄₇	Q₈	1	2	yes	—
24	#125×C₄₇	—	2	1	no	—
25	C₂²⋊Q₈×C₄₇	—	2	2	no	—
26	C₄×Q₈×C₄₇	—	2	2	no	—
27	#117×C₄₇	—	2	1	no	—
28	C₂×C₄²×C₄₇	D₄	1	2	yes	—
29	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₈×C₄₇	—	2	2	no	—
31	#95×C₄₇	—	2	2	no	—
32	C₂×Q₈×C₄₇	—	2	2	no	—
33	C₂²×C₄×C₄₇	—	2	2	no	—
34	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₄₇	#41	1	2	yes	—
36	M₁₆×C₄₇	—	4	2	no	—
37	C₂×C₈×C₄₇	—	4	1	no	—
38	C₄⋊C₄×C₄₇	—	2	2	no	—
39	C₄⋊C₄×C₄₇	—	2	2	no	—
40	C₄⋊C₄×C₄₇	—	4	2	no	—
41	C₂²⋊C₄×C₄₇	—	4	1	no	—
42	C₄²×C₄₇	—	2	2	no	—
43	C₄²×C₄₇	—	4	1	no	—
44	C₄²×C₄₇	SD₁₆	1	4	yes	—
45	C₄²×C₄₇	C₂×D₄	1	2	yes	—
46	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₄₇	#41	1	1	yes	characteristic
48	C₂×C₄₇⋊C₈	—	2	4	no	—
49	C₂×C₄₇⋊C₈	—	4	2	no	—
50	C₂×C₄₇⋊C₈	—	4	4	no	—
51	C₂³×C₄₇	#123	1	1	yes	characteristic
52	Q₈×C₄₇	—	4	4	no	—
53	C₂×C₄×C₄₇	—	2	4	no	—
54	C₂×C₄×C₄₇	—	4	2	no	—
55	C₂×C₄×C₄₇	—	4	4	no	—
56	C₂×C₄×C₄₇	—	4	2	no	—
57	C₂×C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
58	C₂×C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
59	C₂×C₄×C₄₇	#123	1	2	yes	—
60	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
61	C₂×C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
62	C₂×C₄×C₄₇	#137	1	2	yes	—
63	C₈×C₄₇	—	8	1	no	—
64	C₄₇⋊C₈	—	4	4	no	—
65	C₄₇⋊C₈	—	8	2	no	—
66	#10670	—	47	1	no	maximal, Sylow
67	C₂²×C₄₇	—	2	2	no	—
68	C₂²×C₄₇	#399	1	1	yes	characteristic
69	C₂²×C₄₇	#483	1	2	yes	—
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	—	4	4	no	—
73	C₄×C₄₇	—	8	2	no	—
74	C₄×C₄₇	#475	1	2	yes	—
75	C₄×C₄₇	#492	1	1	yes	characteristic
76	C₄×C₄₇	#497	1	1	yes	characteristic
77	#1995	D₄₇	1	1	yes	characteristic
78	#1843	—	47	1	no	—
79	#1570	—	47	2	no	—
80	#1481	—	47	2	no	—
81	#1274	—	47	1	no	—
82	C₂×C₄₇	—	2	2	no	—
83	C₂×C₄₇	#2096	1	1	yes	characteristic
84	C₂×C₄₇	#2098	1	2	yes	—
85	#533	C₂×D₄₇	1	1	yes	characteristic
86	#431	—	47	2	no	—
87	#423	C₂×D₄₇	1	1	yes	characteristic
88	C₂×C₄⋊C₈	—	47	2	no	—
89	C₂×C₄⋊C₈	—	94	2	no	—
90	#376	C₂×D₄₇	1	1	yes	characteristic
91	#339	—	2	2	no	—
92	#334	—	94	4	no	—
93	Q₈⋊C₈	—	94	4	no	—
94	C₄₇	#10670	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	2	1	no	—
96	C₂²×C₈	—	94	2	no	—
97	C₄⋊Q₈	Dic₉₄	1	2	yes	—
98	#125	—	2	1	no	—
99	C₂²⋊Q₈	—	2	2	no	—
100	C₄×Q₈	—	2	2	no	—
101	#117	—	2	1	no	—
102	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
103	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
104	C₄⋊C₈	—	2	2	no	—
105	C₄⋊C₈	—	94	2	no	—
106	C₄⋊C₈	—	94	2	no	—
107	C₄⋊C₈	—	94	2	no	—
108	C₄⋊C₈	—	188	4	no	—
109	C₂²⋊C₈	—	94	2	no	—
110	C₄×C₈	—	47	4	no	—
111	#95	—	2	2	no	—
112	C₂×Q₈	—	2	2	no	—
113	C₂²×C₄	—	2	2	no	—
114	C₂²×C₄	#10625352	1	2	yes	—
115	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
116	M₁₆	—	4	2	no	—
117	C₂×C₈	—	4	1	no	—
118	C₂×C₈	—	94	4	no	—
119	C₂×C₈	—	188	2	no	—
120	C₂×C₈	—	188	4	no	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	2	2	no	—
123	C₄⋊C₄	—	4	2	no	—
124	C₂²⋊C₄	—	4	1	no	—
125	C₄²	—	2	2	no	—
126	C₄²	—	4	1	no	—
127	C₄²	#10625336	1	4	yes	—
128	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
129	C₄²	#10625350	1	1	yes	characteristic
130	C₄²	D₄×D₄₇	1	2	yes	—
131	C₂³	#13094005	1	1	yes	characteristic
132	Q₈	—	4	4	no	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	4	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	#13094001	1	2	yes	—
138	C₂×C₄	#13094003	1	1	yes	characteristic
139	C₂×C₄	#13094006	1	2	yes	—
140	C₂×C₄	#13094020	1	1	yes	characteristic
141	C₂×C₄	C₂×#10625336	1	2	yes	—
142	C₂×C₄	#13094054	1	2	yes	—
143	C₈	—	8	1	no	—
144	C₈	—	376	2	no	—
145	C₈	—	188	4	no	—
146	C₂²	—	2	2	no	—
147	C₂²	#15689059	1	1	yes	characteristic
148	C₂²	#15689451	1	2	yes	—
149	C₄	—	2	2	no	—
150	C₄	—	2	2	no	—
151	C₄	—	4	4	no	—
152	C₄	—	8	2	no	—
153	C₄	#15689408	1	2	yes	—
154	C₄	#15689497	1	1	yes	characteristic
155	C₄	#15689518	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#20244931	1	1	yes	characteristic
158	C₂	#20244943	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20243542

C₂

yes

maximal, characteristic

#20242160

C₂

yes

maximal

#20241791

C₂

yes

maximal

#20241217

C₂

yes

maximal, characteristic

C₄₇×#1995

C₂

yes

maximal, characteristic

#15689176

C₂²

yes

—

C₂×C₄⋊(C₄₇⋊C₈)

—

C₂×C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#15688873

—

Q₈⋊(C₄₇⋊C₈)

—

C₄₇×#533

C₂²

yes

characteristic

C₄₇×#423

C₂²

yes

characteristic

C₄₇×#376

C₂²

yes

characteristic

C₄₇×#339

—

C₂²×C₄₇⋊C₈

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄×C₄₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

Q₈

yes

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

#95×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

SD₁₆

yes

—

C₄²×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#123

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×SD₁₆

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

#10670

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#399

yes

characteristic

C₂²×C₄₇

#483

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#475

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

D₄₇

yes

—

—

—

—

C₂×C₄₇

—

C₂×C₄₇

#2096

yes

characteristic

C₂×C₄₇

#2098

yes

—

#533

C₂×D₄₇

yes

characteristic

#431

—

#423

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

#376

C₂×D₄₇

yes

characteristic

#339

—

#334

—

Q₈⋊C₈

—

C₄₇

#10670

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₉₄

yes

—

#125

—

C₂²⋊Q₈

—

100

C₄×Q₈

—

101

#117

—

102

C₂×C₄²

C₂²⋊D₄₇

yes

—

103

C₂×C₄²

C₂²×D₄₇

yes

characteristic

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

C₄⋊C₈

—

188

—

109

C₂²⋊C₈

—

110

C₄×C₈

—

111

#95

—

112

C₂×Q₈

—

113

C₂²×C₄

—

114

C₂²×C₄

#10625352

yes

—

115

C₂²×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

188

—

120

C₂×C₈

—

188

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

#10625336

yes

—

128

C₄²

C₂×Dic₉₄

yes

characteristic

129

C₄²

#10625350

yes

characteristic

130

C₄²

D₄×D₄₇

yes

—

131

C₂³

#13094005

yes

characteristic

132

Q₈

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#13094001

yes

—

138

C₂×C₄

#13094003

yes

characteristic

139

C₂×C₄

#13094006

yes

—

140

C₂×C₄

#13094020

yes

characteristic

141

C₂×C₄

C₂×#10625336

yes

—

142

C₂×C₄

#13094054

yes

—

143

C₈

—

144

C₈

—

376

—

145

C₈

—

188

—

146

C₂²

—

147

C₂²

#15689059

yes

characteristic

148

C₂²

#15689451

yes

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

#15689408

yes

—

154

C₄

#15689497

yes

characteristic

155

C₄

#15689518

yes

characteristic

156

C₂

—

157

C₂

#20244931

yes

characteristic

158

C₂

#20244943

yes

—

	subgroup	G/H
center	C₂×C₄	#13094020
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1995	C₂
socle	C₂²×C₄₇	#399
prime radical	C₂³×C₄₇	#123
hypercenter	#1995	D₄₇
hypocenter	C₄₇	#10670
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

subgroup

G/H

center

C₂×C₄

#13094020

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1995

C₂

C₂²×C₄₇

C₂³×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1995

N	H ≅ G/N	type
94. C₄₇	#10670	semidirect

H ≅ G/N

type

94. C₄₇

#10670

semidirect

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

subgroup	index
#20243542	2
#20242160	2
#20241791	2
#20241217	2
C₄₇×#1995	2
#10670	47

subgroup

index

C₄₇×#1995

Subgroups — #26121313

Conjugacy classes of subgroups

Complements

Maximal subgroups