Group #26247216 — subgroups

Subgroups — #26247216

Order 12032.

662 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 211 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	C₄×#13094016	C₂	1	2	yes	maximal
2	#20244871	C₂	1	4	yes	maximal
3	#20244727	C₂	1	4	yes	maximal
4	#20243285	C₂	1	4	yes	maximal
5	C₄₇×#1745	C₂	1	1	yes	maximal, characteristic
6	C₂×#13094016	C₂²	1	2	yes	—
7	C₂×#13093958	C₂²	1	4	yes	—
8	C₂×#13093956	C₂²	1	4	yes	—
9	#15689021	C₂²	1	4	yes	—
10	#15689007	—	2	4	no	—
11	#15688989	C₂²	1	2	yes	—
12	C₄×Dic₄₇⋊C₄	—	2	4	no	—
13	C₄×C₄⋊Dic₄₇	C₂²	1	4	yes	—
14	#15688886	C₂²	1	4	yes	—
15	#15688886	C₂²	1	4	yes	—
16	C₂×#97×C₄₇	C₂²	1	2	yes	—
17	C₄³×C₄₇	C₂²	1	1	yes	characteristic
18	C₄₇×#336	C₂²	1	4	yes	—
19	#13094016	—	2	8	no	—
20	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
21	C₂×C₄⋊Dic₄₇	C₂³	1	4	yes	—
22	C₂×Dic₄₇⋊C₄	—	2	4	no	—
23	C₂×Dic₄₇⋊C₄	—	2	4	no	—
24	C₂×C₄×Dic₄₇	—	2	4	no	—
25	#13093958	—	2	8	no	—
26	#13093956	—	2	8	no	—
27	C₄₇⋊#95	—	2	4	no	—
28	C₄₇⋊#95	—	2	4	no	—
29	C₄₇⋊#95	—	2	4	no	—
30	C₂²×C₈×C₄₇	C₂³	1	4	yes	—
31	C₂×C₄²×C₄₇	D₄	1	4	yes	—
32	C₂×C₄²×C₄₇	C₂³	1	2	yes	—
33	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
34	#97×C₄₇	Q₈	1	8	yes	—
35	C₂²×C₄×C₄₇	C₂×D₄	1	4	yes	—
36	C₂²×C₄×C₄₇	C₂×D₄	1	2	yes	—
37	C₂²×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
38	C₂²×Dic₄₇	—	2	4	no	—
39	C₂²×Dic₄₇	—	2	4	no	—
40	C₂×C₈×C₄₇	—	2	4	no	—
41	C₂×C₈×C₄₇	—	2	4	no	—
42	C₂×C₈×C₄₇	C₂×Q₈	1	8	yes	—
43	C₄²×C₄₇	—	2	4	no	—
44	C₄²×C₄₇	—	4	4	no	—
45	C₄²×C₄₇	C₂×Q₈	1	4	yes	—
46	C₄⋊Dic₄₇	—	2	4	no	—
47	C₄⋊Dic₄₇	—	2	4	no	—
48	C₄⋊Dic₄₇	—	2	4	no	—
49	C₄⋊Dic₄₇	#41	1	8	yes	—
50	Dic₄₇⋊C₄	—	2	16	no	—
51	Dic₄₇⋊C₄	—	4	8	no	—
52	C₄×Dic₄₇	—	4	8	no	—
53	C₂³×C₄₇	C₂²×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	—	2	4	no	—
55	C₂×C₄×C₄₇	—	2	4	no	—
56	C₂×C₄×C₄₇	—	2	4	no	—
57	C₂×C₄×C₄₇	—	2	4	no	—
58	C₂×C₄×C₄₇	C₄⋊Q₈	1	4	yes	—
59	C₂×C₄×C₄₇	C₂²×Q₈	1	2	yes	—
60	C₂×C₄×C₄₇	C₂×#41	1	4	yes	—
61	C₂×Dic₄₇	—	2	8	no	—
62	C₂×Dic₄₇	—	4	4	no	—
63	C₂×Dic₄₇	—	4	4	no	—
64	C₂×Dic₄₇	—	4	4	no	—
65	C₂×Dic₄₇	—	4	4	no	—
66	C₂×Dic₄₇	—	4	4	no	—
67	C₈×C₄₇	—	2	8	no	—
68	#23840	—	47	1	no	maximal, Sylow
69	C₂²×C₄₇	#525	1	2	yes	—
70	C₂²×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
71	C₂²×C₄₇	#575	1	4	yes	—
72	C₄×C₄₇	—	2	4	no	—
73	C₄×C₄₇	—	4	4	no	—
74	C₄×C₄₇	#533	1	4	yes	—
75	Dic₄₇	—	4	8	no	—
76	Dic₄₇	—	8	4	no	—
77	C₄×#125	—	47	2	no	—
78	#2085	—	47	4	no	—
79	#2059	—	47	4	no	—
80	#1776	—	47	4	no	—
81	#1745	D₄₇	1	1	yes	characteristic
82	C₂×C₄₇	#2399	1	1	yes	characteristic
83	C₂×C₄₇	#3111	1	4	yes	—
84	C₂×C₄₇	#3166	1	2	yes	—
85	C₂×#125	—	47	2	no	—
86	C₂×#107	—	47	4	no	—
87	C₂×#106	—	47	4	no	—
88	C₂×#97	C₂×D₄₇	1	2	yes	—
89	#389	—	47	4	no	—
90	#387	—	94	4	no	—
91	#382	—	47	2	no	—
92	C₄×C₄⋊C₄	—	47	4	no	—
93	C₄×C₄⋊C₄	—	94	4	no	—
94	C₄³	C₂×D₄₇	1	1	yes	characteristic
95	#340	—	47	4	no	—
96	#340	—	47	4	no	—
97	#336	C₂×D₄₇	1	4	yes	—
98	C₄₇	#23840	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₈	C₂²×D₄₇	1	4	yes	—
100	#125	—	94	8	no	—
101	C₂×C₄⋊C₄	—	47	4	no	—
102	C₂×C₄⋊C₄	—	47	4	no	—
103	C₂×C₄⋊C₄	—	94	4	no	—
104	C₂×C₄⋊C₄	—	94	4	no	—
105	C₂×C₄²	—	94	4	no	—
106	C₂×C₄²	D₁₈₈	1	4	yes	—
107	C₂×C₄²	C₂²×D₄₇	1	2	yes	—
108	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
109	#107	—	94	8	no	—
110	#106	—	94	8	no	—
111	#97	Dic₉₄	1	8	yes	—
112	#95	—	94	4	no	—
113	#95	—	94	4	no	—
114	#95	—	94	4	no	—
115	C₂²×C₄	—	94	4	no	—
116	C₂²×C₄	—	94	4	no	—
117	C₂²×C₄	C₂×D₁₈₈	1	4	yes	—
118	C₂²×C₄	C₂×D₁₈₈	1	2	yes	—
119	C₂²×C₄	C₂³×D₄₇	1	1	yes	characteristic
120	C₂×C₈	—	2	4	no	—
121	C₂×C₈	—	2	4	no	—
122	C₂×C₈	C₂×Dic₉₄	1	8	yes	—
123	C₄⋊C₄	—	47	8	no	—
124	C₄⋊C₄	—	94	16	no	—
125	C₄⋊C₄	—	94	4	no	—
126	C₄⋊C₄	—	94	4	no	—
127	C₄⋊C₄	—	94	4	no	—
128	C₄⋊C₄	—	188	8	no	—
129	C₄²	—	2	4	no	—
130	C₄²	—	4	4	no	—
131	C₄²	—	188	8	no	—
132	C₄²	C₂×Dic₉₄	1	4	yes	—
133	C₂³	C₂²×D₁₈₈	1	1	yes	characteristic
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	2	4	no	—
138	C₂×C₄	—	94	8	no	—
139	C₂×C₄	—	188	4	no	—
140	C₂×C₄	—	188	4	no	—
141	C₂×C₄	—	188	4	no	—
142	C₂×C₄	—	188	4	no	—
143	C₂×C₄	—	188	4	no	—
144	C₂×C₄	#13094024	1	4	yes	—
145	C₂×C₄	C₂²×Dic₉₄	1	2	yes	—
146	C₂×C₄	#41×D₄₇	1	4	yes	—
147	C₈	—	2	8	no	—
148	C₂²	#15689635	1	2	yes	—
149	C₂²	C₂×#13094024	1	1	yes	characteristic
150	C₂²	#15689923	1	4	yes	—
151	C₄	—	2	4	no	—
152	C₄	—	4	4	no	—
153	C₄	—	376	4	no	—
154	C₄	—	188	8	no	—
155	C₄	#15689676	1	4	yes	—
156	C₂	#20246681	1	1	yes	characteristic
157	C₂	#20253499	1	4	yes	—
158	C₂	#20253940	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13094016

C₂

yes

maximal

#20244871

C₂

yes

maximal

#20244727

C₂

yes

maximal

#20243285

C₂

yes

maximal

C₄₇×#1745

C₂

yes

maximal, characteristic

C₂×#13094016

C₂²

yes

—

C₂×#13093958

C₂²

yes

—

C₂×#13093956

C₂²

yes

—

#15689021

C₂²

yes

—

#15689007

—

#15688989

C₂²

yes

—

C₄×Dic₄₇⋊C₄

—

C₄×C₄⋊Dic₄₇

C₂²

yes

—

#15688886

C₂²

yes

—

#15688886

C₂²

yes

—

C₂×#97×C₄₇

C₂²

yes

—

C₄³×C₄₇

C₂²

yes

characteristic

C₄₇×#336

C₂²

yes

—

#13094016

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

C₂×C₄⋊Dic₄₇

C₂³

yes

—

C₂×Dic₄₇⋊C₄

—

C₂×Dic₄₇⋊C₄

—

C₂×C₄×Dic₄₇

—

#13093958

—

#13093956

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

#97×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×Q₈

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

#41

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂²×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊Q₈

yes

—

C₂×C₄×C₄₇

C₂²×Q₈

yes

—

C₂×C₄×C₄₇

C₂×#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

#23840

—

maximal, Sylow

C₂²×C₄₇

#525

yes

—

C₂²×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂²×C₄₇

#575

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#533

yes

—

Dic₄₇

—

Dic₄₇

—

—

—

—

—

D₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#3166

yes

—

C₂×#125

—

C₂×#107

—

C₂×#106

—

C₂×#97

C₂×D₄₇

yes

—

#389

—

#387

—

#382

—

C₄×C₄⋊C₄

—

C₄×C₄⋊C₄

—

C₄³

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

C₄₇

#23840

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²×D₄₇

yes

—

100

#125

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄⋊C₄

—

105

C₂×C₄²

—

106

C₂×C₄²

D₁₈₈

yes

—

107

C₂×C₄²

C₂²×D₄₇

yes

—

108

C₂×C₄²

C₂²×D₄₇

yes

characteristic

109

#107

—

110

#106

—

111

#97

Dic₉₄

yes

—

112

#95

—

113

#95

—

114

#95

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

C₂×D₁₈₈

yes

—

118

C₂²×C₄

C₂×D₁₈₈

yes

—

119

C₂²×C₄

C₂³×D₄₇

yes

characteristic

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂×Dic₉₄

yes

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

188

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

188

—

132

C₄²

C₂×Dic₉₄

yes

—

133

C₂³

C₂²×D₁₈₈

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

188

—

140

C₂×C₄

—

188

—

141

C₂×C₄

—

188

—

142

C₂×C₄

—

188

—

143

C₂×C₄

—

188

—

144

C₂×C₄

#13094024

yes

—

145

C₂×C₄

C₂²×Dic₉₄

yes

—

146

C₂×C₄

#41×D₄₇

yes

—

147

C₈

—

148

C₂²

#15689635

yes

—

149

C₂²

C₂×#13094024

yes

characteristic

150

C₂²

#15689923

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

376

—

154

C₄

—

188

—

155

C₄

#15689676

yes

—

156

C₂

#20246681

yes

characteristic

157

C₂

#20253499

yes

—

158

C₂

#20253940

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₁₈₈
commutator, cosocle	C₂²×C₄×C₄₇	C₂⁴
Frattini	C₂²×C₄	C₂³×D₄₇
Fitting	C₄₇×#1745	C₂
socle, prime radical	C₂³×C₄₇	C₂²×D₄
hypercenter	#1745	D₄₇
hypocenter	C₄₇	#23840
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ #1745

subgroup

G/H

center

C₂³

C₂²×D₁₈₈

commutator, cosocle

C₂²×C₄×C₄₇

C₂⁴

Frattini

C₂²×C₄

C₂³×D₄₇

Fitting

C₄₇×#1745

C₂

socle, prime radical

C₂³×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ #1745

N	H ≅ G/N	type
98. C₄₇	#23840	semidirect

H ≅ G/N

type

98. C₄₇

#23840

semidirect

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

subgroup	index
C₄×#13094016	2
#20244871	2
#20244727	2
#20243285	2
C₄₇×#1745	2
#23840	47

subgroup

index

C₄₇×#1745

Subgroups — #26247216

Conjugacy classes of subgroups

Complements

Maximal subgroups