Group #26103576 — subgroups

Subgroups — #26103576

Order 12032.

432 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20243312	C₂	1	2	yes	maximal
2	C₄₇×#1784	C₂	1	1	yes	maximal, characteristic
3	C₂×#13093943	—	2	2	no	—
4	C₂×#13093943	C₄	1	4	yes	—
5	#15689087	—	2	2	no	—
6	C₂³×Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂×#101×C₄₇	—	2	2	no	—
8	C₄₇×#407	C₄	1	2	yes	—
9	C₂×#10625329	—	2	4	no	—
10	#13093943	—	4	4	no	—
11	#13093943	—	4	4	no	—
12	C₂²⋊(C₄₇⋊C₈)	—	4	4	no	—
13	C₂²×Q₈×C₄₇	—	4	2	no	—
14	C₂²×Q₈×C₄₇	—	4	4	no	—
15	C₂²×Q₈×C₄₇	C₂×C₄	1	2	yes	—
16	C₂²×Q₈×C₄₇	D₄	1	2	yes	—
17	C₂³×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₄×C₄₇	D₄	1	1	yes	characteristic
19	C₂³×C₄×C₄₇	Q₈	1	1	yes	characteristic
20	C₂×M₁₆×C₄₇	—	2	2	no	—
21	#101×C₄₇	—	4	4	no	—
22	C₂²⋊C₈×C₄₇	—	4	2	no	—
23	C₂⁴×C₄₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×Q₈×C₄₇	—	2	4	no	—
25	C₂×Q₈×C₄₇	—	2	4	no	—
26	C₂×Q₈×C₄₇	—	2	4	no	—
27	C₂×Q₈×C₄₇	—	4	2	no	—
28	C₂×Q₈×C₄₇	—	4	2	no	—
29	C₂×Q₈×C₄₇	—	4	2	no	—
30	C₂×Q₈×C₄₇	—	4	16	no	—
31	C₂²×C₄×C₄₇	—	2	2	no	—
32	C₂²×C₄×C₄₇	—	2	4	no	—
33	C₂²×C₄×C₄₇	—	4	2	no	—
34	C₂²×C₄×C₄₇	—	4	1	no	—
35	C₂²×C₄×C₄₇	—	4	2	no	—
36	C₂²×C₄×C₄₇	—	4	1	no	—
37	C₂²×C₄×C₄₇	C₄²	1	1	yes	characteristic
38	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
39	C₂²×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
41	M₁₆×C₄₇	—	4	4	no	—
42	C₂×C₈×C₄₇	—	4	2	no	—
43	#10625329	—	4	4	no	—
44	#10625329	—	4	4	no	—
45	C₂×C₄₇⋊C₈	—	4	4	no	—
46	C₂³×C₄₇	—	2	2	no	—
47	C₂³×C₄₇	—	2	1	no	—
48	C₂³×C₄₇	—	4	2	no	—
49	C₂³×C₄₇	#95	1	1	yes	characteristic
50	Q₈×C₄₇	—	4	8	no	—
51	Q₈×C₄₇	—	4	8	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	1	no	—
56	C₂×C₄×C₄₇	—	2	2	no	—
57	C₂×C₄×C₄₇	—	2	2	no	—
58	C₂×C₄×C₄₇	—	2	2	no	—
59	C₂×C₄×C₄₇	—	2	2	no	—
60	C₂×C₄×C₄₇	—	4	4	no	—
61	C₂×C₄×C₄₇	—	4	4	no	—
62	C₂×C₄×C₄₇	—	4	4	no	—
63	C₂×C₄×C₄₇	#99	1	4	yes	—
64	C₂×C₄×C₄₇	#99	1	2	yes	—
65	C₈×C₄₇	—	8	2	no	—
66	C₄₇⋊C₈	—	8	4	no	—
67	#7315	—	47	1	no	maximal, Sylow
68	C₂²×C₄₇	—	2	4	no	—
69	C₂²×C₄₇	—	2	2	no	—
70	C₂²×C₄₇	—	4	1	no	—
71	C₂²×C₄₇	—	4	2	no	—
72	C₂²×C₄₇	—	4	1	no	—
73	C₂²×C₄₇	—	4	1	no	—
74	C₂²×C₄₇	#342	1	2	yes	—
75	C₂²×C₄₇	#342	1	1	yes	characteristic
76	C₄×C₄₇	—	2	4	no	—
77	C₄×C₄₇	—	2	2	no	—
78	C₄×C₄₇	—	4	1	no	—
79	C₄×C₄₇	—	4	2	no	—
80	#1784	—	47	2	no	—
81	#1784	D₄₇	1	1	yes	characteristic
82	C₂×C₄₇	—	2	2	no	—
83	C₂×C₄₇	—	2	1	no	—
84	C₂×C₄₇	—	4	2	no	—
85	C₂×C₄₇	#1304	1	1	yes	characteristic
86	C₂³×Q₈	C₂×D₄₇	1	1	yes	characteristic
87	C₂×#101	—	2	2	no	—
88	C₂×#101	—	47	4	no	—
89	C₂×#101	—	94	2	no	—
90	#407	—	94	2	no	—
91	#407	Dic₄₇	1	2	yes	—
92	C₄₇	#7315	1	1	yes	characteristic, complemented, Sylow
93	C₂²×Q₈	—	4	4	no	—
94	C₂²×Q₈	—	4	2	no	—
95	C₂²×Q₈	C₄×D₄₇	1	2	yes	—
96	C₂²×Q₈	C₂²⋊D₄₇	1	2	yes	—
97	C₂³×C₄	Dic₉₄	1	1	yes	characteristic
98	C₂³×C₄	D₁₈₈	1	1	yes	characteristic
99	C₂³×C₄	C₂×Dic₄₇	1	1	yes	characteristic
100	C₂×M₁₆	—	2	2	no	—
101	C₂×M₁₆	—	94	4	no	—
102	#101	—	4	4	no	—
103	#101	—	188	4	no	—
104	#101	—	188	4	no	—
105	C₂²⋊C₈	—	4	2	no	—
106	C₂²⋊C₈	—	188	4	no	—
107	C₂⁴	C₄⋊Dic₄₇	1	1	yes	characteristic
108	C₂×Q₈	—	2	4	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	4	2	no	—
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	4	16	no	—
115	C₂²×C₄	—	2	4	no	—
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	4	2	no	—
118	C₂²×C₄	—	4	1	no	—
119	C₂²×C₄	—	4	1	no	—
120	C₂²×C₄	—	4	2	no	—
121	C₂²×C₄	C₄×Dic₄₇	1	1	yes	characteristic
122	C₂²×C₄	Dic₄₇⋊C₄	1	2	yes	—
123	C₂²×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
124	C₂²×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
125	M₁₆	—	4	4	no	—
126	M₁₆	—	188	4	no	—
127	M₁₆	—	188	4	no	—
128	C₂×C₈	—	4	2	no	—
129	C₂×C₈	—	188	4	no	—
130	C₂³	—	2	2	no	—
131	C₂³	—	2	1	no	—
132	C₂³	—	4	2	no	—
133	C₂³	C₄₇⋊#95	1	1	yes	characteristic
134	Q₈	—	4	8	no	—
135	Q₈	—	4	8	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	2	2	no	—
140	C₂×C₄	—	2	2	no	—
141	C₂×C₄	—	2	2	no	—
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	2	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂×C₄	—	4	4	no	—
146	C₂×C₄	—	4	4	no	—
147	C₂×C₄	#13093938	1	4	yes	—
148	C₂×C₄	#13093939	1	2	yes	—
149	C₈	—	8	2	no	—
150	C₈	—	376	4	no	—
151	C₂²	—	2	4	no	—
152	C₂²	—	2	2	no	—
153	C₂²	—	4	1	no	—
154	C₂²	—	4	2	no	—
155	C₂²	—	4	1	no	—
156	C₂²	—	4	1	no	—
157	C₂²	#15688889	1	2	yes	—
158	C₂²	#15688890	1	1	yes	characteristic
159	C₄	—	2	2	no	—
160	C₄	—	2	4	no	—
161	C₄	—	4	1	no	—
162	C₄	—	4	2	no	—
163	C₂	—	2	2	no	—
164	C₂	—	2	1	no	—
165	C₂	—	4	2	no	—
166	C₂	C₄₇⋊#1304	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243312

C₂

yes

maximal

C₄₇×#1784

C₂

yes

maximal, characteristic

C₂×#13093943

—

C₂×#13093943

C₄

yes

—

#15689087

—

C₂³×Q₈×C₄₇

C₂²

yes

characteristic

C₂×#101×C₄₇

—

C₄₇×#407

C₄

yes

—

C₂×#10625329

—

#13093943

—

#13093943

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

—

C₂²×Q₈×C₄₇

C₂×C₄

yes

—

C₂²×Q₈×C₄₇

D₄

yes

—

C₂³×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₄₇

D₄

yes

characteristic

C₂³×C₄×C₄₇

Q₈

yes

characteristic

C₂×M₁₆×C₄₇

—

#101×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

C₄⋊C₄

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₄²

yes

characteristic

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂²×C₄×C₄₇

C₄⋊C₄

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

#95

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

#99

yes

—

C₂×C₄×C₄₇

#99

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#7315

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#342

yes

—

C₂²×C₄₇

#342

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#1784

—

#1784

D₄₇

yes

characteristic

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#1304

yes

characteristic

C₂³×Q₈

C₂×D₄₇

yes

—

—

—

—

Dic₄₇

yes

—

C₄₇

#7315

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₄×D₄₇

yes

—

C₂²×Q₈

C₂²⋊D₄₇

yes

—

C₂³×C₄

Dic₉₄

yes

characteristic

C₂³×C₄

D₁₈₈

yes

characteristic

C₂³×C₄

C₂×Dic₄₇

yes

characteristic

100

C₂×M₁₆

—

101

C₂×M₁₆

—

102

#101

—

103

#101

—

188

—

104

#101

—

188

—

105

C₂²⋊C₈

—

106

C₂²⋊C₈

—

188

—

107

C₂⁴

C₄⋊Dic₄₇

yes

characteristic

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₄×Dic₄₇

yes

characteristic

122

C₂²×C₄

Dic₄₇⋊C₄

yes

—

123

C₂²×C₄

(C₂×D₄₇)⋊C₄

yes

—

124

C₂²×C₄

C₂²⋊Dic₄₇

yes

characteristic

125

M₁₆

—

126

M₁₆

—

188

—

127

M₁₆

—

188

—

128

C₂×C₈

—

129

C₂×C₈

—

188

—

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

C₄₇⋊#95

yes

characteristic

134

Q₈

—

135

Q₈

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

#13093938

yes

—

148

C₂×C₄

#13093939

yes

—

149

C₈

—

150

C₈

—

376

—

151

C₂²

—

152

C₂²

—

153

C₂²

—

154

C₂²

—

155

C₂²

—

156

C₂²

—

157

C₂²

#15688889

yes

—

158

C₂²

#15688890

yes

characteristic

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

163

C₂

—

164

C₂

—

165

C₂

—

166

C₂

C₄₇⋊#1304

yes

characteristic

	subgroup	G/H
center	C₂	C₄₇⋊#1304
commutator	C₂²×C₄×C₄₇	C₄²
Frattini	C₂³×Q₈	C₂×D₄₇
Fitting	C₄₇×#1784	C₂
socle	C₂×C₄₇	#1304
prime radical	C₂⁴×C₄₇	C₄⋊C₄
cosocle	C₂³×Q₈×C₄₇	C₂²
hypercenter	#1784	D₄₇
hypocenter	C₄₇	#7315
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ #1784

subgroup

G/H

center

C₂

C₄₇⋊#1304

commutator

C₂²×C₄×C₄₇

C₄²

Frattini

C₂³×Q₈

C₂×D₄₇

Fitting

C₄₇×#1784

C₂

socle

C₂×C₄₇

#1304

prime radical

C₂⁴×C₄₇

C₄⋊C₄

cosocle

C₂³×Q₈×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ #1784

N	H ≅ G/N	type
92. C₄₇	#7315	semidirect

H ≅ G/N

type

92. C₄₇

#7315

semidirect

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

subgroup	index
#20243312	2
C₄₇×#1784	2
#7315	47

subgroup

index

#20243312

C₄₇×#1784

#7315

Subgroups — #26103576

Conjugacy classes of subgroups

Complements

Maximal subgroups