Group #20256398 — subgroups

Subgroups — #20256398

Order 6016.

384 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 117 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	#15689858	C₂	1	2	yes	maximal, complemented
2	#15689508	C₂	1	2	yes	maximal, complemented
3	#15689500	C₂	1	1	yes	maximal, characteristic
4	#15689341	C₂	1	4	yes	maximal, complemented
5	#15689323	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689205	C₂	1	2	yes	maximal, complemented
8	C₄×Q₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#10625336	C₂²	1	4	yes	—
10	C₂×D₃₇₆	C₂²	1	2	yes	—
11	C₂×D₃₇₆	C₂²	1	2	yes	—
12	#13094021	C₂²	1	2	yes	—
13	#13094013	C₂²	1	4	yes	—
14	#13094012	—	2	2	no	—
15	#13094009	—	2	4	no	—
16	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
17	C₄×D₁₈₈	—	2	2	no	—
18	#13093986	C₂²	1	4	yes	—
19	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₄₇⋊C₈	C₂²	1	2	yes	—
21	#13093944	C₂²	1	4	yes	—
22	C₈×Dic₄₇	C₂²	1	2	yes	—
23	C₂×Q₁₆×C₄₇	C₂²	1	1	yes	characteristic
24	C₄×Q₈×C₄₇	C₂²	1	2	yes	—
25	#107×C₄₇	C₂²	1	1	yes	characteristic
26	#103×C₄₇	C₂²	1	2	yes	—
27	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
29	C₂×D₁₈₈	—	2	2	no	—
30	C₂×D₁₈₈	—	2	2	no	—
31	C₂×D₁₈₈	C₂³	1	4	yes	—
32	C₂×C₄×D₄₇	—	2	4	no	—
33	C₂×Dic₉₄	D₄	1	1	yes	characteristic
34	Q₁₆×C₄₇	—	2	2	no	—
35	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
37	C₄⋊C₄×C₄₇	—	2	2	no	—
38	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
39	C₄²×C₄₇	—	2	2	no	—
40	C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	#10625336	—	4	4	no	—
42	(C₂×D₄₇)⋊C₄	—	2	4	no	—
43	(C₂×D₄₇)⋊C₄	—	2	4	no	—
44	(C₂×D₄₇)⋊C₄	—	2	4	no	—
45	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
46	Dic₄₇⋊C₄	D₄	1	2	yes	—
47	C₄×Dic₄₇	C₂³	1	2	yes	—
48	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
49	D₃₇₆	—	2	4	no	—
50	D₃₇₆	—	4	2	no	—
51	C₂²×D₄₇	—	2	4	no	—
52	Q₈×C₄₇	—	2	2	no	—
53	Q₈×C₄₇	—	2	2	no	—
54	C₂×C₄×C₄₇	—	2	2	no	—
55	C₂×C₄×C₄₇	—	2	2	no	—
56	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
59	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
60	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
61	D₁₈₈	—	2	4	no	—
62	D₁₈₈	—	2	4	no	—
63	D₁₈₈	—	4	4	no	—
64	D₁₈₈	—	8	2	no	—
65	C₄×D₄₇	—	8	4	no	—
66	Dic₉₄	—	2	2	no	—
67	C₈×C₄₇	—	2	1	no	—
68	C₈×C₄₇	#41	1	2	yes	—
69	C₄₇⋊C₈	—	2	2	no	—
70	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
71	C₂×D₄₇	—	4	4	no	—
72	C₂×D₄₇	—	4	4	no	—
73	C₂×D₄₇	—	4	4	no	—
74	C₄×C₄₇	—	2	1	no	—
75	C₄×C₄₇	—	4	2	no	—
76	C₄×C₄₇	—	4	2	no	—
77	C₄×C₄₇	#135	1	2	yes	—
78	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
79	C₄×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
80	Dic₄₇	—	2	2	no	—
81	Dic₄₇	—	2	2	no	—
82	#3396	—	47	1	no	maximal, Sylow
83	C₂×C₄₇	#550	1	1	yes	characteristic
84	C₂×C₄₇	C₂×#135	1	1	yes	characteristic
85	C₂×C₄₇	#576	1	1	yes	characteristic
86	D₄₇	—	8	4	no	—
87	#559	—	47	2	no	—
88	#495	—	47	2	no	—
89	#493	—	47	1	no	—
90	#463	—	47	4	no	—
91	#460	—	47	2	no	—
92	C₈×Q₈	—	47	1	no	—
93	C₄×Q₁₆	D₄₇	1	1	yes	characteristic, complemented
94	C₄×D₈	—	47	2	no	—
95	C₄₇	#3396	1	1	yes	characteristic, complemented, Sylow
96	C₂×Q₁₆	C₂×D₄₇	1	1	yes	characteristic
97	C₂×SD₁₆	—	47	4	no	—
98	C₂×D₈	—	47	2	no	—
99	C₂×D₈	—	47	2	no	—
100	#127	—	47	2	no	—
101	#124	—	47	4	no	—
102	#124	—	94	2	no	—
103	#123	—	94	4	no	—
104	C₄×Q₈	—	47	1	no	—
105	C₄×Q₈	C₂×D₄₇	1	2	yes	—
106	C₄×D₄	—	47	4	no	—
107	C₄×D₄	—	94	2	no	—
108	#107	C₂×D₄₇	1	1	yes	characteristic
109	C₄⋊C₈	—	47	2	no	—
110	C₄⋊C₈	—	47	1	no	—
111	#103	C₂×D₄₇	1	2	yes	—
112	#102	—	47	4	no	—
113	C₄×C₈	—	47	2	no	—
114	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
115	C₂×Q₈	—	47	1	no	—
116	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
117	C₂×D₄	—	47	4	no	—
118	C₂×D₄	—	94	2	no	—
119	C₂×D₄	—	94	2	no	—
120	C₂²×C₄	—	94	4	no	—
121	Q₁₆	—	2	2	no	—
122	SD₁₆	—	188	4	no	—
123	D₈	—	94	4	no	—
124	D₈	—	188	2	no	—
125	C₂×C₈	—	47	2	no	—
126	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
127	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
128	C₄⋊C₄	—	2	2	no	—
129	C₄⋊C₄	—	47	2	no	—
130	C₄⋊C₄	—	47	1	no	—
131	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
132	C₂²⋊C₄	—	94	4	no	—
133	C₂²⋊C₄	—	94	4	no	—
134	C₂²⋊C₄	—	94	4	no	—
135	C₄²	—	2	2	no	—
136	C₄²	—	47	2	no	—
137	C₄²	C₂²×D₄₇	1	1	yes	characteristic
138	C₂³	—	94	4	no	—
139	Q₈	—	2	2	no	—
140	Q₈	—	2	2	no	—
141	Q₈	—	94	2	no	—
142	D₄	—	94	4	no	—
143	D₄	—	94	4	no	—
144	D₄	—	376	2	no	—
145	D₄	—	188	4	no	—
146	C₂×C₄	—	2	2	no	—
147	C₂×C₄	—	2	2	no	—
148	C₂×C₄	—	47	2	no	—
149	C₂×C₄	—	47	2	no	—
150	C₂×C₄	—	376	4	no	—
151	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
152	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
153	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
154	C₈	—	2	1	no	—
155	C₈	—	94	2	no	—
156	C₈	#10625350	1	2	yes	—
157	C₂²	—	188	4	no	—
158	C₂²	—	188	4	no	—
159	C₂²	—	188	4	no	—
160	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
161	C₄	—	2	1	no	—
162	C₄	—	4	2	no	—
163	C₄	—	4	2	no	—
164	C₄	—	94	2	no	—
165	C₄	—	94	2	no	—
166	C₄	#13094047	1	2	yes	—
167	C₄	C₂×#10625350	1	1	yes	characteristic
168	C₄	#13094070	1	1	yes	characteristic
169	C₂	—	376	4	no	—
170	C₂	#15689794	1	1	yes	characteristic
171	C₂	C₂×#13094047	1	1	yes	characteristic
172	C₂	#15689925	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689858

C₂

yes

maximal, complemented

#15689508

C₂

yes

maximal, complemented

#15689500

C₂

yes

maximal, characteristic

#15689341

C₂

yes

maximal, complemented

#15689323

C₂

yes

maximal, complemented

C₈×Dic₉₄

C₂

yes

maximal, characteristic, complemented

#15689205

C₂

yes

maximal, complemented

C₄×Q₁₆×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625336

C₂²

yes

—

C₂×D₃₇₆

C₂²

yes

—

C₂×D₃₇₆

C₂²

yes

—

#13094021

C₂²

yes

—

#13094013

C₂²

yes

—

#13094012

—

#13094009

—

C₄×Dic₉₄

C₂²

yes

characteristic

C₄×D₁₈₈

—

#13093986

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₂²

yes

—

#13093944

C₂²

yes

—

C₈×Dic₄₇

C₂²

yes

—

C₂×Q₁₆×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

characteristic

#103×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

C₂³

yes

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

D₄

yes

characteristic

Q₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625336

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

characteristic

Dic₄₇⋊C₄

D₄

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

D₃₇₆

—

D₃₇₆

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#41

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#135

yes

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

H(5,𝔽₂)

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

—

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

D₄₇

yes

characteristic, complemented

C₄×D₈

—

C₄₇

#3396

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₄₇

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

100

#127

—

101

#124

—

102

#124

—

103

#123

—

104

C₄×Q₈

—

105

C₄×Q₈

C₂×D₄₇

yes

—

106

C₄×D₄

—

107

C₄×D₄

—

108

#107

C₂×D₄₇

yes

characteristic

109

C₄⋊C₈

—

110

C₄⋊C₈

—

111

#103

C₂×D₄₇

yes

—

112

#102

—

113

C₄×C₈

—

114

C₄×C₈

C₂×D₄₇

yes

characteristic

115

C₂×Q₈

—

116

C₂×Q₈

C₂²×D₄₇

yes

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂×D₄

—

120

C₂²×C₄

—

121

Q₁₆

—

122

SD₁₆

—

188

—

123

D₈

—

124

D₈

—

188

—

125

C₂×C₈

—

126

C₂×C₈

C₂²×D₄₇

yes

characteristic

127

C₂×C₈

C₂²×D₄₇

yes

characteristic

128

C₄⋊C₄

—

129

C₄⋊C₄

—

130

C₄⋊C₄

—

131

C₄⋊C₄

C₂²×D₄₇

yes

—

132

C₂²⋊C₄

—

133

C₂²⋊C₄

—

134

C₂²⋊C₄

—

135

C₄²

—

136

C₄²

—

137

C₄²

C₂²×D₄₇

yes

characteristic

138

C₂³

—

139

Q₈

—

140

Q₈

—

141

Q₈

—

142

D₄

—

143

D₄

—

144

D₄

—

376

—

145

D₄

—

188

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

150

C₂×C₄

—

376

—

151

C₂×C₄

D₄×D₄₇

yes

characteristic

152

C₂×C₄

D₄×D₄₇

yes

characteristic

153

C₂×C₄

C₂³×D₄₇

yes

characteristic

154

C₈

—

155

C₈

—

156

C₈

#10625350

yes

—

157

C₂²

—

188

—

158

C₂²

—

188

—

159

C₂²

—

188

—

160

C₂²

C₂×D₄×D₄₇

yes

characteristic

161

C₄

—

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

#13094047

yes

—

167

C₄

C₂×#10625350

yes

characteristic

168

C₄

#13094070

yes

characteristic

169

C₂

—

376

—

170

C₂

#15689794

yes

characteristic

171

C₂

C₂×#13094047

yes

characteristic

172

C₂

#15689925

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄×Q₁₆×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₄×Q₁₆	D₄₇
hypocenter	C₄₇	#3396
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄×Q₁₆×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

hypercenter

C₄×Q₁₆

D₄₇

hypocenter

C₄₇

#3396

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

N	H ≅ G/N	type
1. #15689858	C₂	semidirect
2. #15689508	C₂	semidirect
4. #15689341	C₂	semidirect
5. #15689323	C₂	semidirect
6. C₈×Dic₉₄	C₂	semidirect
7. #15689205	C₂	semidirect
8. C₄×Q₁₆×C₄₇	C₂	semidirect
93. C₄×Q₁₆	D₄₇	semidirect
95. C₄₇	#3396	semidirect

H ≅ G/N

type

1. #15689858

C₂

semidirect

2. #15689508

C₂

semidirect

4. #15689341

C₂

semidirect

5. #15689323

C₂

semidirect

6. C₈×Dic₉₄

C₂

semidirect

7. #15689205

C₂

semidirect

8. C₄×Q₁₆×C₄₇

C₂

semidirect

93. C₄×Q₁₆

D₄₇

semidirect

95. C₄₇

#3396

semidirect

Subgroups — #20256398

Conjugacy classes of subgroups

Complements