Group #20243925 — subgroups

Subgroups — #20243925

Order 6016.

314 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 89 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	#15689697	C₂	1	1	yes	maximal, characteristic
2	#15689141	C₂	1	2	yes	maximal
3	C₂×#13093943	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688879	C₂	1	2	yes	maximal, complemented
5	C₄₇×#519	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	C₂²	1	1	yes	characteristic
7	C₂×#10625329	C₂²	1	2	yes	—
8	#13094022	C₄	1	2	yes	complemented
9	#13094013	C₄	1	2	yes	complemented
10	#13094007	—	2	2	no	—
11	#13094000	—	2	2	no	—
12	#13093983	C₂²	1	1	yes	characteristic
13	#13093952	—	2	4	no	—
14	#13093943	—	2	2	no	—
15	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
16	C₄×Q₈×C₄₇	—	2	4	no	—
17	#117×C₄₇	—	2	2	no	—
18	#117×C₄₇	C₂²	1	2	yes	—
19	C₂×Q₈×C₄₇	—	2	2	no	—
20	C₂×Q₈×C₄₇	—	2	2	no	—
21	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
22	C₂×Q₈×C₄₇	D₄	1	2	yes	—
23	C₂²×C₄×C₄₇	D₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂×C₂²⋊D₄₇	—	2	1	no	—
26	#10625350	—	2	2	no	—
27	#10625350	—	2	2	no	—
28	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₄₇	—	2	1	no	—
30	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₄₇	—	2	4	no	—
32	C₄⋊C₄×C₄₇	—	2	2	no	—
33	C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	—	2	4	no	—
35	C₄²×C₄₇	D₄	1	4	yes	—
36	C₂²⋊Dic₄₇	—	2	1	no	—
37	(C₂×D₄₇)⋊C₄	—	2	4	no	—
38	C₄⋊Dic₄₇	—	2	1	no	—
39	Dic₄₇⋊C₄	—	2	4	no	—
40	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
41	#10625329	—	2	2	no	—
42	#10625329	—	2	2	no	—
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₄₇	—	2	1	no	—
46	Q₈×C₄₇	—	2	4	no	—
47	Q₈×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₄₇	#41	1	2	yes	—
56	C₂²⋊D₄₇	—	4	1	no	—
57	C₂²⋊D₄₇	—	4	1	no	—
58	C₂×Dic₄₇	—	2	2	no	—
59	C₂×Dic₄₇	—	2	1	no	—
60	D₁₈₈	—	2	2	no	—
61	C₄×D₄₇	—	4	2	no	—
62	Dic₉₄	—	2	2	no	—
63	C₄₇⋊C₈	—	4	2	no	—
64	C₂²×C₄₇	—	4	1	no	—
65	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄₇	#120	1	1	yes	characteristic
67	C₂²×C₄₇	#123	1	1	yes	characteristic
68	C₂×D₄₇	—	4	2	no	—
69	C₂×D₄₇	—	4	1	no	—
70	C₄×C₄₇	—	2	4	no	—
71	C₄×C₄₇	—	4	4	no	—
72	C₄×C₄₇	C₄×D₄	1	2	yes	—
73	C₄×C₄₇	#121	1	2	yes	—
74	Dic₄₇	—	4	2	no	—
75	Dic₄₇	—	4	1	no	—
76	#1912	—	47	1	no	maximal, Sylow
77	C₂×C₄₇	—	2	1	no	—
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	—	2	1	no	—
80	C₂×C₄₇	#386	1	1	yes	characteristic
81	D₄₇	—	8	1	no	—
82	#536	—	47	1	no	—
83	#519	D₄₇	1	1	yes	characteristic, complemented
84	#421	—	47	2	no	—
85	C₂×#101	—	47	1	no	—
86	H(3,ℤ/4)	—	47	2	no	—
87	C₄₇	#1912	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	47	1	no	—
89	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
90	C₂×M₁₆	—	47	2	no	—
91	C₄⋊Q₈	—	47	2	no	—
92	#124	—	47	2	no	—
93	#123	—	94	2	no	—
94	C₂²⋊Q₈	—	94	2	no	—
95	C₄×Q₈	—	2	4	no	—
96	#117	—	2	2	no	—
97	#117	—	47	1	no	—
98	#117	C₂×D₄₇	1	2	yes	—
99	#104	—	94	4	no	—
100	#101	—	94	2	no	—
101	#41	—	94	2	no	—
102	#41	—	94	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	47	1	no	—
106	C₂×Q₈	C₄×D₄₇	1	2	yes	—
107	C₂×Q₈	D₁₈₈	1	2	yes	—
108	C₂×D₄	—	47	1	no	—
109	C₂×D₄	—	94	1	no	—
110	C₂²×C₄	—	94	1	no	—
111	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
113	M₁₆	—	94	2	no	—
114	M₁₆	—	94	2	no	—
115	C₂×C₈	—	94	2	no	—
116	C₄⋊C₄	—	2	4	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	94	4	no	—
119	C₄⋊C₄	—	94	1	no	—
120	C₂²⋊C₄	—	2	2	no	—
121	C₂²⋊C₄	—	94	1	no	—
122	C₂²⋊C₄	—	94	4	no	—
123	C₄²	—	2	4	no	—
124	C₄²	—	47	2	no	—
125	C₄²	C₂²⋊D₄₇	1	4	yes	—
126	C₂³	—	94	1	no	—
127	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
128	Q₈	—	2	4	no	—
129	Q₈	—	2	4	no	—
130	Q₈	—	94	2	no	—
131	D₄	—	94	2	no	—
132	D₄	—	188	1	no	—
133	D₄	—	188	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	4	no	—
137	C₂×C₄	—	94	2	no	—
138	C₂×C₄	—	94	1	no	—
139	C₂×C₄	—	188	2	no	—
140	C₂×C₄	(C₂×D₄₇)⋊C₄	1	4	yes	—
141	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
142	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
143	C₂×C₄	#10625350	1	2	yes	—
144	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
145	C₈	—	188	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	188	2	no	—
148	C₂²	—	188	1	no	—
149	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
150	C₂²	#13093993	1	1	yes	characteristic
151	C₂²	#13094007	1	1	yes	characteristic
152	C₄	—	2	4	no	—
153	C₄	—	4	4	no	—
154	C₄	—	188	2	no	—
155	C₄	—	188	1	no	—
156	C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
157	C₄	#13093998	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	376	1	no	—
162	C₂	#15689002	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689697

C₂

yes

maximal, characteristic

#15689141

C₂

yes

maximal

C₂×#13093943

C₂

yes

maximal, characteristic, complemented

#15688879

C₂

yes

maximal, complemented

C₄₇×#519

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

#13094022

C₄

yes

complemented

#13094013

C₄

yes

—

—

C₂²

yes

characteristic

#13093952

—

#13093943

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

#117×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

—

C₂×D₁₈₈

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂×C₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

C₂×C₄

yes

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

Dic₄₇

—

Dic₄₇

—

#1912

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#386

yes

characteristic

D₄₇

—

#536

—

#519

D₄₇

yes

characteristic, complemented

#421

—

C₂×#101

—

H(3,ℤ/4)

—

C₄₇

#1912

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

#117

—

#117

C₂×D₄₇

yes

—

#104

—

100

#101

—

101

#41

—

102

#41

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₄×D₄₇

yes

—

107

C₂×Q₈

D₁₈₈

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₄₇

yes

—

112

C₂²×C₄

C₂²×D₄₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂²⋊D₄₇

yes

—

126

C₂³

—

127

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

188

—

133

D₄

—

188

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

188

—

140

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

141

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

142

C₂×C₄

C₂×D₁₈₈

yes

characteristic

143

C₂×C₄

#10625350

yes

—

144

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

145

C₈

—

188

—

146

C₂²

—

147

C₂²

—

188

—

148

C₂²

—

188

—

149

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

150

C₂²

#13093993

yes

characteristic

151

C₂²

#13094007

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

—

188

—

155

C₄

—

188

—

156

C₄

C₄×C₂²⋊D₄₇

yes

—

157

C₄

#13093998

yes

—

158

C₂

—

159

C₂

—

160

C₂

—

161

C₂

—

376

—

162

C₂

#15689002

yes

characteristic

	subgroup	G/H
center	C₂	#15689002
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#519	C₂
socle	C₂×C₄₇	#386
hypercenter	#519	D₄₇
hypocenter	C₄₇	#1912
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #519

subgroup

G/H

center

C₂

#15689002

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#519

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #519

N	H ≅ G/N	type
3. C₂×#13093943	C₂	semidirect
4. #15688879	C₂	semidirect
5. C₄₇×#519	C₂	semidirect
8. #13094022	C₄	semidirect
9. #13094013	C₄	semidirect
83. #519	D₄₇	semidirect
87. C₄₇	#1912	semidirect

H ≅ G/N

type

3. C₂×#13093943

C₂

semidirect

4. #15688879

C₂

semidirect

5. C₄₇×#519

C₂

semidirect

8. #13094022

C₄

semidirect

9. #13094013

C₄

semidirect

83. #519

D₄₇

semidirect

87. C₄₇

#1912

semidirect

Subgroups — #20243925

Conjugacy classes of subgroups

Complements