Group #20244333 — subgroups

Subgroups — #20244333

Order 6016.

260 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 101 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×#10625329	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688882	C₂	1	4	yes	maximal
4	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
5	C₂×#10625329	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₄₇	—	2	1	no	—
7	C₂×C₄×Dic₄₇	C₂²	1	2	yes	—
8	#13093935	—	2	1	no	—
9	C₄×C₄₇⋊C₈	—	2	1	no	—
10	C₄₇⋊#95	—	2	1	no	—
11	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
12	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
13	#125×C₄₇	C₄	1	2	yes	complemented
14	C₂²⋊Q₈×C₄₇	—	2	1	no	—
15	C₂²⋊Q₈×C₄₇	—	2	1	no	—
16	C₄×Q₈×C₄₇	—	2	1	no	—
17	C₄×Q₈×C₄₇	—	2	1	no	—
18	#117×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	—	2	1	no	—
22	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	—	2	2	no	—
26	C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
31	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
32	C₂²⋊C₄×C₄₇	—	2	2	no	—
33	C₄²×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	D₄	1	1	yes	characteristic
37	C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₄⋊Dic₄₇	—	2	4	no	—
39	C₄×Dic₄₇	—	2	4	no	—
40	#10625329	—	2	2	no	—
41	#10625329	Q₈	1	4	yes	complemented
42	C₂×C₄₇⋊C₈	—	2	2	no	—
43	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₄₇	—	4	2	no	—
45	Q₈×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	4	1	no	—
50	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
57	C₂×C₄×C₄₇	#41	1	2	yes	—
58	C₂×Dic₄₇	—	2	4	no	—
59	C₂×Dic₄₇	—	4	2	no	—
60	C₂×Dic₄₇	—	4	2	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	#124	1	1	yes	characteristic
66	C₂²×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	2	1	no	—
69	C₄×C₄₇	—	2	1	no	—
70	C₄×C₄₇	—	4	2	no	—
71	C₄×C₄₇	—	4	2	no	—
72	C₄×C₄₇	#104	1	2	yes	—
73	C₄×C₄₇	C₄×Q₈	1	2	yes	—
74	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
75	Dic₄₇	—	4	4	no	—
76	#1986	—	47	1	no	maximal, Sylow
77	C₂×C₄₇	—	2	1	no	—
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	#391	1	1	yes	characteristic
80	C₂×C₄₇	C₂×#104	1	1	yes	characteristic
81	C₂×C₄₇	#421	1	1	yes	characteristic
82	#533	D₄₇	1	1	yes	characteristic
83	C₄×M₁₆	—	47	1	no	—
84	C₄×C₄⋊C₄	—	47	1	no	—
85	#339	—	47	4	no	—
86	C₄₇	#1986	1	1	yes	characteristic, complemented, Sylow
87	C₂×M₁₆	—	47	2	no	—
88	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic, complemented
89	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic, complemented
90	#125	Dic₄₇	1	2	yes	complemented
91	C₂²⋊Q₈	—	2	1	no	—
92	C₂²⋊Q₈	—	2	1	no	—
93	C₄×Q₈	—	2	1	no	—
94	C₄×Q₈	—	2	1	no	—
95	#117	C₂×D₄₇	1	2	yes	—
96	C₂×C₄⋊C₄	—	94	1	no	—
97	C₂×C₄²	—	47	2	no	—
98	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
99	#97	—	94	1	no	—
100	C₄×C₈	—	94	1	no	—
101	#95	—	94	1	no	—
102	C₂×Q₈	—	2	1	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂²×C₄	—	94	2	no	—
105	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
106	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
107	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
108	M₁₆	—	47	4	no	—
109	M₁₆	—	94	2	no	—
110	C₂×C₈	—	94	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	94	4	no	—
116	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
117	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₄²	—	2	2	no	—
120	C₄²	—	94	4	no	—
121	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
122	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
123	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
126	Q₈	—	4	2	no	—
127	Q₈	—	4	2	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	94	4	no	—
133	C₂×C₄	—	188	2	no	—
134	C₂×C₄	—	188	2	no	—
135	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
136	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
137	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
138	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
139	C₂×C₄	Q₈×D₄₇	1	2	yes	—
140	C₂×C₄	#10625354	1	2	yes	—
141	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
142	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
143	C₈	—	94	4	no	—
144	C₂²	—	2	1	no	—
145	C₂²	—	2	1	no	—
146	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
147	C₂²	#13094013	1	1	yes	characteristic
148	C₂²	#13094022	1	1	yes	characteristic
149	C₄	—	2	1	no	—
150	C₄	—	2	1	no	—
151	C₄	—	2	1	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	188	4	no	—
155	C₄	#13093951	1	2	yes	—
156	C₄	Q₈×Dic₄₇	1	2	yes	—
157	C₄	#13094000	1	2	yes	—
158	C₂	—	2	1	no	—
159	C₂	—	2	1	no	—
160	C₂	#15689030	1	1	yes	characteristic
161	C₂	C₂×#13093951	1	1	yes	characteristic
162	C₂	#15689140	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10625329

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₄₇

C₂

yes

maximal, characteristic

#15688882

C₂

yes

maximal

C₄₇×#533

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄×Dic₄₇

C₂²

yes

—

#13093935

—

C₄×C₄₇⋊C₈

—

C₄₇⋊#95

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

#125×C₄₇

C₄

yes

complemented

C₂²⋊Q₈×C₄₇

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

Q₈

yes

complemented

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#124

yes

characteristic

C₂²×C₄₇

C₄⋊Q₈

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#104

yes

—

C₄×C₄₇

C₄×Q₈

yes

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

#1986

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

C₄×M₁₆

—

C₄×C₄⋊C₄

—

#339

—

C₄₇

#1986

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

Dic₄₇

yes

characteristic, complemented

C₄⋊Q₈

Dic₄₇

yes

characteristic, complemented

#125

Dic₄₇

yes

complemented

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#97

—

100

C₄×C₈

—

101

#95

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

106

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

107

C₂²×C₄

C₂²×D₄₇

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

C₂×Dic₄₇

yes

—

117

C₄⋊C₄

C₂×Dic₄₇

yes

—

118

C₂²⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

C₂×Dic₄₇

yes

characteristic

122

C₄²

C₂×Dic₄₇

yes

characteristic

123

C₄²

C₂²⋊D₄₇

yes

characteristic

124

C₄²

C₂²⋊D₄₇

yes

characteristic

125

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

126

Q₈

—

127

Q₈

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

188

—

134

C₂×C₄

—

188

—

135

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

136

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

137

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

138

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

139

C₂×C₄

Q₈×D₄₇

yes

—

140

C₂×C₄

#10625354

yes

—

141

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

142

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

147

C₂²

#13094013

yes

characteristic

148

C₂²

#13094022

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

188

—

155

C₄

#13093951

yes

—

156

C₄

Q₈×Dic₄₇

yes

—

157

C₄

#13094000

yes

—

158

C₂

—

159

C₂

—

160

C₂

#15689030

yes

characteristic

161

C₂

C₂×#13093951

yes

characteristic

162

C₂

#15689140

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#533	D₄₇
hypocenter	C₄₇	#1986
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#533

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
11. C₄⋊Q₈×C₄₇	C₄	semidirect
12. C₄⋊Q₈×C₄₇	C₄	semidirect
13. #125×C₄₇	C₄	semidirect
41. #10625329	Q₈	semidirect
86. C₄₇	#1986	semidirect
88. C₄⋊Q₈	Dic₄₇	semidirect
89. C₄⋊Q₈	Dic₄₇	semidirect
90. #125	Dic₄₇	semidirect

H ≅ G/N

type

11. C₄⋊Q₈×C₄₇

C₄

semidirect

12. C₄⋊Q₈×C₄₇

C₄

semidirect

13. #125×C₄₇

C₄

semidirect

41. #10625329

Q₈

semidirect

86. C₄₇

#1986

semidirect

88. C₄⋊Q₈

Dic₄₇

semidirect

89. C₄⋊Q₈

Dic₄₇

semidirect

90. #125

Dic₄₇

semidirect

Subgroups — #20244333

Conjugacy classes of subgroups

Complements