Group #20244541 — subgroups

Subgroups — #20244541

Order 6016.

282 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 71 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	#15689127	C₂	1	2	yes	maximal
2	#15689024	C₂	1	1	yes	maximal, characteristic
3	#15688892	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688882	C₂	1	2	yes	maximal, complemented
5	C₄₇×#544	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
7	C₂×#10625329	C₂²	1	2	yes	—
8	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
9	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
10	#13093944	—	2	2	no	—
11	#13093944	—	2	2	no	—
12	C₄⋊Q₈×C₄₇	—	2	2	no	—
13	#125×C₄₇	—	2	2	no	—
14	C₂²⋊Q₈×C₄₇	—	2	2	no	—
15	#117×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂³×D₄₇	—	2	1	no	—
18	C₂×Q₈×C₄₇	—	2	2	no	—
19	C₂²×C₄×C₄₇	D₄	1	2	yes	—
20	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
21	C₂²×Dic₄₇	—	2	1	no	—
22	C₂×D₁₈₈	—	2	2	no	—
23	C₂×D₁₈₈	—	2	2	no	—
24	C₂×D₁₈₈	—	2	1	no	—
25	C₂×D₁₈₈	Q₈	1	2	yes	complemented
26	C₄⋊C₄×C₄₇	—	2	2	no	—
27	C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
30	C₄⋊C₄×C₄₇	D₄	1	2	yes	—
31	C₂²⋊C₄×C₄₇	—	2	2	no	—
32	C₄²×C₄₇	—	2	2	no	—
33	(C₂×D₄₇)⋊C₄	—	4	4	no	—
34	C₄×Dic₄₇	—	2	2	no	—
35	#10625329	—	2	2	no	—
36	#10625329	—	2	2	no	—
37	C₂×C₄₇⋊C₈	—	2	2	no	—
38	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×D₄₇	—	2	2	no	—
40	C₂²×D₄₇	—	4	2	no	—
41	C₂²×D₄₇	—	4	2	no	—
42	C₂²×D₄₇	—	4	1	no	—
43	C₂²×D₄₇	—	4	1	no	—
44	Q₈×C₄₇	—	8	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#41	1	2	yes	—
52	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	4	1	no	—
55	C₂×Dic₄₇	—	4	1	no	—
56	D₁₈₈	—	2	4	no	—
57	D₁₈₈	—	4	2	no	—
58	C₄₇⋊C₈	—	4	2	no	—
59	C₂²×C₄₇	—	2	1	no	—
60	C₂²×C₄₇	—	2	1	no	—
61	C₂²×C₄₇	#120	1	1	yes	characteristic
62	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
63	C₂²×C₄₇	#124	1	1	yes	characteristic
64	C₂×D₄₇	—	4	4	no	—
65	C₂×D₄₇	—	4	2	no	—
66	C₂×D₄₇	—	8	2	no	—
67	C₂×D₄₇	—	8	1	no	—
68	C₂×D₄₇	—	8	1	no	—
69	C₄×C₄₇	—	4	2	no	—
70	C₄×C₄₇	—	4	2	no	—
71	C₄×C₄₇	—	4	2	no	—
72	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
73	C₄×C₄₇	#124	1	2	yes	—
74	Dic₄₇	—	4	2	no	—
75	#2027	—	47	1	no	maximal, Sylow
76	C₂×C₄₇	—	2	1	no	—
77	C₂×C₄₇	—	2	1	no	—
78	C₂×C₄₇	#393	1	1	yes	characteristic
79	C₂×C₄₇	#453	1	1	yes	characteristic
80	C₂×C₄₇	#454	1	1	yes	characteristic
81	D₄₇	—	8	2	no	—
82	#544	D₄₇	1	1	yes	characteristic, complemented
83	#418	—	47	2	no	—
84	#390	—	47	1	no	—
85	#343	—	47	1	no	—
86	#339	—	47	2	no	—
87	C₄₇	#2027	1	1	yes	characteristic, complemented, Sylow
88	C₂²×D₄	—	47	1	no	—
89	C₂×M₁₆	—	47	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	#125	—	2	2	no	—
92	C₂²⋊Q₈	—	2	2	no	—
93	#117	C₂×D₄₇	1	2	yes	—
94	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
95	C₂×C₂²⋊C₄	—	94	2	no	—
96	C₂×C₄²	—	47	1	no	—
97	#102	—	94	2	no	—
98	#102	—	94	2	no	—
99	C₂⁴	—	94	1	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×D₄	—	47	2	no	—
102	C₂×D₄	—	94	2	no	—
103	C₂×D₄	—	94	1	no	—
104	C₂×D₄	—	94	2	no	—
105	C₂²×C₄	—	94	1	no	—
106	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
107	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
108	M₁₆	—	94	2	no	—
109	M₁₆	—	94	2	no	—
110	C₂×C₈	—	94	2	no	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	4	no	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
115	C₄⋊C₄	C₂²⋊D₄₇	1	2	yes	—
116	C₂²⋊C₄	—	2	2	no	—
117	C₂²⋊C₄	—	188	4	no	—
118	C₄²	—	2	2	no	—
119	C₄²	—	94	2	no	—
120	C₂³	—	94	2	no	—
121	C₂³	—	188	2	no	—
122	C₂³	—	188	2	no	—
123	C₂³	—	188	1	no	—
124	C₂³	—	188	1	no	—
125	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
126	Q₈	—	8	2	no	—
127	D₄	—	94	4	no	—
128	D₄	—	188	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	94	2	no	—
134	C₂×C₄	—	188	1	no	—
135	C₂×C₄	—	188	1	no	—
136	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
137	C₂×C₄	#10625354	1	2	yes	—
138	C₂×C₄	#10625354	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
140	C₈	—	188	2	no	—
141	C₂²	—	2	1	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	376	2	no	—
144	C₂²	—	376	1	no	—
145	C₂²	—	376	1	no	—
146	C₂²	—	188	4	no	—
147	C₂²	—	188	2	no	—
148	C₂²	#13093993	1	1	yes	characteristic
149	C₂²	#13094000	1	1	yes	characteristic
150	C₂²	#13094013	1	1	yes	characteristic
151	C₄	—	4	2	no	—
152	C₄	—	4	2	no	—
153	C₄	—	4	2	no	—
154	C₄	—	188	2	no	—
155	C₄	#13094000	1	2	yes	—
156	C₄	#13094013	1	2	yes	—
157	C₂	—	2	1	no	—
158	C₂	—	2	1	no	—
159	C₂	—	376	2	no	—
160	C₂	#15689034	1	1	yes	characteristic
161	C₂	#15689291	1	1	yes	characteristic
162	C₂	#15689296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689127

C₂

yes

maximal

#15689024

C₂

yes

maximal, characteristic

#15688892

C₂

yes

maximal, characteristic, complemented

#15688882

C₂

yes

maximal, complemented

C₄₇×#544

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

#13093944

—

#13093944

—

C₄⋊Q₈×C₄₇

—

#125×C₄₇

—

C₂²⋊Q₈×C₄₇

—

#117×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

—

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

Q₈

yes

complemented

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

D₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

#124

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

C₄×C₄₇

#124

yes

—

Dic₄₇

—

#2027

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#544

D₄₇

yes

characteristic, complemented

—

—

—

—

C₄₇

#2027

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#102

—

#102

—

C₂⁴

—

100

C₂×Q₈

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂²⋊D₄₇

yes

—

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

C₂²⋊D₄₇

yes

—

115

C₄⋊C₄

C₂²⋊D₄₇

yes

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

188

—

118

C₄²

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

188

—

122

C₂³

—

188

—

123

C₂³

—

188

—

124

C₂³

—

188

—

125

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

126

Q₈

—

127

D₄

—

128

D₄

—

188

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

188

—

135

C₂×C₄

—

188

—

136

C₂×C₄

Q₈×D₄₇

yes

characteristic

137

C₂×C₄

#10625354

yes

—

138

C₂×C₄

#10625354

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

140

C₈

—

188

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

376

—

144

C₂²

—

376

—

145

C₂²

—

376

—

146

C₂²

—

188

—

147

C₂²

—

188

—

148

C₂²

#13093993

yes

characteristic

149

C₂²

#13094000

yes

characteristic

150

C₂²

#13094013

yes

characteristic

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

188

—

155

C₄

#13094000

yes

—

156

C₄

#13094013

yes

—

157

C₂

—

158

C₂

—

159

C₂

—

376

—

160

C₂

#15689034

yes

characteristic

161

C₂

#15689291

yes

characteristic

162

C₂

#15689296

yes

characteristic

	subgroup	G/H
center	C₂²	#13093993
commutator	C₂²×C₄×C₄₇	C₂³
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#544	C₂
socle	C₂²×C₄₇	#120
hypercenter	#544	D₄₇
hypocenter	C₄₇	#2027
LCS	G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#13093993

commutator

C₂²×C₄×C₄₇

C₂³

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#544

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₂²×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
3. #15688892	C₂	semidirect
4. #15688882	C₂	semidirect
5. C₄₇×#544	C₂	semidirect
25. C₂×D₁₈₈	Q₈	semidirect
82. #544	D₄₇	semidirect
87. C₄₇	#2027	semidirect

H ≅ G/N

type

3. #15688892

C₂

semidirect

4. #15688882

C₂

semidirect

5. C₄₇×#544

C₂

semidirect

25. C₂×D₁₈₈

Q₈

semidirect

82. #544

D₄₇

semidirect

87. C₄₇

#2027

semidirect

Subgroups — #20244541

Conjugacy classes of subgroups

Complements