Group #20242216 — subgroups

Subgroups — #20242216

Order 6016.

182 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 63 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×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	#15689200	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689188	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688850	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688850	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₃₇₆	C₄	1	1	yes	characteristic
9	C₂×#10625324	—	2	1	no	—
10	C₂×C₈×D₄₇	—	2	1	no	—
11	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
13	#13093958	C₄	1	1	yes	characteristic
14	#13093946	C₄	1	1	yes	characteristic
15	#13093946	C₄	1	1	yes	characteristic
16	(C₂×D₄₇)⋊C₈	—	2	1	no	—
17	(C₂×D₄₇)⋊C₈	—	2	1	no	—
18	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
23	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
24	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×D₄₇	—	2	1	no	—
26	C₂×C₄×D₄₇	—	2	1	no	—
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	—	2	1	no	—
29	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	(C₂×D₄₇)⋊C₄	—	2	1	no	—
33	(C₂×D₄₇)⋊C₄	—	2	1	no	—
34	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄⋊Dic₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₄₇⋊C₈	—	2	1	no	—
37	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
38	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
39	D₃₇₆	—	2	2	no	—
40	#10625324	—	4	2	no	—
41	C₈×D₄₇	—	4	2	no	—
42	C₂²×D₄₇	—	2	1	no	—
43	C₂²×D₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
47	C₂×Dic₄₇	—	2	1	no	—
48	C₂×Dic₄₇	—	2	1	no	—
49	D₁₈₈	—	2	1	no	—
50	D₁₈₈	—	2	1	no	—
51	D₁₈₈	—	2	1	no	—
52	D₁₈₈	M₁₆	1	2	yes	complemented
53	C₄×D₄₇	—	4	1	no	—
54	C₄×D₄₇	—	4	1	no	—
55	C₄×D₄₇	—	4	1	no	—
56	C₄×D₄₇	—	4	1	no	—
57	C₈×C₄₇	—	2	1	no	—
58	C₈×C₄₇	—	2	1	no	—
59	C₈×C₄₇	—	4	1	no	—
60	C₈×C₄₇	—	4	1	no	—
61	C₄₇⋊C₈	—	2	1	no	—
62	C₄₇⋊C₈	—	4	1	no	—
63	C₄₇⋊C₈	D₈	1	2	yes	complemented
64	C₂²×C₄₇	C₄×D₄	1	1	yes	characteristic
65	C₂×D₄₇	—	2	2	no	—
66	C₂×D₄₇	—	4	1	no	—
67	C₂×D₄₇	—	4	1	no	—
68	C₂×D₄₇	—	4	1	no	—
69	C₂×D₄₇	—	4	1	no	—
70	C₂×D₄₇	—	4	1	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
73	C₄×C₄₇	#131	1	1	yes	characteristic
74	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
75	C₄×C₄₇	#135	1	1	yes	characteristic
76	Dic₄₇	—	4	1	no	—
77	Dic₄₇	—	4	1	no	—
78	#1581	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	#435	1	1	yes	characteristic
80	C₂×C₄₇	C₄×D₈	1	1	yes	characteristic
81	C₂×C₄₇	#444	1	1	yes	characteristic
82	D₄₇	—	4	2	no	—
83	D₄₇	—	8	1	no	—
84	C₄×D₈	—	47	1	no	—
85	#436	—	47	1	no	—
86	C₈×D₄	—	47	1	no	—
87	#335	D₄₇	1	1	yes	characteristic, complemented
88	D₄⋊C₈	—	47	1	no	—
89	D₄⋊C₈	—	47	1	no	—
90	#322	—	47	1	no	—
91	C₄₇	#1581	1	1	yes	characteristic, complemented, Sylow
92	C₂×D₈	—	47	1	no	—
93	C₂×M₁₆	—	94	1	no	—
94	C₂²×C₈	—	94	1	no	—
95	C₄×D₄	—	47	1	no	—
96	C₄×D₄	—	47	1	no	—
97	#107	—	47	1	no	—
98	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
99	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
100	#102	—	47	1	no	—
101	#102	—	47	1	no	—
102	C₂²⋊C₈	—	94	1	no	—
103	C₂²⋊C₈	—	94	1	no	—
104	C₄×C₈	—	47	1	no	—
105	C₄×C₈	—	47	1	no	—
106	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
107	C₂×D₄	—	47	1	no	—
108	C₂×D₄	—	47	1	no	—
109	C₂²×C₄	—	94	1	no	—
110	C₂²×C₄	—	94	1	no	—
111	D₈	—	94	2	no	—
112	M₁₆	—	188	2	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	2	1	no	—
115	C₂×C₈	—	47	1	no	—
116	C₂×C₈	—	47	1	no	—
117	C₂×C₈	—	94	1	no	—
118	C₂×C₈	—	188	2	no	—
119	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
120	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
121	C₄⋊C₄	—	47	1	no	—
122	C₄⋊C₄	—	47	1	no	—
123	C₂²⋊C₄	—	94	1	no	—
124	C₂²⋊C₄	—	94	1	no	—
125	C₄²	C₂²×D₄₇	1	1	yes	characteristic
126	C₂³	—	94	1	no	—
127	C₂³	—	94	1	no	—
128	D₄	—	47	2	no	—
129	D₄	—	94	1	no	—
130	D₄	—	94	1	no	—
131	D₄	—	94	1	no	—
132	C₂×C₄	—	94	1	no	—
133	C₂×C₄	—	94	1	no	—
134	C₂×C₄	—	188	1	no	—
135	C₂×C₄	—	188	1	no	—
136	C₂×C₄	—	188	1	no	—
137	C₂×C₄	—	188	1	no	—
138	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
139	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
140	C₂×C₄	#10625354	1	1	yes	characteristic
141	C₈	—	2	1	no	—
142	C₈	—	2	1	no	—
143	C₈	—	4	1	no	—
144	C₈	—	4	1	no	—
145	C₈	—	47	2	no	—
146	C₈	—	94	1	no	—
147	C₈	—	188	1	no	—
148	C₂²	—	94	2	no	—
149	C₂²	—	188	1	no	—
150	C₂²	—	188	1	no	—
151	C₂²	—	188	1	no	—
152	C₂²	—	188	1	no	—
153	C₂²	—	188	1	no	—
154	C₂²	#13093986	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	188	1	no	—
157	C₄	—	188	1	no	—
158	C₄	M₁₆×D₄₇	1	1	yes	characteristic
159	C₄	#13094030	1	1	yes	characteristic
160	C₄	D₈×D₄₇	1	1	yes	characteristic
161	C₄	#13094047	1	1	yes	characteristic
162	C₂	—	376	1	no	—
163	C₂	—	188	2	no	—
164	C₂	#15689193	1	1	yes	characteristic
165	C₂	#15689205	1	1	yes	characteristic
166	C₂	#15689244	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×D₃₇₆

C₂

yes

maximal, characteristic

#15689200

C₂

yes

maximal, characteristic, complemented

#15689188

C₂

yes

maximal, characteristic, complemented

#15688850

C₂

yes

maximal, characteristic, complemented

#15688850

C₂

yes

maximal, characteristic, complemented

#15688845

C₂

yes

maximal, characteristic, complemented

C₄₇×#335

C₂

yes

maximal, characteristic, complemented

C₂×D₃₇₆

C₄

yes

characteristic

C₂×#10625324

—

C₂×C₈×D₄₇

—

C₄×D₁₈₈

C₂²

yes

characteristic

C₄×D₁₈₈

C₂²

yes

characteristic

#13093958

C₄

yes

characteristic

#13093946

C₄

yes

characteristic

#13093946

C₄

yes

characteristic

(C₂×D₄₇)⋊C₈

—

(C₂×D₄₇)⋊C₈

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₁₈₈

C₂×C₄

yes

characteristic

C₂×D₁₈₈

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂×C₄

yes

characteristic

C₄⋊Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

C₂×C₄₇⋊C₈

D₄

yes

characteristic

D₃₇₆

—

#10625324

—

C₈×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

M₁₆

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₄₇⋊C₈

D₈

yes

complemented

C₂²×C₄₇

C₄×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

#131

yes

characteristic

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

#135

yes

characteristic

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₄×D₈

yes

characteristic

C₂×C₄₇

#444

yes

characteristic

D₄₇

—

D₄₇

—

C₄×D₈

—

#436

—

C₈×D₄

—

#335

D₄₇

yes

characteristic, complemented

D₄⋊C₈

—

D₄⋊C₈

—

#322

—

C₄₇

#1581

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₄×D₄

—

C₄×D₄

—

#107

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

C₄⋊C₈

C₂×D₄₇

yes

characteristic

100

#102

—

101

#102

—

102

C₂²⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

C₂×D₄₇

yes

characteristic

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

D₈

—

112

M₁₆

—

188

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

188

—

119

C₂×C₈

C₄×D₄₇

yes

characteristic

120

C₂×C₈

C₄×D₄₇

yes

characteristic

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

C₂²×D₄₇

yes

characteristic

126

C₂³

—

127

C₂³

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

188

—

135

C₂×C₄

—

188

—

136

C₂×C₄

—

188

—

137

C₂×C₄

—

188

—

138

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

139

C₂×C₄

D₄×D₄₇

yes

characteristic

140

C₂×C₄

#10625354

yes

characteristic

141

C₈

—

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

—

188

—

148

C₂²

—

149

C₂²

—

188

—

150

C₂²

—

188

—

151

C₂²

—

188

—

152

C₂²

—

188

—

153

C₂²

—

188

—

154

C₂²

#13093986

yes

characteristic

155

C₄

—

156

C₄

—

188

—

157

C₄

—

188

—

158

C₄

M₁₆×D₄₇

yes

characteristic

159

C₄

#13094030

yes

characteristic

160

C₄

D₈×D₄₇

yes

characteristic

161

C₄

#13094047

yes

characteristic

162

C₂

—

376

—

163

C₂

—

188

—

164

C₂

#15689193

yes

characteristic

165

C₂

#15689205

yes

characteristic

166

C₂

#15689244

yes

characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#335	C₂
socle	C₂²×C₄₇	C₄×D₄
hypercenter	#335	D₄₇
hypocenter	C₄₇	#1581
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

subgroup

G/H

center

C₂×C₄

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#335

C₂

socle

C₂²×C₄₇

C₄×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #335

N	H ≅ G/N	type
2. #15689200	C₂	semidirect
3. #15689188	C₂	semidirect
4. #15688850	C₂	semidirect
5. #15688850	C₂	semidirect
6. #15688845	C₂	semidirect
7. C₄₇×#335	C₂	semidirect
52. D₁₈₈	M₁₆	semidirect
63. C₄₇⋊C₈	D₈	semidirect
87. #335	D₄₇	semidirect
91. C₄₇	#1581	semidirect

H ≅ G/N

type

2. #15689200

C₂

semidirect

3. #15689188

C₂

semidirect

4. #15688850

C₂

semidirect

5. #15688850

C₂

semidirect

6. #15688845

C₂

semidirect

7. C₄₇×#335

C₂

semidirect

52. D₁₈₈

M₁₆

semidirect

63. C₄₇⋊C₈

D₈

semidirect

87. #335

D₄₇

semidirect

91. C₄₇

#1581

semidirect

Subgroups — #20242216

Conjugacy classes of subgroups

Complements