Group #20243145 — subgroups

Subgroups — #20243145

Order 6016.

436 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 161 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	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₂×D₄₇)⋊C₈	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₄₇	C₂	1	2	yes	maximal, complemented
4	#15688878	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄₇	C₄	1	2	yes	—
7	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
8	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
9	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₄²×D₄₇	—	2	4	no	—
12	(C₂×D₄₇)⋊C₈	C₄	1	8	yes	complemented
13	C₈×Dic₄₇	—	2	4	no	—
14	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×D₄₇	—	2	2	no	—
22	C₂×C₄×D₄₇	—	2	1	no	—
23	C₂×C₄×D₄₇	—	2	1	no	—
24	C₂×C₄×D₄₇	—	2	4	no	—
25	C₂×C₄×D₄₇	—	2	4	no	—
26	C₂×C₄×D₄₇	C₈	1	8	yes	complemented
27	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
31	C₄²×C₄₇	—	2	2	no	—
32	C₄×Dic₄₇	—	2	4	no	—
33	C₄×Dic₄₇	D₄	1	4	yes	—
34	C₂×C₄₇⋊C₈	—	2	2	no	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
37	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
38	C₂²×D₄₇	—	2	1	no	—
39	C₂²×D₄₇	—	2	1	no	—
40	C₂²×D₄₇	—	2	2	no	—
41	C₂²×D₄₇	C₄²	1	2	yes	—
42	C₂²×D₄₇	C₂×C₈	1	4	yes	—
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	C₂×C₈	1	4	yes	—
46	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
49	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
50	C₂×Dic₄₇	—	2	1	no	—
51	C₂×Dic₄₇	—	2	1	no	—
52	C₂×Dic₄₇	C₄²	1	2	yes	—
53	C₂×Dic₄₇	C₂²⋊C₄	1	4	yes	—
54	C₂×Dic₄₇	C₂×C₈	1	4	yes	—
55	C₂×Dic₄₇	M₁₆	1	4	yes	—
56	C₄×D₄₇	—	2	4	no	—
57	C₄×D₄₇	—	2	4	no	—
58	C₄×D₄₇	—	2	16	no	—
59	C₈×C₄₇	—	2	4	no	—
60	C₄₇⋊C₈	—	2	4	no	—
61	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
62	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
63	C₂²×C₄₇	#117	1	1	yes	characteristic
64	C₂²×C₄₇	C₂²×C₈	1	2	yes	—
65	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
66	C₂×D₄₇	—	2	2	no	—
67	C₂×D₄₇	—	2	2	no	—
68	C₂×D₄₇	—	2	4	no	—
69	C₂×D₄₇	—	2	4	no	—
70	C₂×D₄₇	C₄×C₈	1	4	yes	complemented
71	C₄×C₄₇	—	2	4	no	—
72	C₄×C₄₇	C₄×D₄	1	2	yes	—
73	C₄×C₄₇	C₄×D₄	1	2	yes	—
74	Dic₄₇	—	2	2	no	—
75	Dic₄₇	C₂²⋊C₈	1	8	yes	complemented
76	C₄×C₂²⋊C₈	—	47	1	no	maximal, Sylow
77	C₂×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₄₇	C₂×C₄×C₈	1	1	yes	characteristic
79	C₂×C₄₇	C₄×M₁₆	1	1	yes	characteristic
80	C₂×C₄₇	C₂×C₂²⋊C₈	1	2	yes	—
81	C₂×C₄₇	#431	1	2	yes	—
82	D₄₇	—	2	4	no	—
83	C₂²×C₄²	—	47	1	no	—
84	C₂×C₂²⋊C₈	—	47	2	no	—
85	C₂×C₄×C₈	—	47	2	no	—
86	#336	—	47	1	no	—
87	#336	D₄₇	1	1	yes	characteristic, complemented
88	C₄₇	C₄×C₂²⋊C₈	1	1	yes	characteristic, complemented, Sylow
89	C₂³×C₄	—	47	2	no	—
90	C₂³×C₄	—	47	1	no	—
91	C₂²×C₈	—	47	2	no	—
92	C₂²×C₈	C₂×D₄₇	1	2	yes	—
93	C₂×C₄²	—	47	1	no	—
94	C₂×C₄²	—	47	2	no	—
95	C₂×C₄²	—	94	4	no	—
96	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
97	C₂²⋊C₈	—	47	8	no	—
98	C₄×C₈	—	94	4	no	—
99	C₂⁴	—	47	1	no	—
100	C₂²×C₄	—	47	2	no	—
101	C₂²×C₄	—	47	1	no	—
102	C₂²×C₄	—	47	8	no	—
103	C₂²×C₄	—	47	4	no	—
104	C₂²×C₄	—	94	4	no	—
105	C₂²×C₄	—	94	4	no	—
106	C₂²×C₄	—	94	2	no	—
107	C₂²×C₄	—	94	1	no	—
108	C₂²×C₄	—	94	1	no	—
109	C₂²×C₄	C₄×D₄₇	1	2	yes	—
110	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	—	2	2	no	—
113	C₂×C₈	—	47	4	no	—
114	C₂×C₈	—	94	2	no	—
115	C₂×C₈	—	94	2	no	—
116	C₂×C₈	C₄×D₄₇	1	4	yes	complemented
117	C₄²	—	2	2	no	—
118	C₄²	—	47	4	no	—
119	C₄²	—	94	4	no	—
120	C₂³	—	47	2	no	—
121	C₂³	—	47	4	no	—
122	C₂³	—	94	1	no	—
123	C₂³	—	94	1	no	—
124	C₂³	—	94	2	no	—
125	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	47	4	no	—
129	C₂×C₄	—	47	4	no	—
130	C₂×C₄	—	47	4	no	—
131	C₂×C₄	—	47	2	no	—
132	C₂×C₄	—	94	16	no	—
133	C₂×C₄	—	94	1	no	—
134	C₂×C₄	—	94	1	no	—
135	C₂×C₄	—	94	4	no	—
136	C₂×C₄	—	94	4	no	—
137	C₂×C₄	C₈×D₄₇	1	4	yes	—
138	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
139	C₂×C₄	D₄×D₄₇	1	2	yes	—
140	C₂×C₄	#10625352	1	1	yes	characteristic
141	C₂×C₄	#10625354	1	1	yes	characteristic
142	C₈	—	2	4	no	—
143	C₈	—	94	4	no	—
144	C₂²	—	47	4	no	—
145	C₂²	—	94	2	no	—
146	C₂²	—	94	2	no	—
147	C₂²	—	94	4	no	—
148	C₂²	—	94	4	no	—
149	C₂²	C₄²×D₄₇	1	1	yes	characteristic
150	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
151	C₂²	#13093982	1	1	yes	characteristic
152	C₂²	C₂×C₈×D₄₇	1	2	yes	—
153	C₂²	M₁₆×D₄₇	1	2	yes	—
154	C₄	—	2	4	no	—
155	C₄	—	47	8	no	—
156	C₄	—	94	2	no	—
157	C₄	(C₂²⋊D₄₇)⋊C₄	1	2	yes	—
158	C₄	#13093986	1	2	yes	—
159	C₂	—	94	4	no	—
160	C₂	#15688970	1	1	yes	characteristic
161	C₂	C₄×C₈×D₄₇	1	1	yes	characteristic
162	C₂	#15689075	1	1	yes	characteristic
163	C₂	C₂²⋊C₈×D₄₇	1	2	yes	—
164	C₂	#15689175	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic

C₂×(C₂×D₄₇)⋊C₈

C₂

yes

maximal

C₂×C₈×Dic₄₇

C₂

yes

maximal, complemented

#15688878

C₂

yes

maximal, characteristic, complemented

C₄₇×#336

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₄

yes

—

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₄₇

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄²×D₄₇

—

(C₂×D₄₇)⋊C₈

C₄

yes

complemented

C₈×Dic₄₇

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₈

yes

complemented

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

complemented

C₄²×C₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

complemented

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₄²

yes

—

C₂²×D₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₄²

yes

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₂×Dic₄₇

C₂×C₈

yes

—

C₂×Dic₄₇

M₁₆

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#117

yes

characteristic

C₂²×C₄₇

C₂²×C₈

yes

—

C₂²×C₄₇

C₂×M₁₆

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₄×C₈

yes

complemented

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₄×D₄

yes

—

Dic₄₇

—

Dic₄₇

C₂²⋊C₈

yes

complemented

C₄×C₂²⋊C₈

—

maximal, Sylow

C₂×C₄₇

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₄₇

C₂×C₄×C₈

yes

characteristic

C₂×C₄₇

C₄×M₁₆

yes

characteristic

C₂×C₄₇

C₂×C₂²⋊C₈

yes

—

C₂×C₄₇

#431

yes

—

D₄₇

—

C₂²×C₄²

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

#336

—

#336

D₄₇

yes

characteristic, complemented

C₄₇

C₄×C₂²⋊C₈

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

C₄×D₄₇

yes

—

110

C₂²×C₄

C₂²×D₄₇

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

C₄×D₄₇

yes

complemented

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

C₂×C₄×D₄₇

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₈×D₄₇

yes

—

138

C₂×C₄

C₂×C₄×D₄₇

yes

—

139

C₂×C₄

D₄×D₄₇

yes

—

140

C₂×C₄

#10625352

yes

characteristic

141

C₂×C₄

#10625354

yes

characteristic

142

C₈

—

143

C₈

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₄²×D₄₇

yes

characteristic

150

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

151

C₂²

#13093982

yes

characteristic

152

C₂²

C₂×C₈×D₄₇

yes

—

153

C₂²

M₁₆×D₄₇

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

(C₂²⋊D₄₇)⋊C₄

yes

—

158

C₄

#13093986

yes

—

159

C₂

—

160

C₂

#15688970

yes

characteristic

161

C₂

C₄×C₈×D₄₇

yes

characteristic

162

C₂

#15689075

yes

characteristic

163

C₂

C₂²⋊C₈×D₄₇

yes

—

164

C₂

#15689175

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator	C₂×C₄₇	C₂×C₄×C₈
Fitting	C₄₇×#336	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	#336	D₄₇
hypocenter	C₄₇	C₄×C₂²⋊C₈
LCS	G ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator

C₂×C₄₇

C₂×C₄×C₈

Fitting

C₄₇×#336

C₂

socle

C₂³×C₄₇

C₂²×C₄

hypercenter

#336

D₄₇

hypocenter

C₄₇

C₄×C₂²⋊C₈

LCS

G ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
3. C₂×C₈×Dic₄₇	C₂	semidirect
4. #15688878	C₂	semidirect
5. C₄₇×#336	C₂	semidirect
12. (C₂×D₄₇)⋊C₈	C₄	semidirect
26. C₂×C₄×D₄₇	C₈	semidirect
30. C₂×C₈×C₄₇	C₂×C₄	semidirect
36. C₂×C₄₇⋊C₈	C₂×C₄	semidirect
70. C₂×D₄₇	C₄×C₈	semidirect
75. Dic₄₇	C₂²⋊C₈	semidirect
87. #336	D₄₇	semidirect
88. C₄₇	C₄×C₂²⋊C₈	semidirect
116. C₂×C₈	C₄×D₄₇	semidirect

H ≅ G/N

type

3. C₂×C₈×Dic₄₇

C₂

semidirect

4. #15688878

C₂

semidirect

5. C₄₇×#336

C₂

semidirect

12. (C₂×D₄₇)⋊C₈

C₄

semidirect

26. C₂×C₄×D₄₇

C₈

semidirect

30. C₂×C₈×C₄₇

C₂×C₄

semidirect

36. C₂×C₄₇⋊C₈

C₂×C₄

semidirect

70. C₂×D₄₇

C₄×C₈

semidirect

75. Dic₄₇

C₂²⋊C₈

semidirect

87. #336

D₄₇

semidirect

88. C₄₇

C₄×C₂²⋊C₈

semidirect

116. C₂×C₈

C₄×D₄₇

semidirect

Subgroups — #20243145

Conjugacy classes of subgroups

Complements