Group #20243370 — subgroups

Subgroups — #20243370

Order 6016.

312 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 109 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×#13093983	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₄₇⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₂²⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#15688877	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	#13093983	—	2	4	no	—
8	C₂×C₄⋊Dic₄₇	C₄	1	2	yes	—
9	C₂×C₂²⋊Dic₄₇	C₄	1	2	yes	—
10	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
12	Dic₄₇⋊C₈	—	2	4	no	—
13	C₂²⋊(C₄₇⋊C₈)	—	2	2	no	—
14	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂²⋊C₈×C₄₇	—	2	2	no	—
17	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	—	2	2	no	—
19	C₂²×C₄×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	2	yes	—
22	C₂²×C₄×C₄₇	Q₈	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
25	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	—	2	2	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₂²⋊Dic₄₇	—	2	4	no	—
30	C₄⋊Dic₄₇	—	2	4	no	—
31	C₄×Dic₄₇	—	2	2	no	—
32	C₄×Dic₄₇	D₄	1	4	yes	—
33	C₂×C₄₇⋊C₈	—	2	2	no	—
34	C₂×C₄₇⋊C₈	—	2	2	no	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂³×C₄₇	—	2	1	no	—
37	C₂³×C₄₇	—	2	1	no	—
38	C₂³×C₄₇	—	2	2	no	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₄×C₄₇	#41	1	2	yes	—
51	C₂×Dic₄₇	—	2	2	no	—
52	C₂×Dic₄₇	—	2	2	no	—
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	2	2	no	—
55	C₂×Dic₄₇	—	2	2	no	—
56	C₂×Dic₄₇	C₂²⋊C₄	1	4	yes	—
57	C₈×C₄₇	—	4	2	no	—
58	C₄₇⋊C₈	—	4	2	no	—
59	C₂²×C₄₇	—	2	2	no	—
60	C₂²×C₄₇	—	2	2	no	—
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	—	4	2	no	—
63	C₂²×C₄₇	—	4	2	no	—
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄₇	#117	1	1	yes	characteristic
67	C₂²×C₄₇	#131	1	2	yes	—
68	C₂²×C₄₇	#131	1	2	yes	—
69	C₄×C₄₇	—	2	2	no	—
70	C₄×C₄₇	#121	1	2	yes	—
71	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
72	Dic₄₇	—	2	4	no	—
73	Dic₄₇	—	4	2	no	—
74	#1798	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	4	2	no	—
76	C₂×C₄₇	#380	1	1	yes	characteristic
77	C₂×C₄₇	#408	1	2	yes	—
78	C₂×C₄₇	#424	1	1	yes	characteristic
79	C₂×C₄₇	#424	1	1	yes	characteristic
80	C₂×C₄₇	#433	1	2	yes	—
81	C₂×#117	—	47	1	no	—
82	C₂×C₄⋊C₈	—	47	2	no	—
83	C₂×C₂²⋊C₈	—	47	1	no	—
84	C₂×C₂²⋊C₈	D₄₇	1	1	yes	characteristic
85	#336	—	47	2	no	—
86	C₄₇	#1798	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
88	C₂²×C₈	—	47	2	no	—
89	C₂²×C₈	C₂×D₄₇	1	2	yes	—
90	#117	—	94	4	no	—
91	C₂×C₄⋊C₄	—	47	2	no	—
92	C₂×C₂²⋊C₄	—	47	2	no	—
93	C₂×C₄²	—	47	1	no	—
94	C₂×C₄²	—	47	1	no	—
95	C₄⋊C₈	—	94	4	no	—
96	C₂²⋊C₈	—	2	2	no	—
97	C₂²⋊C₈	—	94	2	no	—
98	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	47	2	no	—
102	C₂²×C₄	—	47	2	no	—
103	C₂²×C₄	Dic₉₄	1	2	yes	—
104	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
105	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
106	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	2	2	no	—
109	C₂×C₈	—	2	2	no	—
110	C₂×C₈	—	94	2	no	—
111	C₂×C₈	—	94	2	no	—
112	C₂×C₈	—	94	2	no	—
113	C₄⋊C₄	—	94	4	no	—
114	C₂²⋊C₄	—	94	4	no	—
115	C₄²	—	47	4	no	—
116	C₄²	—	94	2	no	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	2	no	—
120	C₂³	—	2	2	no	—
121	C₂³	Dic₄₇⋊C₄	1	2	yes	—
122	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	47	4	no	—
128	C₂×C₄	—	94	2	no	—
129	C₂×C₄	—	94	2	no	—
130	C₂×C₄	—	94	2	no	—
131	C₂×C₄	—	94	2	no	—
132	C₂×C₄	—	94	2	no	—
133	C₂×C₄	Dic₄₇⋊C₄	1	2	yes	—
134	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
135	C₂×C₄	D₄×D₄₇	1	2	yes	—
136	C₂×C₄	#10625352	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₈	—	4	2	no	—
139	C₈	—	188	2	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	2	2	no	—
143	C₂²	—	4	2	no	—
144	C₂²	—	4	2	no	—
145	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
146	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
147	C₂²	#13093980	1	1	yes	characteristic
148	C₂²	#13094030	1	2	yes	—
149	C₂²	#13094032	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	94	4	no	—
152	C₄	—	188	2	no	—
153	C₄	#13093996	1	2	yes	—
154	C₄	#13094001	1	2	yes	—
155	C₂	—	4	2	no	—
156	C₂	#15688981	1	1	yes	characteristic
157	C₂	#15689088	1	2	yes	—
158	C₂	#15689149	1	1	yes	characteristic
159	C₂	#15689151	1	1	yes	characteristic
160	C₂	#15689181	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093983

C₂

yes

maximal, characteristic

C₂×Dic₄₇⋊C₈

C₂

yes

maximal, complemented

C₂×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

#15688877

C₂

yes

maximal, complemented

C₂×C₂²⋊C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

#13093983

—

C₂×C₄⋊Dic₄₇

C₄

yes

—

C₂×C₂²⋊Dic₄₇

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

—

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#131

yes

—

C₄×C₄₇

—

C₄×C₄₇

#121

yes

—

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

#1798

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#380

yes

characteristic

C₂×C₄₇

#408

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×#117

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₄₇

yes

characteristic

#336

—

C₄₇

#1798

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂⁴

C₄×D₄₇

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

Dic₉₄

yes

—

104

C₂²×C₄

C₄×D₄₇

yes

characteristic

105

C₂²×C₄

C₂²⋊D₄₇

yes

—

106

C₂²×C₄

C₂²×D₄₇

yes

characteristic

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

Dic₄₇⋊C₄

yes

—

122

C₂³

C₂×C₄×D₄₇

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

Dic₄₇⋊C₄

yes

—

134

C₂×C₄

C₂×Dic₉₄

yes

characteristic

135

C₂×C₄

D₄×D₄₇

yes

—

136

C₂×C₄

#10625352

yes

—

137

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

C₈

—

139

C₈

—

188

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

146

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

147

C₂²

#13093980

yes

characteristic

148

C₂²

#13094030

yes

—

149

C₂²

#13094032

yes

—

150

C₄

—

151

C₄

—

152

C₄

—

188

—

153

C₄

#13093996

yes

—

154

C₄

#13094001

yes

—

155

C₂

—

156

C₂

#15688981

yes

characteristic

157

C₂

#15689088

yes

—

158

C₂

#15689149

yes

characteristic

159

C₂

#15689151

yes

characteristic

160

C₂

#15689181

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator, socle	C₂³×C₄₇	C₂²×C₄
Fitting	C₂×C₂²⋊C₈×C₄₇	C₂
hypercenter	C₂×C₂²⋊C₈	D₄₇
hypocenter	C₄₇	#1798
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator, socle

C₂³×C₄₇

C₂²×C₄

Fitting

C₂×C₂²⋊C₈×C₄₇

C₂

hypercenter

C₂×C₂²⋊C₈

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
2. C₂×Dic₄₇⋊C₈	C₂	semidirect
4. #15688877	C₂	semidirect
86. C₄₇	#1798	semidirect

H ≅ G/N

type

2. C₂×Dic₄₇⋊C₈

C₂

semidirect

4. #15688877

C₂

semidirect

86. C₄₇

#1798

semidirect

Subgroups — #20243370

Conjugacy classes of subgroups

Complements