Group #20243154 — subgroups

Subgroups — #20243154

Order 6016.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₂²⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
2	C₂×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688878	C₂	1	2	yes	maximal, complemented
5	C₂×#117×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
8	C₂²⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
9	#13093935	—	2	2	no	—
10	C₄×C₄₇⋊C₈	—	2	2	no	—
11	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
12	#117×C₄₇	—	2	2	no	—
13	#117×C₄₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₄₇	C₄	1	2	yes	—
15	C₂×C₂²⋊C₄×C₄₇	C₄	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	—	2	1	no	—
20	C₂²×C₄×C₄₇	—	2	1	no	—
21	C₂²×C₄×C₄₇	—	2	2	no	—
22	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
25	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₄₇	—	2	4	no	—
28	C₂²⋊C₄×C₄₇	—	2	4	no	—
29	C₄²×C₄₇	—	2	1	no	—
30	C₄²×C₄₇	—	2	1	no	—
31	C₄²×C₄₇	D₄	1	2	yes	—
32	C₄²×C₄₇	D₄	1	2	yes	—
33	C₂×C₄₇⋊C₈	—	2	2	no	—
34	C₂×C₄₇⋊C₈	—	2	2	no	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
38	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
39	C₂³×C₄₇	—	2	1	no	—
40	C₂³×C₄₇	—	2	1	no	—
41	C₂³×C₄₇	—	2	4	no	—
42	C₂³×C₄₇	C₄²	1	2	yes	—
43	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	4	no	—
51	C₂×C₄×C₄₇	C₄²	1	2	yes	—
52	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
53	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	#41	1	2	yes	—
57	C₄₇⋊C₈	—	2	4	no	—
58	C₄₇⋊C₈	—	2	4	no	—
59	C₂²×C₄₇	—	2	2	no	—
60	C₂²×C₄₇	—	2	4	no	—
61	C₂²×C₄₇	—	4	2	no	—
62	C₂²×C₄₇	—	4	2	no	—
63	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	#117	1	1	yes	characteristic
66	C₂²×C₄₇	#131	1	4	yes	—
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	—	4	2	no	—
70	C₄×C₄₇	C₄×D₄	1	2	yes	—
71	C₄×C₄₇	C₄×D₄	1	2	yes	—
72	#1750	—	47	1	no	maximal, Sylow
73	C₂×C₄₇	—	4	2	no	—
74	C₂×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₄₇	#405	1	2	yes	—
76	C₂×C₄₇	#408	1	2	yes	—
77	C₂×C₄₇	#433	1	2	yes	—
78	C₂×#117	D₄₇	1	1	yes	characteristic
79	C₂×C₂²⋊C₈	—	47	2	no	—
80	C₂×#97	—	47	1	no	—
81	C₂×C₄×C₈	—	47	1	no	—
82	#336	—	47	2	no	—
83	C₄₇	#1750	1	1	yes	characteristic, complemented, Sylow
84	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
85	C₂²×C₈	—	47	2	no	—
86	C₂²×C₈	—	47	2	no	—
87	#117	—	2	2	no	—
88	#117	—	2	2	no	—
89	C₂×C₄⋊C₄	Dic₄₇	1	2	yes	—
90	C₂×C₂²⋊C₄	Dic₄₇	1	2	yes	—
91	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
93	C₂²⋊C₈	—	47	8	no	—
94	#97	—	94	2	no	—
95	C₄×C₈	—	94	2	no	—
96	C₂⁴	C₂×Dic₄₇	1	1	yes	characteristic
97	C₂²×C₄	—	2	2	no	—
98	C₂²×C₄	—	2	1	no	—
99	C₂²×C₄	—	2	1	no	—
100	C₂²×C₄	C₄×D₄₇	1	4	yes	—
101	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
102	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
103	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
105	C₂×C₈	—	47	4	no	—
106	C₂×C₈	—	47	4	no	—
107	C₂×C₈	—	94	2	no	—
108	C₂×C₈	—	94	2	no	—
109	C₂×C₈	—	94	2	no	—
110	C₂×C₈	—	94	2	no	—
111	C₄⋊C₄	—	2	4	no	—
112	C₂²⋊C₄	—	2	4	no	—
113	C₄²	—	2	1	no	—
114	C₄²	—	2	1	no	—
115	C₄²	C₂²⋊D₄₇	1	2	yes	—
116	C₄²	C₂²⋊D₄₇	1	2	yes	—
117	C₂³	—	2	1	no	—
118	C₂³	—	2	1	no	—
119	C₂³	—	2	4	no	—
120	C₂³	C₄×Dic₄₇	1	2	yes	—
121	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	2	4	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	2	4	no	—
129	C₂×C₄	C₄×Dic₄₇	1	2	yes	—
130	C₂×C₄	C₂²⋊Dic₄₇	1	4	yes	—
131	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
132	C₂×C₄	#10625350	1	2	yes	—
133	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
135	C₈	—	94	4	no	—
136	C₈	—	94	4	no	—
137	C₂²	—	2	2	no	—
138	C₂²	—	2	4	no	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	2	no	—
141	C₂²	C₂×C₄×Dic₄₇	1	1	yes	characteristic
142	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
143	C₂²	#13093983	1	1	yes	characteristic
144	C₂²	#13094031	1	4	yes	—
145	C₄	—	2	2	no	—
146	C₄	—	2	4	no	—
147	C₄	—	4	2	no	—
148	C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
149	C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
150	C₂	—	4	2	no	—
151	C₂	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
152	C₂	#15689079	1	2	yes	—
153	C₂	#15689090	1	2	yes	—
154	C₂	#15689184	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂²⋊(C₄₇⋊C₈)

C₂

yes

maximal

C₂×#13093935

C₂

yes

maximal, characteristic, complemented

C₂×C₄×C₄₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#15688878

C₂

yes

maximal, complemented

C₂×#117×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂²⋊(C₄₇⋊C₈)

C₄

yes

complemented

#13093935

—

C₄×C₄₇⋊C₈

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

#117×C₄₇

—

#117×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

—

C₂×C₂²⋊C₄×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄²

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

C₄×D₄

yes

—

#1750

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₄₇

#405

yes

—

C₂×C₄₇

#408

yes

—

C₂×C₄₇

#433

yes

—

C₂×#117

D₄₇

yes

characteristic

C₂×C₂²⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#336

—

C₄₇

#1750

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

Dic₄₇

yes

—

C₂×C₂²⋊C₄

Dic₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂⁴

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₄×D₄₇

yes

—

101

C₂²×C₄

C₂×Dic₄₇

yes

—

102

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

103

C₂²×C₄

C₂×Dic₄₇

yes

—

104

C₂²×C₄

C₂²×D₄₇

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂²⋊D₄₇

yes

—

116

C₄²

C₂²⋊D₄₇

yes

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

C₄×Dic₄₇

yes

—

121

C₂³

C₂²×Dic₄₇

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₄×Dic₄₇

yes

—

130

C₂×C₄

C₂²⋊Dic₄₇

yes

—

131

C₂×C₄

C₂×C₄×D₄₇

yes

—

132

C₂×C₄

#10625350

yes

—

133

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

135

C₈

—

136

C₈

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

C₂×C₄×Dic₄₇

yes

characteristic

142

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

143

C₂²

#13093983

yes

characteristic

144

C₂²

#13094031

yes

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₄×C₂²⋊D₄₇

yes

—

149

C₄

C₄×C₂²⋊D₄₇

yes

—

150

C₂

—

151

C₂

C₄×C₂²⋊Dic₄₇

yes

characteristic

152

C₂

#15689079

yes

—

153

C₂

#15689090

yes

—

154

C₂

#15689184

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄²
Fitting	C₂×#117×C₄₇	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	C₂×#117	D₄₇
hypocenter	C₄₇	#1750
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂×C₄²

Fitting

C₂×#117×C₄₇

C₂

socle

C₂³×C₄₇

C₂²×C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ C₂×#117

N	H ≅ G/N	type
2. C₂×#13093935	C₂	semidirect
3. C₂×C₄×C₄₇⋊C₈	C₂	semidirect
4. #15688878	C₂	semidirect
8. C₂²⋊(C₄₇⋊C₈)	C₄	semidirect
83. C₄₇	#1750	semidirect

H ≅ G/N

type

2. C₂×#13093935

C₂

semidirect

3. C₂×C₄×C₄₇⋊C₈

C₂

semidirect

4. #15688878

C₂

semidirect

8. C₂²⋊(C₄₇⋊C₈)

C₄

semidirect

83. C₄₇

#1750

semidirect

Subgroups — #20243154

Conjugacy classes of subgroups

Complements