Group #20246129 — subgroups

Subgroups — #20246129

Order 6016.

444 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 205 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×#13094016	C₂	1	1	yes	maximal, characteristic
2	#15689021	C₂	1	2	yes	maximal
3	#15689020	C₂	1	4	yes	maximal
4	#15689010	C₂	1	4	yes	maximal
5	#15688993	C₂	1	1	yes	maximal, characteristic
6	C₄×C₄⋊Dic₄₇	C₂	1	2	yes	maximal
7	C₄₇×#376	C₂	1	1	yes	maximal, characteristic
8	#13094016	C₄	1	8	yes	complemented
9	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	4	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
15	C₄₇⋊#95	C₂²	1	4	yes	—
16	C₄₇⋊#95	C₂²	1	4	yes	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	#95×C₄₇	C₂²	1	2	yes	—
21	#95×C₄₇	C₂²	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	C₂³	1	4	yes	—
28	C₂²×Dic₄₇	C₂³	1	4	yes	—
29	C₄²×C₄₇	—	2	2	no	—
30	C₄²×C₄₇	C₂×C₄	1	4	yes	—
31	C₄²×C₄₇	Q₈	1	4	yes	—
32	C₄⋊Dic₄₇	—	2	4	no	—
33	C₄⋊Dic₄₇	—	4	4	no	—
34	C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
35	Dic₄₇⋊C₄	—	4	4	no	—
36	Dic₄₇⋊C₄	C₂×C₄	1	16	yes	—
37	C₄×Dic₄₇	—	4	4	no	—
38	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
50	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
51	C₂×C₄×C₄₇	#41	1	2	yes	—
52	C₂×Dic₄₇	—	2	4	no	—
53	C₂×Dic₄₇	—	2	4	no	—
54	C₂×Dic₄₇	—	2	4	no	—
55	C₂×Dic₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	2	4	no	—
57	C₂×Dic₄₇	C₂²×C₄	1	8	yes	—
58	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
59	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
60	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
61	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
62	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
63	C₄×C₄₇	—	2	4	no	—
64	C₄×C₄₇	—	2	4	no	—
65	C₄×C₄₇	—	4	1	no	—
66	C₄×C₄₇	—	4	1	no	—
67	C₄×C₄₇	C₄×Q₈	1	4	yes	—
68	Dic₄₇	—	2	8	no	—
69	Dic₄₇	—	4	4	no	—
70	#2330	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	C₂×C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄₇	#520	1	2	yes	—
73	C₂×C₄₇	#540	1	1	yes	characteristic
74	C₂×C₄₇	#541	1	1	yes	characteristic
75	C₂×C₄₇	#544	1	2	yes	—
76	C₂×#125	—	47	1	no	—
77	#389	—	47	2	no	—
78	#389	—	47	4	no	—
79	#387	—	47	4	no	—
80	#384	—	47	1	no	—
81	C₄×C₄⋊C₄	—	47	2	no	—
82	#376	D₄₇	1	1	yes	characteristic
83	C₄₇	#2330	1	1	yes	characteristic, complemented, Sylow
84	#125	—	47	8	no	—
85	C₂×C₄⋊C₄	—	47	4	no	—
86	C₂×C₄⋊C₄	—	47	2	no	—
87	C₂×C₄⋊C₄	—	47	2	no	—
88	C₂×C₄⋊C₄	—	47	4	no	—
89	C₂×C₄⋊C₄	—	47	4	no	—
90	C₂×C₄²	—	47	4	no	—
91	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
92	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
93	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
94	#95	—	47	4	no	—
95	#95	—	47	4	no	—
96	#95	C₂×D₄₇	1	2	yes	—
97	#95	C₂×D₄₇	1	2	yes	—
98	C₂²×C₄	—	47	4	no	—
99	C₂²×C₄	—	47	4	no	—
100	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
102	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
103	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
104	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
105	C₄⋊C₄	—	47	16	no	—
106	C₄⋊C₄	—	47	8	no	—
107	C₄⋊C₄	—	94	4	no	—
108	C₄⋊C₄	—	188	4	no	—
109	C₄⋊C₄	—	188	4	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	188	4	no	—
112	C₄²	Dic₉₄	1	4	yes	—
113	C₄²	C₄×D₄₇	1	4	yes	—
114	C₂³	C₂³×D₄₇	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	47	8	no	—
124	C₂×C₄	—	94	4	no	—
125	C₂×C₄	—	94	4	no	—
126	C₂×C₄	—	94	4	no	—
127	C₂×C₄	—	94	4	no	—
128	C₂×C₄	—	94	4	no	—
129	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
130	C₂×C₄	C₂×Dic₉₄	1	2	yes	—
131	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
132	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
133	C₂×C₄	#10625350	1	2	yes	—
134	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
135	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
136	C₂²	C₂×#10625350	1	1	yes	characteristic
137	C₂²	#13094070	1	2	yes	—
138	C₂²	#13094071	1	2	yes	—
139	C₄	—	2	4	no	—
140	C₄	—	2	4	no	—
141	C₄	—	4	1	no	—
142	C₄	—	4	1	no	—
143	C₄	—	94	8	no	—
144	C₄	—	188	4	no	—
145	C₄	C₄×Dic₉₄	1	4	yes	—
146	C₂	C₂×C₄×Dic₉₄	1	1	yes	characteristic
147	C₂	#15689603	1	2	yes	—
148	C₂	#15689728	1	1	yes	characteristic
149	C₂	#15689735	1	1	yes	characteristic
150	C₂	#15689751	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094016

C₂

yes

maximal, characteristic

#15689021

C₂

yes

maximal

#15689020

C₂

yes

maximal

#15689010

C₂

yes

maximal

#15688993

C₂

yes

maximal, characteristic

C₄×C₄⋊Dic₄₇

C₂

yes

maximal

C₄₇×#376

C₂

yes

maximal, characteristic

#13094016

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

C₂²

yes

—

#95×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

Q₈

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₂⁴

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

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

#2330

—

maximal, Sylow

C₂×C₄₇

C₂×C₄×Q₈

yes

characteristic

C₂×C₄₇

#520

yes

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

—

—

—

C₄×C₄⋊C₄

—

#376

D₄₇

yes

characteristic

C₄₇

#2330

yes

characteristic, complemented, Sylow

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

#95

C₂×D₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

C₂²×D₄₇

yes

characteristic

101

C₂²×C₄

C₂²×D₄₇

yes

characteristic

102

C₂²×C₄

C₂²×D₄₇

yes

—

103

C₂²×C₄

C₂²×D₄₇

yes

—

104

C₂²×C₄

C₂²×D₄₇

yes

characteristic

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

188

—

109

C₄⋊C₄

—

188

—

110

C₄²

—

111

C₄²

—

188

—

112

C₄²

Dic₉₄

yes

—

113

C₄²

C₄×D₄₇

yes

—

114

C₂³

C₂³×D₄₇

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×Dic₉₄

yes

—

130

C₂×C₄

C₂×Dic₉₄

yes

—

131

C₂×C₄

C₂×C₄×D₄₇

yes

—

132

C₂×C₄

C₂×C₄×D₄₇

yes

—

133

C₂×C₄

#10625350

yes

—

134

C₂²

C₂²×C₄×D₄₇

yes

characteristic

135

C₂²

C₂²×Dic₉₄

yes

characteristic

136

C₂²

C₂×#10625350

yes

characteristic

137

C₂²

#13094070

yes

—

138

C₂²

#13094071

yes

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

188

—

145

C₄

C₄×Dic₉₄

yes

—

146

C₂

C₂×C₄×Dic₉₄

yes

characteristic

147

C₂

#15689603

yes

—

148

C₂

#15689728

yes

characteristic

149

C₂

#15689735

yes

characteristic

150

C₂

#15689751

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#376	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	#376	D₄₇
hypocenter	C₄₇	#2330
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #376

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#376

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #376

N	H ≅ G/N	type
8. #13094016	C₄	semidirect
83. C₄₇	#2330	semidirect

H ≅ G/N

type

8. #13094016

C₄

semidirect

83. C₄₇

#2330

semidirect

Subgroups — #20246129

Conjugacy classes of subgroups

Complements