Group #20241274 — subgroups

Subgroups — #20241274

Order 6016.

246 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 65 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	#15689087	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	C₄×C₂²⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₂×#10625329	C₄	1	2	yes	complemented
5	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
6	C₂×C₄×Dic₄₇	—	2	1	no	—
7	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
8	C₂²⋊(C₄₇⋊C₈)	—	2	1	no	—
9	C₄₇⋊#95	C₄	1	2	yes	complemented
10	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
12	C₂×C₄²×C₄₇	—	2	1	no	—
13	#95×C₄₇	C₄	1	2	yes	complemented
14	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	—	2	2	no	—
16	C₂²×C₄×C₄₇	—	4	2	no	—
17	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	Q₈	1	1	yes	characteristic
23	C₂²×Dic₄₇	—	2	2	no	—
24	C₂²⋊C₄×C₄₇	—	2	4	no	—
25	C₄²×C₄₇	—	4	2	no	—
26	C₂²⋊Dic₄₇	—	2	4	no	—
27	C₄×Dic₄₇	—	4	2	no	—
28	#10625329	—	4	2	no	—
29	C₂×C₄₇⋊C₈	—	2	2	no	—
30	C₂³×C₄₇	—	2	2	no	—
31	C₂³×C₄₇	—	2	2	no	—
32	C₂³×C₄₇	—	4	1	no	—
33	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
34	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂³×C₄₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	—	2	2	no	—
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	2	4	no	—
40	C₂×C₄×C₄₇	—	4	1	no	—
41	C₂×C₄×C₄₇	—	4	1	no	—
42	C₂×C₄×C₄₇	—	4	1	no	—
43	C₂×C₄×C₄₇	—	4	2	no	—
44	C₂×C₄×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	C₄²	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₄⋊C₄	1	1	yes	characteristic
49	C₂×Dic₄₇	—	2	4	no	—
50	C₂×Dic₄₇	—	4	2	no	—
51	C₂×Dic₄₇	—	4	2	no	—
52	C₄₇⋊C₈	—	4	2	no	—
53	C₂²×C₄₇	—	2	1	no	—
54	C₂²×C₄₇	—	2	1	no	—
55	C₂²×C₄₇	—	2	1	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	—	2	1	no	—
59	C₂²×C₄₇	—	2	1	no	—
60	C₂²×C₄₇	—	4	4	no	—
61	C₂²×C₄₇	#95	1	1	yes	characteristic
62	C₂²×C₄₇	#104	1	2	yes	—
63	C₂²×C₄₇	#104	1	2	yes	—
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	2	1	no	—
67	C₄×C₄₇	—	4	4	no	—
68	C₄×C₄₇	#99	1	2	yes	—
69	Dic₄₇	—	4	4	no	—
70	#1297	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	—	2	2	no	—
72	C₂×C₄₇	—	2	2	no	—
73	C₂×C₄₇	—	4	1	no	—
74	C₂×C₄₇	#339	1	1	yes	characteristic
75	C₂×C₄₇	#339	1	1	yes	characteristic
76	C₂×C₄₇	#342	1	1	yes	characteristic
77	#407	—	47	1	no	—
78	C₄×C₂²⋊C₄	—	47	1	no	—
79	C₄×C₂²⋊C₄	D₄₇	1	1	yes	characteristic
80	C₄₇	#1297	1	1	yes	characteristic, complemented, Sylow
81	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
82	C₂×M₁₆	—	47	2	no	—
83	C₂×C₂²⋊C₄	—	2	1	no	—
84	C₂×C₂²⋊C₄	—	94	1	no	—
85	C₂×C₄²	—	2	1	no	—
86	C₂×C₄²	—	94	1	no	—
87	C₂²⋊C₈	—	94	1	no	—
88	C₂²⋊C₈	—	94	1	no	—
89	#95	—	47	2	no	—
90	#95	Dic₄₇	1	2	yes	complemented
91	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	2	no	—
93	C₂²×C₄	—	4	2	no	—
94	C₂²×C₄	—	94	2	no	—
95	C₂²×C₄	Dic₉₄	1	1	yes	characteristic
96	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
97	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
98	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
99	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
100	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
101	M₁₆	—	188	2	no	—
102	C₂×C₈	—	94	2	no	—
103	C₂²⋊C₄	—	2	4	no	—
104	C₂²⋊C₄	—	94	4	no	—
105	C₄²	—	4	2	no	—
106	C₄²	—	188	2	no	—
107	C₂³	—	2	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	4	1	no	—
110	C₂³	Dic₄₇⋊C₄	1	1	yes	characteristic
111	C₂³	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
112	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	4	1	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	94	4	no	—
123	C₂×C₄	—	188	2	no	—
124	C₂×C₄	—	188	2	no	—
125	C₂×C₄	C₄×Dic₄₇	1	1	yes	characteristic
126	C₂×C₄	Dic₄₇⋊C₄	1	1	yes	characteristic
127	C₂×C₄	C₄⋊Dic₄₇	1	1	yes	characteristic
128	C₂×C₄	(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
129	C₈	—	188	2	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	4	4	no	—
138	C₂²	C₄₇⋊#95	1	1	yes	characteristic
139	C₂²	#13093951	1	2	yes	—
140	C₂²	#13093952	1	2	yes	—
141	C₄	—	2	1	no	—
142	C₄	—	2	1	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	—	188	4	no	—
146	C₄	#13093938	1	2	yes	—
147	C₂	—	2	2	no	—
148	C₂	—	2	2	no	—
149	C₂	—	4	1	no	—
150	C₂	#15688882	1	1	yes	characteristic
151	C₂	#15688883	1	1	yes	characteristic
152	C₂	#15688889	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689087

C₂

yes

maximal, characteristic

C₄×C₂²⋊Dic₄₇

C₂

yes

maximal, characteristic

C₄×C₂²⋊C₄×C₄₇

C₂

yes

maximal, characteristic

C₂×#10625329

C₄

yes

complemented

C₂×C₂²⋊Dic₄₇

—

C₂×C₄×Dic₄₇

—

C₂²⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

C₄

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

—

#95×C₄₇

C₄

yes

complemented

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

characteristic

C₂²×Dic₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₂²⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂³×C₄₇

C₄⋊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

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄²

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₄⋊C₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#95

yes

characteristic

C₂²×C₄₇

#104

yes

—

C₂²×C₄₇

#104

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#99

yes

—

Dic₄₇

—

#1297

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₄×C₂²⋊C₄

—

C₄×C₂²⋊C₄

D₄₇

yes

characteristic

C₄₇

#1297

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#95

—

#95

Dic₄₇

yes

complemented

C₂⁴

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₉₄

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

100

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

101

M₁₆

—

188

—

102

C₂×C₈

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₄²

—

106

C₄²

—

188

—

107

C₂³

—

108

C₂³

—

109

C₂³

—

110

C₂³

Dic₄₇⋊C₄

yes

characteristic

111

C₂³

(C₂×D₄₇)⋊C₄

yes

characteristic

112

C₂³

C₂²⋊Dic₄₇

yes

characteristic

113

C₂×C₄

—

114

C₂×C₄

—

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

—

188

—

124

C₂×C₄

—

188

—

125

C₂×C₄

C₄×Dic₄₇

yes

characteristic

126

C₂×C₄

Dic₄₇⋊C₄

yes

characteristic

127

C₂×C₄

C₄⋊Dic₄₇

yes

characteristic

128

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

characteristic

129

C₈

—

188

—

130

C₂²

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

C₄₇⋊#95

yes

characteristic

139

C₂²

#13093951

yes

—

140

C₂²

#13093952

yes

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

188

—

146

C₄

#13093938

yes

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

#15688882

yes

characteristic

151

C₂

#15688883

yes

characteristic

152

C₂

#15688889

yes

characteristic

	subgroup	G/H
center	C₂×C₄	(C₂×D₄₇)⋊C₄
commutator	C₂×C₄×C₄₇	C₄²
Frattini	C₂³×C₄	C₂×D₄₇
Fitting	C₄×C₂²⋊C₄×C₄₇	C₂
socle	C₂²×C₄₇	#95
hypercenter	C₄×C₂²⋊C₄	D₄₇
hypocenter	C₄₇	#1297
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₄×C₂²⋊C₄

subgroup

G/H

center

C₂×C₄

(C₂×D₄₇)⋊C₄

commutator

C₂×C₄×C₄₇

C₄²

Frattini

C₂³×C₄

C₂×D₄₇

Fitting

C₄×C₂²⋊C₄×C₄₇

C₂

socle

C₂²×C₄₇

#95

hypercenter

C₄×C₂²⋊C₄

D₄₇

hypocenter

C₄₇

#1297

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₄×C₂²⋊C₄

N	H ≅ G/N	type
4. C₂×#10625329	C₄	semidirect
9. C₄₇⋊#95	C₄	semidirect
13. #95×C₄₇	C₄	semidirect
80. C₄₇	#1297	semidirect
90. #95	Dic₄₇	semidirect

H ≅ G/N

type

4. C₂×#10625329

C₄

semidirect

9. C₄₇⋊#95

C₄

semidirect

13. #95×C₄₇

C₄

semidirect

80. C₄₇

#1297

semidirect

90. #95

Dic₄₇

semidirect

Subgroups — #20241274

Conjugacy classes of subgroups

Complements