Group #20245077 — subgroups

Subgroups — #20245077

Order 6016.

230 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 73 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×#13093943	C₂	1	2	yes	maximal, complemented
2	#15688922	C₂	1	2	yes	maximal, complemented
3	#15688920	C₂	1	2	yes	maximal, complemented
4	C₄₇×#536	C₂	1	1	yes	maximal, characteristic
5	C₂×#10625329	—	2	2	no	—
6	#13093943	—	2	2	no	—
7	#13093943	C₂²	1	4	yes	—
8	C₂×#41×C₄₇	C₄	1	1	yes	characteristic
9	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
10	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic
11	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
12	#124×C₄₇	C₄	1	1	yes	characteristic
13	#124×C₄₇	C₂²	1	1	yes	characteristic
14	#123×C₄₇	—	2	2	no	—
15	C₂²⋊Q₈×C₄₇	—	2	2	no	—
16	#117×C₄₇	C₄	1	1	yes	characteristic
17	#41×C₄₇	—	4	2	no	—
18	C₂×Q₈×C₄₇	—	4	2	no	—
19	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂×Q₈×C₄₇	D₄	1	2	yes	—
22	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₄×C₄₇	D₄	1	2	yes	—
28	C₄⋊C₄×C₄₇	—	2	1	no	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	C₂²⋊C₄×C₄₇	—	2	1	no	—
32	C₂²⋊C₄×C₄₇	—	2	2	no	—
33	C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	#10625329	—	2	4	no	—
37	#10625329	—	4	2	no	—
38	C₂×C₄₇⋊C₈	—	4	2	no	—
39	C₂³×C₄₇	—	2	1	no	—
40	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
41	Q₈×C₄₇	—	4	1	no	—
42	Q₈×C₄₇	—	4	2	no	—
43	Q₈×C₄₇	—	4	1	no	—
44	Q₈×C₄₇	—	4	1	no	—
45	D₄×C₄₇	—	4	1	no	—
46	D₄×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	1	no	—
51	C₂×C₄×C₄₇	—	2	1	no	—
52	C₂×C₄×C₄₇	—	2	1	no	—
53	C₂×C₄×C₄₇	—	4	2	no	—
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
58	C₄₇⋊C₈	—	4	4	no	—
59	C₂²×C₄₇	—	4	1	no	—
60	C₂²×C₄₇	—	4	2	no	—
61	C₂²×C₄₇	—	4	1	no	—
62	C₂²×C₄₇	#99	1	2	yes	—
63	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₄×C₄₇	—	2	2	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	2	2	no	—
67	C₄×C₄₇	—	2	1	no	—
68	C₄×C₄₇	—	4	1	no	—
69	C₄×C₄₇	—	4	1	no	—
70	C₄×C₄₇	—	4	1	no	—
71	#2133	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	—	2	1	no	—
73	C₂×C₄₇	—	2	2	no	—
74	C₂×C₄₇	—	8	1	no	—
75	C₂×C₄₇	C₂×#99	1	1	yes	characteristic
76	#536	D₄₇	1	1	yes	characteristic
77	C₂×#101	—	47	2	no	—
78	#356	—	47	2	no	—
79	#355	—	47	2	no	—
80	C₄₇	#2133	1	1	yes	characteristic, complemented, Sylow
81	C₂×#41	Dic₄₇	1	1	yes	characteristic
82	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
83	C₂×M₁₆	—	94	2	no	—
84	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic
85	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
86	#124	Dic₄₇	1	1	yes	characteristic
87	#124	C₂×D₄₇	1	1	yes	characteristic
88	#123	—	2	2	no	—
89	C₂²⋊Q₈	—	2	2	no	—
90	#117	Dic₄₇	1	1	yes	characteristic
91	#101	—	47	4	no	—
92	#101	—	94	2	no	—
93	#41	—	4	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
96	C₂×Q₈	C₂×Dic₄₇	1	1	yes	characteristic
97	C₂×Q₈	C₂²⋊D₄₇	1	2	yes	—
98	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	C₂×Dic₄₇	1	1	yes	characteristic
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
103	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
104	M₁₆	—	94	4	no	—
105	M₁₆	—	188	2	no	—
106	C₂×C₈	—	188	2	no	—
107	C₄⋊C₄	—	2	1	no	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₂²⋊C₄	—	2	1	no	—
111	C₂²⋊C₄	—	2	2	no	—
112	C₂²⋊C₄	—	2	2	no	—
113	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
114	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
115	C₂³	—	2	1	no	—
116	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
117	Q₈	—	4	1	no	—
118	Q₈	—	4	2	no	—
119	Q₈	—	4	1	no	—
120	Q₈	—	4	1	no	—
121	D₄	—	4	1	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
131	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
132	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
133	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
134	C₈	—	188	4	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	#13093939	1	2	yes	—
139	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
140	C₄	—	2	2	no	—
141	C₄	—	2	1	no	—
142	C₄	—	2	2	no	—
143	C₄	—	2	1	no	—
144	C₄	—	4	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₂	—	2	1	no	—
148	C₂	—	2	2	no	—
149	C₂	—	8	1	no	—
150	C₂	C₂×#13093939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093943

C₂

yes

maximal, complemented

#15688922

C₂

yes

maximal, complemented

#15688920

C₂

yes

maximal, complemented

C₄₇×#536

C₂

yes

maximal, characteristic

C₂×#10625329

—

#13093943

—

#13093943

C₂²

yes

—

C₂×#41×C₄₇

C₄

yes

characteristic

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄⋊Q₈×C₄₇

C₄

yes

characteristic

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

#124×C₄₇

C₄

yes

characteristic

#124×C₄₇

C₂²

yes

characteristic

#123×C₄₇

—

C₂²⋊Q₈×C₄₇

—

#117×C₄₇

C₄

yes

characteristic

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×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

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#99

yes

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

#2133

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#99

yes

characteristic

#536

D₄₇

yes

—

—

—

C₄₇

#2133

yes

characteristic, complemented, Sylow

C₂×#41

Dic₄₇

yes

characteristic

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₄⋊Q₈

Dic₄₇

yes

characteristic

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#124

Dic₄₇

yes

characteristic

#124

C₂×D₄₇

yes

characteristic

#123

—

C₂²⋊Q₈

—

#117

Dic₄₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₄₇

yes

characteristic

C₂×Q₈

C₂×Dic₄₇

yes

characteristic

C₂×Q₈

C₂²⋊D₄₇

yes

—

C₂×Q₈

C₂²×D₄₇

yes

characteristic

C₂×D₄

—

100

C₂×D₄

C₂×Dic₄₇

yes

characteristic

101

C₂²×C₄

—

102

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

103

C₂²×C₄

C₂²⋊D₄₇

yes

—

104

M₁₆

—

105

M₁₆

—

188

—

106

C₂×C₈

—

188

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₂²⋊C₄

—

113

C₄²

C₂×Dic₄₇

yes

characteristic

114

C₄²

C₂×Dic₄₇

yes

characteristic

115

C₂³

—

116

C₂³

C₂²⋊Dic₄₇

yes

characteristic

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²⋊Dic₄₇

yes

—

131

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

132

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

133

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

134

C₈

—

188

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13093939

yes

—

139

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₂

—

148

C₂

—

149

C₂

—

150

C₂

C₂×#13093939

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#13093939
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂×Q₈	C₂²×D₄₇
Fitting	C₄₇×#536	C₂
socle	C₂×C₄₇	C₂×#99
hypercenter	#536	D₄₇
hypocenter	C₄₇	#2133
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×Q₈ ◁ #536

subgroup

G/H

center

C₂

C₂×#13093939

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂×Q₈

C₂²×D₄₇

Fitting

C₄₇×#536

C₂

socle

C₂×C₄₇

C₂×#99

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂³ ◁ C₂²×Q₈ ◁ #536

N	H ≅ G/N	type
1. C₂×#13093943	C₂	semidirect
2. #15688922	C₂	semidirect
3. #15688920	C₂	semidirect
80. C₄₇	#2133	semidirect

H ≅ G/N

type

1. C₂×#13093943

C₂

semidirect

2. #15688922

C₂

semidirect

3. #15688920

C₂

semidirect

80. C₄₇

#2133

semidirect

Subgroups — #20245077

Conjugacy classes of subgroups

Complements