Group #20253791 — subgroups

Subgroups — #20253791

Order 6016.

392 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 392 conjugacy classes of subgroups — too many to draw. Only its 133 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	#15689673	C₂	1	1	yes	maximal, characteristic
2	#15689493	C₂	1	2	yes	maximal, complemented
3	#15689489	C₂	1	2	yes	maximal, complemented
4	#15689480	C₂	1	4	yes	maximal, complemented
5	#15689122	C₂	1	4	yes	maximal, complemented
6	C₄×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#125×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625350	C₂²	1	2	yes	—
9	C₂×#10625329	C₂²	1	2	yes	—
10	#13094024	C₂²	1	1	yes	characteristic
11	#13094021	C₂²	1	1	yes	characteristic
12	#13094012	C₂²	1	2	yes	—
13	#13093998	—	2	2	no	—
14	C₄×D₁₈₈	—	2	2	no	—
15	#13093947	C₂²	1	8	yes	—
16	#13093944	C₂²	1	8	yes	—
17	#13093935	C₂²	1	2	yes	—
18	C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
19	#125×C₄₇	—	2	2	no	—
20	#125×C₄₇	C₂²	1	2	yes	—
21	#125×C₄₇	C₂²	1	2	yes	—
22	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
23	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	—	2	2	no	—
26	C₂²×C₄×C₄₇	D₄	1	2	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂×C₂²⋊D₄₇	—	2	2	no	—
29	#10625350	—	2	8	no	—
30	C₂×D₁₈₈	—	2	2	no	—
31	C₂×D₁₈₈	C₂³	1	2	yes	—
32	C₂×C₄×D₄₇	—	2	2	no	—
33	C₂×Dic₉₄	C₂³	1	2	yes	—
34	C₄⋊C₄×C₄₇	—	2	4	no	—
35	C₄⋊C₄×C₄₇	—	2	4	no	—
36	C₄⋊C₄×C₄₇	—	2	2	no	—
37	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
38	C₄²×C₄₇	D₄	1	2	yes	—
39	C₄²×C₄₇	C₂³	1	1	yes	characteristic
40	C₄²×C₄₇	C₂³	1	1	yes	characteristic
41	(C₂×D₄₇)⋊C₄	—	2	4	no	—
42	C₄⋊Dic₄₇	—	2	2	no	—
43	#10625329	—	2	4	no	—
44	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂²×D₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	—	4	4	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	4	yes	—
54	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂²⋊D₄₇	—	4	4	no	—
56	C₂×Dic₄₇	—	2	2	no	—
57	D₁₈₈	—	2	4	no	—
58	D₁₈₈	—	4	4	no	—
59	C₄×D₄₇	—	4	4	no	—
60	Dic₉₄	—	2	4	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	#124	1	2	yes	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₄₇	—	4	2	no	—
66	C₂×D₄₇	—	4	4	no	—
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	—	4	4	no	—
70	C₄×C₄₇	#124	1	2	yes	—
71	C₄×C₄₇	C₂×#41	1	2	yes	—
72	Dic₄₇	—	4	2	no	—
73	#3142	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	—	2	2	no	—
75	C₂×C₄₇	C₂×#124	1	1	yes	characteristic
76	C₂×C₄₇	#575	1	2	yes	—
77	D₄₇	—	8	2	no	—
78	#532	—	47	1	no	—
79	C₂×#125	D₄₇	1	1	yes	characteristic, complemented
80	#491	—	47	2	no	—
81	#490	—	47	2	no	—
82	#488	—	47	4	no	—
83	#417	—	47	4	no	—
84	C₄×M₁₆	—	47	1	no	—
85	C₄₇	#3142	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	47	2	no	—
87	C₂×M₁₆	—	47	2	no	—
88	C₄⋊Q₈	—	47	1	no	—
89	#127	—	47	1	no	—
90	#125	—	2	2	no	—
91	#125	C₂×D₄₇	1	2	yes	—
92	#125	C₂×D₄₇	1	2	yes	—
93	#124	—	47	2	no	—
94	#121	—	94	2	no	—
95	C₄×D₄	—	94	2	no	—
96	C₂×C₄⋊C₄	—	2	2	no	—
97	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
98	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
99	#103	—	47	8	no	—
100	#102	—	47	8	no	—
101	#97	—	47	2	no	—
102	C₄×C₈	—	47	2	no	—
103	#41	—	94	8	no	—
104	C₂×Q₈	—	47	2	no	—
105	C₂×D₄	—	47	2	no	—
106	C₂×D₄	—	94	2	no	—
107	C₂×D₄	—	94	2	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	94	2	no	—
110	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
112	M₁₆	—	94	4	no	—
113	C₂×C₈	—	47	4	no	—
114	C₄⋊C₄	—	2	4	no	—
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	94	2	no	—
118	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
119	C₂²⋊C₄	—	94	4	no	—
120	C₄²	C₂²⋊D₄₇	1	2	yes	—
121	C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	C₂³	—	94	2	no	—
124	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
125	Q₈	—	94	4	no	—
126	D₄	—	94	4	no	—
127	D₄	—	188	4	no	—
128	D₄	—	188	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	4	2	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	94	2	no	—
133	C₂×C₄	—	188	4	no	—
134	C₂×C₄	#10625354	1	4	yes	—
135	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
136	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
137	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
139	C₈	—	94	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	188	2	no	—
142	C₂²	—	188	4	no	—
143	C₂²	#13094013	1	2	yes	—
144	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	4	4	no	—
148	C₄	—	188	2	no	—
149	C₄	#13094013	1	2	yes	—
150	C₄	C₂×#10625354	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	376	2	no	—
153	C₂	C₂×#13094013	1	1	yes	characteristic
154	C₂	#15689921	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689673

C₂

yes

maximal, characteristic

#15689493

C₂

yes

maximal, complemented

#15689489

C₂

yes

maximal, complemented

#15689480

C₂

yes

maximal, complemented

#15689122

C₂

yes

maximal, complemented

C₄×#10625329

C₂

yes

maximal, characteristic, complemented

C₂×#125×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

—

C₂×#10625329

C₂²

yes

—

#13094024

C₂²

yes

characteristic

#13094021

C₂²

yes

characteristic

#13094012

C₂²

yes

—

#13093998

—

C₄×D₁₈₈

—

#13093947

C₂²

yes

—

#13093944

C₂²

yes

—

#13093935

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

—

#125×C₄₇

—

#125×C₄₇

C₂²

yes

—

#125×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

C₂×D₁₈₈

—

C₂×D₁₈₈

C₂³

yes

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

#124

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#124

yes

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

#3142

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#124

yes

characteristic

C₂×C₄₇

#575

yes

—

D₄₇

—

#532

—

C₂×#125

D₄₇

yes

characteristic, complemented

—

—

—

—

C₄×M₁₆

—

C₄₇

#3142

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

—

#125

—

#125

C₂×D₄₇

yes

—

#125

C₂×D₄₇

yes

—

#124

—

#121

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#103

—

100

#102

—

101

#97

—

102

C₄×C₈

—

103

#41

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₄₇

yes

—

111

C₂²×C₄

C₂²×D₄₇

yes

characteristic

112

M₁₆

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

C₂²×D₄₇

yes

—

119

C₂²⋊C₄

—

120

C₄²

C₂²⋊D₄₇

yes

—

121

C₄²

C₂²×D₄₇

yes

characteristic

122

C₄²

C₂²×D₄₇

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

188

—

128

D₄

—

188

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

188

—

134

C₂×C₄

#10625354

yes

—

135

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

136

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

137

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

138

C₂×C₄

C₂³×D₄₇

yes

characteristic

139

C₈

—

140

C₂²

—

141

C₂²

—

188

—

142

C₂²

—

188

—

143

C₂²

#13094013

yes

—

144

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

188

—

149

C₄

#13094013

yes

—

150

C₄

C₂×#10625354

yes

—

151

C₂

—

152

C₂

—

376

—

153

C₂

C₂×#13094013

yes

characteristic

154

C₂

#15689921

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×#125×C₄₇	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	C₂×#125	D₄₇
hypocenter	C₄₇	#3142
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×#125×C₄₇

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #15689493	C₂	semidirect
3. #15689489	C₂	semidirect
4. #15689480	C₂	semidirect
5. #15689122	C₂	semidirect
6. C₄×#10625329	C₂	semidirect
7. C₂×#125×C₄₇	C₂	semidirect
79. C₂×#125	D₄₇	semidirect
85. C₄₇	#3142	semidirect

H ≅ G/N

type

2. #15689493

C₂

semidirect

3. #15689489

C₂

semidirect

4. #15689480

C₂

semidirect

5. #15689122

C₂

semidirect

6. C₄×#10625329

C₂

semidirect

7. C₂×#125×C₄₇

C₂

semidirect

79. C₂×#125

D₄₇

semidirect

85. C₄₇

#3142

semidirect

Subgroups — #20253791

Conjugacy classes of subgroups

Complements