Group #20243199 — subgroups

Subgroups — #20243199

Order 6016.

320 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₄×#10625350	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093952	C₂	1	2	yes	maximal
3	C₄×#10625329	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688883	C₂	1	2	yes	maximal, complemented
5	C₄₇×#376	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	C₂²	1	1	yes	characteristic
7	C₂×#10625329	C₂²	1	2	yes	—
8	C₄×Dic₉₄	C₄	1	2	yes	complemented
9	C₄×D₁₈₈	C₄	1	2	yes	complemented
10	C₄×C₂²⋊D₄₇	—	2	2	no	—
11	#13093983	C₂²	1	1	yes	characteristic
12	#13093981	—	2	1	no	—
13	C₄²×D₄₇	—	2	1	no	—
14	#13093952	C₄	1	8	yes	complemented
15	#13093935	—	2	1	no	—
16	C₄×C₄₇⋊C₈	—	2	1	no	—
17	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	#95×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	—	2	2	no	—
21	C₂²×C₄×C₄₇	D₄	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	C₂×C₂²⋊D₄₇	—	2	1	no	—
24	#10625350	—	2	2	no	—
25	#10625350	C₂×C₄	1	4	yes	—
26	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₄₇	—	2	1	no	—
28	C₂×C₄×D₄₇	—	2	2	no	—
29	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₄₇	—	2	2	no	—
31	C₄²×C₄₇	C₂×C₄	1	2	yes	—
32	C₄²×C₄₇	C₂×C₄	1	4	yes	complemented
33	C₄²×C₄₇	D₄	1	1	yes	characteristic
34	C₄²×C₄₇	D₄	1	1	yes	characteristic
35	C₂²⋊Dic₄₇	—	2	1	no	—
36	(C₂×D₄₇)⋊C₄	—	2	2	no	—
37	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
38	Dic₄₇⋊C₄	—	2	2	no	—
39	C₄×Dic₄₇	—	2	2	no	—
40	C₄×Dic₄₇	—	2	1	no	—
41	#10625329	—	2	2	no	—
42	#10625329	C₂×C₄	1	4	yes	complemented
43	C₂×C₄₇⋊C₈	—	2	2	no	—
44	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
45	C₂²×D₄₇	—	2	1	no	—
46	C₂×C₄×C₄₇	—	2	4	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	—	4	4	no	—
49	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
53	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	#41	1	2	yes	—
55	C₂²⋊D₄₇	—	2	4	no	—
56	C₂×Dic₄₇	—	2	1	no	—
57	C₂×Dic₄₇	—	2	2	no	—
58	D₁₈₈	—	2	1	no	—
59	D₁₈₈	C₄²	1	2	yes	—
60	C₄×D₄₇	—	2	4	no	—
61	C₄×D₄₇	—	4	4	no	—
62	Dic₉₄	—	2	1	no	—
63	Dic₉₄	C₄²	1	2	yes	—
64	C₄₇⋊C₈	—	2	4	no	—
65	C₂²×C₄₇	—	2	2	no	—
66	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₇	C₄×D₄	1	2	yes	—
68	C₂×D₄₇	—	2	2	no	—
69	C₂×D₄₇	—	4	2	no	—
70	C₄×C₄₇	—	2	2	no	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	—	4	4	no	—
73	C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
74	C₄×C₄₇	#117	1	1	yes	characteristic
75	C₄×C₄₇	C₄×D₄	1	2	yes	—
76	Dic₄₇	—	2	2	no	—
77	Dic₄₇	—	4	2	no	—
78	#1759	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	—	2	2	no	—
80	C₂×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
81	C₂×C₄₇	#421	1	2	yes	—
82	D₄₇	—	4	2	no	—
83	C₄×#41	—	47	1	no	—
84	C₂×#104	—	47	2	no	—
85	C₄×M₁₆	—	47	1	no	—
86	#376	D₄₇	1	1	yes	characteristic, complemented
87	#339	—	47	2	no	—
88	C₄₇	#1759	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	47	1	no	—
90	C₂×M₁₆	—	47	2	no	—
91	C₄×Q₈	—	47	2	no	—
92	C₄×D₄	—	47	2	no	—
93	C₄×D₄	—	94	2	no	—
94	#117	—	47	1	no	—
95	#117	—	94	1	no	—
96	C₂×C₄²	—	94	1	no	—
97	C₂×C₄²	C₂×D₄₇	1	2	yes	—
98	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
99	#104	—	47	8	no	—
100	#97	—	94	1	no	—
101	C₄×C₈	—	94	1	no	—
102	#95	—	2	2	no	—
103	#41	—	47	4	no	—
104	#41	—	94	2	no	—
105	C₂×Q₈	—	47	1	no	—
106	C₂×D₄	—	47	1	no	—
107	C₂×D₄	—	94	1	no	—
108	C₂²×C₄	—	2	2	no	—
109	C₂²×C₄	—	94	1	no	—
110	C₂²×C₄	—	94	2	no	—
111	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
113	M₁₆	—	47	4	no	—
114	M₁₆	—	94	2	no	—
115	C₂×C₈	—	94	2	no	—
116	C₄⋊C₄	—	47	2	no	—
117	C₄⋊C₄	—	94	2	no	—
118	C₂²⋊C₄	—	94	1	no	—
119	C₂²⋊C₄	—	94	2	no	—
120	C₄²	—	2	2	no	—
121	C₄²	—	94	2	no	—
122	C₄²	—	94	1	no	—
123	C₄²	C₄×D₄₇	1	2	yes	—
124	C₄²	C₄×D₄₇	1	4	yes	complemented
125	C₄²	D₁₈₈	1	1	yes	characteristic
126	C₄²	D₁₈₈	1	1	yes	characteristic
127	C₂³	—	94	1	no	—
128	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
129	Q₈	—	47	2	no	—
130	Q₈	—	94	1	no	—
131	D₄	—	47	2	no	—
132	D₄	—	94	4	no	—
133	D₄	—	94	1	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	4	2	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	94	1	no	—
138	C₂×C₄	—	94	4	no	—
139	C₂×C₄	—	94	2	no	—
140	C₂×C₄	—	188	4	no	—
141	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
142	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
143	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
144	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
145	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
146	C₂×C₄	#10625350	1	2	yes	—
147	C₈	—	94	4	no	—
148	C₂²	—	2	2	no	—
149	C₂²	—	94	2	no	—
150	C₂²	—	188	2	no	—
151	C₂²	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
152	C₂²	C₄×C₂²⋊D₄₇	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	4	no	—
156	C₄	—	94	2	no	—
157	C₄	—	188	2	no	—
158	C₄	C₄²×D₄₇	1	1	yes	characteristic
159	C₄	#13093981	1	1	yes	characteristic
160	C₄	C₄×D₁₈₈	1	2	yes	—
161	C₂	—	2	2	no	—
162	C₂	—	188	2	no	—
163	C₂	C₄×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
164	C₂	#15689141	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10625350

C₂

yes

maximal, characteristic

C₂×#13093952

C₂

yes

maximal

C₄×#10625329

C₂

yes

maximal, characteristic, complemented

#15688883

C₂

yes

maximal, complemented

C₄₇×#376

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

C₄×Dic₉₄

C₄

yes

complemented

C₄×D₁₈₈

C₄

yes

complemented

C₄×C₂²⋊D₄₇

—

#13093983

C₂²

yes

characteristic

#13093981

—

C₄²×D₄₇

—

#13093952

C₄

yes

complemented

#13093935

—

C₄×C₄₇⋊C₈

—

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

—

#10625350

—

#10625350

C₂×C₄

yes

—

C₂×D₁₈₈

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×Dic₉₄

C₂×C₄

yes

characteristic

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

complemented

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₂²⋊Dic₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

C₂×C₄

yes

complemented

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

C₄²

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

Dic₉₄

C₄²

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₄×D₄

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄²

yes

characteristic

C₄×C₄₇

#117

yes

characteristic

C₄×C₄₇

C₄×D₄

yes

—

Dic₄₇

—

Dic₄₇

—

#1759

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₄×C₂²⋊C₄

yes

characteristic

C₂×C₄₇

#421

yes

—

D₄₇

—

C₄×#41

—

C₂×#104

—

C₄×M₁₆

—

#376

D₄₇

yes

characteristic, complemented

#339

—

C₄₇

#1759

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#104

—

100

#97

—

101

C₄×C₈

—

102

#95

—

103

#41

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₄₇

yes

—

112

C₂²×C₄

C₂²×D₄₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₄²

C₄×D₄₇

yes

—

124

C₄²

C₄×D₄₇

yes

complemented

125

C₄²

D₁₈₈

yes

characteristic

126

C₄²

D₁₈₈

yes

characteristic

127

C₂³

—

128

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

188

—

141

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

142

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

143

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

144

C₂×C₄

C₂×C₄×D₄₇

yes

—

145

C₂×C₄

C₂×D₁₈₈

yes

characteristic

146

C₂×C₄

#10625350

yes

—

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

188

—

151

C₂²

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

152

C₂²

C₄×C₂²⋊D₄₇

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

188

—

158

C₄

C₄²×D₄₇

yes

characteristic

159

C₄

#13093981

yes

characteristic

160

C₄

C₄×D₁₈₈

yes

—

161

C₂

—

162

C₂

—

188

—

163

C₂

C₄×(C₂×D₄₇)⋊C₄

yes

characteristic

164

C₂

#15689141

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₁₈₈
commutator	C₄×C₄₇	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#376	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#376	D₄₇
hypocenter	C₄₇	#1759
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

subgroup

G/H

center

C₂×C₄

C₂×D₁₈₈

commutator

C₄×C₄₇

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#376

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #376

N	H ≅ G/N	type
3. C₄×#10625329	C₂	semidirect
4. #15688883	C₂	semidirect
5. C₄₇×#376	C₂	semidirect
8. C₄×Dic₉₄	C₄	semidirect
9. C₄×D₁₈₈	C₄	semidirect
14. #13093952	C₄	semidirect
32. C₄²×C₄₇	C₂×C₄	semidirect
42. #10625329	C₂×C₄	semidirect
86. #376	D₄₇	semidirect
88. C₄₇	#1759	semidirect
124. C₄²	C₄×D₄₇	semidirect

H ≅ G/N

type

3. C₄×#10625329

C₂

semidirect

4. #15688883

C₂

semidirect

5. C₄₇×#376

C₂

semidirect

8. C₄×Dic₉₄

C₄

semidirect

9. C₄×D₁₈₈

C₄

semidirect

14. #13093952

C₄

semidirect

32. C₄²×C₄₇

C₂×C₄

semidirect

42. #10625329

C₂×C₄

semidirect

86. #376

D₄₇

semidirect

88. C₄₇

#1759

semidirect

124. C₄²

C₄×D₄₇

semidirect

Subgroups — #20243199

Conjugacy classes of subgroups

Complements