Group #20252040 — subgroups

Subgroups — #20252040

Order 6016.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	#15689182	C₂	1	4	yes	maximal, complemented
3	#15689178	C₂	1	4	yes	maximal, complemented
4	#15689151	C₂	1	2	yes	maximal, complemented
5	#15689091	C₂	1	2	yes	maximal
6	C₂×#13093935	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625350	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
10	C₄×Dic₉₄	C₄	1	2	yes	—
11	C₄×D₁₈₈	C₄	1	2	yes	—
12	C₄×C₂²⋊D₄₇	C₄	1	4	yes	—
13	#13093983	C₂²	1	1	yes	characteristic
14	#13093981	C₂²	1	2	yes	—
15	C₄²×D₄₇	C₂²	1	2	yes	—
16	Dic₄₇⋊C₈	C₂²	1	8	yes	—
17	(C₂×D₄₇)⋊C₈	C₂²	1	8	yes	—
18	#13093935	C₂²	1	2	yes	—
19	#13093935	C₂²	1	2	yes	—
20	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
22	#97×C₄₇	C₂²	1	2	yes	—
23	C₄×C₈×C₄₇	C₂²	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₄₇	C₂×C₄	1	2	yes	—
27	#10625350	—	2	4	no	—
28	C₂×D₁₈₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₄₇	C₂³	1	2	yes	—
31	C₂×Dic₉₄	C₂×C₄	1	1	yes	characteristic
32	M₁₆×C₄₇	—	2	4	no	—
33	C₂×C₈×C₄₇	C₂³	1	4	yes	—
34	C₄²×C₄₇	C₂×C₄	1	2	yes	—
35	C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₄₇	C₂×C₄	1	2	yes	—
38	(C₂×D₄₇)⋊C₄	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₄₇	C₂×C₄	1	2	yes	—
40	Dic₄₇⋊C₄	C₂×C₄	1	4	yes	—
41	C₄×Dic₄₇	C₂×C₄	1	4	yes	—
42	C₄×Dic₄₇	C₂³	1	2	yes	—
43	C₂×C₄₇⋊C₈	—	2	4	no	—
44	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
45	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	#41	1	4	yes	—
52	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₄₇	—	2	4	no	—
54	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
55	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
56	D₁₈₈	—	2	2	no	—
57	C₄×D₄₇	—	2	4	no	—
58	C₄×D₄₇	—	4	4	no	—
59	Dic₉₄	—	2	2	no	—
60	C₈×C₄₇	—	2	4	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	#117	1	2	yes	—
64	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₄₇	—	2	2	no	—
66	C₂×D₄₇	—	2	4	no	—
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	#117	1	2	yes	—
70	C₄×C₄₇	C₂×#41	1	2	yes	—
71	Dic₄₇	—	2	2	no	—
72	Dic₄₇	—	2	4	no	—
73	#2923	—	47	1	no	maximal, Sylow
74	C₂×C₄₇	—	2	2	no	—
75	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
76	C₂×C₄₇	#568	1	2	yes	—
77	D₄₇	—	4	2	no	—
78	C₄×#41	—	47	1	no	—
79	#433	—	47	4	no	—
80	#432	—	47	4	no	—
81	#424	—	47	2	no	—
82	#408	—	47	2	no	—
83	C₄×M₁₆	D₄₇	1	1	yes	characteristic, complemented
84	C₂×#97	—	47	1	no	—
85	C₄₇	#2923	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	47	1	no	—
87	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
88	C₂²×C₈	—	47	2	no	—
89	C₄×Q₈	—	47	2	no	—
90	C₄×D₄	—	47	4	no	—
91	C₄×D₄	—	47	2	no	—
92	#117	—	47	1	no	—
93	#117	—	47	2	no	—
94	C₂×C₄²	—	47	2	no	—
95	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
96	C₄⋊C₈	—	47	8	no	—
97	C₂²⋊C₈	—	47	8	no	—
98	#97	—	47	2	no	—
99	#97	—	47	2	no	—
100	#97	C₂×D₄₇	1	2	yes	—
101	C₄×C₈	C₂×D₄₇	1	2	yes	—
102	#41	—	94	4	no	—
103	C₂×Q₈	—	47	1	no	—
104	C₂×D₄	—	47	2	no	—
105	C₂×D₄	—	47	1	no	—
106	C₂²×C₄	—	47	2	no	—
107	C₂²×C₄	—	47	4	no	—
108	C₂²×C₄	C₄×D₄₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	47	4	no	—
112	C₂×C₈	—	94	4	no	—
113	C₂×C₈	C₂²×D₄₇	1	4	yes	—
114	C₄⋊C₄	—	47	4	no	—
115	C₄⋊C₄	—	47	2	no	—
116	C₂²⋊C₄	—	47	2	no	—
117	C₂²⋊C₄	—	47	4	no	—
118	C₄²	—	47	4	no	—
119	C₄²	—	47	2	no	—
120	C₄²	C₄×D₄₇	1	2	yes	—
121	C₄²	C₂²×D₄₇	1	1	yes	characteristic
122	C₄²	C₂²×D₄₇	1	1	yes	characteristic
123	C₂³	—	47	2	no	—
124	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
125	Q₈	—	94	2	no	—
126	D₄	—	94	4	no	—
127	D₄	—	94	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	47	4	no	—
130	C₂×C₄	—	47	2	no	—
131	C₂×C₄	—	94	4	no	—
132	C₂×C₄	—	188	4	no	—
133	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂×C₄	#10625350	1	4	yes	—
137	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	94	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	94	2	no	—
142	C₂²	—	94	4	no	—
143	C₂²	#13093981	1	2	yes	—
144	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	94	2	no	—
148	C₄	—	94	4	no	—
149	C₄	#13093981	1	2	yes	—
150	C₄	C₂×#10625350	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	188	2	no	—
153	C₂	C₂×#13093981	1	1	yes	characteristic
154	C₂	#15689889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10625350

C₂

yes

maximal, characteristic

#15689182

C₂

yes

maximal, complemented

#15689178

C₂

yes

maximal, complemented

#15689151

C₂

yes

maximal, complemented

#15689091

C₂

yes

maximal

C₂×#13093935

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₄×Dic₉₄

C₄

yes

—

C₄×D₁₈₈

C₄

yes

—

C₄×C₂²⋊D₄₇

C₄

yes

—

#13093983

C₂²

yes

characteristic

#13093981

C₂²

yes

—

C₄²×D₄₇

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

(C₂×D₄₇)⋊C₈

C₂²

yes

—

#13093935

C₂²

yes

—

#13093935

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#97×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₂²⋊D₄₇

C₂×C₄

yes

—

#10625350

—

C₂×D₁₈₈

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

C₂×Dic₉₄

C₂×C₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

C₂×C₄

yes

—

(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

C₂²×C₄

yes

—

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

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#117

yes

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

Dic₄₇

—

#2923

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#117

yes

characteristic

C₂×C₄₇

#568

yes

—

D₄₇

—

C₄×#41

—

—

—

—

—

C₄×M₁₆

D₄₇

yes

characteristic, complemented

C₂×#97

—

C₄₇

#2923

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

100

#97

C₂×D₄₇

yes

—

101

C₄×C₈

C₂×D₄₇

yes

—

102

#41

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₄₇

yes

—

109

C₂²×C₄

C₂²×D₄₇

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₄₇

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₄₇

yes

—

121

C₄²

C₂²×D₄₇

yes

characteristic

122

C₄²

C₂²×D₄₇

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₄×D₄₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

188

—

133

C₂×C₄

C₂×C₄×D₄₇

yes

—

134

C₂×C₄

C₂×C₄×D₄₇

yes

—

135

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

136

C₂×C₄

#10625350

yes

—

137

C₂×C₄

C₂³×D₄₇

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#13093981

yes

—

144

C₂²

C₂²×C₄×D₄₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#13093981

yes

—

150

C₄

C₂×#10625350

yes

—

151

C₂

—

152

C₂

—

188

—

153

C₂

C₂×#13093981

yes

characteristic

154

C₂

#15689889

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄×M₁₆×C₄₇	C₂
hypercenter	C₄×M₁₆	D₄₇
hypocenter	C₄₇	#2923
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄×M₁₆×C₄₇

C₂

hypercenter

C₄×M₁₆

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #15689182	C₂	semidirect
3. #15689178	C₂	semidirect
4. #15689151	C₂	semidirect
6. C₂×#13093935	C₂	semidirect
7. C₄×M₁₆×C₄₇	C₂	semidirect
83. C₄×M₁₆	D₄₇	semidirect
85. C₄₇	#2923	semidirect

H ≅ G/N

type

2. #15689182

C₂

semidirect

3. #15689178

C₂

semidirect

4. #15689151

C₂

semidirect

6. C₂×#13093935

C₂

semidirect

7. C₄×M₁₆×C₄₇

C₂

semidirect

83. C₄×M₁₆

D₄₇

semidirect

85. C₄₇

#2923

semidirect

Subgroups — #20252040

Conjugacy classes of subgroups

Complements