Group #20252025 — subgroups

Subgroups — #20252025

Order 6016.

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

normal subgroups only

This group has 392 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	C₂×#13093981	C₂	1	1	yes	maximal, characteristic
2	#15689183	C₂	1	8	yes	maximal, complemented
3	#15689147	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689086	C₂	1	2	yes	maximal
5	M₁₆×Dic₄₇	C₂	1	2	yes	maximal, complemented
6	C₄₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
8	C₂×#10625329	C₂²	1	2	yes	—
9	#13093981	C₄	1	4	yes	—
10	#13093981	C₂²	1	4	yes	—
11	C₂×Dic₄₇⋊C₄	C₄	1	2	yes	—
12	C₂×(C₂×D₄₇)⋊C₄	C₄	1	2	yes	—
13	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	complemented
15	(C₂×D₄₇)⋊C₈	C₂²	1	8	yes	—
16	#13093934	C₂²	1	4	yes	complemented
17	C₈×Dic₄₇	C₂²	1	4	yes	complemented
18	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₄₇	C₂²	1	4	yes	complemented
21	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₄₇	—	2	2	no	—
27	C₂×C₄×D₄₇	—	2	2	no	—
28	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
29	C₂×C₄×D₄₇	C₂³	1	2	yes	—
30	M₁₆×C₄₇	—	2	4	no	—
31	C₂×C₈×C₄₇	C₂³	1	4	yes	—
32	C₄²×C₄₇	C₂×C₄	1	2	yes	—
33	C₄²×C₄₇	C₂³	1	2	yes	—
34	(C₂×D₄₇)⋊C₄	C₂×C₄	1	8	yes	—
35	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
36	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
37	C₄×Dic₄₇	C₂³	1	2	yes	—
38	#10625329	—	2	4	no	—
39	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
40	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂²×D₄₇	—	2	2	no	—
42	C₂²×D₄₇	—	4	2	no	—
43	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#41	1	2	yes	—
48	C₂×C₄×C₄₇	#41	1	2	yes	—
49	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₄₇	—	2	2	no	—
51	C₂×Dic₄₇	—	4	2	no	—
52	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
53	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
54	C₄×D₄₇	—	2	4	no	—
55	C₄×D₄₇	—	2	2	no	—
56	C₄×D₄₇	—	2	2	no	—
57	C₈×C₄₇	—	2	4	no	—
58	C₄₇⋊C₈	—	2	4	no	—
59	C₂²×C₄₇	—	2	2	no	—
60	C₂²×C₄₇	#117	1	2	yes	—
61	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
62	C₂×D₄₇	—	2	2	no	—
63	C₂×D₄₇	—	2	4	no	—
64	C₂×D₄₇	—	4	4	no	—
65	C₄×C₄₇	—	2	4	no	—
66	C₄×C₄₇	#117	1	2	yes	—
67	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
68	C₄×C₄₇	C₂×#41	1	1	yes	characteristic
69	Dic₄₇	—	2	2	no	—
70	Dic₄₇	—	2	4	no	—
71	#2921	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	—	2	2	no	—
73	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
74	C₂×C₄₇	#568	1	2	yes	—
75	D₄₇	—	4	2	no	—
76	C₂×#117	—	47	1	no	—
77	#433	—	47	8	no	—
78	#423	—	47	1	no	—
79	#423	D₄₇	1	1	yes	characteristic, complemented
80	#407	—	47	2	no	—
81	C₄×M₁₆	—	47	2	no	—
82	C₄₇	#2921	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₄	—	47	1	no	—
84	C₂×M₁₆	—	47	2	no	—
85	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
86	#117	—	47	4	no	—
87	#117	—	47	4	no	—
88	C₂×C₄⋊C₄	—	47	2	no	—
89	C₂×C₂²⋊C₄	—	47	2	no	—
90	C₂×C₄²	—	47	1	no	—
91	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
92	C₄⋊C₈	—	47	4	no	—
93	C₄⋊C₈	C₂×D₄₇	1	4	yes	complemented
94	C₂²⋊C₈	—	47	8	no	—
95	#97	—	47	4	no	—
96	C₄×C₈	—	47	4	no	—
97	C₂⁴	—	47	1	no	—
98	C₂²×C₄	—	47	1	no	—
99	C₂²×C₄	—	47	2	no	—
100	C₂²×C₄	—	47	2	no	—
101	C₂²×C₄	—	47	2	no	—
102	C₂²×C₄	—	94	2	no	—
103	C₂²×C₄	—	94	2	no	—
104	C₂²×C₄	C₄×D₄₇	1	2	yes	—
105	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
106	M₁₆	—	2	4	no	—
107	M₁₆	—	94	4	no	—
108	C₂×C₈	—	47	4	no	—
109	C₂×C₈	C₂²×D₄₇	1	4	yes	—
110	C₄⋊C₄	—	47	8	no	—
111	C₂²⋊C₄	—	47	8	no	—
112	C₄²	—	47	2	no	—
113	C₄²	—	47	2	no	—
114	C₄²	C₄×D₄₇	1	2	yes	—
115	C₄²	C₂²×D₄₇	1	2	yes	—
116	C₂³	—	47	2	no	—
117	C₂³	—	94	2	no	—
118	C₂³	—	188	2	no	—
119	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	47	4	no	—
122	C₂×C₄	—	47	2	no	—
123	C₂×C₄	—	94	2	no	—
124	C₂×C₄	—	94	4	no	—
125	C₂×C₄	—	94	2	no	—
126	C₂×C₄	—	94	2	no	—
127	C₂×C₄	—	188	2	no	—
128	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
129	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂×C₄	#10625352	1	2	yes	—
131	C₂×C₄	#10625354	1	2	yes	—
132	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
133	C₈	—	2	4	no	—
134	C₈	—	94	4	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	94	2	no	—
137	C₂²	—	94	4	no	—
138	C₂²	—	188	4	no	—
139	C₂²	#13093982	1	2	yes	—
140	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	94	2	no	—
143	C₄	—	94	4	no	—
144	C₄	#13093982	1	2	yes	—
145	C₄	C₂×#10625352	1	1	yes	characteristic
146	C₄	C₂×#10625354	1	1	yes	characteristic
147	C₂	—	2	2	no	—
148	C₂	—	188	2	no	—
149	C₂	C₂×#13093982	1	1	yes	characteristic
150	C₂	#15689889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093981

C₂

yes

maximal, characteristic

#15689183

C₂

yes

maximal, complemented

#15689147

C₂

yes

maximal, characteristic, complemented

#15689086

C₂

yes

maximal

M₁₆×Dic₄₇

C₂

yes

maximal, complemented

C₄₇×#423

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

#13093981

C₄

yes

—

#13093981

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₄

yes

—

C₂×(C₂×D₄₇)⋊C₄

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

(C₂×D₄₇)⋊C₈

C₂²

yes

—

#13093934

C₂²

yes

complemented

C₈×Dic₄₇

C₂²

yes

complemented

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

complemented

C₂³×D₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

C₂³

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂³

yes

—

(C₂×D₄₇)⋊C₄

C₂×C₄

yes

—

Dic₄₇⋊C₄

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂³

yes

—

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#117

yes

—

C₄×C₄₇

C₂×#41

yes

characteristic

C₄×C₄₇

C₂×#41

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2921

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#117

yes

characteristic

C₂×C₄₇

#568

yes

—

D₄₇

—

—

—

—

D₄₇

yes

characteristic, complemented

#407

—

C₄×M₁₆

—

C₄₇

#2921

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

complemented

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₄×D₄₇

yes

—

105

C₂²×C₄

C₂²×D₄₇

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

C₂×C₈

—

109

C₂×C₈

C₂²×D₄₇

yes

—

110

C₄⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₄²

C₄×D₄₇

yes

—

115

C₄²

C₂²×D₄₇

yes

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

188

—

119

C₂³

C₂×C₄×D₄₇

yes

characteristic

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

188

—

128

C₂×C₄

C₂×C₄×D₄₇

yes

—

129

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

130

C₂×C₄

#10625352

yes

—

131

C₂×C₄

#10625354

yes

—

132

C₂×C₄

C₂³×D₄₇

yes

characteristic

133

C₈

—

134

C₈

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

188

—

139

C₂²

#13093982

yes

—

140

C₂²

C₂²×C₄×D₄₇

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#13093982

yes

—

145

C₄

C₂×#10625352

yes

characteristic

146

C₄

C₂×#10625354

yes

characteristic

147

C₂

—

148

C₂

—

188

—

149

C₂

C₂×#13093982

yes

characteristic

150

C₂

#15689889

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#423	C₂
hypercenter	#423	D₄₇
hypocenter	C₄₇	#2921
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#423

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
2. #15689183	C₂	semidirect
3. #15689147	C₂	semidirect
5. M₁₆×Dic₄₇	C₂	semidirect
6. C₄₇×#423	C₂	semidirect
14. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
16. #13093934	C₂²	semidirect
17. C₈×Dic₄₇	C₂²	semidirect
20. C₄⋊C₈×C₄₇	C₂²	semidirect
79. #423	D₄₇	semidirect
82. C₄₇	#2921	semidirect
93. C₄⋊C₈	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689183

C₂

semidirect

3. #15689147

C₂

semidirect

5. M₁₆×Dic₄₇

C₂

semidirect

6. C₄₇×#423

C₂

semidirect

14. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

16. #13093934

C₂²

semidirect

17. C₈×Dic₄₇

C₂²

semidirect

20. C₄⋊C₈×C₄₇

C₂²

semidirect

79. #423

D₄₇

semidirect

82. C₄₇

#2921

semidirect

93. C₄⋊C₈

C₂×D₄₇

semidirect

Subgroups — #20252025

Conjugacy classes of subgroups

Complements