Group #20252023 — subgroups

Subgroups — #20252023

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×#13093983	C₂	1	1	yes	maximal, characteristic
2	#15689181	C₂	1	8	yes	maximal, complemented
3	#15689146	C₂	1	2	yes	maximal, complemented
4	#15689087	C₂	1	1	yes	maximal, characteristic
5	M₁₆×Dic₄₇	C₂	1	2	yes	maximal, complemented
6	C₄₇×#407	C₂	1	1	yes	maximal, characteristic
7	C₂×#10625329	C₂²	1	2	yes	—
8	#13093983	C₄	1	4	yes	—
9	#13093983	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₄₇	C₄	1	2	yes	—
11	C₂×C₂²⋊Dic₄₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₈	C₂²	1	8	yes	complemented
15	C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
16	#13093934	C₂²	1	4	yes	complemented
17	C₈×Dic₄₇	C₂²	1	4	yes	complemented
18	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂²⋊C₈×C₄₇	C₂²	1	4	yes	—
21	C₂⁴×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	—	2	4	no	—
23	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
28	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
29	M₁₆×C₄₇	—	2	4	no	—
30	C₂×C₈×C₄₇	C₂³	1	4	yes	—
31	C₂²⋊Dic₄₇	C₂×C₄	1	8	yes	—
32	C₄⋊Dic₄₇	C₂×C₄	1	8	yes	—
33	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
34	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
35	C₄×Dic₄₇	C₂³	1	2	yes	—
36	C₄×Dic₄₇	C₂³	1	2	yes	—
37	#10625329	—	2	4	no	—
38	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	—	4	2	no	—
41	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
42	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₄₇	—	2	2	no	—
44	C₂×C₄×C₄₇	—	2	4	no	—
45	C₂×C₄×C₄₇	—	2	4	no	—
46	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	#41	1	4	yes	—
49	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
50	C₂×Dic₄₇	—	4	2	no	—
51	C₂×Dic₄₇	—	4	2	no	—
52	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
53	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
54	C₈×C₄₇	—	2	4	no	—
55	C₄₇⋊C₈	—	2	4	no	—
56	C₂²×C₄₇	—	2	2	no	—
57	C₂²×C₄₇	—	2	2	no	—
58	C₂²×C₄₇	—	2	4	no	—
59	C₂²×C₄₇	—	4	4	no	—
60	C₂²×C₄₇	#117	1	2	yes	—
61	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₄₇	—	2	2	no	—
63	C₄×C₄₇	#117	1	2	yes	—
64	C₄×C₄₇	C₂×#41	1	2	yes	—
65	Dic₄₇	—	2	4	no	—
66	Dic₄₇	—	2	4	no	—
67	#2921	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	2	2	no	—
69	C₂×C₄₇	—	4	2	no	—
70	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
71	C₂×C₄₇	#568	1	2	yes	—
72	C₂×#117	—	47	1	no	—
73	#433	—	47	8	no	—
74	#423	—	47	2	no	—
75	#407	—	47	1	no	—
76	#407	D₄₇	1	1	yes	characteristic
77	C₄×M₁₆	—	47	2	no	—
78	C₄₇	#2921	1	1	yes	characteristic, complemented, Sylow
79	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
80	C₂×M₁₆	—	47	2	no	—
81	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
82	#117	—	47	4	no	—
83	#117	—	47	4	no	—
84	C₂×C₄⋊C₄	—	47	2	no	—
85	C₂×C₂²⋊C₄	—	47	2	no	—
86	C₂×C₄²	—	47	1	no	—
87	C₂×C₄²	—	47	1	no	—
88	C₄⋊C₈	—	47	8	no	—
89	C₂²⋊C₈	—	47	4	no	—
90	C₂²⋊C₈	C₂×D₄₇	1	4	yes	—
91	#97	—	47	4	no	—
92	C₄×C₈	—	47	4	no	—
93	C₂⁴	C₄×D₄₇	1	1	yes	characteristic
94	C₂²×C₄	—	2	4	no	—
95	C₂²×C₄	—	47	2	no	—
96	C₂²×C₄	—	47	2	no	—
97	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
98	C₂²×C₄	C₄×D₄₇	1	2	yes	—
99	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
100	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	M₁₆	—	2	4	no	—
102	M₁₆	—	94	4	no	—
103	C₂×C₈	—	47	4	no	—
104	C₂×C₈	C₂²×D₄₇	1	4	yes	—
105	C₄⋊C₄	—	47	8	no	—
106	C₂²⋊C₄	—	47	8	no	—
107	C₄²	—	47	2	no	—
108	C₄²	—	47	2	no	—
109	C₄²	—	47	2	no	—
110	C₄²	—	47	2	no	—
111	C₂³	—	2	2	no	—
112	C₂³	—	4	2	no	—
113	C₂³	C₂×C₄×D₄₇	1	2	yes	—
114	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	47	4	no	—
119	C₂×C₄	—	47	4	no	—
120	C₂×C₄	—	188	2	no	—
121	C₂×C₄	—	188	2	no	—
122	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
123	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
124	C₂×C₄	#10625352	1	4	yes	—
125	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
126	C₈	—	2	4	no	—
127	C₈	—	94	4	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	4	4	no	—
132	C₂²	#13093980	1	2	yes	—
133	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	94	4	no	—
136	C₄	—	94	4	no	—
137	C₄	#13093980	1	2	yes	—
138	C₄	C₂×#10625352	1	2	yes	—
139	C₂	—	2	2	no	—
140	C₂	—	4	2	no	—
141	C₂	C₂×#13093980	1	1	yes	characteristic
142	C₂	#15689889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093983

C₂

yes

maximal, characteristic

#15689181

C₂

yes

maximal, complemented

#15689146

C₂

yes

maximal, complemented

#15689087

C₂

yes

maximal, characteristic

M₁₆×Dic₄₇

C₂

yes

maximal, complemented

C₄₇×#407

C₂

yes

maximal, characteristic

C₂×#10625329

C₂²

yes

—

#13093983

C₄

yes

—

#13093983

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₄

yes

—

C₂×C₂²⋊Dic₄₇

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₂²

yes

complemented

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093934

C₂²

yes

complemented

C₈×Dic₄₇

C₂²

yes

complemented

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂²⋊C₈×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂³

yes

—

C₄×Dic₄₇

C₂³

yes

—

#10625329

—

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

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

#41

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#117

yes

—

C₄×C₄₇

C₂×#41

yes

—

Dic₄₇

—

Dic₄₇

—

#2921

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×#117

yes

characteristic

C₂×C₄₇

#568

yes

—

—

—

—

—

D₄₇

yes

characteristic

C₄×M₁₆

—

C₄₇

#2921

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₄₇

yes

—

#97

—

C₄×C₈

—

C₂⁴

C₄×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

C₂²×C₄

C₄×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

100

C₂²×C₄

C₂²×D₄₇

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

C₂×C₈

—

104

C₂×C₈

C₂²×D₄₇

yes

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₄²

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₂³

—

112

C₂³

—

113

C₂³

C₂×C₄×D₄₇

yes

—

114

C₂³

C₂×C₄×D₄₇

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

188

—

121

C₂×C₄

—

188

—

122

C₂×C₄

C₂×C₄×D₄₇

yes

—

123

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

124

C₂×C₄

#10625352

yes

—

125

C₂×C₄

C₂³×D₄₇

yes

characteristic

126

C₈

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#13093980

yes

—

133

C₂²

C₂²×C₄×D₄₇

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

#13093980

yes

—

138

C₄

C₂×#10625352

yes

—

139

C₂

—

140

C₂

—

141

C₂

C₂×#13093980

yes

characteristic

142

C₂

#15689889

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₄₇×#407	C₂
hypercenter	#407	D₄₇
hypocenter	C₄₇	#2921
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ #407

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₄₇×#407

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ #407

N	H ≅ G/N	type
2. #15689181	C₂	semidirect
3. #15689146	C₂	semidirect
5. M₁₆×Dic₄₇	C₂	semidirect
14. Dic₄₇⋊C₈	C₂²	semidirect
16. #13093934	C₂²	semidirect
17. C₈×Dic₄₇	C₂²	semidirect
78. C₄₇	#2921	semidirect

H ≅ G/N

type

2. #15689181

C₂

semidirect

3. #15689146

C₂

semidirect

5. M₁₆×Dic₄₇

C₂

semidirect

14. Dic₄₇⋊C₈

C₂²

semidirect

16. #13093934

C₂²

semidirect

17. C₈×Dic₄₇

C₂²

semidirect

78. C₄₇

#2921

semidirect

Subgroups — #20252023

Conjugacy classes of subgroups

Complements