Group #20251950 — subgroups

Subgroups — #20251950

Order 6016.

484 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 213 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×M₁₆×D₄₇	C₂	1	2	yes	maximal
2	C₂×#13093981	C₂	1	1	yes	maximal, characteristic
3	#15689148	C₂	1	8	yes	maximal, complemented
4	#15689147	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689146	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	C₂×#10625324	C₂²	1	4	yes	—
10	M₁₆×D₄₇	—	2	8	no	—
11	C₂×C₈×D₄₇	C₂²	1	4	yes	—
12	#13093981	C₄	1	4	yes	—
13	#13093981	C₂²	1	4	yes	—
14	C₂×Dic₄₇⋊C₄	C₄	1	2	yes	—
15	C₂×(C₂×D₄₇)⋊C₄	C₄	1	2	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
17	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	complemented
18	Dic₄₇⋊C₈	C₂²	1	8	yes	complemented
19	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₄₇	C₂²	1	4	yes	complemented
22	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
26	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₄₇	D₄	1	4	yes	—
29	C₂×C₄×D₄₇	Q₈	1	4	yes	—
30	C₂×C₄×D₄₇	C₂³	1	2	yes	—
31	M₁₆×C₄₇	—	2	4	no	—
32	C₂×C₈×C₄₇	C₂³	1	4	yes	—
33	C₄²×C₄₇	C₂×C₄	1	2	yes	—
34	C₄²×C₄₇	C₂³	1	2	yes	—
35	(C₂×D₄₇)⋊C₄	C₂×C₄	1	8	yes	—
36	Dic₄₇⋊C₄	C₂×C₄	1	8	yes	—
37	C₄×Dic₄₇	C₂×C₄	1	2	yes	—
38	C₄×Dic₄₇	C₂³	1	2	yes	—
39	#10625329	—	2	4	no	—
40	C₂×C₄₇⋊C₈	C₂³	1	4	yes	—
41	#10625324	—	2	8	no	—
42	C₈×D₄₇	—	2	8	no	—
43	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×D₄₇	—	4	2	no	—
45	C₂²×D₄₇	C₄⋊C₄	1	4	yes	—
46	C₂²×D₄₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₄₇	—	4	2	no	—
54	C₂×Dic₄₇	C₄⋊C₄	1	4	yes	—
55	C₂×Dic₄₇	C₂²×C₄	1	4	yes	—
56	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
57	C₄×D₄₇	C₄⋊C₄	1	8	yes	—
58	C₄×D₄₇	C₂×D₄	1	4	yes	—
59	C₄×D₄₇	C₂×Q₈	1	4	yes	—
60	C₈×C₄₇	—	2	4	no	—
61	C₄₇⋊C₈	—	2	4	no	—
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₄₇	—	2	4	no	—
66	C₂×D₄₇	—	4	4	no	—
67	C₂×D₄₇	C₂×C₄⋊C₄	1	4	yes	—
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
72	Dic₄₇	—	2	4	no	—
73	Dic₄₇	C₂×C₄⋊C₄	1	4	yes	—
74	#2905	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	—	2	2	no	—
76	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄₇	#568	1	2	yes	—
78	D₄₇	—	2	4	no	—
79	C₂²×M₁₆	—	47	2	no	—
80	C₂×#117	—	47	1	no	—
81	#424	—	47	8	no	—
82	#423	—	47	2	no	—
83	#423	—	47	1	no	—
84	#423	D₄₇	1	1	yes	characteristic, complemented
85	C₄₇	#2905	1	1	yes	characteristic, complemented, Sylow
86	C₂³×C₄	—	47	1	no	—
87	C₂×M₁₆	—	47	2	no	—
88	C₂×M₁₆	—	47	4	no	—
89	C₂×M₁₆	—	94	8	no	—
90	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
91	C₂²×C₈	—	47	4	no	—
92	#117	—	47	4	no	—
93	#117	—	47	4	no	—
94	C₂×C₄⋊C₄	—	47	2	no	—
95	C₂×C₂²⋊C₄	—	47	2	no	—
96	C₂×C₄²	—	47	1	no	—
97	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
98	C₄⋊C₈	—	47	8	no	—
99	C₄⋊C₈	—	47	4	no	—
100	C₄⋊C₈	C₂×D₄₇	1	4	yes	complemented
101	C₂⁴	—	47	1	no	—
102	C₂²×C₄	—	47	1	no	—
103	C₂²×C₄	—	47	2	no	—
104	C₂²×C₄	—	47	4	no	—
105	C₂²×C₄	—	47	4	no	—
106	C₂²×C₄	—	47	2	no	—
107	C₂²×C₄	—	47	2	no	—
108	C₂²×C₄	C₄×D₄₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	94	8	no	—
112	M₁₆	—	94	4	no	—
113	C₂×C₈	—	47	4	no	—
114	C₂×C₈	—	94	8	no	—
115	C₂×C₈	C₂²×D₄₇	1	4	yes	—
116	C₄⋊C₄	—	47	8	no	—
117	C₂²⋊C₄	—	47	8	no	—
118	C₄²	—	47	2	no	—
119	C₄²	—	47	2	no	—
120	C₄²	C₄×D₄₇	1	2	yes	—
121	C₄²	C₂²×D₄₇	1	2	yes	—
122	C₂³	—	47	2	no	—
123	C₂³	—	47	4	no	—
124	C₂³	—	188	2	no	—
125	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	47	4	no	—
128	C₂×C₄	—	47	2	no	—
129	C₂×C₄	—	47	8	no	—
130	C₂×C₄	—	47	4	no	—
131	C₂×C₄	—	47	4	no	—
132	C₂×C₄	—	47	4	no	—
133	C₂×C₄	—	188	2	no	—
134	C₂×C₄	C₂×C₄×D₄₇	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
136	C₂×C₄	D₄×D₄₇	1	2	yes	—
137	C₂×C₄	Q₈×D₄₇	1	2	yes	—
138	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	94	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	47	4	no	—
143	C₂²	—	94	4	no	—
144	C₂²	—	188	4	no	—
145	C₂²	C₄⋊C₄×D₄₇	1	2	yes	—
146	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	47	4	no	—
149	C₄	—	94	4	no	—
150	C₄	C₄⋊C₄×D₄₇	1	2	yes	—
151	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
152	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	94	4	no	—
155	C₂	C₂×C₄⋊C₄×D₄₇	1	1	yes	characteristic
156	C₂	#15689889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₄₇

C₂

yes

maximal

C₂×#13093981

C₂

yes

maximal, characteristic

#15689148

C₂

yes

maximal, complemented

#15689147

C₂

yes

maximal, characteristic, complemented

#15689146

C₂

yes

maximal, complemented

C₄₇×#423

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

C₂×#10625324

C₂²

yes

—

M₁₆×D₄₇

—

C₂×C₈×D₄₇

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

Dic₄₇⋊C₈

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

—

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

Q₈

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

—

#10625324

—

C₈×D₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

C₄⋊C₄

yes

—

C₂²×D₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₄⋊C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₄×D₄₇

C₄⋊C₄

yes

—

C₄×D₄₇

C₂×D₄

yes

—

C₄×D₄₇

C₂×Q₈

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂²×Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

C₂×C₄⋊C₄

yes

—

#2905

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

#568

yes

—

D₄₇

—

C₂²×M₁₆

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#2905

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₄₇

yes

complemented

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₄₇

yes

—

109

C₂²×C₄

C₂²×D₄₇

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²×D₄₇

yes

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₄₇

yes

—

121

C₄²

C₂²×D₄₇

yes

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

188

—

125

C₂³

C₂×C₄×D₄₇

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

188

—

134

C₂×C₄

C₂×C₄×D₄₇

yes

—

135

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

136

C₂×C₄

D₄×D₄₇

yes

—

137

C₂×C₄

Q₈×D₄₇

yes

—

138

C₂×C₄

C₂³×D₄₇

yes

characteristic

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

188

—

145

C₂²

C₄⋊C₄×D₄₇

yes

—

146

C₂²

C₂²×C₄×D₄₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₄⋊C₄×D₄₇

yes

—

151

C₄

C₂×D₄×D₄₇

yes

characteristic

152

C₄

C₂×Q₈×D₄₇

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

C₂×C₄⋊C₄×D₄₇

yes

characteristic

156

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	#2905
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
3. #15689148	C₂	semidirect
4. #15689147	C₂	semidirect
5. #15689146	C₂	semidirect
6. C₄₇×#423	C₂	semidirect
17. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
18. Dic₄₇⋊C₈	C₂²	semidirect
21. C₄⋊C₈×C₄₇	C₂²	semidirect
84. #423	D₄₇	semidirect
85. C₄₇	#2905	semidirect
100. C₄⋊C₈	C₂×D₄₇	semidirect

H ≅ G/N

type

3. #15689148

C₂

semidirect

4. #15689147

C₂

semidirect

5. #15689146

C₂

semidirect

6. C₄₇×#423

C₂

semidirect

17. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

18. Dic₄₇⋊C₈

C₂²

semidirect

21. C₄⋊C₈×C₄₇

C₂²

semidirect

84. #423

D₄₇

semidirect

85. C₄₇

#2905

semidirect

100. C₄⋊C₈

C₂×D₄₇

semidirect

Subgroups — #20251950

Conjugacy classes of subgroups

Complements