Group #20253893 — subgroups

Subgroups — #500×D₄₇ #20253893

Order 6016.

424 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 165 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⋊Q₈×D₄₇	C₂	1	2	yes	maximal
2	#15689528	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689527	C₂	1	4	yes	maximal, complemented
4	#15689526	C₂	1	2	yes	maximal, complemented
5	#107×D₄₇	C₂	1	4	yes	maximal
6	C₄×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
7	C₄₇×#500	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₄₇	—	2	2	no	—
9	C₂×C₈×D₄₇	C₂²	1	2	yes	—
10	#13094024	C₂²	1	2	yes	—
11	#13094023	—	2	2	no	—
12	#13094023	C₂²	1	4	yes	—
13	#13094022	—	2	2	no	—
14	C₄⋊C₄×D₄₇	—	2	2	no	—
15	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
16	C₄²×D₄₇	C₂²	1	1	yes	characteristic
17	#13093958	C₂²	1	4	yes	—
18	#13093957	C₂²	1	8	yes	—
19	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
20	C₈×Dic₄₇	C₂²	1	2	yes	—
21	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
22	#107×C₄₇	C₂²	1	4	yes	—
23	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₄₇	—	2	2	no	—
25	Q₈×D₄₇	—	4	8	no	—
26	C₂×C₄×D₄₇	—	2	4	no	—
27	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
28	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
29	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₉₄	—	2	2	no	—
31	C₂×Dic₉₄	—	2	4	no	—
32	C₂×C₈×C₄₇	C₂³	1	2	yes	—
33	C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
35	C₄²×C₄₇	C₂³	1	1	yes	characteristic
36	C₄⋊Dic₄₇	—	2	2	no	—
37	C₄⋊Dic₄₇	C₂³	1	4	yes	—
38	Dic₄₇⋊C₄	—	2	4	no	—
39	Dic₄₇⋊C₄	—	2	4	no	—
40	C₄×Dic₄₇	D₄	1	1	yes	characteristic
41	C₄×Dic₄₇	D₄	1	1	yes	characteristic
42	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
43	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
44	C₈×D₄₇	Q₈	1	8	yes	complemented
45	C₂²×D₄₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₄₇	—	4	4	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
51	C₂×Dic₄₇	—	2	4	no	—
52	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₄₇	C₂×D₄	1	1	yes	characteristic
55	C₄×D₄₇	—	4	8	no	—
56	C₄×D₄₇	D₈	1	4	yes	—
57	C₄×D₄₇	Q₁₆	1	4	yes	—
58	C₄×D₄₇	C₂×Q₈	1	4	yes	—
59	Dic₉₄	—	4	4	no	—
60	Dic₉₄	—	4	8	no	—
61	C₈×C₄₇	C₂×Q₈	1	4	yes	complemented
62	C₄₇⋊C₈	C₂×Q₈	1	4	yes	complemented
63	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₄₇	C₄⋊Q₈	1	2	yes	—
65	C₂×D₄₇	C₂×D₈	1	2	yes	—
66	C₂×D₄₇	C₂×Q₁₆	1	2	yes	—
67	C₄×C₄₇	—	4	4	no	—
68	C₄×C₄₇	C₂×D₈	1	2	yes	—
69	C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
70	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
71	Dic₄₇	—	4	4	no	—
72	Dic₄₇	C₄⋊Q₈	1	2	yes	—
73	Dic₄₇	C₂×D₈	1	2	yes	—
74	Dic₄₇	C₂×Q₁₆	1	2	yes	—
75	C₂×#500	—	47	1	no	maximal, Sylow
76	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
77	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
78	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
79	D₄₇	#500	1	4	yes	direct product factor
80	C₂×C₄⋊Q₈	—	47	2	no	—
81	#500	—	47	2	no	—
82	#500	—	47	4	no	—
83	#500	—	47	1	no	—
84	#500	D₄₇	1	1	yes	characteristic, direct product factor
85	C₂×#107	—	47	4	no	—
86	C₂×C₄×C₈	—	47	1	no	—
87	C₄₇	C₂×#500	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	94	2	no	—
89	C₂²×C₈	—	47	2	no	—
90	C₄⋊Q₈	—	47	4	no	—
91	C₄⋊Q₈	—	47	2	no	—
92	C₄⋊Q₈	—	94	2	no	—
93	C₄⋊Q₈	—	94	2	no	—
94	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
95	C₂×C₄⋊C₄	—	47	4	no	—
96	C₂×C₄⋊C₄	—	94	2	no	—
97	C₂×C₄²	—	47	1	no	—
98	#107	—	47	4	no	—
99	#107	—	47	8	no	—
100	#107	C₂×D₄₇	1	4	yes	—
101	C₄×C₈	—	47	2	no	—
102	C₄×C₈	—	47	1	no	—
103	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	94	4	no	—
106	C₂×Q₈	—	94	2	no	—
107	C₂×Q₈	—	188	8	no	—
108	C₂²×C₄	—	47	1	no	—
109	C₂²×C₄	—	47	1	no	—
110	C₂²×C₄	—	47	1	no	—
111	C₂²×C₄	—	94	4	no	—
112	C₂×C₈	—	47	2	no	—
113	C₂×C₈	—	47	8	no	—
114	C₂×C₈	C₂²×D₄₇	1	2	yes	—
115	C₄⋊C₄	—	2	2	no	—
116	C₄⋊C₄	—	47	4	no	—
117	C₄⋊C₄	—	94	4	no	—
118	C₄⋊C₄	—	94	4	no	—
119	C₄⋊C₄	—	94	2	no	—
120	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
121	C₄²	—	47	1	no	—
122	C₄²	—	47	1	no	—
123	C₄²	—	47	1	no	—
124	C₄²	C₂²×D₄₇	1	1	yes	characteristic
125	C₂³	—	47	1	no	—
126	Q₈	—	4	4	no	—
127	Q₈	—	188	8	no	—
128	Q₈	—	188	4	no	—
129	C₂×C₄	—	2	4	no	—
130	C₂×C₄	—	47	1	no	—
131	C₂×C₄	—	47	1	no	—
132	C₂×C₄	—	47	1	no	—
133	C₂×C₄	—	47	4	no	—
134	C₂×C₄	—	47	4	no	—
135	C₂×C₄	—	47	4	no	—
136	C₂×C₄	—	94	4	no	—
137	C₂×C₄	—	188	8	no	—
138	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
139	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
140	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
141	C₈	—	47	4	no	—
142	C₈	Q₈×D₄₇	1	4	yes	complemented
143	C₂²	—	47	2	no	—
144	C₂²	—	47	2	no	—
145	C₂²	—	47	2	no	—
146	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
147	C₄	—	4	4	no	—
148	C₄	—	47	2	no	—
149	C₄	—	47	2	no	—
150	C₄	—	47	2	no	—
151	C₄	—	188	4	no	—
152	C₄	D₈×D₄₇	1	2	yes	—
153	C₄	Q₁₆×D₄₇	1	2	yes	—
154	C₄	C₂×Q₈×D₄₇	1	2	yes	—
155	C₂	—	47	4	no	—
156	C₂	C₄⋊Q₈×D₄₇	1	1	yes	characteristic
157	C₂	C₂×D₈×D₄₇	1	1	yes	characteristic
158	C₂	C₂×Q₁₆×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊Q₈×D₄₇

C₂

yes

maximal

#15689528

C₂

yes

maximal, characteristic, complemented

#15689527

C₂

yes

maximal, complemented

#15689526

C₂

yes

maximal, complemented

#107×D₄₇

C₂

yes

maximal

C₄×C₈×D₄₇

C₂

yes

maximal, characteristic

C₄₇×#500

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₄₇

—

C₂×C₈×D₄₇

C₂²

yes

—

#13094024

C₂²

yes

—

#13094023

—

#13094023

C₂²

yes

—

#13094022

—

C₄⋊C₄×D₄₇

—

C₄⋊C₄×D₄₇

C₂²

yes

—

C₄²×D₄₇

C₂²

yes

characteristic

#13093958

C₂²

yes

—

#13093957

C₂²

yes

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₈×Dic₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

Q₈×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

C₂³

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

C₄×Dic₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

C₈×D₄₇

Q₈

yes

complemented

C₂²×D₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

C₂×D₄

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

D₈

yes

—

C₄×D₄₇

Q₁₆

yes

—

C₄×D₄₇

C₂×Q₈

yes

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

C₂×Q₈

yes

complemented

C₄₇⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂×D₄₇

C₄⋊Q₈

yes

—

C₂×D₄₇

C₂×D₈

yes

—

C₂×D₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

C₂²×Q₈

yes

—

Dic₄₇

—

Dic₄₇

C₄⋊Q₈

yes

—

Dic₄₇

C₂×D₈

yes

—

Dic₄₇

C₂×Q₁₆

yes

—

C₂×#500

—

maximal, Sylow

C₂×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₇

C₂²×D₈

yes

characteristic

C₂×C₄₇

C₂²×Q₁₆

yes

characteristic

D₄₇

#500

yes

direct product factor

C₂×C₄⋊Q₈

—

—

—

—

D₄₇

yes

characteristic, direct product factor

C₂×#107

—

C₂×C₄×C₈

—

C₄₇

C₂×#500

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

#107

—

100

#107

C₂×D₄₇

yes

—

101

C₄×C₈

—

102

C₄×C₈

—

103

C₄×C₈

C₂×D₄₇

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

188

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂²×D₄₇

yes

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

C₂²×D₄₇

yes

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²×D₄₇

yes

characteristic

125

C₂³

—

126

Q₈

—

127

Q₈

—

188

—

128

Q₈

—

188

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

188

—

138

C₂×C₄

D₄×D₄₇

yes

characteristic

139

C₂×C₄

D₄×D₄₇

yes

characteristic

140

C₂×C₄

C₂³×D₄₇

yes

characteristic

141

C₈

—

142

C₈

Q₈×D₄₇

yes

complemented

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

C₂×D₄×D₄₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

188

—

152

C₄

D₈×D₄₇

yes

—

153

C₄

Q₁₆×D₄₇

yes

—

154

C₄

C₂×Q₈×D₄₇

yes

—

155

C₂

—

156

C₂

C₄⋊Q₈×D₄₇

yes

characteristic

157

C₂

C₂×D₈×D₄₇

yes

characteristic

158

C₂

C₂×Q₁₆×D₄₇

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#500	C₂
socle	C₂²×C₄₇	C₂²×D₄
hypercenter	#500	D₄₇
hypocenter	C₄₇	C₂×#500
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

C₂×D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₄₇×#500

C₂

socle

C₂²×C₄₇

C₂²×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
2. #15689528	C₂	semidirect
3. #15689527	C₂	semidirect
4. #15689526	C₂	semidirect
7. C₄₇×#500	C₂	semidirect
44. C₈×D₄₇	Q₈	semidirect
61. C₈×C₄₇	C₂×Q₈	semidirect
62. C₄₇⋊C₈	C₂×Q₈	semidirect
79. D₄₇	#500	direct (N × H)
84. #500	D₄₇	direct (N × H)
87. C₄₇	C₂×#500	semidirect
142. C₈	Q₈×D₄₇	semidirect

H ≅ G/N

type

2. #15689528

C₂

semidirect

3. #15689527

C₂

semidirect

4. #15689526

C₂

semidirect

7. C₄₇×#500

C₂

semidirect

44. C₈×D₄₇

Q₈

semidirect

61. C₈×C₄₇

C₂×Q₈

semidirect

62. C₄₇⋊C₈

C₂×Q₈

semidirect

79. D₄₇

#500

direct (N × H)

84. #500

D₄₇

direct (N × H)

87. C₄₇

C₂×#500

semidirect

142. C₈

Q₈×D₄₇

semidirect

Subgroups — #500×D₄₇ #20253893

Conjugacy classes of subgroups

Complements

Subgroups — #500×D47 #20253893

Conjugacy classes of subgroups

Complements

Subgroups — #500×D₄₇ #20253893