Group #20256271 — subgroups

Subgroups — #20256271

Order 6016.

336 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 133 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	#15689853	C₂	1	2	yes	maximal
2	#15689527	C₂	1	2	yes	maximal
3	#15689417	C₂	1	4	yes	maximal
4	#15689415	C₂	1	2	yes	maximal
5	C₈×Dic₉₄	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₈₈	C₂	1	1	yes	maximal, characteristic
7	#15689217	C₂	1	2	yes	maximal
8	C₄₇×#500	C₂	1	1	yes	maximal, characteristic
9	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic
10	#13094023	—	2	2	no	—
11	#13094023	C₂²	1	4	yes	—
12	#13094022	—	2	4	no	—
13	C₄×Dic₉₄	C₂²	1	2	yes	—
14	C₄×Dic₉₄	C₂²	1	1	yes	characteristic
15	#13093989	—	2	2	no	—
16	#13093989	C₂²	1	4	yes	—
17	#13093958	C₂²	1	1	yes	characteristic
18	#13093957	C₂²	1	4	yes	—
19	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
20	Dic₄₇⋊C₈	C₂²	1	2	yes	—
21	#13093949	C₂²	1	2	yes	—
22	#13093947	C₂²	1	4	yes	—
23	C₈×Dic₄₇	C₂²	1	2	yes	—
24	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
25	#107×C₄₇	C₂²	1	2	yes	—
26	#107×C₄₇	C₂²	1	2	yes	—
27	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
28	C₂×Q₈×C₄₇	—	2	2	no	—
29	C₂×Dic₉₄	—	2	4	no	—
30	C₂×Dic₉₄	D₄	1	1	yes	characteristic
31	C₂×Dic₉₄	C₂³	1	2	yes	—
32	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
36	C₄²×C₄₇	C₂³	1	1	yes	characteristic
37	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
38	C₄⋊Dic₄₇	C₂³	1	2	yes	—
39	Dic₄₇⋊C₄	—	2	2	no	—
40	Dic₄₇⋊C₄	—	2	4	no	—
41	Dic₄₇⋊C₄	—	2	4	no	—
42	Dic₄₇⋊C₄	D₄	1	2	yes	—
43	C₄×Dic₄₇	—	2	2	no	—
44	C₄×Dic₄₇	—	2	4	no	—
45	C₄×Dic₄₇	C₂³	1	2	yes	—
46	C₂×C₄₇⋊C₈	C₂³	1	2	yes	—
47	Dic₁₈₈	Q₈	1	4	yes	complemented
48	Q₈×C₄₇	—	8	2	no	—
49	C₂×C₄×C₄₇	—	2	4	no	—
50	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	—	2	2	no	—
55	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
56	C₂×Dic₄₇	C₂×D₄	1	2	yes	—
57	Dic₉₄	—	2	2	no	—
58	Dic₉₄	—	8	4	no	—
59	Dic₉₄	Q₁₆	1	4	yes	complemented
60	Dic₉₄	C₂×Q₈	1	4	yes	—
61	C₈×C₄₇	—	2	1	no	—
62	C₈×C₄₇	C₂×Q₈	1	2	yes	—
63	C₄₇⋊C₈	—	2	2	no	—
64	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	—	4	4	no	—
67	C₄×C₄₇	C₂×Q₁₆	1	2	yes	—
68	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₄₇	—	2	4	no	—
71	Dic₄₇	—	2	2	no	—
72	Dic₄₇	—	4	2	no	—
73	Dic₄₇	C₂×Q₁₆	1	4	yes	—
74	Q₈×Q₁₆	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	D₄×Q₈	1	1	yes	characteristic
76	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
77	C₂×C₄₇	#576	1	1	yes	characteristic
78	Q₈²	—	47	2	no	—
79	#500	—	47	2	no	—
80	#500	D₄₇	1	1	yes	characteristic
81	#477	—	47	2	no	—
82	#477	—	47	4	no	—
83	C₈×Q₈	—	47	1	no	—
84	C₄×Q₁₆	—	47	2	no	—
85	C₄×Q₁₆	—	47	1	no	—
86	C₄₇	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
87	C₂×Q₁₆	—	47	1	no	—
88	C₄⋊Q₈	—	47	4	no	—
89	C₄⋊Q₈	—	94	2	no	—
90	C₄⋊Q₈	—	94	4	no	—
91	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
92	C₄×Q₈	—	47	1	no	—
93	C₄×Q₈	—	47	4	no	—
94	C₄×Q₈	—	47	2	no	—
95	C₄×Q₈	—	94	2	no	—
96	#107	—	47	1	no	—
97	#107	—	47	4	no	—
98	#107	C₂×D₄₇	1	2	yes	—
99	#107	C₂×D₄₇	1	2	yes	—
100	C₄⋊C₈	—	47	2	no	—
101	C₄⋊C₈	—	47	1	no	—
102	#103	—	47	2	no	—
103	#103	—	47	4	no	—
104	C₄×C₈	—	47	2	no	—
105	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
106	C₂×Q₈	—	2	2	no	—
107	C₂×Q₈	—	47	1	no	—
108	C₂×Q₈	—	47	2	no	—
109	C₂×Q₈	—	94	4	no	—
110	Q₁₆	—	47	4	no	—
111	C₂×C₈	—	47	2	no	—
112	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
113	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
114	C₄⋊C₄	—	2	2	no	—
115	C₄⋊C₄	—	47	2	no	—
116	C₄⋊C₄	—	47	1	no	—
117	C₄⋊C₄	—	47	2	no	—
118	C₄⋊C₄	—	94	4	no	—
119	C₄⋊C₄	—	94	4	no	—
120	C₄⋊C₄	—	94	2	no	—
121	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
122	C₄²	—	47	2	no	—
123	C₄²	—	94	4	no	—
124	C₄²	—	94	2	no	—
125	C₄²	C₂²×D₄₇	1	1	yes	characteristic
126	Q₈	—	8	2	no	—
127	Q₈	—	47	4	no	—
128	Q₈	—	47	4	no	—
129	Q₈	—	94	2	no	—
130	Q₈	—	376	4	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	47	2	no	—
133	C₂×C₄	—	47	2	no	—
134	C₂×C₄	—	94	2	no	—
135	C₂×C₄	—	94	2	no	—
136	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
137	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
138	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
139	C₈	—	2	1	no	—
140	C₈	—	94	2	no	—
141	C₈	Q₈×D₄₇	1	2	yes	—
142	C₂²	C₂×D₄×D₄₇	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	4	4	no	—
145	C₄	—	47	4	no	—
146	C₄	—	94	4	no	—
147	C₄	—	94	2	no	—
148	C₄	—	188	2	no	—
149	C₄	Q₁₆×D₄₇	1	2	yes	—
150	C₄	C₂×Q₈×D₄₇	1	1	yes	characteristic
151	C₄	#13094072	1	1	yes	characteristic
152	C₂	#15689788	1	1	yes	characteristic
153	C₂	C₂×Q₁₆×D₄₇	1	1	yes	characteristic
154	C₂	#15689924	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689853

C₂

yes

maximal

#15689527

C₂

yes

maximal

#15689417

C₂

yes

maximal

#15689415

C₂

yes

maximal

C₈×Dic₉₄

C₂

yes

maximal, characteristic

C₄×Dic₁₈₈

C₂

yes

maximal, characteristic

#15689217

C₂

yes

maximal

C₄₇×#500

C₂

yes

maximal, characteristic

C₂×Dic₁₈₈

C₂²

yes

characteristic

#13094023

—

#13094023

C₂²

yes

—

#13094022

—

C₄×Dic₉₄

C₂²

yes

—

C₄×Dic₉₄

C₂²

yes

characteristic

#13093989

—

#13093989

C₂²

yes

—

#13093958

C₂²

yes

characteristic

#13093957

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₂²

yes

—

#13093949

C₂²

yes

—

#13093947

C₂²

yes

—

C₈×Dic₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

—

#107×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

D₄

yes

characteristic

C₂×Dic₉₄

C₂³

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂³

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊Dic₄₇

D₄

yes

characteristic

C₄⋊Dic₄₇

C₂³

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

D₄

yes

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂³

yes

—

C₂×C₄₇⋊C₈

C₂³

yes

—

Dic₁₈₈

Q₈

yes

complemented

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×Dic₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

C₂×D₄

yes

—

Dic₉₄

—

Dic₉₄

—

Dic₉₄

Q₁₆

yes

complemented

Dic₉₄

C₂×Q₈

yes

—

C₈×C₄₇

—

C₈×C₄₇

C₂×Q₈

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

C₂²×Q₈

yes

characteristic

C₄×C₄₇

2₋¹⁺⁴

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

C₂×Q₁₆

yes

—

Q₈×Q₁₆

—

maximal, Sylow

C₂×C₄₇

D₄×Q₈

yes

characteristic

C₂×C₄₇

C₂²×Q₁₆

yes

characteristic

C₂×C₄₇

#576

yes

characteristic

Q₈²

—

#500

—

#500

D₄₇

yes

characteristic

#477

—

#477

—

C₈×Q₈

—

C₄×Q₁₆

—

C₄×Q₁₆

—

C₄₇

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#107

—

#107

—

#107

C₂×D₄₇

yes

—

#107

C₂×D₄₇

yes

—

100

C₄⋊C₈

—

101

C₄⋊C₈

—

102

#103

—

103

#103

—

104

C₄×C₈

—

105

C₄×C₈

C₂×D₄₇

yes

characteristic

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

Q₁₆

—

111

C₂×C₈

—

112

C₂×C₈

C₂²×D₄₇

yes

characteristic

113

C₂×C₈

C₂²×D₄₇

yes

characteristic

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

C₂²×D₄₇

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

C₂²×D₄₇

yes

characteristic

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

Q₈

—

376

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

D₄×D₄₇

yes

characteristic

137

C₂×C₄

D₄×D₄₇

yes

characteristic

138

C₂×C₄

C₂³×D₄₇

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

Q₈×D₄₇

yes

—

142

C₂²

C₂×D₄×D₄₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

188

—

149

C₄

Q₁₆×D₄₇

yes

—

150

C₄

C₂×Q₈×D₄₇

yes

characteristic

151

C₄

#13094072

yes

characteristic

152

C₂

#15689788

yes

characteristic

153

C₂

C₂×Q₁₆×D₄₇

yes

characteristic

154

C₂

#15689924

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	Q₈×Q₁₆
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

hypercenter

#500

D₄₇

hypocenter

C₄₇

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
47. Dic₁₈₈	Q₈	semidirect
59. Dic₉₄	Q₁₆	semidirect
86. C₄₇	Q₈×Q₁₆	semidirect

H ≅ G/N

type

47. Dic₁₈₈

Q₈

semidirect

59. Dic₉₄

Q₁₆

semidirect

86. C₄₇

Q₈×Q₁₆

semidirect

Subgroups — #20256271

Conjugacy classes of subgroups

Complements