Group #20243026 — subgroups

Subgroups — #20243026

Order 6016.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 57 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	#15689495	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689474	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689469	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689469	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689469	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#493	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625335	—	2	2	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13094024	C₂²	1	1	yes	characteristic
11	#13093949	—	2	2	no	—
12	#13093945	—	2	2	no	—
13	#13093945	—	2	2	no	—
14	#13093945	—	2	2	no	—
15	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×D₈×C₄₇	—	2	2	no	—
18	#127×C₄₇	C₂²	1	1	yes	characteristic
19	#127×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×D₄×C₄₇	—	2	1	no	—
22	C₂×D₄×C₄₇	—	2	1	no	—
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	1	no	—
26	C₂×D₄×C₄₇	—	2	1	no	—
27	C₂×Dic₉₄	—	2	1	no	—
28	C₂×Dic₉₄	—	2	1	no	—
29	D₈×C₄₇	—	4	4	no	—
30	C₂×C₈×C₄₇	D₄	1	2	yes	—
31	C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#10625335	—	4	4	no	—
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₄⋊Dic₄₇	—	2	1	no	—
36	C₄⋊Dic₄₇	—	2	1	no	—
37	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
38	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
39	C₂³×C₄₇	—	4	1	no	—
40	C₂³×C₄₇	—	4	1	no	—
41	D₄×C₄₇	—	4	2	no	—
42	D₄×C₄₇	—	4	2	no	—
43	D₄×C₄₇	—	4	2	no	—
44	D₄×C₄₇	—	4	2	no	—
45	D₄×C₄₇	—	4	2	no	—
46	D₄×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×Dic₄₇	—	4	1	no	—
51	C₂×Dic₄₇	—	4	1	no	—
52	Dic₉₄	—	4	2	no	—
53	Dic₉₄	—	4	2	no	—
54	C₈×C₄₇	—	2	2	no	—
55	C₄₇⋊C₈	—	2	2	no	—
56	C₄₇⋊C₈	—	2	2	no	—
57	C₂²×C₄₇	—	8	1	no	—
58	C₂²×C₄₇	—	8	1	no	—
59	C₂²×C₄₇	—	8	1	no	—
60	C₂²×C₄₇	—	8	1	no	—
61	C₂²×C₄₇	—	8	1	no	—
62	C₂²×C₄₇	—	8	1	no	—
63	C₂²×C₄₇	#127	1	1	yes	characteristic
64	C₄×C₄₇	#135	1	2	yes	—
65	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
66	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
67	Dic₄₇	—	8	1	no	—
68	Dic₄₇	—	8	1	no	—
69	#1717	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	—	16	1	no	—
71	C₂×C₄₇	—	16	1	no	—
72	C₂×C₄₇	#495	1	1	yes	characteristic
73	C₂×C₄₇	#496	1	1	yes	characteristic
74	C₂×C₄₇	#496	1	1	yes	characteristic
75	#493	D₄₇	1	1	yes	characteristic
76	#492	—	47	1	no	—
77	#487	—	47	1	no	—
78	#486	—	47	1	no	—
79	#486	—	47	1	no	—
80	#486	—	47	1	no	—
81	#322	—	47	1	no	—
82	C₄₇	#1717	1	1	yes	characteristic, complemented, Sylow
83	C₂×SD₁₆	—	94	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	47	1	no	—
86	C₄⋊Q₈	—	47	1	no	—
87	#127	C₂×D₄₇	1	1	yes	characteristic
88	#127	C₂×D₄₇	1	1	yes	characteristic
89	#103	—	94	2	no	—
90	#102	—	94	2	no	—
91	#102	—	94	2	no	—
92	#102	—	94	2	no	—
93	C₄×C₈	—	47	1	no	—
94	C₄×C₈	—	47	1	no	—
95	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
96	C₂×Q₈	—	94	1	no	—
97	C₂×Q₈	—	94	1	no	—
98	C₂×D₄	—	2	1	no	—
99	C₂×D₄	—	2	1	no	—
100	C₂×D₄	—	2	1	no	—
101	C₂×D₄	—	2	1	no	—
102	C₂×D₄	—	2	1	no	—
103	C₂×D₄	—	2	1	no	—
104	SD₁₆	—	188	4	no	—
105	D₈	—	4	4	no	—
106	C₂×C₈	—	47	2	no	—
107	C₂×C₈	—	47	2	no	—
108	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
109	C₄⋊C₄	—	94	1	no	—
110	C₄⋊C₄	—	94	1	no	—
111	C₄⋊C₄	—	94	1	no	—
112	C₄⋊C₄	—	94	1	no	—
113	C₄²	C₂²×D₄₇	1	1	yes	characteristic
114	C₂³	—	4	1	no	—
115	C₂³	—	4	1	no	—
116	Q₈	—	188	2	no	—
117	Q₈	—	188	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	4	2	no	—
122	D₄	—	4	2	no	—
123	D₄	—	4	2	no	—
124	C₂×C₄	—	188	1	no	—
125	C₂×C₄	—	188	1	no	—
126	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
127	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	94	2	no	—
131	C₈	—	94	2	no	—
132	C₂²	—	8	1	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	8	1	no	—
136	C₂²	—	8	1	no	—
137	C₂²	—	8	1	no	—
138	C₂²	#13094020	1	1	yes	characteristic
139	C₄	—	376	1	no	—
140	C₄	—	376	1	no	—
141	C₄	#13094043	1	2	yes	—
142	C₄	#13094048	1	2	yes	—
143	C₄	#13094050	1	2	yes	—
144	C₂	—	16	1	no	—
145	C₂	—	16	1	no	—
146	C₂	#15689506	1	1	yes	characteristic
147	C₂	#15689510	1	1	yes	characteristic
148	C₂	#15689511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689495

C₂

yes

maximal, characteristic, complemented

#15689474

C₂

yes

maximal, characteristic, complemented

#15689469

C₂

yes

maximal, characteristic, complemented

#15689469

C₂

yes

maximal, characteristic, complemented

#15689469

C₂

yes

maximal, characteristic, complemented

#15688845

C₂

yes

maximal, characteristic, complemented

C₄₇×#493

C₂

yes

maximal, characteristic

C₂×#10625335

—

#13094024

C₂²

yes

characteristic

#13094024

C₂²

yes

—

—

—

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×D₈×C₄₇

—

#127×C₄₇

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625335

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#127

yes

characteristic

C₄×C₄₇

#135

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

Dic₄₇

—

Dic₄₇

—

#1717

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

—

—

—

—

—

—

C₄₇

#1717

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

C₂×D₄₇

yes

characteristic

#127

C₂×D₄₇

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

103

C₂×D₄

—

104

SD₁₆

—

188

—

105

D₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₄₇

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂²×D₄₇

yes

characteristic

114

C₂³

—

115

C₂³

—

116

Q₈

—

188

—

117

Q₈

—

188

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

188

—

125

C₂×C₄

—

188

—

126

C₂×C₄

D₄×D₄₇

yes

characteristic

127

C₂×C₄

D₄×D₄₇

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13094020

yes

characteristic

139

C₄

—

376

—

140

C₄

—

376

—

141

C₄

#13094043

yes

—

142

C₄

#13094048

yes

—

143

C₄

#13094050

yes

—

144

C₂

—

145

C₂

—

146

C₂

#15689506

yes

characteristic

147

C₂

#15689510

yes

characteristic

148

C₂

#15689511

yes

characteristic

	subgroup	G/H
center	C₂²	#13094020
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#493	C₂
socle	C₂²×C₄₇	#127
hypercenter	#493	D₄₇
hypocenter	C₄₇	#1717
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#13094020

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#493

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #15689495	C₂	semidirect
2. #15689474	C₂	semidirect
3. #15689469	C₂	semidirect
4. #15689469	C₂	semidirect
5. #15689469	C₂	semidirect
6. #15688845	C₂	semidirect
82. C₄₇	#1717	semidirect

H ≅ G/N

type

1. #15689495

C₂

semidirect

2. #15689474

C₂

semidirect

3. #15689469

C₂

semidirect

4. #15689469

C₂

semidirect

5. #15689469

C₂

semidirect

6. #15688845

C₂

semidirect

82. C₄₇

#1717

semidirect

Subgroups — #20243026

Conjugacy classes of subgroups

Complements