Group #20242744 — subgroups

Subgroups — #20242744

Order 6016.

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

normal subgroups only

This group has 204 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	#15689500	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689319	C₂	1	2	yes	maximal, complemented
3	C₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688866	C₂	1	2	yes	maximal, complemented
5	C₄₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₃₇₆	—	2	2	no	—
7	C₂×D₃₇₆	—	2	1	no	—
8	C₂×D₃₇₆	—	2	1	no	—
9	C₂×C₈×D₄₇	—	2	1	no	—
10	#13094021	C₂²	1	2	yes	—
11	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
12	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
13	#13093946	—	2	2	no	—
14	(C₂×D₄₇)⋊C₈	—	2	1	no	—
15	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×D₁₈₈	—	2	2	no	—
18	C₂×D₁₈₈	—	2	2	no	—
19	C₂×D₁₈₈	—	2	2	no	—
20	C₂×D₁₈₈	D₄	1	1	yes	characteristic
21	C₂×C₄×D₄₇	—	2	1	no	—
22	C₂×C₈×C₄₇	—	2	2	no	—
23	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₄²×C₄₇	C₂³	1	1	yes	characteristic
26	(C₂×D₄₇)⋊C₄	—	2	1	no	—
27	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
28	C₂×C₄₇⋊C₈	—	2	1	no	—
29	D₃₇₆	—	4	2	no	—
30	D₃₇₆	—	4	4	no	—
31	D₃₇₆	—	4	2	no	—
32	C₈×D₄₇	—	4	2	no	—
33	C₂²×D₄₇	—	2	1	no	—
34	C₂²×D₄₇	—	4	2	no	—
35	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
38	C₂×Dic₄₇	—	2	1	no	—
39	D₁₈₈	—	2	1	no	—
40	D₁₈₈	—	4	2	no	—
41	D₁₈₈	—	4	2	no	—
42	D₁₈₈	—	4	2	no	—
43	D₁₈₈	—	4	2	no	—
44	D₁₈₈	—	8	2	no	—
45	D₁₈₈	D₈	1	2	yes	complemented
46	C₄×D₄₇	—	4	1	no	—
47	C₄×D₄₇	—	4	1	no	—
48	C₈×C₄₇	—	2	1	no	—
49	C₈×C₄₇	—	4	2	no	—
50	C₈×C₄₇	D₈	1	2	yes	complemented
51	C₄₇⋊C₈	—	4	1	no	—
52	C₂²×C₄₇	#121	1	1	yes	characteristic
53	C₂×D₄₇	—	2	2	no	—
54	C₂×D₄₇	—	4	1	no	—
55	C₂×D₄₇	—	4	1	no	—
56	C₂×D₄₇	—	8	2	no	—
57	C₂×D₄₇	—	8	2	no	—
58	C₂×D₄₇	—	8	2	no	—
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
61	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
62	C₄×C₄₇	#135	1	1	yes	characteristic
63	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
64	Dic₄₇	—	4	1	no	—
65	#1667	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	#459	1	1	yes	characteristic
67	C₂×C₄₇	#466	1	1	yes	characteristic
68	C₂×C₄₇	#472	1	1	yes	characteristic
69	D₄₇	—	4	2	no	—
70	D₄₇	—	16	2	no	—
71	#493	—	47	1	no	—
72	#459	—	47	2	no	—
73	C₈×D₄	—	47	1	no	—
74	#335	D₄₇	1	1	yes	characteristic, complemented
75	#331	—	47	2	no	—
76	C₄₇	#1667	1	1	yes	characteristic, complemented, Sylow
77	C₂×D₈	—	94	2	no	—
78	C₂×D₈	—	94	1	no	—
79	C₂×D₈	—	94	1	no	—
80	C₂²×C₈	—	94	1	no	—
81	#127	—	47	2	no	—
82	C₄×D₄	—	47	1	no	—
83	C₄⋊C₈	—	47	1	no	—
84	C₄⋊C₈	C₂×D₄₇	1	2	yes	—
85	#102	—	94	2	no	—
86	C₂²⋊C₈	—	94	1	no	—
87	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
88	C₂×D₄	—	47	1	no	—
89	C₂×D₄	—	94	2	no	—
90	C₂×D₄	—	94	2	no	—
91	C₂×D₄	—	94	2	no	—
92	C₂²×C₄	—	94	1	no	—
93	D₈	—	188	2	no	—
94	D₈	—	188	4	no	—
95	D₈	—	188	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	94	1	no	—
98	C₂×C₈	—	188	2	no	—
99	C₂×C₈	D₁₈₈	1	1	yes	characteristic
100	C₂×C₈	D₁₈₈	1	1	yes	characteristic
101	C₄⋊C₄	—	47	1	no	—
102	C₂²⋊C₄	—	94	1	no	—
103	C₄²	C₂²×D₄₇	1	1	yes	characteristic
104	C₂³	—	94	1	no	—
105	C₂³	—	188	2	no	—
106	D₄	—	47	2	no	—
107	D₄	—	94	1	no	—
108	D₄	—	376	2	no	—
109	D₄	—	188	2	no	—
110	D₄	—	188	2	no	—
111	D₄	—	188	2	no	—
112	D₄	—	188	2	no	—
113	C₂×C₄	—	94	1	no	—
114	C₂×C₄	—	188	1	no	—
115	C₂×C₄	—	188	1	no	—
116	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
117	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
118	C₂×C₄	#10625354	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	4	2	no	—
121	C₈	—	188	1	no	—
122	C₈	D₃₇₆	1	2	yes	complemented
123	C₂²	—	94	2	no	—
124	C₂²	—	376	2	no	—
125	C₂²	—	376	2	no	—
126	C₂²	—	376	2	no	—
127	C₂²	—	188	1	no	—
128	C₂²	—	188	1	no	—
129	C₂²	#13093999	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	188	1	no	—
132	C₄	D₈×D₄₇	1	1	yes	characteristic
133	C₄	C₂×D₃₇₆	1	1	yes	characteristic
134	C₄	#13094047	1	1	yes	characteristic
135	C₄	#13094052	1	1	yes	characteristic
136	C₂	—	188	2	no	—
137	C₂	—	752	2	no	—
138	C₂	#15689319	1	1	yes	characteristic
139	C₂	#15689360	1	1	yes	characteristic
140	C₂	#15689394	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689500

C₂

yes

maximal, characteristic, complemented

#15689319

C₂

yes

maximal, complemented

C₈×D₁₈₈

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, complemented

C₄₇×#335

C₂

yes

maximal, characteristic, complemented

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×C₈×D₄₇

—

#13094021

C₂²

yes

—

C₄×D₁₈₈

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093946

—

(C₂×D₄₇)⋊C₈

—

C₄⋊C₈×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

—

C₈×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

D₈

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

D₈

yes

complemented

C₄₇⋊C₈

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×D₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

D₄₇

—

#493

—

#459

—

C₈×D₄

—

#335

D₄₇

yes

characteristic, complemented

#331

—

C₄₇

#1667

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

#127

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

188

—

D₈

—

188

—

D₈

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

D₁₈₈

yes

characteristic

100

C₂×C₈

D₁₈₈

yes

characteristic

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

C₂²×D₄₇

yes

characteristic

104

C₂³

—

105

C₂³

—

188

—

106

D₄

—

107

D₄

—

108

D₄

—

376

—

109

D₄

—

188

—

110

D₄

—

188

—

111

D₄

—

188

—

112

D₄

—

188

—

113

C₂×C₄

—

114

C₂×C₄

—

188

—

115

C₂×C₄

—

188

—

116

C₂×C₄

C₂×D₁₈₈

yes

characteristic

117

C₂×C₄

D₄×D₄₇

yes

characteristic

118

C₂×C₄

#10625354

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

188

—

122

C₈

D₃₇₆

yes

complemented

123

C₂²

—

124

C₂²

—

376

—

125

C₂²

—

376

—

126

C₂²

—

376

—

127

C₂²

—

188

—

128

C₂²

—

188

—

129

C₂²

#13093999

yes

characteristic

130

C₄

—

131

C₄

—

188

—

132

C₄

D₈×D₄₇

yes

characteristic

133

C₄

C₂×D₃₇₆

yes

characteristic

134

C₄

#13094047

yes

characteristic

135

C₄

#13094052

yes

characteristic

136

C₂

—

188

—

137

C₂

—

752

—

138

C₂

#15689319

yes

characteristic

139

C₂

#15689360

yes

characteristic

140

C₂

#15689394

yes

characteristic

	subgroup	G/H
center	C₂²	#13093999
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#335	C₂
socle	C₂²×C₄₇	#121
hypercenter	#335	D₄₇
hypocenter	C₄₇	#1667
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #335

subgroup

G/H

center

C₂²

#13093999

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#335

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #335

N	H ≅ G/N	type
1. #15689500	C₂	semidirect
2. #15689319	C₂	semidirect
3. C₈×D₁₈₈	C₂	semidirect
4. #15688866	C₂	semidirect
5. C₄₇×#335	C₂	semidirect
45. D₁₈₈	D₈	semidirect
50. C₈×C₄₇	D₈	semidirect
74. #335	D₄₇	semidirect
76. C₄₇	#1667	semidirect
122. C₈	D₃₇₆	semidirect

H ≅ G/N

type

1. #15689500

C₂

semidirect

2. #15689319

C₂

semidirect

3. C₈×D₁₈₈

C₂

semidirect

4. #15688866

C₂

semidirect

5. C₄₇×#335

C₂

semidirect

45. D₁₈₈

D₈

semidirect

50. C₈×C₄₇

D₈

semidirect

74. #335

D₄₇

semidirect

76. C₄₇

#1667

semidirect

122. C₈

D₃₇₆

semidirect

Subgroups — #20242744

Conjugacy classes of subgroups

Complements