Group #20245991 — subgroups

Subgroups — #20245991

Order 6016.

512 conjugacy classes of subgroups; only the 229 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 229 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	#15689009	C₂	1	8	yes	maximal, complemented
2	#15688986	C₂	1	2	yes	maximal
3	#15688983	C₂	1	2	yes	maximal
4	C₄×C₂²⋊Dic₄₇	C₂	1	2	yes	maximal
5	C₂²×C₄⋊C₄×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
7	C₂×C₂²⋊Dic₄₇	C₂²	1	4	yes	—
8	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
9	C₄₇⋊#95	C₂²	1	4	yes	—
10	C₄₇⋊#95	C₂²	1	4	yes	—
11	C₄₇⋊#95	C₂²	1	8	yes	—
12	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₇	C₄	1	8	yes	complemented
16	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
17	C₂⁴×C₄₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	—	2	4	no	—
19	C₂²×C₄×C₄₇	—	4	2	no	—
20	C₂²×C₄×C₄₇	—	4	2	no	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₄₇	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
27	C₂²×Dic₄₇	C₂³	1	8	yes	—
28	C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
29	C₂²⋊Dic₄₇	—	2	8	no	—
30	C₄⋊Dic₄₇	—	4	4	no	—
31	C₄×Dic₄₇	—	4	4	no	—
32	C₂³×C₄₇	—	2	2	no	—
33	C₂³×C₄₇	—	2	2	no	—
34	C₂³×C₄₇	C₂²×C₄	1	2	yes	—
35	C₂³×C₄₇	#41	1	4	yes	—
36	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
37	C₂×C₄×C₄₇	—	2	8	no	—
38	C₂×C₄×C₄₇	—	2	4	no	—
39	C₂×C₄×C₄₇	—	2	4	no	—
40	C₂×C₄×C₄₇	—	2	4	no	—
41	C₂×C₄×C₄₇	—	4	4	no	—
42	C₂×C₄×C₄₇	—	4	4	no	—
43	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
46	C₂×Dic₄₇	—	2	8	no	—
47	C₂×Dic₄₇	—	2	8	no	—
48	C₂×Dic₄₇	—	2	8	no	—
49	C₂²×C₄₇	—	2	2	no	—
50	C₂²×C₄₇	—	2	2	no	—
51	C₂²×C₄₇	—	2	8	no	—
52	C₂²×C₄₇	#117	1	4	yes	—
53	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
54	C₂²×C₄₇	C₂×#41	1	2	yes	—
55	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
56	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
57	C₄×C₄₇	—	2	4	no	—
58	C₄×C₄₇	—	2	4	no	—
59	C₄×C₄₇	—	2	4	no	—
60	Dic₄₇	—	4	8	no	—
61	#2313	—	47	1	no	maximal, Sylow
62	C₂×C₄₇	—	2	4	no	—
63	C₂×C₄₇	C₂×#117	1	1	yes	characteristic
64	C₂×C₄₇	#518	1	1	yes	characteristic
65	C₂×C₄₇	#519	1	1	yes	characteristic
66	C₂×C₄₇	#539	1	4	yes	—
67	C₂²×C₄⋊C₄	D₄₇	1	1	yes	characteristic
68	#387	—	47	8	no	—
69	#381	—	47	2	no	—
70	#380	—	47	2	no	—
71	C₄×C₂²⋊C₄	—	47	2	no	—
72	C₄₇	#2313	1	1	yes	characteristic, complemented, Sylow
73	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
74	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
75	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
76	C₂×C₄⋊C₄	—	47	4	no	—
77	C₂×C₄⋊C₄	Dic₄₇	1	8	yes	complemented
78	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
79	C₂×C₂²⋊C₄	—	47	4	no	—
80	C₂×C₄²	—	47	4	no	—
81	#95	—	47	4	no	—
82	#95	—	47	4	no	—
83	#95	—	47	8	no	—
84	C₂⁴	C₂²×D₄₇	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	2	no	—
88	C₂²×C₄	—	47	8	no	—
89	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
90	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
91	C₂²×C₄	C₂×Dic₄₇	1	4	yes	—
92	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
93	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
94	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
95	C₄⋊C₄	—	188	4	no	—
96	C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
97	C₂²⋊C₄	—	94	8	no	—
98	C₄²	—	188	4	no	—
99	C₂³	—	2	2	no	—
100	C₂³	—	2	2	no	—
101	C₂³	#10625350	1	4	yes	—
102	C₂³	C₂²×Dic₄₇	1	2	yes	—
103	C₂³	C₂³×D₄₇	1	1	yes	characteristic
104	C₂×C₄	—	2	8	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	—	4	4	no	—
110	C₂×C₄	—	94	8	no	—
111	C₂×C₄	—	94	8	no	—
112	C₂×C₄	—	94	8	no	—
113	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
114	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
115	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	8	no	—
119	C₂²	#13093983	1	4	yes	—
120	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
121	C₂²	C₂×#10625350	1	2	yes	—
122	C₂²	#13094069	1	2	yes	—
123	C₂²	#13094072	1	2	yes	—
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	—	2	4	no	—
127	C₄	—	188	8	no	—
128	C₂	—	2	4	no	—
129	C₂	C₂×#13093983	1	1	yes	characteristic
130	C₂	#15689598	1	1	yes	characteristic
131	C₂	#15689601	1	1	yes	characteristic
132	C₂	#15689720	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689009

C₂

yes

maximal, complemented

#15688986

C₂

yes

maximal

#15688983

C₂

yes

maximal

C₄×C₂²⋊Dic₄₇

C₂

yes

maximal

C₂²×C₄⋊C₄×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂⁴×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂²×C₄

yes

—

C₂³×C₄₇

#41

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#117

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

—

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2313

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂²×C₄⋊C₄

D₄₇

yes

—

—

—

C₄×C₂²⋊C₄

—

C₄₇

#2313

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

#95

—

#95

—

#95

—

C₂⁴

C₂²×D₄₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

188

—

C₄⋊C₄

C₂×Dic₄₇

yes

—

C₂²⋊C₄

—

C₄²

—

188

—

C₂³

—

100

C₂³

—

101

C₂³

#10625350

yes

—

102

C₂³

C₂²×Dic₄₇

yes

—

103

C₂³

C₂³×D₄₇

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×Dic₄₇

yes

—

114

C₂×C₄

C₂²×Dic₄₇

yes

—

115

C₂×C₄

C₂²×Dic₄₇

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#13093983

yes

—

120

C₂²

C₂³×Dic₄₇

yes

characteristic

121

C₂²

C₂×#10625350

yes

—

122

C₂²

#13094069

yes

—

123

C₂²

#13094072

yes

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

188

—

128

C₂

—

129

C₂

C₂×#13093983

yes

characteristic

130

C₂

#15689598

yes

characteristic

131

C₂

#15689601

yes

characteristic

132

C₂

#15689720

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂²×C₄⋊C₄×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂²×C₄⋊C₄	D₄₇
hypocenter	C₄₇	#2313
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂²×C₄⋊C₄×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₂²×C₄⋊C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂²×C₄⋊C₄

N	H ≅ G/N	type
1. #15689009	C₂	semidirect
15. C₂×C₄⋊C₄×C₄₇	C₄	semidirect
72. C₄₇	#2313	semidirect
77. C₂×C₄⋊C₄	Dic₄₇	semidirect

H ≅ G/N

type

1. #15689009

C₂

semidirect

15. C₂×C₄⋊C₄×C₄₇

C₄

semidirect

72. C₄₇

#2313

semidirect

77. C₂×C₄⋊C₄

Dic₄₇

semidirect

Subgroups — #20245991

Conjugacy classes of subgroups

Complements