Group #20246574 — subgroups

Subgroups — #20246574

Order 6016.

408 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 408 conjugacy classes of subgroups — too many to draw. Only its 213 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	#15689022	C₂	1	4	yes	maximal
2	#15689009	C₂	1	4	yes	maximal
3	#15689009	C₂	1	4	yes	maximal
4	#15688989	C₂	1	2	yes	maximal
5	C₂×#125×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
9	C₂×C₄×Dic₄₇	C₂²	1	4	yes	—
10	C₄₇⋊#95	C₂²	1	4	yes	—
11	C₄₇⋊#95	C₂²	1	4	yes	—
12	C₄₇⋊#95	C₂²	1	4	yes	—
13	C₄₇⋊#95	C₂²	1	4	yes	—
14	#125×C₄₇	C₄	1	8	yes	complemented
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₄₇	C₂²	1	4	yes	—
17	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
20	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
21	C₂²×Dic₄₇	C₂³	1	4	yes	—
22	C₂²×Dic₄₇	C₂³	1	4	yes	—
23	C₄⋊C₄×C₄₇	C₂×C₄	1	8	yes	—
24	C₄⋊C₄×C₄₇	C₂×C₄	1	16	yes	—
25	C₄²×C₄₇	C₂×C₄	1	4	yes	—
26	C₄⋊Dic₄₇	—	4	2	no	—
27	C₄⋊Dic₄₇	—	4	4	no	—
28	C₄⋊Dic₄₇	—	4	2	no	—
29	C₄×Dic₄₇	—	4	4	no	—
30	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
31	C₂×C₄×C₄₇	—	2	4	no	—
32	C₂×C₄×C₄₇	—	2	2	no	—
33	C₂×C₄×C₄₇	—	2	2	no	—
34	C₂×C₄×C₄₇	—	2	4	no	—
35	C₂×C₄×C₄₇	—	2	1	no	—
36	C₂×C₄×C₄₇	—	2	1	no	—
37	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
38	C₂×C₄×C₄₇	C₂²×C₄	1	4	yes	—
39	C₂×C₄×C₄₇	C₂²×C₄	1	8	yes	—
40	C₂×Dic₄₇	—	2	4	no	—
41	C₂×Dic₄₇	—	2	4	no	—
42	C₂×Dic₄₇	—	2	4	no	—
43	C₂×Dic₄₇	—	2	4	no	—
44	C₂×Dic₄₇	—	2	4	no	—
45	C₂×Dic₄₇	—	2	4	no	—
46	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
47	C₂²×C₄₇	H(5,𝔽₂)	1	2	yes	—
48	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	—
49	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
50	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
51	C₄×C₄₇	—	2	2	no	—
52	C₄×C₄₇	—	2	8	no	—
53	C₄×C₄₇	—	2	4	no	—
54	Dic₄₇	—	4	4	no	—
55	Dic₄₇	—	4	4	no	—
56	#2382	—	47	1	no	maximal, Sylow
57	C₂×C₄₇	#519	1	1	yes	characteristic
58	C₂×C₄₇	#520	1	2	yes	—
59	C₂×C₄₇	#562	1	1	yes	characteristic
60	C₂×C₄₇	#563	1	2	yes	—
61	C₂×C₄₇	#564	1	1	yes	characteristic
62	C₂×#125	D₄₇	1	1	yes	characteristic
63	#389	—	47	4	no	—
64	#387	—	47	4	no	—
65	#387	—	47	4	no	—
66	#382	—	47	2	no	—
67	C₄₇	#2382	1	1	yes	characteristic, complemented, Sylow
68	#125	Dic₄₇	1	8	yes	complemented
69	C₂×C₄⋊C₄	—	47	2	no	—
70	C₂×C₄⋊C₄	—	47	2	no	—
71	C₂×C₄⋊C₄	—	47	4	no	—
72	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
73	C₂×C₄⋊C₄	C₂×D₄₇	1	4	yes	—
74	C₂×C₄²	—	47	4	no	—
75	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
76	#95	—	47	4	no	—
77	#95	—	47	4	no	—
78	#95	—	47	4	no	—
79	#95	—	47	4	no	—
80	C₂²×C₄	—	47	4	no	—
81	C₂²×C₄	—	47	4	no	—
82	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
84	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
85	C₄⋊C₄	—	188	4	no	—
86	C₄⋊C₄	—	188	2	no	—
87	C₄⋊C₄	—	188	2	no	—
88	C₄⋊C₄	C₂×Dic₄₇	1	8	yes	—
89	C₄⋊C₄	C₂×Dic₄₇	1	16	yes	—
90	C₄²	—	188	4	no	—
91	C₄²	C₂×Dic₄₇	1	4	yes	—
92	C₂³	C₂³×D₄₇	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	94	4	no	—
100	C₂×C₄	—	94	4	no	—
101	C₂×C₄	—	94	4	no	—
102	C₂×C₄	—	94	4	no	—
103	C₂×C₄	—	94	4	no	—
104	C₂×C₄	—	94	4	no	—
105	C₂×C₄	C₂²×Dic₄₇	1	2	yes	—
106	C₂×C₄	C₂²×Dic₄₇	1	4	yes	—
107	C₂×C₄	C₂²×Dic₄₇	1	8	yes	—
108	C₂²	C₂³×Dic₄₇	1	1	yes	characteristic
109	C₂²	#13094070	1	2	yes	—
110	C₂²	#13094071	1	2	yes	—
111	C₂²	#13094072	1	1	yes	characteristic
112	C₂²	#13094072	1	1	yes	characteristic
113	C₄	—	2	2	no	—
114	C₄	—	2	8	no	—
115	C₄	—	2	4	no	—
116	C₄	—	188	4	no	—
117	C₄	—	188	4	no	—
118	C₂	#15689601	1	1	yes	characteristic
119	C₂	#15689606	1	2	yes	—
120	C₂	#15689871	1	1	yes	characteristic
121	C₂	#15689877	1	2	yes	—
122	C₂	C₄₇⋊#564	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689022

C₂

yes

maximal

#15689009

C₂

yes

maximal

#15689009

C₂

yes

maximal

#15688989

C₂

yes

maximal

C₂×#125×C₄₇

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

#125×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#562

yes

characteristic

C₂×C₄₇

#563

yes

—

C₂×C₄₇

#564

yes

characteristic

C₂×#125

D₄₇

yes

—

—

—

—

C₄₇

#2382

yes

characteristic, complemented, Sylow

#125

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄⋊C₄

C₂×Dic₄₇

yes

—

C₄⋊C₄

C₂×Dic₄₇

yes

—

C₄²

—

188

—

C₄²

C₂×Dic₄₇

yes

—

C₂³

C₂³×D₄₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂²×Dic₄₇

yes

—

106

C₂×C₄

C₂²×Dic₄₇

yes

—

107

C₂×C₄

C₂²×Dic₄₇

yes

—

108

C₂²

C₂³×Dic₄₇

yes

characteristic

109

C₂²

#13094070

yes

—

110

C₂²

#13094071

yes

—

111

C₂²

#13094072

yes

characteristic

112

C₂²

#13094072

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

188

—

117

C₄

—

188

—

118

C₂

#15689601

yes

characteristic

119

C₂

#15689606

yes

—

120

C₂

#15689871

yes

characteristic

121

C₂

#15689877

yes

—

122

C₂

C₄₇⋊#564

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×#125×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×#125	D₄₇
hypocenter	C₄₇	#2382
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂×#125×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
14. #125×C₄₇	C₄	semidirect
67. C₄₇	#2382	semidirect
68. #125	Dic₄₇	semidirect

H ≅ G/N

type

14. #125×C₄₇

C₄

semidirect

67. C₄₇

#2382

semidirect

68. #125

Dic₄₇

semidirect

Subgroups — #20246574

Conjugacy classes of subgroups

Complements