Group #20245105 — subgroups

Subgroups — #20245105

Order 6016.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×D₃₇₆	C₂	1	1	yes	maximal, characteristic
2	#15689161	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688925	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₁₈₈	—	2	1	no	—
6	C₂×D₃₇₆	—	2	1	no	—
7	C₂×D₃₇₆	—	2	2	no	—
8	C₂×D₃₇₆	C₄	1	4	yes	complemented
9	C₂×D₃₇₆	C₂²	1	2	yes	—
10	#13093983	—	2	1	no	—
11	#13093958	C₂²	1	2	yes	complemented
12	#13093956	—	2	1	no	—
13	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
14	M₃₂×C₄₇	—	2	2	no	—
15	C₂×C₁₆×C₄₇	C₂²	1	2	yes	complemented
16	C₂³×D₄₇	—	4	1	no	—
17	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
18	C₂×D₁₈₈	—	2	4	no	—
19	C₂×D₁₈₈	—	2	2	no	—
20	C₂×D₁₈₈	—	4	1	no	—
21	C₂×D₁₈₈	—	4	2	no	—
22	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
24	C₂×C₈×C₄₇	D₄	1	2	yes	—
25	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²⋊Dic₄₇	—	4	1	no	—
27	C₄⋊Dic₄₇	—	2	2	no	—
28	C₄×Dic₄₇	—	4	1	no	—
29	D₃₇₆	—	2	4	no	—
30	D₃₇₆	—	2	2	no	—
31	D₃₇₆	C₂×C₄	1	4	yes	—
32	C₁₆×C₄₇	—	2	2	no	—
33	C₂³×C₄₇	D₈	1	1	yes	characteristic
34	C₂²×D₄₇	—	4	4	no	—
35	C₂²×D₄₇	—	4	2	no	—
36	C₂²×D₄₇	—	8	4	no	—
37	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₄₇	D₈	1	1	yes	characteristic
39	C₂×C₄×C₄₇	SD₁₆	1	2	yes	—
40	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×Dic₄₇	—	4	2	no	—
42	D₁₈₈	—	2	4	no	—
43	D₁₈₈	—	4	4	no	—
44	D₁₈₈	—	4	2	no	—
45	C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
46	C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₄₇	—	2	2	no	—
49	C₂²×C₄₇	#102	1	2	yes	—
50	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
51	C₂×D₄₇	—	4	4	no	—
52	C₂×D₄₇	—	8	8	no	—
53	C₂×D₄₇	—	8	4	no	—
54	C₄×C₄₇	#102	1	2	yes	—
55	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
57	Dic₄₇	—	8	2	no	—
58	#2140	—	47	1	no	maximal, Sylow
59	C₂×C₄₇	—	2	2	no	—
60	C₂×C₄₇	C₂×#102	1	1	yes	characteristic
61	C₂×C₄₇	#509	1	2	yes	—
62	D₄₇	—	8	4	no	—
63	C₂²×D₈	—	47	1	no	—
64	C₂×M₃₂	D₄₇	1	1	yes	characteristic, complemented
65	#427	—	47	1	no	—
66	#357	—	47	4	no	—
67	C₄₇	#2140	1	1	yes	characteristic, complemented, Sylow
68	C₂²×D₄	—	94	1	no	—
69	C₂×D₈	—	47	4	no	—
70	C₂×D₈	—	47	2	no	—
71	C₂×D₈	—	94	1	no	—
72	C₂×D₈	—	94	2	no	—
73	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
74	#117	—	94	1	no	—
75	M₃₂	—	2	2	no	—
76	C₂×C₁₆	C₂×D₄₇	1	2	yes	complemented
77	#107	—	47	2	no	—
78	#106	—	94	1	no	—
79	C₂⁴	—	188	1	no	—
80	C₂×D₄	—	94	4	no	—
81	C₂×D₄	—	94	2	no	—
82	C₂×D₄	—	188	2	no	—
83	C₂×D₄	—	188	1	no	—
84	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
85	D₈	—	47	4	no	—
86	D₈	—	94	4	no	—
87	D₈	—	94	2	no	—
88	C₂×C₈	C₄×D₄₇	1	2	yes	—
89	C₂×C₈	D₁₈₈	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
91	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
92	C₄⋊C₄	—	94	2	no	—
93	C₂²⋊C₄	—	188	1	no	—
94	C₄²	—	188	1	no	—
95	C₁₆	—	2	2	no	—
96	C₂³	—	376	4	no	—
97	C₂³	—	188	4	no	—
98	C₂³	—	188	2	no	—
99	C₂³	D₃₇₆	1	1	yes	characteristic
100	D₄	—	94	4	no	—
101	D₄	—	188	4	no	—
102	D₄	—	188	2	no	—
103	C₂×C₄	—	188	2	no	—
104	C₂×C₄	#10625325	1	2	yes	—
105	C₂×C₄	D₃₇₆	1	1	yes	characteristic
106	C₂×C₄	(C₂×D₄₇)⋊C₄	1	2	yes	—
107	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
108	C₈	(C₂×D₄₇)⋊C₄	1	2	yes	—
109	C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
110	C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	376	8	no	—
113	C₂²	—	376	4	no	—
114	C₂²	—	188	4	no	—
115	C₂²	#13093946	1	2	yes	—
116	C₂²	C₂×D₃₇₆	1	1	yes	characteristic
117	C₄	—	376	2	no	—
118	C₄	#13093946	1	2	yes	—
119	C₄	C₂×(C₂×D₄₇)⋊C₄	1	1	yes	characteristic
120	C₄	C₂×#10625325	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	376	4	no	—
123	C₂	C₂×#13093946	1	1	yes	characteristic
124	C₂	#15689561	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₃₇₆

C₂

yes

maximal, characteristic

#15689161

C₂

yes

maximal, characteristic, complemented

#15688925

C₂

yes

maximal, complemented

C₂×M₃₂×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₁₈₈

—

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×D₃₇₆

C₄

yes

complemented

C₂×D₃₇₆

C₂²

yes

—

#13093983

—

#13093958

C₂²

yes

complemented

#13093956

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

M₃₂×C₄₇

—

C₂×C₁₆×C₄₇

C₂²

yes

complemented

C₂³×D₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

D₃₇₆

—

D₃₇₆

—

D₃₇₆

C₂×C₄

yes

—

C₁₆×C₄₇

—

C₂³×C₄₇

D₈

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

D₈

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

C₂²⋊C₄

yes

—

C₈×C₄₇

C₂²×C₄

yes

characteristic

C₈×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄₇

—

C₂²×C₄₇

#102

yes

—

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#102

yes

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

Dic₄₇

—

#2140

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#102

yes

characteristic

C₂×C₄₇

#509

yes

—

D₄₇

—

C₂²×D₈

—

C₂×M₃₂

D₄₇

yes

characteristic, complemented

#427

—

#357

—

C₄₇

#2140

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

#117

—

M₃₂

—

C₂×C₁₆

C₂×D₄₇

yes

complemented

#107

—

#106

—

C₂⁴

—

188

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂²×C₄

D₁₈₈

yes

characteristic

D₈

—

D₈

—

D₈

—

C₂×C₈

C₄×D₄₇

yes

—

C₂×C₈

D₁₈₈

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

—

C₂×C₈

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

188

—

C₄²

—

188

—

C₁₆

—

C₂³

—

376

—

C₂³

—

188

—

C₂³

—

188

—

C₂³

D₃₇₆

yes

characteristic

100

D₄

—

101

D₄

—

188

—

102

D₄

—

188

—

103

C₂×C₄

—

188

—

104

C₂×C₄

#10625325

yes

—

105

C₂×C₄

D₃₇₆

yes

characteristic

106

C₂×C₄

(C₂×D₄₇)⋊C₄

yes

—

107

C₂×C₄

C₂×D₁₈₈

yes

characteristic

108

C₈

(C₂×D₄₇)⋊C₄

yes

—

109

C₈

C₂×C₄×D₄₇

yes

characteristic

110

C₈

C₂×C₂²⋊D₄₇

yes

characteristic

111

C₂²

—

112

C₂²

—

376

—

113

C₂²

—

376

—

114

C₂²

—

188

—

115

C₂²

#13093946

yes

—

116

C₂²

C₂×D₃₇₆

yes

characteristic

117

C₄

—

376

—

118

C₄

#13093946

yes

—

119

C₄

C₂×(C₂×D₄₇)⋊C₄

yes

characteristic

120

C₄

C₂×#10625325

yes

characteristic

121

C₂

—

122

C₂

—

376

—

123

C₂

C₂×#13093946

yes

characteristic

124

C₂

#15689561

yes

—

	subgroup	G/H
center	C₂²	C₂×D₃₇₆
commutator	C₈×C₄₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₂×M₃₂×C₄₇	C₂
socle	C₂²×C₄₇	C₂×D₈
hypercenter	C₂×M₃₂	D₄₇
hypocenter	C₄₇	#2140
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

subgroup

G/H

center

C₂²

C₂×D₃₇₆

commutator

C₈×C₄₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₂×M₃₂×C₄₇

C₂

socle

C₂²×C₄₇

C₂×D₈

hypercenter

C₂×M₃₂

D₄₇

hypocenter

C₄₇

#2140

LCS

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×M₃₂

N	H ≅ G/N	type
2. #15689161	C₂	semidirect
3. #15688925	C₂	semidirect
4. C₂×M₃₂×C₄₇	C₂	semidirect
8. C₂×D₃₇₆	C₄	semidirect
11. #13093958	C₂²	semidirect
15. C₂×C₁₆×C₄₇	C₂²	semidirect
64. C₂×M₃₂	D₄₇	semidirect
67. C₄₇	#2140	semidirect
76. C₂×C₁₆	C₂×D₄₇	semidirect

H ≅ G/N

type

2. #15689161

C₂

semidirect

3. #15688925

C₂

semidirect

4. C₂×M₃₂×C₄₇

C₂

semidirect

8. C₂×D₃₇₆

C₄

semidirect

11. #13093958

C₂²

semidirect

15. C₂×C₁₆×C₄₇

C₂²

semidirect

64. C₂×M₃₂

D₄₇

semidirect

67. C₄₇

#2140

semidirect

76. C₂×C₁₆

C₂×D₄₇

semidirect

Subgroups — #20245105

Conjugacy classes of subgroups

Complements