Group #20243434 — subgroups

Subgroups — #20243434

Order 6016.

260 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 105 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	#15689161	C₂	1	2	yes	maximal, complemented
2	#15689097	C₂	1	2	yes	maximal
3	#15688880	C₂	1	2	yes	maximal, complemented
4	C₂×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
5	#13093983	C₂²	1	4	yes	—
6	#13093958	—	2	2	no	—
7	#13093956	—	2	2	no	—
8	#13093939	—	2	4	no	—
9	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
10	#131×C₄₇	—	2	2	no	—
11	#131×C₄₇	—	2	2	no	—
12	C₂×M₁₆×C₄₇	C₄	1	2	yes	complemented
13	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₄₇	C₄	1	2	yes	complemented
15	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
16	#41×C₄₇	—	2	2	no	—
17	#41×C₄₇	—	2	2	no	—
18	C₂×Q₈×C₄₇	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₄₇	D₄	1	2	yes	—
20	C₂×D₄×C₄₇	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
23	M₁₆×C₄₇	—	2	2	no	—
24	M₁₆×C₄₇	—	2	4	no	—
25	C₂×C₈×C₄₇	—	2	2	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
28	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	D₄	1	2	yes	—
31	C₂×C₈×C₄₇	D₄	1	2	yes	—
32	C₂²⋊Dic₄₇	—	2	4	no	—
33	C₄⋊Dic₄₇	—	2	4	no	—
34	C₄×Dic₄₇	—	2	4	no	—
35	C₂³×C₄₇	C₄⋊C₄	1	2	yes	—
36	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₄₇	—	2	2	no	—
38	D₄×C₄₇	—	2	4	no	—
39	D₄×C₄₇	—	2	2	no	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	2	2	no	—
42	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	#41	1	2	yes	—
47	C₂×Dic₄₇	—	4	4	no	—
48	C₈×C₄₇	—	2	2	no	—
49	C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₈×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₈×C₄₇	C₂²⋊C₄	1	2	yes	—
52	C₂²×C₄₇	—	2	2	no	—
53	C₂²×C₄₇	—	4	1	no	—
54	C₂²×C₄₇	—	4	2	no	—
55	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₄₇	#121	1	2	yes	—
57	C₄×C₄₇	—	2	2	no	—
58	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₄×C₄₇	#117	1	1	yes	characteristic
60	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
61	Dic₄₇	—	8	4	no	—
62	#1812	—	47	1	no	maximal, Sylow
63	C₂×C₄₇	—	2	2	no	—
64	C₂×C₄₇	—	2	1	no	—
65	C₂×C₄₇	—	4	2	no	—
66	C₂×C₄₇	#380	1	1	yes	characteristic
67	C₂×#131	D₄₇	1	1	yes	characteristic
68	#427	—	47	2	no	—
69	#410	—	47	2	no	—
70	H(3,ℤ/4)	—	47	2	no	—
71	C₄₇	#1812	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
73	#131	—	2	2	no	—
74	#131	—	2	2	no	—
75	C₂×M₁₆	Dic₄₇	1	2	yes	complemented
76	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
77	C₂²×C₈	Dic₄₇	1	2	yes	complemented
78	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
79	#117	—	47	4	no	—
80	#107	—	94	2	no	—
81	#106	—	94	2	no	—
82	#99	—	94	4	no	—
83	#41	—	2	2	no	—
84	#41	—	2	2	no	—
85	C₂×Q₈	Dic₉₄	1	1	yes	characteristic
86	C₂×D₄	Dic₉₄	1	1	yes	characteristic
87	C₂×D₄	D₁₈₈	1	2	yes	—
88	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×Dic₄₇	1	2	yes	—
95	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
96	C₂×C₈	C₂×Dic₄₇	1	1	yes	characteristic
97	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
98	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
99	C₄⋊C₄	—	94	4	no	—
100	C₂²⋊C₄	—	94	4	no	—
101	C₄²	—	94	4	no	—
102	C₂³	C₄⋊Dic₄₇	1	2	yes	—
103	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
104	Q₈	—	2	2	no	—
105	D₄	—	2	4	no	—
106	D₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	188	4	no	—
110	C₂×C₄	C₄⋊Dic₄₇	1	2	yes	—
111	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
112	C₂×C₄	#10625350	1	2	yes	—
113	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
114	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
115	C₈	—	2	2	no	—
116	C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
117	C₈	C₂²⋊Dic₄₇	1	1	yes	characteristic
118	C₈	C₂²⋊Dic₄₇	1	2	yes	—
119	C₂²	—	2	2	no	—
120	C₂²	—	4	1	no	—
121	C₂²	—	4	2	no	—
122	C₂²	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
123	C₂²	#13093998	1	2	yes	—
124	C₄	—	2	2	no	—
125	C₄	—	376	4	no	—
126	C₄	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
127	C₄	#13093983	1	1	yes	characteristic
128	C₄	#13094003	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	2	1	no	—
131	C₂	—	4	2	no	—
132	C₂	#15688983	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689161

C₂

yes

maximal, complemented

#15689097

C₂

yes

maximal

#15688880

C₂

yes

maximal, complemented

C₂×#131×C₄₇

C₂

yes

maximal, characteristic

#13093983

C₂²

yes

—

#13093958

—

#13093956

—

#13093939

—

C₂×#41×C₄₇

C₂²

yes

characteristic

#131×C₄₇

—

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₄

yes

complemented

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₄

yes

complemented

C₂²×C₈×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

Q₈

yes

characteristic

C₂×D₄×C₄₇

D₄

yes

—

C₂×D₄×C₄₇

Q₈

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

D₄

yes

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

C₂³×C₄₇

C₄⋊C₄

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₈×C₄₇

C₂²⋊C₄

yes

characteristic

C₈×C₄₇

C₂²⋊C₄

yes

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

#117

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

#1812

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#380

yes

characteristic

C₂×#131

D₄₇

yes

—

—

—

C₄₇

#1812

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

Dic₄₇

yes

complemented

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

Dic₄₇

yes

complemented

C₂²×C₈

C₂×D₄₇

yes

—

—

—

—

—

—

C₂×Q₈

Dic₉₄

yes

characteristic

C₂×D₄

Dic₉₄

yes

characteristic

C₂×D₄

D₁₈₈

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₄₇

yes

—

C₂×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₈

C₂×Dic₄₇

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

—

C₂×C₈

C₂²⋊D₄₇

yes

—

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₂³

C₄⋊Dic₄₇

yes

—

103

C₂³

C₂×D₁₈₈

yes

characteristic

104

Q₈

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

188

—

110

C₂×C₄

C₄⋊Dic₄₇

yes

—

111

C₂×C₄

C₂×Dic₉₄

yes

characteristic

112

C₂×C₄

#10625350

yes

—

113

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

114

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

115

C₈

—

116

C₈

C₂²⋊Dic₄₇

yes

characteristic

117

C₈

C₂²⋊Dic₄₇

yes

characteristic

118

C₈

C₂²⋊Dic₄₇

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

C₂×C₄⋊Dic₄₇

yes

characteristic

123

C₂²

#13093998

yes

—

124

C₄

—

125

C₄

—

376

—

126

C₄

C₂×C₂²⋊Dic₄₇

yes

characteristic

127

C₄

#13093983

yes

characteristic

128

C₄

#13094003

yes

—

129

C₂

—

130

C₂

—

131

C₂

—

132

C₂

#15688983

yes

characteristic

	subgroup	G/H
center	C₄	#13093983
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×#131×C₄₇	C₂
socle	C₂×C₄₇	#380
hypercenter	C₂×#131	D₄₇
hypocenter	C₄₇	#1812
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

subgroup

G/H

center

C₄

#13093983

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×#131×C₄₇

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ C₂×#131

N	H ≅ G/N	type
1. #15689161	C₂	semidirect
3. #15688880	C₂	semidirect
12. C₂×M₁₆×C₄₇	C₄	semidirect
14. C₂²×C₈×C₄₇	C₄	semidirect
71. C₄₇	#1812	semidirect
75. C₂×M₁₆	Dic₄₇	semidirect
77. C₂²×C₈	Dic₄₇	semidirect

H ≅ G/N

type

1. #15689161

C₂

semidirect

3. #15688880

C₂

semidirect

12. C₂×M₁₆×C₄₇

C₄

semidirect

14. C₂²×C₈×C₄₇

C₄

semidirect

71. C₄₇

#1812

semidirect

75. C₂×M₁₆

Dic₄₇

semidirect

77. C₂²×C₈

Dic₄₇

semidirect

Subgroups — #20243434

Conjugacy classes of subgroups

Complements