Group #20283847 — subgroups

Subgroups — #20283847

Order 6784.

394 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 221 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	Q₈×#88322	C₂	1	2	yes	maximal, complemented
2	#16810701	C₂	1	6	yes	maximal, complemented
3	#16810676	C₂	1	3	yes	maximal, complemented
4	#16810647	C₂	1	3	yes	maximal, complemented
5	C₄×#41×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	6	yes	—
7	C₄⋊#88322	C₂²	1	3	yes	—
8	C₄⋊#88322	C₂²	1	6	yes	—
9	C₄⋊#88322	C₂²	1	1	yes	characteristic
10	C₂²⋊#88322	C₂²	1	6	yes	—
11	C₄×#88322	C₂²	1	6	yes	—
12	C₂×#41×C₅₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×C₅₃	C₄	1	1	yes	characteristic
14	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₄×D₄×C₅₃	C₄	1	3	yes	—
16	C₄×D₄×C₅₃	C₂²	1	3	yes	—
17	#117×C₅₃	C₄	1	3	yes	—
18	C₂×C₄²×C₅₃	C₂²	1	3	yes	—
19	#41×C₅₃	—	2	4	no	—
20	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×D₄×C₅₃	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	3	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	3	yes	—
25	C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
26	C₄⋊C₄×C₅₃	C₂×C₄	1	3	yes	—
27	C₂²⋊C₄×C₅₃	C₂×C₄	1	6	yes	—
28	C₄²×C₅₃	C₂×C₄	1	3	yes	—
29	C₄²×C₅₃	C₂×C₄	1	3	yes	—
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄²×C₅₃	C₂³	1	3	yes	—
32	#11719917	—	4	6	no	—
33	C₂×#88322	C₂³	1	2	yes	—
34	C₂×#88322	C₂³	1	6	yes	—
35	C₂³×C₅₃	C₂²×C₄	1	3	yes	—
36	Q₈×C₅₃	M₁₆	1	4	yes	complemented
37	D₄×C₅₃	—	2	6	no	—
38	C₂×C₄×C₅₃	—	2	6	no	—
39	C₂×C₄×C₅₃	—	2	6	no	—
40	C₂×C₄×C₅₃	—	2	3	no	—
41	C₂×C₄×C₅₃	—	2	3	no	—
42	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	3	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
48	#88322	—	2	6	no	—
49	#88322	—	2	2	no	—
50	C₂²×C₅₃	—	2	3	no	—
51	C₂²×C₅₃	—	2	3	no	—
52	C₂²×C₅₃	—	2	3	no	—
53	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₅₃	—	2	3	no	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	3	no	—
57	C₄×C₅₃	C₂×M₁₆	1	6	yes	—
58	C₄×C₅₃	#131	1	2	yes	—
59	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
60	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
61	#2991	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	4	3	no	—
63	C₂×C₅₃	#520	1	1	yes	characteristic
64	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
65	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
66	C₄×#41	D₅₃	1	1	yes	characteristic
67	C₈×Q₈	—	53	2	no	—
68	#436	—	53	6	no	—
69	#431	—	53	3	no	—
70	#423	—	53	3	no	—
71	C₅₃	#2991	1	1	yes	characteristic, complemented, Sylow
72	C₂×#41	Dic₅₃	1	1	yes	characteristic
73	C₂×M₁₆	—	53	6	no	—
74	C₄×Q₈	Dic₅₃	1	1	yes	characteristic
75	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
76	C₄×D₄	Dic₅₃	1	3	yes	—
77	C₄×D₄	C₂×D₅₃	1	3	yes	—
78	#117	Dic₅₃	1	3	yes	—
79	C₂×C₄²	C₂×D₅₃	1	3	yes	—
80	C₄⋊C₈	—	53	3	no	—
81	C₄⋊C₈	—	53	6	no	—
82	C₄⋊C₈	—	53	1	no	—
83	C₂²⋊C₈	—	53	6	no	—
84	C₄×C₈	—	53	6	no	—
85	#41	—	2	4	no	—
86	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂×D₄	C₂×Dic₅₃	1	3	yes	—
88	C₂²×C₄	C₂×Dic₅₃	1	3	yes	—
89	C₂²×C₄	C₂×Dic₅₃	1	3	yes	—
90	C₂²×C₄	C₂²×D₅₃	1	3	yes	—
91	M₁₆	—	212	6	no	—
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	53	6	no	—
94	C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
95	C₄⋊C₄	C₂×Dic₅₃	1	3	yes	—
96	C₂²⋊C₄	C₂×Dic₅₃	1	6	yes	—
97	C₄²	C₂×Dic₅₃	1	3	yes	—
98	C₄²	C₂×Dic₅₃	1	3	yes	—
99	C₄²	C₂²×D₅₃	1	3	yes	—
100	C₄²	C₂²×D₅₃	1	1	yes	characteristic
101	C₂³	C₂²×Dic₅₃	1	3	yes	—
102	Q₈	#11719917	1	4	yes	complemented
103	D₄	—	2	6	no	—
104	C₂×C₄	—	2	6	no	—
105	C₂×C₄	—	2	6	no	—
106	C₂×C₄	—	2	3	no	—
107	C₂×C₄	—	2	3	no	—
108	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₅₃	1	3	yes	—
110	C₂×C₄	C₂²×Dic₅₃	1	3	yes	—
111	C₂×C₄	C₂²×Dic₅₃	1	3	yes	—
112	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
113	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
114	C₈	—	106	6	no	—
115	C₈	—	106	2	no	—
116	C₂²	—	2	3	no	—
117	C₂²	—	2	3	no	—
118	C₂²	—	2	3	no	—
119	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	3	no	—
123	C₄	C₂×#11719917	1	6	yes	—
124	C₄	#13133224	1	2	yes	—
125	C₄	#13133263	1	1	yes	characteristic
126	C₄	#13133264	1	1	yes	characteristic
127	C₂	—	4	3	no	—
128	C₂	#16811106	1	1	yes	characteristic
129	C₂	C₂²×#11719917	1	1	yes	characteristic
130	C₂	C₂×#13133224	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#88322

C₂

yes

maximal, complemented

#16810701

C₂

yes

maximal, complemented

#16810676

C₂

yes

maximal, complemented

#16810647

C₂

yes

maximal, complemented

C₄×#41×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

C₂²⋊#88322

C₂²

yes

—

C₄×#88322

C₂²

yes

—

C₂×#41×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

C₄

yes

characteristic

C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₄

yes

—

C₄×D₄×C₅₃

C₂²

yes

—

#117×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

—

#41×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

C₂³

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

—

Q₈×C₅₃

M₁₆

yes

complemented

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

#131

yes

—

C₄×C₅₃

H(5,𝔽₂)

yes

characteristic

C₄×C₅₃

2₋¹⁺⁴

yes

characteristic

#2991

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#520

yes

characteristic

C₂×C₅₃

C₂²×M₁₆

yes

characteristic

C₂×C₅₃

C₂×#131

yes

characteristic

C₄×#41

D₅₃

yes

characteristic

C₈×Q₈

—

#436

—

#431

—

#423

—

C₅₃

#2991

yes

characteristic, complemented, Sylow

C₂×#41

Dic₅₃

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

Dic₅₃

yes

characteristic

C₄×Q₈

C₂×D₅₃

yes

characteristic

C₄×D₄

Dic₅₃

yes

—

C₄×D₄

C₂×D₅₃

yes

—

#117

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×Dic₅₃

yes

characteristic

C₂×D₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

100

C₄²

C₂²×D₅₃

yes

characteristic

101

C₂³

C₂²×Dic₅₃

yes

—

102

Q₈

#11719917

yes

complemented

103

D₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

109

C₂×C₄

C₂²×Dic₅₃

yes

—

110

C₂×C₄

C₂²×Dic₅₃

yes

—

111

C₂×C₄

C₂²×Dic₅₃

yes

—

112

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

113

C₂×C₄

C₂³×D₅₃

yes

characteristic

114

C₈

—

106

—

115

C₈

—

106

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

C₂³×Dic₅₃

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

C₂×#11719917

yes

—

124

C₄

#13133224

yes

—

125

C₄

#13133263

yes

characteristic

126

C₄

#13133264

yes

characteristic

127

C₂

—

128

C₂

#16811106

yes

characteristic

129

C₂

C₂²×#11719917

yes

characteristic

130

C₂

C₂×#13133224

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄×#41×C₅₃	C₂
hypercenter	C₄×#41	D₅₃
hypocenter	C₅₃	#2991
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₄×#41×C₅₃

C₂

hypercenter

C₄×#41

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₄×#41

N	H ≅ G/N	type
1. Q₈×#88322	C₂	semidirect
2. #16810701	C₂	semidirect
3. #16810676	C₂	semidirect
4. #16810647	C₂	semidirect
36. Q₈×C₅₃	M₁₆	semidirect
71. C₅₃	#2991	semidirect
102. Q₈	#11719917	semidirect

H ≅ G/N

type

1. Q₈×#88322

C₂

semidirect

2. #16810701

C₂

semidirect

3. #16810676

C₂

semidirect

4. #16810647

C₂

semidirect

36. Q₈×C₅₃

M₁₆

semidirect

71. C₅₃

#2991

semidirect

102. Q₈

#11719917

semidirect

Subgroups — #20283847

Conjugacy classes of subgroups

Complements