Group #20283353 — subgroups

Subgroups — #20283353

Order 6784.

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

normal subgroups only

This group has 392 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	#16810684	C₂	1	8	yes	maximal, complemented
2	#16810647	C₂	1	2	yes	maximal, complemented
3	#16810587	C₂	1	2	yes	maximal
4	C₄×#11719917	C₂	1	2	yes	maximal, complemented
5	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11719917	C₂²	1	4	yes	—
7	C₄⋊#88322	C₂²	1	8	yes	complemented
8	C₂²⋊#88322	C₂²	1	8	yes	—
9	#13133128	C₂²	1	4	yes	complemented
10	C₄×#88322	C₂²	1	4	yes	complemented
11	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	#117×C₅₃	C₄	1	4	yes	—
13	#117×C₅₃	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	—	2	4	no	—
20	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
25	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
26	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
27	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
28	C₄²×C₅₃	C₂×C₄	1	2	yes	—
29	C₄²×C₅₃	C₂×C₄	1	2	yes	—
30	C₄²×C₅₃	C₂³	1	2	yes	—
31	C₄²×C₅₃	C₂³	1	2	yes	—
32	#11719917	—	2	8	no	—
33	C₂×#88322	C₂³	1	8	yes	—
34	C₂³×C₅₃	—	2	2	no	—
35	C₂³×C₅₃	—	4	2	no	—
36	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
37	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	—	2	4	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
45	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
46	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	#41	1	4	yes	—
48	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
49	#88322	—	2	8	no	—
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	—	2	4	no	—
53	C₂²×C₅₃	—	4	4	no	—
54	C₂²×C₅₃	#117	1	2	yes	—
55	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	2	4	no	—
58	C₄×C₅₃	—	2	4	no	—
59	C₄×C₅₃	#117	1	2	yes	—
60	C₄×C₅₃	C₂×#41	1	2	yes	—
61	#2921	—	53	1	no	maximal, Sylow
62	C₂×C₅₃	—	2	2	no	—
63	C₂×C₅₃	—	4	2	no	—
64	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
65	C₂×C₅₃	#568	1	2	yes	—
66	C₂×#117	D₅₃	1	1	yes	characteristic
67	#433	—	53	8	no	—
68	#423	—	53	2	no	—
69	#407	—	53	2	no	—
70	C₄×M₁₆	—	53	2	no	—
71	C₅₃	#2921	1	1	yes	characteristic, complemented, Sylow
72	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
73	C₂×M₁₆	—	53	4	no	—
74	#117	Dic₅₃	1	4	yes	—
75	#117	C₂×D₅₃	1	4	yes	—
76	C₂×C₄⋊C₄	Dic₅₃	1	2	yes	—
77	C₂×C₂²⋊C₄	Dic₅₃	1	2	yes	—
78	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
79	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
80	C₄⋊C₈	—	53	8	no	—
81	C₂²⋊C₈	—	53	8	no	—
82	#97	—	53	4	no	—
83	C₄×C₈	—	53	4	no	—
84	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂²×C₄	—	2	4	no	—
86	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
88	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
89	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
91	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
92	M₁₆	—	106	8	no	—
93	C₂×C₈	—	53	8	no	—
94	C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
95	C₂²⋊C₄	C₂×Dic₅₃	1	8	yes	—
96	C₄²	C₂×Dic₅₃	1	2	yes	—
97	C₄²	C₂×Dic₅₃	1	2	yes	—
98	C₄²	C₂²×D₅₃	1	2	yes	—
99	C₄²	C₂²×D₅₃	1	2	yes	—
100	C₂³	—	2	2	no	—
101	C₂³	—	4	2	no	—
102	C₂³	C₂²×Dic₅₃	1	2	yes	—
103	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	#11719945	1	4	yes	—
110	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
111	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
112	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
113	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
114	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
115	C₈	—	106	8	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	4	no	—
119	C₂²	—	4	4	no	—
120	C₂²	#13133176	1	2	yes	—
121	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	4	no	—
125	C₄	#13133176	1	2	yes	—
126	C₄	C₂×#11719945	1	2	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	4	2	no	—
129	C₂	C₂×#13133176	1	1	yes	characteristic
130	C₂	#16811390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810684

C₂

yes

maximal, complemented

#16810647

C₂

yes

maximal, complemented

#16810587

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, complemented

C₂×#117×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₄⋊#88322

C₂²

yes

complemented

C₂²⋊#88322

C₂²

yes

—

#13133128

C₂²

yes

complemented

C₄×#88322

C₂²

yes

complemented

C₂³×C₄×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₄

yes

—

#117×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₂²⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

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

—

C₄²×C₅₃

C₂³

yes

—

#11719917

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

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

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#117

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#117

yes

—

C₄×C₅₃

C₂×#41

yes

—

#2921

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#117

yes

characteristic

C₂×C₅₃

#568

yes

—

C₂×#117

D₅₃

yes

—

—

—

C₄×M₁₆

—

C₅₃

#2921

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

#117

Dic₅₃

yes

—

#117

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

106

—

C₂×C₈

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

100

C₂³

—

101

C₂³

—

102

C₂³

C₂²×Dic₅₃

yes

—

103

C₂³

C₂²×Dic₅₃

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

#11719945

yes

—

110

C₂×C₄

C₂²×Dic₅₃

yes

—

111

C₂×C₄

C₂²×Dic₅₃

yes

—

112

C₂×C₄

C₂²×Dic₅₃

yes

—

113

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

114

C₂×C₄

C₂³×D₅₃

yes

characteristic

115

C₈

—

106

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

#13133176

yes

—

121

C₂²

C₂³×Dic₅₃

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#13133176

yes

—

126

C₄

C₂×#11719945

yes

—

127

C₂

—

128

C₂

—

129

C₂

C₂×#13133176

yes

characteristic

130

C₂

#16811390

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₂×#117×C₅₃	C₂
hypercenter	C₂×#117	D₅₃
hypocenter	C₅₃	#2921
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#117

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₅₃

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₂×#117×C₅₃

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₄ ◁ C₂×#117

N	H ≅ G/N	type
1. #16810684	C₂	semidirect
2. #16810647	C₂	semidirect
4. C₄×#11719917	C₂	semidirect
7. C₄⋊#88322	C₂²	semidirect
9. #13133128	C₂²	semidirect
10. C₄×#88322	C₂²	semidirect
71. C₅₃	#2921	semidirect

H ≅ G/N

type

1. #16810684

C₂

semidirect

2. #16810647

C₂

semidirect

4. C₄×#11719917

C₂

semidirect

7. C₄⋊#88322

C₂²

semidirect

9. #13133128

C₂²

semidirect

10. C₄×#88322

C₂²

semidirect

71. C₅₃

#2921

semidirect

Subgroups — #20283353

Conjugacy classes of subgroups

Complements