Group #20283279 — subgroups

Subgroups — #20283279

Order 6784.

484 conjugacy classes of subgroups; only the 293 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 293 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×#11719917	C₂	1	2	yes	maximal
2	#16810650	C₂	1	8	yes	maximal, complemented
3	#16810647	C₂	1	4	yes	maximal, complemented
4	C₂×#117×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	—	2	8	no	—
6	C₂×#11719917	C₂²	1	4	yes	—
7	C₂×#11719917	C₂²	1	4	yes	—
8	C₂²×#88322	C₂²	1	4	yes	—
9	C₄⋊#88322	C₂²	1	16	yes	complemented
10	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
11	#117×C₅₃	C₄	1	4	yes	—
12	#117×C₅₃	C₂²	1	4	yes	—
13	C₂×C₄⋊C₄×C₅₃	C₄	1	2	yes	—
14	C₂×C₂²⋊C₄×C₅₃	C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂²	1	2	yes	—
16	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₅₃	D₄	1	4	yes	—
21	C₂²×C₄×C₅₃	Q₈	1	4	yes	—
22	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
24	C₄⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
25	C₂²⋊C₄×C₅₃	C₂×C₄	1	8	yes	—
26	C₄²×C₅₃	C₂×C₄	1	4	yes	—
27	C₄²×C₅₃	C₂³	1	4	yes	—
28	#11719917	—	2	8	no	—
29	#11719917	—	2	8	no	—
30	C₂×#88322	—	2	8	no	—
31	C₂×#88322	C₂³	1	8	yes	—
32	C₂³×C₅₃	—	4	2	no	—
33	C₂³×C₅₃	C₄⋊C₄	1	4	yes	—
34	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
36	C₂×C₄×C₅₃	—	4	4	no	—
37	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
38	C₂×C₄×C₅₃	C₄⋊C₄	1	8	yes	—
39	C₂×C₄×C₅₃	C₂²×C₄	1	8	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×Q₈	1	4	yes	—
46	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
47	#88322	—	2	8	no	—
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	—	2	4	no	—
50	C₂²×C₅₃	—	4	4	no	—
51	C₂²×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
52	C₂²×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
53	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
54	C₄×C₅₃	—	2	8	no	—
55	C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₄×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
57	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
59	#2905	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	—	2	4	no	—
62	C₂×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₅₃	#568	1	2	yes	—
64	C₂²×M₁₆	—	53	2	no	—
65	C₂×#117	D₅₃	1	1	yes	characteristic
66	#424	—	53	8	no	—
67	#423	—	53	4	no	—
68	C₅₃	#2905	1	1	yes	characteristic, complemented, Sylow
69	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
70	C₂×M₁₆	—	53	4	no	—
71	C₂×M₁₆	—	53	4	no	—
72	C₂×M₁₆	—	106	8	no	—
73	C₂²×C₈	—	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	2	yes	—
79	C₄⋊C₈	—	53	16	no	—
80	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
81	C₂²×C₄	Dic₁₀₆	1	4	yes	—
82	C₂²×C₄	D₂₁₂	1	4	yes	—
83	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
84	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
85	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
86	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
88	M₁₆	—	106	8	no	—
89	M₁₆	—	106	8	no	—
90	C₂×C₈	—	53	8	no	—
91	C₂×C₈	—	106	8	no	—
92	C₄⋊C₄	C₂×Dic₅₃	1	8	yes	—
93	C₂²⋊C₄	C₂×Dic₅₃	1	8	yes	—
94	C₄²	C₂×Dic₅₃	1	4	yes	—
95	C₄²	C₂²×D₅₃	1	4	yes	—
96	C₂³	—	4	2	no	—
97	C₂³	C₄⋊Dic₅₃	1	4	yes	—
98	C₂³	C₂²×Dic₅₃	1	2	yes	—
99	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₂×C₄	—	4	4	no	—
101	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
102	C₂×C₄	C₄⋊Dic₅₃	1	8	yes	—
103	C₂×C₄	C₂×Dic₁₀₆	1	2	yes	—
104	C₂×C₄	C₂×Dic₁₀₆	1	4	yes	—
105	C₂×C₄	C₂×D₂₁₂	1	4	yes	—
106	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
107	C₂×C₄	C₂²×Dic₅₃	1	8	yes	—
108	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
109	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
110	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
111	C₈	—	106	8	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	2	4	no	—
114	C₂²	—	4	4	no	—
115	C₂²	C₂×C₄⋊Dic₅₃	1	4	yes	—
116	C₂²	C₂×C₄⋊Dic₅₃	1	2	yes	—
117	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
118	C₄	—	2	8	no	—
119	C₄	C₂×C₄⋊Dic₅₃	1	4	yes	—
120	C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
121	C₄	C₂²×D₂₁₂	1	1	yes	characteristic
122	C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	2	4	no	—
125	C₂	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
126	C₂	#16811390	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719917

C₂

yes

maximal

#16810650

C₂

yes

maximal, complemented

#16810647

C₂

yes

maximal, complemented

C₂×#117×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

—

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

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

—

C₂⁴×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂³

yes

—

—

—

—

C₂³

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂³×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂×C₄⋊C₄

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₄×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

C₂²×Q₈

yes

characteristic

#2905

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₅₃

#568

yes

—

C₂²×M₁₆

—

C₂×#117

D₅₃

yes

characteristic

#424

—

#423

—

C₅₃

#2905

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₅₃

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

#117

Dic₅₃

yes

—

#117

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₂²⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

—

C₄⋊C₈

—

C₂⁴

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

Dic₁₀₆

yes

—

C₂²×C₄

D₂₁₂

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

C₂×Dic₅₃

yes

—

C₂²⋊C₄

C₂×Dic₅₃

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

—

C₂³

C₄⋊Dic₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

C₄⋊Dic₅₃

yes

—

102

C₂×C₄

C₄⋊Dic₅₃

yes

—

103

C₂×C₄

C₂×Dic₁₀₆

yes

—

104

C₂×C₄

C₂×Dic₁₀₆

yes

—

105

C₂×C₄

C₂×D₂₁₂

yes

—

106

C₂×C₄

C₂×D₂₁₂

yes

—

107

C₂×C₄

C₂²×Dic₅₃

yes

—

108

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

109

C₂×C₄

C₂²×Dic₅₃

yes

—

110

C₂×C₄

C₂³×D₅₃

yes

characteristic

111

C₈

—

106

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊Dic₅₃

yes

—

116

C₂²

C₂×C₄⋊Dic₅₃

yes

—

117

C₂²

C₂³×Dic₅₃

yes

characteristic

118

C₄

—

119

C₄

C₂×C₄⋊Dic₅₃

yes

—

120

C₄

C₂×C₄⋊Dic₅₃

yes

—

121

C₄

C₂²×D₂₁₂

yes

characteristic

122

C₄

C₂²×Dic₁₀₆

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

C₂²×C₄⋊Dic₅₃

yes

characteristic

126

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	#2905
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
2. #16810650	C₂	semidirect
3. #16810647	C₂	semidirect
9. C₄⋊#88322	C₂²	semidirect
68. C₅₃	#2905	semidirect

H ≅ G/N

type

2. #16810650

C₂

semidirect

3. #16810647

C₂

semidirect

9. C₄⋊#88322

C₂²

semidirect

68. C₅₃

#2905

semidirect

Subgroups — #20283279

Conjugacy classes of subgroups

Complements