Group #27298826 — subgroups

Subgroups — #27298826

Order 13568.

316 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20276970	C₂	1	2	yes	maximal
2	#20276967	C₂	1	2	yes	maximal, complemented
3	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
4	#20272632	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1709	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133163	—	2	4	no	—
7	C₂×#13133161	—	2	4	no	—
8	#16811028	—	2	2	no	—
9	#16811003	C₂²	1	2	yes	—
10	C₄×#11719908	C₂²	1	2	yes	—
11	C₅₃×#494	C₂²	1	1	yes	characteristic
12	C₅₃×#493	C₂²	1	1	yes	characteristic
13	C₅₃×#486	—	2	2	no	—
14	C₈²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₂	—	2	2	no	—
16	C₂×Dic₂₁₂	—	2	2	no	—
17	#13133217	—	2	2	no	—
18	#13133163	—	4	8	no	—
19	#13133161	—	4	8	no	—
20	C₂×#11719908	D₄	1	4	yes	—
21	#13133151	—	4	4	no	—
22	C₂×Q₁₆×C₅₃	—	2	2	no	—
23	C₂×D₈×C₅₃	—	2	2	no	—
24	C₄⋊Q₈×C₅₃	—	2	1	no	—
25	#127×C₅₃	—	2	1	no	—
26	#102×C₅₃	—	4	4	no	—
27	C₄×C₈×C₅₃	D₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	—	4	1	no	—
30	C₂×D₄×C₅₃	—	4	1	no	—
31	C₂×D₄×C₅₃	—	4	2	no	—
32	C₂×Dic₁₀₆	—	4	2	no	—
33	Q₁₆×C₅₃	—	4	4	no	—
34	D₈×C₅₃	—	4	4	no	—
35	C₂×C₈×C₅₃	D₈	1	4	yes	—
36	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
37	C₄⋊C₄×C₅₃	—	4	2	no	—
38	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₅₃	—	4	4	no	—
40	Dic₂₁₂	—	4	4	no	—
41	Dic₂₁₂	—	4	4	no	—
42	#11719908	D₈	1	8	yes	—
43	C₂³×C₅₃	—	8	1	no	—
44	Q₈×C₅₃	—	8	2	no	—
45	D₄×C₅₃	—	8	2	no	—
46	D₄×C₅₃	—	8	4	no	—
47	C₂×C₄×C₅₃	—	8	1	no	—
48	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
50	C₂×Dic₅₃	—	8	2	no	—
51	Dic₁₀₆	—	8	2	no	—
52	Dic₁₀₆	—	8	2	no	—
53	C₈×C₅₃	—	2	4	no	—
54	C₈×C₅₃	C₂×D₈	1	4	yes	—
55	#12520	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	16	1	no	—
57	C₂²×C₅₃	—	16	2	no	—
58	C₂²×C₅₃	#493	1	1	yes	characteristic
59	C₄×C₅₃	—	16	1	no	—
60	C₄×C₅₃	#493	1	2	yes	—
61	C₄×C₅₃	#510	1	4	yes	—
62	Dic₅₃	—	16	2	no	—
63	#2248	—	53	2	no	—
64	#2247	—	53	2	no	—
65	#1713	—	53	1	no	—
66	#1709	D₅₃	1	1	yes	characteristic
67	#1311	—	53	1	no	—
68	C₂×C₅₃	—	32	1	no	—
69	C₂×C₅₃	#1712	1	1	yes	characteristic
70	C₂×C₅₃	#2251	1	2	yes	—
71	C₂×Q₃₂	—	106	4	no	—
72	C₂×SD₃₂	—	106	4	no	—
73	#500	—	106	2	no	—
74	#494	—	53	2	no	—
75	#494	C₂×D₅₃	1	1	yes	characteristic
76	#493	C₂×D₅₃	1	1	yes	characteristic
77	#486	—	2	2	no	—
78	C₄×C₁₆	—	53	2	no	—
79	C₈²	C₂×D₅₃	1	1	yes	characteristic
80	C₅₃	#12520	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	106	2	no	—
83	C₂×Q₁₆	—	106	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	2	1	no	—
86	C₄⋊Q₈	—	106	2	no	—
87	#127	—	2	1	no	—
88	Q₃₂	—	212	8	no	—
89	SD₃₂	—	212	8	no	—
90	C₂×C₁₆	—	53	4	no	—
91	#107	—	212	4	no	—
92	#102	—	4	4	no	—
93	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
94	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
95	C₂×Q₈	—	4	1	no	—
96	C₂×Q₈	—	212	2	no	—
97	C₂×D₄	—	4	1	no	—
98	C₂×D₄	—	4	2	no	—
99	Q₁₆	—	4	4	no	—
100	Q₁₆	—	212	4	no	—
101	Q₁₆	—	212	4	no	—
102	D₈	—	4	4	no	—
103	C₂×C₈	#11719922	1	4	yes	—
104	C₂×C₈	D₄×D₅₃	1	2	yes	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	212	4	no	—
107	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	C₁₆	—	53	8	no	—
109	C₂³	—	8	1	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	424	2	no	—
112	Q₈	—	424	2	no	—
113	D₄	—	8	4	no	—
114	D₄	—	8	2	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	424	2	no	—
117	C₂×C₄	#13133213	1	1	yes	characteristic
118	C₂×C₄	C₂×#11719922	1	2	yes	—
119	C₈	—	2	4	no	—
120	C₈	D₈×D₅₃	1	4	yes	—
121	C₂²	—	16	1	no	—
122	C₂²	—	16	2	no	—
123	C₂²	#16810999	1	1	yes	characteristic
124	C₄	—	16	1	no	—
125	C₄	—	848	2	no	—
126	C₄	#16810998	1	2	yes	—
127	C₄	#16811063	1	4	yes	—
128	C₂	—	32	1	no	—
129	C₂	#20274329	1	1	yes	characteristic
130	C₂	#20276987	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276970

C₂

yes

maximal

#20276967

C₂

yes

maximal, complemented

#20274333

C₂

yes

maximal, characteristic, complemented

#20272632

C₂

yes

maximal, characteristic, complemented

C₅₃×#1709

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

C₄×#11719908

C₂²

yes

—

C₅₃×#494

C₂²

yes

characteristic

C₅₃×#493

C₂²

yes

characteristic

C₅₃×#486

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

—

—

—

D₄

yes

—

#13133151

—

C₂×Q₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#127×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

#11719908

D₈

yes

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×D₈

yes

—

#12520

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#493

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#493

yes

—

C₄×C₅₃

#510

yes

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic

#1311

—

C₂×C₅₃

—

C₂×C₅₃

#1712

yes

characteristic

C₂×C₅₃

#2251

yes

—

C₂×Q₃₂

—

106

—

C₂×SD₃₂

—

106

—

#500

—

106

—

#494

—

#494

C₂×D₅₃

yes

characteristic

#493

C₂×D₅₃

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12520

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×D₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#127

—

Q₃₂

—

212

—

SD₃₂

—

212

—

C₂×C₁₆

—

#107

—

212

—

#102

—

C₄×C₈

C₂²⋊D₅₃

yes

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

100

Q₁₆

—

212

—

101

Q₁₆

—

212

—

102

D₈

—

103

C₂×C₈

#11719922

yes

—

104

C₂×C₈

D₄×D₅₃

yes

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

212

—

107

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

108

C₁₆

—

109

C₂³

—

110

Q₈

—

111

Q₈

—

424

—

112

Q₈

—

424

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

424

—

117

C₂×C₄

#13133213

yes

characteristic

118

C₂×C₄

C₂×#11719922

yes

—

119

C₈

—

120

C₈

D₈×D₅₃

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#16810999

yes

characteristic

124

C₄

—

125

C₄

—

848

—

126

C₄

#16810998

yes

—

127

C₄

#16811063

yes

—

128

C₂

—

129

C₂

#20274329

yes

characteristic

130

C₂

#20276987

yes

—

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1709	C₂
socle	C₂²×C₅₃	#493
prime radical	C₅₃×#493	C₂²
hypercenter	#1709	D₅₃
hypocenter	C₅₃	#12520
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1709

C₂

socle

C₂²×C₅₃

#493

prime radical

C₅₃×#493

C₂²

D₅₃

C₅₃

G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
2. #20276967	C₂	semidirect
3. #20274333	C₂	semidirect
4. #20272632	C₂	semidirect
80. C₅₃	#12520	semidirect

H ≅ G/N

type

2. #20276967

C₂

semidirect

3. #20274333

C₂

semidirect

4. #20272632

C₂

semidirect

80. C₅₃

#12520

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20276970	2
#20276967	2
#20274333	2
#20272632	2
C₅₃×#1709	2
#12520	53

subgroup

index

C₅₃×#1709

Subgroups — #27298826

Conjugacy classes of subgroups

Complements

Maximal subgroups