Group #27298777 — subgroups

Subgroups — #27298777

Order 13568.

306 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 95 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	#20276938	C₂	1	4	yes	maximal, complemented
2	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#11719908	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1712	C₂	1	1	yes	maximal, characteristic
5	#16811028	—	2	1	no	—
6	#16811028	C₂²	1	2	yes	—
7	#16811003	—	2	1	no	—
8	#16810424	—	2	8	no	—
9	C₄×#11719908	C₂²	1	2	yes	—
10	C₅₃×#493	—	2	1	no	—
11	C₅₃×#493	—	2	1	no	—
12	C₅₃×#493	C₂²	1	2	yes	—
13	C₈²×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₂₁₂	—	4	2	no	—
15	#13133217	—	2	2	no	—
16	C₂×#11719908	D₄	1	4	yes	—
17	#13133151	—	2	4	no	—
18	#13133151	—	4	2	no	—
19	C₂×D₈×C₅₃	—	2	4	no	—
20	C₂×D₈×C₅₃	—	4	2	no	—
21	C₂×D₈×C₅₃	—	4	2	no	—
22	#127×C₅₃	—	2	2	no	—
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×D₄×C₅₃	—	4	2	no	—
27	C₂×D₄×C₅₃	—	4	2	no	—
28	C₂×D₄×C₅₃	—	4	2	no	—
29	C₂×Dic₁₀₆	—	4	2	no	—
30	D₈×C₅₃	—	4	8	no	—
31	D₈×C₅₃	—	8	4	no	—
32	D₈×C₅₃	—	8	4	no	—
33	C₂×C₈×C₅₃	D₈	1	2	yes	—
34	C₂×C₈×C₅₃	D₈	1	2	yes	—
35	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
36	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅₃	—	4	2	no	—
38	C₄⋊Dic₅₃	—	4	2	no	—
39	Dic₂₁₂	—	8	4	no	—
40	#11719908	—	2	4	no	—
41	C₂³×C₅₃	—	8	2	no	—
42	D₄×C₅₃	—	8	4	no	—
43	D₄×C₅₃	—	8	4	no	—
44	D₄×C₅₃	—	8	4	no	—
45	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₅₃	—	8	2	no	—
49	Dic₁₀₆	—	8	4	no	—
50	C₈×C₅₃	D₁₆	1	4	yes	—
51	C₈×C₅₃	SD₃₂	1	4	yes	—
52	C₈×C₅₃	#135	1	4	yes	—
53	#12509	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	16	2	no	—
55	C₂²×C₅₃	—	16	2	no	—
56	C₂²×C₅₃	—	16	2	no	—
57	C₂²×C₅₃	#493	1	1	yes	characteristic
58	C₄×C₅₃	#495	1	2	yes	—
59	C₄×C₅₃	C₂×D₁₆	1	2	yes	—
60	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
61	Dic₅₃	—	16	2	no	—
62	#2240	—	53	4	no	—
63	#1713	—	53	1	no	—
64	#1712	D₅₃	1	1	yes	characteristic
65	C₈×C₁₆	—	53	1	no	—
66	C₂×C₅₃	—	32	2	no	—
67	C₂×C₅₃	#1709	1	1	yes	characteristic
68	C₂×C₅₃	#2246	1	1	yes	characteristic
69	C₂×C₅₃	#2248	1	1	yes	characteristic
70	#500	—	53	2	no	—
71	#500	—	106	1	no	—
72	#494	—	106	1	no	—
73	#493	—	2	1	no	—
74	#493	—	2	1	no	—
75	#493	C₂×D₅₃	1	2	yes	—
76	#357	—	106	8	no	—
77	C₄×C₁₆	—	53	2	no	—
78	C₈²	C₂×D₅₃	1	1	yes	characteristic
79	C₅₃	#12509	1	1	yes	characteristic, complemented, Sylow
80	C₂×Q₁₆	—	212	2	no	—
81	C₂×D₈	—	2	4	no	—
82	C₂×D₈	—	4	2	no	—
83	C₂×D₈	—	4	2	no	—
84	C₄⋊Q₈	—	106	2	no	—
85	#127	—	2	2	no	—
86	C₂×C₁₆	—	53	4	no	—
87	#107	—	106	4	no	—
88	#107	—	212	2	no	—
89	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
90	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
91	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
92	C₂×Q₈	—	212	2	no	—
93	C₂×D₄	—	4	2	no	—
94	C₂×D₄	—	4	2	no	—
95	C₂×D₄	—	4	2	no	—
96	Q₁₆	—	424	4	no	—
97	D₈	—	4	8	no	—
98	D₈	—	8	4	no	—
99	D₈	—	8	4	no	—
100	C₂×C₈	#11719922	1	2	yes	—
101	C₂×C₈	#11719922	1	2	yes	—
102	C₂×C₈	D₄×D₅₃	1	2	yes	—
103	C₄⋊C₄	—	212	2	no	—
104	C₄⋊C₄	—	212	2	no	—
105	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
106	C₁₆	—	106	4	no	—
107	C₂³	—	8	2	no	—
108	Q₈	—	424	4	no	—
109	D₄	—	8	4	no	—
110	D₄	—	8	4	no	—
111	D₄	—	8	4	no	—
112	C₂×C₄	—	424	2	no	—
113	C₂×C₄	#13133213	1	1	yes	characteristic
114	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
115	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
116	C₈	#13133158	1	4	yes	—
117	C₈	#13133161	1	4	yes	—
118	C₈	#13133236	1	4	yes	—
119	C₂²	—	16	2	no	—
120	C₂²	—	16	2	no	—
121	C₂²	—	16	2	no	—
122	C₂²	#16810999	1	1	yes	characteristic
123	C₄	—	848	2	no	—
124	C₄	#16811006	1	2	yes	—
125	C₄	C₂×#13133158	1	2	yes	—
126	C₄	C₂×#13133161	1	2	yes	—
127	C₂	—	32	2	no	—
128	C₂	#20274320	1	1	yes	characteristic
129	C₂	#20276964	1	1	yes	characteristic
130	C₂	#20276970	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276938

C₂

yes

maximal, complemented

#20274333

C₂

yes

maximal, characteristic, complemented

C₈×#11719908

C₂

yes

maximal, characteristic, complemented

C₅₃×#1712

C₂

yes

maximal, characteristic

#16811028

—

#16811028

C₂²

yes

—

#16811003

—

#16810424

—

C₄×#11719908

C₂²

yes

—

C₅₃×#493

—

C₅₃×#493

—

C₅₃×#493

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

#13133217

—

C₂×#11719908

D₄

yes

—

#13133151

—

#13133151

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

#127×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

D₁₆

yes

—

C₈×C₅₃

SD₃₂

yes

—

C₈×C₅₃

#135

yes

—

#12509

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#493

yes

characteristic

C₄×C₅₃

#495

yes

—

C₄×C₅₃

C₂×D₁₆

yes

—

C₄×C₅₃

C₂×SD₃₂

yes

—

Dic₅₃

—

#2240

—

#1713

—

#1712

D₅₃

yes

characteristic

C₈×C₁₆

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

106

—

#494

—

106

—

#493

—

#493

—

#493

C₂×D₅₃

yes

—

#357

—

106

—

C₄×C₁₆

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12509

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

212

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

106

—

#127

—

C₂×C₁₆

—

#107

—

106

—

#107

—

212

—

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

212

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

424

—

D₈

—

D₈

—

D₈

—

100

C₂×C₈

#11719922

yes

—

101

C₂×C₈

#11719922

yes

—

102

C₂×C₈

D₄×D₅₃

yes

—

103

C₄⋊C₄

—

212

—

104

C₄⋊C₄

—

212

—

105

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

106

C₁₆

—

106

—

107

C₂³

—

108

Q₈

—

424

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

424

—

113

C₂×C₄

#13133213

yes

characteristic

114

C₂×C₄

C₂×#11719922

yes

characteristic

115

C₂×C₄

C₂×#11719922

yes

characteristic

116

C₈

#13133158

yes

—

117

C₈

#13133161

yes

—

118

C₈

#13133236

yes

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

122

C₂²

#16810999

yes

characteristic

123

C₄

—

848

—

124

C₄

#16811006

yes

—

125

C₄

C₂×#13133158

yes

—

126

C₄

C₂×#13133161

yes

—

127

C₂

—

128

C₂

#20274320

yes

characteristic

129

C₂

#20276964

yes

characteristic

130

C₂

#20276970

yes

characteristic

	subgroup	G/H
center	C₂²	#16810999
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#1712	C₂
socle	C₂²×C₅₃	#493
hypercenter	#1712	D₅₃
hypocenter	C₅₃	#12509
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#16810999

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting, prime radical

C₅₃×#1712

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #20276938	C₂	semidirect
2. #20274333	C₂	semidirect
3. C₈×#11719908	C₂	semidirect
79. C₅₃	#12509	semidirect

H ≅ G/N

type

1. #20276938

C₂

semidirect

2. #20274333

C₂

semidirect

3. C₈×#11719908

C₂

semidirect

79. C₅₃

#12509

semidirect

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

subgroup	index
#20276938	2
#20274333	2
C₈×#11719908	2
C₅₃×#1712	2
#12509	53

subgroup

index

C₅₃×#1712

Subgroups — #27298777

Conjugacy classes of subgroups

Complements

Maximal subgroups