Group #27298812 — subgroups

Subgroups — #27298812

Order 13568.

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

normal subgroups only

This group has 330 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, complemented
2	#20276968	C₂	1	1	yes	maximal, characteristic, complemented
3	#20274320	C₂	1	2	yes	maximal, complemented
4	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2246	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	—	2	2	no	—
7	C₂×#13133161	—	2	4	no	—
8	#16811003	C₂²	1	2	yes	—
9	#16810969	—	2	4	no	—
10	C₄×#11719908	C₂²	1	1	yes	characteristic
11	C₈×#88322	C₂²	1	1	yes	characteristic
12	C₂×D₁₆×C₅₃	—	2	2	no	—
13	C₅₃×#493	C₂²	1	2	yes	—
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Dic₂₁₂	—	2	2	no	—
16	C₂×Dic₂₁₂	—	2	2	no	—
17	#13133217	—	2	2	no	—
18	Dic₄₂₄	—	4	4	no	—
19	#13133161	—	4	8	no	—
20	C₂×#11719908	D₄	1	2	yes	—
21	#13133138	—	4	8	no	—
22	C₄×#88322	D₄	1	2	yes	—
23	C₂×D₈×C₅₃	—	2	2	no	—
24	C₂×D₈×C₅₃	—	2	2	no	—
25	#127×C₅₃	—	2	2	no	—
26	D₁₆×C₅₃	—	4	4	no	—
27	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×D₄×C₅₃	—	4	2	no	—
30	C₂×D₄×C₅₃	—	4	4	no	—
31	C₂×Dic₁₀₆	—	4	2	no	—
32	D₈×C₅₃	—	4	4	no	—
33	D₈×C₅₃	—	4	4	no	—
34	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅₃	—	4	4	no	—
38	C₂×#88322	D₈	1	4	yes	—
39	Dic₂₁₂	—	4	4	no	—
40	Dic₂₁₂	—	4	4	no	—
41	C₁₆×C₅₃	D₈	1	4	yes	—
42	#11719908	D₈	1	4	yes	—
43	C₂³×C₅₃	—	8	2	no	—
44	D₄×C₅₃	—	8	2	no	—
45	D₄×C₅₃	—	8	2	no	—
46	D₄×C₅₃	—	8	8	no	—
47	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
49	C₂×Dic₅₃	—	8	2	no	—
50	Dic₁₀₆	—	8	2	no	—
51	Dic₁₀₆	—	8	2	no	—
52	C₈×C₅₃	C₂×D₈	1	2	yes	—
53	C₈×C₅₃	C₂×D₈	1	2	yes	—
54	#88322	—	2	4	no	—
55	#12518	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	16	2	no	—
57	C₂²×C₅₃	—	16	4	no	—
58	C₂²×C₅₃	#493	1	1	yes	characteristic
59	C₄×C₅₃	#493	1	1	yes	characteristic
60	C₄×C₅₃	#493	1	1	yes	characteristic
61	C₄×C₅₃	#508	1	4	yes	—
62	Dic₅₃	—	16	2	no	—
63	#2248	—	53	2	no	—
64	#2247	—	53	1	no	—
65	#2246	D₅₃	1	1	yes	characteristic
66	#1709	—	53	2	no	—
67	C₈×C₁₆	—	53	1	no	—
68	C₂×C₅₃	—	32	2	no	—
69	C₂×C₅₃	#1712	1	1	yes	characteristic
70	C₂×C₅₃	#2249	1	2	yes	—
71	C₂×Q₃₂	—	106	2	no	—
72	C₂×SD₃₂	—	106	4	no	—
73	C₂×D₁₆	—	2	2	no	—
74	#494	—	53	2	no	—
75	#493	C₂×D₅₃	1	2	yes	—
76	#486	—	106	4	no	—
77	C₄×C₁₆	—	53	1	no	—
78	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
79	C₈²	—	53	1	no	—
80	C₅₃	#12518	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	106	2	no	—
82	C₂×Q₁₆	—	106	2	no	—
83	C₂×D₈	—	2	2	no	—
84	C₂×D₈	—	2	2	no	—
85	C₄⋊Q₈	—	106	2	no	—
86	#127	—	2	2	no	—
87	Q₃₂	—	212	4	no	—
88	SD₃₂	—	212	8	no	—
89	D₁₆	—	4	4	no	—
90	C₂×C₁₆	—	53	2	no	—
91	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
92	#102	—	212	8	no	—
93	C₄×C₈	—	53	2	no	—
94	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
95	C₂×Q₈	—	212	2	no	—
96	C₂×D₄	—	4	2	no	—
97	C₂×D₄	—	4	4	no	—
98	Q₁₆	—	212	4	no	—
99	Q₁₆	—	212	4	no	—
100	D₈	—	4	4	no	—
101	D₈	—	4	4	no	—
102	C₂×C₈	—	53	4	no	—
103	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
105	C₄⋊C₄	—	212	4	no	—
106	C₄²	D₄×D₅₃	1	1	yes	characteristic
107	C₁₆	—	53	4	no	—
108	C₁₆	#11719922	1	4	yes	—
109	C₂³	—	8	2	no	—
110	Q₈	—	424	2	no	—
111	Q₈	—	424	2	no	—
112	D₄	—	8	8	no	—
113	D₄	—	8	2	no	—
114	D₄	—	8	2	no	—
115	C₂×C₄	—	424	2	no	—
116	C₂×C₄	#13133213	1	1	yes	characteristic
117	C₂×C₄	D₈×D₅₃	1	2	yes	—
118	C₈	—	106	4	no	—
119	C₈	D₈×D₅₃	1	2	yes	—
120	C₈	C₂×#11719922	1	2	yes	—
121	C₂²	—	16	2	no	—
122	C₂²	—	16	4	no	—
123	C₂²	#16810998	1	1	yes	characteristic
124	C₄	—	848	2	no	—
125	C₄	#16810998	1	1	yes	characteristic
126	C₄	#16810999	1	1	yes	characteristic
127	C₄	#16811052	1	4	yes	—
128	C₂	—	32	2	no	—
129	C₂	#20274329	1	1	yes	characteristic
130	C₂	#20276973	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276970

C₂

yes

maximal, complemented

#20276968

C₂

yes

maximal, characteristic, complemented

#20274320

C₂

yes

maximal, complemented

C₁₆×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#2246

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

—

C₂×#13133161

—

#16811003

C₂²

yes

—

#16810969

—

C₄×#11719908

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₂×D₁₆×C₅₃

—

C₅₃×#493

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

—

Dic₄₂₄

—

#13133161

—

C₂×#11719908

D₄

yes

—

#13133138

—

C₄×#88322

D₄

yes

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

#127×C₅₃

—

D₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₂×#88322

D₈

yes

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

D₈

yes

—

#11719908

D₈

yes

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂×D₈

yes

—

C₈×C₅₃

C₂×D₈

yes

—

#88322

—

#12518

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

#2248

—

#2247

—

#2246

D₅₃

yes

characteristic

#1709

—

C₈×C₁₆

—

C₂×C₅₃

—

C₂×C₅₃

#1712

yes

characteristic

C₂×C₅₃

#2249

yes

—

C₂×Q₃₂

—

106

—

C₂×SD₃₂

—

106

—

C₂×D₁₆

—

#494

—

#493

C₂×D₅₃

yes

—

#486

—

106

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

—

C₅₃

#12518

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

106

—

#127

—

Q₃₂

—

212

—

SD₃₂

—

212

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₃

yes

—

#102

—

212

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

212

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

212

—

Q₁₆

—

212

—

100

D₈

—

101

D₈

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₅₃

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

105

C₄⋊C₄

—

212

—

106

C₄²

D₄×D₅₃

yes

characteristic

107

C₁₆

—

108

C₁₆

#11719922

yes

—

109

C₂³

—

110

Q₈

—

424

—

111

Q₈

—

424

—

112

D₄

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

424

—

116

C₂×C₄

#13133213

yes

characteristic

117

C₂×C₄

D₈×D₅₃

yes

—

118

C₈

—

106

—

119

C₈

D₈×D₅₃

yes

—

120

C₈

C₂×#11719922

yes

—

121

C₂²

—

122

C₂²

—

123

C₂²

#16810998

yes

characteristic

124

C₄

—

848

—

125

C₄

#16810998

yes

characteristic

126

C₄

#16810999

yes

characteristic

127

C₄

#16811052

yes

—

128

C₂

—

129

C₂

#20274329

yes

characteristic

130

C₂

#20276973

yes

—

	subgroup	G/H
center	C₂²	#16810998
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting, prime radical	C₅₃×#2246	C₂
socle	C₂²×C₅₃	#493
hypercenter	#2246	D₅₃
hypocenter	C₅₃	#12518
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

subgroup

G/H

center

C₂²

#16810998

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting, prime radical

C₅₃×#2246

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2246

N	H ≅ G/N	type
1. #20276970	C₂	semidirect
2. #20276968	C₂	semidirect
3. #20274320	C₂	semidirect
4. C₁₆×#88322	C₂	semidirect
80. C₅₃	#12518	semidirect

H ≅ G/N

type

1. #20276970

C₂

semidirect

2. #20276968

C₂

semidirect

3. #20274320

C₂

semidirect

4. C₁₆×#88322

C₂

semidirect

80. C₅₃

#12518

semidirect

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

subgroup	index
#20276970	2
#20276968	2
#20274320	2
C₁₆×#88322	2
C₅₃×#2246	2
#12518	53

subgroup

index

#20276970

#20276968

#20274320

C₁₆×#88322

C₅₃×#2246

#12518

Subgroups — #27298812

Conjugacy classes of subgroups

Complements

Maximal subgroups