Group #27270749 — subgroups

Subgroups — #27270749

Order 13568.

140 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 57 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	C₈×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	#20272633	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1367	C₂	1	1	yes	maximal, characteristic
4	C₈×Dic₁₀₆	—	2	1	no	—
5	C₄×Dic₂₁₂	C₄	1	1	yes	characteristic
6	C₄×#11719908	—	2	1	no	—
7	#16810376	C₄	1	1	yes	characteristic
8	#16810355	—	2	1	no	—
9	C₅₃×C₄⋊C₁₆	—	2	1	no	—
10	C₈²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×Dic₂₁₂	C₈	1	1	yes	characteristic
12	C₄×Dic₁₀₆	—	2	1	no	—
13	C₂×#11719908	—	2	2	no	—
14	#13133151	C₈	1	1	yes	characteristic
15	C₄⋊#88322	—	2	1	no	—
16	#13133146	—	4	1	no	—
17	#13133142	—	2	1	no	—
18	C₈×Dic₅₃	—	4	1	no	—
19	C₂×C₁₆×C₅₃	—	4	1	no	—
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₂×Dic₁₀₆	—	2	1	no	—
24	C₂×C₈×C₅₃	—	2	1	no	—
25	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
26	C₂×C₈×C₅₃	M₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	SD₁₆	1	1	yes	characteristic
29	C₄²×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₄⋊Dic₅₃	—	2	1	no	—
31	#11719919	—	4	1	no	—
32	C₄×Dic₅₃	—	4	1	no	—
33	C₂×#88322	—	4	1	no	—
34	Dic₂₁₂	—	2	1	no	—
35	Dic₂₁₂	C₁₆	1	2	yes	complemented
36	C₁₆×C₅₃	—	8	1	no	—
37	#11719908	—	4	2	no	—
38	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
40	C₂×C₄×C₅₃	#104	1	1	yes	characteristic
41	C₂×Dic₅₃	—	4	1	no	—
42	C₂×Dic₅₃	—	4	1	no	—
43	Dic₁₀₆	—	2	2	no	—
44	Dic₁₀₆	—	4	1	no	—
45	C₈×C₅₃	—	2	2	no	—
46	C₈×C₅₃	—	2	1	no	—
47	C₈×C₅₃	—	2	1	no	—
48	C₈×C₅₃	—	2	1	no	—
49	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
50	C₈×C₅₃	M₃₂	1	1	yes	characteristic
51	#88322	—	8	1	no	—
52	#7071	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
56	C₄×C₅₃	#350	1	1	yes	characteristic
57	C₄×C₅₃	#360	1	1	yes	characteristic
58	C₄×C₅₃	#362	1	1	yes	characteristic
59	Dic₅₃	—	4	2	no	—
60	Dic₅₃	—	8	1	no	—
61	C₈×Q₁₆	—	53	1	no	—
62	#1367	D₅₃	1	1	yes	characteristic
63	#1312	—	53	1	no	—
64	C₂×C₅₃	D₄⋊C₁₆	1	1	yes	characteristic
65	C₂×C₅₃	#1334	1	1	yes	characteristic
66	C₂×C₅₃	#1336	1	1	yes	characteristic
67	C₈×Q₈	—	106	1	no	—
68	C₄×Q₁₆	—	53	1	no	—
69	C₄⋊C₁₆	—	2	1	no	—
70	C₄×C₁₆	—	106	1	no	—
71	#335	—	53	1	no	—
72	Q₈⋊C₈	—	106	1	no	—
73	C₈²	C₂×D₅₃	1	1	yes	characteristic
74	C₅₃	#7071	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	53	1	no	—
76	C₄×Q₈	—	106	1	no	—
77	C₂×C₁₆	—	4	1	no	—
78	C₂×C₁₆	—	106	2	no	—
79	#107	—	53	1	no	—
80	C₄⋊C₈	—	106	1	no	—
81	C₄⋊C₈	—	212	1	no	—
82	#103	—	106	1	no	—
83	C₄×C₈	—	212	1	no	—
84	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
85	C₄×C₈	D₂₁₂	1	1	yes	characteristic
86	C₄×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
87	C₂×Q₈	—	106	1	no	—
88	Q₁₆	—	53	2	no	—
89	Q₁₆	—	106	1	no	—
90	C₂×C₈	—	2	1	no	—
91	C₂×C₈	—	212	1	no	—
92	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
93	C₂×C₈	#11719912	1	1	yes	characteristic
94	C₂×C₈	#11719922	1	1	yes	characteristic
95	C₂×C₈	#11719924	1	1	yes	characteristic
96	C₄⋊C₄	—	106	1	no	—
97	C₄⋊C₄	—	212	1	no	—
98	C₄²	—	212	1	no	—
99	C₄²	#11719921	1	1	yes	characteristic
100	C₁₆	—	8	1	no	—
101	C₁₆	—	212	2	no	—
102	Q₈	—	106	2	no	—
103	Q₈	—	212	1	no	—
104	C₂×C₄	—	212	1	no	—
105	C₂×C₄	—	212	1	no	—
106	C₂×C₄	#13133129	1	1	yes	characteristic
107	C₂×C₄	#13133137	1	1	yes	characteristic
108	C₂×C₄	#13133143	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	2	1	no	—
113	C₈	—	424	1	no	—
114	C₈	C₁₆×D₅₃	1	1	yes	characteristic
115	C₈	#13133156	1	1	yes	characteristic
116	C₂²	#16810350	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	424	1	no	—
119	C₄	—	212	2	no	—
120	C₄	#16810402	1	1	yes	characteristic
121	C₄	#16810408	1	1	yes	characteristic
122	C₄	#16810432	1	1	yes	characteristic
123	C₄	#16810438	1	1	yes	characteristic
124	C₂	#20272669	1	1	yes	characteristic
125	C₂	#20272684	1	1	yes	characteristic
126	C₂	#20272690	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₂₁₂

C₂

yes

maximal, characteristic

#20272633

C₂

yes

maximal, characteristic

C₅₃×#1367

C₂

yes

maximal, characteristic

C₈×Dic₁₀₆

—

C₄×Dic₂₁₂

C₄

yes

characteristic

C₄×#11719908

—

#16810376

C₄

yes

characteristic

#16810355

—

C₅₃×C₄⋊C₁₆

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

C₈

yes

characteristic

C₄×Dic₁₀₆

—

C₂×#11719908

—

#13133151

C₈

yes

—

—

—

C₈×Dic₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

characteristic

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

characteristic

C₄²×C₅₃

C₂²⋊C₄

yes

characteristic

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

C₂×#88322

—

Dic₂₁₂

—

Dic₂₁₂

C₁₆

yes

complemented

C₁₆×C₅₃

—

#11719908

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×C₁₆

yes

characteristic

C₈×C₅₃

M₃₂

yes

—

—

C₂²×C₅₃

D₄⋊C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊C₁₆

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

C₈×Q₁₆

—

#1367

D₅₃

yes

characteristic

#1312

—

C₂×C₅₃

D₄⋊C₁₆

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₈×Q₈

—

106

—

C₄×Q₁₆

—

C₄⋊C₁₆

—

C₄×C₁₆

—

106

—

#335

—

Q₈⋊C₈

—

106

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#7071

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

#107

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

#103

—

106

—

C₄×C₈

—

212

—

C₄×C₈

C₄×D₅₃

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

C₂²⋊D₅₃

yes

characteristic

C₂×Q₈

—

106

—

Q₁₆

—

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

C₈×D₅₃

yes

C₂×C₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

—

212

—

C₄²

#11719921

yes

characteristic

100

C₁₆

—

101

C₁₆

—

212

—

102

Q₈

—

106

—

103

Q₈

—

212

—

104

C₂×C₄

—

212

—

105

C₂×C₄

—

212

—

106

C₂×C₄

#13133129

yes

characteristic

107

C₂×C₄

#13133137

yes

characteristic

108

C₂×C₄

#13133143

yes

characteristic

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

424

—

114

C₈

C₁₆×D₅₃

yes

characteristic

115

C₈

#13133156

yes

characteristic

116

C₂²

#16810350

yes

characteristic

117

C₄

—

118

C₄

—

424

—

119

C₄

—

212

—

120

C₄

#16810402

yes

characteristic

121

C₄

#16810408

yes

characteristic

122

C₄

#16810432

yes

characteristic

123

C₄

#16810438

yes

characteristic

124

C₂

#20272669

yes

characteristic

125

C₂

#20272684

yes

characteristic

126

C₂

#20272690

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133137
commutator	C₈×C₅₃	C₂×C₁₆
Frattini	C₈²	C₂×D₅₃
Fitting	C₅₃×#1367	C₂
socle, prime radical	C₂²×C₅₃	D₄⋊C₈
cosocle	C₈²×C₅₃	C₂²
hypercenter	#1367	D₅₃
hypocenter	C₅₃	#7071
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

subgroup

G/H

center

C₂×C₄

#13133137

commutator

C₈×C₅₃

C₂×C₁₆

Frattini

C₈²

C₂×D₅₃

Fitting

C₅₃×#1367

C₂

socle, prime radical

C₂²×C₅₃

D₄⋊C₈

cosocle

C₈²×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈² ◁ #1367

N	H ≅ G/N	type
35. Dic₂₁₂	C₁₆	semidirect
74. C₅₃	#7071	semidirect

H ≅ G/N

type

35. Dic₂₁₂

C₁₆

semidirect

74. C₅₃

#7071

semidirect

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

subgroup	index
C₈×Dic₂₁₂	2
#20272633	2
C₅₃×#1367	2
#7071	53

subgroup

index

C₈×Dic₂₁₂

#20272633

C₅₃×#1367

#7071

Subgroups — #27270749

Conjugacy classes of subgroups

Complements

Maximal subgroups