Group #27311254 — subgroups

Subgroups — #27311254

Order 13568.

3998 conjugacy classes of subgroups; only the 651 normal subgroups are shown.

normal subgroups only

This group has 3998 conjugacy classes of subgroups — too many to draw. Only its 651 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	C₂×#16810486	C₂	1	6	yes	maximal
2	#20272986	C₂	1	8	yes	maximal, complemented
3	C₂⁴×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂²⋊Dic₅₃	C₂²	1	4	yes	—
5	#16810486	—	2	24	no	—
6	C₂×#13133124	C₂²	1	24	yes	—
7	C₂³×Q₈×C₅₃	—	2	24	no	—
8	C₂³×Q₈×C₅₃	C₄	1	8	yes	complemented
9	C₂³×Q₈×C₅₃	C₂²	1	4	yes	—
10	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
11	C₂³×Dic₅₃	C₂³	1	8	yes	—
12	C₂×C₂²⋊Dic₅₃	—	2	24	no	—
13	#13133124	—	2	48	no	—
14	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
15	C₂²×Q₈×C₅₃	—	2	128	no	—
16	C₂²×Q₈×C₅₃	—	2	96	no	—
17	C₂²×Q₈×C₅₃	—	2	48	no	—
18	C₂²×Q₈×C₅₃	C₂×C₄	1	16	yes	—
19	C₂³×C₄×C₅₃	—	2	24	no	—
20	C₂³×C₄×C₅₃	C₂×C₄	1	12	yes	—
21	C₂³×C₄×C₅₃	D₄	1	24	yes	—
22	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
23	C₂⁴×C₅₃	—	2	8	no	—
24	C₂⁴×C₅₃	C₂²×C₄	1	2	yes	—
25	C₂⁴×C₅₃	C₂×D₄	1	12	yes	—
26	C₂⁴×C₅₃	C₂⁴	1	1	yes	characteristic
27	C₂×Q₈×C₅₃	—	2	128	no	—
28	C₂×Q₈×C₅₃	—	2	256	no	—
29	C₂×Q₈×C₅₃	—	2	96	no	—
30	C₂²×C₄×C₅₃	—	2	48	no	—
31	C₂²×C₄×C₅₃	—	2	96	no	—
32	C₂²×C₄×C₅₃	—	2	24	no	—
33	C₂²×C₄×C₅₃	C₂²⋊C₄	1	48	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	24	yes	—
36	C₂²×Dic₅₃	—	2	24	no	—
37	C₂²×Dic₅₃	—	2	32	no	—
38	C₂²⋊Dic₅₃	—	4	16	no	—
39	C₂³×C₅₃	—	2	8	no	—
40	C₂³×C₅₃	—	2	48	no	—
41	C₂³×C₅₃	C₂×C₂²⋊C₄	1	12	yes	—
42	C₂³×C₅₃	#120	1	16	yes	—
43	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
44	C₂³×C₅₃	C₂²×D₄	1	6	yes	—
45	C₂³×C₅₃	2₋¹⁺⁴	1	8	yes	—
46	Q₈×C₅₃	—	2	128	no	—
47	C₂×C₄×C₅₃	—	2	24	no	—
48	C₂×C₄×C₅₃	—	2	48	no	—
49	C₂×C₄×C₅₃	—	2	96	no	—
50	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	24	yes	—
51	C₂×Dic₅₃	—	4	8	no	—
52	C₂×Dic₅₃	—	4	48	no	—
53	#14336	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	2	24	no	—
55	C₂²×C₅₃	—	2	32	no	—
56	C₂²×C₅₃	#379	1	8	yes	—
57	C₂²×C₅₃	C₂²×C₂²⋊C₄	1	3	yes	—
58	C₂²×C₅₃	#519	1	4	yes	—
59	C₂²×C₅₃	C₂×#120	1	4	yes	—
60	C₂²×C₅₃	#536	1	24	yes	—
61	C₄×C₅₃	—	2	24	no	—
62	Dic₅₃	—	8	8	no	—
63	C₂⁴×Q₈	D₅₃	1	1	yes	characteristic
64	C₂×#381	—	53	6	no	—
65	#1446	—	53	8	no	—
66	C₂×C₅₃	—	2	8	no	—
67	C₂×C₅₃	C₂×#379	1	1	yes	characteristic
68	C₂×C₅₃	#2310	1	6	yes	—
69	C₂×C₅₃	#2614	1	8	yes	—
70	C₂³×Q₈	—	2	24	no	—
71	C₂³×Q₈	Dic₅₃	1	8	yes	complemented
72	C₂³×Q₈	C₂×D₅₃	1	4	yes	—
73	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
74	C₂²×C₂²⋊C₄	—	53	4	no	—
75	#381	—	106	24	no	—
76	C₂×#95	—	53	24	no	—
77	C₅₃	#14336	1	1	yes	characteristic, complemented, Sylow
78	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
79	C₂²×Q₈	—	2	128	no	—
80	C₂²×Q₈	—	2	48	no	—
81	C₂²×Q₈	—	2	96	no	—
82	C₂²×Q₈	C₂×Dic₅₃	1	16	yes	—
83	C₂³×C₄	—	2	24	no	—
84	C₂³×C₄	—	53	8	no	—
85	C₂³×C₄	C₂×Dic₅₃	1	12	yes	—
86	C₂³×C₄	C₂²⋊D₅₃	1	24	yes	—
87	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
88	C₂×C₂²⋊C₄	—	106	24	no	—
89	#95	—	106	48	no	—
90	C₂⁴	—	2	8	no	—
91	C₂⁴	C₂²×Dic₅₃	1	2	yes	—
92	C₂⁴	C₂×C₂²⋊D₅₃	1	12	yes	—
93	C₂⁴	C₂³×D₅₃	1	1	yes	characteristic
94	C₂×Q₈	—	2	128	no	—
95	C₂×Q₈	—	2	256	no	—
96	C₂×Q₈	—	2	96	no	—
97	C₂²×C₄	—	2	48	no	—
98	C₂²×C₄	—	2	96	no	—
99	C₂²×C₄	—	2	24	no	—
100	C₂²×C₄	—	106	24	no	—
101	C₂²×C₄	—	106	32	no	—
102	C₂²×C₄	C₂²⋊Dic₅₃	1	48	yes	—
103	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
104	C₂²×C₄	C₂×C₂²⋊D₅₃	1	24	yes	—
105	C₂²⋊C₄	—	212	16	no	—
106	C₂³	—	2	8	no	—
107	C₂³	—	2	48	no	—
108	C₂³	C₂×C₂²⋊Dic₅₃	1	12	yes	—
109	C₂³	#13133186	1	16	yes	—
110	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
111	C₂³	C₂²×C₂²⋊D₅₃	1	6	yes	—
112	C₂³	#13133265	1	8	yes	—
113	Q₈	—	2	128	no	—
114	C₂×C₄	—	2	24	no	—
115	C₂×C₄	—	2	48	no	—
116	C₂×C₄	—	2	96	no	—
117	C₂×C₄	—	212	8	no	—
118	C₂×C₄	—	212	48	no	—
119	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	24	yes	—
120	C₂²	—	2	24	no	—
121	C₂²	—	2	32	no	—
122	C₂²	#16810479	1	8	yes	—
123	C₂²	C₂²×C₂²⋊Dic₅₃	1	3	yes	—
124	C₂²	#16811101	1	4	yes	—
125	C₂²	C₂×#13133186	1	4	yes	—
126	C₂²	#16811197	1	24	yes	—
127	C₄	—	2	24	no	—
128	C₄	—	424	8	no	—
129	C₂	—	2	8	no	—
130	C₂	C₂×#16810479	1	1	yes	characteristic
131	C₂	#20277301	1	6	yes	—
132	C₂	#20280434	1	8	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810486

C₂

yes

maximal

#20272986

C₂

yes

maximal, complemented

C₂⁴×Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×C₂²⋊Dic₅₃

C₂²

yes

—

#16810486

—

C₂×#13133124

C₂²

yes

—

C₂³×Q₈×C₅₃

—

C₂³×Q₈×C₅₃

C₄

yes

complemented

C₂³×Q₈×C₅₃

C₂²

yes

—

C₂⁴×C₄×C₅₃

C₂²

yes

—

C₂³×Dic₅₃

C₂³

yes

—

C₂×C₂²⋊Dic₅₃

—

#13133124

—

C₂⁵×C₅₃

C₂³

yes

characteristic

C₂²×Q₈×C₅₃

—

128

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

—

C₂³×C₄×C₅₃

—

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂³×C₄×C₅₃

D₄

yes

—

C₂³×C₄×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

—

C₂⁴×C₅₃

C₂²×C₄

yes

—

C₂⁴×C₅₃

C₂×D₄

yes

—

C₂⁴×C₅₃

C₂⁴

yes

characteristic

C₂×Q₈×C₅₃

—

128

—

C₂×Q₈×C₅₃

—

256

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²⋊Dic₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂³×C₅₃

#120

yes

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂³×C₅₃

C₂²×D₄

yes

—

C₂³×C₅₃

2₋¹⁺⁴

yes

—

Q₈×C₅₃

—

128

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#14336

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#379

yes

—

C₂²×C₅₃

C₂²×C₂²⋊C₄

yes

—

C₂²×C₅₃

#519

yes

—

C₂²×C₅₃

C₂×#120

yes

—

C₂²×C₅₃

#536

yes

—

C₄×C₅₃

—

Dic₅₃

—

C₂⁴×Q₈

D₅₃

yes

characteristic

C₂×#381

—

#1446

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#379

yes

characteristic

C₂×C₅₃

#2310

yes

—

C₂×C₅₃

#2614

yes

—

C₂³×Q₈

—

C₂³×Q₈

Dic₅₃

yes

complemented

C₂³×Q₈

C₂×D₅₃

yes

—

C₂⁴×C₄

C₂×D₅₃

yes

—

C₂²×C₂²⋊C₄

—

#381

—

106

—

C₂×#95

—

C₅₃

#14336

yes

characteristic, complemented, Sylow

C₂⁵

C₂²×D₅₃

yes

characteristic

C₂²×Q₈

—

128

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂×Dic₅₃

yes

—

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₂×Dic₅₃

yes

—

C₂³×C₄

C₂²⋊D₅₃

yes

—

C₂³×C₄

C₂²×D₅₃

yes

—

C₂×C₂²⋊C₄

—

106

—

#95

—

106

—

C₂⁴

—

C₂⁴

C₂²×Dic₅₃

yes

—

C₂⁴

C₂×C₂²⋊D₅₃

yes

—

C₂⁴

C₂³×D₅₃

yes

characteristic

C₂×Q₈

—

128

—

C₂×Q₈

—

256

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

106

—

101

C₂²×C₄

—

106

—

102

C₂²×C₄

C₂²⋊Dic₅₃

yes

—

103

C₂²×C₄

C₂²×Dic₅₃

yes

—

104

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

105

C₂²⋊C₄

—

212

—

106

C₂³

—

107

C₂³

—

108

C₂³

C₂×C₂²⋊Dic₅₃

yes

—

109

C₂³

#13133186

yes

—

110

C₂³

C₂³×Dic₅₃

yes

characteristic

111

C₂³

C₂²×C₂²⋊D₅₃

yes

—

112

C₂³

#13133265

yes

—

113

Q₈

—

128

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

212

—

118

C₂×C₄

—

212

—

119

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

#16810479

yes

—

123

C₂²

C₂²×C₂²⋊Dic₅₃

yes

—

124

C₂²

#16811101

yes

—

125

C₂²

C₂×#13133186

yes

—

126

C₂²

#16811197

yes

—

127

C₄

—

128

C₄

—

424

—

129

C₂

—

130

C₂

C₂×#16810479

yes

characteristic

131

C₂

#20277301

yes

—

132

C₂

#20280434

yes

—

	subgroup	G/H
center, Frattini	C₂⁴	C₂³×D₅₃
commutator	C₂³×C₅₃	C₂³×C₄
Fitting	C₂⁴×Q₈×C₅₃	C₂
socle, cosocle	C₂⁴×C₅₃	C₂⁴
prime radical	C₂⁵×C₅₃	C₂³
hypercenter	C₂⁴×Q₈	D₅₃
hypocenter	C₅₃	#14336
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

subgroup

G/H

center, Frattini

C₂⁴

C₂³×D₅₃

commutator

C₂³×C₅₃

C₂³×C₄

Fitting

C₂⁴×Q₈×C₅₃

C₂

socle, cosocle

C₂⁴×C₅₃

C₂⁴

prime radical

C₂⁵×C₅₃

C₂³

hypercenter

C₂⁴×Q₈

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂⁴ ◁ C₂⁴×Q₈

N	H ≅ G/N	type
2. #20272986	C₂	semidirect
8. C₂³×Q₈×C₅₃	C₄	semidirect
71. C₂³×Q₈	Dic₅₃	semidirect
77. C₅₃	#14336	semidirect

H ≅ G/N

type

2. #20272986

C₂

semidirect

8. C₂³×Q₈×C₅₃

C₄

semidirect

71. C₂³×Q₈

Dic₅₃

semidirect

77. C₅₃

#14336

semidirect

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

subgroup	index
C₂×#16810486	2
#20272986	2
C₂⁴×Q₈×C₅₃	2
#14336	53

subgroup

index

C₂×#16810486

#20272986

C₂⁴×Q₈×C₅₃

#14336

Subgroups — #27311254

Conjugacy classes of subgroups

Complements

Maximal subgroups