Group #27298828 — subgroups

Subgroups — #27298828

Order 13568.

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

normal subgroups only

This group has 262 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	#20276990	C₂	1	1	yes	maximal, characteristic
2	#20276988	C₂	1	2	yes	maximal
3	#20274332	C₂	1	2	yes	maximal
4	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2252	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16811026	—	2	2	no	—
8	#16811002	—	2	2	no	—
9	#16810449	—	2	2	no	—
10	#16810448	—	2	4	no	—
11	C₄×#11719908	C₂²	1	1	yes	characteristic
12	C₈×#88322	C₂²	1	1	yes	characteristic
13	C₅₃×#500	C₂²	1	2	yes	—
14	C₅₃×#366	—	2	2	no	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#11719925	—	4	4	no	—
17	#13133217	—	2	2	no	—
18	C₂×#11719908	D₄	1	2	yes	—
19	#13133151	—	2	2	no	—
20	#13133151	—	2	2	no	—
21	#13133150	—	4	4	no	—
22	C₄×#88322	D₄	1	1	yes	characteristic
23	C₄×#88322	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
26	#107×C₅₃	—	2	2	no	—
27	#107×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	—	4	2	no	—
30	C₂×Dic₁₀₆	—	4	2	no	—
31	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₄⋊C₄×C₅₃	—	4	2	no	—
34	C₄⋊C₄×C₅₃	—	4	2	no	—
35	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
36	#11719925	—	8	8	no	—
37	C₄⋊Dic₅₃	—	4	2	no	—
38	C₄⋊Dic₅₃	—	4	2	no	—
39	C₂×#88322	D₈	1	2	yes	—
40	C₂×#88322	D₈	1	2	yes	—
41	C₁₆×C₅₃	Q₁₆	1	4	yes	complemented
42	#11719908	Q₁₆	1	4	yes	complemented
43	Q₈×C₅₃	—	8	4	no	—
44	C₂×C₄×C₅₃	—	8	2	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	C₂×Q₁₆	1	2	yes	—
51	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
52	#88322	D₁₆	1	4	yes	—
53	#88322	Q₃₂	1	4	yes	—
54	#12521	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	#493	1	1	yes	characteristic
56	C₄×C₅₃	—	16	2	no	—
57	C₄×C₅₃	#494	1	1	yes	characteristic
58	C₄×C₅₃	#494	1	1	yes	characteristic
59	C₄×C₅₃	C₂×D₁₆	1	2	yes	—
60	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
61	Dic₅₃	—	16	2	no	—
62	#2252	—	53	2	no	—
63	#2252	—	53	1	no	—
64	#2252	D₅₃	1	1	yes	characteristic
65	#1713	—	53	2	no	—
66	C₈×C₁₆	—	53	1	no	—
67	C₂×C₅₃	#1713	1	1	yes	characteristic
68	C₂×C₅₃	#2246	1	1	yes	characteristic
69	C₂×C₅₃	#2247	1	1	yes	characteristic
70	#500	—	53	2	no	—
71	#500	—	106	2	no	—
72	#500	C₂×D₅₃	1	2	yes	—
73	#494	—	106	2	no	—
74	#366	—	2	2	no	—
75	#366	—	106	4	no	—
76	#366	—	106	2	no	—
77	C₄×C₁₆	—	53	1	no	—
78	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
79	C₈²	—	53	1	no	—
80	C₅₃	#12521	1	1	yes	characteristic, complemented, Sylow
81	C₂×Q₁₆	—	212	4	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	106	2	no	—
84	C₂×C₁₆	—	53	2	no	—
85	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
86	#107	—	2	2	no	—
87	#107	—	2	2	no	—
88	#107	—	106	2	no	—
89	#107	—	106	2	no	—
90	#107	—	212	4	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	—	53	1	no	—
93	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	212	2	no	—
96	Q₁₆	—	424	8	no	—
97	C₂×C₈	—	53	2	no	—
98	C₂×C₈	—	53	2	no	—
99	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	4	2	no	—
103	C₄⋊C₄	—	212	2	no	—
104	C₄⋊C₄	—	212	2	no	—
105	C₄²	D₄×D₅₃	1	1	yes	characteristic
106	C₁₆	—	53	4	no	—
107	C₁₆	#11719925	1	4	yes	complemented
108	Q₈	—	8	4	no	—
109	Q₈	—	424	4	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	424	2	no	—
112	C₂×C₄	#13133213	1	1	yes	characteristic
113	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
114	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
115	C₈	—	53	4	no	—
116	C₈	—	53	4	no	—
117	C₈	Q₁₆×D₅₃	1	2	yes	—
118	C₈	C₂×#11719925	1	2	yes	—
119	C₂²	#16810998	1	1	yes	characteristic
120	C₄	—	16	2	no	—
121	C₄	—	848	2	no	—
122	C₄	#16811001	1	1	yes	characteristic
123	C₄	#16811002	1	1	yes	characteristic
124	C₄	D₁₆×D₅₃	1	2	yes	—
125	C₄	Q₃₂×D₅₃	1	2	yes	—
126	C₂	#20274332	1	1	yes	characteristic
127	C₂	#20276963	1	1	yes	characteristic
128	C₂	#20276966	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276990

C₂

yes

maximal, characteristic

#20276988

C₂

yes

maximal

#20274332

C₂

yes

maximal

C₁₆×#88322

C₂

yes

maximal, characteristic

C₅₃×#2252

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₅₃×#500

C₂²

yes

—

C₅₃×#366

—

C₄×C₁₆×C₅₃

C₂²

yes

—

—

D₄

yes

—

—

—

—

D₄

yes

characteristic

C₄×#88322

D₄

yes

characteristic

C₄⋊Q₈×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

#107×C₅₃

—

#107×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₈

yes

—

C₂×#88322

D₈

yes

—

C₁₆×C₅₃

Q₁₆

yes

complemented

#11719908

Q₁₆

yes

complemented

Q₈×C₅₃

—

C₂×C₄×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

C₂×Q₁₆

yes

—

C₈×C₅₃

C₂×Q₁₆

yes

—

#88322

D₁₆

yes

—

#88322

Q₃₂

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×D₁₆

yes

—

C₄×C₅₃

C₂×Q₃₂

yes

—

Dic₅₃

—

#2252

—

#2252

—

#2252

D₅₃

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

106

—

#500

C₂×D₅₃

yes

—

#494

—

106

—

#366

—

#366

—

106

—

#366

—

106

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

—

C₅₃

#12521

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

212

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₃

yes

—

#107

—

#107

—

#107

—

106

—

#107

—

106

—

#107

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

Q₁₆

—

424

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₅₃

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

212

—

104

C₄⋊C₄

—

212

—

105

C₄²

D₄×D₅₃

yes

characteristic

106

C₁₆

—

107

C₁₆

#11719925

yes

complemented

108

Q₈

—

109

Q₈

—

424

—

110

C₂×C₄

—

111

C₂×C₄

—

424

—

112

C₂×C₄

#13133213

yes

characteristic

113

C₂×C₄

D₈×D₅₃

yes

characteristic

114

C₂×C₄

D₈×D₅₃

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

Q₁₆×D₅₃

yes

—

118

C₈

C₂×#11719925

yes

—

119

C₂²

#16810998

yes

characteristic

120

C₄

—

121

C₄

—

848

—

122

C₄

#16811001

yes

characteristic

123

C₄

#16811002

yes

characteristic

124

C₄

D₁₆×D₅₃

yes

—

125

C₄

Q₃₂×D₅₃

yes

—

126

C₂

#20274332

yes

characteristic

127

C₂

#20276963

yes

characteristic

128

C₂

#20276966

yes

characteristic

	subgroup	G/H
center	C₂²	#16810998
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2252	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#2252	D₅₃
hypocenter	C₅₃	#12521
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16810998

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#2252

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
41. C₁₆×C₅₃	Q₁₆	semidirect
42. #11719908	Q₁₆	semidirect
80. C₅₃	#12521	semidirect
107. C₁₆	#11719925	semidirect

H ≅ G/N

type

41. C₁₆×C₅₃

Q₁₆

semidirect

42. #11719908

Q₁₆

semidirect

80. C₅₃

#12521

semidirect

107. C₁₆

#11719925

semidirect

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

subgroup	index
#20276990	2
#20276988	2
#20274332	2
C₁₆×#88322	2
C₅₃×#2252	2
#12521	53

subgroup

index

#20276990

#20276988

#20274332

C₁₆×#88322

C₅₃×#2252

#12521

Subgroups — #27298828

Conjugacy classes of subgroups

Complements

Maximal subgroups