Group #27298835 — subgroups

Subgroups — #27298835

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	#20276997	C₂	1	1	yes	maximal, characteristic
2	#20276995	C₂	1	2	yes	maximal
3	#20274332	C₂	1	2	yes	maximal
4	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2254	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16811026	—	2	2	no	—
8	#16811002	—	2	2	no	—
9	#16810451	—	2	2	no	—
10	#16810450	—	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×#367	—	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	2	yes	—
23	C₄⋊Q₈×C₅₃	—	2	2	no	—
24	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
25	#107×C₅₃	—	2	2	no	—
26	#107×C₅₃	—	2	2	no	—
27	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₅₃	—	4	2	no	—
29	C₂×Dic₁₀₆	—	4	2	no	—
30	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
32	C₄⋊C₄×C₅₃	—	4	2	no	—
33	C₄⋊C₄×C₅₃	—	4	2	no	—
34	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
35	#11719925	—	8	8	no	—
36	C₄⋊Dic₅₃	—	4	2	no	—
37	C₄⋊Dic₅₃	—	4	2	no	—
38	C₂×#88322	D₈	1	4	yes	—
39	C₁₆×C₅₃	Q₁₆	1	4	yes	complemented
40	#11719908	Q₁₆	1	4	yes	complemented
41	Q₈×C₅₃	—	8	4	no	—
42	C₂×C₄×C₅₃	—	8	2	no	—
43	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
45	C₂×Dic₅₃	—	8	2	no	—
46	Dic₁₀₆	—	8	4	no	—
47	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
48	C₈×C₅₃	C₂×Q₁₆	1	2	yes	—
49	#88322	SD₃₂	1	8	yes	—
50	#12523	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	#493	1	1	yes	characteristic
52	C₄×C₅₃	—	16	2	no	—
53	C₄×C₅₃	#494	1	1	yes	characteristic
54	C₄×C₅₃	#494	1	1	yes	characteristic
55	C₄×C₅₃	C₂×SD₃₂	1	4	yes	—
56	Dic₅₃	—	16	2	no	—
57	#2254	—	53	2	no	—
58	#2254	—	53	1	no	—
59	#2254	D₅₃	1	1	yes	characteristic
60	#1713	—	53	2	no	—
61	C₈×C₁₆	—	53	1	no	—
62	C₂×C₅₃	#1713	1	1	yes	characteristic
63	C₂×C₅₃	#2248	1	2	yes	—
64	#500	—	53	2	no	—
65	#500	—	106	2	no	—
66	#500	C₂×D₅₃	1	2	yes	—
67	#494	—	106	2	no	—
68	#367	—	2	2	no	—
69	#367	—	106	4	no	—
70	#367	—	106	2	no	—
71	C₄×C₁₆	—	53	1	no	—
72	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
73	C₈²	—	53	1	no	—
74	C₅₃	#12523	1	1	yes	characteristic, complemented, Sylow
75	C₂×Q₁₆	—	212	4	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	C₄⋊Q₈	—	106	2	no	—
78	C₂×C₁₆	—	53	2	no	—
79	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
80	#107	—	2	2	no	—
81	#107	—	2	2	no	—
82	#107	—	106	2	no	—
83	#107	—	106	2	no	—
84	#107	—	212	4	no	—
85	C₄×C₈	—	53	2	no	—
86	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
87	C₂×Q₈	—	4	2	no	—
88	C₂×Q₈	—	212	2	no	—
89	Q₁₆	—	424	8	no	—
90	C₂×C₈	—	53	4	no	—
91	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	212	2	no	—
96	C₄⋊C₄	—	212	2	no	—
97	C₄²	D₄×D₅₃	1	1	yes	characteristic
98	C₁₆	—	53	4	no	—
99	C₁₆	#11719925	1	4	yes	complemented
100	Q₈	—	8	4	no	—
101	Q₈	—	424	4	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	424	2	no	—
104	C₂×C₄	#13133213	1	1	yes	characteristic
105	C₂×C₄	D₈×D₅₃	1	2	yes	—
106	C₈	—	53	8	no	—
107	C₈	Q₁₆×D₅₃	1	2	yes	—
108	C₈	C₂×#11719925	1	2	yes	—
109	C₂²	#16810998	1	1	yes	characteristic
110	C₄	—	16	2	no	—
111	C₄	—	848	2	no	—
112	C₄	#16811001	1	1	yes	characteristic
113	C₄	#16811002	1	1	yes	characteristic
114	C₄	SD₃₂×D₅₃	1	4	yes	—
115	C₂	#20274332	1	1	yes	characteristic
116	C₂	#20276969	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276997

C₂

yes

maximal, characteristic

#20276995

C₂

yes

maximal

#20274332

C₂

yes

maximal

C₁₆×#88322

C₂

yes

maximal, characteristic

C₅₃×#2254

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₅₃×#500

C₂²

yes

—

C₅₃×#367

—

C₄×C₁₆×C₅₃

C₂²

yes

—

—

D₄

yes

—

—

—

—

D₄

yes

—

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

—

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

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

C₂×Q₁₆

yes

—

#88322

SD₃₂

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×SD₃₂

yes

—

Dic₅₃

—

#2254

—

#2254

—

#2254

D₅₃

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂×C₅₃

#1713

yes

characteristic

C₂×C₅₃

#2248

yes

—

#500

—

#500

—

106

—

#500

C₂×D₅₃

yes

—

#494

—

106

—

#367

—

#367

—

106

—

#367

—

106

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

—

C₅₃

#12523

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

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

Q₁₆

—

424

—

C₂×C₈

—

C₂×C₈

D₄×D₅₃

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

212

—

C₄⋊C₄

—

212

—

C₄²

D₄×D₅₃

yes

characteristic

C₁₆

—

C₁₆

#11719925

yes

complemented

100

Q₈

—

101

Q₈

—

424

—

102

C₂×C₄

—

103

C₂×C₄

—

424

—

104

C₂×C₄

#13133213

yes

characteristic

105

C₂×C₄

D₈×D₅₃

yes

—

106

C₈

—

107

C₈

Q₁₆×D₅₃

yes

—

108

C₈

C₂×#11719925

yes

—

109

C₂²

#16810998

yes

characteristic

110

C₄

—

111

C₄

—

848

—

112

C₄

#16811001

yes

characteristic

113

C₄

#16811002

yes

characteristic

114

C₄

SD₃₂×D₅₃

yes

—

115

C₂

#20274332

yes

characteristic

116

C₂

#20276969

yes

—

	subgroup	G/H
center	C₂²	#16810998
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2254	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#2254	D₅₃
hypocenter	C₅₃	#12523
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

subgroup

G/H

center

C₂²

#16810998

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#2254

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2254

N	H ≅ G/N	type
39. C₁₆×C₅₃	Q₁₆	semidirect
40. #11719908	Q₁₆	semidirect
74. C₅₃	#12523	semidirect
99. C₁₆	#11719925	semidirect

H ≅ G/N

type

39. C₁₆×C₅₃

Q₁₆

semidirect

40. #11719908

Q₁₆

semidirect

74. C₅₃

#12523

semidirect

99. C₁₆

#11719925

semidirect

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

subgroup	index
#20276997	2
#20276995	2
#20274332	2
C₁₆×#88322	2
C₅₃×#2254	2
#12523	53

subgroup

index

#20276997

#20276995

#20274332

C₁₆×#88322

C₅₃×#2254

#12523

Subgroups — #27298835

Conjugacy classes of subgroups

Complements

Maximal subgroups