Group #27298805 — subgroups

Subgroups — #27298805

Order 13568.

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

normal subgroups only

This group has 310 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	#20276968	C₂	1	1	yes	maximal, characteristic
2	#20276967	C₂	1	2	yes	maximal
3	#20274331	C₂	1	2	yes	maximal
4	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2247	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	—	2	2	no	—
7	C₂×#13133163	—	2	4	no	—
8	#16811026	—	2	4	no	—
9	#16811003	C₂²	1	2	yes	—
10	C₄×#11719908	C₂²	1	1	yes	characteristic
11	C₈×#88322	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₅₃	—	2	2	no	—
13	C₅₃×#494	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	#13133163	—	4	8	no	—
20	C₂×#11719908	D₄	1	2	yes	—
21	#13133150	—	4	8	no	—
22	C₄×#88322	D₄	1	2	yes	—
23	C₂×Q₁₆×C₅₃	—	2	2	no	—
24	C₂×Q₁₆×C₅₃	—	2	2	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	Q₃₂×C₅₃	—	4	4	no	—
27	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	—	4	2	no	—
30	C₂×Dic₁₀₆	—	4	2	no	—
31	Q₁₆×C₅₃	—	4	4	no	—
32	Q₁₆×C₅₃	—	4	4	no	—
33	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	4	4	no	—
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	Q₈×C₅₃	—	8	2	no	—
44	Q₈×C₅₃	—	8	2	no	—
45	C₂×C₄×C₅₃	—	8	2	no	—
46	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
48	C₂×Dic₅₃	—	8	2	no	—
49	Dic₁₀₆	—	8	2	no	—
50	Dic₁₀₆	—	8	2	no	—
51	C₈×C₅₃	C₂×D₈	1	2	yes	—
52	C₈×C₅₃	C₂×D₈	1	2	yes	—
53	#88322	Q₃₂	1	8	yes	complemented
54	#12516	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	#493	1	1	yes	characteristic
56	C₄×C₅₃	—	16	2	no	—
57	C₄×C₅₃	#493	1	1	yes	characteristic
58	C₄×C₅₃	#493	1	1	yes	characteristic
59	C₄×C₅₃	C₂×Q₃₂	1	4	yes	—
60	Dic₅₃	—	16	2	no	—
61	#2247	—	53	2	no	—
62	#2247	—	53	1	no	—
63	#2247	D₅₃	1	1	yes	characteristic
64	#1713	—	53	2	no	—
65	C₈×C₁₆	—	53	1	no	—
66	C₂×C₅₃	#1712	1	1	yes	characteristic
67	C₂×C₅₃	#2247	1	2	yes	—
68	C₂×Q₃₂	—	2	2	no	—
69	C₂×Q₃₂	—	106	4	no	—
70	C₂×Q₃₂	—	106	2	no	—
71	#500	—	106	4	no	—
72	#494	—	53	2	no	—
73	#494	C₂×D₅₃	1	2	yes	—
74	C₄×C₁₆	—	53	1	no	—
75	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
76	C₈²	—	53	1	no	—
77	C₅₃	#12516	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	2	2	no	—
79	C₂×Q₁₆	—	2	2	no	—
80	C₂×Q₁₆	—	106	2	no	—
81	C₂×Q₁₆	—	106	2	no	—
82	C₄⋊Q₈	—	2	2	no	—
83	C₄⋊Q₈	—	106	2	no	—
84	Q₃₂	—	4	4	no	—
85	Q₃₂	—	212	8	no	—
86	Q₃₂	—	212	4	no	—
87	C₂×C₁₆	—	53	2	no	—
88	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
89	#107	—	212	8	no	—
90	C₄×C₈	—	53	2	no	—
91	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
92	C₂×Q₈	—	4	2	no	—
93	C₂×Q₈	—	212	2	no	—
94	Q₁₆	—	4	4	no	—
95	Q₁₆	—	4	4	no	—
96	Q₁₆	—	212	4	no	—
97	Q₁₆	—	212	4	no	—
98	C₂×C₈	—	53	4	no	—
99	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
100	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
101	C₄⋊C₄	—	4	4	no	—
102	C₄⋊C₄	—	212	4	no	—
103	C₄²	D₄×D₅₃	1	1	yes	characteristic
104	C₁₆	—	53	4	no	—
105	C₁₆	#11719922	1	4	yes	—
106	Q₈	—	8	2	no	—
107	Q₈	—	8	2	no	—
108	Q₈	—	424	2	no	—
109	Q₈	—	424	2	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	2	yes	—
114	C₈	—	53	8	no	—
115	C₈	D₈×D₅₃	1	2	yes	—
116	C₈	C₂×#11719922	1	2	yes	—
117	C₂²	#16810998	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	848	2	no	—
120	C₄	#16810998	1	1	yes	characteristic
121	C₄	#16810999	1	1	yes	characteristic
122	C₄	Q₃₂×D₅₃	1	4	yes	—
123	C₂	#20274329	1	1	yes	characteristic
124	C₂	#20276966	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276968

C₂

yes

maximal, characteristic

#20276967

C₂

yes

maximal

#20274331

C₂

yes

maximal

C₁₆×#88322

C₂

yes

maximal, characteristic

C₅₃×#2247

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

—

C₂×#13133163

—

#16811026

—

#16811003

C₂²

yes

—

C₄×#11719908

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₂×Q₃₂×C₅₃

—

C₅₃×#494

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

—

Dic₄₂₄

—

#13133163

—

C₂×#11719908

D₄

yes

—

#13133150

—

C₄×#88322

D₄

yes

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₄⋊Q₈×C₅₃

—

Q₃₂×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Dic₁₀₆

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₂×#88322

D₈

yes

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

D₈

yes

—

#11719908

D₈

yes

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×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

Q₃₂

yes

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

—

Dic₅₃

—

#2247

—

#2247

—

#2247

D₅₃

yes

characteristic

#1713

—

C₈×C₁₆

—

C₂×C₅₃

#1712

yes

characteristic

C₂×C₅₃

#2247

yes

—

C₂×Q₃₂

—

C₂×Q₃₂

—

106

—

C₂×Q₃₂

—

106

—

#500

—

106

—

#494

—

#494

C₂×D₅₃

yes

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

—

C₅₃

#12516

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

Q₃₂

—

Q₃₂

—

212

—

Q₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₃

yes

—

#107

—

212

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

212

—

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

D₄×D₅₃

yes

characteristic

100

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

101

C₄⋊C₄

—

102

C₄⋊C₄

—

212

—

103

C₄²

D₄×D₅₃

yes

characteristic

104

C₁₆

—

105

C₁₆

#11719922

yes

—

106

Q₈

—

107

Q₈

—

108

Q₈

—

424

—

109

Q₈

—

424

—

110

C₂×C₄

—

111

C₂×C₄

—

424

—

112

C₂×C₄

#13133213

yes

characteristic

113

C₂×C₄

D₈×D₅₃

yes

—

114

C₈

—

115

C₈

D₈×D₅₃

yes

—

116

C₈

C₂×#11719922

yes

—

117

C₂²

#16810998

yes

characteristic

118

C₄

—

119

C₄

—

848

—

120

C₄

#16810998

yes

characteristic

121

C₄

#16810999

yes

characteristic

122

C₄

Q₃₂×D₅₃

yes

—

123

C₂

#20274329

yes

characteristic

124

C₂

#20276966

yes

—

	subgroup	G/H
center	C₂²	#16810998
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#2247	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	#2247	D₅₃
hypocenter	C₅₃	#12516
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16810998

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#2247

C₂

C₂²×C₅₃

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
53. #88322	Q₃₂	semidirect
77. C₅₃	#12516	semidirect

H ≅ G/N

type

53. #88322

Q₃₂

semidirect

77. C₅₃

#12516

semidirect

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

subgroup	index
#20276968	2
#20276967	2
#20274331	2
C₁₆×#88322	2
C₅₃×#2247	2
#12516	53

subgroup

index

#20276968

#20276967

#20274331

C₁₆×#88322

C₅₃×#2247

#12516

Subgroups — #27298805

Conjugacy classes of subgroups

Complements

Maximal subgroups