Group #27298747 — subgroups

Subgroups — #27298747

Order 13568.

262 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 87 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	#20276942	C₂	1	2	yes	maximal
3	#20274309	C₂	1	2	yes	maximal
4	C₁₆×#88322	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#2247	C₂	1	1	yes	maximal, characteristic
6	#16811028	C₂²	1	2	yes	—
7	#16811019	—	2	2	no	—
8	#16810975	—	2	2	no	—
9	#16810449	—	2	2	no	—
10	#16810427	—	2	4	no	—
11	C₄×#11719908	C₂²	1	1	yes	characteristic
12	C₈×#88322	C₂²	1	1	yes	characteristic
13	C₂×Q₃₂×C₅₃	—	2	2	no	—
14	C₅₃×#494	C₂²	1	2	yes	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#13133217	—	2	2	no	—
17	C₂×#11719908	D₄	1	2	yes	—
18	#13133151	—	2	2	no	—
19	#13133151	—	2	2	no	—
20	#13133148	—	4	4	no	—
21	#13133140	—	4	4	no	—
22	C₄×#88322	D₄	1	1	yes	characteristic
23	C₄×#88322	D₄	1	1	yes	characteristic
24	C₂×Q₁₆×C₅₃	—	2	2	no	—
25	C₂×Q₁₆×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	Q₃₂×C₅₃	—	4	4	no	—
28	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
29	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Q₈×C₅₃	—	4	2	no	—
31	C₂×Dic₁₀₆	—	4	2	no	—
32	Q₁₆×C₅₃	—	4	4	no	—
33	Q₁₆×C₅₃	—	4	4	no	—
34	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₄⋊C₄×C₅₃	—	4	2	no	—
38	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₄⋊Dic₅₃	—	4	2	no	—
40	C₄⋊Dic₅₃	—	4	2	no	—
41	C₂×#88322	D₈	1	2	yes	—
42	C₂×#88322	D₈	1	2	yes	—
43	C₁₆×C₅₃	SD₁₆	1	4	yes	—
44	#11719908	—	2	2	no	—
45	Q₈×C₅₃	—	8	2	no	—
46	Q₈×C₅₃	—	8	2	no	—
47	C₂×C₄×C₅₃	—	8	2	no	—
48	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₅₃	—	8	2	no	—
52	Dic₁₀₆	—	8	4	no	—
53	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
54	C₈×C₅₃	#135	1	2	yes	—
55	#88322	—	2	2	no	—
56	#88322	Q₃₂	1	4	yes	complemented
57	#12504	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	#493	1	1	yes	characteristic
59	C₄×C₅₃	—	16	2	no	—
60	C₄×C₅₃	#492	1	1	yes	characteristic
61	C₄×C₅₃	#495	1	1	yes	characteristic
62	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
63	C₄×C₅₃	#508	1	2	yes	—
64	Dic₅₃	—	16	2	no	—
65	#2252	—	53	1	no	—
66	#2247	D₅₃	1	1	yes	characteristic
67	#2241	—	53	2	no	—
68	#1707	—	53	2	no	—
69	C₈×C₁₆	—	53	1	no	—
70	C₂×C₅₃	#1706	1	1	yes	characteristic
71	C₂×C₅₃	#2247	1	1	yes	characteristic
72	C₂×C₅₃	#2249	1	1	yes	characteristic
73	C₂×Q₃₂	—	2	2	no	—
74	#500	—	53	2	no	—
75	#498	—	106	2	no	—
76	#494	C₂×D₅₃	1	2	yes	—
77	#487	—	106	2	no	—
78	#366	—	106	2	no	—
79	#358	—	106	4	no	—
80	C₄×C₁₆	—	53	1	no	—
81	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
82	C₈²	—	53	1	no	—
83	C₅₃	#12504	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₁₆	—	2	2	no	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	106	2	no	—
88	Q₃₂	—	4	4	no	—
89	C₂×C₁₆	—	53	2	no	—
90	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
91	#107	—	106	2	no	—
92	#107	—	106	2	no	—
93	#106	—	212	4	no	—
94	#103	—	212	4	no	—
95	C₄×C₈	—	53	1	no	—
96	C₄×C₈	—	53	1	no	—
97	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	212	2	no	—
100	Q₁₆	—	4	4	no	—
101	Q₁₆	—	4	4	no	—
102	C₂×C₈	—	53	2	no	—
103	C₂×C₈	—	53	2	no	—
104	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	212	2	no	—
109	C₄⋊C₄	—	212	2	no	—
110	C₄²	D₄×D₅₃	1	1	yes	characteristic
111	C₁₆	—	106	2	no	—
112	C₁₆	#11719923	1	4	yes	—
113	Q₈	—	8	2	no	—
114	Q₈	—	8	2	no	—
115	Q₈	—	424	4	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	424	2	no	—
118	C₂×C₄	#13133213	1	1	yes	characteristic
119	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
120	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
121	C₈	—	53	4	no	—
122	C₈	—	106	2	no	—
123	C₈	C₂×#11719923	1	2	yes	—
124	C₈	#13133236	1	2	yes	—
125	C₂²	#16810998	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	848	2	no	—
128	C₄	#16810995	1	1	yes	characteristic
129	C₄	#16811006	1	1	yes	characteristic
130	C₄	Q₃₂×D₅₃	1	2	yes	—
131	C₄	#16811056	1	2	yes	—
132	C₂	#20274304	1	1	yes	characteristic
133	C₂	#20276966	1	1	yes	characteristic
134	C₂	#20276977	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276990

C₂

yes

maximal, characteristic

#20276942

C₂

yes

maximal

#20274309

C₂

yes

maximal

C₁₆×#88322

C₂

yes

maximal, characteristic

C₅₃×#2247

C₂

yes

maximal, characteristic

#16811028

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₂×Q₃₂×C₅₃

—

C₅₃×#494

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

#13133217

—

C₂×#11719908

D₄

yes

—

—

—

—

—

D₄

yes

characteristic

C₄×#88322

D₄

yes

characteristic

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

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₈

yes

—

C₂×#88322

D₈

yes

—

C₁₆×C₅₃

SD₁₆

yes

—

#11719908

—

Q₈×C₅₃

—

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×SD₁₆

yes

—

C₈×C₅₃

#135

yes

—

#88322

—

#88322

Q₃₂

yes

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

—

C₄×C₅₃

#508

yes

—

Dic₅₃

—

#2252

—

#2247

D₅₃

yes

characteristic

#2241

—

#1707

—

C₈×C₁₆

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

#500

—

#498

—

106

—

#494

C₂×D₅₃

yes

—

#487

—

106

—

#366

—

106

—

#358

—

106

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

—

C₅₃

#12504

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₅₃

yes

—

#107

—

106

—

#107

—

106

—

#106

—

212

—

#103

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

212

—

100

Q₁₆

—

101

Q₁₆

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₅₃

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

212

—

109

C₄⋊C₄

—

212

—

110

C₄²

D₄×D₅₃

yes

characteristic

111

C₁₆

—

106

—

112

C₁₆

#11719923

yes

—

113

Q₈

—

114

Q₈

—

115

Q₈

—

424

—

116

C₂×C₄

—

117

C₂×C₄

—

424

—

118

C₂×C₄

#13133213

yes

characteristic

119

C₂×C₄

D₈×D₅₃

yes

characteristic

120

C₂×C₄

D₈×D₅₃

yes

characteristic

121

C₈

—

122

C₈

—

106

—

123

C₈

C₂×#11719923

yes

—

124

C₈

#13133236

yes

—

125

C₂²

#16810998

yes

characteristic

126

C₄

—

127

C₄

—

848

—

128

C₄

#16810995

yes

characteristic

129

C₄

#16811006

yes

characteristic

130

C₄

Q₃₂×D₅₃

yes

—

131

C₄

#16811056

yes

—

132

C₂

#20274304

yes

characteristic

133

C₂

#20276966

yes

characteristic

134

C₂

#20276977

yes

characteristic

	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	#12504
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
56. #88322	Q₃₂	semidirect
83. C₅₃	#12504	semidirect

H ≅ G/N

type

56. #88322

Q₃₂

semidirect

83. C₅₃

#12504

semidirect

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

subgroup	index
#20276990	2
#20276942	2
#20274309	2
C₁₆×#88322	2
C₅₃×#2247	2
#12504	53

subgroup

index

#20276990

#20276942

#20274309

C₁₆×#88322

C₅₃×#2247

#12504

Subgroups — #27298747

Conjugacy classes of subgroups

Complements

Maximal subgroups