Group #27298750 — subgroups

Subgroups — #27298750

Order 13568.

262 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 99 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	#20276968	C₂	1	1	yes	maximal, characteristic
3	#20276943	C₂	1	2	yes	maximal
4	#20274312	C₂	1	2	yes	maximal
5	C₈×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₄₂₄	—	2	2	no	—
7	#16811028	C₂²	1	2	yes	—
8	#16811021	—	2	2	no	—
9	#16811003	C₂²	1	2	yes	—
10	#16810974	—	2	2	no	—
11	#16810449	—	2	2	no	—
12	#16810428	—	2	4	no	—
13	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₈²×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₁₂	—	2	2	no	—
17	C₂×Dic₂₁₂	—	2	2	no	—
18	#13133217	—	2	2	no	—
19	#13133217	—	2	2	no	—
20	Dic₄₂₄	—	4	4	no	—
21	#13133151	—	2	2	no	—
22	#13133151	—	2	2	no	—
23	#13133149	—	4	4	no	—
24	#13133142	—	4	4	no	—
25	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
26	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
27	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₀₆	—	4	2	no	—
31	C₂×Dic₁₀₆	—	4	2	no	—
32	C₂×C₈×C₅₃	D₈	1	2	yes	—
33	C₂×C₈×C₅₃	D₈	1	2	yes	—
34	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₄⋊Dic₅₃	—	4	2	no	—
38	C₄⋊Dic₅₃	—	4	2	no	—
39	C₄⋊Dic₅₃	—	4	2	no	—
40	C₄⋊Dic₅₃	—	4	2	no	—
41	Dic₂₁₂	—	4	4	no	—
42	Dic₂₁₂	—	4	4	no	—
43	C₁₆×C₅₃	—	2	2	no	—
44	C₁₆×C₅₃	SD₁₆	1	4	yes	—
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	C₂×Dic₅₃	—	8	2	no	—
50	Dic₁₀₆	—	8	2	no	—
51	Dic₁₀₆	—	8	2	no	—
52	Dic₁₀₆	—	8	4	no	—
53	C₈×C₅₃	—	2	2	no	—
54	C₈×C₅₃	Q₃₂	1	4	yes	complemented
55	C₈×C₅₃	C₂×SD₁₆	1	2	yes	—
56	C₈×C₅₃	#135	1	2	yes	—
57	#12504	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	#493	1	1	yes	characteristic
59	C₄×C₅₃	#492	1	1	yes	characteristic
60	C₄×C₅₃	#495	1	1	yes	characteristic
61	C₄×C₅₃	C₂×Q₃₂	1	2	yes	—
62	C₄×C₅₃	#508	1	2	yes	—
63	Dic₅₃	—	16	2	no	—
64	Dic₅₃	—	16	2	no	—
65	#2252	—	53	1	no	—
66	#2247	—	53	1	no	—
67	#2241	—	53	2	no	—
68	#1707	—	53	2	no	—
69	C₈×C₁₆	D₅₃	1	1	yes	characteristic
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	—	106	2	no	—
74	#500	—	53	2	no	—
75	#498	—	106	2	no	—
76	#494	—	53	2	no	—
77	#487	—	106	2	no	—
78	#366	—	106	2	no	—
79	#358	—	106	4	no	—
80	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
81	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
82	C₈²	C₂×D₅₃	1	1	yes	characteristic
83	C₅₃	#12504	1	1	yes	characteristic, complemented, Sylow
84	C₂×Q₁₆	—	106	2	no	—
85	C₂×Q₁₆	—	106	2	no	—
86	C₄⋊Q₈	—	106	2	no	—
87	C₄⋊Q₈	—	106	2	no	—
88	Q₃₂	—	212	4	no	—
89	C₂×C₁₆	D₂₁₂	1	2	yes	—
90	C₂×C₁₆	D₂₁₂	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	D₂₁₂	1	1	yes	characteristic
96	C₄×C₈	D₂₁₂	1	1	yes	characteristic
97	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×Q₈	—	212	2	no	—
99	C₂×Q₈	—	212	2	no	—
100	Q₁₆	—	212	4	no	—
101	Q₁₆	—	212	4	no	—
102	C₂×C₈	D₄₂₄	1	2	yes	—
103	C₂×C₈	D₄₂₄	1	2	yes	—
104	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
105	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
106	C₄⋊C₄	—	212	2	no	—
107	C₄⋊C₄	—	212	2	no	—
108	C₄⋊C₄	—	212	2	no	—
109	C₄⋊C₄	—	212	2	no	—
110	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
111	C₁₆	—	2	2	no	—
112	C₁₆	#11719913	1	4	yes	—
113	Q₈	—	424	4	no	—
114	Q₈	—	424	2	no	—
115	Q₈	—	424	2	no	—
116	C₂×C₄	—	424	2	no	—
117	C₂×C₄	—	424	2	no	—
118	C₂×C₄	#13133214	1	1	yes	characteristic
119	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
120	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	Dic₄₂₄	1	4	yes	complemented
123	C₈	C₂×#11719913	1	2	yes	—
124	C₈	#13133239	1	2	yes	—
125	C₂²	#16811000	1	1	yes	characteristic
126	C₄	—	848	2	no	—
127	C₄	—	848	2	no	—
128	C₄	#16810996	1	1	yes	characteristic
129	C₄	#16811007	1	1	yes	characteristic
130	C₄	C₂×Dic₄₂₄	1	2	yes	—
131	C₄	#16811055	1	2	yes	—
132	C₂	#20274307	1	1	yes	characteristic
133	C₂	#20276968	1	1	yes	characteristic
134	C₂	#20276976	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276990

C₂

yes

maximal, characteristic

#20276968

C₂

yes

maximal, characteristic

#20276943

C₂

yes

maximal

#20274312

C₂

yes

maximal

C₈×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×Dic₄₂₄

—

#16811028

C₂²

yes

—

#16811021

—

#16811003

C₂²

yes

—

#16810974

—

#16810449

—

#16810428

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

#13133217

—

#13133217

—

Dic₄₂₄

—

—

—

—

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

C₁₆×C₅₃

—

C₁₆×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

Q₃₂

yes

complemented

C₈×C₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

#135

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×Q₃₂

yes

—

C₄×C₅₃

#508

yes

—

Dic₅₃

—

Dic₅₃

—

—

—

—

—

C₈×C₁₆

D₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×Q₃₂

—

106

—

#500

—

#498

—

106

—

#494

—

#487

—

106

—

#366

—

106

—

#358

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12504

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

Q₃₂

—

212

—

C₂×C₁₆

D₂₁₂

yes

—

C₂×C₁₆

D₂₁₂

yes

—

#107

—

106

—

#107

—

106

—

#106

—

212

—

#103

—

212

—

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

D₂₁₂

yes

characteristic

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

100

Q₁₆

—

212

—

101

Q₁₆

—

212

—

102

C₂×C₈

D₄₂₄

yes

—

103

C₂×C₈

D₄₂₄

yes

—

104

C₂×C₈

C₂×D₂₁₂

yes

characteristic

105

C₂×C₈

C₂×D₂₁₂

yes

characteristic

106

C₄⋊C₄

—

212

—

107

C₄⋊C₄

—

212

—

108

C₄⋊C₄

—

212

—

109

C₄⋊C₄

—

212

—

110

C₄²

C₂×D₂₁₂

yes

characteristic

111

C₁₆

—

112

C₁₆

#11719913

yes

—

113

Q₈

—

424

—

114

Q₈

—

424

—

115

Q₈

—

424

—

116

C₂×C₄

—

424

—

117

C₂×C₄

—

424

—

118

C₂×C₄

#13133214

yes

characteristic

119

C₂×C₄

C₂×D₄₂₄

yes

characteristic

120

C₂×C₄

C₂×D₄₂₄

yes

characteristic

121

C₈

—

122

C₈

Dic₄₂₄

yes

complemented

123

C₈

C₂×#11719913

yes

—

124

C₈

#13133239

yes

—

125

C₂²

#16811000

yes

characteristic

126

C₄

—

848

—

127

C₄

—

848

—

128

C₄

#16810996

yes

characteristic

129

C₄

#16811007

yes

characteristic

130

C₄

C₂×Dic₄₂₄

yes

—

131

C₄

#16811055

yes

—

132

C₂

#20274307

yes

characteristic

133

C₂

#20276968

yes

characteristic

134

C₂

#20276976

yes

characteristic

	subgroup	G/H
center	C₂²	#16811000
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₈×C₁₆×C₅₃	C₂
socle, prime radical	C₂²×C₅₃	#493
hypercenter	C₈×C₁₆	D₅₃
hypocenter	C₅₃	#12504
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

subgroup

G/H

center

C₂²

#16811000

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₈×C₁₆×C₅₃

C₂

socle, prime radical

C₂²×C₅₃

#493

hypercenter

C₈×C₁₆

D₅₃

hypocenter

C₅₃

#12504

LCS

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

N	H ≅ G/N	type
54. C₈×C₅₃	Q₃₂	semidirect
83. C₅₃	#12504	semidirect
122. C₈	Dic₄₂₄	semidirect

H ≅ G/N

type

54. C₈×C₅₃

Q₃₂

semidirect

83. C₅₃

#12504

semidirect

122. C₈

Dic₄₂₄

semidirect

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

subgroup	index
#20276990	2
#20276968	2
#20276943	2
#20274312	2
C₈×C₁₆×C₅₃	2
#12504	53

subgroup

index

C₈×C₁₆×C₅₃

Subgroups — #27298750

Conjugacy classes of subgroups

Complements

Maximal subgroups