Group #28310060 — subgroups

Subgroups — #28310060

Order 13568.

214 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 78 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	#20290308	C₂	1	2	yes	maximal
2	#20289739	C₂	1	1	yes	maximal, characteristic
3	#20289675	C₂	1	2	yes	maximal
4	#20289631	C₂	1	1	yes	maximal, characteristic
5	#20276586	C₂	1	1	yes	maximal, characteristic
6	Q₈×#88324	—	2	2	no	—
7	#16811512	C₂²	1	1	yes	characteristic
8	#16811484	C₂²	1	2	yes	—
9	Q₈⋊#88324	—	2	2	no	—
10	#16811464	C₂²	1	1	yes	characteristic
11	Q₁₆×Dic₅₃	C₂²	1	2	yes	—
12	#16810451	C₄	1	1	yes	characteristic
13	#16810427	C₄	1	2	yes	—
14	#16810398	C₂²	1	1	yes	characteristic
15	C₂×Q₃₂×C₅₃	C₄	1	1	yes	characteristic
16	C₂×#11719910	—	2	1	no	—
17	C₄⋊#88324	—	2	2	no	—
18	C₄⋊#88324	—	4	2	no	—
19	C₄×#88324	—	4	2	no	—
20	C₄×#88324	D₄	1	2	yes	—
21	Q₈×Dic₅₃	—	2	2	no	—
22	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
23	#13133151	C₂×C₄	1	2	yes	—
24	#13133141	—	2	2	no	—
25	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
26	C₂×Q₁₆×C₅₃	C₂×C₄	1	2	yes	—
27	Q₃₂×C₅₃	C₈	1	4	yes	complemented
28	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×Q₈×C₅₃	—	2	2	no	—
30	C₂×#88324	—	2	2	no	—
31	C₂×#88324	—	4	2	no	—
32	Q₁₆×C₅₃	—	2	2	no	—
33	Q₁₆×C₅₃	C₂×C₈	1	4	yes	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊Dic₅₃	—	2	2	no	—
36	C₄⋊Dic₅₃	—	4	2	no	—
37	C₄×Dic₅₃	—	4	2	no	—
38	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×#88322	#41	1	1	yes	characteristic
40	#11719910	—	4	1	no	—
41	C₁₆×C₅₃	C₂×C₈	1	2	yes	—
42	#11719908	—	2	1	no	—
43	Q₈×C₅₃	—	2	4	no	—
44	Q₈×C₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
47	C₂×Dic₅₃	—	4	2	no	—
48	C₂×Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
49	#88324	—	4	2	no	—
50	#88324	—	8	2	no	—
51	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₅₃	#131	1	1	yes	characteristic
53	#88322	—	2	1	no	—
54	#12066	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	#442	1	1	yes	characteristic
56	C₄×C₅₃	—	4	4	no	—
57	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
58	C₄×C₅₃	#444	1	1	yes	characteristic
59	Dic₅₃	—	2	2	no	—
60	Dic₅₃	—	8	2	no	—
61	#2176	—	53	1	no	—
62	#1579	—	53	2	no	—
63	#1368	—	53	1	no	—
64	#1334	—	53	2	no	—
65	#1313	—	53	1	no	—
66	C₂×C₅₃	#1580	1	1	yes	characteristic
67	C₂×C₅₃	#2180	1	2	yes	—
68	C₂×Q₃₂	#6157	1	1	yes	characteristic
69	C₈×Q₈	—	106	2	no	—
70	C₄×Q₁₆	—	53	2	no	—
71	#367	—	53	1	no	—
72	#358	—	53	2	no	—
73	#346	—	53	1	no	—
74	#346	—	53	1	no	—
75	#334	—	53	2	no	—
76	Q₈⋊C₈	—	106	2	no	—
77	#322	—	53	1	no	—
78	C₅₃	#12066	1	1	yes	characteristic, complemented, Sylow
79	C₂×Q₁₆	C₂×#6157	1	2	yes	—
80	C₄×Q₈	—	106	2	no	—
81	Q₃₂	#88324	1	4	yes	complemented
82	C₂×C₁₆	—	53	1	no	—
83	C₂×C₁₆	—	106	1	no	—
84	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
85	#107	—	53	2	no	—
86	C₄⋊C₈	—	106	2	no	—
87	C₄⋊C₈	—	212	2	no	—
88	#103	—	106	2	no	—
89	C₄×C₈	—	53	1	no	—
90	C₄×C₈	—	53	2	no	—
91	C₄×C₈	—	212	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	C₂×#88324	1	4	yes	—
95	C₂×C₈	—	53	1	no	—
96	C₂×C₈	—	106	2	no	—
97	C₂×C₈	—	212	2	no	—
98	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
99	C₄⋊C₄	—	106	2	no	—
100	C₄⋊C₄	—	212	2	no	—
101	C₄²	—	53	1	no	—
102	C₄²	—	212	2	no	—
103	C₁₆	—	106	1	no	—
104	C₁₆	—	212	1	no	—
105	C₁₆	C₂×#88324	1	2	yes	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	53	2	no	—
110	C₂×C₄	—	212	2	no	—
111	C₂×C₄	D₄×#6157	1	1	yes	characteristic
112	C₈	—	106	1	no	—
113	C₈	—	424	2	no	—
114	C₈	—	212	2	no	—
115	C₈	C₂²×#88324	1	1	yes	characteristic
116	C₈	#13133304	1	1	yes	characteristic
117	C₂²	#16811664	1	1	yes	characteristic
118	C₄	—	4	4	no	—
119	C₄	—	106	2	no	—
120	C₄	—	424	2	no	—
121	C₄	D₄×#88324	1	1	yes	characteristic
122	C₄	#16811670	1	1	yes	characteristic
123	C₂	#20290311	1	1	yes	characteristic
124	C₂	#20291122	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290308

C₂

yes

maximal

#20289739

C₂

yes

maximal, characteristic

#20289675

C₂

yes

maximal

#20289631

C₂

yes

maximal, characteristic

#20276586

C₂

yes

maximal, characteristic

Q₈×#88324

—

#16811512

C₂²

yes

characteristic

#16811484

C₂²

yes

—

Q₈⋊#88324

—

#16811464

C₂²

yes

characteristic

Q₁₆×Dic₅₃

C₂²

yes

—

#16810451

C₄

yes

characteristic

#16810427

C₄

yes

—

#16810398

C₂²

yes

characteristic

C₂×Q₃₂×C₅₃

C₄

yes

—

—

—

—

D₄

yes

—

Q₈×Dic₅₃

—

C₂×#11719908

C₂×C₄

yes

characteristic

#13133151

C₂×C₄

yes

—

#13133141

—

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×Q₁₆×C₅₃

C₂×C₄

yes

—

Q₃₂×C₅₃

C₈

yes

complemented

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×#88324

—

C₂×#88324

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×D₄

yes

yes

—

C₁₆×C₅₃

C₂×C₈

yes

—

#11719908

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

AGL(1,ℤ/8)

yes

—

#88324

—

#88324

—

C₈×C₅₃

C₂²×C₈

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

#444

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

—

—

—

—

C₂×C₅₃

#1580

yes

characteristic

C₂×C₅₃

#2180

yes

—

C₂×Q₃₂

#6157

yes

characteristic

C₈×Q₈

—

106

—

C₄×Q₁₆

—

—

—

—

—

—

Q₈⋊C₈

—

106

—

#322

—

C₅₃

#12066

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×#6157

yes

—

C₄×Q₈

—

106

—

Q₃₂

#88324

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×#6157

yes

characteristic

#107

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

#103

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

212

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×#88324

yes

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

C₂²×#6157

yes

characteristic

C₄⋊C₄

—

106

—

100

C₄⋊C₄

—

212

—

101

C₄²

—

102

C₄²

—

212

—

103

C₁₆

—

106

—

104

C₁₆

—

212

—

105

C₁₆

C₂×#88324

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

212

—

111

C₂×C₄

D₄×#6157

yes

characteristic

112

C₈

—

106

—

113

C₈

—

424

—

114

C₈

—

212

—

115

C₈

C₂²×#88324

yes

characteristic

116

C₈

#13133304

yes

characteristic

117

C₂²

#16811664

yes

characteristic

118

C₄

—

119

C₄

—

106

—

120

C₄

—

424

—

121

C₄

D₄×#88324

yes

characteristic

122

C₄

#16811670

yes

characteristic

123

C₂

#20290311

yes

characteristic

124

C₂

#20291122

yes

—

	subgroup	G/H
center	C₂²	#16811664
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂×Q₃₂×C₅₃	C₄
socle, prime radical	C₂²×C₅₃	#442
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂×Q₃₂	#6157
hypocenter	C₅₃	#12066
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

#16811664

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₂×Q₃₂×C₅₃

C₄

socle, prime radical

C₂²×C₅₃

#442

cosocle

C₈×Dic₅₃

C₂³

C₂×Q₃₂

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
27. Q₃₂×C₅₃	C₈	semidirect
78. C₅₃	#12066	semidirect
81. Q₃₂	#88324	semidirect

H ≅ G/N

type

27. Q₃₂×C₅₃

C₈

semidirect

78. C₅₃

#12066

semidirect

81. Q₃₂

#88324

semidirect

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

subgroup	index
#20290308	2
#20289739	2
#20289675	2
#20289631	2
#20276586	2
#12066	53

subgroup

index

Subgroups — #28310060

Conjugacy classes of subgroups

Complements

Maximal subgroups