Group #28309835 — subgroups

Subgroups — #28309835

Order 13568.

234 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 112 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	#20290262	C₂	1	2	yes	maximal
2	#20289741	C₂	1	4	yes	maximal, complemented
3	C₂×#16810398	C₂	1	1	yes	maximal, characteristic
4	#16811639	—	2	2	no	—
5	#16811486	—	2	2	no	—
6	#16811484	C₂²	1	4	yes	—
7	C₂²×#11719908	C₄	1	1	yes	characteristic
8	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
9	#16810398	C₄	1	2	yes	—
10	#16810398	C₂²	1	2	yes	—
11	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
12	C₄⋊#88324	—	2	4	no	—
13	C₂²⋊#88324	—	4	2	no	—
14	C₄×#88324	—	4	2	no	—
15	C₂²×#88322	Q₈	1	1	yes	characteristic
16	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
17	C₂×#11719908	C₈	1	4	yes	complemented
18	C₂×#11719908	C₂×C₄	1	2	yes	—
19	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
20	C₈×Dic₅₃	D₄	1	1	yes	characteristic
21	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
22	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	—	2	2	no	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂²×Dic₅₃	SD₁₆	1	2	yes	—
28	C₂×#88324	—	4	4	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
31	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
32	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄×Dic₅₃	SD₁₆	1	2	yes	—
35	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
36	C₂×#88322	—	2	2	no	—
37	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
38	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
39	C₁₆×C₅₃	—	2	2	no	—
40	#11719908	C₂×C₈	1	4	yes	—
41	C₂³×C₅₃	#106	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
43	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
44	C₂×C₄×C₅₃	#108	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×Dic₅₃	—	4	2	no	—
47	C₂×Dic₅₃	#106	1	2	yes	—
48	C₂×Dic₅₃	C₂×SD₁₆	1	2	yes	—
49	#88324	—	8	4	no	—
50	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
51	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
52	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
53	#88322	—	2	1	no	—
54	#88322	—	2	1	no	—
55	#11999	—	53	1	no	maximal, Sylow
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	#334	1	2	yes	—
58	C₂²×C₅₃	C₂×#106	1	1	yes	characteristic
59	C₄×C₅₃	#334	1	2	yes	—
60	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
61	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
62	Dic₅₃	—	2	2	no	—
63	Dic₅₃	—	2	2	no	—
64	C₂×#346	—	53	1	no	—
65	#1564	—	53	2	no	—
66	#1368	—	53	4	no	—
67	C₂×C₅₃	—	2	2	no	—
68	C₂×C₅₃	C₂×#334	1	1	yes	characteristic
69	C₂×C₅₃	#2155	1	2	yes	—
70	C₂²×C₁₆	—	53	1	no	—
71	C₂²×C₁₆	#6157	1	1	yes	characteristic
72	#431	—	106	2	no	—
73	C₂×C₄×C₈	—	53	1	no	—
74	#346	—	53	2	no	—
75	#346	—	53	2	no	—
76	#335	—	106	2	no	—
77	#334	—	53	4	no	—
78	C₅₃	#11999	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₈	—	53	1	no	—
80	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
81	C₂×C₄²	—	53	1	no	—
82	C₂×C₁₆	—	2	2	no	—
83	C₂×C₁₆	—	53	2	no	—
84	C₂×C₁₆	—	53	4	no	—
85	C₂×C₁₆	C₂×#6157	1	2	yes	—
86	C₄⋊C₈	—	106	4	no	—
87	C₂²⋊C₈	—	212	2	no	—
88	C₄×C₈	—	53	1	no	—
89	C₄×C₈	—	53	1	no	—
90	C₄×C₈	—	53	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	—	212	2	no	—
93	C₂²×C₄	—	53	2	no	—
94	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
95	C₂×C₈	—	53	1	no	—
96	C₂×C₈	—	53	1	no	—
97	C₂×C₈	—	106	2	no	—
98	C₂×C₈	—	212	4	no	—
99	C₂×C₈	D₅₃⋊C₈	1	2	yes	—
100	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
101	C₂×C₈	#11719940	1	2	yes	—
102	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
103	C₄²	—	53	2	no	—
104	C₄²	—	53	1	no	—
105	C₄²	—	53	1	no	—
106	C₁₆	—	2	2	no	—
107	C₁₆	—	53	4	no	—
108	C₂³	#13133285	1	1	yes	characteristic
109	C₂×C₄	—	53	2	no	—
110	C₂×C₄	—	53	2	no	—
111	C₂×C₄	—	212	2	no	—
112	C₂×C₄	#13133284	1	2	yes	—
113	C₂×C₄	#13133285	1	1	yes	characteristic
114	C₂×C₄	#13133288	1	2	yes	—
115	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
116	C₈	—	106	1	no	—
117	C₈	—	106	1	no	—
118	C₈	—	424	4	no	—
119	C₈	#13133284	1	2	yes	—
120	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
121	C₈	C₂×#11719940	1	1	yes	characteristic
122	C₂²	—	2	2	no	—
123	C₂²	#16811485	1	2	yes	—
124	C₂²	C₂×#13133285	1	1	yes	characteristic
125	C₄	—	106	2	no	—
126	C₄	—	106	2	no	—
127	C₄	#16811485	1	2	yes	—
128	C₄	C₂×#13133284	1	1	yes	characteristic
129	C₄	C₂×#13133288	1	1	yes	characteristic
130	C₂	—	2	2	no	—
131	C₂	C₂×#16811485	1	1	yes	characteristic
132	C₂	#20291062	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290262

C₂

yes

maximal

#20289741

C₂

yes

maximal, complemented

C₂×#16810398

C₂

yes

maximal, characteristic

#16811639

—

#16811486

—

#16811484

C₂²

yes

—

C₂²×#11719908

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810398

C₄

yes

—

#16810398

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₄

yes

—

—

—

C₂²×#88322

Q₈

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×#11719908

C₈

yes

complemented

C₂×#11719908

C₂×C₄

yes

—

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

D₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₅₃

SD₁₆

yes

—

C₂×#88324

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

SD₁₆

yes

—

C₄×Dic₅₃

C₂×D₄

yes

characteristic

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₁₆×C₅₃

—

#11719908

C₂×C₈

yes

—

C₂³×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

#106

yes

characteristic

C₂×C₄×C₅₃

#108

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#106

yes

—

C₂×Dic₅₃

C₂×SD₁₆

yes

—

#88324

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#334

yes

—

C₂²×C₅₃

C₂×#106

yes

characteristic

C₄×C₅₃

#334

yes

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×#108

yes

characteristic

Dic₅₃

—

Dic₅₃

—

C₂×#346

—

#1564

—

#1368

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#334

yes

characteristic

C₂×C₅₃

#2155

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

#6157

yes

characteristic

#431

—

106

—

C₂×C₄×C₈

—

#346

—

#346

—

#335

—

106

—

#334

—

C₅₃

#11999

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6157

yes

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

212

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

D₅₃⋊C₈

yes

—

100

C₂×C₈

C₄⋊#6157

yes

characteristic

101

C₂×C₈

#11719940

yes

—

102

C₂×C₈

C₂²×#6157

yes

characteristic

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₁₆

—

107

C₁₆

—

108

C₂³

#13133285

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

212

—

112

C₂×C₄

#13133284

yes

—

113

C₂×C₄

#13133285

yes

characteristic

114

C₂×C₄

#13133288

yes

—

115

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

116

C₈

—

106

—

117

C₈

—

106

—

118

C₈

—

424

—

119

C₈

#13133284

yes

—

120

C₈

C₂×D₅₃⋊C₈

yes

characteristic

121

C₈

C₂×#11719940

yes

characteristic

122

C₂²

—

123

C₂²

#16811485

yes

—

124

C₂²

C₂×#13133285

yes

characteristic

125

C₄

—

106

—

126

C₄

—

106

—

127

C₄

#16811485

yes

—

128

C₄

C₂×#13133284

yes

characteristic

129

C₄

C₂×#13133288

yes

characteristic

130

C₂

—

131

C₂

C₂×#16811485

yes

characteristic

132

C₂

#20291062

yes

—

	subgroup	G/H
center	C₂×C₄	#13133285
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#106
prime radical	C₂³×C₅₃	#106
cosocle	C₈×Dic₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#11999
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂×C₄

#13133285

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₂²×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₈×Dic₅₃

C₂³

C₂²×C₁₆

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #20289741	C₂	semidirect
17. C₂×#11719908	C₈	semidirect
78. C₅₃	#11999	semidirect

H ≅ G/N

type

2. #20289741

C₂

semidirect

17. C₂×#11719908

C₈

semidirect

78. C₅₃

#11999

semidirect

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

subgroup	index
#20290262	2
#20289741	2
C₂×#16810398	2
#11999	53

subgroup

index

Subgroups — #28309835

Conjugacy classes of subgroups

Complements

Maximal subgroups