Group #28310070 — subgroups

Subgroups — #28310070

Order 13568.

246 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 246 conjugacy classes of subgroups — too many to draw. Only its 70 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	#20290313	C₂	1	2	yes	maximal, complemented
2	#20289740	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289674	C₂	1	2	yes	maximal, complemented
4	#20289631	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276590	C₂	1	1	yes	maximal, characteristic
6	#16811651	—	2	2	no	—
7	#16811512	C₂²	1	1	yes	characteristic
8	#16811484	C₂²	1	2	yes	—
9	#16811471	—	2	2	no	—
10	#16811464	C₂²	1	1	yes	characteristic
11	C₂×D₈₄₈	C₄	1	1	yes	characteristic
12	#16810705	C₂²	1	2	yes	—
13	#16810423	C₄	1	2	yes	—
14	#16810398	C₂²	1	1	yes	characteristic
15	C₅₃×#367	C₄	1	1	yes	characteristic
16	C₂×D₅₃⋊C₈	—	4	2	no	—
17	C₂×#11719910	—	2	1	no	—
18	C₄⋊#88324	—	2	2	no	—
19	#13133274	—	4	2	no	—
20	C₄×#88324	D₄	1	2	yes	—
21	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
22	#13133179	—	2	2	no	—
23	D₈₄₈	C₈	1	4	yes	complemented
24	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
25	#13133137	—	2	2	no	—
26	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	#107×C₅₃	C₂×C₄	1	2	yes	—
29	C₂×D₂₁₂	—	2	2	no	—
30	C₂×C₄×D₅₃	—	4	2	no	—
31	C₂×#88324	—	2	2	no	—
32	C₂×#88324	—	4	2	no	—
33	D₅₃⋊C₈	—	8	4	no	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	#11719921	—	4	2	no	—
37	C₄×Dic₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×#88322	#41	1	1	yes	characteristic
39	D₄₂₄	—	2	2	no	—
40	D₄₂₄	C₂×C₈	1	4	yes	complemented
41	#11719910	—	4	1	no	—
42	C₁₆×C₅₃	C₂×C₈	1	2	yes	complemented
43	#11719908	—	2	1	no	—
44	C₂²×D₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	—	4	2	no	—
46	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
47	C₂×Dic₅₃	AGL(1,ℤ/8)	1	2	yes	—
48	D₂₁₂	—	2	4	no	—
49	D₂₁₂	—	4	2	no	—
50	C₄×D₅₃	—	8	2	no	—
51	C₄×D₅₃	—	8	2	no	—
52	#88324	—	4	2	no	—
53	#88324	—	8	2	no	—
54	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₅₃	#131	1	1	yes	characteristic
56	#88322	—	2	1	no	—
57	#12067	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	#442	1	1	yes	characteristic
59	C₂×D₅₃	—	4	4	no	—
60	C₂×D₅₃	—	8	2	no	—
61	C₂×D₅₃	—	8	2	no	—
62	C₄×C₅₃	—	8	2	no	—
63	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
64	C₄×C₅₃	#444	1	1	yes	characteristic
65	Dic₅₃	—	2	2	no	—
66	#2177	—	53	1	no	—
67	#1580	—	53	2	no	—
68	#1368	—	53	1	no	—
69	#1333	—	53	2	no	—
70	#1313	—	53	1	no	—
71	C₂×C₅₃	#1580	1	1	yes	characteristic
72	C₂×C₅₃	AGL(1,ℤ/16)	1	2	yes	—
73	D₅₃	—	8	4	no	—
74	C₂×D₁₆	—	53	1	no	—
75	C₄×D₈	—	53	2	no	—
76	C₈×D₄	—	106	2	no	—
77	#367	#6157	1	1	yes	characteristic
78	#357	—	53	2	no	—
79	#346	—	53	1	no	—
80	#346	—	53	1	no	—
81	#334	—	53	2	no	—
82	D₄⋊C₈	—	106	2	no	—
83	#322	—	53	1	no	—
84	C₅₃	#12067	1	1	yes	characteristic, complemented, Sylow
85	C₂×D₈	—	53	2	no	—
86	C₂²×C₈	—	212	2	no	—
87	C₄×D₄	—	106	2	no	—
88	D₁₆	—	53	4	no	—
89	C₂×C₁₆	—	53	1	no	—
90	C₂×C₁₆	—	106	1	no	—
91	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
92	#107	C₂×#6157	1	2	yes	—
93	C₄⋊C₈	—	106	2	no	—
94	#102	—	106	2	no	—
95	C₂²⋊C₈	—	212	2	no	—
96	C₄×C₈	—	53	1	no	—
97	C₄×C₈	—	53	2	no	—
98	C₂×D₄	—	106	2	no	—
99	C₂²×C₄	—	212	2	no	—
100	D₈	—	53	4	no	—
101	D₈	—	106	2	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	106	2	no	—
104	C₂×C₈	—	424	4	no	—
105	C₂×C₈	—	212	2	no	—
106	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
107	C₄⋊C₄	—	2	2	no	—
108	C₂²⋊C₄	—	212	2	no	—
109	C₄²	—	53	1	no	—
110	C₁₆	—	106	1	no	—
111	C₁₆	—	212	1	no	—
112	C₁₆	D₅₃⋊C₈	1	2	yes	complemented
113	C₂³	—	212	2	no	—
114	D₄	—	106	4	no	—
115	D₄	—	212	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	53	2	no	—
118	C₂×C₄	—	424	2	no	—
119	C₂×C₄	—	424	2	no	—
120	C₂×C₄	D₄×#6157	1	1	yes	characteristic
121	C₈	—	106	1	no	—
122	C₈	—	424	2	no	—
123	C₈	—	212	2	no	—
124	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
125	C₈	#13133305	1	1	yes	characteristic
126	C₂²	—	424	2	no	—
127	C₂²	—	424	2	no	—
128	C₂²	—	212	4	no	—
129	C₂²	#16811664	1	1	yes	characteristic
130	C₄	—	8	2	no	—
131	C₄	—	106	2	no	—
132	C₄	#16811651	1	1	yes	characteristic
133	C₄	#16811669	1	1	yes	characteristic
134	C₂	—	424	4	no	—
135	C₂	#20290313	1	1	yes	characteristic
136	C₂	#20291126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290313

C₂

yes

maximal, complemented

#20289740

C₂

yes

maximal, characteristic, complemented

#20289674

C₂

yes

maximal, complemented

#20289631

C₂

yes

maximal, characteristic, complemented

#20276590

C₂

yes

maximal, characteristic

#16811651

—

#16811512

C₂²

yes

characteristic

#16811484

C₂²

yes

—

#16811471

—

#16811464

C₂²

yes

characteristic

C₂×D₈₄₈

C₄

yes

characteristic

#16810705

C₂²

yes

—

#16810423

C₄

yes

—

#16810398

C₂²

yes

characteristic

C₅₃×#367

C₄

yes

characteristic

C₂×D₅₃⋊C₈

—

—

—

—

D₄

yes

—

C₂×D₄₂₄

C₂×C₄

yes

—

#13133179

—

D₈₄₈

C₈

yes

complemented

C₂×#11719908

C₂×C₄

yes

characteristic

#13133137

—

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

#107×C₅₃

C₂×C₄

yes

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×#88324

—

C₂×#88324

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

#11719921

—

C₄×Dic₅₃

C₂×D₄

yes

yes

D₄₂₄

—

D₄₂₄

C₂×C₈

yes

complemented

#11719910

—

C₁₆×C₅₃

C₂×C₈

yes

complemented

#11719908

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×Dic₅₃

AGL(1,ℤ/8)

yes

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

#88324

—

C₈×C₅₃

C₂²×C₈

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

#444

yes

characteristic

Dic₅₃

—

—

—

—

—

—

C₂×C₅₃

#1580

yes

characteristic

C₂×C₅₃

AGL(1,ℤ/16)

yes

—

D₅₃

—

C₂×D₁₆

—

C₄×D₈

—

C₈×D₄

—

106

—

yes

—

—

—

—

D₄⋊C₈

—

106

—

#322

—

C₅₃

#12067

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂²×C₈

—

212

—

C₄×D₄

—

106

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

yes

yes

—

C₄⋊C₈

—

106

—

#102

—

106

—

C₂²⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

—

C₂×D₄

—

106

—

C₂²×C₄

—

212

—

100

D₈

—

101

D₈

—

106

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

424

—

105

C₂×C₈

—

212

—

106

C₂×C₈

C₂²×#6157

yes

characteristic

107

C₄⋊C₄

—

108

C₂²⋊C₄

—

212

—

109

C₄²

—

110

C₁₆

—

106

—

111

C₁₆

—

212

—

112

C₁₆

D₅₃⋊C₈

yes

complemented

113

C₂³

—

212

—

114

D₄

—

106

—

115

D₄

—

212

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

424

—

119

C₂×C₄

—

424

—

120

C₂×C₄

D₄×#6157

yes

characteristic

121

C₈

—

106

—

122

C₈

—

424

—

123

C₈

—

212

—

124

C₈

C₂×D₅₃⋊C₈

yes

characteristic

125

C₈

#13133305

yes

characteristic

126

C₂²

—

424

—

127

C₂²

—

424

—

128

C₂²

—

212

—

129

C₂²

#16811664

yes

characteristic

130

C₄

—

131

C₄

—

106

—

132

C₄

#16811651

yes

characteristic

133

C₄

#16811669

yes

characteristic

134

C₂

—

424

—

135

C₂

#20290313

yes

characteristic

136

C₂

#20291126

yes

—

	subgroup	G/H
center	C₂²	#16811664
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#367	C₄
socle	C₂²×C₅₃	#442
prime radical	C₂×D₈₄₈	C₄
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#367	#6157
hypocenter	C₅₃	#12067
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

subgroup

G/H

center

C₂²

#16811664

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#367

C₄

socle

C₂²×C₅₃

#442

prime radical

C₂×D₈₄₈

C₄

cosocle

C₈×Dic₅₃

C₂³

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

N	H ≅ G/N	type
1. #20290313	C₂	semidirect
2. #20289740	C₂	semidirect
3. #20289674	C₂	semidirect
4. #20289631	C₂	semidirect
23. D₈₄₈	C₈	semidirect
40. D₄₂₄	C₂×C₈	semidirect
42. C₁₆×C₅₃	C₂×C₈	semidirect
84. C₅₃	#12067	semidirect
112. C₁₆	D₅₃⋊C₈	semidirect

H ≅ G/N

type

1. #20290313

C₂

semidirect

2. #20289740

C₂

semidirect

3. #20289674

C₂

semidirect

4. #20289631

C₂

semidirect

23. D₈₄₈

C₈

semidirect

40. D₄₂₄

C₂×C₈

semidirect

42. C₁₆×C₅₃

C₂×C₈

semidirect

84. C₅₃

#12067

semidirect

112. C₁₆

D₅₃⋊C₈

semidirect

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

subgroup	index
#20290313	2
#20289740	2
#20289674	2
#20289631	2
#20276590	2
#12067	53

subgroup

index

Subgroups — #28310070

Conjugacy classes of subgroups

Complements

Maximal subgroups