Group #28312640 — subgroups

Subgroups — #28312640

Order 13568.

184 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 92 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	#20291138	C₂	1	2	yes	maximal, complemented
2	#20289874	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289832	C₂	1	2	yes	maximal, complemented
4	#20289822	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276552	C₂	1	1	yes	maximal, characteristic
6	#16811555	—	2	2	no	—
7	#16811555	C₂²	1	2	yes	—
8	C₂×#13133267	—	2	2	no	—
9	C₂×#13133267	C₂²	1	1	yes	characteristic
10	C₂×#13133267	C₂²	1	1	yes	characteristic
11	#16811040	C₂²	1	2	yes	—
12	#16810444	C₄	1	1	yes	characteristic
13	#16810407	C₄	1	2	yes	—
14	C₄×#11719908	C₂²	1	1	yes	characteristic
15	C₅₃×#443	C₄	1	1	yes	characteristic
16	#13133267	—	2	2	no	—
17	#13133267	—	2	1	no	—
18	#13133267	D₄	1	2	yes	—
19	#13133157	—	2	2	no	—
20	#13133157	C₂×C₄	1	2	yes	—
21	C₂×#11719908	—	2	2	no	—
22	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
23	C₂×#11719908	C₂³	1	1	yes	characteristic
24	#135×C₅₃	C₈	1	1	yes	characteristic
25	#131×C₅₃	C₂×C₄	1	2	yes	—
26	#108×C₅₃	C₈	1	1	yes	characteristic
27	#104×C₅₃	C₈	1	2	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#41×C₅₃	C₂×C₈	1	2	yes	—
30	Q₁₆×C₅₃	C₁₆	1	1	yes	characteristic
31	SD₁₆×C₅₃	C₁₆	1	2	yes	—
32	D₈×C₅₃	C₁₆	1	1	yes	characteristic
33	M₁₆×C₅₃	—	2	2	no	—
34	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
39	#11719908	—	2	2	no	—
40	#11719908	—	2	1	no	—
41	#11719908	C₂×D₄	1	1	yes	characteristic
42	#11719908	#41	1	1	yes	characteristic
43	Q₈×C₅₃	C₂×C₁₆	1	2	yes	—
44	D₄×C₅₃	—	2	2	no	—
45	D₄×C₅₃	C₂×C₁₆	1	2	yes	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
53	C₈×C₅₃	#131	1	1	yes	characteristic
54	#13648	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	#504	1	1	yes	characteristic
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	—	4	1	no	—
59	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
60	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
61	#2258	—	53	2	no	—
62	#2170	—	53	1	no	—
63	#1422	—	53	1	no	—
64	#1401	—	53	2	no	—
65	#1397	—	53	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	—	4	2	no	—
68	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
69	#504	—	53	2	no	—
70	#443	#6157	1	1	yes	characteristic
71	M₆₄	—	53	2	no	—
72	M₆₄	—	106	2	no	—
73	C₂×C₃₂	—	53	1	no	—
74	C₂×C₃₂	—	53	1	no	—
75	C₂×C₃₂	—	106	2	no	—
76	#364	—	53	1	no	—
77	#350	—	53	2	no	—
78	C₄×C₁₆	—	53	1	no	—
79	C₅₃	#13648	1	1	yes	characteristic, complemented, Sylow
80	#135	#88324	1	1	yes	characteristic
81	#131	C₂×#6157	1	2	yes	—
82	M₃₂	—	53	2	no	—
83	M₃₂	—	106	2	no	—
84	C₂×C₁₆	—	53	1	no	—
85	C₂×C₁₆	—	53	1	no	—
86	C₂×C₁₆	—	106	2	no	—
87	#108	#88324	1	1	yes	characteristic
88	#104	#88324	1	2	yes	—
89	C₄×C₈	C₂×#6157	1	1	yes	characteristic
90	C₃₂	—	53	2	no	—
91	C₃₂	—	106	1	no	—
92	C₃₂	—	106	2	no	—
93	#41	C₂×#88324	1	2	yes	—
94	Q₁₆	#11719910	1	1	yes	characteristic
95	SD₁₆	#11719910	1	2	yes	—
96	D₈	#11719910	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	C₂×#88324	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×#88324	1	1	yes	characteristic
101	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
102	C₄²	C₂×#88324	1	1	yes	characteristic
103	C₁₆	—	53	1	no	—
104	C₁₆	—	53	1	no	—
105	C₁₆	—	106	2	no	—
106	C₁₆	—	106	1	no	—
107	Q₈	C₂×#11719910	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	C₂×#11719910	1	2	yes	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	C₂×#11719910	1	1	yes	characteristic
115	C₈	C₂×#11719910	1	1	yes	characteristic
116	C₈	D₄×#6157	1	1	yes	characteristic
117	C₈	#13133304	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#16811681	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	4	1	no	—
122	C₄	D₄×#88324	1	1	yes	characteristic
123	C₄	C₂²×#11719910	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	2	no	—
126	C₂	D₄×#11719910	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291138

C₂

yes

maximal, complemented

#20289874

C₂

yes

maximal, characteristic, complemented

#20289832

C₂

yes

maximal, complemented

#20289822

C₂

yes

maximal, characteristic, complemented

#20276552

C₂

yes

maximal, characteristic

#16811555

—

#16811555

C₂²

yes

—

C₂×#13133267

—

C₂×#13133267

C₂²

yes

characteristic

C₂×#13133267

C₂²

yes

characteristic

#16811040

C₂²

yes

—

#16810444

C₄

yes

characteristic

#16810407

C₄

yes

—

C₄×#11719908

C₂²

yes

characteristic

C₅₃×#443

C₄

yes

—

—

D₄

yes

—

#13133157

—

#13133157

C₂×C₄

yes

—

C₂×#11719908

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂³

yes

characteristic

#135×C₅₃

C₈

yes

characteristic

#131×C₅₃

C₂×C₄

yes

—

#108×C₅₃

C₈

yes

characteristic

#104×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

#41×C₅₃

C₂×C₈

yes

—

Q₁₆×C₅₃

C₁₆

yes

characteristic

SD₁₆×C₅₃

C₁₆

yes

—

D₈×C₅₃

C₁₆

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

—

—

C₂×D₄

yes

yes

Q₈×C₅₃

C₂×C₁₆

yes

—

D₄×C₅₃

—

D₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×C₁₆

yes

characteristic

C₈×C₅₃

C₂×C₁₆

yes

characteristic

C₈×C₅₃

C₄×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

#504

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

C₂²×C₁₆

yes

—

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

D₄×C₁₆

yes

—

yes

M₆₄

—

M₆₄

—

106

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

106

—

#364

—

#350

—

C₄×C₁₆

—

C₅₃

#13648

yes

characteristic, complemented, Sylow

yes

yes

—

M₃₂

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

yes

yes

—

C₄×C₈

C₂×#6157

yes

characteristic

C₃₂

—

C₃₂

—

106

—

C₃₂

—

106

—

#41

C₂×#88324

yes

—

Q₁₆

#11719910

yes

characteristic

SD₁₆

#11719910

yes

—

D₈

#11719910

yes

characteristic

M₁₆

—

M₁₆

C₂×#88324

yes

—

C₂×C₈

—

100

C₂×C₈

C₂×#88324

yes

characteristic

101

C₂×C₈

C₂²×#6157

yes

characteristic

102

C₄²

C₂×#88324

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

—

106

—

106

C₁₆

—

106

—

107

Q₈

C₂×#11719910

yes

—

108

D₄

—

109

D₄

C₂×#11719910

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×#88324

yes

characteristic

113

C₈

—

114

C₈

C₂×#11719910

yes

characteristic

115

C₈

C₂×#11719910

yes

characteristic

116

C₈

D₄×#6157

yes

characteristic

117

C₈

#13133304

yes

characteristic

118

C₂²

—

119

C₂²

#16811681

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

D₄×#88324

yes

characteristic

123

C₄

C₂²×#11719910

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

D₄×#11719910

yes

characteristic

	subgroup	G/H
center	C₈	D₄×#6157
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#443	C₄
socle	C₂×C₅₃	D₄×C₁₆
prime radical	#135×C₅₃	C₈
cosocle	C₂×#11719908	C₂³
hypercenter	#443	#6157
hypocenter	C₅₃	#13648
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

subgroup

G/H

center

C₈

D₄×#6157

commutator

C₄×C₅₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#443

C₄

socle

C₂×C₅₃

D₄×C₁₆

prime radical

#135×C₅₃

C₈

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ #443

N	H ≅ G/N	type
1. #20291138	C₂	semidirect
2. #20289874	C₂	semidirect
3. #20289832	C₂	semidirect
4. #20289822	C₂	semidirect
79. C₅₃	#13648	semidirect

H ≅ G/N

type

1. #20291138

C₂

semidirect

2. #20289874

C₂

semidirect

3. #20289832

C₂

semidirect

4. #20289822

C₂

semidirect

79. C₅₃

#13648

semidirect

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

subgroup	index
#20291138	2
#20289874	2
#20289832	2
#20289822	2
#20276552	2
#13648	53

subgroup

index

Subgroups — #28312640

Conjugacy classes of subgroups

Complements

Maximal subgroups