Group #28312591 — subgroups

Subgroups — #28312591

Order 13568.

198 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 124 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	C₂×#16811555	C₂	1	2	yes	maximal
2	#20289874	C₂	1	4	yes	maximal, complemented
3	#20276312	C₂	1	1	yes	maximal, characteristic
4	#16811555	—	2	2	no	—
5	#16811555	C₂²	1	4	yes	—
6	C₂×#13133267	—	2	2	no	—
7	C₂×#13133157	C₄	1	1	yes	characteristic
8	C₂²×#11719908	C₂²	1	1	yes	characteristic
9	#16810399	C₄	1	2	yes	—
10	C₄×#11719908	C₂²	1	2	yes	—
11	C₅₃×#405	C₄	1	1	yes	characteristic
12	#13133267	—	2	4	no	—
13	#13133157	—	2	2	no	—
14	C₂×#11719908	C₂×C₄	1	2	yes	—
15	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
16	C₂×#11719908	D₄	1	2	yes	—
17	C₂×#11719908	Q₈	1	2	yes	—
18	C₂×#11719908	C₂³	1	1	yes	characteristic
19	C₂×M₁₆×C₅₃	C₈	1	1	yes	characteristic
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	#117×C₅₃	C₈	1	1	yes	characteristic
22	#97×C₅₃	C₈	1	2	yes	—
23	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
25	M₁₆×C₅₃	C₁₆	1	4	yes	—
26	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
29	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
30	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
31	C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂²⋊C₄×C₅₃	—	2	1	no	—
33	C₄²×C₅₃	C₂×C₈	1	2	yes	—
34	#11719908	—	2	2	no	—
35	#11719908	C₄⋊C₄	1	2	yes	—
36	#11719908	C₂×D₄	1	1	yes	characteristic
37	#11719908	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₃	M₃₂	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
41	C₂×C₄×C₅₃	C₂×C₁₆	1	2	yes	—
42	C₂×C₄×C₅₃	M₃₂	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
44	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
45	C₈×C₅₃	C₂×C₁₆	1	4	yes	—
46	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
48	#13630	—	53	1	no	maximal, Sylow
49	C₂²×C₅₃	—	4	1	no	—
50	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
51	C₂²×C₅₃	C₂×M₃₂	1	1	yes	characteristic
52	C₄×C₅₃	—	4	2	no	—
53	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
54	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
56	C₂×M₆₄	—	53	2	no	—
57	#2108	—	53	1	no	—
58	#1422	—	53	4	no	—
59	C₂×C₅₃	—	2	1	no	—
60	C₂×C₅₃	—	2	2	no	—
61	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
62	C₂×M₃₂	—	53	1	no	—
63	C₂²×C₁₆	—	53	1	no	—
64	#405	#6157	1	1	yes	characteristic
65	M₆₄	—	53	4	no	—
66	M₆₄	—	106	2	no	—
67	C₂×C₃₂	—	106	2	no	—
68	#346	—	53	2	no	—
69	C₄×C₁₆	—	53	2	no	—
70	C₅₃	#13630	1	1	yes	characteristic, complemented, Sylow
71	C₂×M₁₆	#88324	1	1	yes	characteristic
72	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
73	#117	#88324	1	1	yes	characteristic
74	M₃₂	—	106	2	no	—
75	C₂×C₁₆	—	53	2	no	—
76	C₂×C₁₆	—	53	2	no	—
77	C₂×C₁₆	—	53	1	no	—
78	C₂×C₁₆	—	53	1	no	—
79	C₂×C₁₆	—	53	2	no	—
80	#97	#88324	1	2	yes	—
81	C₄×C₈	C₂×#6157	1	2	yes	—
82	C₃₂	—	106	4	no	—
83	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
84	M₁₆	#11719910	1	4	yes	—
85	C₂×C₈	#11719936	1	2	yes	—
86	C₂×C₈	C₄⋊#6157	1	2	yes	—
87	C₂×C₈	C₂×#88324	1	1	yes	characteristic
88	C₂×C₈	C₂×#88324	1	2	yes	—
89	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
90	C₄⋊C₄	—	2	1	no	—
91	C₂²⋊C₄	—	2	1	no	—
92	C₄²	C₂×#88324	1	2	yes	—
93	C₁₆	—	53	1	no	—
94	C₁₆	—	53	2	no	—
95	C₁₆	—	53	1	no	—
96	C₁₆	—	106	2	no	—
97	C₂³	#13133291	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	C₄⋊#88324	1	2	yes	—
100	C₂×C₄	C₂×#11719910	1	2	yes	—
101	C₂×C₄	#13133291	1	1	yes	characteristic
102	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
103	C₈	C₄⋊#88324	1	2	yes	—
104	C₈	C₂×#11719910	1	4	yes	—
105	C₈	C₂×C₄⋊#6157	1	1	yes	characteristic
106	C₈	C₂×#11719936	1	1	yes	characteristic
107	C₂²	—	4	1	no	—
108	C₂²	C₄⋊#11719910	1	2	yes	—
109	C₂²	C₂×#13133291	1	1	yes	characteristic
110	C₄	—	4	2	no	—
111	C₄	C₄⋊#11719910	1	2	yes	—
112	C₄	C₂×C₄⋊#88324	1	1	yes	characteristic
113	C₄	C₂²×#11719910	1	1	yes	characteristic
114	C₂	—	2	1	no	—
115	C₂	—	2	2	no	—
116	C₂	C₂×C₄⋊#11719910	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811555

C₂

yes

maximal

#20289874

C₂

yes

maximal, complemented

#20276312

C₂

yes

maximal, characteristic

#16811555

—

#16811555

C₂²

yes

—

C₂×#13133267

—

C₂×#13133157

C₄

yes

characteristic

C₂²×#11719908

C₂²

yes

characteristic

#16810399

C₄

yes

—

C₄×#11719908

C₂²

yes

—

C₅₃×#405

C₄

yes

—

—

C₂×C₄

yes

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

D₄

yes

—

C₂×#11719908

Q₈

yes

—

C₂×#11719908

C₂³

yes

characteristic

C₂×M₁₆×C₅₃

C₈

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

#117×C₅₃

C₈

yes

characteristic

#97×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

M₁₆×C₅₃

C₁₆

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

—

#11719908

—

#11719908

C₄⋊C₄

yes

—

#11719908

C₂×D₄

yes

characteristic

#11719908

C₂×Q₈

yes

characteristic

C₂³×C₅₃

M₃₂

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

M₃₂

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₂×C₁₆

yes

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

characteristic

#13630

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₁₆

yes

—

C₂²×C₅₃

C₂×M₃₂

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×M₆₄

—

#2108

—

#1422

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×M₃₂

—

C₂²×C₁₆

—

#405

#6157

yes

characteristic

M₆₄

—

M₆₄

—

106

—

C₂×C₃₂

—

106

—

#346

—

C₄×C₁₆

—

C₅₃

#13630

yes

characteristic, complemented, Sylow

C₂×M₁₆

yes

C₂²×C₈

yes

yes

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

#97

#88324

yes

—

C₄×C₈

C₂×#6157

yes

—

C₃₂

—

106

—

C₂²×C₄

C₂×#88324

yes

characteristic

M₁₆

#11719910

yes

—

C₂×C₈

#11719936

yes

—

C₂×C₈

C₄⋊#6157

yes

—

C₂×C₈

C₂×#88324

yes

characteristic

C₂×C₈

C₂×#88324

yes

—

C₂×C₈

C₂²×#6157

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂×#88324

yes

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

106

—

C₂³

#13133291

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄⋊#88324

yes

—

100

C₂×C₄

C₂×#11719910

yes

—

101

C₂×C₄

#13133291

yes

characteristic

102

C₂×C₄

C₂²×#88324

yes

characteristic

103

C₈

C₄⋊#88324

yes

—

104

C₈

C₂×#11719910

yes

—

105

C₈

C₂×C₄⋊#6157

yes

characteristic

106

C₈

C₂×#11719936

yes

characteristic

107

C₂²

—

108

C₂²

C₄⋊#11719910

yes

—

109

C₂²

C₂×#13133291

yes

characteristic

110

C₄

—

111

C₄

C₄⋊#11719910

yes

—

112

C₄

C₂×C₄⋊#88324

yes

characteristic

113

C₄

C₂²×#11719910

yes

characteristic

114

C₂

—

115

C₂

—

116

C₂

C₂×C₄⋊#11719910

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₄⋊#6157
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#405	C₄
socle	C₂×C₅₃	C₂×C₄⋊C₁₆
prime radical	C₂³×C₅₃	M₃₂
cosocle	C₂×#11719908	C₂³
hypercenter	#405	#6157
hypocenter	C₅₃	#13630
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

subgroup

G/H

center

C₈

C₂×C₄⋊#6157

commutator

C₄×C₅₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#405

C₄

socle

C₂×C₅₃

C₂×C₄⋊C₁₆

prime radical

C₂³×C₅₃

M₃₂

C₂³

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #405

N	H ≅ G/N	type
2. #20289874	C₂	semidirect
70. C₅₃	#13630	semidirect

H ≅ G/N

type

2. #20289874

C₂

semidirect

70. C₅₃

#13630

semidirect

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

subgroup	index
C₂×#16811555	2
#20289874	2
#20276312	2
#13630	53

subgroup

index

Subgroups — #28312591

Conjugacy classes of subgroups

Complements

Maximal subgroups