Group #28307773 — subgroups

Subgroups — #28307773

Order 13568.

322 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 160 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	C₄×#13133292	C₂	1	2	yes	maximal
2	#20289628	C₂	1	4	yes	maximal, complemented
3	C₈²×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133292	C₄	1	4	yes	—
5	#16811512	C₄	1	4	yes	—
6	C₄×#11719910	C₂²	1	4	yes	—
7	C₄×C₈×D₅₃	C₄	1	2	yes	—
8	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₈×#88322	C₄	1	1	yes	characteristic
10	C₈×#88322	C₂²	1	2	yes	—
11	C₈²×C₅₃	C₄	1	1	yes	characteristic
12	#13133292	—	2	8	no	—
13	C₂×#11719910	C₂×C₄	1	8	yes	—
14	C₂×C₈×D₅₃	C₈	1	4	yes	—
15	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
16	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
17	C₄×#88322	C₂×C₄	1	2	yes	—
18	C₄×#88322	C₂³	1	1	yes	characteristic
19	C₈×Dic₅₃	C₈	1	4	yes	—
20	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
24	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
25	C₂×C₈×C₅₃	C₄²	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
27	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
29	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
30	C₂×#88322	C₂×C₈	1	4	yes	—
31	C₂×#88322	C₂²×C₄	1	2	yes	—
32	C₈×D₅₃	—	2	8	no	—
33	C₈×D₅₃	M₁₆	1	8	yes	—
34	#11719910	—	2	8	no	—
35	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
38	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
39	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
40	C₄×D₅₃	—	2	4	no	—
41	C₄×D₅₃	#97	1	4	yes	—
42	C₈×C₅₃	#97	1	4	yes	—
43	C₈×C₅₃	M₃₂	1	8	yes	complemented
44	#88322	—	2	4	no	—
45	#88322	C₂×M₁₆	1	4	yes	—
46	#11402	—	53	1	no	maximal, Sylow
47	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
48	C₂×D₅₃	—	2	2	no	—
49	C₂×D₅₃	#322	1	2	yes	—
50	C₄×C₅₃	C₂×#97	1	2	yes	—
51	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
52	Dic₅₃	—	2	2	no	—
53	Dic₅₃	#322	1	2	yes	—
54	C₄×M₃₂	—	53	2	no	—
55	C₂×C₈²	—	53	1	no	—
56	#1312	—	53	4	no	—
57	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
58	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
59	D₅₃	—	2	2	no	—
60	C₂×M₃₂	—	53	4	no	—
61	C₂×C₄×C₈	—	53	1	no	—
62	C₂×C₄×C₈	—	53	2	no	—
63	#346	—	53	4	no	—
64	C₄×C₁₆	—	53	4	no	—
65	C₈²	—	53	2	no	—
66	C₈²	—	53	1	no	—
67	C₈²	#6157	1	1	yes	characteristic
68	C₅₃	#11402	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₈	—	53	2	no	—
70	C₂²×C₈	—	53	4	no	—
71	C₂×C₄²	—	53	1	no	—
72	M₃₂	—	106	8	no	—
73	C₂×C₁₆	—	53	8	no	—
74	C₄×C₈	—	53	2	no	—
75	C₄×C₈	—	53	4	no	—
76	C₄×C₈	—	53	2	no	—
77	C₄×C₈	—	53	1	no	—
78	C₄×C₈	C₂×#6157	1	1	yes	characteristic
79	C₄×C₈	C₂×#6157	1	2	yes	—
80	C₂²×C₄	—	53	1	no	—
81	C₂²×C₄	—	53	2	no	—
82	C₂×C₈	—	53	2	no	—
83	C₂×C₈	—	53	8	no	—
84	C₂×C₈	—	53	4	no	—
85	C₂×C₈	—	106	8	no	—
86	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
87	C₂×C₈	C₄×#6157	1	2	yes	—
88	C₄²	—	53	1	no	—
89	C₄²	—	53	2	no	—
90	C₄²	C₂²×#6157	1	1	yes	characteristic
91	C₁₆	—	106	8	no	—
92	C₂³	—	53	1	no	—
93	C₂×C₄	—	53	1	no	—
94	C₂×C₄	—	53	4	no	—
95	C₂×C₄	—	53	2	no	—
96	C₂×C₄	—	106	4	no	—
97	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
98	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
99	C₈	—	53	4	no	—
100	C₈	—	106	4	no	—
101	C₈	#13133271	1	4	yes	—
102	C₈	#13133292	1	8	yes	complemented
103	C₂²	—	53	2	no	—
104	C₂²	—	106	2	no	—
105	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
106	C₄	—	53	2	no	—
107	C₄	—	106	2	no	—
108	C₄	C₂×#13133271	1	2	yes	—
109	C₄	C₂×#13133292	1	4	yes	—
110	C₂	—	106	2	no	—
111	C₂	#20289909	1	1	yes	characteristic
112	C₂	C₄×#13133292	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133292

C₂

yes

maximal

#20289628

C₂

yes

maximal, complemented

C₈²×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133292

C₄

yes

—

#16811512

C₄

yes

—

C₄×#11719910

C₂²

yes

—

C₄×C₈×D₅₃

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₈×#88322

C₄

yes

characteristic

C₈×#88322

C₂²

yes

—

C₈²×C₅₃

C₄

yes

characteristic

#13133292

—

C₂×#11719910

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₈

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

—

C₄²×D₅₃

C₂×C₄

yes

characteristic

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

characteristic

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₄²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

C₂²×C₄

yes

—

C₈×D₅₃

—

C₈×D₅₃

M₁₆

yes

—

#11719910

—

C₂²×D₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₄×C₈

yes

—

C₂×Dic₅₃

C₄×C₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

#97

yes

—

C₈×C₅₃

#97

yes

—

C₈×C₅₃

M₃₂

yes

complemented

#88322

—

#88322

C₂×M₁₆

yes

—

#11402

—

maximal, Sylow

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

#322

yes

—

C₄×C₅₃

C₂×#97

yes

—

C₄×C₅₃

C₂×M₃₂

yes

—

Dic₅₃

—

Dic₅₃

#322

yes

—

C₄×M₃₂

—

C₂×C₈²

—

#1312

—

C₂×C₅₃

C₂×#322

yes

characteristic

C₂×C₅₃

C₄×M₃₂

yes

—

D₅₃

—

C₂×M₃₂

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₈²

—

C₈²

—

C₈²

#6157

yes

characteristic

C₅₃

#11402

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

106

—

C₂×C₁₆

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

C₄×C₈

C₂×#6157

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

D₅₃⋊C₈

yes

—

C₂×C₈

C₄×#6157

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×#6157

yes

characteristic

C₁₆

—

106

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

C₂×C₄×#6157

yes

characteristic

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

C₈

—

100

C₈

—

106

—

101

C₈

#13133271

yes

—

102

C₈

#13133292

yes

complemented

103

C₂²

—

104

C₂²

—

106

—

105

C₂²

C₄×D₅₃⋊C₈

yes

characteristic

106

C₄

—

107

C₄

—

106

—

108

C₄

C₂×#13133271

yes

—

109

C₄

C₂×#13133292

yes

—

110

C₂

—

106

—

111

C₂

#20289909

yes

characteristic

112

C₂

C₄×#13133292

yes

—

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂×C₄×C₈
Fitting	C₈²×C₅₃	C₄
prime radical	C₂²×D₅₃	C₄×C₈
cosocle	C₄×#88322	C₂³
hypercenter	C₈²	#6157
hypocenter	C₅₃	#11402
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₈²

subgroup

G/H

center, Frattini

C₄²

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂×C₄×C₈

Fitting

C₈²×C₅₃

C₄

prime radical

C₂²×D₅₃

C₄×C₈

C₂³

C₈²

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ C₈²

N	H ≅ G/N	type
2. #20289628	C₂	semidirect
43. C₈×C₅₃	M₃₂	semidirect
68. C₅₃	#11402	semidirect
102. C₈	#13133292	semidirect

H ≅ G/N

type

2. #20289628

C₂

semidirect

43. C₈×C₅₃

M₃₂

semidirect

68. C₅₃

#11402

semidirect

102. C₈

#13133292

semidirect

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

subgroup	index
C₄×#13133292	2
#20289628	2
C₈²×D₅₃	2
#11402	53

subgroup

index

C₄×#13133292

#20289628

C₈²×D₅₃

#11402

Subgroups — #28307773

Conjugacy classes of subgroups

Complements

Maximal subgroups