Group #20274568 — subgroups

Subgroups — #20274568

Order 6784.

292 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄⋊#88322	C₂	1	2	yes	maximal
2	C₂×#13133128	C₂	1	2	yes	maximal
3	#16810378	C₂	1	2	yes	maximal
4	C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	4	yes	—
6	C₄⋊#88322	C₄	1	8	yes	complemented
7	#13133128	—	2	4	no	—
8	C₂×C₄²×C₅₃	C₄	1	2	yes	—
9	C₂×C₄²×C₅₃	C₄	1	2	yes	—
10	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
18	C₄²×C₅₃	—	2	4	no	—
19	C₄²×C₅₃	—	2	4	no	—
20	C₄²×C₅₃	—	2	2	no	—
21	C₄²×C₅₃	C₂×C₄	1	4	yes	—
22	C₄²×C₅₃	D₄	1	2	yes	—
23	C₄²×C₅₃	Q₈	1	2	yes	—
24	C₂×#88322	—	2	4	no	—
25	C₂×#88322	—	2	4	no	—
26	C₂×#88322	C₂×C₄	1	8	yes	—
27	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₅₃	—	2	2	no	—
29	C₂×C₄×C₅₃	—	2	2	no	—
30	C₂×C₄×C₅₃	—	2	2	no	—
31	C₂×C₄×C₅₃	—	2	2	no	—
32	C₂×C₄×C₅₃	—	2	2	no	—
33	C₂×C₄×C₅₃	C₄²	1	2	yes	—
34	C₂×C₄×C₅₃	C₄²	1	2	yes	—
35	C₂×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
36	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
37	C₂×C₄×C₅₃	M₁₆	1	4	yes	—
38	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	#41	1	2	yes	—
42	#88322	—	2	8	no	—
43	C₂²×C₅₃	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₅₃	#117	1	1	yes	characteristic
46	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
47	C₂²×C₅₃	C₂×M₁₆	1	2	yes	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	—	2	4	no	—
50	C₄×C₅₃	—	2	4	no	—
51	C₄×C₅₃	#97	1	4	yes	complemented
52	C₄×C₅₃	C₄×D₄	1	2	yes	—
53	C₄×C₅₃	C₄×Q₈	1	2	yes	—
54	#1767	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	C₄×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₅₃	C₂×#97	1	1	yes	characteristic
57	C₂×C₅₃	C₄×M₁₆	1	1	yes	characteristic
58	C₂×C₅₃	#423	1	2	yes	—
59	C₂×C₅₃	#432	1	2	yes	—
60	C₂×C₄⋊C₈	—	53	2	no	—
61	C₂×#97	—	53	2	no	—
62	C₄³	D₅₃	1	1	yes	characteristic
63	#336	—	53	2	no	—
64	C₅₃	#1767	1	1	yes	characteristic, complemented, Sylow
65	C₂²×C₈	—	53	4	no	—
66	C₂×C₄²	Dic₅₃	1	2	yes	—
67	C₂×C₄²	Dic₅₃	1	2	yes	—
68	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
71	C₄⋊C₈	—	53	8	no	—
72	#97	—	106	4	no	—
73	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
74	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
75	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
76	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
77	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
78	C₂×C₈	—	53	8	no	—
79	C₂×C₈	—	106	4	no	—
80	C₂×C₈	—	106	4	no	—
81	C₄²	—	2	2	no	—
82	C₄²	—	2	4	no	—
83	C₄²	—	2	4	no	—
84	C₄²	Dic₁₀₆	1	2	yes	—
85	C₄²	C₄×D₅₃	1	4	yes	—
86	C₄²	D₂₁₂	1	2	yes	—
87	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	#11719917	1	4	yes	—
94	C₂×C₄	#11719917	1	4	yes	—
95	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
96	C₂×C₄	C₄×Dic₅₃	1	2	yes	—
97	C₂×C₄	C₄⋊Dic₅₃	1	4	yes	—
98	C₂×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
99	C₂×C₄	C₂×C₄×D₅₃	1	2	yes	—
100	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
101	C₂×C₄	#11719945	1	2	yes	—
102	C₈	—	106	8	no	—
103	C₂²	C₂×C₄×Dic₅₃	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
105	C₂²	#13133176	1	1	yes	characteristic
106	C₂²	C₂×#11719917	1	2	yes	—
107	C₂²	C₂×#11719917	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	4	no	—
111	C₄	#13133128	1	4	yes	complemented
112	C₄	C₄×D₂₁₂	1	2	yes	—
113	C₄	C₄×Dic₁₀₆	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
115	C₂	C₂×#13133128	1	1	yes	characteristic
116	C₂	C₄×#11719917	1	1	yes	characteristic
117	C₂	#16810647	1	2	yes	—
118	C₂	#16810680	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#88322

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal

#16810378

C₂

yes

maximal

C₄³×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₄⋊#88322

C₄

yes

complemented

#13133128

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

C₂×C₄

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₄²

yes

—

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

#88322

—

C₂²×C₅₃

C₂×C₄²

yes

characteristic

C₂²×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₅₃

#117

yes

characteristic

C₂²×C₅₃

C₂×M₁₆

yes

—

C₂²×C₅₃

C₂×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#97

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

C₄×Q₈

yes

—

#1767

—

maximal, Sylow

C₂×C₅₃

C₄×C₄⋊C₄

yes

characteristic

C₂×C₅₃

C₂×#97

yes

characteristic

C₂×C₅₃

C₄×M₁₆

yes

characteristic

C₂×C₅₃

#423

yes

—

C₂×C₅₃

#432

yes

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₄³

D₅₃

yes

characteristic

#336

—

C₅₃

#1767

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

#97

—

106

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₁₀₆

yes

—

C₄²

C₄×D₅₃

yes

—

C₄²

D₂₁₂

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#11719917

yes

—

C₂×C₄

#11719917

yes

—

C₂×C₄

C₄×Dic₅₃

yes

—

C₂×C₄

C₄×Dic₅₃

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂×C₄

C₂×C₄×D₅₃

yes

—

100

C₂×C₄

C₂×D₂₁₂

yes

characteristic

101

C₂×C₄

#11719945

yes

—

102

C₈

—

106

—

103

C₂²

C₂×C₄×Dic₅₃

yes

characteristic

104

C₂²

C₂×C₄⋊Dic₅₃

yes

characteristic

105

C₂²

#13133176

yes

characteristic

106

C₂²

C₂×#11719917

yes

—

107

C₂²

C₂×#11719917

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#13133128

yes

complemented

112

C₄

C₄×D₂₁₂

yes

—

113

C₄

C₄×Dic₁₀₆

yes

—

114

C₂

C₄×C₄⋊Dic₅₃

yes

characteristic

115

C₂

C₂×#13133128

yes

characteristic

116

C₂

C₄×#11719917

yes

characteristic

117

C₂

#16810647

yes

—

118

C₂

#16810680

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂×C₄²
Fitting	C₄³×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₄³	D₅₃
hypocenter	C₅₃	#1767
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₅₃

commutator

C₂²×C₅₃

C₂×C₄²

Fitting

C₄³×C₅₃

C₂

socle

C₂³×C₅₃

C₂²×C₄

hypercenter

C₄³

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
6. C₄⋊#88322	C₄	semidirect
51. C₄×C₅₃	#97	semidirect
64. C₅₃	#1767	semidirect
111. C₄	#13133128	semidirect

H ≅ G/N

type

6. C₄⋊#88322

C₄

semidirect

51. C₄×C₅₃

#97

semidirect

64. C₅₃

#1767

semidirect

111. C₄

#13133128

semidirect

Subgroups — #20274568

Conjugacy classes of subgroups

Complements