Group #20285056 — subgroups

Subgroups — C₂×#16810993 #20285056

Order 6784.

384 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 149 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×#13133217	C₂	1	1	yes	maximal, characteristic
2	#16810993	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13133140	C₂	1	4	yes	maximal
4	C₂×#13133128	C₂	1	1	yes	maximal, characteristic
5	C₂×#125×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	C₂²×#88322	C₂²	1	2	yes	—
8	#13133217	—	2	2	no	—
9	#13133217	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₅₃	—	2	2	no	—
11	#13133140	C₂²	1	16	yes	—
12	#13133128	C₂²	1	4	yes	—
13	#125×C₅₃	—	2	2	no	—
14	#125×C₅₃	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
16	C₂×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₅₃	—	2	2	no	—
19	C₂²×C₄×C₅₃	D₄	1	2	yes	—
20	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
21	C₂²×Dic₅₃	—	2	2	no	—
22	C₂×Dic₁₀₆	—	2	8	no	—
23	C₂×Dic₁₀₆	—	2	2	no	—
24	C₂×Dic₁₀₆	C₂³	1	4	yes	—
25	C₄⋊C₄×C₅₃	—	2	8	no	—
26	C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₄⋊C₄×C₅₃	C₂³	1	4	yes	—
28	C₄²×C₅₃	D₄	1	2	yes	—
29	C₄²×C₅₃	C₂³	1	2	yes	—
30	C₄⋊Dic₅₃	—	2	8	no	—
31	C₂×#88322	—	2	4	no	—
32	C₂×#88322	C₂³	1	4	yes	—
33	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₅₃	—	2	4	no	—
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	—	4	4	no	—
37	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
38	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#41	1	4	yes	—
40	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
41	C₂×Dic₅₃	—	2	4	no	—
42	C₂×Dic₅₃	—	4	4	no	—
43	Dic₁₀₆	—	2	8	no	—
44	Dic₁₀₆	—	4	4	no	—
45	#88322	—	2	4	no	—
46	C₂²×C₅₃	#124	1	2	yes	—
47	C₂²×C₅₃	#137	1	4	yes	—
48	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₅₃	—	2	4	no	—
50	C₄×C₅₃	—	4	4	no	—
51	C₄×C₅₃	#124	1	2	yes	—
52	C₄×C₅₃	C₂×#41	1	2	yes	—
53	Dic₅₃	—	4	4	no	—
54	C₂×#491	—	53	1	no	maximal, Sylow
55	C₂×C₅₃	#491	1	4	yes	complemented
56	C₂×C₅₃	C₂×#124	1	1	yes	characteristic
57	C₂×C₅₃	C₂×#137	1	2	yes	—
58	C₂×C₄⋊Q₈	—	53	1	no	—
59	C₂×#125	D₅₃	1	1	yes	characteristic
60	#491	—	53	8	no	—
61	C₂×#103	—	53	4	no	—
62	C₂×#97	—	53	1	no	—
63	C₅₃	C₂×#491	1	1	yes	characteristic, complemented, Sylow
64	C₂²×Q₈	—	53	2	no	—
65	C₂²×C₈	—	53	2	no	—
66	C₄⋊Q₈	—	53	4	no	—
67	C₄⋊Q₈	—	106	2	no	—
68	#125	—	2	2	no	—
69	#125	C₂×D₅₃	1	4	yes	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	106	2	no	—
72	C₂×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
73	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
74	#103	—	53	16	no	—
75	#97	—	53	4	no	—
76	C₂×Q₈	—	53	4	no	—
77	C₂×Q₈	—	106	2	no	—
78	C₂×Q₈	—	106	8	no	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	106	2	no	—
81	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
82	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
83	C₂×C₈	—	53	4	no	—
84	C₂×C₈	—	106	4	no	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	106	8	no	—
88	C₄⋊C₄	C₂²×D₅₃	1	4	yes	—
89	C₄²	C₂²⋊D₅₃	1	2	yes	—
90	C₄²	C₂²×D₅₃	1	2	yes	—
91	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
92	Q₈	—	106	8	no	—
93	Q₈	—	212	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	4	4	no	—
97	C₂×C₄	—	106	4	no	—
98	C₂×C₄	—	212	4	no	—
99	C₂×C₄	#11719949	1	4	yes	—
100	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
101	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
102	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
103	C₈	—	106	4	no	—
104	C₂²	#13133206	1	2	yes	—
105	C₂²	#13133247	1	4	yes	—
106	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	212	4	no	—
110	C₄	#13133206	1	2	yes	—
111	C₄	C₂×#11719949	1	2	yes	—
112	C₂	#16810993	1	4	yes	direct product factor
113	C₂	C₂×#13133206	1	1	yes	characteristic
114	C₂	C₂×#13133247	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133217

C₂

yes

maximal, characteristic

#16810993

C₂

yes

maximal, direct product factor

C₂×#13133140

C₂

yes

maximal

C₂×#13133128

C₂

yes

maximal, characteristic

C₂×#125×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133217

—

#13133217

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

#13133140

C₂²

yes

—

#13133128

C₂²

yes

—

#125×C₅₃

—

#125×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

—

C₂²×C₅₃

#124

yes

—

C₂²×C₅₃

#137

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#124

yes

—

C₄×C₅₃

C₂×#41

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₄⋊Q₈

—

C₂×#125

D₅₃

yes

characteristic

#491

—

C₂×#103

—

C₂×#97

—

C₅₃

C₂×#491

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

#125

—

#125

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄⋊C₄

C₂×D₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

#97

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

C₂²×D₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

Q₈

—

106

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

106

—

C₂×C₄

—

212

—

C₂×C₄

#11719949

yes

—

100

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

101

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

102

C₂×C₄

C₂³×D₅₃

yes

characteristic

103

C₈

—

106

—

104

C₂²

#13133206

yes

—

105

C₂²

#13133247

yes

—

106

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

212

—

110

C₄

#13133206

yes

—

111

C₄

C₂×#11719949

yes

—

112

C₂

#16810993

yes

direct product factor

113

C₂

C₂×#13133206

yes

characteristic

114

C₂

C₂×#13133247

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×#125×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×#125	D₅₃
hypocenter	C₅₃	C₂×#491
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×#125×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
2. #16810993	C₂	direct (N × H)
55. C₂×C₅₃	#491	semidirect
63. C₅₃	C₂×#491	semidirect
112. C₂	#16810993	direct (N × H)

H ≅ G/N

type

2. #16810993

C₂

direct (N × H)

55. C₂×C₅₃

#491

semidirect

63. C₅₃

C₂×#491

semidirect

112. C₂

#16810993

direct (N × H)

Subgroups — C₂×#16810993 #20285056

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810993 #20285056

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810993 #20285056