Group #20285130 — subgroups

Subgroups — C₂×#16811002 #20285130

Order 6784.

552 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 197 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	C₂²×#11719925	C₂	1	4	yes	maximal
2	C₂×#13133217	C₂	1	1	yes	maximal, characteristic
3	#16811002	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₀₆	C₂²	1	2	yes	—
7	C₂×#11719925	—	2	16	no	—
8	C₂×#11719925	C₂²	1	16	yes	—
9	C₂²×#88322	C₂²	1	2	yes	—
10	#13133217	—	2	2	no	—
11	#13133217	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₅₃	—	2	2	no	—
13	C₄×#88322	C₂²	1	4	yes	—
14	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
15	C₄⋊Q₈×C₅₃	—	2	2	no	—
16	C₄⋊Q₈×C₅₃	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
18	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₅₃	—	2	2	no	—
20	C₂×Q₈×C₅₃	—	2	8	no	—
21	C₂×Q₈×C₅₃	C₂³	1	4	yes	—
22	C₂²×C₄×C₅₃	—	2	2	no	—
23	C₂²×C₄×C₅₃	D₄	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
25	C₂²×Dic₅₃	—	2	2	no	—
26	C₂×Dic₁₀₆	—	2	8	no	—
27	C₂×Dic₁₀₆	—	2	2	no	—
28	C₂×Dic₁₀₆	C₂³	1	4	yes	—
29	C₄⋊C₄×C₅₃	—	2	8	no	—
30	C₄²×C₅₃	D₄	1	2	yes	—
31	C₄²×C₅₃	C₂³	1	2	yes	—
32	#11719925	—	2	32	no	—
33	C₄⋊Dic₅₃	—	2	8	no	—
34	C₂×#88322	D₄	1	8	yes	—
35	C₂×#88322	C₂³	1	4	yes	—
36	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₅₃	—	2	8	no	—
38	Q₈×C₅₃	—	4	4	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	—	4	4	no	—
41	C₂×C₄×C₅₃	Q₁₆	1	8	yes	complemented
42	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	4	yes	—
45	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
46	C₂×Dic₅₃	—	2	4	no	—
47	C₂×Dic₅₃	—	4	4	no	—
48	Dic₁₀₆	—	2	8	no	—
49	Dic₁₀₆	—	4	4	no	—
50	#88322	C₂×D₄	1	8	yes	—
51	C₂²×C₅₃	#127	1	2	yes	—
52	C₂²×C₅₃	C₂×Q₁₆	1	4	yes	—
53	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
54	C₄×C₅₃	—	4	4	no	—
55	C₄×C₅₃	#127	1	2	yes	—
56	C₄×C₅₃	C₂×Q₁₆	1	8	yes	complemented
57	C₄×C₅₃	C₂²×D₄	1	2	yes	—
58	Dic₅₃	—	4	4	no	—
59	C₂×#494	—	53	1	no	maximal, Sylow
60	C₂×C₅₃	#494	1	4	yes	complemented
61	C₂×C₅₃	C₂×#127	1	1	yes	characteristic
62	C₂×C₅₃	C₂²×Q₁₆	1	2	yes	—
63	C₂²×Q₁₆	—	53	4	no	—
64	C₂×C₄⋊Q₈	—	53	1	no	—
65	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
66	#494	—	53	8	no	—
67	C₂×C₄×C₈	—	53	1	no	—
68	C₅₃	C₂×#494	1	1	yes	characteristic, complemented, Sylow
69	C₂²×Q₈	—	53	2	no	—
70	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
71	C₂×Q₁₆	—	53	16	no	—
72	C₂×Q₁₆	—	106	16	no	—
73	C₂²×C₈	—	53	2	no	—
74	C₄⋊Q₈	—	2	2	no	—
75	C₄⋊Q₈	—	53	4	no	—
76	C₄⋊Q₈	—	106	2	no	—
77	C₄⋊Q₈	C₂×D₅₃	1	4	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄⋊C₄	—	106	2	no	—
80	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
81	C₄×C₈	—	53	4	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	8	no	—
84	C₂×Q₈	—	53	4	no	—
85	C₂×Q₈	—	106	2	no	—
86	C₂×Q₈	—	106	8	no	—
87	C₂×Q₈	C₂²×D₅₃	1	4	yes	—
88	C₂²×C₄	—	2	2	no	—
89	C₂²×C₄	—	106	2	no	—
90	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
91	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
92	Q₁₆	—	106	32	no	—
93	C₂×C₈	—	53	8	no	—
94	C₂×C₈	—	53	4	no	—
95	C₄⋊C₄	—	2	8	no	—
96	C₄⋊C₄	—	106	8	no	—
97	C₄²	C₂²⋊D₅₃	1	2	yes	—
98	C₄²	C₂²×D₅₃	1	2	yes	—
99	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
100	Q₈	—	2	8	no	—
101	Q₈	—	4	4	no	—
102	Q₈	—	106	8	no	—
103	Q₈	—	212	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	4	4	no	—
106	C₂×C₄	—	106	4	no	—
107	C₂×C₄	—	212	4	no	—
108	C₂×C₄	#11719925	1	8	yes	complemented
109	C₂×C₄	D₄×D₅₃	1	4	yes	—
110	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
111	C₂×C₄	C₂×C₂²⋊D₅₃	1	4	yes	—
112	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
113	C₈	—	53	8	no	—
114	C₂²	#13133213	1	2	yes	—
115	C₂²	C₂×#11719925	1	4	yes	—
116	C₂²	C₂²×C₂²⋊D₅₃	1	1	yes	characteristic
117	C₄	—	4	4	no	—
118	C₄	—	212	4	no	—
119	C₄	#13133213	1	2	yes	—
120	C₄	C₂×#11719925	1	8	yes	complemented
121	C₄	C₂×D₄×D₅₃	1	2	yes	—
122	C₂	#16811002	1	4	yes	direct product factor
123	C₂	C₂×#13133213	1	1	yes	characteristic
124	C₂	C₂²×#11719925	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719925

C₂

yes

maximal

C₂×#13133217

C₂

yes

maximal, characteristic

#16811002

C₂

yes

maximal, direct product factor

C₂×C₄×#88322

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

C₂²

yes

—

C₂×#11719925

—

C₂×#11719925

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

#13133217

—

#13133217

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₄×#88322

C₂²

yes

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₂²

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂³

yes

—

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

—

C₄²×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

—

#11719925

—

C₄⋊Dic₅₃

—

C₂×#88322

D₄

yes

—

C₂×#88322

C₂³

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

Q₁₆

yes

complemented

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

#88322

C₂×D₄

yes

—

C₂²×C₅₃

#127

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₂²×C₅₃

C₂²×D₄

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#127

yes

—

C₄×C₅₃

C₂×Q₁₆

yes

complemented

C₄×C₅₃

C₂²×D₄

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₂²×Q₁₆

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

#494

—

C₂×C₄×C₈

—

C₅₃

C₂×#494

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₅₃

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

C₂²×D₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

Q₁₆

—

106

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄²

C₂²⋊D₅₃

yes

—

C₄²

C₂²×D₅₃

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

106

—

103

Q₈

—

212

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

106

—

107

C₂×C₄

—

212

—

108

C₂×C₄

#11719925

yes

complemented

109

C₂×C₄

D₄×D₅₃

yes

—

110

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

111

C₂×C₄

C₂×C₂²⋊D₅₃

yes

—

112

C₂×C₄

C₂³×D₅₃

yes

characteristic

113

C₈

—

114

C₂²

#13133213

yes

—

115

C₂²

C₂×#11719925

yes

—

116

C₂²

C₂²×C₂²⋊D₅₃

yes

characteristic

117

C₄

—

118

C₄

—

212

—

119

C₄

#13133213

yes

—

120

C₄

C₂×#11719925

yes

complemented

121

C₄

C₂×D₄×D₅₃

yes

—

122

C₂

#16811002

yes

direct product factor

123

C₂

C₂×#13133213

yes

characteristic

124

C₂

C₂²×#11719925

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	C₂×#494
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

C₂×#494

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
3. #16811002	C₂	direct (N × H)
41. C₂×C₄×C₅₃	Q₁₆	semidirect
56. C₄×C₅₃	C₂×Q₁₆	semidirect
60. C₂×C₅₃	#494	semidirect
68. C₅₃	C₂×#494	semidirect
108. C₂×C₄	#11719925	semidirect
120. C₄	C₂×#11719925	semidirect
122. C₂	#16811002	direct (N × H)

H ≅ G/N

type

3. #16811002

C₂

direct (N × H)

41. C₂×C₄×C₅₃

Q₁₆

semidirect

56. C₄×C₅₃

C₂×Q₁₆

semidirect

60. C₂×C₅₃

#494

semidirect

68. C₅₃

C₂×#494

semidirect

108. C₂×C₄

#11719925

semidirect

120. C₄

C₂×#11719925

semidirect

122. C₂

#16811002

direct (N × H)

Subgroups — C₂×#16811002 #20285130

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16811002 #20285130

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16811002 #20285130