Group #20276418 — subgroups

Subgroups — C₂×#16810428 #20276418

Order 6784.

232 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 101 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×Dic₂₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133151	C₂	1	1	yes	maximal, characteristic
3	#16810428	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₀₆	—	2	1	no	—
6	C₂×Dic₂₁₂	—	2	2	no	—
7	C₂×Dic₂₁₂	—	2	1	no	—
8	C₂×Dic₂₁₂	C₄	1	4	yes	complemented
9	C₂×Dic₂₁₂	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₅₃	—	2	1	no	—
11	#13133151	—	2	1	no	—
12	#13133151	C₂²	1	2	yes	—
13	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×C₅₃	—	2	2	no	—
15	C₂×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₂²×C₄×C₅₃	D₄	1	1	yes	characteristic
17	C₂²×Dic₅₃	—	4	1	no	—
18	C₂²×Dic₅₃	—	4	1	no	—
19	C₂×Dic₁₀₆	—	2	4	no	—
20	C₂×Dic₁₀₆	—	2	2	no	—
21	C₂×Dic₁₀₆	—	4	1	no	—
22	C₂×Dic₁₀₆	—	4	2	no	—
23	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×C₈×C₅₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	D₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₄⋊Dic₅₃	—	2	2	no	—
28	C₄⋊Dic₅₃	—	4	1	no	—
29	Dic₂₁₂	—	2	4	no	—
30	Dic₂₁₂	—	2	2	no	—
31	Dic₂₁₂	C₂×C₄	1	4	yes	complemented
32	C₁₆×C₅₃	—	2	2	no	—
33	C₂³×C₅₃	D₈	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
35	C₂×C₄×C₅₃	D₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	SD₁₆	1	2	yes	—
37	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₅₃	—	4	4	no	—
39	C₂×Dic₅₃	—	4	2	no	—
40	C₂×Dic₅₃	—	4	2	no	—
41	C₂×Dic₅₃	—	8	2	no	—
42	Dic₁₀₆	—	2	4	no	—
43	Dic₁₀₆	—	4	4	no	—
44	Dic₁₀₆	—	4	2	no	—
45	C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
46	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₅₃	#102	1	2	yes	—
49	C₂²×C₅₃	SD₃₂	1	2	yes	—
50	C₂²×C₅₃	Q₃₂	1	2	yes	—
51	C₂²×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₄×C₅₃	#102	1	2	yes	—
53	C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
54	C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
55	Dic₅₃	—	4	4	no	—
56	Dic₅₃	—	8	2	no	—
57	C₂×#358	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	#358	1	4	yes	complemented
59	C₂×C₅₃	C₂×#102	1	1	yes	characteristic
60	C₂×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
61	C₂×C₅₃	C₂×Q₃₂	1	1	yes	characteristic
62	C₂²×Q₁₆	—	53	1	no	—
63	C₂²×C₁₆	D₅₃	1	1	yes	characteristic
64	C₂×#107	—	53	1	no	—
65	#358	—	53	4	no	—
66	C₅₃	C₂×#358	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₈	—	106	1	no	—
68	C₂×Q₁₆	—	53	4	no	—
69	C₂×Q₁₆	—	53	2	no	—
70	C₂×Q₁₆	—	106	1	no	—
71	C₂×Q₁₆	—	106	2	no	—
72	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
73	C₂×C₄⋊C₄	—	106	1	no	—
74	C₂×C₁₆	—	2	2	no	—
75	C₂×C₁₆	C₂×D₅₃	1	2	yes	—
76	#107	—	53	2	no	—
77	#107	—	106	1	no	—
78	C₂×Q₈	—	106	2	no	—
79	C₂×Q₈	—	106	4	no	—
80	C₂×Q₈	—	212	2	no	—
81	C₂×Q₈	—	212	1	no	—
82	C₂²×C₄	—	212	1	no	—
83	C₂²×C₄	—	212	1	no	—
84	C₂²×C₄	D₂₁₂	1	1	yes	characteristic
85	Q₁₆	—	53	4	no	—
86	Q₁₆	—	106	2	no	—
87	Q₁₆	—	106	4	no	—
88	C₂×C₈	C₄×D₅₃	1	2	yes	—
89	C₂×C₈	D₂₁₂	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
91	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
92	C₄⋊C₄	—	106	2	no	—
93	C₄⋊C₄	—	212	1	no	—
94	C₁₆	—	2	2	no	—
95	C₂³	D₄₂₄	1	1	yes	characteristic
96	Q₈	—	106	4	no	—
97	Q₈	—	212	4	no	—
98	Q₈	—	212	2	no	—
99	C₂×C₄	—	424	2	no	—
100	C₂×C₄	—	212	2	no	—
101	C₂×C₄	—	212	4	no	—
102	C₂×C₄	—	212	2	no	—
103	C₂×C₄	#11719913	1	2	yes	—
104	C₂×C₄	D₄₂₄	1	1	yes	characteristic
105	C₂×C₄	#11719921	1	2	yes	—
106	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
107	C₈	#11719921	1	2	yes	—
108	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
109	C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
110	C₂²	#13133139	1	2	yes	—
111	C₂²	#13133162	1	2	yes	—
112	C₂²	Dic₄₂₄	1	2	yes	—
113	C₂²	C₂×D₄₂₄	1	1	yes	characteristic
114	C₄	—	424	2	no	—
115	C₄	—	212	4	no	—
116	C₄	#13133139	1	2	yes	—
117	C₄	C₂×#11719921	1	1	yes	characteristic
118	C₄	C₂×#11719913	1	1	yes	characteristic
119	C₂	#16810428	1	4	yes	direct product factor
120	C₂	C₂×#13133139	1	1	yes	characteristic
121	C₂	C₂×#13133162	1	1	yes	characteristic
122	C₂	C₂×Dic₄₂₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₁₂

C₂

yes

maximal, characteristic

C₂×#13133151

C₂

yes

maximal, characteristic

#16810428

C₂

yes

maximal, direct product factor

C₂²×C₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₆

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

C₄

yes

complemented

C₂×Dic₂₁₂

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

#13133151

—

#13133151

C₂²

yes

—

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

D₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

D₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

Dic₂₁₂

C₂×C₄

yes

complemented

C₁₆×C₅₃

—

C₂³×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

D₈

yes

characteristic

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

C₂²⋊C₄

yes

—

C₈×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₅₃

#102

yes

—

C₂²×C₅₃

SD₃₂

yes

—

C₂²×C₅₃

Q₃₂

yes

—

C₂²×C₅₃

C₂×D₈

yes

characteristic

C₄×C₅₃

#102

yes

—

C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₅₃

C₂×SD₁₆

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

C₂×SD₃₂

yes

characteristic

C₂×C₅₃

C₂×Q₃₂

yes

characteristic

C₂²×Q₁₆

—

C₂²×C₁₆

D₅₃

yes

characteristic

C₂×#107

—

#358

—

C₅₃

C₂×#358

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

106

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

106

—

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄⋊C₄

—

106

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

—

#107

—

#107

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

212

—

C₂×Q₈

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

—

212

—

C₂²×C₄

D₂₁₂

yes

characteristic

Q₁₆

—

Q₁₆

—

106

—

Q₁₆

—

106

—

C₂×C₈

C₄×D₅₃

yes

—

C₂×C₈

D₂₁₂

yes

characteristic

C₂×C₈

C₂²⋊D₅₃

yes

—

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₁₆

—

C₂³

D₄₂₄

yes

characteristic

Q₈

—

106

—

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

424

—

100

C₂×C₄

—

212

—

101

C₂×C₄

—

212

—

102

C₂×C₄

—

212

—

103

C₂×C₄

#11719913

yes

—

104

C₂×C₄

D₄₂₄

yes

characteristic

105

C₂×C₄

#11719921

yes

—

106

C₂×C₄

C₂×D₂₁₂

yes

characteristic

107

C₈

#11719921

yes

—

108

C₈

C₂×C₄×D₅₃

yes

characteristic

109

C₈

C₂×C₂²⋊D₅₃

yes

characteristic

110

C₂²

#13133139

yes

—

111

C₂²

#13133162

yes

—

112

C₂²

Dic₄₂₄

yes

—

113

C₂²

C₂×D₄₂₄

yes

characteristic

114

C₄

—

424

—

115

C₄

—

212

—

116

C₄

#13133139

yes

—

117

C₄

C₂×#11719921

yes

characteristic

118

C₄

C₂×#11719913

yes

characteristic

119

C₂

#16810428

yes

direct product factor

120

C₂

C₂×#13133139

yes

characteristic

121

C₂

C₂×#13133162

yes

characteristic

122

C₂

C₂×Dic₄₂₄

yes

characteristic

	subgroup	G/H
center	C₂³	D₄₂₄
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂²×C₁₆×C₅₃	C₂
socle	C₂³×C₅₃	D₈
hypercenter	C₂²×C₁₆	D₅₃
hypocenter	C₅₃	C₂×#358
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂³

D₄₂₄

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂²×C₁₆×C₅₃

C₂

socle

C₂³×C₅₃

D₈

hypercenter

C₂²×C₁₆

D₅₃

hypocenter

C₅₃

C₂×#358

LCS

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
3. #16810428	C₂	direct (N × H)
8. C₂×Dic₂₁₂	C₄	semidirect
31. Dic₂₁₂	C₂×C₄	semidirect
58. C₂×C₅₃	#358	semidirect
66. C₅₃	C₂×#358	semidirect
119. C₂	#16810428	direct (N × H)

H ≅ G/N

type

3. #16810428

C₂

direct (N × H)

8. C₂×Dic₂₁₂

C₄

semidirect

31. Dic₂₁₂

C₂×C₄

semidirect

58. C₂×C₅₃

#358

semidirect

66. C₅₃

C₂×#358

semidirect

119. C₂

#16810428

direct (N × H)

Subgroups — C₂×#16810428 #20276418

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16810428 #20276418

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16810428 #20276418