Group #20265375 — subgroups

Subgroups — #20265375

Order 6464.

128 conjugacy classes of subgroups; only the 43 normal subgroups are shown.

normal subgroups only

This group has 128 conjugacy classes of subgroups — too many to draw. Only its 43 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×#13108061	C₂	1	1	yes	maximal, characteristic, complemented
2	#15692251	C₂	1	1	yes	maximal, characteristic
3	Q₈×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#15692184	C₂	1	1	yes	maximal, characteristic, complemented
5	#15692177	C₂	1	1	yes	maximal, characteristic, complemented
6	#15692176	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×SD₁₆×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₄₀₄	C₂²	1	1	yes	characteristic
9	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
10	C₂×C₂²⋊D₁₀₁	—	2	1	no	—
11	#13108061	—	2	2	no	—
12	C₂×#11717579	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₁	—	2	1	no	—
14	C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
15	C₄×Dic₁₀₁	—	2	1	no	—
16	C₄×Dic₁₀₁	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₁₀₁	C₂²	1	1	yes	characteristic
18	C₂×D₄×C₁₀₁	C₂²	1	1	yes	characteristic
19	SD₁₆×C₁₀₁	—	2	2	no	—
20	C₂×C₈×C₁₀₁	C₂²	1	1	yes	characteristic
21	C₂³×C₁₀₁	—	2	1	no	—
22	C₂²×D₁₀₁	—	2	1	no	—
23	Q₈×C₁₀₁	—	2	1	no	—
24	Q₈×C₁₀₁	D₄	1	2	yes	complemented
25	D₄×C₁₀₁	—	2	1	no	—
26	D₄×C₁₀₁	—	2	1	no	—
27	C₂×C₄×C₁₀₁	—	2	1	no	—
28	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
29	C₂²⋊D₁₀₁	—	4	2	no	—
30	C₂²⋊D₁₀₁	—	4	2	no	—
31	C₂×Dic₁₀₁	—	2	1	no	—
32	C₂×Dic₁₀₁	D₄	1	1	yes	characteristic
33	C₂×Dic₁₀₁	D₄	1	1	yes	characteristic
34	D₄₀₄	—	2	1	no	—
35	D₄₀₄	—	2	1	no	—
36	C₈×C₁₀₁	—	2	1	no	—
37	#11717579	—	2	1	no	—
38	C₂²×C₁₀₁	—	4	1	no	—
39	C₂²×C₁₀₁	—	4	1	no	—
40	C₂²×C₁₀₁	—	4	1	no	—
41	C₂²×C₁₀₁	C₂×D₄	1	1	yes	characteristic
42	C₂×D₁₀₁	—	4	1	no	—
43	C₂×D₁₀₁	—	4	1	no	—
44	C₂×D₁₀₁	—	4	1	no	—
45	C₄×C₁₀₁	—	2	2	no	—
46	C₄×C₁₀₁	C₂×D₄	1	1	yes	characteristic
47	C₄×C₁₀₁	#41	1	1	yes	characteristic
48	Dic₁₀₁	—	2	1	no	—
49	Dic₁₀₁	—	4	1	no	—
50	Dic₁₀₁	SD₁₆	1	2	yes	complemented
51	C₂×C₁₀₁	—	8	1	no	—
52	C₂×C₁₀₁	#121	1	1	yes	characteristic
53	C₂×C₁₀₁	C₂×SD₁₆	1	1	yes	characteristic
54	C₂×C₁₀₁	AGL(1,ℤ/8)	1	1	yes	characteristic
55	D₁₀₁	—	8	1	no	—
56	C₁₀₁	#460	1	1	yes	characteristic, complemented, Sylow
57	#460	—	101	1	no	maximal, Sylow
58	C₂×SD₁₆	—	101	1	no	—
59	C₂×SD₁₆	D₁₀₁	1	1	yes	characteristic, complemented
60	#127	—	101	1	no	—
61	C₄×Q₈	—	101	1	no	—
62	C₄⋊C₈	—	101	1	no	—
63	#102	—	101	1	no	—
64	#102	—	101	1	no	—
65	C₂×Q₈	C₂×D₁₀₁	1	1	yes	characteristic
66	C₂×D₄	—	101	1	no	—
67	C₂×D₄	—	202	1	no	—
68	C₂×D₄	—	202	1	no	—
69	C₂×D₄	C₂×D₁₀₁	1	1	yes	characteristic
70	SD₁₆	—	2	2	no	—
71	SD₁₆	—	202	2	no	—
72	C₂×C₈	—	101	1	no	—
73	C₂×C₈	C₂×D₁₀₁	1	1	yes	characteristic
74	C₄⋊C₄	—	101	1	no	—
75	C₄⋊C₄	—	202	1	no	—
76	C₄²	—	101	1	no	—
77	C₄²	—	202	1	no	—
78	C₂³	—	2	1	no	—
79	C₂³	—	202	1	no	—
80	Q₈	—	2	1	no	—
81	Q₈	C₂²⋊D₁₀₁	1	2	yes	complemented
82	D₄	—	2	1	no	—
83	D₄	—	2	1	no	—
84	D₄	—	404	2	no	—
85	D₄	—	404	2	no	—
86	D₄	—	202	1	no	—
87	D₄	—	202	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	101	1	no	—
90	C₂×C₄	—	101	1	no	—
91	C₂×C₄	—	202	1	no	—
92	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
93	C₈	—	2	1	no	—
94	C₈	—	202	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	1	no	—
97	C₂²	—	4	1	no	—
98	C₂²	—	404	1	no	—
99	C₂²	—	404	1	no	—
100	C₂²	—	404	1	no	—
101	C₂²	D₄×D₁₀₁	1	1	yes	characteristic
102	C₄	—	2	2	no	—
103	C₄	—	101	2	no	—
104	C₄	—	404	1	no	—
105	C₄	—	202	1	no	—
106	C₄	C₂×C₂²⋊D₁₀₁	1	1	yes	characteristic
107	C₄	#13108073	1	1	yes	characteristic
108	C₂	—	8	1	no	—
109	C₂	—	808	1	no	—
110	C₂	#15692227	1	1	yes	characteristic
111	C₂	SD₁₆×D₁₀₁	1	1	yes	characteristic
112	C₂	#15692282	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13108061

C₂

yes

maximal, characteristic, complemented

#15692251

C₂

yes

maximal, characteristic

Q₈×Dic₁₀₁

C₂

yes

maximal, characteristic, complemented

#15692184

C₂

yes

maximal, characteristic, complemented

#15692177

C₂

yes

maximal, characteristic, complemented

#15692176

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×D₄₀₄

C₂²

yes

characteristic

C₂×C₂²⋊D₁₀₁

—

C₂×C₂²⋊D₁₀₁

—

#13108061

—

C₂×#11717579

C₂²

yes

characteristic

C₄⋊Dic₁₀₁

—

C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₄×Dic₁₀₁

—

C₄×Dic₁₀₁

C₂²

yes

characteristic

C₂×Q₈×C₁₀₁

C₂²

yes

characteristic

C₂×D₄×C₁₀₁

C₂²

yes

characteristic

SD₁₆×C₁₀₁

—

C₂×C₈×C₁₀₁

C₂²

yes

characteristic

C₂³×C₁₀₁

—

C₂²×D₁₀₁

—

Q₈×C₁₀₁

—

Q₈×C₁₀₁

D₄

yes

complemented

D₄×C₁₀₁

—

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂²⋊D₁₀₁

—

C₂²⋊D₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

D₄

yes

characteristic

C₂×Dic₁₀₁

D₄

yes

characteristic

D₄₀₄

—

D₄₀₄

—

C₈×C₁₀₁

—

#11717579

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×D₄

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

C₂×D₄

yes

characteristic

C₄×C₁₀₁

#41

yes

characteristic

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

SD₁₆

yes

complemented

C₂×C₁₀₁

—

C₂×C₁₀₁

#121

yes

characteristic

C₂×C₁₀₁

C₂×SD₁₆

yes

characteristic

C₂×C₁₀₁

AGL(1,ℤ/8)

yes

characteristic

D₁₀₁

—

C₁₀₁

#460

yes

characteristic, complemented, Sylow

#460

—

101

maximal, Sylow

C₂×SD₁₆

—

101

—

C₂×SD₁₆

D₁₀₁

yes

characteristic, complemented

#127

—

101

—

C₄×Q₈

—

101

—

C₄⋊C₈

—

101

—

#102

—

101

—

#102

—

101

—

C₂×Q₈

C₂×D₁₀₁

yes

characteristic

C₂×D₄

—

101

—

C₂×D₄

—

202

—

C₂×D₄

—

202

—

C₂×D₄

C₂×D₁₀₁

yes

characteristic

SD₁₆

—

SD₁₆

—

202

—

C₂×C₈

—

101

—

C₂×C₈

C₂×D₁₀₁

yes

characteristic

C₄⋊C₄

—

101

—

C₄⋊C₄

—

202

—

C₄²

—

101

—

C₄²

—

202

—

C₂³

—

C₂³

—

202

—

Q₈

—

Q₈

C₂²⋊D₁₀₁

yes

complemented

D₄

—

D₄

—

D₄

—

404

—

D₄

—

404

—

D₄

—

202

—

D₄

—

202

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

202

—

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

C₈

—

C₈

—

202

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

404

—

C₂²

—

404

—

100

C₂²

—

404

—

101

C₂²

D₄×D₁₀₁

yes

characteristic

102

C₄

—

103

C₄

—

101

—

104

C₄

—

404

—

105

C₄

—

202

—

106

C₄

C₂×C₂²⋊D₁₀₁

yes

characteristic

107

C₄

#13108073

yes

characteristic

108

C₂

—

109

C₂

—

808

—

110

C₂

#15692227

yes

characteristic

111

C₂

SD₁₆×D₁₀₁

yes

characteristic

112

C₂

#15692282

yes

characteristic

	subgroup	G/H
center	C₂²	D₄×D₁₀₁
commutator	C₂×C₄×C₁₀₁	C₂³
Frattini	C₂×C₄	C₂²×D₁₀₁
Fitting	C₂×SD₁₆×C₁₀₁	C₂
socle	C₂²×C₁₀₁	C₂×D₄
hypercenter	C₂×SD₁₆	D₁₀₁
hypocenter	C₁₀₁	#460
LCS	G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

subgroup

G/H

center

C₂²

D₄×D₁₀₁

commutator

C₂×C₄×C₁₀₁

C₂³

Frattini

C₂×C₄

C₂²×D₁₀₁

Fitting

C₂×SD₁₆×C₁₀₁

C₂

socle

C₂²×C₁₀₁

C₂×D₄

hypercenter

C₂×SD₁₆

D₁₀₁

hypocenter

C₁₀₁

#460

LCS

G ▷ C₂×C₄×C₁₀₁ ▷ C₂×C₁₀₁ ▷ C₁₀₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×SD₁₆

N	H ≅ G/N	type
1. C₂×#13108061	C₂	semidirect
3. Q₈×Dic₁₀₁	C₂	semidirect
4. #15692184	C₂	semidirect
5. #15692177	C₂	semidirect
6. #15692176	C₂	semidirect
7. C₂×SD₁₆×C₁₀₁	C₂	semidirect
24. Q₈×C₁₀₁	D₄	semidirect
50. Dic₁₀₁	SD₁₆	semidirect
56. C₁₀₁	#460	semidirect
59. C₂×SD₁₆	D₁₀₁	semidirect
81. Q₈	C₂²⋊D₁₀₁	semidirect

H ≅ G/N

type

1. C₂×#13108061

C₂

semidirect

3. Q₈×Dic₁₀₁

C₂

semidirect

4. #15692184

C₂

semidirect

5. #15692177

C₂

semidirect

6. #15692176

C₂

semidirect

7. C₂×SD₁₆×C₁₀₁

C₂

semidirect

24. Q₈×C₁₀₁

D₄

semidirect

50. Dic₁₀₁

SD₁₆

semidirect

56. C₁₀₁

#460

semidirect

59. C₂×SD₁₆

D₁₀₁

semidirect

81. Q₈

C₂²⋊D₁₀₁

semidirect

Subgroups — #20265375

Conjugacy classes of subgroups

Complements