Group #20265901 — subgroups

Subgroups — #20265901

Order 6464.

212 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 99 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	#15692250	C₂	1	4	yes	maximal, complemented
2	#15692244	C₂	1	2	yes	maximal
3	#15692235	C₂	1	2	yes	maximal, complemented
4	Q₈×Dic₁₀₁	C₂	1	2	yes	maximal, complemented
5	C₄×Dic₂₀₂	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×D₄₀₄	C₂	1	1	yes	maximal, characteristic
7	#15692213	C₂	1	2	yes	maximal, complemented
8	C₄⋊Q₈×C₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×Dic₂₀₂	C₂²	1	1	yes	characteristic
10	C₂×D₄₀₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₁₀₁	C₂²	1	2	yes	—
12	C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₁₀₁	C₂²	1	1	yes	characteristic
14	C₄⋊Dic₁₀₁	C₂²	1	4	yes	—
15	#13108053	C₂²	1	2	yes	—
16	#13108051	C₂²	1	4	yes	—
17	#13108051	C₂²	1	2	yes	—
18	#13108051	C₂²	1	4	yes	—
19	C₄×Dic₁₀₁	C₂²	1	4	yes	—
20	C₄×Dic₁₀₁	C₂²	1	2	yes	—
21	C₂×Q₈×C₁₀₁	C₂²	1	2	yes	—
22	C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
23	C₄⋊C₄×C₁₀₁	C₂²	1	2	yes	—
24	C₄²×C₁₀₁	C₂²	1	1	yes	characteristic
25	C₂²×D₁₀₁	C₂³	1	2	yes	—
26	Q₈×C₁₀₁	—	2	4	no	—
27	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₀₁	C₂³	1	4	yes	—
29	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
30	C₂×C₄×C₁₀₁	C₂³	1	1	yes	characteristic
31	C₂×Dic₁₀₁	C₂³	1	2	yes	—
32	C₂×Dic₁₀₁	C₂³	1	2	yes	—
33	C₂×Dic₁₀₁	C₂³	1	2	yes	—
34	D₄₀₄	—	2	2	no	—
35	C₄×D₁₀₁	—	2	4	no	—
36	Dic₂₀₂	—	2	2	no	—
37	C₂²×C₁₀₁	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₀₁	—	2	2	no	—
39	C₂×D₁₀₁	—	2	2	no	—
40	C₂×D₁₀₁	—	2	2	no	—
41	C₄×C₁₀₁	—	2	4	no	—
42	C₄×C₁₀₁	—	2	1	no	—
43	C₄×C₁₀₁	#41	1	2	yes	—
44	C₄×C₁₀₁	#41	1	2	yes	—
45	Dic₁₀₁	—	2	2	no	—
46	Dic₁₀₁	—	2	2	no	—
47	Dic₁₀₁	—	2	2	no	—
48	C₂×C₁₀₁	C₂×#41	1	1	yes	characteristic
49	C₂×C₁₀₁	C₂×#41	1	1	yes	characteristic
50	C₂×C₁₀₁	2₋¹⁺⁴	1	1	yes	characteristic
51	D₁₀₁	—	4	2	no	—
52	C₁₀₁	#556	1	1	yes	characteristic, complemented, Sylow
53	#556	—	101	1	no	maximal, Sylow
54	C₄⋊Q₈	D₁₀₁	1	1	yes	characteristic, complemented
55	#126	—	101	4	no	—
56	#124	—	101	2	no	—
57	C₂²⋊Q₈	—	101	2	no	—
58	C₄×Q₈	—	101	2	no	—
59	C₄×Q₈	—	101	1	no	—
60	C₄×D₄	—	101	1	no	—
61	#117	—	101	2	no	—
62	C₂×Q₈	—	101	1	no	—
63	C₂×Q₈	C₂×D₁₀₁	1	2	yes	—
64	C₂×D₄	—	101	1	no	—
65	C₂²×C₄	—	101	2	no	—
66	C₄⋊C₄	—	101	2	no	—
67	C₄⋊C₄	—	101	1	no	—
68	C₄⋊C₄	—	101	1	no	—
69	C₄⋊C₄	—	101	4	no	—
70	C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
71	C₄⋊C₄	C₂×D₁₀₁	1	2	yes	—
72	C₂²⋊C₄	—	101	4	no	—
73	C₂²⋊C₄	—	101	2	no	—
74	C₂²⋊C₄	—	101	4	no	—
75	C₄²	—	101	4	no	—
76	C₄²	—	101	2	no	—
77	C₄²	C₂×D₁₀₁	1	1	yes	characteristic
78	C₂³	—	101	2	no	—
79	Q₈	—	2	4	no	—
80	Q₈	—	202	2	no	—
81	D₄	—	202	2	no	—
82	C₂×C₄	—	101	2	no	—
83	C₂×C₄	—	101	2	no	—
84	C₂×C₄	—	101	2	no	—
85	C₂×C₄	—	202	4	no	—
86	C₂×C₄	C₂²×D₁₀₁	1	4	yes	—
87	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
88	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
89	C₂×C₄	C₂²×D₁₀₁	1	1	yes	characteristic
90	C₂²	—	202	2	no	—
91	C₂²	—	202	2	no	—
92	C₂²	—	202	2	no	—
93	C₂²	C₂³×D₁₀₁	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	2	1	no	—
96	C₄	—	202	2	no	—
97	C₄	—	202	2	no	—
98	C₄	—	202	2	no	—
99	C₄	#13108073	1	2	yes	—
100	C₄	#13108075	1	2	yes	—
101	C₂	—	404	2	no	—
102	C₂	C₂×#13108073	1	1	yes	characteristic
103	C₂	C₂×#13108075	1	1	yes	characteristic
104	C₂	#15692303	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15692250

C₂

yes

maximal, complemented

#15692244

C₂

yes

maximal

#15692235

C₂

yes

maximal, complemented

Q₈×Dic₁₀₁

C₂

yes

maximal, complemented

C₄×Dic₂₀₂

C₂

yes

maximal, characteristic, complemented

C₄×D₄₀₄

C₂

yes

maximal, characteristic

#15692213

C₂

yes

maximal, complemented

C₄⋊Q₈×C₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₀₂

C₂²

yes

characteristic

C₂×D₄₀₄

C₂²

yes

characteristic

C₂×C₄×D₁₀₁

C₂²

yes

—

C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₄⋊Dic₁₀₁

C₂²

yes

characteristic

C₄⋊Dic₁₀₁

C₂²

yes

—

#13108053

C₂²

yes

—

#13108051

C₂²

yes

—

#13108051

C₂²

yes

—

#13108051

C₂²

yes

—

C₄×Dic₁₀₁

C₂²

yes

—

C₄×Dic₁₀₁

C₂²

yes

—

C₂×Q₈×C₁₀₁

C₂²

yes

—

C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₄⋊C₄×C₁₀₁

C₂²

yes

—

C₄²×C₁₀₁

C₂²

yes

characteristic

C₂²×D₁₀₁

C₂³

yes

—

Q₈×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×C₄×C₁₀₁

C₂³

yes

characteristic

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

D₄₀₄

—

C₄×D₁₀₁

—

Dic₂₀₂

—

C₂²×C₁₀₁

C₂⁴

yes

characteristic

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₂×D₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

—

C₄×C₁₀₁

#41

yes

—

C₄×C₁₀₁

#41

yes

—

Dic₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

C₂×#41

yes

characteristic

C₂×C₁₀₁

C₂×#41

yes

characteristic

C₂×C₁₀₁

2₋¹⁺⁴

yes

characteristic

D₁₀₁

—

C₁₀₁

#556

yes

characteristic, complemented, Sylow

#556

—

101

maximal, Sylow

C₄⋊Q₈

D₁₀₁

yes

characteristic, complemented

#126

—

101

—

#124

—

101

—

C₂²⋊Q₈

—

101

—

C₄×Q₈

—

101

—

C₄×Q₈

—

101

—

C₄×D₄

—

101

—

#117

—

101

—

C₂×Q₈

—

101

—

C₂×Q₈

C₂×D₁₀₁

yes

—

C₂×D₄

—

101

—

C₂²×C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

—

101

—

C₄⋊C₄

C₂×D₁₀₁

yes

—

C₄⋊C₄

C₂×D₁₀₁

yes

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₄²

—

101

—

C₄²

—

101

—

C₄²

C₂×D₁₀₁

yes

characteristic

C₂³

—

101

—

Q₈

—

Q₈

—

202

—

D₄

—

202

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

202

—

C₂×C₄

C₂²×D₁₀₁

yes

—

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

C₂×C₄

C₂²×D₁₀₁

yes

characteristic

C₂²

—

202

—

C₂²

—

202

—

C₂²

—

202

—

C₂²

C₂³×D₁₀₁

yes

characteristic

C₄

—

C₄

—

C₄

—

202

—

C₄

—

202

—

C₄

—

202

—

C₄

#13108073

yes

—

100

C₄

#13108075

yes

—

101

C₂

—

404

—

102

C₂

C₂×#13108073

yes

characteristic

103

C₂

C₂×#13108075

yes

characteristic

104

C₂

#15692303

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₁₀₁
commutator, socle	C₂²×C₁₀₁	C₂⁴
Fitting	C₄⋊Q₈×C₁₀₁	C₂
hypercenter	C₄⋊Q₈	D₁₀₁
hypocenter	C₁₀₁	#556
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₁₀₁

commutator, socle

C₂²×C₁₀₁

C₂⁴

Fitting

C₄⋊Q₈×C₁₀₁

C₂

hypercenter

C₄⋊Q₈

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #15692250	C₂	semidirect
3. #15692235	C₂	semidirect
4. Q₈×Dic₁₀₁	C₂	semidirect
5. C₄×Dic₂₀₂	C₂	semidirect
7. #15692213	C₂	semidirect
8. C₄⋊Q₈×C₁₀₁	C₂	semidirect
52. C₁₀₁	#556	semidirect
54. C₄⋊Q₈	D₁₀₁	semidirect

H ≅ G/N

type

1. #15692250

C₂

semidirect

3. #15692235

C₂

semidirect

4. Q₈×Dic₁₀₁

C₂

semidirect

5. C₄×Dic₂₀₂

C₂

semidirect

7. #15692213

C₂

semidirect

8. C₄⋊Q₈×C₁₀₁

C₂

semidirect

52. C₁₀₁

#556

semidirect

54. C₄⋊Q₈

D₁₀₁

semidirect

Subgroups — #20265901

Conjugacy classes of subgroups

Complements