Group #20265678 — subgroups

Subgroups — C₂×#15692236 #20265678

Order 6464.

342 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂³×Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
2	#15692236	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13108053	C₂	1	2	yes	maximal, complemented
4	C₂×C₂²⋊Dic₁₀₁	C₂	1	2	yes	maximal, complemented
5	C₂×C₂²⋊Dic₁₀₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₄×C₁₀₁	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₀₁	—	2	4	no	—
8	C₂²×Dic₁₀₁	C₂²	1	2	yes	—
9	C₂²×Dic₁₀₁	C₂²	1	2	yes	—
10	C₂²×Dic₁₀₁	C₂²	1	4	yes	complemented
11	#13108053	C₂²	1	8	yes	complemented
12	C₂²⋊Dic₁₀₁	C₂²	1	4	yes	complemented
13	C₂²⋊Dic₁₀₁	C₂²	1	8	yes	complemented
14	C₂⁴×C₁₀₁	C₂²	1	1	yes	characteristic
15	C₂⁴×C₁₀₁	C₂²	1	1	yes	characteristic
16	C₂×D₄×C₁₀₁	—	2	4	no	—
17	C₂×D₄×C₁₀₁	C₂²	1	4	yes	—
18	C₂²×C₄×C₁₀₁	C₂²	1	1	yes	characteristic
19	C₂³×C₁₀₁	—	2	4	no	—
20	C₂³×C₁₀₁	—	2	2	no	—
21	C₂³×C₁₀₁	—	2	4	no	—
22	C₂³×C₁₀₁	D₄	1	4	yes	—
23	C₂³×C₁₀₁	C₂³	1	2	yes	—
24	C₂³×C₁₀₁	C₂³	1	2	yes	—
25	C₂³×C₁₀₁	C₂³	1	1	yes	characteristic
26	D₄×C₁₀₁	—	2	8	no	—
27	C₂×C₄×C₁₀₁	—	2	2	no	—
28	C₂×C₄×C₁₀₁	C₂³	1	2	yes	—
29	C₂×Dic₁₀₁	—	2	4	no	—
30	C₂×Dic₁₀₁	—	2	8	no	—
31	C₂×Dic₁₀₁	—	2	4	no	—
32	C₂×Dic₁₀₁	C₂³	1	4	yes	—
33	C₂×Dic₁₀₁	C₂³	1	4	yes	—
34	C₂²×C₁₀₁	—	2	8	no	—
35	C₂²×C₁₀₁	—	2	4	no	—
36	C₂²×C₁₀₁	—	2	2	no	—
37	C₂²×C₁₀₁	—	2	4	no	—
38	C₂²×C₁₀₁	—	4	4	no	—
39	C₂²×C₁₀₁	C₂×D₄	1	4	yes	—
40	C₂²×C₁₀₁	C₂×D₄	1	2	yes	—
41	C₂²×C₁₀₁	#41	1	4	yes	—
42	C₂²×C₁₀₁	C₂⁴	1	1	yes	characteristic
43	C₄×C₁₀₁	—	2	2	no	—
44	Dic₁₀₁	—	2	4	no	—
45	Dic₁₀₁	—	2	4	no	—
46	C₂×C₁₀₁	—	2	4	no	—
47	C₂×C₁₀₁	—	4	2	no	—
48	C₂×C₁₀₁	#123	1	4	yes	complemented
49	C₂×C₁₀₁	C₂²×D₄	1	1	yes	characteristic
50	C₂×C₁₀₁	C₂×#41	1	2	yes	—
51	C₁₀₁	C₂×#123	1	1	yes	characteristic, complemented, Sylow
52	C₂×#123	—	101	1	no	maximal, Sylow
53	C₂²×D₄	D₁₀₁	1	1	yes	characteristic
54	C₂³×C₄	—	101	1	no	—
55	#123	—	101	8	no	—
56	C₂×C₄⋊C₄	—	101	2	no	—
57	C₂×C₂²⋊C₄	—	101	1	no	—
58	C₂×C₂²⋊C₄	—	101	2	no	—
59	C₂⁴	C₂×D₁₀₁	1	1	yes	characteristic
60	C₂⁴	C₂×D₁₀₁	1	1	yes	characteristic
61	C₂×D₄	—	2	4	no	—
62	C₂×D₄	C₂×D₁₀₁	1	4	yes	—
63	C₂²×C₄	—	101	4	no	—
64	C₂²×C₄	—	101	2	no	—
65	C₂²×C₄	—	101	2	no	—
66	C₂²×C₄	—	202	4	no	—
67	C₂²×C₄	C₂×D₁₀₁	1	1	yes	characteristic
68	C₄⋊C₄	—	101	8	no	—
69	C₂²⋊C₄	—	101	4	no	—
70	C₂²⋊C₄	—	101	8	no	—
71	C₂³	—	2	4	no	—
72	C₂³	—	2	2	no	—
73	C₂³	—	2	4	no	—
74	C₂³	C₂²⋊D₁₀₁	1	4	yes	—
75	C₂³	C₂²×D₁₀₁	1	2	yes	—
76	C₂³	C₂²×D₁₀₁	1	2	yes	—
77	C₂³	C₂²×D₁₀₁	1	1	yes	characteristic
78	D₄	—	2	8	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	101	4	no	—
81	C₂×C₄	—	101	4	no	—
82	C₂×C₄	—	202	8	no	—
83	C₂×C₄	—	202	4	no	—
84	C₂×C₄	—	202	4	no	—
85	C₂×C₄	C₂²×D₁₀₁	1	2	yes	—
86	C₂²	—	2	4	no	—
87	C₂²	—	2	8	no	—
88	C₂²	—	2	2	no	—
89	C₂²	—	2	4	no	—
90	C₂²	—	4	4	no	—
91	C₂²	C₂×C₂²⋊D₁₀₁	1	4	yes	—
92	C₂²	C₂×C₂²⋊D₁₀₁	1	2	yes	—
93	C₂²	#13108073	1	4	yes	—
94	C₂²	C₂³×D₁₀₁	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	202	4	no	—
97	C₄	—	202	4	no	—
98	C₂	—	2	4	no	—
99	C₂	—	4	2	no	—
100	C₂	#15692236	1	4	yes	direct product factor
101	C₂	C₂²×C₂²⋊D₁₀₁	1	1	yes	characteristic
102	C₂	C₂×#13108073	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Dic₁₀₁

C₂

yes

maximal, characteristic, complemented

#15692236

C₂

yes

maximal, direct product factor

C₂×#13108053

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₀₁

C₂

yes

maximal, complemented

C₂×C₂²⋊Dic₁₀₁

C₂

yes

maximal, characteristic, complemented

C₂²×D₄×C₁₀₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₀₁

—

C₂²×Dic₁₀₁

C₂²

yes

—

C₂²×Dic₁₀₁

C₂²

yes

—

C₂²×Dic₁₀₁

C₂²

yes

complemented

#13108053

C₂²

yes

complemented

C₂²⋊Dic₁₀₁

C₂²

yes

complemented

C₂²⋊Dic₁₀₁

C₂²

yes

complemented

C₂⁴×C₁₀₁

C₂²

yes

characteristic

C₂⁴×C₁₀₁

C₂²

yes

characteristic

C₂×D₄×C₁₀₁

—

C₂×D₄×C₁₀₁

C₂²

yes

—

C₂²×C₄×C₁₀₁

C₂²

yes

characteristic

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

—

C₂³×C₁₀₁

D₄

yes

—

C₂³×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂³

yes

—

C₂³×C₁₀₁

C₂³

yes

characteristic

D₄×C₁₀₁

—

C₂×C₄×C₁₀₁

—

C₂×C₄×C₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂×Dic₁₀₁

C₂³

yes

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

—

C₂²×C₁₀₁

C₂×D₄

yes

—

C₂²×C₁₀₁

C₂×D₄

yes

—

C₂²×C₁₀₁

#41

yes

—

C₂²×C₁₀₁

C₂⁴

yes

characteristic

C₄×C₁₀₁

—

Dic₁₀₁

—

Dic₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

—

C₂×C₁₀₁

#123

yes

complemented

C₂×C₁₀₁

C₂²×D₄

yes

characteristic

C₂×C₁₀₁

C₂×#41

yes

—

C₁₀₁

C₂×#123

yes

characteristic, complemented, Sylow

C₂×#123

—

101

maximal, Sylow

C₂²×D₄

D₁₀₁

yes

characteristic

C₂³×C₄

—

101

—

#123

—

101

—

C₂×C₄⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂×C₂²⋊C₄

—

101

—

C₂⁴

C₂×D₁₀₁

yes

characteristic

C₂⁴

C₂×D₁₀₁

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂×D₁₀₁

yes

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

101

—

C₂²×C₄

—

202

—

C₂²×C₄

C₂×D₁₀₁

yes

characteristic

C₄⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂²⋊C₄

—

101

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₂²⋊D₁₀₁

yes

—

C₂³

C₂²×D₁₀₁

yes

—

C₂³

C₂²×D₁₀₁

yes

—

C₂³

C₂²×D₁₀₁

yes

characteristic

D₄

—

C₂×C₄

—

C₂×C₄

—

101

—

C₂×C₄

—

101

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

—

202

—

C₂×C₄

C₂²×D₁₀₁

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊D₁₀₁

yes

—

C₂²

C₂×C₂²⋊D₁₀₁

yes

—

C₂²

#13108073

yes

—

C₂²

C₂³×D₁₀₁

yes

characteristic

C₄

—

C₄

—

202

—

C₄

—

202

—

C₂

—

C₂

—

100

C₂

#15692236

yes

direct product factor

101

C₂

C₂²×C₂²⋊D₁₀₁

yes

characteristic

102

C₂

C₂×#13108073

yes

—

	subgroup	G/H
center	C₂³	C₂²×D₁₀₁
commutator	C₂²×C₁₀₁	C₂⁴
Frattini	C₂²	C₂³×D₁₀₁
Fitting	C₂²×D₄×C₁₀₁	C₂
socle	C₂³×C₁₀₁	C₂³
hypercenter	C₂²×D₄	D₁₀₁
hypocenter	C₁₀₁	C₂×#123
LCS	G ▷ C₂²×C₁₀₁ ▷ C₁₀₁
UCS	C₁ ◁ C₂³ ◁ C₂²×D₄

subgroup

G/H

center

C₂³

C₂²×D₁₀₁

commutator

C₂²×C₁₀₁

C₂⁴

Frattini

C₂²

C₂³×D₁₀₁

Fitting

C₂²×D₄×C₁₀₁

C₂

socle

C₂³×C₁₀₁

C₂³

hypercenter

C₂²×D₄

D₁₀₁

C₁₀₁

G ▷ C₂²×C₁₀₁ ▷ C₁₀₁

C₁ ◁ C₂³ ◁ C₂²×D₄

N	H ≅ G/N	type
1. C₂³×Dic₁₀₁	C₂	semidirect
2. #15692236	C₂	direct (N × H)
3. C₂×#13108053	C₂	semidirect
4. C₂×C₂²⋊Dic₁₀₁	C₂	semidirect
5. C₂×C₂²⋊Dic₁₀₁	C₂	semidirect
10. C₂²×Dic₁₀₁	C₂²	semidirect
11. #13108053	C₂²	semidirect
12. C₂²⋊Dic₁₀₁	C₂²	semidirect
13. C₂²⋊Dic₁₀₁	C₂²	semidirect
48. C₂×C₁₀₁	#123	semidirect
51. C₁₀₁	C₂×#123	semidirect
100. C₂	#15692236	direct (N × H)

H ≅ G/N

type

1. C₂³×Dic₁₀₁

C₂

semidirect

2. #15692236

C₂

direct (N × H)

3. C₂×#13108053

C₂

semidirect

4. C₂×C₂²⋊Dic₁₀₁

C₂

semidirect

5. C₂×C₂²⋊Dic₁₀₁

C₂

semidirect

10. C₂²×Dic₁₀₁

C₂²

semidirect

11. #13108053

C₂²

semidirect

12. C₂²⋊Dic₁₀₁

C₂²

semidirect

13. C₂²⋊Dic₁₀₁

C₂²

semidirect

48. C₂×C₁₀₁

#123

semidirect

51. C₁₀₁

C₂×#123

semidirect

100. C₂

#15692236

direct (N × H)

Subgroups — C₂×#15692236 #20265678

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15692236 #20265678

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15692236 #20265678