Group #22581415 — subgroups

Subgroups — C₂×C₄×D₅₅₆ #22581415

Order 8896.

426 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 183 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×D₅₅₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×D₁₃₉	C₂	1	2	yes	maximal, complemented
3	C₄×D₅₅₆	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊Dic₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×(C₂×D₁₃₉)⋊C₄	C₂	1	2	yes	maximal, complemented
6	C₂×C₄²×C₁₃₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₃₉	C₂²	1	2	yes	—
8	C₂×D₅₅₆	C₄	1	8	yes	direct product factor
9	C₂×D₅₅₆	C₂²	1	4	yes	—
10	C₂²×Dic₁₃₉	C₂²	1	2	yes	—
11	C₂×C₄×D₁₃₉	—	2	8	no	—
12	C₂×C₄×D₁₃₉	C₂²	1	8	yes	complemented
13	C₄⋊Dic₁₃₉	C₂²	1	4	yes	complemented
14	(C₂×D₁₃₉)⋊C₄	C₂²	1	8	yes	complemented
15	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×C₁₃₉	C₂²	1	1	yes	characteristic
18	C₄²×C₁₃₉	C₂²	1	4	yes	complemented
19	C₂²×D₁₃₉	—	2	4	no	—
20	C₂²×D₁₃₉	—	2	4	no	—
21	C₂²×D₁₃₉	C₂×C₄	1	8	yes	complemented
22	C₂²×D₁₃₉	C₂³	1	4	yes	—
23	D₅₅₆	C₂×C₄	1	16	yes	direct product factor
24	C₂×Dic₁₃₉	—	2	4	no	—
25	C₂×Dic₁₃₉	C₂³	1	4	yes	—
26	C₄×D₁₃₉	—	2	16	no	—
27	C₂³×C₁₃₉	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₁₃₉	—	2	2	no	—
29	C₂×C₄×C₁₃₉	C₂×C₄	1	4	yes	complemented
30	C₂×C₄×C₁₃₉	D₄	1	4	yes	complemented
31	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
32	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
33	C₂×C₄×C₁₃₉	C₂³	1	2	yes	—
34	C₂²×C₁₃₉	C₂²×C₄	1	2	yes	—
35	C₂²×C₁₃₉	C₂×D₄	1	2	yes	—
36	C₂²×C₁₃₉	#41	1	2	yes	—
37	C₂²×C₁₃₉	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₃₉	—	2	16	no	—
39	C₂×D₁₃₉	—	2	4	no	—
40	C₂×D₁₃₉	—	2	4	no	—
41	C₂×D₁₃₉	C₂²×C₄	1	8	yes	complemented
42	C₄×C₁₃₉	—	2	2	no	—
43	C₄×C₁₃₉	C₂²×C₄	1	4	yes	complemented
44	C₄×C₁₃₉	C₂×D₄	1	4	yes	complemented
45	Dic₁₃₉	—	2	4	no	—
46	C₂×C₁₃₉	C₄×D₄	1	4	yes	complemented
47	C₂×C₁₃₉	C₂³×C₄	1	1	yes	characteristic
48	C₂×C₁₃₉	C₂²×D₄	1	1	yes	characteristic
49	C₂×C₁₃₉	C₂×#41	1	1	yes	characteristic
50	D₁₃₉	—	2	8	no	—
51	C₁₃₉	C₂×C₄×D₄	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₄×D₄	—	139	1	no	maximal, Sylow
53	C₂²×D₄	—	139	1	no	—
54	C₂³×C₄	—	139	2	no	—
55	C₄×D₄	—	139	8	no	—
56	C₂×C₄⋊C₄	—	139	1	no	—
57	C₂×C₂²⋊C₄	—	139	2	no	—
58	C₂×C₄²	D₁₃₉	1	1	yes	characteristic, complemented
59	C₂⁴	—	139	2	no	—
60	C₂×D₄	—	139	8	no	—
61	C₂×D₄	—	139	4	no	—
62	C₂²×C₄	—	278	8	no	—
63	C₂²×C₄	—	139	2	no	—
64	C₂²×C₄	—	139	8	no	—
65	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
66	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
67	C₂²×C₄	C₂×D₁₃₉	1	1	yes	characteristic
68	C₄⋊C₄	—	139	4	no	—
69	C₂²⋊C₄	—	139	8	no	—
70	C₄²	C₂×D₁₃₉	1	4	yes	complemented
71	C₂³	—	278	4	no	—
72	C₂³	—	278	4	no	—
73	C₂³	—	139	8	no	—
74	C₂³	—	139	4	no	—
75	C₂³	C₂²×D₁₃₉	1	1	yes	characteristic
76	D₄	—	139	16	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	278	4	no	—
79	C₂×C₄	—	278	16	no	—
80	C₂×C₄	—	139	4	no	—
81	C₂×C₄	C₄×D₁₃₉	1	4	yes	complemented
82	C₂×C₄	D₅₅₆	1	4	yes	direct product factor
83	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
84	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
85	C₂×C₄	C₂²×D₁₃₉	1	2	yes	—
86	C₂²	—	278	16	no	—
87	C₂²	—	278	4	no	—
88	C₂²	—	278	4	no	—
89	C₂²	—	139	8	no	—
90	C₂²	C₂×C₄×D₁₃₉	1	2	yes	—
91	C₂²	C₂×D₅₅₆	1	2	yes	—
92	C₂²	#14555785	1	2	yes	—
93	C₂²	C₂³×D₁₃₉	1	1	yes	characteristic
94	C₄	—	2	2	no	—
95	C₄	—	278	4	no	—
96	C₄	C₂×C₄×D₁₃₉	1	4	yes	complemented
97	C₄	C₂×D₅₅₆	1	4	yes	direct product factor
98	C₂	—	278	8	no	—
99	C₂	C₄×D₅₅₆	1	4	yes	direct product factor
100	C₂	C₂²×C₄×D₁₃₉	1	1	yes	characteristic
101	C₂	C₂²×D₅₅₆	1	1	yes	characteristic
102	C₂	C₂×#14555785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₅₆

C₂

yes

maximal, characteristic

C₂²×C₄×D₁₃₉

C₂

yes

maximal, complemented

C₄×D₅₅₆

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂×(C₂×D₁₃₉)⋊C₄

C₂

yes

maximal, complemented

C₂×C₄²×C₁₃₉

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₃₉

C₂²

yes

—

C₂×D₅₅₆

C₄

yes

direct product factor

C₂×D₅₅₆

C₂²

yes

—

C₂²×Dic₁₃₉

C₂²

yes

—

C₂×C₄×D₁₃₉

—

C₂×C₄×D₁₃₉

C₂²

yes

complemented

C₄⋊Dic₁₃₉

C₂²

yes

complemented

(C₂×D₁₃₉)⋊C₄

C₂²

yes

complemented

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₂²×C₄×C₁₃₉

C₂²

yes

characteristic

C₄²×C₁₃₉

C₂²

yes

complemented

C₂²×D₁₃₉

—

C₂²×D₁₃₉

—

C₂²×D₁₃₉

C₂×C₄

yes

complemented

C₂²×D₁₃₉

C₂³

yes

—

D₅₅₆

C₂×C₄

yes

direct product factor

C₂×Dic₁₃₉

—

C₂×Dic₁₃₉

C₂³

yes

—

C₄×D₁₃₉

—

C₂³×C₁₃₉

C₂³

yes

characteristic

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂×C₄

yes

complemented

C₂×C₄×C₁₃₉

D₄

yes

complemented

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂×C₄×C₁₃₉

C₂³

yes

—

C₂²×C₁₃₉

C₂²×C₄

yes

—

C₂²×C₁₃₉

C₂×D₄

yes

—

C₂²×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

C₂⁴

yes

characteristic

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

—

C₂×D₁₃₉

C₂²×C₄

yes

complemented

C₄×C₁₃₉

—

C₄×C₁₃₉

C₂²×C₄

yes

complemented

C₄×C₁₃₉

C₂×D₄

yes

complemented

Dic₁₃₉

—

C₂×C₁₃₉

C₄×D₄

yes

complemented

C₂×C₁₃₉

C₂³×C₄

yes

characteristic

C₂×C₁₃₉

C₂²×D₄

yes

characteristic

C₂×C₁₃₉

C₂×#41

yes

characteristic

D₁₃₉

—

C₁₃₉

C₂×C₄×D₄

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

—

139

maximal, Sylow

C₂²×D₄

—

139

—

C₂³×C₄

—

139

—

C₄×D₄

—

139

—

C₂×C₄⋊C₄

—

139

—

C₂×C₂²⋊C₄

—

139

—

C₂×C₄²

D₁₃₉

yes

characteristic, complemented

C₂⁴

—

139

—

C₂×D₄

—

139

—

C₂×D₄

—

139

—

C₂²×C₄

—

278

—

C₂²×C₄

—

139

—

C₂²×C₄

—

139

—

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₂²×C₄

C₂×D₁₃₉

yes

characteristic

C₄⋊C₄

—

139

—

C₂²⋊C₄

—

139

—

C₄²

C₂×D₁₃₉

yes

complemented

C₂³

—

278

—

C₂³

—

278

—

C₂³

—

139

—

C₂³

—

139

—

C₂³

C₂²×D₁₃₉

yes

characteristic

D₄

—

139

—

C₂×C₄

—

C₂×C₄

—

278

—

C₂×C₄

—

278

—

C₂×C₄

—

139

—

C₂×C₄

C₄×D₁₃₉

yes

complemented

C₂×C₄

D₅₅₆

yes

direct product factor

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂×C₄

C₂²×D₁₃₉

yes

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

278

—

C₂²

—

139

—

C₂²

C₂×C₄×D₁₃₉

yes

—

C₂²

C₂×D₅₅₆

yes

—

C₂²

#14555785

yes

—

C₂²

C₂³×D₁₃₉

yes

characteristic

C₄

—

C₄

—

278

—

C₄

C₂×C₄×D₁₃₉

yes

complemented

C₄

C₂×D₅₅₆

yes

direct product factor

C₂

—

278

—

C₂

C₄×D₅₅₆

yes

direct product factor

100

C₂

C₂²×C₄×D₁₃₉

yes

characteristic

101

C₂

C₂²×D₅₅₆

yes

characteristic

102

C₂

C₂×#14555785

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₁₃₉
commutator	C₂×C₁₃₉	C₂³×C₄
Frattini	C₂²	C₂³×D₁₃₉
Fitting	C₂×C₄²×C₁₃₉	C₂
socle	C₂³×C₁₃₉	C₂³
hypercenter	C₂×C₄²	D₁₃₉
hypocenter	C₁₃₉	C₂×C₄×D₄
LCS	G ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

subgroup

G/H

center

C₂²×C₄

C₂×D₁₃₉

commutator

C₂×C₁₃₉

C₂³×C₄

Frattini

C₂²

C₂³×D₁₃₉

Fitting

C₂×C₄²×C₁₃₉

C₂

socle

C₂³×C₁₃₉

C₂³

hypercenter

C₂×C₄²

D₁₃₉

hypocenter

C₁₃₉

C₂×C₄×D₄

LCS

G ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄²

N	H ≅ G/N	type
2. C₂²×C₄×D₁₃₉	C₂	semidirect
3. C₄×D₅₅₆	C₂	direct (N × H)
4. C₂×C₄⋊Dic₁₃₉	C₂	semidirect
5. C₂×(C₂×D₁₃₉)⋊C₄	C₂	semidirect
6. C₂×C₄²×C₁₃₉	C₂	semidirect
8. C₂×D₅₅₆	C₄	direct (N × H)
12. C₂×C₄×D₁₃₉	C₂²	semidirect
13. C₄⋊Dic₁₃₉	C₂²	semidirect
14. (C₂×D₁₃₉)⋊C₄	C₂²	semidirect
18. C₄²×C₁₃₉	C₂²	semidirect
21. C₂²×D₁₃₉	C₂×C₄	semidirect
23. D₅₅₆	C₂×C₄	direct (N × H)
29. C₂×C₄×C₁₃₉	C₂×C₄	semidirect
30. C₂×C₄×C₁₃₉	D₄	semidirect
41. C₂×D₁₃₉	C₂²×C₄	semidirect
43. C₄×C₁₃₉	C₂²×C₄	semidirect
44. C₄×C₁₃₉	C₂×D₄	semidirect
46. C₂×C₁₃₉	C₄×D₄	semidirect
51. C₁₃₉	C₂×C₄×D₄	semidirect
58. C₂×C₄²	D₁₃₉	semidirect
70. C₄²	C₂×D₁₃₉	semidirect
81. C₂×C₄	C₄×D₁₃₉	semidirect
82. C₂×C₄	D₅₅₆	direct (N × H)
96. C₄	C₂×C₄×D₁₃₉	semidirect
97. C₄	C₂×D₅₅₆	direct (N × H)
99. C₂	C₄×D₅₅₆	direct (N × H)

H ≅ G/N

type

2. C₂²×C₄×D₁₃₉

C₂

semidirect

3. C₄×D₅₅₆

C₂

direct (N × H)

4. C₂×C₄⋊Dic₁₃₉

C₂

semidirect

5. C₂×(C₂×D₁₃₉)⋊C₄

C₂

semidirect

6. C₂×C₄²×C₁₃₉

C₂

semidirect

8. C₂×D₅₅₆

C₄

direct (N × H)

12. C₂×C₄×D₁₃₉

C₂²

semidirect

13. C₄⋊Dic₁₃₉

C₂²

semidirect

14. (C₂×D₁₃₉)⋊C₄

C₂²

semidirect

18. C₄²×C₁₃₉

C₂²

semidirect

21. C₂²×D₁₃₉

C₂×C₄

semidirect

23. D₅₅₆

C₂×C₄

direct (N × H)

29. C₂×C₄×C₁₃₉

C₂×C₄

semidirect

30. C₂×C₄×C₁₃₉

D₄

semidirect

41. C₂×D₁₃₉

C₂²×C₄

semidirect

43. C₄×C₁₃₉

C₂²×C₄

semidirect

44. C₄×C₁₃₉

C₂×D₄

semidirect

46. C₂×C₁₃₉

C₄×D₄

semidirect

51. C₁₃₉

C₂×C₄×D₄

semidirect

58. C₂×C₄²

D₁₃₉

semidirect

70. C₄²

C₂×D₁₃₉

semidirect

81. C₂×C₄

C₄×D₁₃₉

semidirect

82. C₂×C₄

D₅₅₆

direct (N × H)

96. C₄

C₂×C₄×D₁₃₉

semidirect

97. C₄

C₂×D₅₅₆

direct (N × H)

99. C₂

C₄×D₅₅₆

direct (N × H)

Subgroups — C₂×C₄×D₅₅₆ #22581415

Conjugacy classes of subgroups

Complements

Subgroups — C2×C4×D556 #22581415

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄×D₅₅₆ #22581415