Group #36712593 — subgroups

Subgroups — C₂²×C₄²×Dic₈₉ #36712593

Order 22784.

8094 conjugacy classes of subgroups; only the 5547 normal subgroups are shown.

normal subgroups only

This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 5547 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×C₄×Dic₈₉	C₂	1	6	yes	maximal
2	C₂×C₄²×Dic₈₉	C₂	1	24	yes	maximal, direct product factor
3	C₂³×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₂⁴×Dic₈₉	C₂²	1	4	yes	—
5	C₂²×C₄×Dic₈₉	C₄	1	96	yes	direct product factor
6	C₂²×C₄×Dic₈₉	C₂²	1	72	yes	—
7	C₄²×Dic₈₉	C₂²	1	64	yes	direct product factor
8	C₂⁴×C₄×C₈₉	C₂²	1	3	yes	—
9	C₂²×C₄²×C₈₉	C₄	1	16	yes	complemented
10	C₂²×C₄²×C₈₉	C₂²	1	12	yes	—
11	C₂³×Dic₈₉	C₂×C₄	1	96	yes	—
12	C₂³×Dic₈₉	C₂³	1	24	yes	—
13	C₂×C₄×Dic₈₉	C₂×C₄	1	576	yes	direct product factor
14	C₂×C₄×Dic₈₉	C₂³	1	96	yes	—
15	C₂⁵×C₈₉	C₂³	1	1	yes	characteristic
16	C₂³×C₄×C₈₉	C₂×C₄	1	48	yes	—
17	C₂³×C₄×C₈₉	C₂×C₄	1	24	yes	—
18	C₂³×C₄×C₈₉	C₂³	1	18	yes	—
19	C₂×C₄²×C₈₉	C₂×C₄	1	96	yes	complemented
20	C₂×C₄²×C₈₉	C₂³	1	16	yes	—
21	C₂²×Dic₈₉	C₄²	1	256	yes	direct product factor
22	C₂²×Dic₈₉	C₂²×C₄	1	288	yes	—
23	C₂²×Dic₈₉	C₂⁴	1	16	yes	—
24	C₄×Dic₈₉	C₂²×C₄	1	384	yes	direct product factor
25	C₂⁴×C₈₉	C₂²×C₄	1	24	yes	—
26	C₂⁴×C₈₉	C₂²×C₄	1	4	yes	—
27	C₂⁴×C₈₉	C₂⁴	1	3	yes	—
28	C₂²×C₄×C₈₉	C₄²	1	192	yes	complemented
29	C₂²×C₄×C₈₉	C₂²×C₄	1	144	yes	—
30	C₂²×C₄×C₈₉	C₂²×C₄	1	72	yes	—
31	C₂²×C₄×C₈₉	C₂⁴	1	12	yes	—
32	C₄²×C₈₉	C₂²×C₄	1	64	yes	complemented
33	C₂³×C₈₉	C₂×C₄²	1	64	yes	—
34	C₂³×C₈₉	C₂×C₄²	1	48	yes	—
35	C₂³×C₈₉	C₂³×C₄	1	36	yes	—
36	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
37	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
38	C₂×C₄×C₈₉	C₂×C₄²	1	288	yes	complemented
39	C₂×C₄×C₈₉	C₂³×C₄	1	48	yes	—
40	C₂×C₄×C₈₉	C₂³×C₄	1	24	yes	—
41	C₂×Dic₈₉	C₂×C₄²	1	384	yes	direct product factor
42	C₂×Dic₈₉	C₂³×C₄	1	96	yes	—
43	C₂²×C₈₉	C₄³	1	64	yes	complemented
44	C₂²×C₈₉	C₂²×C₄²	1	48	yes	—
45	C₂²×C₈₉	C₂²×C₄²	1	36	yes	—
46	C₂²×C₈₉	C₂⁴×C₄	1	6	yes	—
47	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
48	C₄×C₈₉	C₂²×C₄²	1	48	yes	complemented
49	Dic₈₉	C₂²×C₄²	1	64	yes	direct product factor
50	C₂²×C₄³	—	89	1	no	maximal, Sylow
51	C₂×C₈₉	C₂×C₄³	1	24	yes	complemented
52	C₂×C₈₉	C₂³×C₄²	1	4	yes	—
53	C₂×C₈₉	C₂³×C₄²	1	3	yes	—
54	C₂³×C₄²	—	89	6	no	—
55	C₂³×C₄²	D₈₉	1	1	yes	characteristic
56	C₂×C₄³	—	89	24	no	—
57	C₈₉	C₂²×C₄³	1	1	yes	characteristic, complemented, Sylow
58	C₂⁴×C₄	—	89	4	no	—
59	C₂⁴×C₄	C₂×D₈₉	1	3	yes	—
60	C₂²×C₄²	—	89	96	no	—
61	C₂²×C₄²	—	89	72	no	—
62	C₂²×C₄²	Dic₈₉	1	16	yes	direct product factor
63	C₂²×C₄²	C₂×D₈₉	1	12	yes	—
64	C₄³	—	89	64	no	—
65	C₂⁵	C₂²×D₈₉	1	1	yes	characteristic
66	C₂³×C₄	—	89	96	no	—
67	C₂³×C₄	—	89	24	no	—
68	C₂³×C₄	C₄×D₈₉	1	48	yes	—
69	C₂³×C₄	C₂×Dic₈₉	1	24	yes	—
70	C₂³×C₄	C₂²×D₈₉	1	18	yes	—
71	C₂×C₄²	—	89	576	no	—
72	C₂×C₄²	—	89	96	no	—
73	C₂×C₄²	C₂×Dic₈₉	1	96	yes	direct product factor
74	C₂×C₄²	C₂²×D₈₉	1	16	yes	—
75	C₂⁴	C₂×C₄×D₈₉	1	24	yes	—
76	C₂⁴	C₂²×Dic₈₉	1	4	yes	—
77	C₂⁴	C₂³×D₈₉	1	3	yes	—
78	C₂²×C₄	—	89	288	no	—
79	C₂²×C₄	—	89	16	no	—
80	C₂²×C₄	—	89	256	no	—
81	C₂²×C₄	C₄×Dic₈₉	1	192	yes	direct product factor
82	C₂²×C₄	C₂×C₄×D₈₉	1	144	yes	—
83	C₂²×C₄	C₂²×Dic₈₉	1	72	yes	—
84	C₂²×C₄	C₂³×D₈₉	1	12	yes	—
85	C₄²	—	89	384	no	—
86	C₄²	C₂²×Dic₈₉	1	64	yes	direct product factor
87	C₂³	C₄²×D₈₉	1	64	yes	—
88	C₂³	C₂×C₄×Dic₈₉	1	48	yes	—
89	C₂³	C₂²×C₄×D₈₉	1	36	yes	—
90	C₂³	C₂³×Dic₈₉	1	6	yes	—
91	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
92	C₂×C₄	—	89	96	no	—
93	C₂×C₄	—	89	384	no	—
94	C₂×C₄	C₂×C₄×Dic₈₉	1	288	yes	direct product factor
95	C₂×C₄	C₂²×C₄×D₈₉	1	48	yes	—
96	C₂×C₄	C₂³×Dic₈₉	1	24	yes	—
97	C₂²	C₄²×Dic₈₉	1	64	yes	direct product factor
98	C₂²	C₂×C₄²×D₈₉	1	48	yes	—
99	C₂²	C₂²×C₄×Dic₈₉	1	36	yes	—
100	C₂²	C₂³×C₄×D₈₉	1	6	yes	—
101	C₂²	C₂⁴×Dic₈₉	1	1	yes	characteristic
102	C₄	—	89	64	no	—
103	C₄	C₂²×C₄×Dic₈₉	1	48	yes	direct product factor
104	C₂	C₂×C₄²×Dic₈₉	1	24	yes	direct product factor
105	C₂	C₂²×C₄²×D₈₉	1	4	yes	—
106	C₂	C₂³×C₄×Dic₈₉	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄×Dic₈₉

C₂

yes

maximal

C₂×C₄²×Dic₈₉

C₂

yes

maximal, direct product factor

C₂³×C₄²×C₈₉

C₂

yes

maximal, characteristic

C₂⁴×Dic₈₉

C₂²

yes

—

C₂²×C₄×Dic₈₉

C₄

yes

direct product factor

C₂²×C₄×Dic₈₉

C₂²

yes

—

C₄²×Dic₈₉

C₂²

yes

direct product factor

C₂⁴×C₄×C₈₉

C₂²

yes

—

C₂²×C₄²×C₈₉

C₄

yes

complemented

C₂²×C₄²×C₈₉

C₂²

yes

—

C₂³×Dic₈₉

C₂×C₄

yes

—

C₂³×Dic₈₉

C₂³

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

576

yes

direct product factor

C₂×C₄×Dic₈₉

C₂³

yes

—

C₂⁵×C₈₉

C₂³

yes

characteristic

C₂³×C₄×C₈₉

C₂×C₄

yes

—

C₂³×C₄×C₈₉

C₂×C₄

yes

—

C₂³×C₄×C₈₉

C₂³

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

complemented

C₂×C₄²×C₈₉

C₂³

yes

—

C₂²×Dic₈₉

C₄²

256

yes

direct product factor

C₂²×Dic₈₉

C₂²×C₄

288

yes

—

C₂²×Dic₈₉

C₂⁴

yes

—

C₄×Dic₈₉

C₂²×C₄

384

yes

direct product factor

C₂⁴×C₈₉

C₂²×C₄

yes

—

C₂⁴×C₈₉

C₂²×C₄

yes

—

C₂⁴×C₈₉

C₂⁴

yes

—

C₂²×C₄×C₈₉

C₄²

192

yes

complemented

C₂²×C₄×C₈₉

C₂²×C₄

144

yes

—

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

—

C₄²×C₈₉

C₂²×C₄

yes

complemented

C₂³×C₈₉

C₂×C₄²

yes

—

C₂³×C₈₉

C₂×C₄²

yes

—

C₂³×C₈₉

C₂³×C₄

yes

—

C₂³×C₈₉

C₂³×C₄

yes

—

C₂³×C₈₉

C₂⁵

yes

characteristic

C₂×C₄×C₈₉

C₂×C₄²

288

yes

complemented

C₂×C₄×C₈₉

C₂³×C₄

yes

—

C₂×C₄×C₈₉

C₂³×C₄

yes

—

C₂×Dic₈₉

C₂×C₄²

384

yes

direct product factor

C₂×Dic₈₉

C₂³×C₄

yes

—

C₂²×C₈₉

C₄³

yes

complemented

C₂²×C₈₉

C₂²×C₄²

yes

—

C₂²×C₈₉

C₂²×C₄²

yes

—

C₂²×C₈₉

C₂⁴×C₄

yes

—

C₂²×C₈₉

C₂⁴×C₄

yes

characteristic

C₄×C₈₉

C₂²×C₄²

yes

complemented

Dic₈₉

C₂²×C₄²

yes

direct product factor

C₂²×C₄³

—

maximal, Sylow

C₂×C₈₉

C₂×C₄³

yes

complemented

C₂×C₈₉

C₂³×C₄²

yes

—

C₂×C₈₉

C₂³×C₄²

yes

—

C₂³×C₄²

—

C₂³×C₄²

D₈₉

yes

characteristic

C₂×C₄³

—

C₈₉

C₂²×C₄³

yes

characteristic, complemented, Sylow

C₂⁴×C₄

—

C₂⁴×C₄

C₂×D₈₉

yes

—

C₂²×C₄²

—

C₂²×C₄²

—

C₂²×C₄²

Dic₈₉

yes

direct product factor

C₂²×C₄²

C₂×D₈₉

yes

—

C₄³

—

C₂⁵

C₂²×D₈₉

yes

characteristic

C₂³×C₄

—

C₂³×C₄

—

C₂³×C₄

C₄×D₈₉

yes

—

C₂³×C₄

C₂×Dic₈₉

yes

—

C₂³×C₄

C₂²×D₈₉

yes

—

C₂×C₄²

—

576

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₈₉

yes

direct product factor

C₂×C₄²

C₂²×D₈₉

yes

—

C₂⁴

C₂×C₄×D₈₉

yes

—

C₂⁴

C₂²×Dic₈₉

yes

—

C₂⁴

C₂³×D₈₉

yes

—

C₂²×C₄

—

288

—

C₂²×C₄

—

C₂²×C₄

—

256

—

C₂²×C₄

C₄×Dic₈₉

192

yes

direct product factor

C₂²×C₄

C₂×C₄×D₈₉

144

yes

—

C₂²×C₄

C₂²×Dic₈₉

yes

—

C₂²×C₄

C₂³×D₈₉

yes

—

C₄²

—

384

—

C₄²

C₂²×Dic₈₉

yes

direct product factor

C₂³

C₄²×D₈₉

yes

—

C₂³

C₂×C₄×Dic₈₉

yes

—

C₂³

C₂²×C₄×D₈₉

yes

—

C₂³

C₂³×Dic₈₉

yes

—

C₂³

C₂⁴×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

384

—

C₂×C₄

C₂×C₄×Dic₈₉

288

yes

direct product factor

C₂×C₄

C₂²×C₄×D₈₉

yes

—

C₂×C₄

C₂³×Dic₈₉

yes

—

C₂²

C₄²×Dic₈₉

yes

direct product factor

C₂²

C₂×C₄²×D₈₉

yes

—

C₂²

C₂²×C₄×Dic₈₉

yes

—

100

C₂²

C₂³×C₄×D₈₉

yes

—

101

C₂²

C₂⁴×Dic₈₉

yes

characteristic

102

C₄

—

103

C₄

C₂²×C₄×Dic₈₉

yes

direct product factor

104

C₂

C₂×C₄²×Dic₈₉

yes

direct product factor

105

C₂

C₂²×C₄²×D₈₉

yes

—

106

C₂

C₂³×C₄×Dic₈₉

yes

—

	subgroup	G/H
center, hypercenter	C₂³×C₄²	D₈₉
commutator, hypocenter	C₈₉	C₂²×C₄³
Frattini	C₂³	C₂⁴×D₈₉
Fitting	C₂³×C₄²×C₈₉	C₂
socle, prime radical	C₂⁵×C₈₉	C₂³
cosocle	C₂³×C₈₉	C₂⁵
LCS	G ▷ C₈₉
UCS	C₁ ◁ C₂³×C₄²

subgroup

G/H

center, hypercenter

C₂³×C₄²

D₈₉

commutator, hypocenter

C₈₉

C₂²×C₄³

Frattini

C₂³

C₂⁴×D₈₉

Fitting

C₂³×C₄²×C₈₉

C₂

socle, prime radical

C₂⁵×C₈₉

C₂³

cosocle

C₂³×C₈₉

C₂⁵

LCS

G ▷ C₈₉

UCS

C₁ ◁ C₂³×C₄²

N	H ≅ G/N	type
2. C₂×C₄²×Dic₈₉	C₂	direct (N × H)
5. C₂²×C₄×Dic₈₉	C₄	direct (N × H)
7. C₄²×Dic₈₉	C₂²	direct (N × H)
9. C₂²×C₄²×C₈₉	C₄	semidirect
13. C₂×C₄×Dic₈₉	C₂×C₄	direct (N × H)
19. C₂×C₄²×C₈₉	C₂×C₄	semidirect
21. C₂²×Dic₈₉	C₄²	direct (N × H)
24. C₄×Dic₈₉	C₂²×C₄	direct (N × H)
28. C₂²×C₄×C₈₉	C₄²	semidirect
32. C₄²×C₈₉	C₂²×C₄	semidirect
38. C₂×C₄×C₈₉	C₂×C₄²	semidirect
41. C₂×Dic₈₉	C₂×C₄²	direct (N × H)
43. C₂²×C₈₉	C₄³	semidirect
48. C₄×C₈₉	C₂²×C₄²	semidirect
49. Dic₈₉	C₂²×C₄²	direct (N × H)
51. C₂×C₈₉	C₂×C₄³	semidirect
57. C₈₉	C₂²×C₄³	semidirect
62. C₂²×C₄²	Dic₈₉	direct (N × H)
73. C₂×C₄²	C₂×Dic₈₉	direct (N × H)
81. C₂²×C₄	C₄×Dic₈₉	direct (N × H)
86. C₄²	C₂²×Dic₈₉	direct (N × H)
94. C₂×C₄	C₂×C₄×Dic₈₉	direct (N × H)
97. C₂²	C₄²×Dic₈₉	direct (N × H)
103. C₄	C₂²×C₄×Dic₈₉	direct (N × H)
104. C₂	C₂×C₄²×Dic₈₉	direct (N × H)

H ≅ G/N

type

2. C₂×C₄²×Dic₈₉

C₂

direct (N × H)

5. C₂²×C₄×Dic₈₉

C₄

direct (N × H)

7. C₄²×Dic₈₉

C₂²

direct (N × H)

9. C₂²×C₄²×C₈₉

C₄

semidirect

13. C₂×C₄×Dic₈₉

C₂×C₄

direct (N × H)

19. C₂×C₄²×C₈₉

C₂×C₄

semidirect

21. C₂²×Dic₈₉

C₄²

direct (N × H)

24. C₄×Dic₈₉

C₂²×C₄

direct (N × H)

28. C₂²×C₄×C₈₉

C₄²

semidirect

32. C₄²×C₈₉

C₂²×C₄

semidirect

38. C₂×C₄×C₈₉

C₂×C₄²

semidirect

41. C₂×Dic₈₉

C₂×C₄²

direct (N × H)

43. C₂²×C₈₉

C₄³

semidirect

48. C₄×C₈₉

C₂²×C₄²

semidirect

49. Dic₈₉

C₂²×C₄²

direct (N × H)

51. C₂×C₈₉

C₂×C₄³

semidirect

57. C₈₉

C₂²×C₄³

semidirect

62. C₂²×C₄²

Dic₈₉

direct (N × H)

73. C₂×C₄²

C₂×Dic₈₉

direct (N × H)

81. C₂²×C₄

C₄×Dic₈₉

direct (N × H)

86. C₄²

C₂²×Dic₈₉

direct (N × H)

94. C₂×C₄

C₂×C₄×Dic₈₉

direct (N × H)

97. C₂²

C₄²×Dic₈₉

direct (N × H)

103. C₄

C₂²×C₄×Dic₈₉

direct (N × H)

104. C₂

C₂×C₄²×Dic₈₉

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂³×C₄×Dic₈₉	2
C₂×C₄²×Dic₈₉	2
C₂³×C₄²×C₈₉	2
C₂²×C₄³	89

subgroup

index

C₂³×C₄×Dic₈₉

C₂×C₄²×Dic₈₉

C₂³×C₄²×C₈₉

C₂²×C₄³

Subgroups — C₂²×C₄²×Dic₈₉ #36712593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C42×Dic89 #36712593

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₄²×Dic₈₉ #36712593