Group #36712592 — subgroups

Subgroups — C₂×C₄³×D₈₉ #36712592

Order 22784.

8094 conjugacy classes of subgroups; only the 4683 normal subgroups are shown.

normal subgroups only

This group has 8094 conjugacy classes of subgroups — too many to draw. Only its 4683 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×C₄²×D₈₉	C₂	1	7	yes	maximal
2	C₂×C₄²×Dic₈₉	C₂	1	7	yes	maximal, complemented
3	C₄³×D₈₉	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₄³×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×C₄×D₈₉	C₂²	1	7	yes	—
6	C₂²×C₄×Dic₈₉	C₂²	1	21	yes	—
7	C₂×C₄²×D₈₉	C₄	1	112	yes	direct product factor
8	C₂×C₄²×D₈₉	C₂²	1	56	yes	—
9	C₄²×Dic₈₉	C₂²	1	56	yes	complemented
10	C₂²×C₄²×C₈₉	C₂²	1	7	yes	—
11	C₄³×C₈₉	C₂²	1	8	yes	complemented
12	C₂⁴×D₈₉	C₂³	1	1	yes	characteristic
13	C₂³×Dic₈₉	C₂³	1	7	yes	—
14	C₂²×C₄×D₈₉	C₂×C₄	1	168	yes	—
15	C₂²×C₄×D₈₉	C₂³	1	28	yes	—
16	C₂×C₄×Dic₈₉	C₂×C₄	1	168	yes	complemented
17	C₂×C₄×Dic₈₉	C₂³	1	84	yes	—
18	C₄²×D₈₉	C₂×C₄	1	448	yes	direct product factor
19	C₂³×C₄×C₈₉	C₂³	1	7	yes	—
20	C₂×C₄²×C₈₉	C₂×C₄	1	56	yes	complemented
21	C₂×C₄²×C₈₉	C₂³	1	28	yes	—
22	C₂³×D₈₉	C₂²×C₄	1	28	yes	—
23	C₂³×D₈₉	C₂⁴	1	2	yes	—
24	C₂²×Dic₈₉	C₂²×C₄	1	84	yes	—
25	C₂²×Dic₈₉	C₂⁴	1	14	yes	—
26	C₂×C₄×D₈₉	C₄²	1	448	yes	direct product factor
27	C₂×C₄×D₈₉	C₂²×C₄	1	336	yes	—
28	C₄×Dic₈₉	C₂²×C₄	1	336	yes	complemented
29	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
30	C₂²×C₄×C₈₉	C₂²×C₄	1	84	yes	—
31	C₂²×C₄×C₈₉	C₂⁴	1	14	yes	—
32	C₄²×C₈₉	C₂²×C₄	1	112	yes	complemented
33	C₂³×C₈₉	C₂³×C₄	1	14	yes	—
34	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
35	C₂²×D₈₉	C₂×C₄²	1	112	yes	—
36	C₂²×D₈₉	C₂³×C₄	1	28	yes	—
37	C₂×C₄×C₈₉	C₂×C₄²	1	112	yes	complemented
38	C₂×C₄×C₈₉	C₂³×C₄	1	84	yes	—
39	C₂×Dic₈₉	C₂×C₄²	1	112	yes	complemented
40	C₂×Dic₈₉	C₂³×C₄	1	84	yes	—
41	C₄×D₈₉	C₂×C₄²	1	448	yes	direct product factor
42	C₂²×C₈₉	C₂²×C₄²	1	28	yes	—
43	C₂²×C₈₉	C₂⁴×C₄	1	7	yes	—
44	C₂×D₈₉	C₄³	1	64	yes	direct product factor
45	C₂×D₈₉	C₂²×C₄²	1	56	yes	—
46	C₄×C₈₉	C₂²×C₄²	1	56	yes	complemented
47	Dic₈₉	C₂²×C₄²	1	56	yes	complemented
48	C₂²×C₄³	—	89	1	no	maximal, Sylow
49	C₂×C₈₉	C₂×C₄³	1	8	yes	complemented
50	C₂×C₈₉	C₂³×C₄²	1	7	yes	—
51	D₈₉	C₂×C₄³	1	16	yes	direct product factor
52	C₂³×C₄²	—	89	7	no	—
53	C₂×C₄³	—	89	7	no	—
54	C₂×C₄³	—	89	16	no	—
55	C₂×C₄³	D₈₉	1	1	yes	characteristic, direct product factor
56	C₈₉	C₂²×C₄³	1	1	yes	characteristic, complemented, Sylow
57	C₂⁴×C₄	—	89	7	no	—
58	C₂²×C₄²	—	89	21	no	—
59	C₂²×C₄²	—	89	112	no	—
60	C₂²×C₄²	—	89	56	no	—
61	C₂²×C₄²	C₂×D₈₉	1	7	yes	—
62	C₄³	—	89	56	no	—
63	C₄³	C₂×D₈₉	1	8	yes	direct product factor
64	C₂⁵	—	89	1	no	—
65	C₂³×C₄	—	89	7	no	—
66	C₂³×C₄	—	89	168	no	—
67	C₂³×C₄	—	89	28	no	—
68	C₂³×C₄	C₂²×D₈₉	1	7	yes	—
69	C₂×C₄²	—	89	168	no	—
70	C₂×C₄²	—	89	84	no	—
71	C₂×C₄²	—	89	448	no	—
72	C₂×C₄²	C₄×D₈₉	1	56	yes	direct product factor
73	C₂×C₄²	C₂²×D₈₉	1	28	yes	—
74	C₂⁴	—	89	28	no	—
75	C₂⁴	—	89	2	no	—
76	C₂⁴	C₂³×D₈₉	1	1	yes	characteristic
77	C₂²×C₄	—	89	84	no	—
78	C₂²×C₄	—	89	14	no	—
79	C₂²×C₄	—	89	336	no	—
80	C₂²×C₄	—	89	448	no	—
81	C₂²×C₄	C₂×C₄×D₈₉	1	84	yes	—
82	C₂²×C₄	C₂³×D₈₉	1	14	yes	—
83	C₄²	—	89	336	no	—
84	C₄²	C₂×C₄×D₈₉	1	112	yes	direct product factor
85	C₂³	—	89	28	no	—
86	C₂³	—	89	112	no	—
87	C₂³	C₂²×C₄×D₈₉	1	14	yes	—
88	C₂³	C₂⁴×D₈₉	1	1	yes	characteristic
89	C₂×C₄	—	89	84	no	—
90	C₂×C₄	—	89	112	no	—
91	C₂×C₄	—	89	448	no	—
92	C₂×C₄	C₄²×D₈₉	1	112	yes	direct product factor
93	C₂×C₄	C₂²×C₄×D₈₉	1	84	yes	—
94	C₂²	—	89	64	no	—
95	C₂²	—	89	56	no	—
96	C₂²	C₂×C₄²×D₈₉	1	28	yes	—
97	C₂²	C₂³×C₄×D₈₉	1	7	yes	—
98	C₄	—	89	56	no	—
99	C₄	C₂×C₄²×D₈₉	1	56	yes	direct product factor
100	C₂	—	89	16	no	—
101	C₂	C₄³×D₈₉	1	8	yes	direct product factor
102	C₂	C₂²×C₄²×D₈₉	1	7	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄²×D₈₉

C₂

yes

maximal

C₂×C₄²×Dic₈₉

C₂

yes

maximal, complemented

C₄³×D₈₉

C₂

yes

maximal, direct product factor

C₂×C₄³×C₈₉

C₂

yes

maximal, characteristic, complemented

C₂³×C₄×D₈₉

C₂²

yes

—

C₂²×C₄×Dic₈₉

C₂²

yes

—

C₂×C₄²×D₈₉

C₄

112

yes

direct product factor

C₂×C₄²×D₈₉

C₂²

yes

—

C₄²×Dic₈₉

C₂²

yes

complemented

C₂²×C₄²×C₈₉

C₂²

yes

—

C₄³×C₈₉

C₂²

yes

complemented

C₂⁴×D₈₉

C₂³

yes

characteristic

C₂³×Dic₈₉

C₂³

yes

—

C₂²×C₄×D₈₉

C₂×C₄

168

yes

—

C₂²×C₄×D₈₉

C₂³

yes

—

C₂×C₄×Dic₈₉

C₂×C₄

168

yes

complemented

C₂×C₄×Dic₈₉

C₂³

yes

—

C₄²×D₈₉

C₂×C₄

448

yes

direct product factor

C₂³×C₄×C₈₉

C₂³

yes

—

C₂×C₄²×C₈₉

C₂×C₄

yes

complemented

C₂×C₄²×C₈₉

C₂³

yes

—

C₂³×D₈₉

C₂²×C₄

yes

—

C₂³×D₈₉

C₂⁴

yes

—

C₂²×Dic₈₉

C₂²×C₄

yes

—

C₂²×Dic₈₉

C₂⁴

yes

—

C₂×C₄×D₈₉

C₄²

448

yes

direct product factor

C₂×C₄×D₈₉

C₂²×C₄

336

yes

—

C₄×Dic₈₉

C₂²×C₄

336

yes

complemented

C₂⁴×C₈₉

C₂⁴

yes

characteristic

C₂²×C₄×C₈₉

C₂²×C₄

yes

—

C₂²×C₄×C₈₉

C₂⁴

yes

—

C₄²×C₈₉

C₂²×C₄

112

yes

complemented

C₂³×C₈₉

C₂³×C₄

yes

—

C₂³×C₈₉

C₂⁵

yes

characteristic

C₂²×D₈₉

C₂×C₄²

112

yes

—

C₂²×D₈₉

C₂³×C₄

yes

—

C₂×C₄×C₈₉

C₂×C₄²

112

yes

complemented

C₂×C₄×C₈₉

C₂³×C₄

yes

—

C₂×Dic₈₉

C₂×C₄²

112

yes

complemented

C₂×Dic₈₉

C₂³×C₄

yes

—

C₄×D₈₉

C₂×C₄²

448

yes

direct product factor

C₂²×C₈₉

C₂²×C₄²

yes

—

C₂²×C₈₉

C₂⁴×C₄

yes

—

C₂×D₈₉

C₄³

yes

direct product factor

C₂×D₈₉

C₂²×C₄²

yes

—

C₄×C₈₉

C₂²×C₄²

yes

complemented

Dic₈₉

C₂²×C₄²

yes

complemented

C₂²×C₄³

—

maximal, Sylow

C₂×C₈₉

C₂×C₄³

yes

complemented

C₂×C₈₉

C₂³×C₄²

yes

—

D₈₉

C₂×C₄³

yes

direct product factor

C₂³×C₄²

—

C₂×C₄³

—

C₂×C₄³

—

C₂×C₄³

D₈₉

yes

characteristic, direct product factor

C₈₉

C₂²×C₄³

yes

characteristic, complemented, Sylow

C₂⁴×C₄

—

C₂²×C₄²

—

C₂²×C₄²

—

112

—

C₂²×C₄²

—

C₂²×C₄²

C₂×D₈₉

yes

—

C₄³

—

C₄³

C₂×D₈₉

yes

direct product factor

C₂⁵

—

C₂³×C₄

—

C₂³×C₄

—

168

—

C₂³×C₄

—

C₂³×C₄

C₂²×D₈₉

yes

—

C₂×C₄²

—

168

—

C₂×C₄²

—

C₂×C₄²

—

448

—

C₂×C₄²

C₄×D₈₉

yes

direct product factor

C₂×C₄²

C₂²×D₈₉

yes

—

C₂⁴

—

C₂⁴

—

C₂⁴

C₂³×D₈₉

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

336

—

C₂²×C₄

—

448

—

C₂²×C₄

C₂×C₄×D₈₉

yes

—

C₂²×C₄

C₂³×D₈₉

yes

—

C₄²

—

336

—

C₄²

C₂×C₄×D₈₉

112

yes

direct product factor

C₂³

—

C₂³

—

112

—

C₂³

C₂²×C₄×D₈₉

yes

—

C₂³

C₂⁴×D₈₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

112

—

C₂×C₄

—

448

—

C₂×C₄

C₄²×D₈₉

112

yes

direct product factor

C₂×C₄

C₂²×C₄×D₈₉

yes

—

C₂²

—

C₂²

—

C₂²

C₂×C₄²×D₈₉

yes

—

C₂²

C₂³×C₄×D₈₉

yes

—

C₄

—

C₄

C₂×C₄²×D₈₉

yes

direct product factor

100

C₂

—

101

C₂

C₄³×D₈₉

yes

direct product factor

102

C₂

C₂²×C₄²×D₈₉

yes

—

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

subgroup

G/H

center, hypercenter

C₂×C₄³

D₈₉

commutator, hypocenter

C₈₉

C₂²×C₄³

Frattini

C₂³

C₂⁴×D₈₉

Fitting

C₂×C₄³×C₈₉

C₂

socle

C₂⁴×C₈₉

C₂⁴

prime radical

C₂⁴×D₈₉

C₂³

cosocle

C₂³×C₈₉

C₂⁵

LCS

G ▷ C₈₉

UCS

C₁ ◁ C₂×C₄³

N	H ≅ G/N	type
2. C₂×C₄²×Dic₈₉	C₂	semidirect
3. C₄³×D₈₉	C₂	direct (N × H)
4. C₂×C₄³×C₈₉	C₂	semidirect
7. C₂×C₄²×D₈₉	C₄	direct (N × H)
9. C₄²×Dic₈₉	C₂²	semidirect
11. C₄³×C₈₉	C₂²	semidirect
16. C₂×C₄×Dic₈₉	C₂×C₄	semidirect
18. C₄²×D₈₉	C₂×C₄	direct (N × H)
20. C₂×C₄²×C₈₉	C₂×C₄	semidirect
26. C₂×C₄×D₈₉	C₄²	direct (N × H)
28. C₄×Dic₈₉	C₂²×C₄	semidirect
32. C₄²×C₈₉	C₂²×C₄	semidirect
37. C₂×C₄×C₈₉	C₂×C₄²	semidirect
39. C₂×Dic₈₉	C₂×C₄²	semidirect
41. C₄×D₈₉	C₂×C₄²	direct (N × H)
44. C₂×D₈₉	C₄³	direct (N × H)
46. C₄×C₈₉	C₂²×C₄²	semidirect
47. Dic₈₉	C₂²×C₄²	semidirect
49. C₂×C₈₉	C₂×C₄³	semidirect
51. D₈₉	C₂×C₄³	direct (N × H)
55. C₂×C₄³	D₈₉	direct (N × H)
56. C₈₉	C₂²×C₄³	semidirect
63. C₄³	C₂×D₈₉	direct (N × H)
72. C₂×C₄²	C₄×D₈₉	direct (N × H)
84. C₄²	C₂×C₄×D₈₉	direct (N × H)
92. C₂×C₄	C₄²×D₈₉	direct (N × H)
99. C₄	C₂×C₄²×D₈₉	direct (N × H)
101. C₂	C₄³×D₈₉	direct (N × H)

H ≅ G/N

type

2. C₂×C₄²×Dic₈₉

C₂

semidirect

3. C₄³×D₈₉

C₂

direct (N × H)

4. C₂×C₄³×C₈₉

C₂

semidirect

7. C₂×C₄²×D₈₉

C₄

direct (N × H)

9. C₄²×Dic₈₉

C₂²

semidirect

11. C₄³×C₈₉

C₂²

semidirect

16. C₂×C₄×Dic₈₉

C₂×C₄

semidirect

18. C₄²×D₈₉

C₂×C₄

direct (N × H)

20. C₂×C₄²×C₈₉

C₂×C₄

semidirect

26. C₂×C₄×D₈₉

C₄²

direct (N × H)

28. C₄×Dic₈₉

C₂²×C₄

semidirect

32. C₄²×C₈₉

C₂²×C₄

semidirect

37. C₂×C₄×C₈₉

C₂×C₄²

semidirect

39. C₂×Dic₈₉

C₂×C₄²

semidirect

41. C₄×D₈₉

C₂×C₄²

direct (N × H)

44. C₂×D₈₉

C₄³

direct (N × H)

46. C₄×C₈₉

C₂²×C₄²

semidirect

47. Dic₈₉

C₂²×C₄²

semidirect

49. C₂×C₈₉

C₂×C₄³

semidirect

51. D₈₉

C₂×C₄³

direct (N × H)

55. C₂×C₄³

D₈₉

direct (N × H)

56. C₈₉

C₂²×C₄³

semidirect

63. C₄³

C₂×D₈₉

direct (N × H)

72. C₂×C₄²

C₄×D₈₉

direct (N × H)

84. C₄²

C₂×C₄×D₈₉

direct (N × H)

92. C₂×C₄

C₄²×D₈₉

direct (N × H)

99. C₄

C₂×C₄²×D₈₉

direct (N × H)

101. C₂

C₄³×D₈₉

direct (N × H)

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

subgroup	index
C₂²×C₄²×D₈₉	2
C₂×C₄²×Dic₈₉	2
C₄³×D₈₉	2
C₂×C₄³×C₈₉	2
C₂²×C₄³	89

subgroup

index

C₂²×C₄²×D₈₉

C₂×C₄²×Dic₈₉

C₄³×D₈₉

C₂×C₄³×C₈₉

C₂²×C₄³

Subgroups — C₂×C₄³×D₈₉ #36712592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C43×D89 #36712592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₄³×D₈₉ #36712592