Group #34077510 — subgroups

Subgroups — C₂²×C₈×Dic₁₅₁ #34077510

Order 19328.

760 conjugacy classes of subgroups; only the 549 normal subgroups are shown.

normal subgroups only

This group has 760 conjugacy classes of subgroups — too many to draw. Only its 549 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×C₁₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₁₅₁	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Dic₁₅₁	C₄	1	2	yes	—
6	C₂²×C₁₅₁⋊C₈	C₄	1	8	yes	complemented
7	C₂²×C₁₅₁⋊C₈	C₂²	1	6	yes	—
8	C₂×C₄×Dic₁₅₁	C₄	1	6	yes	—
9	C₂×C₄×Dic₁₅₁	C₂²	1	6	yes	—
10	C₈×Dic₁₅₁	C₂²	1	16	yes	direct product factor
11	C₂³×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₁₅₁	C₄	1	8	yes	complemented
13	C₂²×C₈×C₁₅₁	C₂²	1	6	yes	—
14	C₂²×Dic₁₅₁	C₈	1	16	yes	direct product factor
15	C₂²×Dic₁₅₁	C₂×C₄	1	12	yes	—
16	C₂×C₁₅₁⋊C₈	C₂×C₄	1	24	yes	complemented
17	C₂×C₁₅₁⋊C₈	C₂³	1	4	yes	—
18	C₄×Dic₁₅₁	C₂×C₄	1	12	yes	—
19	C₄×Dic₁₅₁	C₂³	1	4	yes	—
20	C₂⁴×C₁₅₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₁	C₂×C₄	1	3	yes	—
22	C₂²×C₄×C₁₅₁	C₂×C₄	1	4	yes	—
23	C₂²×C₄×C₁₅₁	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₁₅₁	C₂³	1	3	yes	—
25	C₂×C₈×C₁₅₁	C₂×C₄	1	24	yes	complemented
26	C₂×C₈×C₁₅₁	C₂³	1	4	yes	—
27	C₂×Dic₁₅₁	C₂×C₈	1	48	yes	direct product factor
28	C₂×Dic₁₅₁	C₂²×C₄	1	8	yes	—
29	C₁₅₁⋊C₈	C₂²×C₄	1	8	yes	complemented
30	C₂³×C₁₅₁	C₄²	1	4	yes	—
31	C₂³×C₁₅₁	C₂×C₈	1	8	yes	—
32	C₂³×C₁₅₁	C₂²×C₄	1	3	yes	—
33	C₂×C₄×C₁₅₁	C₄²	1	12	yes	—
34	C₂×C₄×C₁₅₁	C₂²×C₄	1	3	yes	—
35	C₂×C₄×C₁₅₁	C₂²×C₄	1	6	yes	—
36	C₂×C₄×C₁₅₁	C₂²×C₄	1	6	yes	—
37	C₂×C₄×C₁₅₁	C₂⁴	1	1	yes	characteristic
38	C₈×C₁₅₁	C₂²×C₄	1	8	yes	complemented
39	C₂²×C₁₅₁	C₄×C₈	1	16	yes	complemented
40	C₂²×C₁₅₁	C₂×C₄²	1	6	yes	—
41	C₂²×C₁₅₁	C₂²×C₈	1	12	yes	—
42	C₂²×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
43	C₄×C₁₅₁	C₂×C₄²	1	6	yes	—
44	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
45	C₄×C₁₅₁	C₂³×C₄	1	1	yes	characteristic
46	Dic₁₅₁	C₂²×C₈	1	16	yes	direct product factor
47	C₂×C₁₅₁	C₂×C₄×C₈	1	12	yes	complemented
48	C₂×C₁₅₁	C₂²×C₄²	1	1	yes	characteristic
49	C₂×C₁₅₁	C₂³×C₈	1	2	yes	—
50	C₁₅₁	C₂²×C₄×C₈	1	1	yes	characteristic, complemented, Sylow
51	C₂²×C₄×C₈	—	151	1	no	maximal, Sylow
52	C₂³×C₈	—	151	1	no	—
53	C₂³×C₈	D₁₅₁	1	1	yes	characteristic
54	C₂²×C₄²	—	151	1	no	—
55	C₂×C₄×C₈	—	151	12	no	—
56	C₂³×C₄	—	151	2	no	—
57	C₂³×C₄	C₂×D₁₅₁	1	1	yes	characteristic
58	C₂²×C₈	—	151	8	no	—
59	C₂²×C₈	—	151	6	no	—
60	C₂²×C₈	Dic₁₅₁	1	8	yes	direct product factor
61	C₂²×C₈	C₂×D₁₅₁	1	6	yes	—
62	C₂×C₄²	—	151	6	no	—
63	C₂×C₄²	—	151	6	no	—
64	C₄×C₈	—	151	16	no	—
65	C₂⁴	C₄×D₁₅₁	1	1	yes	characteristic
66	C₂²×C₄	—	151	16	no	—
67	C₂²×C₄	—	151	12	no	—
68	C₂²×C₄	C₄×D₁₅₁	1	3	yes	—
69	C₂²×C₄	C₄×D₁₅₁	1	4	yes	—
70	C₂²×C₄	C₂×Dic₁₅₁	1	4	yes	—
71	C₂²×C₄	C₂²×D₁₅₁	1	3	yes	—
72	C₂×C₈	—	151	24	no	—
73	C₂×C₈	—	151	4	no	—
74	C₂×C₈	C₂×Dic₁₅₁	1	24	yes	direct product factor
75	C₂×C₈	C₂²×D₁₅₁	1	4	yes	—
76	C₄²	—	151	12	no	—
77	C₄²	—	151	4	no	—
78	C₂³	C₄×Dic₁₅₁	1	4	yes	—
79	C₂³	C₈×D₁₅₁	1	8	yes	—
80	C₂³	C₂×C₄×D₁₅₁	1	3	yes	—
81	C₂×C₄	—	151	48	no	—
82	C₂×C₄	—	151	8	no	—
83	C₂×C₄	C₄×Dic₁₅₁	1	12	yes	—
84	C₂×C₄	C₂×C₄×D₁₅₁	1	3	yes	—
85	C₂×C₄	C₂×C₄×D₁₅₁	1	6	yes	—
86	C₂×C₄	C₂²×Dic₁₅₁	1	6	yes	—
87	C₂×C₄	C₂³×D₁₅₁	1	1	yes	characteristic
88	C₈	—	151	8	no	—
89	C₈	C₂²×Dic₁₅₁	1	8	yes	direct product factor
90	C₂²	C₈×Dic₁₅₁	1	16	yes	direct product factor
91	C₂²	C₂×C₄×Dic₁₅₁	1	6	yes	—
92	C₂²	C₂×C₈×D₁₅₁	1	12	yes	—
93	C₂²	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
94	C₄	—	151	16	no	—
95	C₄	C₂×C₄×Dic₁₅₁	1	6	yes	—
96	C₄	C₂²×C₄×D₁₅₁	1	1	yes	characteristic
97	C₄	C₂³×Dic₁₅₁	1	1	yes	characteristic
98	C₂	C₂×C₈×Dic₁₅₁	1	12	yes	direct product factor
99	C₂	C₂²×C₄×Dic₁₅₁	1	1	yes	characteristic
100	C₂	C₂²×C₈×D₁₅₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₁₅₁⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₄×Dic₁₅₁

C₂

yes

maximal, characteristic

C₂×C₈×Dic₁₅₁

C₂

yes

maximal, direct product factor

C₂³×C₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂³×Dic₁₅₁

C₄

yes

—

C₂²×C₁₅₁⋊C₈

C₄

yes

complemented

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₄

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

—

C₈×Dic₁₅₁

C₂²

yes

direct product factor

C₂³×C₄×C₁₅₁

C₂²

yes

characteristic

C₂²×C₈×C₁₅₁

C₄

yes

complemented

C₂²×C₈×C₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

C₈

yes

direct product factor

C₂²×Dic₁₅₁

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

complemented

C₂×C₁₅₁⋊C₈

C₂³

yes

—

C₄×Dic₁₅₁

C₂×C₄

yes

—

C₄×Dic₁₅₁

C₂³

yes

—

C₂⁴×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

—

C₂×C₈×C₁₅₁

C₂×C₄

yes

complemented

C₂×C₈×C₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂×C₈

yes

direct product factor

C₂×Dic₁₅₁

C₂²×C₄

yes

—

C₁₅₁⋊C₈

C₂²×C₄

yes

complemented

C₂³×C₁₅₁

C₄²

yes

—

C₂³×C₁₅₁

C₂×C₈

yes

—

C₂³×C₁₅₁

C₂²×C₄

yes

—

C₂×C₄×C₁₅₁

C₄²

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

—

C₂×C₄×C₁₅₁

C₂⁴

yes

characteristic

C₈×C₁₅₁

C₂²×C₄

yes

complemented

C₂²×C₁₅₁

C₄×C₈

yes

complemented

C₂²×C₁₅₁

C₂×C₄²

yes

—

C₂²×C₁₅₁

C₂²×C₈

yes

—

C₂²×C₁₅₁

C₂³×C₄

yes

characteristic

C₄×C₁₅₁

C₂×C₄²

yes

—

C₄×C₁₅₁

C₂³×C₄

yes

characteristic

C₄×C₁₅₁

C₂³×C₄

yes

characteristic

Dic₁₅₁

C₂²×C₈

yes

direct product factor

C₂×C₁₅₁

C₂×C₄×C₈

yes

complemented

C₂×C₁₅₁

C₂²×C₄²

yes

characteristic

C₂×C₁₅₁

C₂³×C₈

yes

—

C₁₅₁

C₂²×C₄×C₈

yes

characteristic, complemented, Sylow

C₂²×C₄×C₈

—

151

maximal, Sylow

C₂³×C₈

—

151

—

C₂³×C₈

D₁₅₁

yes

characteristic

C₂²×C₄²

—

151

—

C₂×C₄×C₈

—

151

—

C₂³×C₄

—

151

—

C₂³×C₄

C₂×D₁₅₁

yes

characteristic

C₂²×C₈

—

151

—

C₂²×C₈

—

151

—

C₂²×C₈

Dic₁₅₁

yes

direct product factor

C₂²×C₈

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

C₂×C₄²

—

151

—

C₄×C₈

—

151

—

C₂⁴

C₄×D₁₅₁

yes

characteristic

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₄×D₁₅₁

yes

—

C₂²×C₄

C₄×D₁₅₁

yes

—

C₂²×C₄

C₂×Dic₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

C₂×Dic₁₅₁

yes

direct product factor

C₂×C₈

C₂²×D₁₅₁

yes

—

C₄²

—

151

—

C₄²

—

151

—

C₂³

C₄×Dic₁₅₁

yes

—

C₂³

C₈×D₁₅₁

yes

—

C₂³

C₂×C₄×D₁₅₁

yes

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₄×Dic₁₅₁

yes

—

C₂×C₄

C₂×C₄×D₁₅₁

yes

—

C₂×C₄

C₂×C₄×D₁₅₁

yes

—

C₂×C₄

C₂²×Dic₁₅₁

yes

—

C₂×C₄

C₂³×D₁₅₁

yes

characteristic

C₈

—

151

—

C₈

C₂²×Dic₁₅₁

yes

direct product factor

C₂²

C₈×Dic₁₅₁

yes

direct product factor

C₂²

C₂×C₄×Dic₁₅₁

yes

—

C₂²

C₂×C₈×D₁₅₁

yes

—

C₂²

C₂²×C₄×D₁₅₁

yes

characteristic

C₄

—

151

—

C₄

C₂×C₄×Dic₁₅₁

yes

—

C₄

C₂²×C₄×D₁₅₁

yes

characteristic

C₄

C₂³×Dic₁₅₁

yes

characteristic

C₂

C₂×C₈×Dic₁₅₁

yes

direct product factor

C₂

C₂²×C₄×Dic₁₅₁

yes

characteristic

100

C₂

C₂²×C₈×D₁₅₁

yes

—

	subgroup	G/H
center, hypercenter	C₂³×C₈	D₁₅₁
commutator, hypocenter	C₁₅₁	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×D₁₅₁
Fitting	C₂³×C₈×C₁₅₁	C₂
socle, prime radical	C₂⁴×C₁₅₁	C₂×C₄
cosocle	C₂×C₄×C₁₅₁	C₂⁴
LCS	G ▷ C₁₅₁
UCS	C₁ ◁ C₂³×C₈

subgroup

G/H

center, hypercenter

C₂³×C₈

D₁₅₁

commutator, hypocenter

C₁₅₁

C₂²×C₄×C₈

Frattini

C₂×C₄

C₂³×D₁₅₁

Fitting

C₂³×C₈×C₁₅₁

C₂

socle, prime radical

C₂⁴×C₁₅₁

C₂×C₄

cosocle

C₂×C₄×C₁₅₁

C₂⁴

LCS

G ▷ C₁₅₁

UCS

C₁ ◁ C₂³×C₈

N	H ≅ G/N	type
3. C₂×C₈×Dic₁₅₁	C₂	direct (N × H)
6. C₂²×C₁₅₁⋊C₈	C₄	semidirect
10. C₈×Dic₁₅₁	C₂²	direct (N × H)
12. C₂²×C₈×C₁₅₁	C₄	semidirect
14. C₂²×Dic₁₅₁	C₈	direct (N × H)
16. C₂×C₁₅₁⋊C₈	C₂×C₄	semidirect
25. C₂×C₈×C₁₅₁	C₂×C₄	semidirect
27. C₂×Dic₁₅₁	C₂×C₈	direct (N × H)
29. C₁₅₁⋊C₈	C₂²×C₄	semidirect
38. C₈×C₁₅₁	C₂²×C₄	semidirect
39. C₂²×C₁₅₁	C₄×C₈	semidirect
46. Dic₁₅₁	C₂²×C₈	direct (N × H)
47. C₂×C₁₅₁	C₂×C₄×C₈	semidirect
50. C₁₅₁	C₂²×C₄×C₈	semidirect
60. C₂²×C₈	Dic₁₅₁	direct (N × H)
74. C₂×C₈	C₂×Dic₁₅₁	direct (N × H)
89. C₈	C₂²×Dic₁₅₁	direct (N × H)
90. C₂²	C₈×Dic₁₅₁	direct (N × H)
98. C₂	C₂×C₈×Dic₁₅₁	direct (N × H)

H ≅ G/N

type

3. C₂×C₈×Dic₁₅₁

C₂

direct (N × H)

6. C₂²×C₁₅₁⋊C₈

C₄

semidirect

10. C₈×Dic₁₅₁

C₂²

direct (N × H)

12. C₂²×C₈×C₁₅₁

C₄

semidirect

14. C₂²×Dic₁₅₁

C₈

direct (N × H)

16. C₂×C₁₅₁⋊C₈

C₂×C₄

semidirect

25. C₂×C₈×C₁₅₁

C₂×C₄

semidirect

27. C₂×Dic₁₅₁

C₂×C₈

direct (N × H)

29. C₁₅₁⋊C₈

C₂²×C₄

semidirect

38. C₈×C₁₅₁

C₂²×C₄

semidirect

39. C₂²×C₁₅₁

C₄×C₈

semidirect

46. Dic₁₅₁

C₂²×C₈

direct (N × H)

47. C₂×C₁₅₁

C₂×C₄×C₈

semidirect

50. C₁₅₁

C₂²×C₄×C₈

semidirect

60. C₂²×C₈

Dic₁₅₁

direct (N × H)

74. C₂×C₈

C₂×Dic₁₅₁

direct (N × H)

89. C₈

C₂²×Dic₁₅₁

direct (N × H)

90. C₂²

C₈×Dic₁₅₁

direct (N × H)

98. C₂

C₂×C₈×Dic₁₅₁

direct (N × H)

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

subgroup	index
C₂³×C₁₅₁⋊C₈	2
C₂²×C₄×Dic₁₅₁	2
C₂×C₈×Dic₁₅₁	2
C₂³×C₈×C₁₅₁	2
C₂²×C₄×C₈	151

subgroup

index

C₂³×C₁₅₁⋊C₈

C₂²×C₄×Dic₁₅₁

C₂×C₈×Dic₁₅₁

C₂³×C₈×C₁₅₁

C₂²×C₄×C₈

151

Subgroups — C₂²×C₈×Dic₁₅₁ #34077510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×C8×Dic151 #34077510

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₈×Dic₁₅₁ #34077510