Group #23752707 — subgroups

Subgroups — C₂×#18012715 #23752707

Order 9664.

322 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 135 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×C₄×D₁₅₁	C₂	1	1	yes	maximal, characteristic
2	#18012715	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₄⋊Dic₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₁₅₁⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂×(C₂×D₁₅₁)⋊C₄	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₁₅₁	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₅₁	C₂²	1	2	yes	—
9	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₁₅₁	—	2	4	no	—
11	C₂×C₄×D₁₅₁	C₂²	1	4	yes	—
12	C₄⋊Dic₁₅₁	C₂²	1	4	yes	complemented
13	Dic₁₅₁⋊C₄	C₂²	1	8	yes	complemented
14	(C₂×D₁₅₁)⋊C₄	C₂²	1	8	yes	—
15	C₂×Q₈×C₁₅₁	—	2	4	no	—
16	C₂×Q₈×C₁₅₁	C₂²	1	4	yes	complemented
17	C₂²×C₄×C₁₅₁	C₂²	1	2	yes	—
18	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
19	C₂²×D₁₅₁	—	2	2	no	—
20	C₂²×D₁₅₁	—	2	2	no	—
21	C₂²×D₁₅₁	Q₈	1	4	yes	complemented
22	C₂²×D₁₅₁	C₂³	1	2	yes	—
23	C₂×Dic₁₅₁	—	2	2	no	—
24	C₂×Dic₁₅₁	—	2	4	no	—
25	C₂×Dic₁₅₁	C₂³	1	4	yes	—
26	C₂×Dic₁₅₁	C₂³	1	2	yes	—
27	C₄×D₁₅₁	—	2	8	no	—
28	C₂³×C₁₅₁	C₂³	1	1	yes	characteristic
29	Q₈×C₁₅₁	—	2	8	no	—
30	C₂×C₄×C₁₅₁	—	2	4	no	—
31	C₂×C₄×C₁₅₁	D₄	1	4	yes	—
32	C₂×C₄×C₁₅₁	C₂³	1	4	yes	—
33	C₂×C₄×C₁₅₁	C₂³	1	2	yes	—
34	C₂²×C₁₅₁	C₂×D₄	1	2	yes	—
35	C₂²×C₁₅₁	C₂×Q₈	1	2	yes	—
36	C₂²×C₁₅₁	#41	1	2	yes	—
37	C₂²×C₁₅₁	C₂⁴	1	1	yes	characteristic
38	C₂×D₁₅₁	—	2	8	no	—
39	C₂×D₁₅₁	—	2	2	no	—
40	C₂×D₁₅₁	—	2	2	no	—
41	C₂×D₁₅₁	C₂×Q₈	1	4	yes	complemented
42	C₄×C₁₅₁	—	2	4	no	—
43	C₄×C₁₅₁	C₂×D₄	1	4	yes	—
44	Dic₁₅₁	—	2	2	no	—
45	Dic₁₅₁	—	2	4	no	—
46	C₂×C₁₅₁	C₂²⋊Q₈	1	4	yes	complemented
47	C₂×C₁₅₁	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₁₅₁	C₂²×Q₈	1	1	yes	characteristic
49	C₂×C₁₅₁	C₂×#41	1	1	yes	characteristic
50	D₁₅₁	—	2	4	no	—
51	C₁₅₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
52	C₂×C₂²⋊Q₈	—	151	1	no	maximal, Sylow
53	C₂²×Q₈	D₁₅₁	1	1	yes	characteristic, complemented
54	C₂³×C₄	—	151	1	no	—
55	C₂²⋊Q₈	—	151	8	no	—
56	C₂×C₄⋊C₄	—	151	2	no	—
57	C₂×C₄⋊C₄	—	151	1	no	—
58	C₂×C₂²⋊C₄	—	151	2	no	—
59	C₂⁴	—	151	1	no	—
60	C₂×Q₈	—	2	4	no	—
61	C₂×Q₈	C₂×D₁₅₁	1	4	yes	complemented
62	C₂²×C₄	—	302	4	no	—
63	C₂²×C₄	—	151	2	no	—
64	C₂²×C₄	—	151	1	no	—
65	C₂²×C₄	—	151	4	no	—
66	C₂²×C₄	C₂×D₁₅₁	1	2	yes	—
67	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
68	C₄⋊C₄	—	151	8	no	—
69	C₄⋊C₄	—	151	4	no	—
70	C₂²⋊C₄	—	151	8	no	—
71	C₂³	—	302	2	no	—
72	C₂³	—	302	2	no	—
73	C₂³	—	151	4	no	—
74	C₂³	—	151	2	no	—
75	C₂³	C₂²×D₁₅₁	1	1	yes	characteristic
76	Q₈	—	2	8	no	—
77	C₂×C₄	—	2	4	no	—
78	C₂×C₄	—	302	2	no	—
79	C₂×C₄	—	302	4	no	—
80	C₂×C₄	—	302	8	no	—
81	C₂×C₄	—	151	4	no	—
82	C₂×C₄	—	151	2	no	—
83	C₂×C₄	C₂²⋊D₁₅₁	1	4	yes	—
84	C₂×C₄	C₂²×D₁₅₁	1	4	yes	—
85	C₂×C₄	C₂²×D₁₅₁	1	2	yes	—
86	C₂²	—	302	8	no	—
87	C₂²	—	302	2	no	—
88	C₂²	—	302	2	no	—
89	C₂²	—	151	4	no	—
90	C₂²	C₂×C₂²⋊D₁₅₁	1	2	yes	—
91	C₂²	Q₈×D₁₅₁	1	2	yes	—
92	C₂²	#14559005	1	2	yes	—
93	C₂²	C₂³×D₁₅₁	1	1	yes	characteristic
94	C₄	—	2	4	no	—
95	C₄	—	302	2	no	—
96	C₄	—	302	4	no	—
97	C₄	C₂×C₂²⋊D₁₅₁	1	4	yes	—
98	C₂	—	302	4	no	—
99	C₂	#18012715	1	4	yes	direct product factor
100	C₂	C₂²×C₂²⋊D₁₅₁	1	1	yes	characteristic
101	C₂	C₂×Q₈×D₁₅₁	1	1	yes	characteristic
102	C₂	C₂×#14559005	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄×D₁₅₁

C₂

yes

maximal, characteristic

#18012715

C₂

yes

maximal, direct product factor

C₂×C₄⋊Dic₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₁₅₁⋊C₄

C₂

yes

maximal, complemented

C₂×(C₂×D₁₅₁)⋊C₄

C₂

yes

maximal

C₂²×Q₈×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

C₂²

yes

characteristic

C₂×C₄×D₁₅₁

—

C₂×C₄×D₁₅₁

C₂²

yes

—

C₄⋊Dic₁₅₁

C₂²

yes

complemented

Dic₁₅₁⋊C₄

C₂²

yes

complemented

(C₂×D₁₅₁)⋊C₄

C₂²

yes

—

C₂×Q₈×C₁₅₁

—

C₂×Q₈×C₁₅₁

C₂²

yes

complemented

C₂²×C₄×C₁₅₁

C₂²

yes

—

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

Q₈

yes

complemented

C₂²×D₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₂³

yes

—

C₂×Dic₁₅₁

C₂³

yes

—

C₄×D₁₅₁

—

C₂³×C₁₅₁

C₂³

yes

characteristic

Q₈×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

D₄

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂×C₄×C₁₅₁

C₂³

yes

—

C₂²×C₁₅₁

C₂×D₄

yes

—

C₂²×C₁₅₁

C₂×Q₈

yes

—

C₂²×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

C₂⁴

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

C₂×Q₈

yes

complemented

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

C₂²⋊Q₈

yes

complemented

C₂×C₁₅₁

C₂²×D₄

yes

characteristic

C₂×C₁₅₁

C₂²×Q₈

yes

characteristic

C₂×C₁₅₁

C₂×#41

yes

characteristic

D₁₅₁

—

C₁₅₁

C₂×C₂²⋊Q₈

yes

characteristic, complemented, Sylow

C₂×C₂²⋊Q₈

—

151

maximal, Sylow

C₂²×Q₈

D₁₅₁

yes

characteristic, complemented

C₂³×C₄

—

151

—

C₂²⋊Q₈

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₂²⋊C₄

—

151

—

C₂⁴

—

151

—

C₂×Q₈

—

C₂×Q₈

C₂×D₁₅₁

yes

complemented

C₂²×C₄

—

302

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

—

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

151

—

C₂³

—

151

—

C₂³

C₂²×D₁₅₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₂²⋊D₁₅₁

yes

—

C₂×C₄

C₂²×D₁₅₁

yes

—

C₂×C₄

C₂²×D₁₅₁

yes

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

151

—

C₂²

C₂×C₂²⋊D₁₅₁

yes

—

C₂²

Q₈×D₁₅₁

yes

—

C₂²

#14559005

yes

—

C₂²

C₂³×D₁₅₁

yes

characteristic

C₄

—

C₄

—

302

—

C₄

—

302

—

C₄

C₂×C₂²⋊D₁₅₁

yes

—

C₂

—

302

—

C₂

#18012715

yes

direct product factor

100

C₂

C₂²×C₂²⋊D₁₅₁

yes

characteristic

101

C₂

C₂×Q₈×D₁₅₁

yes

characteristic

102

C₂

C₂×#14559005

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×D₁₅₁
commutator, cosocle	C₂²×C₁₅₁	C₂⁴
Frattini	C₂²	C₂³×D₁₅₁
Fitting	C₂²×Q₈×C₁₅₁	C₂
socle	C₂³×C₁₅₁	C₂³
prime radical	C₂³×D₁₅₁	C₂²
hypercenter	C₂²×Q₈	D₁₅₁
hypocenter	C₁₅₁	C₂×C₂²⋊Q₈
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center

C₂³

C₂²×D₁₅₁

commutator, cosocle

C₂²×C₁₅₁

C₂⁴

Frattini

C₂²

C₂³×D₁₅₁

Fitting

C₂²×Q₈×C₁₅₁

C₂

socle

C₂³×C₁₅₁

C₂³

prime radical

C₂³×D₁₅₁

C₂²

hypercenter

C₂²×Q₈

D₁₅₁

hypocenter

C₁₅₁

C₂×C₂²⋊Q₈

LCS

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #18012715	C₂	direct (N × H)
3. C₂×C₄⋊Dic₁₅₁	C₂	semidirect
4. C₂×Dic₁₅₁⋊C₄	C₂	semidirect
6. C₂²×Q₈×C₁₅₁	C₂	semidirect
12. C₄⋊Dic₁₅₁	C₂²	semidirect
13. Dic₁₅₁⋊C₄	C₂²	semidirect
16. C₂×Q₈×C₁₅₁	C₂²	semidirect
21. C₂²×D₁₅₁	Q₈	semidirect
41. C₂×D₁₅₁	C₂×Q₈	semidirect
46. C₂×C₁₅₁	C₂²⋊Q₈	semidirect
51. C₁₅₁	C₂×C₂²⋊Q₈	semidirect
53. C₂²×Q₈	D₁₅₁	semidirect
61. C₂×Q₈	C₂×D₁₅₁	semidirect
99. C₂	#18012715	direct (N × H)

H ≅ G/N

type

2. #18012715

C₂

direct (N × H)

3. C₂×C₄⋊Dic₁₅₁

C₂

semidirect

4. C₂×Dic₁₅₁⋊C₄

C₂

semidirect

6. C₂²×Q₈×C₁₅₁

C₂

semidirect

12. C₄⋊Dic₁₅₁

C₂²

semidirect

13. Dic₁₅₁⋊C₄

C₂²

semidirect

16. C₂×Q₈×C₁₅₁

C₂²

semidirect

21. C₂²×D₁₅₁

Q₈

semidirect

41. C₂×D₁₅₁

C₂×Q₈

semidirect

46. C₂×C₁₅₁

C₂²⋊Q₈

semidirect

51. C₁₅₁

C₂×C₂²⋊Q₈

semidirect

53. C₂²×Q₈

D₁₅₁

semidirect

61. C₂×Q₈

C₂×D₁₅₁

semidirect

99. C₂

#18012715

direct (N × H)

Subgroups — C₂×#18012715 #23752707

Conjugacy classes of subgroups

Complements

Subgroups — C2×#18012715 #23752707

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#18012715 #23752707