Group #23752237 — subgroups

Subgroups — #23752237

Order 9664.

114 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 114 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₂×#14558987	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×D₁₅₁	C₂	1	1	yes	maximal, characteristic
3	#18012696	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₁₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₁₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	Dic₁₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₁₅₁	C₂²	1	1	yes	characteristic
9	#14558987	—	2	2	no	—
10	C₂×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₅₁⋊C₈	C₂²	1	1	yes	characteristic
12	C₈×D₁₅₁	—	2	2	no	—
13	Dic₁₅₁⋊C₄	C₄	1	1	yes	characteristic
14	Dic₁₅₁⋊C₄	C₄	1	1	yes	characteristic
15	(C₂×D₁₅₁)⋊C₄	C₄	1	1	yes	characteristic
16	(C₂×D₁₅₁)⋊C₄	C₄	1	1	yes	characteristic
17	C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
18	C₂×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×C₁₅₁	C₂²	1	1	yes	characteristic
20	C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
21	C₂²×D₁₅₁	C₂×C₄	1	1	yes	characteristic
22	C₂×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
24	C₂×Dic₁₅₁	C₂×C₄	1	1	yes	characteristic
25	C₄×D₁₅₁	D₄	1	2	yes	—
26	C₄×D₁₅₁	Q₈	1	2	yes	—
27	C₁₅₁⋊C₈	—	2	1	no	—
28	C₁₅₁⋊C₈	—	2	1	no	—
29	C₂×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
32	C₈×C₁₅₁	—	2	1	no	—
33	C₈×C₁₅₁	—	2	1	no	—
34	C₂²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×D₁₅₁	—	2	1	no	—
36	C₂×D₁₅₁	—	2	1	no	—
37	C₂×D₁₅₁	C₄⋊C₄	1	2	yes	—
38	C₄×C₁₅₁	—	2	1	no	—
39	C₄×C₁₅₁	—	2	1	no	—
40	C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
41	C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
42	Dic₁₅₁	—	2	1	no	—
43	Dic₁₅₁	—	2	1	no	—
44	Dic₁₅₁	C₄⋊C₄	1	2	yes	—
45	C₂×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₁₅₁	#131	1	1	yes	characteristic
47	C₂×C₁₅₁	#131	1	1	yes	characteristic
48	D₁₅₁	—	2	2	no	—
49	C₁₅₁	#424	1	1	yes	characteristic, complemented, Sylow
50	#424	—	151	1	no	maximal, Sylow
51	C₂×M₁₆	—	151	1	no	—
52	C₂²×C₈	—	151	1	no	—
53	#117	—	151	1	no	—
54	C₄⋊C₈	—	151	1	no	—
55	C₄⋊C₈	—	151	1	no	—
56	C₄⋊C₈	—	151	1	no	—
57	C₄⋊C₈	D₁₅₁	1	1	yes	characteristic, complemented
58	C₂²×C₄	—	151	1	no	—
59	M₁₆	—	302	2	no	—
60	C₂×C₈	—	302	2	no	—
61	C₂×C₈	—	151	1	no	—
62	C₂×C₈	—	151	1	no	—
63	C₂×C₈	C₂×D₁₅₁	1	1	yes	characteristic
64	C₂×C₈	C₂×D₁₅₁	1	1	yes	characteristic
65	C₄⋊C₄	—	151	1	no	—
66	C₄⋊C₄	—	151	1	no	—
67	C₂²⋊C₄	—	151	1	no	—
68	C₂²⋊C₄	—	151	1	no	—
69	C₄²	—	151	1	no	—
70	C₄²	C₂×D₁₅₁	1	1	yes	characteristic
71	C₂³	—	151	1	no	—
72	C₂×C₄	—	151	1	no	—
73	C₂×C₄	—	151	1	no	—
74	C₂×C₄	—	151	1	no	—
75	C₂×C₄	—	151	2	no	—
76	C₂×C₄	—	151	2	no	—
77	C₂×C₄	C₄×D₁₅₁	1	1	yes	characteristic
78	C₂×C₄	C₄×D₁₅₁	1	1	yes	characteristic
79	C₂×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	2	1	no	—
82	C₈	—	302	1	no	—
83	C₈	—	302	1	no	—
84	C₂²	—	302	1	no	—
85	C₂²	—	302	1	no	—
86	C₂²	—	151	2	no	—
87	C₂²	C₂×C₄×D₁₅₁	1	1	yes	characteristic
88	C₄	—	2	1	no	—
89	C₄	—	2	1	no	—
90	C₄	—	302	1	no	—
91	C₄	—	302	1	no	—
92	C₄	—	151	2	no	—
93	C₄	D₄×D₁₅₁	1	1	yes	characteristic
94	C₄	Q₈×D₁₅₁	1	1	yes	characteristic
95	C₂	—	302	2	no	—
96	C₂	C₄⋊C₄×D₁₅₁	1	1	yes	characteristic
97	C₂	#18012745	1	1	yes	characteristic
98	C₂	#18012747	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14558987

C₂

yes

maximal, characteristic

C₂×C₈×D₁₅₁

C₂

yes

maximal, characteristic

#18012696

C₂

yes

maximal, characteristic

C₄⋊(C₁₅₁⋊C₈)

C₂

yes

maximal, characteristic, complemented

Dic₁₅₁⋊C₈

C₂

yes

maximal, characteristic, complemented

Dic₁₅₁⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄⋊C₈×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂×C₄×D₁₅₁

C₂²

yes

characteristic

#14558987

—

C₂×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₂×C₁₅₁⋊C₈

C₂²

yes

characteristic

C₈×D₁₅₁

—

Dic₁₅₁⋊C₄

C₄

yes

characteristic

Dic₁₅₁⋊C₄

C₄

yes

characteristic

(C₂×D₁₅₁)⋊C₄

C₄

yes

characteristic

(C₂×D₁₅₁)⋊C₄

C₄

yes

characteristic

C₄×Dic₁₅₁

C₂²

yes

characteristic

C₂×C₈×C₁₅₁

C₂²

yes

characteristic

C₂×C₈×C₁₅₁

C₂²

yes

characteristic

C₄²×C₁₅₁

C₂²

yes

characteristic

C₂²×D₁₅₁

C₂×C₄

yes

characteristic

C₂×Dic₁₅₁

C₂×C₄

yes

characteristic

C₂×Dic₁₅₁

C₂×C₄

yes

characteristic

C₂×Dic₁₅₁

C₂×C₄

yes

characteristic

C₄×D₁₅₁

D₄

yes

—

C₄×D₁₅₁

Q₈

yes

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂³

yes

characteristic

C₈×C₁₅₁

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

C₄⋊C₄

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₄×C₁₅₁

C₂×Q₈

yes

characteristic

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

C₄⋊C₄

yes

—

C₂×C₁₅₁

C₂×C₄⋊C₄

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

D₁₅₁

—

C₁₅₁

#424

yes

characteristic, complemented, Sylow

#424

—

151

maximal, Sylow

C₂×M₁₆

—

151

—

C₂²×C₈

—

151

—

#117

—

151

—

C₄⋊C₈

—

151

—

C₄⋊C₈

—

151

—

C₄⋊C₈

—

151

—

C₄⋊C₈

D₁₅₁

yes

characteristic, complemented

C₂²×C₄

—

151

—

M₁₆

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

151

—

C₂×C₈

—

151

—

C₂×C₈

C₂×D₁₅₁

yes

characteristic

C₂×C₈

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

—

151

—

C₄⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₂²⋊C₄

—

151

—

C₄²

—

151

—

C₄²

C₂×D₁₅₁

yes

characteristic

C₂³

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₄×D₁₅₁

yes

characteristic

C₂×C₄

C₄×D₁₅₁

yes

characteristic

C₂×C₄

C₂²×D₁₅₁

yes

characteristic

C₈

—

C₈

—

C₈

—

302

—

C₈

—

302

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

151

—

C₂²

C₂×C₄×D₁₅₁

yes

characteristic

C₄

—

C₄

—

C₄

—

302

—

C₄

—

302

—

C₄

—

151

—

C₄

D₄×D₁₅₁

yes

characteristic

C₄

Q₈×D₁₅₁

yes

characteristic

C₂

—

302

—

C₂

C₄⋊C₄×D₁₅₁

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₁₅₁
commutator, socle	C₂²×C₁₅₁	C₂²×C₄
Fitting	C₄⋊C₈×C₁₅₁	C₂
prime radical	C₂²×D₁₅₁	C₂×C₄
cosocle	C₂×C₄×C₁₅₁	C₂³
hypercenter	C₄⋊C₈	D₁₅₁
hypocenter	C₁₅₁	#424
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×D₁₅₁

commutator, socle

C₂²×C₁₅₁

C₂²×C₄

Fitting

C₄⋊C₈×C₁₅₁

C₂

prime radical

C₂²×D₁₅₁

C₂×C₄

cosocle

C₂×C₄×C₁₅₁

C₂³

hypercenter

C₄⋊C₈

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂×C₄ ◁ C₄⋊C₈

N	H ≅ G/N	type
4. C₄⋊(C₁₅₁⋊C₈)	C₂	semidirect
5. Dic₁₅₁⋊C₈	C₂	semidirect
6. Dic₁₅₁⋊C₈	C₂	semidirect
7. C₄⋊C₈×C₁₅₁	C₂	semidirect
49. C₁₅₁	#424	semidirect
57. C₄⋊C₈	D₁₅₁	semidirect

H ≅ G/N

type

4. C₄⋊(C₁₅₁⋊C₈)

C₂

semidirect

5. Dic₁₅₁⋊C₈

C₂

semidirect

6. Dic₁₅₁⋊C₈

C₂

semidirect

7. C₄⋊C₈×C₁₅₁

C₂

semidirect

49. C₁₅₁

#424

semidirect

57. C₄⋊C₈

D₁₅₁

semidirect

Subgroups — #23752237

Conjugacy classes of subgroups

Complements