Group #23752138 — subgroups

Subgroups — #23752138

Order 9664.

170 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 57 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×(C₂×D₁₅₁)⋊C₄	C₂	1	2	yes	maximal
2	C₂×(C₂×D₁₅₁)⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₁₅₁⋊#95	C₂	1	2	yes	maximal, complemented
4	C₁₅₁⋊#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₄⋊C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₁₅₁	C₂²	1	1	yes	characteristic
7	C₂²×Dic₁₅₁	C₂²	1	2	yes	—
8	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
9	(C₂×D₁₅₁)⋊C₄	—	2	4	no	—
10	(C₂×D₁₅₁)⋊C₄	—	2	2	no	—
11	C₂²×C₄×C₁₅₁	C₂²	1	2	yes	—
12	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₁₅₁	—	2	2	no	—
14	C₂²×D₁₅₁	—	2	1	no	—
15	C₂²×D₁₅₁	—	2	1	no	—
16	C₂²×D₁₅₁	—	2	2	no	—
17	C₂²×D₁₅₁	—	2	2	no	—
18	C₂²×D₁₅₁	D₄	1	2	yes	—
19	C₂×Dic₁₅₁	—	2	2	no	—
20	C₂×Dic₁₅₁	—	2	1	no	—
21	C₂×Dic₁₅₁	—	2	2	no	—
22	C₂×Dic₁₅₁	—	2	2	no	—
23	C₂×Dic₁₅₁	—	2	2	no	—
24	C₂³×C₁₅₁	C₂³	1	1	yes	characteristic
25	C₂×C₄×C₁₅₁	—	2	2	no	—
26	C₂×C₄×C₁₅₁	—	2	2	no	—
27	C₂×C₄×C₁₅₁	—	2	2	no	—
28	C₂×C₄×C₁₅₁	—	2	2	no	—
29	C₂×C₄×C₁₅₁	D₄	1	2	yes	—
30	C₂²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
32	C₂²×C₁₅₁	#41	1	1	yes	characteristic
33	C₂²×C₁₅₁	#41	1	2	yes	—
34	C₂²×C₁₅₁	#41	1	2	yes	—
35	C₂×D₁₅₁	—	2	2	no	—
36	C₂×D₁₅₁	—	2	4	no	—
37	C₂×D₁₅₁	—	4	1	no	—
38	C₂×D₁₅₁	—	4	1	no	—
39	C₂×D₁₅₁	—	4	2	no	—
40	C₄×C₁₅₁	—	2	2	no	—
41	C₄×C₁₅₁	—	4	2	no	—
42	Dic₁₅₁	—	4	2	no	—
43	Dic₁₅₁	—	4	1	no	—
44	C₂×C₁₅₁	#121	1	1	yes	characteristic
45	C₂×C₁₅₁	#123	1	1	yes	characteristic
46	C₂×C₁₅₁	#123	1	2	yes	—
47	C₂×C₁₅₁	#124	1	1	yes	characteristic
48	C₂×C₁₅₁	#126	1	2	yes	—
49	D₁₅₁	—	4	2	no	—
50	C₁₅₁	#397	1	1	yes	characteristic, complemented, Sylow
51	#397	—	151	1	no	maximal, Sylow
52	C₂×C₄⋊C₄	D₁₅₁	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	—	151	2	no	—
54	C₂×C₂²⋊C₄	—	151	1	no	—
55	#95	—	151	2	no	—
56	#95	—	151	1	no	—
57	C₂⁴	—	151	1	no	—
58	C₂²×C₄	—	151	2	no	—
59	C₂²×C₄	—	151	1	no	—
60	C₂²×C₄	C₂×D₁₅₁	1	2	yes	—
61	C₂²×C₄	C₂×D₁₅₁	1	1	yes	characteristic
62	C₄⋊C₄	—	2	2	no	—
63	C₂²⋊C₄	—	302	4	no	—
64	C₂²⋊C₄	—	302	2	no	—
65	C₂³	—	302	1	no	—
66	C₂³	—	302	1	no	—
67	C₂³	—	302	2	no	—
68	C₂³	—	302	2	no	—
69	C₂³	—	151	2	no	—
70	C₂³	C₂²×D₁₅₁	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	—	2	2	no	—
73	C₂×C₄	—	2	2	no	—
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	302	2	no	—
76	C₂×C₄	—	302	1	no	—
77	C₂×C₄	—	302	2	no	—
78	C₂×C₄	—	302	2	no	—
79	C₂×C₄	—	302	2	no	—
80	C₂×C₄	C₂²⋊D₁₅₁	1	2	yes	—
81	C₂²	—	302	2	no	—
82	C₂²	—	302	4	no	—
83	C₂²	—	604	1	no	—
84	C₂²	—	604	1	no	—
85	C₂²	—	604	2	no	—
86	C₂²	D₄×D₁₅₁	1	1	yes	characteristic
87	C₂²	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
88	C₂²	#14559003	1	1	yes	characteristic
89	C₂²	#14559004	1	2	yes	—
90	C₂²	#14559005	1	2	yes	—
91	C₄	—	2	2	no	—
92	C₄	—	4	2	no	—
93	C₄	—	604	2	no	—
94	C₄	—	604	1	no	—
95	C₂	—	604	2	no	—
96	C₂	#18012710	1	1	yes	characteristic
97	C₂	#18012722	1	1	yes	characteristic
98	C₂	#18012724	1	2	yes	—
99	C₂	#18012728	1	1	yes	characteristic
100	C₂	#18012734	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×(C₂×D₁₅₁)⋊C₄

C₂

yes

maximal

C₂×(C₂×D₁₅₁)⋊C₄

C₂

yes

maximal, characteristic

C₁₅₁⋊#95

C₂

yes

maximal, complemented

C₁₅₁⋊#95

C₂

yes

maximal, characteristic, complemented

C₂×C₄⋊C₄×C₁₅₁

C₂

yes

maximal, characteristic, complemented

C₂³×D₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

C₂²

yes

characteristic

(C₂×D₁₅₁)⋊C₄

—

(C₂×D₁₅₁)⋊C₄

—

C₂²×C₄×C₁₅₁

C₂²

yes

—

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

—

C₂²×D₁₅₁

D₄

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂³×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

D₄

yes

—

C₂²×C₁₅₁

C₂×D₄

yes

characteristic

C₂²×C₁₅₁

C₂×D₄

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

—

C₂²×C₁₅₁

#41

yes

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₂×D₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

—

C₂×C₁₅₁

#124

yes

characteristic

C₂×C₁₅₁

#126

yes

—

D₁₅₁

—

C₁₅₁

#397

yes

characteristic, complemented, Sylow

#397

—

151

maximal, Sylow

C₂×C₄⋊C₄

D₁₅₁

yes

characteristic, complemented

C₂×C₂²⋊C₄

—

151

—

C₂×C₂²⋊C₄

—

151

—

#95

—

151

—

#95

—

151

—

C₂⁴

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

—

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

302

—

C₂²⋊C₄

—

302

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

302

—

C₂³

—

151

—

C₂³

C₂²×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

C₂²⋊D₁₅₁

yes

—

C₂²

—

302

—

C₂²

—

302

—

C₂²

—

604

—

C₂²

—

604

—

C₂²

—

604

—

C₂²

D₄×D₁₅₁

yes

characteristic

C₂²

C₂×C₂²⋊D₁₅₁

yes

C₂²

yes

C₂²

yes

—

C₂²

#14559005

yes

—

C₄

—

C₄

—

C₄

—

604

—

C₄

—

604

—

C₂

—

604

—

C₂

yes

C₂

yes

C₂

yes

—

C₂

#18012728

yes

characteristic

100

C₂

#18012734

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₅₁
commutator, socle, cosocle	C₂³×C₁₅₁	C₂³
Fitting	C₂×C₄⋊C₄×C₁₅₁	C₂
prime radical	C₂³×D₁₅₁	C₂²
hypercenter	C₂×C₄⋊C₄	D₁₅₁
hypocenter	C₁₅₁	#397
LCS	G ▷ C₂³×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₅₁

commutator, socle, cosocle

C₂³×C₁₅₁

C₂³

Fitting

C₂×C₄⋊C₄×C₁₅₁

C₂

prime radical

C₂³×D₁₅₁

C₂²

hypercenter

C₂×C₄⋊C₄

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. C₁₅₁⋊#95	C₂	semidirect
4. C₁₅₁⋊#95	C₂	semidirect
5. C₂×C₄⋊C₄×C₁₅₁	C₂	semidirect
50. C₁₅₁	#397	semidirect
52. C₂×C₄⋊C₄	D₁₅₁	semidirect

H ≅ G/N

type

3. C₁₅₁⋊#95

C₂

semidirect

4. C₁₅₁⋊#95

C₂

semidirect

5. C₂×C₄⋊C₄×C₁₅₁

C₂

semidirect

50. C₁₅₁

#397

semidirect

52. C₂×C₄⋊C₄

D₁₅₁

semidirect

Subgroups — #23752138

Conjugacy classes of subgroups

Complements