Group #23752111 — subgroups

Subgroups — #23752111

Order 9664.

170 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 91 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×C₄⋊Dic₁₅₁	C₂	1	2	yes	maximal
2	C₂×C₄⋊Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Dic₁₅₁	C₂	1	1	yes	maximal, characteristic
4	C₁₅₁⋊#95	C₂	1	2	yes	maximal
5	C₂×C₄⋊C₄×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₅₁	C₂²	1	2	yes	—
7	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
8	C₂²×Dic₁₅₁	C₂²	1	1	yes	characteristic
9	C₄⋊Dic₁₅₁	—	2	4	no	—
10	C₄⋊Dic₁₅₁	—	2	2	no	—
11	C₄×Dic₁₅₁	—	2	2	no	—
12	C₂²×C₄×C₁₅₁	C₂²	1	2	yes	—
13	C₂²×C₄×C₁₅₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₁₅₁	C₄	1	4	yes	complemented
15	C₂×Dic₁₅₁	—	2	1	no	—
16	C₂×Dic₁₅₁	—	2	1	no	—
17	C₂×Dic₁₅₁	—	2	1	no	—
18	C₂×Dic₁₅₁	—	2	1	no	—
19	C₂×Dic₁₅₁	—	2	2	no	—
20	C₂×Dic₁₅₁	—	2	2	no	—
21	C₂×Dic₁₅₁	—	2	2	no	—
22	C₂×Dic₁₅₁	D₄	1	2	yes	—
23	C₂×Dic₁₅₁	Q₈	1	2	yes	—
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	C₂×C₄	1	4	yes	—
28	C₂×C₄×C₁₅₁	C₂×C₄	1	2	yes	—
29	C₂×C₄×C₁₅₁	D₄	1	2	yes	—
30	C₂×C₄×C₁₅₁	Q₈	1	2	yes	—
31	C₂²×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
33	C₂²×C₁₅₁	C₂×D₄	1	1	yes	characteristic
34	C₂²×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
35	C₂²×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₁₅₁	#41	1	1	yes	characteristic
37	C₂²×C₁₅₁	#41	1	1	yes	characteristic
38	C₄×C₁₅₁	—	2	4	no	—
39	C₄×C₁₅₁	C₄⋊C₄	1	4	yes	complemented
40	Dic₁₅₁	—	2	2	no	—
41	Dic₁₅₁	—	2	2	no	—
42	Dic₁₅₁	—	4	2	no	—
43	C₂×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₁₅₁	C₄×D₄	1	1	yes	characteristic
45	C₂×C₁₅₁	C₄×Q₈	1	1	yes	characteristic
46	C₂×C₁₅₁	#121	1	1	yes	characteristic
47	C₂×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
48	C₂×C₁₅₁	#125	1	1	yes	characteristic
49	C₂×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
50	C₁₅₁	#389	1	1	yes	characteristic, complemented, Sylow
51	#389	—	151	1	no	maximal, Sylow
52	C₂×C₄⋊C₄	—	151	2	no	—
53	C₂×C₄⋊C₄	—	151	1	no	—
54	C₂×C₄⋊C₄	D₁₅₁	1	1	yes	characteristic
55	C₂×C₄²	—	151	1	no	—
56	#95	—	151	2	no	—
57	C₂²×C₄	—	151	2	no	—
58	C₂²×C₄	—	151	1	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	—	302	4	no	—
63	C₄⋊C₄	—	302	2	no	—
64	C₄⋊C₄	Dic₁₅₁	1	4	yes	complemented
65	C₄²	—	302	2	no	—
66	C₂³	C₂²×D₁₅₁	1	1	yes	characteristic
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	302	1	no	—
70	C₂×C₄	—	302	1	no	—
71	C₂×C₄	—	302	1	no	—
72	C₂×C₄	—	302	1	no	—
73	C₂×C₄	—	302	2	no	—
74	C₂×C₄	—	302	2	no	—
75	C₂×C₄	—	302	2	no	—
76	C₂×C₄	—	151	2	no	—
77	C₂×C₄	—	151	2	no	—
78	C₂×C₄	C₂×Dic₁₅₁	1	4	yes	—
79	C₂×C₄	C₂×Dic₁₅₁	1	2	yes	—
80	C₂×C₄	D₆₀₄	1	2	yes	—
81	C₂×C₄	Dic₃₀₂	1	2	yes	—
82	C₂²	C₂²×Dic₁₅₁	1	1	yes	characteristic
83	C₂²	D₄×D₁₅₁	1	1	yes	characteristic
84	C₂²	C₂×D₆₀₄	1	1	yes	characteristic
85	C₂²	Q₈×D₁₅₁	1	1	yes	characteristic
86	C₂²	C₂×Dic₃₀₂	1	1	yes	characteristic
87	C₂²	#14559003	1	1	yes	characteristic
88	C₂²	#14559005	1	1	yes	characteristic
89	C₄	—	2	4	no	—
90	C₄	—	302	2	no	—
91	C₄	—	302	2	no	—
92	C₄	—	604	2	no	—
93	C₄	C₄⋊Dic₁₅₁	1	4	yes	complemented
94	C₂	C₂×C₄⋊Dic₁₅₁	1	1	yes	characteristic
95	C₂	D₄×Dic₁₅₁	1	1	yes	characteristic
96	C₂	Q₈×Dic₁₅₁	1	1	yes	characteristic
97	C₂	#18012714	1	1	yes	characteristic
98	C₂	#18012719	1	1	yes	characteristic
99	C₂	#18012730	1	1	yes	characteristic
100	C₂	#18012738	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Dic₁₅₁

C₂

yes

maximal

C₂×C₄⋊Dic₁₅₁

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₅₁

C₂

yes

maximal, characteristic

C₁₅₁⋊#95

C₂

yes

maximal

C₂×C₄⋊C₄×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×Dic₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

C₂²

yes

characteristic

C₂²×Dic₁₅₁

C₂²

yes

characteristic

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₄×Dic₁₅₁

—

C₂²×C₄×C₁₅₁

C₂²

yes

—

C₂²×C₄×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₄×C₁₅₁

C₄

yes

complemented

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

D₄

yes

—

C₂×Dic₁₅₁

Q₈

yes

—

C₂³×C₁₅₁

C₂³

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×C₄

yes

—

C₂×C₄×C₁₅₁

C₂×C₄

yes

—

C₂×C₄×C₁₅₁

D₄

yes

—

C₂×C₄×C₁₅₁

Q₈

yes

—

C₂²×C₁₅₁

C₂²×C₄

yes

characteristic

C₂²×C₁₅₁

C₂×D₄

yes

characteristic

C₂²×C₁₅₁

C₂×D₄

yes

characteristic

C₂²×C₁₅₁

C₂×Q₈

yes

characteristic

C₂²×C₁₅₁

C₂×Q₈

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₄×C₁₅₁

—

C₄×C₁₅₁

C₄⋊C₄

yes

complemented

Dic₁₅₁

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂×C₁₅₁

C₄×D₄

yes

characteristic

C₂×C₁₅₁

C₄×Q₈

yes

characteristic

C₂×C₁₅₁

#121

yes

characteristic

C₂×C₁₅₁

C₂²⋊Q₈

yes

characteristic

C₂×C₁₅₁

#125

yes

characteristic

C₂×C₁₅₁

C₄⋊Q₈

yes

characteristic

C₁₅₁

#389

yes

characteristic, complemented, Sylow

#389

—

151

maximal, Sylow

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

D₁₅₁

yes

characteristic

C₂×C₄²

—

151

—

#95

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₂×D₁₅₁

yes

—

C₂²×C₄

C₂×D₁₅₁

yes

characteristic

C₄⋊C₄

—

302

—

C₄⋊C₄

—

302

—

C₄⋊C₄

Dic₁₅₁

yes

complemented

C₄²

—

302

—

C₂³

C₂²×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

—

151

—

C₂×C₄

C₂×Dic₁₅₁

yes

—

C₂×C₄

C₂×Dic₁₅₁

yes

—

C₂×C₄

D₆₀₄

yes

—

C₂×C₄

Dic₃₀₂

yes

—

C₂²

C₂²×Dic₁₅₁

yes

characteristic

C₂²

D₄×D₁₅₁

yes

characteristic

C₂²

C₂×D₆₀₄

yes

characteristic

C₂²

Q₈×D₁₅₁

yes

characteristic

C₂²

C₂×Dic₃₀₂

yes

C₂²

yes

C₂²

yes

C₄

—

C₄

—

302

—

C₄

—

302

—

C₄

—

604

—

C₄

C₄⋊Dic₁₅₁

yes

complemented

C₂

C₂×C₄⋊Dic₁₅₁

yes

characteristic

C₂

D₄×Dic₁₅₁

yes

characteristic

C₂

Q₈×Dic₁₅₁

yes

C₂

yes

C₂

yes

C₂

yes

100

C₂

#18012738

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂²×D₁₅₁
commutator	C₂²×C₁₅₁	C₂²×C₄
Fitting	C₂×C₄⋊C₄×C₁₅₁	C₂
socle, prime radical, cosocle	C₂³×C₁₅₁	C₂³
hypercenter	C₂×C₄⋊C₄	D₁₅₁
hypocenter	C₁₅₁	#389
LCS	G ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂³

C₂²×D₁₅₁

commutator

C₂²×C₁₅₁

C₂²×C₄

Fitting

C₂×C₄⋊C₄×C₁₅₁

C₂

socle, prime radical, cosocle

C₂³×C₁₅₁

C₂³

hypercenter

C₂×C₄⋊C₄

D₁₅₁

C₁₅₁

G ▷ C₂²×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
14. C₄⋊C₄×C₁₅₁	C₄	semidirect
39. C₄×C₁₅₁	C₄⋊C₄	semidirect
50. C₁₅₁	#389	semidirect
64. C₄⋊C₄	Dic₁₅₁	semidirect
93. C₄	C₄⋊Dic₁₅₁	semidirect

H ≅ G/N

type

14. C₄⋊C₄×C₁₅₁

C₄

semidirect

39. C₄×C₁₅₁

C₄⋊C₄

semidirect

50. C₁₅₁

#389

semidirect

64. C₄⋊C₄

Dic₁₅₁

semidirect

93. C₄

C₄⋊Dic₁₅₁

semidirect

Subgroups — #23752111

Conjugacy classes of subgroups

Complements