Group #34068438 — subgroups

Subgroups — #34068438

Order 19328.

172 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 49 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	#23752622	C₂	1	2	yes	maximal
2	#23752582	C₂	1	1	yes	maximal, characteristic
3	#23751954	C₂	1	1	yes	maximal, characteristic
4	#23751954	C₂	1	2	yes	maximal
5	#564×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	#18012739	C₂²	1	1	yes	characteristic
7	#18012672	—	2	2	no	—
8	#18012670	—	2	2	no	—
9	#18012662	—	2	2	no	—
10	#18012650	C₂²	1	1	yes	characteristic
11	#18012650	C₂²	1	2	yes	—
12	#125×C₁₅₁	—	2	2	no	—
13	#125×C₁₅₁	—	2	2	no	—
14	#125×C₁₅₁	—	2	2	no	—
15	#125×C₁₅₁	C₂²	1	1	yes	characteristic
16	#125×C₁₅₁	C₂²	1	2	yes	—
17	C₂×Dic₃₀₂	—	2	1	no	—
18	C₂×C₁₅₁⋊C₈	—	2	2	no	—
19	C₂×C₁₅₁⋊C₈	—	2	1	no	—
20	C₄⋊Dic₁₅₁	—	2	2	no	—
21	C₄⋊C₄×C₁₅₁	—	2	2	no	—
22	C₄⋊C₄×C₁₅₁	—	2	2	no	—
23	C₄⋊C₄×C₁₅₁	—	2	2	no	—
24	C₄⋊C₄×C₁₅₁	—	2	2	no	—
25	C₄⋊C₄×C₁₅₁	—	2	2	no	—
26	C₄⋊C₄×C₁₅₁	—	2	2	no	—
27	C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
28	C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₅₁	D₄	1	2	yes	—
30	C₄²×C₁₅₁	—	2	2	no	—
31	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
32	Dic₃₀₂	—	4	2	no	—
33	C₂×Dic₁₅₁	—	4	1	no	—
34	C₁₅₁⋊C₈	—	2	4	no	—
35	C₁₅₁⋊C₈	—	4	1	no	—
36	C₂×C₄×C₁₅₁	—	2	2	no	—
37	C₂×C₄×C₁₅₁	—	2	2	no	—
38	C₂×C₄×C₁₅₁	—	2	2	no	—
39	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
40	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
41	C₂²×C₁₅₁	#120	1	1	yes	characteristic
42	C₄×C₁₅₁	—	2	1	no	—
43	C₄×C₁₅₁	—	2	2	no	—
44	C₄×C₁₅₁	—	4	2	no	—
45	C₄×C₁₅₁	—	4	2	no	—
46	C₄×C₁₅₁	—	4	2	no	—
47	Dic₁₅₁	—	8	1	no	—
48	C₂×C₁₅₁	#454	1	1	yes	characteristic
49	C₂×C₁₅₁	#456	1	2	yes	—
50	C₁₅₁	#1663	1	1	yes	characteristic, complemented, Sylow
51	#1663	—	151	1	no	maximal, Sylow
52	#564	D₁₅₁	1	1	yes	characteristic
53	#501	—	151	2	no	—
54	#491	—	151	1	no	—
55	#330	—	151	2	no	—
56	#330	—	151	1	no	—
57	C₄⋊Q₈	—	151	1	no	—
58	#125	—	2	2	no	—
59	#125	—	2	2	no	—
60	#125	—	2	2	no	—
61	#125	C₂×D₁₅₁	1	1	yes	characteristic
62	#125	C₂×D₁₅₁	1	2	yes	—
63	#107	—	302	2	no	—
64	#106	—	302	2	no	—
65	#103	—	302	2	no	—
66	#97	—	151	1	no	—
67	#97	—	151	2	no	—
68	C₂×Q₈	—	302	1	no	—
69	C₂×C₈	—	302	2	no	—
70	C₂×C₈	—	302	1	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	2	no	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	302	2	no	—
78	C₄⋊C₄	C₂²⋊D₁₅₁	1	2	yes	—
79	C₄⋊C₄	C₂²⋊D₁₅₁	1	2	yes	—
80	C₄⋊C₄	C₂²⋊D₁₅₁	1	2	yes	—
81	C₄²	—	2	2	no	—
82	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
83	Q₈	—	604	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	604	1	no	—
88	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	2	yes	—
89	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
90	C₈	—	302	4	no	—
91	C₈	—	604	1	no	—
92	C₂²	#18012708	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	2	2	no	—
95	C₄	—	4	2	no	—
96	C₄	—	4	2	no	—
97	C₄	—	4	2	no	—
98	C₄	—	1208	1	no	—
99	C₂	#23752385	1	1	yes	characteristic
100	C₂	#23752395	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#23752622

C₂

yes

maximal

#23752582

C₂

yes

maximal, characteristic

#23751954

C₂

yes

maximal, characteristic

#23751954

C₂

yes

maximal

#564×C₁₅₁

C₂

yes

maximal, characteristic

#18012739

C₂²

yes

—

—

—

C₂²

yes

characteristic

#18012650

C₂²

yes

—

#125×C₁₅₁

—

#125×C₁₅₁

—

#125×C₁₅₁

—

#125×C₁₅₁

C₂²

yes

characteristic

#125×C₁₅₁

C₂²

yes

—

C₂×Dic₃₀₂

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄⋊Dic₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

—

C₄²×C₁₅₁

C₂³

yes

characteristic

Dic₃₀₂

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂²×C₁₅₁

#120

yes

characteristic

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

#454

yes

characteristic

C₂×C₁₅₁

#456

yes

—

C₁₅₁

#1663

yes

characteristic, complemented, Sylow

#1663

—

151

maximal, Sylow

#564

D₁₅₁

yes

characteristic

#501

—

151

—

#491

—

151

—

#330

—

151

—

#330

—

151

—

C₄⋊Q₈

—

151

—

—

—

—

C₂×D₁₅₁

yes

characteristic

#125

C₂×D₁₅₁

yes

—

#107

—

302

—

#106

—

302

—

#103

—

302

—

#97

—

151

—

#97

—

151

—

C₂×Q₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

C₄⋊C₄

C₂²⋊D₁₅₁

yes

—

C₄²

—

C₄²

C₂²×D₁₅₁

yes

characteristic

Q₈

—

604

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

604

—

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₈

—

302

—

C₈

—

604

—

C₂²

#18012708

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

1208

—

C₂

#23752385

yes

characteristic

100

C₂

#23752395

yes

—

	subgroup	G/H
center	C₂²	#18012708
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#564×C₁₅₁	C₂
socle, prime radical	C₂²×C₁₅₁	#120
hypercenter	#564	D₁₅₁
hypocenter	C₁₅₁	#1663
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

subgroup

G/H

center

C₂²

#18012708

commutator, cosocle

C₄²×C₁₅₁

C₂³

Frattini

C₄²

C₂²×D₁₅₁

Fitting

#564×C₁₅₁

C₂

C₂²×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #564

N	H ≅ G/N	type
50. C₁₅₁	#1663	semidirect

H ≅ G/N

type

50. C₁₅₁

#1663

semidirect

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

subgroup	index
#23752622	2
#23752582	2
#23751954	2
#564×C₁₅₁	2
#1663	151

subgroup

index

#564×C₁₅₁

151

Subgroups — #34068438

Conjugacy classes of subgroups

Complements

Maximal subgroups