Group #34069263 — subgroups

Subgroups — #34069263

Order 19328.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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⋊(C₁₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₁₅₁⋊C₈	C₂	1	4	yes	maximal
3	#23752058	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₅₁	C₂²	1	2	yes	—
7	C₄⋊(C₁₅₁⋊C₈)	—	2	2	no	—
8	Dic₁₅₁⋊C₈	—	2	8	no	—
9	C₁₅₁⋊#95	C₄	1	2	yes	—
10	C₁₅₁⋊#95	C₄	1	2	yes	—
11	C₂²×C₈×C₁₅₁	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₅₁	—	2	2	no	—
14	C₂²×Dic₁₅₁	C₂×C₄	1	4	yes	—
15	C₂×C₁₅₁⋊C₈	—	2	4	no	—
16	C₂×C₁₅₁⋊C₈	—	2	2	no	—
17	C₄×Dic₁₅₁	D₄	1	4	yes	—
18	C₄×Dic₁₅₁	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₅₁	—	2	2	no	—
23	C₂×C₈×C₁₅₁	—	2	4	no	—
24	C₄²×C₁₅₁	D₄	1	2	yes	—
25	C₄²×C₁₅₁	Q₈	1	2	yes	—
26	C₂×Dic₁₅₁	—	2	4	no	—
27	C₂×Dic₁₅₁	—	2	4	no	—
28	C₂×Dic₁₅₁	C₄⋊C₄	1	8	yes	—
29	C₁₅₁⋊C₈	—	4	2	no	—
30	C₂³×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₁	—	2	2	no	—
32	C₂×C₄×C₁₅₁	—	2	2	no	—
33	C₂×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₅₁	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₅₁	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₁₅₁	—	4	2	no	—
40	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₅₁	#131	1	2	yes	—
43	C₂²×C₁₅₁	#131	1	2	yes	—
44	C₄×C₁₅₁	—	2	2	no	—
45	C₄×C₁₅₁	—	2	2	no	—
46	C₄×C₁₅₁	#127	1	1	yes	characteristic
47	C₄×C₁₅₁	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₁₅₁	C₄⋊Q₈	1	2	yes	—
49	Dic₁₅₁	—	2	8	no	—
50	C₂×C₁₅₁	#384	1	1	yes	characteristic
51	C₂×C₁₅₁	#424	1	4	yes	—
52	C₂×C₁₅₁	#424	1	1	yes	characteristic
53	C₂×C₁₅₁	#424	1	1	yes	characteristic
54	C₁₅₁	#1843	1	1	yes	characteristic, complemented, Sylow
55	#1843	—	151	1	no	maximal, Sylow
56	C₂×C₄⋊C₈	—	151	4	no	—
57	C₂×C₄⋊C₈	—	151	1	no	—
58	C₂×C₄⋊C₈	D₁₅₁	1	1	yes	characteristic
59	#376	—	151	1	no	—
60	C₂²×C₈	—	151	2	no	—
61	C₂²×C₈	C₂×D₁₅₁	1	2	yes	—
62	C₂×C₄²	—	151	2	no	—
63	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	302	8	no	—
66	C₄⋊C₈	—	302	2	no	—
67	#95	—	151	2	no	—
68	#95	—	151	2	no	—
69	C₂²×C₄	—	151	4	no	—
70	C₂²×C₄	C₄×D₁₅₁	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₁₅₁	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	302	4	no	—
76	C₂×C₈	—	302	2	no	—
77	C₄²	—	151	4	no	—
78	C₄²	—	151	4	no	—
79	C₄²	C₂²⋊D₁₅₁	1	2	yes	—
80	C₄²	Dic₃₀₂	1	2	yes	—
81	C₂³	C₂×C₄×D₁₅₁	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	302	4	no	—
85	C₂×C₄	—	302	4	no	—
86	C₂×C₄	—	151	8	no	—
87	C₂×C₄	Dic₁₅₁⋊C₄	1	2	yes	—
88	C₂×C₄	Dic₁₅₁⋊C₄	1	2	yes	—
89	C₂×C₄	D₄×D₁₅₁	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₁₅₁	1	2	yes	—
92	C₂×C₄	C₂×Dic₃₀₂	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	604	2	no	—
95	C₂²	C₄⋊C₄×D₁₅₁	1	2	yes	—
96	C₂²	C₂×Dic₁₅₁⋊C₄	1	1	yes	characteristic
97	C₂²	#18012745	1	2	yes	—
98	C₂²	#18012747	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	302	8	no	—
102	C₄	#18012735	1	1	yes	characteristic
103	C₄	#18012737	1	1	yes	characteristic
104	C₄	#18012738	1	2	yes	—
105	C₂	#23752081	1	1	yes	characteristic
106	C₂	#23752237	1	4	yes	—
107	C₂	#23752238	1	1	yes	characteristic
108	C₂	#23752240	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₅₁⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₁₅₁⋊C₈

C₂

yes

maximal

#23752058

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₅₁

C₂²

yes

—

C₄⋊(C₁₅₁⋊C₈)

—

Dic₁₅₁⋊C₈

—

C₁₅₁⋊#95

C₄

yes

—

C₁₅₁⋊#95

C₄

yes

—

C₂²×C₈×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₄⋊C₈×C₁₅₁

—

C₂²×Dic₁₅₁

C₂×C₄

yes

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄×Dic₁₅₁

D₄

yes

—

C₄×Dic₁₅₁

Q₈

yes

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₂×C₈×C₁₅₁

—

C₂×C₈×C₁₅₁

—

C₄²×C₁₅₁

D₄

yes

—

C₄²×C₁₅₁

Q₈

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

C₄⋊C₄

yes

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

—

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₈×C₁₅₁

—

C₂²×C₁₅₁

C₂×C₄⋊C₄

yes

—

C₂²×C₁₅₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₅₁

#131

yes

—

C₂²×C₁₅₁

#131

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#127

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

characteristic

C₄×C₁₅₁

C₄⋊Q₈

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

#384

yes

characteristic

C₂×C₁₅₁

#424

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1843

—

151

maximal, Sylow

C₂×C₄⋊C₈

—

151

—

C₂×C₄⋊C₈

—

151

—

C₂×C₄⋊C₈

D₁₅₁

yes

characteristic

#376

—

151

—

C₂²×C₈

—

151

—

C₂²×C₈

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

302

—

C₄⋊C₈

—

302

—

#95

—

151

—

#95

—

151

—

C₂²×C₄

—

151

—

C₂²×C₄

C₄×D₁₅₁

yes

characteristic

C₂²×C₄

C₄×D₁₅₁

yes

characteristic

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₄²

—

151

—

C₄²

—

151

—

C₄²

C₂²⋊D₁₅₁

yes

—

C₄²

Dic₃₀₂

yes

—

C₂³

C₂×C₄×D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

151

—

C₂×C₄

Dic₁₅₁⋊C₄

yes

—

C₂×C₄

Dic₁₅₁⋊C₄

yes

—

C₂×C₄

D₄×D₁₅₁

yes

—

C₂×C₄

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₂×C₄

Q₈×D₁₅₁

yes

—

C₂×C₄

C₂×Dic₃₀₂

yes

characteristic

C₈

—

C₈

—

604

—

C₂²

C₄⋊C₄×D₁₅₁

yes

—

C₂²

C₂×Dic₁₅₁⋊C₄

yes

characteristic

C₂²

#18012745

yes

—

C₂²

#18012747

yes

—

C₄

—

100

C₄

—

101

C₄

—

302

—

102

C₄

#18012735

yes

characteristic

103

C₄

#18012737

yes

characteristic

104

C₄

#18012738

yes

—

105

C₂

#23752081

yes

characteristic

106

C₂

#23752237

yes

—

107

C₂

#23752238

yes

characteristic

108

C₂

#23752240

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₅₁
commutator, socle, prime radical	C₂³×C₁₅₁	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₅₁	C₂
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₅₁
hypocenter	C₁₅₁	#1843
LCS	G ▷ C₂³×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₅₁

commutator, socle, prime radical

C₂³×C₁₅₁

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₅₁

C₂

cosocle

C₂²×C₄×C₁₅₁

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₅₁

C₁₅₁

G ▷ C₂³×C₁₅₁ ▷ C₁₅₁

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
54. C₁₅₁	#1843	semidirect

H ≅ G/N

type

54. C₁₅₁

#1843

semidirect

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

subgroup	index
C₂×C₄⋊(C₁₅₁⋊C₈)	2
C₂×Dic₁₅₁⋊C₈	2
#23752058	2
C₂×C₄⋊C₈×C₁₅₁	2
#1843	151

subgroup

index

C₂×C₄⋊(C₁₅₁⋊C₈)

C₂×Dic₁₅₁⋊C₈

#23752058

C₂×C₄⋊C₈×C₁₅₁

#1843

151

Subgroups — #34069263

Conjugacy classes of subgroups

Complements

Maximal subgroups