Group #34070686 — subgroups

Subgroups — #34070686

Order 19328.

158 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 59 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	#23752261	C₂	1	1	yes	maximal, characteristic
2	#23752235	C₂	1	1	yes	maximal, characteristic
3	#23751982	C₂	1	2	yes	maximal
4	#23751971	C₂	1	2	yes	maximal
5	H(3,ℤ/4)×C₁₅₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14558988	C₂²	1	2	yes	—
7	C₂×#14558988	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₅₁	C₂²	1	1	yes	characteristic
9	#18012675	—	2	2	no	—
10	Dic₁₅₁⋊C₈	—	2	2	no	—
11	C₂×M₁₆×C₁₅₁	C₂²	1	1	yes	characteristic
12	#117×C₁₅₁	C₂²	1	2	yes	—
13	C₂²×Dic₁₅₁	—	2	1	no	—
14	#14558988	—	2	2	no	—
15	#14558988	—	2	2	no	—
16	#14558988	—	2	2	no	—
17	C₂×C₁₅₁⋊C₈	—	2	1	no	—
18	C₂×C₁₅₁⋊C₈	—	2	2	no	—
19	C₄×Dic₁₅₁	—	2	1	no	—
20	C₄×Dic₁₅₁	Q₈	1	1	yes	characteristic
21	C₄×Dic₁₅₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₁₅₁	—	2	2	no	—
24	C₂×C₈×C₁₅₁	D₄	1	2	yes	—
25	C₄⋊C₄×C₁₅₁	—	2	2	no	—
26	C₂²⋊C₄×C₁₅₁	—	2	2	no	—
27	C₄²×C₁₅₁	—	2	2	no	—
28	C₂×Dic₁₅₁	—	2	1	no	—
29	C₂×Dic₁₅₁	—	2	1	no	—
30	C₂×Dic₁₅₁	—	4	2	no	—
31	C₁₅₁⋊C₈	—	4	1	no	—
32	C₁₅₁⋊C₈	—	4	2	no	—
33	C₂³×C₁₅₁	#41	1	1	yes	characteristic
34	C₂×C₄×C₁₅₁	—	4	2	no	—
35	C₂×C₄×C₁₅₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₅₁	#41	1	2	yes	—
38	C₂×C₄×C₁₅₁	#41	1	2	yes	—
39	C₈×C₁₅₁	—	2	2	no	—
40	C₂²×C₁₅₁	—	4	1	no	—
41	C₂²×C₁₅₁	C₂²⋊Q₈	1	1	yes	characteristic
42	C₂²×C₁₅₁	#126	1	2	yes	—
43	C₄×C₁₅₁	—	8	2	no	—
44	C₄×C₁₅₁	#123	1	1	yes	characteristic
45	C₄×C₁₅₁	#124	1	1	yes	characteristic
46	C₄×C₁₅₁	#125	1	2	yes	—
47	Dic₁₅₁	—	2	2	no	—
48	Dic₁₅₁	—	4	1	no	—
49	C₂×C₁₅₁	—	2	1	no	—
50	C₂×C₁₅₁	—	2	2	no	—
51	C₂×C₁₅₁	#400	1	1	yes	characteristic
52	C₁₅₁	#2102	1	1	yes	characteristic, complemented, Sylow
53	#2102	—	151	1	no	maximal, Sylow
54	#430	—	151	1	no	—
55	#423	—	151	1	no	—
56	#344	—	151	2	no	—
57	#339	—	151	2	no	—
58	H(3,ℤ/4)	D₁₅₁	1	1	yes	characteristic
59	C₂×M₁₆	—	151	2	no	—
60	C₂×M₁₆	—	151	1	no	—
61	C₂×M₁₆	C₂×D₁₅₁	1	1	yes	characteristic
62	#117	C₂×D₁₅₁	1	2	yes	—
63	C₂×C₄²	—	151	1	no	—
64	#108	—	302	2	no	—
65	C₄⋊C₈	—	302	2	no	—
66	C₂²×C₄	—	302	1	no	—
67	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	302	2	no	—
70	M₁₆	—	302	2	no	—
71	M₁₆	—	302	2	no	—
72	C₂×C₈	—	302	1	no	—
73	C₂×C₈	—	302	2	no	—
74	C₂×C₈	D₆₀₄	1	2	yes	—
75	C₄⋊C₄	—	2	2	no	—
76	C₂²⋊C₄	—	2	2	no	—
77	C₄²	—	2	2	no	—
78	C₄²	—	302	1	no	—
79	C₄²	—	151	1	no	—
80	C₄²	—	151	1	no	—
81	C₂³	#14559003	1	1	yes	characteristic
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	302	1	no	—
84	C₂×C₄	—	302	1	no	—
85	C₂×C₄	—	604	2	no	—
86	C₂×C₄	C₂×D₆₀₄	1	1	yes	characteristic
87	C₂×C₄	Q₈×D₁₅₁	1	1	yes	characteristic
88	C₂×C₄	#14559003	1	2	yes	—
89	C₂×C₄	#14559004	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	604	1	no	—
92	C₈	—	604	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#18012719	1	1	yes	characteristic
95	C₂²	#18012732	1	2	yes	—
96	C₄	—	8	2	no	—
97	C₄	—	302	2	no	—
98	C₄	—	604	1	no	—
99	C₄	#18012723	1	1	yes	characteristic
100	C₄	#18012727	1	1	yes	characteristic
101	C₄	#18012729	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	—	2	1	no	—
104	C₂	#23752152	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#23752261

C₂

yes

maximal, characteristic

#23752235

C₂

yes

maximal, characteristic

#23751982

C₂

yes

maximal

#23751971

C₂

yes

maximal

H(3,ℤ/4)×C₁₅₁

C₂

yes

maximal, characteristic

C₂×#14558988

C₂²

yes

—

C₂×#14558988

C₂²

yes

characteristic

C₂×C₄×Dic₁₅₁

C₂²

yes

characteristic

#18012675

—

Dic₁₅₁⋊C₈

—

C₂×M₁₆×C₁₅₁

C₂²

yes

characteristic

#117×C₁₅₁

C₂²

yes

—

C₂²×Dic₁₅₁

—

#14558988

—

#14558988

—

#14558988

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₄×Dic₁₅₁

—

C₄×Dic₁₅₁

Q₈

yes

characteristic

C₄×Dic₁₅₁

Q₈

yes

characteristic

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

M₁₆×C₁₅₁

—

C₂×C₈×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₂²⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

#41

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₈×C₁₅₁

—

C₂²×C₁₅₁

—

C₂²×C₁₅₁

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₅₁

#126

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

C₄×C₁₅₁

yes

—

Dic₁₅₁

—

Dic₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

—

C₂×C₁₅₁

#400

yes

characteristic

C₁₅₁

#2102

yes

characteristic, complemented, Sylow

#2102

—

151

maximal, Sylow

#430

—

151

—

#423

—

151

—

#344

—

151

—

#339

—

151

—

H(3,ℤ/4)

D₁₅₁

yes

characteristic

C₂×M₁₆

—

151

—

C₂×M₁₆

—

151

—

C₂×M₁₆

C₂×D₁₅₁

yes

characteristic

#117

C₂×D₁₅₁

yes

—

C₂×C₄²

—

151

—

#108

—

302

—

C₄⋊C₈

—

302

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

M₁₆

—

M₁₆

—

302

—

M₁₆

—

302

—

M₁₆

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

D₆₀₄

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

302

—

C₄²

—

151

—

C₄²

—

151

—

C₂³

#14559003

yes

characteristic

C₂×C₄

—

C₂×C₄

—

302

—

C₂×C₄

—

302

—

C₂×C₄

—

604

—

C₂×C₄

C₂×D₆₀₄

yes

characteristic

C₂×C₄

Q₈×D₁₅₁

yes

characteristic

C₂×C₄

#14559003

yes

—

C₂×C₄

#14559004

yes

—

C₈

—

C₈

—

604

—

C₈

—

604

—

C₂²

—

C₂²

#18012719

yes

characteristic

C₂²

#18012732

yes

—

C₄

—

C₄

—

302

—

C₄

—

604

—

C₄

#18012723

yes

characteristic

100

C₄

#18012727

yes

characteristic

101

C₄

#18012729

yes

—

102

C₂

—

103

C₂

—

104

C₂

#23752152

yes

characteristic

	subgroup	G/H
center	C₄	#18012727
commutator, cosocle	C₂²×C₄×C₁₅₁	C₂³
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	H(3,ℤ/4)×C₁₅₁	C₂
socle	C₂×C₁₅₁	#400
prime radical	C₂³×C₁₅₁	#41
hypercenter	H(3,ℤ/4)	D₁₅₁
hypocenter	C₁₅₁	#2102
LCS	G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#18012727

commutator, cosocle

C₂²×C₄×C₁₅₁

C₂³

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

H(3,ℤ/4)×C₁₅₁

C₂

C₂×C₁₅₁

C₂³×C₁₅₁

D₁₅₁

C₁₅₁

G ▷ C₂²×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
52. C₁₅₁	#2102	semidirect

H ≅ G/N

type

52. C₁₅₁

#2102

semidirect

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

subgroup	index
#23752261	2
#23752235	2
#23751982	2
#23751971	2
H(3,ℤ/4)×C₁₅₁	2
#2102	151

subgroup

index

H(3,ℤ/4)×C₁₅₁

151

Subgroups — #34070686

Conjugacy classes of subgroups

Complements

Maximal subgroups