Group #34069243 — subgroups

Subgroups — #34069243

Order 19328.

224 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 101 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×#18012650	C₂	1	1	yes	maximal, characteristic
2	#23752082	C₂	1	1	yes	maximal, characteristic
3	#23751974	C₂	1	4	yes	maximal
4	#382×C₁₅₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₅₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₅₁	—	2	2	no	—
7	C₂×C₄⋊Dic₁₅₁	C₂²	1	2	yes	—
8	#18012650	C₄	1	4	yes	complemented
9	C₂×C₄⋊C₄×C₁₅₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
11	#95×C₁₅₁	—	2	2	no	—
12	C₂²×Dic₁₅₁	—	2	2	no	—
13	C₂×C₁₅₁⋊C₈	—	2	2	no	—
14	C₂×C₁₅₁⋊C₈	—	2	2	no	—
15	C₂×C₁₅₁⋊C₈	C₂×C₄	1	4	yes	—
16	C₄⋊Dic₁₅₁	—	2	4	no	—
17	C₄⋊Dic₁₅₁	—	4	4	no	—
18	C₂²×C₄×C₁₅₁	—	2	2	no	—
19	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
20	C₂²×C₄×C₁₅₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₁₅₁	—	2	4	no	—
22	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
23	C₄²×C₁₅₁	Q₈	1	2	yes	—
24	C₂×Dic₁₅₁	—	4	2	no	—
25	C₂×Dic₁₅₁	—	4	4	no	—
26	C₁₅₁⋊C₈	—	2	4	no	—
27	C₂³×C₁₅₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₁₅₁	—	2	2	no	—
29	C₂×C₄×C₁₅₁	—	2	2	no	—
30	C₂×C₄×C₁₅₁	—	4	2	no	—
31	C₂×C₄×C₁₅₁	—	4	2	no	—
32	C₂×C₄×C₁₅₁	—	4	2	no	—
33	C₂×C₄×C₁₅₁	C₄⋊C₄	1	4	yes	—
34	C₂×C₄×C₁₅₁	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₅₁	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₁₅₁	#41	1	2	yes	—
37	C₂×C₄×C₁₅₁	#41	1	2	yes	—
38	C₂²×C₁₅₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₁₅₁	#124	1	1	yes	characteristic
40	C₂²×C₁₅₁	#124	1	1	yes	characteristic
41	C₂²×C₁₅₁	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₁₅₁	#137	1	2	yes	—
43	C₄×C₁₅₁	—	2	4	no	—
44	C₄×C₁₅₁	—	8	2	no	—
45	C₄×C₁₅₁	#117	1	2	yes	—
46	C₄×C₁₅₁	#125	1	2	yes	—
47	Dic₁₅₁	—	8	2	no	—
48	C₂×C₁₅₁	#382	1	1	yes	characteristic
49	C₂×C₁₅₁	#428	1	2	yes	—
50	C₂×C₁₅₁	#489	1	2	yes	—
51	C₂×C₁₅₁	#490	1	1	yes	characteristic
52	C₂×C₁₅₁	#491	1	1	yes	characteristic
53	C₁₅₁	#1836	1	1	yes	characteristic, complemented, Sylow
54	#1836	—	151	1	no	maximal, Sylow
55	C₂×#97	—	151	1	no	—
56	#384	—	151	1	no	—
57	#382	D₁₅₁	1	1	yes	characteristic
58	#340	—	151	4	no	—
59	C₂²×C₈	—	151	2	no	—
60	C₂×C₄⋊C₄	—	302	2	no	—
61	C₂×C₄⋊C₄	—	151	2	no	—
62	C₂×C₄⋊C₄	C₂×D₁₅₁	1	2	yes	—
63	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
64	#97	—	151	4	no	—
65	#95	—	2	2	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	302	2	no	—
68	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
69	C₂²×C₄	C₂²×D₁₅₁	1	1	yes	characteristic
70	C₂×C₈	—	302	2	no	—
71	C₂×C₈	—	302	2	no	—
72	C₂×C₈	—	151	4	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	302	4	no	—
75	C₄⋊C₄	—	604	4	no	—
76	C₄²	C₄×D₁₅₁	1	2	yes	—
77	C₄²	Dic₃₀₂	1	2	yes	—
78	C₂³	C₂×C₂²⋊D₁₅₁	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	4	2	no	—
82	C₂×C₄	—	4	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	604	2	no	—
85	C₂×C₄	—	604	4	no	—
86	C₂×C₄	Dic₁₅₁⋊C₄	1	4	yes	—
87	C₂×C₄	C₂×C₄×D₁₅₁	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₃₀₂	1	1	yes	characteristic
89	C₂×C₄	#14559003	1	2	yes	—
90	C₂×C₄	#14559005	1	2	yes	—
91	C₈	—	302	4	no	—
92	C₂²	C₂×Dic₁₅₁⋊C₄	1	1	yes	characteristic
93	C₂²	#18012726	1	1	yes	characteristic
94	C₂²	#18012728	1	1	yes	characteristic
95	C₂²	#18012764	1	2	yes	—
96	C₂²	#18012769	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	8	2	no	—
99	C₄	—	1208	2	no	—
100	C₄	#18012697	1	2	yes	—
101	C₄	#18012730	1	2	yes	—
102	C₂	#23752077	1	1	yes	characteristic
103	C₂	#23752255	1	2	yes	—
104	C₂	#23752571	1	2	yes	—
105	C₂	#23752578	1	1	yes	characteristic
106	C₂	#23752582	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#18012650

C₂

yes

maximal, characteristic

#23752082

C₂

yes

maximal, characteristic

#23751974

C₂

yes

maximal

#382×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₅₁

—

C₂×C₄⋊Dic₁₅₁

C₂²

yes

—

#18012650

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₁

C₂²

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

#95×C₁₅₁

—

C₂²×Dic₁₅₁

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₁₅₁

—

C₄⋊Dic₁₅₁

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

D₄

yes

—

C₂²×C₄×C₁₅₁

C₂³

yes

characteristic

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

Q₈

yes

—

C₂×Dic₁₅₁

—

C₂×Dic₁₅₁

—

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₅₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₅₁

#41

yes

—

C₂×C₄×C₁₅₁

#41

yes

—

C₂²×C₁₅₁

C₂×C₄⋊C₄

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

—

C₂²×C₁₅₁

#137

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#117

yes

—

C₄×C₁₅₁

#125

yes

—

Dic₁₅₁

—

C₂×C₁₅₁

#382

yes

characteristic

C₂×C₁₅₁

#428

yes

—

C₂×C₁₅₁

#489

yes

—

C₂×C₁₅₁

yes

C₂×C₁₅₁

yes

C₁₅₁

yes

characteristic, complemented, Sylow

#1836

—

151

maximal, Sylow

C₂×#97

—

151

—

#384

—

151

—

#382

D₁₅₁

yes

characteristic

#340

—

151

—

C₂²×C₈

—

151

—

C₂×C₄⋊C₄

—

302

—

C₂×C₄⋊C₄

—

151

—

C₂×C₄⋊C₄

C₂×D₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

#97

—

151

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

302

—

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

C₂²×C₄

C₂²×D₁₅₁

yes

characteristic

C₂×C₈

—

302

—

C₂×C₈

—

302

—

C₂×C₈

—

151

—

C₄⋊C₄

—

C₄⋊C₄

—

302

—

C₄⋊C₄

—

604

—

C₄²

C₄×D₁₅₁

yes

—

C₄²

Dic₃₀₂

yes

—

C₂³

C₂×C₂²⋊D₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

604

—

C₂×C₄

—

604

—

C₂×C₄

Dic₁₅₁⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₁₅₁

yes

characteristic

C₂×C₄

C₂×Dic₃₀₂

yes

characteristic

C₂×C₄

#14559003

yes

—

C₂×C₄

#14559005

yes

—

C₈

—

302

—

C₂²

C₂×Dic₁₅₁⋊C₄

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#18012769

yes

—

C₄

—

C₄

—

C₄

—

1208

—

100

C₄

#18012697

yes

—

101

C₄

#18012730

yes

—

102

C₂

#23752077

yes

characteristic

103

C₂

#23752255

yes

—

104

C₂

#23752571

yes

—

105

C₂

#23752578

yes

characteristic

106

C₂

#23752582

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₅₁
commutator	C₂×C₄×C₁₅₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₅₁
Fitting	#382×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂×D₄
cosocle	C₂²×C₄×C₁₅₁	C₂³
hypercenter	#382	D₁₅₁
hypocenter	C₁₅₁	#1836
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₅₁

commutator

C₂×C₄×C₁₅₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₅₁

Fitting

#382×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂×D₄

cosocle

C₂²×C₄×C₁₅₁

C₂³

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #382

N	H ≅ G/N	type
8. #18012650	C₄	semidirect
53. C₁₅₁	#1836	semidirect

H ≅ G/N

type

8. #18012650

C₄

semidirect

53. C₁₅₁

#1836

semidirect

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

subgroup	index
C₂×#18012650	2
#23752082	2
#23751974	2
#382×C₁₅₁	2
#1836	151

subgroup

index

#382×C₁₅₁

151

Subgroups — #34069243

Conjugacy classes of subgroups

Complements

Maximal subgroups