Group #34066993 — subgroups

Subgroups — #34066993

Order 19328.

188 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 105 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	2	yes	maximal
2	#384×C₁₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₅₁⋊C₈	—	2	2	no	—
4	C₄⋊(C₁₅₁⋊C₈)	—	2	2	no	—
5	C₄⋊(C₁₅₁⋊C₈)	C₄	1	4	yes	complemented
6	C₂×C₄⋊C₄×C₁₅₁	—	2	2	no	—
7	C₂×C₄⋊C₄×C₁₅₁	C₄	1	2	yes	—
8	C₂×C₄²×C₁₅₁	C₂²	1	1	yes	characteristic
9	C₂×C₁₅₁⋊C₈	—	2	4	no	—
10	C₂×C₁₅₁⋊C₈	—	4	2	no	—
11	C₂×C₁₅₁⋊C₈	—	4	2	no	—
12	C₂²×C₄×C₁₅₁	—	2	2	no	—
13	C₂²×C₄×C₁₅₁	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₁₅₁	D₄	1	2	yes	—
15	C₄⋊C₄×C₁₅₁	—	4	2	no	—
16	C₄⋊C₄×C₁₅₁	—	4	2	no	—
17	C₄⋊C₄×C₁₅₁	—	4	2	no	—
18	C₄²×C₁₅₁	C₂×C₄	1	2	yes	—
19	C₄²×C₁₅₁	D₄	1	1	yes	characteristic
20	C₄²×C₁₅₁	Q₈	1	1	yes	characteristic
21	C₁₅₁⋊C₈	—	4	4	no	—
22	C₂³×C₁₅₁	C₂²⋊C₄	1	1	yes	characteristic
23	C₂×C₄×C₁₅₁	—	2	1	no	—
24	C₂×C₄×C₁₅₁	—	2	1	no	—
25	C₂×C₄×C₁₅₁	—	4	2	no	—
26	C₂×C₄×C₁₅₁	—	4	4	no	—
27	C₂×C₄×C₁₅₁	C₄²	1	1	yes	characteristic
28	C₂×C₄×C₁₅₁	C₂²⋊C₄	1	2	yes	—
29	C₂×C₄×C₁₅₁	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₁₅₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₅₁	D₈	1	2	yes	—
32	C₂×C₄×C₁₅₁	SD₁₆	1	2	yes	—
33	C₂×C₄×C₁₅₁	SD₁₆	1	2	yes	—
34	C₂×C₄×C₁₅₁	Q₁₆	1	2	yes	—
35	C₂²×C₁₅₁	#95	1	1	yes	characteristic
36	C₂²×C₁₅₁	#100	1	1	yes	characteristic
37	C₂²×C₁₅₁	#101	1	1	yes	characteristic
38	C₂²×C₁₅₁	#102	1	2	yes	—
39	C₂²×C₁₅₁	#103	1	2	yes	—
40	C₄×C₁₅₁	—	2	1	no	—
41	C₄×C₁₅₁	—	2	1	no	—
42	C₄×C₁₅₁	—	8	2	no	—
43	C₄×C₁₅₁	#102	1	2	yes	—
44	C₄×C₁₅₁	#103	1	2	yes	—
45	C₄×C₁₅₁	#106	1	2	yes	—
46	C₄×C₁₅₁	#107	1	2	yes	—
47	C₂×C₁₅₁	#331	1	1	yes	characteristic
48	C₂×C₁₅₁	#332	1	2	yes	—
49	C₂×C₁₅₁	#333	1	1	yes	characteristic
50	C₂×C₁₅₁	#340	1	2	yes	—
51	C₂×C₁₅₁	#343	1	1	yes	characteristic
52	C₁₅₁	#1296	1	1	yes	characteristic, complemented, Sylow
53	#1296	—	151	1	no	maximal, Sylow
54	C₂×C₄⋊C₈	—	151	2	no	—
55	#384	D₁₅₁	1	1	yes	characteristic
56	C₂²×C₈	—	302	2	no	—
57	C₂×C₄⋊C₄	—	2	2	no	—
58	C₂×C₄⋊C₄	Dic₁₅₁	1	2	yes	—
59	C₂×C₄²	C₂×D₁₅₁	1	1	yes	characteristic
60	C₄⋊C₈	—	302	2	no	—
61	C₄⋊C₈	—	151	4	no	—
62	C₂²×C₄	—	2	2	no	—
63	C₂²×C₄	C₂×Dic₁₅₁	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₁₅₁	1	2	yes	—
65	C₂×C₈	—	302	4	no	—
66	C₂×C₈	—	604	2	no	—
67	C₂×C₈	—	604	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄⋊C₄	—	4	2	no	—
70	C₄⋊C₄	—	4	2	no	—
71	C₄²	C₄×D₁₅₁	1	2	yes	—
72	C₄²	D₆₀₄	1	1	yes	characteristic
73	C₄²	Dic₃₀₂	1	1	yes	characteristic
74	C₂³	C₂²⋊Dic₁₅₁	1	1	yes	characteristic
75	C₂×C₄	—	2	1	no	—
76	C₂×C₄	—	2	1	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	4	4	no	—
79	C₂×C₄	C₄×Dic₁₅₁	1	1	yes	characteristic
80	C₂×C₄	(C₂×D₁₅₁)⋊C₄	1	2	yes	—
81	C₂×C₄	Dic₁₅₁⋊C₄	1	2	yes	—
82	C₂×C₄	C₄⋊Dic₁₅₁	1	1	yes	characteristic
83	C₂×C₄	#14558989	1	2	yes	—
84	C₂×C₄	#14558991	1	2	yes	—
85	C₂×C₄	#14558992	1	2	yes	—
86	C₂×C₄	#14558994	1	2	yes	—
87	C₈	—	604	4	no	—
88	C₂²	C₁₅₁⋊#95	1	1	yes	characteristic
89	C₂²	#18012656	1	1	yes	characteristic
90	C₂²	#18012658	1	1	yes	characteristic
91	C₂²	#18012660	1	2	yes	—
92	C₂²	#18012663	1	2	yes	—
93	C₄	—	2	1	no	—
94	C₄	—	2	1	no	—
95	C₄	—	8	2	no	—
96	C₄	#18012659	1	2	yes	—
97	C₄	#18012662	1	2	yes	—
98	C₄	#18012670	1	2	yes	—
99	C₄	#18012672	1	2	yes	—
100	C₂	#23751956	1	1	yes	characteristic
101	C₂	#23751958	1	2	yes	—
102	C₂	#23751961	1	1	yes	characteristic
103	C₂	#23751974	1	2	yes	—
104	C₂	#23751981	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₁₅₁⋊C₈)

C₂

yes

maximal

#384×C₁₅₁

C₂

yes

maximal, characteristic

C₂²×C₁₅₁⋊C₈

—

C₄⋊(C₁₅₁⋊C₈)

—

C₄⋊(C₁₅₁⋊C₈)

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₅₁

—

C₂×C₄⋊C₄×C₁₅₁

C₄

yes

—

C₂×C₄²×C₁₅₁

C₂²

yes

characteristic

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂×C₁₅₁⋊C₈

—

C₂²×C₄×C₁₅₁

—

C₂²×C₄×C₁₅₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₅₁

D₄

yes

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄⋊C₄×C₁₅₁

—

C₄²×C₁₅₁

C₂×C₄

yes

—

C₄²×C₁₅₁

D₄

yes

characteristic

C₄²×C₁₅₁

Q₈

yes

characteristic

C₁₅₁⋊C₈

—

C₂³×C₁₅₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

—

C₂×C₄×C₁₅₁

C₄²

yes

characteristic

C₂×C₄×C₁₅₁

C₂²⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

—

C₂×C₄×C₁₅₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₅₁

D₈

yes

—

C₂×C₄×C₁₅₁

SD₁₆

yes

—

C₂×C₄×C₁₅₁

SD₁₆

yes

—

C₂×C₄×C₁₅₁

Q₁₆

yes

—

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

C₂²×C₁₅₁

yes

—

C₂²×C₁₅₁

#103

yes

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

—

C₄×C₁₅₁

#102

yes

—

C₄×C₁₅₁

#103

yes

—

C₄×C₁₅₁

#106

yes

—

C₄×C₁₅₁

#107

yes

—

C₂×C₁₅₁

#331

yes

characteristic

C₂×C₁₅₁

#332

yes

—

C₂×C₁₅₁

#333

yes

characteristic

C₂×C₁₅₁

#340

yes

—

C₂×C₁₅₁

#343

yes

characteristic

C₁₅₁

#1296

yes

characteristic, complemented, Sylow

#1296

—

151

maximal, Sylow

C₂×C₄⋊C₈

—

151

—

#384

D₁₅₁

yes

characteristic

C₂²×C₈

—

302

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₁₅₁

yes

—

C₂×C₄²

C₂×D₁₅₁

yes

characteristic

C₄⋊C₈

—

302

—

C₄⋊C₈

—

151

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₁₅₁

yes

characteristic

C₂²×C₄

C₂²⋊D₁₅₁

yes

—

C₂×C₈

—

302

—

C₂×C₈

—

604

—

C₂×C₈

—

604

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₁₅₁

yes

—

C₄²

D₆₀₄

yes

characteristic

C₄²

Dic₃₀₂

yes

characteristic

C₂³

C₂²⋊Dic₁₅₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₅₁

yes

characteristic

C₂×C₄

(C₂×D₁₅₁)⋊C₄

yes

—

C₂×C₄

Dic₁₅₁⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₁₅₁

yes

characteristic

C₂×C₄

#14558989

yes

—

C₂×C₄

#14558991

yes

—

C₂×C₄

#14558992

yes

—

C₂×C₄

#14558994

yes

—

C₈

—

604

—

C₂²

C₁₅₁⋊#95

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#18012663

yes

—

C₄

—

C₄

—

C₄

—

C₄

#18012659

yes

—

C₄

#18012662

yes

—

C₄

#18012670

yes

—

C₄

#18012672

yes

—

100

C₂

#23751956

yes

characteristic

101

C₂

#23751958

yes

—

102

C₂

#23751961

yes

characteristic

103

C₂

#23751974

yes

—

104

C₂

#23751981

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₁₅₁
commutator	C₂×C₄×C₁₅₁	C₄²
Frattini	C₂×C₄²	C₂×D₁₅₁
Fitting	#384×C₁₅₁	C₂
socle, prime radical	C₂³×C₁₅₁	C₂²⋊C₄
cosocle	C₂×C₄²×C₁₅₁	C₂²
hypercenter	#384	D₁₅₁
hypocenter	C₁₅₁	#1296
LCS	G ▷ C₂×C₄×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂²⋊Dic₁₅₁

commutator

C₂×C₄×C₁₅₁

C₄²

Frattini

C₂×C₄²

C₂×D₁₅₁

Fitting

#384×C₁₅₁

C₂

socle, prime radical

C₂³×C₁₅₁

C₂²⋊C₄

cosocle

C₂×C₄²×C₁₅₁

C₂²

D₁₅₁

C₁₅₁

G ▷ C₂×C₄×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
5. C₄⋊(C₁₅₁⋊C₈)	C₄	semidirect
52. C₁₅₁	#1296	semidirect

H ≅ G/N

type

5. C₄⋊(C₁₅₁⋊C₈)

C₄

semidirect

52. C₁₅₁

#1296

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
#384×C₁₅₁	2
#1296	151

subgroup

index

C₂×C₄⋊(C₁₅₁⋊C₈)

#384×C₁₅₁

#1296

151

Subgroups — #34066993

Conjugacy classes of subgroups

Complements

Maximal subgroups