Group #22586212 — subgroups

Subgroups — #22586212

Order 9088.

138 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17986056	C₂	1	1	yes	maximal, characteristic
2	#17986056	C₂	1	1	yes	maximal, characteristic
3	C₄×M₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
5	C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
6	C₂²⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
8	#14556039	—	2	1	no	—
9	#14556039	—	2	1	no	—
10	C₂×M₁₆×C₇₁	—	2	1	no	—
11	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
12	#97×C₇₁	—	2	1	no	—
13	C₄×C₈×C₇₁	C₄	1	2	yes	—
14	C₂×C₇₁⋊C₈	—	2	2	no	—
15	C₂×C₇₁⋊C₈	—	2	2	no	—
16	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
19	M₁₆×C₇₁	—	2	4	no	—
20	C₂×C₈×C₇₁	—	2	2	no	—
21	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₄²×C₇₁	D₄	1	1	yes	characteristic
24	C₄²×C₇₁	D₄	1	1	yes	characteristic
25	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₇₁	—	2	2	no	—
27	C₂×C₄×C₇₁	—	4	1	no	—
28	C₂×C₄×C₇₁	—	4	1	no	—
29	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
35	C₈×C₇₁	—	2	4	no	—
36	C₇₁⋊C₈	—	4	2	no	—
37	C₇₁⋊C₈	—	4	2	no	—
38	C₂²×C₇₁	—	2	1	no	—
39	C₂²×C₇₁	—	2	1	no	—
40	C₂²×C₇₁	—	2	1	no	—
41	C₂²×C₇₁	#95	1	1	yes	characteristic
42	C₄×C₇₁	—	2	1	no	—
43	C₄×C₇₁	—	2	1	no	—
44	C₄×C₇₁	—	2	1	no	—
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	—	2	1	no	—
47	C₄×C₇₁	#104	1	2	yes	—
48	C₂×C₇₁	—	4	1	no	—
49	C₂×C₇₁	#338	1	1	yes	characteristic
50	C₂×C₇₁	#339	1	1	yes	characteristic
51	C₂×C₇₁	#344	1	1	yes	characteristic
52	#1291	—	71	1	no	maximal, Sylow
53	C₇₁	#1291	1	1	yes	characteristic, complemented, Sylow
54	#432	—	71	1	no	—
55	#432	—	71	1	no	—
56	C₄×M₁₆	D₇₁	1	1	yes	characteristic
57	C₂×M₁₆	—	2	1	no	—
58	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
59	C₄⋊C₈	—	71	2	no	—
60	C₄⋊C₈	—	142	1	no	—
61	C₂²⋊C₈	—	71	2	no	—
62	C₂²⋊C₈	—	142	1	no	—
63	#97	—	2	1	no	—
64	#97	—	142	1	no	—
65	#97	—	142	1	no	—
66	C₄×C₈	Dic₇₁	1	2	yes	—
67	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
68	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
69	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
70	M₁₆	—	2	4	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	142	2	no	—
73	C₂×C₈	—	142	2	no	—
74	C₄²	C₄×D₇₁	1	1	yes	characteristic
75	C₄²	D₂₈₄	1	1	yes	characteristic
76	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
77	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
78	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	—	4	1	no	—
82	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
83	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
84	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
85	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
86	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
87	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
88	C₈	—	2	4	no	—
89	C₈	—	284	2	no	—
90	C₈	—	284	2	no	—
91	C₂²	—	2	1	no	—
92	C₂²	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	C₇₁⋊#95	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	2	1	no	—
97	C₄	—	2	1	no	—
98	C₄	—	2	1	no	—
99	C₄	—	2	1	no	—
100	C₄	#14556056	1	2	yes	—
101	C₂	—	4	1	no	—
102	C₂	C₇₁⋊#338	1	1	yes	characteristic
103	C₂	#17985759	1	1	yes	characteristic
104	C₂	#17985770	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986056

C₂

yes

maximal, characteristic

#17986056

C₂

yes

maximal, characteristic

C₄×M₁₆×C₇₁

C₂

yes

maximal, characteristic

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

—

C₂²⋊(C₇₁⋊C₈)

—

C₂²⋊(C₇₁⋊C₈)

C₄

yes

—

#14556039

—

#14556039

—

C₂×M₁₆×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

—

C₄×C₈×C₇₁

C₄

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#95

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#104

yes

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#432

—

#432

—

C₄×M₁₆

D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₂²⋊C₈

—

C₂²⋊C₈

—

142

—

#97

—

#97

—

142

—

#97

—

142

—

C₄×C₈

Dic₇₁

yes

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄²

C₄×D₇₁

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₄²

C₂×Dic₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₂³

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₈

—

C₈

—

284

—

C₈

—

284

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₇₁⋊#95

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

C₄

—

100

C₄

#14556056

yes

—

101

C₂

—

102

C₂

C₇₁⋊#338

yes

characteristic

103

C₂

#17985759

yes

characteristic

104

C₂

#17985770

yes

characteristic

	subgroup	G/H
center	C₂²	C₇₁⋊#95
commutator	C₂×C₄×C₇₁	C₄²
Frattini	C₂×C₄²	C₂×D₇₁
Fitting	C₄×M₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#95
hypercenter	C₄×M₁₆	D₇₁
hypocenter	C₇₁	#1291
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₇₁⋊#95

commutator

C₂×C₄×C₇₁

C₄²

Frattini

C₂×C₄²

C₂×D₇₁

Fitting

C₄×M₁₆×C₇₁

C₂

socle

C₂²×C₇₁

#95

hypercenter

C₄×M₁₆

D₇₁

hypocenter

C₇₁

#1291

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
53. C₇₁	#1291	semidirect

H ≅ G/N

type

53. C₇₁

#1291

semidirect

Subgroups — #22586212

Conjugacy classes of subgroups

Complements