Group #22586504 — subgroups

Subgroups — #22586504

Order 9088.

110 conjugacy classes of subgroups; only the 23 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 23 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	#17986371	C₂	1	1	yes	maximal, characteristic
2	#17985792	C₂	1	1	yes	maximal, characteristic
3	#353×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765853	—	2	1	no	—
5	C₂×#11765853	—	2	1	no	—
6	#14556128	C₄	1	1	yes	characteristic, complemented
7	#14556043	—	2	1	no	—
8	C₄×C₇₁⋊C₈	C₄	1	1	yes	characteristic, complemented
9	#127×C₇₁	C₂²	1	1	yes	characteristic
10	#99×C₇₁	—	2	1	no	—
11	C₂×Dic₁₄₂	—	2	1	no	—
12	#11765853	—	4	2	no	—
13	#11765853	—	4	2	no	—
14	C₂×C₇₁⋊C₈	—	2	1	no	—
15	C₄⋊Dic₇₁	—	4	1	no	—
16	C₂²⋊Dic₇₁	—	4	1	no	—
17	C₂×D₄×C₇₁	—	4	1	no	—
18	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
19	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
20	C₂²⋊C₄×C₇₁	—	4	1	no	—
21	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂³×C₇₁	—	2	1	no	—
23	C₂³×C₇₁	—	2	1	no	—
24	D₄×C₇₁	—	4	1	no	—
25	D₄×C₇₁	—	4	1	no	—
26	D₄×C₇₁	—	8	2	no	—
27	C₂×C₄×C₇₁	—	2	1	no	—
28	C₂×C₄×C₇₁	—	8	1	no	—
29	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×Dic₇₁	—	4	1	no	—
31	C₂×Dic₇₁	—	8	1	no	—
32	Dic₁₄₂	—	4	1	no	—
33	Dic₁₄₂	—	4	1	no	—
34	C₇₁⋊C₈	—	2	2	no	—
35	C₂²×C₇₁	—	4	1	no	—
36	C₂²×C₇₁	—	4	1	no	—
37	C₂²×C₇₁	—	8	1	no	—
38	C₂²×C₇₁	—	8	1	no	—
39	C₂²×C₇₁	#99	1	1	yes	characteristic
40	C₄×C₇₁	—	2	2	no	—
41	C₄×C₇₁	—	2	1	no	—
42	C₄×C₇₁	—	16	1	no	—
43	Dic₇₁	—	8	1	no	—
44	Dic₇₁	—	16	1	no	—
45	C₂×C₇₁	—	2	1	no	—
46	C₂×C₇₁	—	8	1	no	—
47	C₂×C₇₁	—	8	1	no	—
48	C₂×C₇₁	#353	1	1	yes	characteristic
49	#1409	—	71	1	no	maximal, Sylow
50	C₇₁	#1409	1	1	yes	characteristic, complemented, Sylow
51	#492	—	71	1	no	—
52	#353	—	71	1	no	—
53	#353	D₇₁	1	1	yes	characteristic
54	C₂×SD₁₆	—	142	1	no	—
55	C₂×SD₁₆	—	142	1	no	—
56	C₄⋊Q₈	—	71	1	no	—
57	#127	C₂×D₇₁	1	1	yes	characteristic
58	#99	—	2	1	no	—
59	#99	—	142	1	no	—
60	C₄×C₈	—	71	1	no	—
61	C₂×Q₈	—	142	1	no	—
62	C₂×D₄	—	4	1	no	—
63	C₂×D₄	D₂₈₄	1	1	yes	characteristic
64	C₂×D₄	C₂²⋊D₇₁	1	1	yes	characteristic
65	SD₁₆	—	284	2	no	—
66	SD₁₆	—	284	2	no	—
67	C₂×C₈	—	142	1	no	—
68	C₄⋊C₄	—	284	1	no	—
69	C₂²⋊C₄	—	4	1	no	—
70	C₂²⋊C₄	—	284	1	no	—
71	C₄²	C₄×D₇₁	1	1	yes	characteristic
72	C₂³	—	2	1	no	—
73	C₂³	—	2	1	no	—
74	Q₈	—	284	1	no	—
75	Q₈	—	284	1	no	—
76	D₄	—	4	1	no	—
77	D₄	—	4	1	no	—
78	D₄	—	8	2	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	8	1	no	—
81	C₂×C₄	—	284	1	no	—
82	C₂×C₄	—	568	1	no	—
83	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
84	C₈	—	142	2	no	—
85	C₂²	—	4	1	no	—
86	C₂²	—	4	1	no	—
87	C₂²	—	8	1	no	—
88	C₂²	—	8	1	no	—
89	C₂²	#14556042	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	—	16	1	no	—
93	C₄	—	568	1	no	—
94	C₄	—	1136	1	no	—
95	C₂	—	2	1	no	—
96	C₂	—	8	1	no	—
97	C₂	—	8	1	no	—
98	C₂	#17985791	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986371

C₂

yes

maximal, characteristic

#17985792

C₂

yes

maximal, characteristic

#353×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765853

—

C₂×#11765853

—

#14556128

C₄

yes

characteristic, complemented

#14556043

—

C₄×C₇₁⋊C₈

C₄

yes

characteristic, complemented

#127×C₇₁

C₂²

yes

characteristic

#99×C₇₁

—

C₂×Dic₁₄₂

—

#11765853

—

#11765853

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂³×C₇₁

—

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#99

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#492

—

#353

—

#353

D₇₁

yes

characteristic

C₂×SD₁₆

—

142

—

C₂×SD₁₆

—

142

—

C₄⋊Q₈

—

#127

C₂×D₇₁

yes

characteristic

#99

—

#99

—

142

—

C₄×C₈

—

C₂×Q₈

—

142

—

C₂×D₄

—

C₂×D₄

D₂₈₄

yes

characteristic

C₂×D₄

C₂²⋊D₇₁

yes

characteristic

SD₁₆

—

284

—

SD₁₆

—

284

—

C₂×C₈

—

142

—

C₄⋊C₄

—

284

—

C₂²⋊C₄

—

C₂²⋊C₄

—

284

—

C₄²

C₄×D₇₁

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

284

—

Q₈

—

284

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

568

—

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₈

—

142

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14556042

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

568

—

C₄

—

1136

—

C₂

—

C₂

—

C₂

—

C₂

#17985791

yes

characteristic

	subgroup	G/H
center	C₂	#17985791
commutator	C₄²×C₇₁	C₂×C₄
Frattini	#127	C₂×D₇₁
Fitting	#353×C₇₁	C₂
socle	C₂×C₇₁	#353
hypercenter	#353	D₇₁
hypocenter	C₇₁	#1409
LCS	G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #353

subgroup

G/H

center

C₂

#17985791

commutator

C₄²×C₇₁

C₂×C₄

Frattini

#127

C₂×D₇₁

Fitting

#353×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ #127 ◁ #353

N	H ≅ G/N	type
6. #14556128	C₄	semidirect
8. C₄×C₇₁⋊C₈	C₄	semidirect
50. C₇₁	#1409	semidirect

H ≅ G/N

type

6. #14556128

C₄

semidirect

8. C₄×C₇₁⋊C₈

C₄

semidirect

50. C₇₁

#1409

semidirect

Subgroups — #22586504

Conjugacy classes of subgroups

Complements