Group #22586353 — subgroups

Subgroups — #22586353

Order 9088.

116 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 116 conjugacy classes of subgroups — too many to draw. Only its 39 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	#17986240	C₂	1	1	yes	maximal, characteristic
2	#17985762	C₂	1	1	yes	maximal, characteristic
3	C₂²⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×Dic₂₈₄	C₄	1	1	yes	characteristic, complemented
5	#14556107	—	2	1	no	—
6	C₂×C₄⋊Dic₇₁	—	2	1	no	—
7	#14556062	C₄	1	1	yes	characteristic, complemented
8	#14556053	—	2	1	no	—
9	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂×C₁₆×C₇₁	—	2	1	no	—
11	C₂×Dic₁₄₂	—	2	1	no	—
12	C₂²×Dic₇₁	—	4	1	no	—
13	Dic₂₈₄	—	4	1	no	—
14	C₄⋊Dic₇₁	—	2	2	no	—
15	C₄⋊Dic₇₁	—	2	1	no	—
16	C₄⋊Dic₇₁	—	4	1	no	—
17	Dic₇₁⋊C₄	—	4	1	no	—
18	C₂²⋊Dic₇₁	—	4	1	no	—
19	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₇₁	—	4	1	no	—
21	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
23	C₁₆×C₇₁	—	4	1	no	—
24	C₂³×C₇₁	D₈	1	1	yes	characteristic
25	C₂×C₄×C₇₁	—	2	1	no	—
26	C₂×C₄×C₇₁	—	2	1	no	—
27	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
29	C₂×Dic₇₁	—	4	2	no	—
30	C₂×Dic₇₁	—	4	1	no	—
31	C₂×Dic₇₁	—	4	1	no	—
32	C₂×Dic₇₁	—	8	1	no	—
33	C₂×Dic₇₁	—	8	1	no	—
34	Dic₁₄₂	—	4	1	no	—
35	Dic₁₄₂	—	4	1	no	—
36	C₈×C₇₁	—	2	1	no	—
37	C₈×C₇₁	—	2	1	no	—
38	C₂²×C₇₁	—	2	1	no	—
39	C₂²×C₇₁	—	2	1	no	—
40	C₂²×C₇₁	#102	1	1	yes	characteristic
41	C₂²×C₇₁	SD₃₂	1	1	yes	characteristic
42	C₂²×C₇₁	Q₃₂	1	1	yes	characteristic
43	C₄×C₇₁	—	2	1	no	—
44	C₄×C₇₁	#99	1	1	yes	characteristic
45	C₄×C₇₁	#104	1	1	yes	characteristic
46	Dic₇₁	—	8	2	no	—
47	Dic₇₁	—	8	1	no	—
48	Dic₇₁	—	8	1	no	—
49	C₂×C₇₁	—	2	1	no	—
50	C₂×C₇₁	—	2	1	no	—
51	C₂×C₇₁	#327	1	1	yes	characteristic
52	C₂×C₇₁	#358	1	1	yes	characteristic
53	C₂×C₇₁	#360	1	1	yes	characteristic
54	#1348	—	71	1	no	maximal, Sylow
55	C₇₁	#1348	1	1	yes	characteristic, complemented, Sylow
56	#467	—	71	1	no	—
57	C₂²⋊C₁₆	D₇₁	1	1	yes	characteristic
58	#340	—	71	1	no	—
59	C₂×Q₁₆	—	71	1	no	—
60	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
61	C₂²⋊Q₈	—	142	1	no	—
62	C₂×C₄⋊C₄	—	142	1	no	—
63	C₂×C₁₆	—	2	1	no	—
64	#107	—	71	1	no	—
65	#103	—	142	1	no	—
66	C₂×Q₈	—	142	1	no	—
67	C₂²×C₄	—	284	1	no	—
68	C₂²×C₄	D₂₈₄	1	1	yes	characteristic
69	Q₁₆	—	284	1	no	—
70	C₂×C₈	—	4	1	no	—
71	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
72	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
73	C₄⋊C₄	—	284	1	no	—
74	C₄⋊C₄	—	284	1	no	—
75	C₄⋊C₄	—	142	2	no	—
76	C₄⋊C₄	—	142	1	no	—
77	C₂²⋊C₄	—	284	1	no	—
78	C₁₆	—	4	1	no	—
79	C₂³	D₅₆₈	1	1	yes	characteristic
80	Q₈	—	284	1	no	—
81	Q₈	—	284	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	284	2	no	—
85	C₂×C₄	—	284	1	no	—
86	C₂×C₄	—	284	1	no	—
87	C₂×C₄	—	568	1	no	—
88	C₂×C₄	—	568	1	no	—
89	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#11765854	1	1	yes	characteristic
91	C₈	—	2	1	no	—
92	C₈	—	2	1	no	—
93	C₂²	—	2	1	no	—
94	C₂²	—	2	1	no	—
95	C₂²	#14556050	1	1	yes	characteristic
96	C₂²	#14556073	1	1	yes	characteristic
97	C₂²	Dic₅₆₈	1	1	yes	characteristic
98	C₄	—	2	1	no	—
99	C₄	—	568	2	no	—
100	C₄	—	568	1	no	—
101	C₄	—	568	1	no	—
102	C₄	#14556042	1	1	yes	characteristic
103	C₄	#14556054	1	1	yes	characteristic
104	C₂	—	2	1	no	—
105	C₂	—	2	1	no	—
106	C₂	#17985734	1	1	yes	characteristic
107	C₂	#17985804	1	1	yes	characteristic
108	C₂	#17985810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986240

C₂

yes

maximal, characteristic

#17985762

C₂

yes

maximal, characteristic

C₂²⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₄

yes

characteristic, complemented

#14556107

—

C₂×C₄⋊Dic₇₁

—

#14556062

C₄

yes

characteristic, complemented

#14556053

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

Dic₂₈₄

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₂²⋊Dic₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#102

yes

characteristic

C₂²×C₇₁

SD₃₂

yes

characteristic

C₂²×C₇₁

Q₃₂

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#467

—

C₂²⋊C₁₆

D₇₁

yes

characteristic

#340

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂²⋊Q₈

—

142

—

C₂×C₄⋊C₄

—

142

—

C₂×C₁₆

—

#107

—

#103

—

142

—

C₂×Q₈

—

142

—

C₂²×C₄

—

284

—

C₂²×C₄

D₂₈₄

yes

characteristic

Q₁₆

—

284

—

C₂×C₈

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

C₁₆

—

C₂³

D₅₆₈

yes

characteristic

Q₈

—

284

—

Q₈

—

284

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

568

—

C₂×C₄

—

568

—

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

#11765854

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

yes

C₂²

yes

C₂²

Dic₅₆₈

yes

characteristic

C₄

—

C₄

—

568

—

100

C₄

—

568

—

101

C₄

—

568

—

102

C₄

#14556042

yes

characteristic

103

C₄

#14556054

yes

characteristic

104

C₂

—

105

C₂

—

106

C₂

#17985734

yes

characteristic

107

C₂

#17985804

yes

characteristic

108

C₂

#17985810

yes

characteristic

	subgroup	G/H
center	C₂²	#14556050
commutator	C₂×C₈×C₇₁	C₂×C₄
Frattini	C₂²×C₈	C₂×D₇₁
Fitting	C₂²⋊C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#102
hypercenter	C₂²⋊C₁₆	D₇₁
hypocenter	C₇₁	#1348
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#14556050

commutator

C₂×C₈×C₇₁

C₂×C₄

Frattini

C₂²×C₈

C₂×D₇₁

Fitting

C₂²⋊C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

#102

hypercenter

C₂²⋊C₁₆

D₇₁

hypocenter

C₇₁

#1348

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
4. C₂×Dic₂₈₄	C₄	semidirect
7. #14556062	C₄	semidirect
55. C₇₁	#1348	semidirect

H ≅ G/N

type

4. C₂×Dic₂₈₄

C₄

semidirect

7. #14556062

C₄

semidirect

55. C₇₁

#1348

semidirect

Subgroups — #22586353

Conjugacy classes of subgroups

Complements