Group #22590428 — subgroups

Subgroups — #22590428

Order 9088.

154 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 51 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×C₁₆×D₇₁	C₂	1	1	yes	maximal, characteristic
2	#17986237	C₂	1	2	yes	maximal
3	#17985825	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985803	C₂	1	2	yes	maximal, complemented
5	C₂×Q₃₂×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
7	#14556104	—	2	2	no	—
8	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
9	C₁₆×D₇₁	—	2	2	no	—
10	#14556062	C₂²	1	2	yes	—
11	#14556052	—	2	2	no	—
12	C₂×Q₁₆×C₇₁	C₂²	1	2	yes	—
13	Q₃₂×C₇₁	—	2	2	no	—
14	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
16	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
17	C₈×D₇₁	—	2	1	no	—
18	C₈×D₇₁	—	2	1	no	—
19	C₄⋊Dic₇₁	—	2	2	no	—
20	Dic₇₁⋊C₄	—	4	2	no	—
21	(C₂×D₇₁)⋊C₄	—	4	2	no	—
22	C₇₁⋊C₁₆	—	2	1	no	—
23	C₂×Q₈×C₇₁	—	2	2	no	—
24	Q₁₆×C₇₁	—	2	2	no	—
25	Q₁₆×C₇₁	—	2	2	no	—
26	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
27	C₁₆×C₇₁	D₄	1	2	yes	—
28	C₂²×D₇₁	D₈	1	1	yes	characteristic
29	Q₈×C₇₁	—	4	2	no	—
30	Q₈×C₇₁	—	4	2	no	—
31	C₂×C₄×C₇₁	—	4	2	no	—
32	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₇₁	—	4	2	no	—
34	C₂×Dic₇₁	D₈	1	1	yes	characteristic
35	C₄×D₇₁	—	2	1	no	—
36	C₄×D₇₁	—	2	1	no	—
37	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
38	C₈×C₇₁	#41	1	1	yes	characteristic
39	C₇₁⋊C₈	—	2	1	no	—
40	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
41	C₂×D₇₁	—	2	1	no	—
42	C₂×D₇₁	—	2	1	no	—
43	C₂×D₇₁	Q₃₂	1	2	yes	complemented
44	C₄×C₇₁	—	8	2	no	—
45	C₄×C₇₁	#121	1	1	yes	characteristic
46	C₄×C₇₁	#135	1	1	yes	characteristic
47	Dic₇₁	—	2	1	no	—
48	Dic₇₁	—	8	2	no	—
49	C₂×C₇₁	#466	1	1	yes	characteristic
50	C₂×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
51	C₂×C₇₁	#508	1	1	yes	characteristic
52	D₇₁	—	2	2	no	—
53	#2216	—	71	1	no	maximal, Sylow
54	C₇₁	#2216	1	1	yes	characteristic, complemented, Sylow
55	C₂×Q₃₂	D₇₁	1	1	yes	characteristic, complemented
56	C₂²×C₁₆	—	71	1	no	—
57	#467	—	71	2	no	—
58	#366	—	71	1	no	—
59	#358	—	71	2	no	—
60	C₂×Q₁₆	C₂×D₇₁	1	2	yes	—
61	C₂²×C₈	—	71	1	no	—
62	C₂²⋊Q₈	—	142	2	no	—
63	Q₃₂	—	2	2	no	—
64	C₂×C₁₆	—	71	1	no	—
65	C₂×C₁₆	—	142	2	no	—
66	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
67	#107	—	71	2	no	—
68	#103	—	142	2	no	—
69	C₂×Q₈	—	2	2	no	—
70	C₂²×C₄	—	71	1	no	—
71	Q₁₆	—	2	2	no	—
72	Q₁₆	—	2	2	no	—
73	C₂×C₈	—	71	1	no	—
74	C₂×C₈	—	142	1	no	—
75	C₂×C₈	—	142	1	no	—
76	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
77	C₄⋊C₄	—	284	2	no	—
78	C₄⋊C₄	—	142	2	no	—
79	C₂²⋊C₄	—	284	2	no	—
80	C₁₆	—	142	1	no	—
81	C₁₆	C₂²⋊D₇₁	1	2	yes	—
82	C₂³	—	71	1	no	—
83	Q₈	—	4	2	no	—
84	Q₈	—	4	2	no	—
85	C₂×C₄	—	4	2	no	—
86	C₂×C₄	—	284	2	no	—
87	C₂×C₄	—	71	1	no	—
88	C₂×C₄	—	142	1	no	—
89	C₂×C₄	—	142	1	no	—
90	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
91	C₈	—	142	1	no	—
92	C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
93	C₈	#11765864	1	1	yes	characteristic
94	C₂²	—	71	2	no	—
95	C₂²	—	142	1	no	—
96	C₂²	—	142	1	no	—
97	C₂²	D₈×D₇₁	1	1	yes	characteristic
98	C₄	—	8	2	no	—
99	C₄	—	568	2	no	—
100	C₄	—	142	1	no	—
101	C₄	#14556099	1	1	yes	characteristic
102	C₄	#14556147	1	1	yes	characteristic
103	C₂	—	142	2	no	—
104	C₂	#17986232	1	1	yes	characteristic
105	C₂	Q₃₂×D₇₁	1	1	yes	characteristic
106	C₂	#17986432	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₁₆×D₇₁

C₂

yes

maximal, characteristic

#17986237

C₂

yes

maximal

#17985825

C₂

yes

maximal, characteristic, complemented

#17985803

C₂

yes

maximal, complemented

C₂×Q₃₂×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇₁

C₂²

yes

characteristic

#14556104

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₇₁

—

#14556062

C₂²

yes

—

#14556052

—

C₂×Q₁₆×C₇₁

C₂²

yes

—

Q₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₄×D₇₁

D₄

yes

characteristic

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₈×D₇₁

—

C₈×D₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₁₆×C₇₁

D₄

yes

—

C₂²×D₇₁

D₈

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

Q₃₂

yes

complemented

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#466

yes

characteristic

C₂×C₇₁

C₂×Q₃₂

yes

characteristic

C₂×C₇₁

#508

yes

characteristic

D₇₁

—

#2216

—

maximal, Sylow

C₇₁

#2216

yes

characteristic, complemented, Sylow

C₂×Q₃₂

D₇₁

yes

characteristic, complemented

C₂²×C₁₆

—

#467

—

#366

—

#358

—

C₂×Q₁₆

C₂×D₇₁

yes

—

C₂²×C₈

—

C₂²⋊Q₈

—

142

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

#107

—

#103

—

142

—

C₂×Q₈

—

C₂²×C₄

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

C₁₆

—

142

—

C₁₆

C₂²⋊D₇₁

yes

—

C₂³

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

D₄×D₇₁

yes

characteristic

C₈

—

142

—

C₈

C₂×C₂²⋊D₇₁

yes

characteristic

C₈

#11765864

yes

characteristic

C₂²

—

C₂²

—

142

—

C₂²

—

142

—

C₂²

D₈×D₇₁

yes

characteristic

C₄

—

C₄

—

568

—

100

C₄

—

142

—

101

C₄

#14556099

yes

characteristic

102

C₄

#14556147

yes

characteristic

103

C₂

—

142

—

104

C₂

#17986232

yes

characteristic

105

C₂

Q₃₂×D₇₁

yes

characteristic

106

C₂

#17986432

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	C₂×Q₃₂×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	C₂×Q₃₂	D₇₁
hypocenter	C₇₁	#2216
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

C₂×Q₃₂×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

hypercenter

C₂×Q₃₂

D₇₁

hypocenter

C₇₁

#2216

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×Q₃₂

N	H ≅ G/N	type
3. #17985825	C₂	semidirect
4. #17985803	C₂	semidirect
5. C₂×Q₃₂×C₇₁	C₂	semidirect
43. C₂×D₇₁	Q₃₂	semidirect
54. C₇₁	#2216	semidirect
55. C₂×Q₃₂	D₇₁	semidirect

H ≅ G/N

type

3. #17985825

C₂

semidirect

4. #17985803

C₂

semidirect

5. C₂×Q₃₂×C₇₁

C₂

semidirect

43. C₂×D₇₁

Q₃₂

semidirect

54. C₇₁

#2216

semidirect

55. C₂×Q₃₂

D₇₁

semidirect

Subgroups — #22590428

Conjugacy classes of subgroups

Complements