Group #22587731 — subgroups

Subgroups — #22587731

Order 9088.

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

normal subgroups only

This group has 156 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	#17986206	C₂	1	2	yes	maximal
2	C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
3	#17985752	C₂	1	1	yes	maximal, characteristic
4	#17985745	C₂	1	2	yes	maximal
5	#500×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765855	—	2	2	no	—
7	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
8	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	Dic₇₁⋊C₈	—	2	1	no	—
11	#14556052	—	2	2	no	—
12	C₈×Dic₇₁	—	2	1	no	—
13	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
14	#107×C₇₁	—	2	1	no	—
15	#107×C₇₁	—	2	1	no	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
18	#11765855	—	4	4	no	—
19	C₂×C₇₁⋊C₈	—	2	1	no	—
20	C₂×C₇₁⋊C₈	—	2	2	no	—
21	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
22	Dic₇₁⋊C₄	—	2	1	no	—
23	C₄×Dic₇₁	—	2	1	no	—
24	C₂×Q₈×C₇₁	—	2	2	no	—
25	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₄⋊C₄×C₇₁	—	2	2	no	—
28	C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₄²×C₇₁	C₂³	1	1	yes	characteristic
30	Q₈×C₇₁	—	4	2	no	—
31	Q₈×C₇₁	—	4	2	no	—
32	C₂×C₄×C₇₁	—	4	2	no	—
33	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
36	C₂×Dic₇₁	—	2	1	no	—
37	C₂×Dic₇₁	—	2	1	no	—
38	Dic₁₄₂	—	2	1	no	—
39	Dic₁₄₂	D₈	1	2	yes	—
40	C₈×C₇₁	—	2	1	no	—
41	C₈×C₇₁	Q₁₆	1	2	yes	complemented
42	C₇₁⋊C₈	—	4	2	no	—
43	C₇₁⋊C₈	—	4	1	no	—
44	C₂²×C₇₁	#121	1	1	yes	characteristic
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	—	8	2	no	—
47	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×C₇₁	#135	1	1	yes	characteristic
50	C₄×C₇₁	#137	1	1	yes	characteristic
51	Dic₇₁	—	2	2	no	—
52	Dic₇₁	—	4	1	no	—
53	C₂×C₇₁	#462	1	1	yes	characteristic
54	C₂×C₇₁	#466	1	1	yes	characteristic
55	C₂×C₇₁	#473	1	1	yes	characteristic
56	#1674	—	71	1	no	maximal, Sylow
57	C₇₁	#1674	1	1	yes	characteristic, complemented, Sylow
58	#500	D₇₁	1	1	yes	characteristic
59	#462	—	71	2	no	—
60	C₈×Q₈	—	71	1	no	—
61	#335	—	71	1	no	—
62	#332	—	71	2	no	—
63	C₂×Q₁₆	—	142	2	no	—
64	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
65	C₄×Q₈	—	71	1	no	—
66	#107	—	2	1	no	—
67	#107	—	2	1	no	—
68	C₄⋊C₈	—	71	1	no	—
69	C₄⋊C₈	—	71	2	no	—
70	C₄⋊C₈	—	142	1	no	—
71	#103	—	142	2	no	—
72	C₄×C₈	—	142	1	no	—
73	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
74	C₂×Q₈	—	2	2	no	—
75	C₂×Q₈	—	71	1	no	—
76	Q₁₆	—	284	4	no	—
77	C₂×C₈	—	142	2	no	—
78	C₂×C₈	—	142	1	no	—
79	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	71	1	no	—
84	C₄⋊C₄	—	142	1	no	—
85	C₄²	—	142	1	no	—
86	C₄²	C₂²×D₇₁	1	1	yes	characteristic
87	Q₈	—	4	2	no	—
88	Q₈	—	4	2	no	—
89	Q₈	—	71	2	no	—
90	Q₈	—	142	1	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	142	1	no	—
93	C₂×C₄	—	142	1	no	—
94	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
95	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
96	C₂×C₄	#11765864	1	1	yes	characteristic
97	C₈	—	2	1	no	—
98	C₈	—	284	2	no	—
99	C₈	—	284	1	no	—
100	C₈	#11765855	1	2	yes	complemented
101	C₂²	#14556099	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	8	2	no	—
104	C₄	—	284	1	no	—
105	C₄	—	142	2	no	—
106	C₄	D₈×D₇₁	1	1	yes	characteristic
107	C₄	C₂×#11765855	1	1	yes	characteristic
108	C₄	#14556147	1	1	yes	characteristic
109	C₄	#14556159	1	1	yes	characteristic
110	C₂	#17986206	1	1	yes	characteristic
111	C₂	#17986232	1	1	yes	characteristic
112	C₂	#17986271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986206

C₂

yes

maximal

C₈×Dic₁₄₂

C₂

yes

maximal, characteristic

#17985752

C₂

yes

maximal, characteristic

#17985745

C₂

yes

maximal

#500×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765855

—

C₄×Dic₁₄₂

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

Dic₇₁⋊C₈

—

#14556052

—

C₈×Dic₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

characteristic

#11765855

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

C₂³

yes

characteristic

Q₈×C₇₁

—

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

D₈

yes

—

C₈×C₇₁

—

C₈×C₇₁

Q₁₆

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#500

D₇₁

yes

characteristic

#462

—

C₈×Q₈

—

#335

—

#332

—

C₂×Q₁₆

—

142

—

C₄⋊Q₈

C₂×D₇₁

yes

—

C₄×Q₈

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

#103

—

142

—

C₄×C₈

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

284

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄²

—

142

—

C₄²

C₂²×D₇₁

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

142

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

D₄×D₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

#11765864

yes

characteristic

C₈

—

C₈

—

284

—

C₈

—

284

—

100

C₈

#11765855

yes

complemented

101

C₂²

#14556099

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

284

—

105

C₄

—

142

—

106

C₄

D₈×D₇₁

yes

characteristic

107

C₄

C₂×#11765855

yes

characteristic

108

C₄

#14556147

yes

characteristic

109

C₄

#14556159

yes

characteristic

110

C₂

#17986206

yes

characteristic

111

C₂

#17986232

yes

characteristic

112

C₂

#17986271

yes

characteristic

	subgroup	G/H
center	C₂²	#14556099
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#500×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#500	D₇₁
hypocenter	C₇₁	#1674
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #500

subgroup

G/H

center

C₂²

#14556099

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#500×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #500

N	H ≅ G/N	type
41. C₈×C₇₁	Q₁₆	semidirect
57. C₇₁	#1674	semidirect
100. C₈	#11765855	semidirect

H ≅ G/N

type

41. C₈×C₇₁

Q₁₆

semidirect

57. C₇₁

#1674

semidirect

100. C₈

#11765855

semidirect

Subgroups — #22587731

Conjugacy classes of subgroups

Complements