Group #22587831 — subgroups

Subgroups — #22587831

Order 9088.

152 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 152 conjugacy classes of subgroups — too many to draw. Only its 55 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	#17986281	C₂	1	2	yes	maximal
2	#17986129	C₂	1	1	yes	maximal, characteristic
3	#17985750	C₂	1	1	yes	maximal, characteristic
4	#17985745	C₂	1	2	yes	maximal
5	#494×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
7	#14556059	—	2	2	no	—
8	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	Dic₇₁⋊C₈	—	2	1	no	—
11	#14556051	—	2	2	no	—
12	#14556038	—	2	1	no	—
13	C₂×Q₁₆×C₇₁	—	2	1	no	—
14	C₂×Q₁₆×C₇₁	—	2	1	no	—
15	C₄⋊Q₈×C₇₁	C₂²	1	2	yes	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
18	C₂×C₇₁⋊C₈	—	2	1	no	—
19	C₂×C₇₁⋊C₈	—	2	2	no	—
20	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
21	Dic₇₁⋊C₄	—	2	1	no	—
22	C₄×Dic₇₁	—	2	1	no	—
23	C₂×Q₈×C₇₁	—	2	2	no	—
24	Q₁₆×C₇₁	—	4	2	no	—
25	Q₁₆×C₇₁	—	4	2	no	—
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₇₁	—	2	2	no	—
29	C₄⋊C₄×C₇₁	—	2	2	no	—
30	C₄²×C₇₁	C₂³	1	1	yes	characteristic
31	Q₈×C₇₁	—	4	2	no	—
32	Q₈×C₇₁	—	4	2	no	—
33	C₂×C₄×C₇₁	—	4	2	no	—
34	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
37	C₂×Dic₇₁	—	2	1	no	—
38	C₂×Dic₇₁	—	2	1	no	—
39	Dic₁₄₂	—	2	2	no	—
40	C₈×C₇₁	—	2	1	no	—
41	C₈×C₇₁	SD₁₆	1	2	yes	—
42	C₇₁⋊C₈	—	4	1	no	—
43	C₇₁⋊C₈	—	4	2	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×SD₁₆	1	1	yes	characteristic
48	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₄×C₇₁	#137	1	2	yes	—
50	Dic₇₁	—	4	1	no	—
51	Dic₇₁	—	4	1	no	—
52	C₂×C₇₁	#460	1	1	yes	characteristic
53	C₂×C₇₁	#470	1	1	yes	characteristic
54	C₂×C₇₁	#473	1	1	yes	characteristic
55	#1690	—	71	1	no	maximal, Sylow
56	C₇₁	#1690	1	1	yes	characteristic, complemented, Sylow
57	#494	D₇₁	1	1	yes	characteristic
58	#475	—	71	2	no	—
59	#446	—	71	1	no	—
60	#334	—	71	1	no	—
61	#332	—	71	2	no	—
62	C₂×Q₁₆	—	2	1	no	—
63	C₂×Q₁₆	—	2	1	no	—
64	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
65	C₄×Q₈	—	71	1	no	—
66	#106	—	142	2	no	—
67	C₄⋊C₈	—	71	1	no	—
68	C₄⋊C₈	—	71	2	no	—
69	C₄⋊C₈	—	142	1	no	—
70	#103	—	142	2	no	—
71	#97	—	142	1	no	—
72	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
73	C₂×Q₈	—	2	2	no	—
74	C₂×Q₈	—	71	1	no	—
75	Q₁₆	—	4	2	no	—
76	Q₁₆	—	4	2	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	—	142	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	142	1	no	—
92	C₂×C₄	—	142	1	no	—
93	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
94	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
95	C₂×C₄	#11765864	1	1	yes	characteristic
96	C₈	—	2	1	no	—
97	C₈	—	284	2	no	—
98	C₈	—	284	1	no	—
99	C₈	#11765853	1	2	yes	—
100	C₂²	#14556099	1	1	yes	characteristic
101	C₄	—	2	1	no	—
102	C₄	—	8	2	no	—
103	C₄	—	284	1	no	—
104	C₄	—	284	1	no	—
105	C₄	C₂×#11765853	1	1	yes	characteristic
106	C₄	#14556154	1	1	yes	characteristic
107	C₄	#14556160	1	2	yes	—
108	C₂	#17986196	1	1	yes	characteristic
109	C₂	#17986254	1	1	yes	characteristic
110	C₂	#17986271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986281

C₂

yes

maximal

#17986129

C₂

yes

maximal, characteristic

#17985750

C₂

yes

maximal, characteristic

#17985745

C₂

yes

maximal

#494×C₇₁

C₂

yes

maximal, characteristic

C₄×Dic₁₄₂

C₂²

yes

characteristic

#14556059

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

Dic₇₁⋊C₈

—

#14556051

—

#14556038

—

C₂×Q₁₆×C₇₁

—

C₂×Q₁₆×C₇₁

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

D₄

yes

characteristic

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₄

yes

characteristic

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×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

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#494

D₇₁

yes

—

—

—

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₇₁

yes

—

C₄×Q₈

—

#106

—

142

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

#103

—

142

—

#97

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

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

—

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

—

C₈

#11765853

yes

—

100

C₂²

#14556099

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

—

284

—

104

C₄

—

284

—

105

C₄

C₂×#11765853

yes

characteristic

106

C₄

#14556154

yes

characteristic

107

C₄

#14556160

yes

—

108

C₂

#17986196

yes

characteristic

109

C₂

#17986254

yes

characteristic

110

C₂

#17986271

yes

characteristic

	subgroup	G/H
center	C₂²	#14556099
commutator	C₄²×C₇₁	C₂³
Frattini	C₄²	C₂²×D₇₁
Fitting	#494×C₇₁	C₂
socle	C₂²×C₇₁	#121
hypercenter	#494	D₇₁
hypocenter	C₇₁	#1690
LCS	G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#14556099

commutator

C₄²×C₇₁

C₂³

Frattini

C₄²

C₂²×D₇₁

Fitting

#494×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₄²×C₇₁ ▷ C₂²×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
56. C₇₁	#1690	semidirect

H ≅ G/N

type

56. C₇₁

#1690

semidirect

Subgroups — #22587831

Conjugacy classes of subgroups

Complements