Group #22587110 — subgroups

Subgroups — #22587110

Order 9088.

168 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 168 conjugacy classes of subgroups — too many to draw. Only its 89 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	#17986052	C₂	1	1	yes	maximal, characteristic
2	#17986023	C₂	1	1	yes	maximal, characteristic
3	#17985749	C₂	1	4	yes	maximal, complemented
4	C₂×C₄⋊C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	—	2	1	no	—
6	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₄⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
8	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
9	C₂²⋊(C₇₁⋊C₈)	C₄	1	2	yes	—
10	C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
11	C₂²×C₈×C₇₁	—	2	1	no	—
12	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₇₁	—	2	1	no	—
14	C₄⋊C₈×C₇₁	C₂²	1	2	yes	—
15	#11765849	—	4	2	no	—
16	C₂×C₇₁⋊C₈	—	2	2	no	—
17	C₂×C₇₁⋊C₈	C₂×C₄	1	4	yes	—
18	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
20	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
21	C₂×C₈×C₇₁	—	2	2	no	—
22	C₂×C₈×C₇₁	—	4	2	no	—
23	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₇₁	D₄	1	1	yes	characteristic
25	C₄²×C₇₁	Q₈	1	1	yes	characteristic
26	C₄²×C₇₁	C₂³	1	1	yes	characteristic
27	C₂³×C₇₁	C₄⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₁	—	2	2	no	—
29	C₂×C₄×C₇₁	—	4	1	no	—
30	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
34	C₂×C₄×C₇₁	SD₁₆	1	2	yes	—
35	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
38	C₈×C₇₁	—	4	2	no	—
39	C₇₁⋊C₈	—	2	4	no	—
40	C₇₁⋊C₈	—	4	2	no	—
41	C₂²×C₇₁	—	2	1	no	—
42	C₂²×C₇₁	—	2	1	no	—
43	C₂²×C₇₁	#106	1	2	yes	—
44	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	—	2	1	no	—
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	#106	1	2	yes	—
49	C₄×C₇₁	#131	1	2	yes	—
50	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
51	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
52	C₂×C₇₁	—	2	2	no	—
53	C₂×C₇₁	#424	1	1	yes	characteristic
54	C₂×C₇₁	C₂×#106	1	1	yes	characteristic
55	C₂×C₇₁	#430	1	1	yes	characteristic
56	#1570	—	71	1	no	maximal, Sylow
57	C₇₁	#1570	1	1	yes	characteristic, complemented, Sylow
58	#431	—	71	1	no	—
59	#423	—	71	1	no	—
60	C₂×C₄⋊C₈	D₇₁	1	1	yes	characteristic
61	#334	—	71	4	no	—
62	C₂×M₁₆	—	142	1	no	—
63	C₂²×C₈	—	2	1	no	—
64	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
65	C₄⋊C₈	—	2	1	no	—
66	C₄⋊C₈	—	71	2	no	—
67	C₄⋊C₈	—	71	2	no	—
68	C₄⋊C₈	—	142	1	no	—
69	C₄⋊C₈	C₂×D₇₁	1	2	yes	—
70	C₂²⋊C₈	—	71	2	no	—
71	C₄×C₈	—	71	2	no	—
72	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
73	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
74	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
75	M₁₆	—	284	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	4	2	no	—
78	C₂×C₈	—	71	4	no	—
79	C₂×C₈	—	142	2	no	—
80	C₄²	Dic₁₄₂	1	1	yes	characteristic
81	C₄²	C₄×D₇₁	1	1	yes	characteristic
82	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
83	C₄²	C₂²×D₇₁	1	1	yes	characteristic
84	C₂³	Dic₇₁⋊C₄	1	1	yes	characteristic
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	4	1	no	—
87	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
88	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
89	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
90	C₂×C₄	#11765852	1	2	yes	—
91	C₂×C₄	#11765853	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
95	C₈	—	4	2	no	—
96	C₈	—	284	2	no	—
97	C₈	—	142	4	no	—
98	C₂²	—	2	1	no	—
99	C₂²	—	2	1	no	—
100	C₂²	#14556059	1	2	yes	—
101	C₂²	C₂×Dic₇₁⋊C₄	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	2	1	no	—
104	C₄	—	2	1	no	—
105	C₄	#14556059	1	2	yes	—
106	C₄	#14556134	1	2	yes	—
107	C₄	C₂×#11765852	1	1	yes	characteristic
108	C₄	C₂×#11765853	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	#17986025	1	1	yes	characteristic
111	C₂	C₂×#14556059	1	1	yes	characteristic
112	C₂	#17986047	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986052

C₂

yes

maximal, characteristic

#17986023

C₂

yes

maximal, characteristic

#17985749

C₂

yes

maximal, complemented

C₂×C₄⋊C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

C₄⋊(C₇₁⋊C₈)

—

C₄⋊(C₇₁⋊C₈)

C₄

yes

—

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

C₂²⋊(C₇₁⋊C₈)

C₄

yes

—

C₄×C₇₁⋊C₈

C₂²

yes

—

C₂²×C₈×C₇₁

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

—

C₄⋊C₈×C₇₁

C₂²

yes

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

Q₈

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

Q₈

yes

characteristic

C₄²×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

SD₁₆

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#106

yes

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#106

yes

—

C₄×C₇₁

#131

yes

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#431

—

#423

—

C₂×C₄⋊C₈

D₇₁

yes

characteristic

#334

—

C₂×M₁₆

—

142

—

C₂²×C₈

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

142

—

C₄⋊C₈

C₂×D₇₁

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

M₁₆

—

284

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₄²

Dic₁₄₂

yes

characteristic

C₄²

C₄×D₇₁

yes

characteristic

C₄²

C₂²⋊D₇₁

yes

characteristic

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

#11765852

yes

—

C₂×C₄

#11765853

yes

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₈

—

C₈

—

284

—

C₈

—

142

—

C₂²

—

C₂²

—

100

C₂²

#14556059

yes

—

101

C₂²

C₂×Dic₇₁⋊C₄

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

#14556059

yes

—

106

C₄

#14556134

yes

—

107

C₄

C₂×#11765852

yes

characteristic

108

C₄

C₂×#11765853

yes

characteristic

109

C₂

—

110

C₂

#17986025

yes

characteristic

111

C₂

C₂×#14556059

yes

characteristic

112

C₂

#17986047

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₄²	C₂²×D₇₁
Fitting	C₂×C₄⋊C₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	C₂×C₄⋊C₈	D₇₁
hypocenter	C₇₁	#1570
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₄²

C₂²×D₇₁

Fitting

C₂×C₄⋊C₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

hypercenter

C₂×C₄⋊C₈

D₇₁

hypocenter

C₇₁

#1570

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
3. #17985749	C₂	semidirect
57. C₇₁	#1570	semidirect

H ≅ G/N

type

3. #17985749

C₂

semidirect

57. C₇₁

#1570

semidirect

Subgroups — #22587110

Conjugacy classes of subgroups

Complements