Group #22587850 — subgroups

Subgroups — #22587850

Order 9088.

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

normal subgroups only

This group has 166 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	#17986201	C₂	1	2	yes	maximal
2	#17986078	C₂	1	1	yes	maximal, characteristic
3	#17985750	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985745	C₂	1	2	yes	maximal, complemented
5	#500×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11765852	—	2	2	no	—
7	C₂×#11765848	—	2	1	no	—
8	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
9	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
10	C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
11	#14556052	—	2	2	no	—
12	(C₂×D₇₁)⋊C₈	—	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×D₂₈₄	D₄	1	1	yes	characteristic
18	C₂×C₄×D₇₁	—	2	1	no	—
19	#11765852	—	4	4	no	—
20	#11765848	—	4	2	no	—
21	C₂×C₇₁⋊C₈	—	2	1	no	—
22	C₂×C₇₁⋊C₈	—	2	2	no	—
23	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
24	(C₂×D₇₁)⋊C₄	—	2	1	no	—
25	C₂×Q₈×C₇₁	—	2	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	C₂²×D₇₁	—	2	1	no	—
32	Q₈×C₇₁	—	4	2	no	—
33	Q₈×C₇₁	—	4	2	no	—
34	C₂×C₄×C₇₁	—	4	2	no	—
35	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
38	C₂×Dic₇₁	—	2	1	no	—
39	D₂₈₄	—	2	2	no	—
40	C₄×D₇₁	—	4	1	no	—
41	C₄×D₇₁	—	4	1	no	—
42	C₈×C₇₁	—	2	1	no	—
43	C₈×C₇₁	SD₁₆	1	2	yes	complemented
44	C₇₁⋊C₈	—	4	1	no	—
45	C₇₁⋊C₈	—	4	2	no	—
46	C₂²×C₇₁	#121	1	1	yes	characteristic
47	C₂×D₇₁	—	4	1	no	—
48	C₂×D₇₁	—	4	1	no	—
49	C₂×D₇₁	—	4	1	no	—
50	C₄×C₇₁	—	2	1	no	—
51	C₄×C₇₁	—	8	2	no	—
52	C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
53	C₄×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
54	C₄×C₇₁	#137	1	1	yes	characteristic
55	Dic₇₁	—	4	1	no	—
56	C₂×C₇₁	#461	1	1	yes	characteristic
57	C₂×C₇₁	#469	1	1	yes	characteristic
58	C₂×C₇₁	#473	1	1	yes	characteristic
59	D₇₁	—	8	1	no	—
60	#1693	—	71	1	no	maximal, Sylow
61	C₇₁	#1693	1	1	yes	characteristic, complemented, Sylow
62	#500	D₇₁	1	1	yes	characteristic, complemented
63	#461	—	71	2	no	—
64	#436	—	71	1	no	—
65	#334	—	71	1	no	—
66	#332	—	71	2	no	—
67	C₂×SD₁₆	—	142	2	no	—
68	C₂×M₁₆	—	142	1	no	—
69	C₄⋊Q₈	C₂×D₇₁	1	2	yes	—
70	C₄×D₄	—	71	1	no	—
71	#107	—	2	1	no	—
72	#107	—	2	1	no	—
73	C₄⋊C₈	—	71	1	no	—
74	C₄⋊C₈	—	71	2	no	—
75	#103	—	142	2	no	—
76	C₂²⋊C₈	—	142	1	no	—
77	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
78	C₂×Q₈	—	2	2	no	—
79	C₂×D₄	—	71	1	no	—
80	C₂²×C₄	—	142	1	no	—
81	SD₁₆	—	284	4	no	—
82	M₁₆	—	284	2	no	—
83	C₂×C₈	—	142	2	no	—
84	C₂×C₈	—	142	1	no	—
85	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	71	1	no	—
90	C₂²⋊C₄	—	142	1	no	—
91	C₄²	C₂²×D₇₁	1	1	yes	characteristic
92	C₂³	—	142	1	no	—
93	Q₈	—	4	2	no	—
94	Q₈	—	4	2	no	—
95	D₄	—	142	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	284	1	no	—
98	C₂×C₄	—	284	1	no	—
99	C₂×C₄	—	142	1	no	—
100	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
102	C₂×C₄	#11765864	1	1	yes	characteristic
103	C₈	—	2	1	no	—
104	C₈	—	284	1	no	—
105	C₈	—	284	2	no	—
106	C₈	#11765852	1	2	yes	complemented
107	C₂²	—	284	1	no	—
108	C₂²	—	284	1	no	—
109	C₂²	—	284	1	no	—
110	C₂²	#14556099	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	8	2	no	—
113	C₄	—	284	1	no	—
114	C₄	C₂×#11765852	1	1	yes	characteristic
115	C₄	#14556152	1	2	yes	—
116	C₄	#14556159	1	1	yes	characteristic
117	C₂	—	568	1	no	—
118	C₂	#17986201	1	1	yes	characteristic
119	C₂	#17986249	1	1	yes	characteristic
120	C₂	#17986271	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986201

C₂

yes

maximal

#17986078

C₂

yes

maximal, characteristic

#17985750

C₂

yes

maximal, characteristic, complemented

#17985745

C₂

yes

maximal, complemented

#500×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×#11765852

—

C₂×#11765848

—

C₄×D₂₈₄

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₂²

yes

—

#14556052

—

(C₂×D₇₁)⋊C₈

—

C₄⋊Q₈×C₇₁

C₂²

yes

—

#107×C₇₁

—

#107×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×D₂₈₄

D₄

yes

characteristic

C₂×C₄×D₇₁

—

#11765852

—

#11765848

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₄⋊Dic₇₁

D₄

yes

characteristic

(C₂×D₇₁)⋊C₄

—

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

C₂²×D₇₁

—

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

—

D₂₈₄

—

C₄×D₇₁

—

C₄×D₇₁

—

C₈×C₇₁

—

C₈×C₇₁

SD₁₆

yes

complemented

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

#121

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂×SD₁₆

yes

characteristic

C₄×C₇₁

AGL(1,ℤ/8)

yes

—

C₄×C₇₁

#137

yes

characteristic

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1693

—

maximal, Sylow

C₇₁

#1693

yes

characteristic, complemented, Sylow

#500

D₇₁

yes

characteristic, complemented

—

—

—

—

C₂×SD₁₆

—

142

—

C₂×M₁₆

—

142

—

C₄⋊Q₈

C₂×D₇₁

yes

—

C₄×D₄

—

#107

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

142

—

C₂²⋊C₈

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

142

—

SD₁₆

—

284

—

M₁₆

—

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

—

C₂²⋊C₄

—

142

—

C₄²

C₂²×D₇₁

yes

characteristic

C₂³

—

142

—

Q₈

—

Q₈

—

D₄

—

142

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

142

—

100

C₂×C₄

D₄×D₇₁

yes

characteristic

101

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

102

C₂×C₄

#11765864

yes

characteristic

103

C₈

—

104

C₈

—

284

—

105

C₈

—

284

—

106

C₈

#11765852

yes

complemented

107

C₂²

—

284

—

108

C₂²

—

284

—

109

C₂²

—

284

—

110

C₂²

#14556099

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

284

—

114

C₄

C₂×#11765852

yes

characteristic

115

C₄

#14556152

yes

—

116

C₄

#14556159

yes

characteristic

117

C₂

—

568

—

118

C₂

#17986201

yes

characteristic

119

C₂

#17986249

yes

characteristic

120

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	#1693
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
3. #17985750	C₂	semidirect
4. #17985745	C₂	semidirect
5. #500×C₇₁	C₂	semidirect
43. C₈×C₇₁	SD₁₆	semidirect
61. C₇₁	#1693	semidirect
62. #500	D₇₁	semidirect
106. C₈	#11765852	semidirect

H ≅ G/N

type

3. #17985750

C₂

semidirect

4. #17985745

C₂

semidirect

5. #500×C₇₁

C₂

semidirect

43. C₈×C₇₁

SD₁₆

semidirect

61. C₇₁

#1693

semidirect

62. #500

D₇₁

semidirect

106. C₈

#11765852

semidirect

Subgroups — #22587850

Conjugacy classes of subgroups

Complements