Group #22590417 — subgroups

Subgroups — #22590417

Order 9088.

170 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 170 conjugacy classes of subgroups — too many to draw. Only its 47 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	#17986437	C₂	1	2	yes	maximal, complemented
2	#17986376	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986117	C₂	1	1	yes	maximal, characteristic, complemented
4	#17985813	C₂	1	2	yes	maximal, complemented
5	#364×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	#14556150	C₂²	1	1	yes	characteristic
7	C₂×D₅₆₈	—	2	1	no	—
8	C₂×D₅₆₈	C₂²	1	2	yes	—
9	#14556134	—	2	1	no	—
10	#14556125	—	2	1	no	—
11	#14556073	—	2	2	no	—
12	D₁₁₃₆	—	2	2	no	—
13	#14556064	C₂²	1	1	yes	characteristic
14	#14556056	—	2	1	no	—
15	M₃₂×C₇₁	C₂²	1	2	yes	—
16	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
17	#11765865	—	2	1	no	—
18	C₂×D₂₈₄	—	2	2	no	—
19	C₂×D₂₈₄	—	4	2	no	—
20	Dic₂₈₄	D₄	1	1	yes	characteristic, complemented
21	#11765854	—	2	1	no	—
22	D₅₆₈	—	2	2	no	—
23	D₅₆₈	—	2	2	no	—
24	D₅₆₈	—	4	1	no	—
25	D₅₆₈	—	4	1	no	—
26	D₅₆₈	D₄	1	1	yes	characteristic, complemented
27	#11765849	—	2	1	no	—
28	#11765848	—	4	1	no	—
29	C₈×D₇₁	—	4	1	no	—
30	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₄²×C₇₁	D₄	1	1	yes	characteristic
33	C₁₆×C₇₁	—	2	2	no	—
34	C₂²×D₇₁	—	4	2	no	—
35	C₂×C₄×C₇₁	—	2	1	no	—
36	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
37	C₂²⋊D₇₁	—	4	1	no	—
38	D₂₈₄	—	2	1	no	—
39	D₂₈₄	—	4	2	no	—
40	D₂₈₄	—	4	2	no	—
41	D₂₈₄	—	8	4	no	—
42	C₄×D₇₁	—	4	1	no	—
43	Dic₁₄₂	—	2	1	no	—
44	C₈×C₇₁	D₈	1	1	yes	characteristic
45	C₈×C₇₁	D₈	1	1	yes	characteristic
46	C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
47	C₈×C₇₁	#41	1	1	yes	characteristic
48	C₇₁⋊C₈	—	4	1	no	—
49	C₂²×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	C₂×D₇₁	—	4	1	no	—
51	C₂×D₇₁	—	8	4	no	—
52	C₂×D₇₁	—	8	2	no	—
53	C₄×C₇₁	—	2	2	no	—
54	C₄×C₇₁	#121	1	1	yes	characteristic
55	C₄×C₇₁	C₂×D₈	1	1	yes	characteristic
56	Dic₇₁	—	4	1	no	—
57	C₂×C₇₁	—	2	1	no	—
58	C₂×C₇₁	#459	1	1	yes	characteristic
59	D₇₁	—	8	1	no	—
60	D₇₁	—	16	2	no	—
61	#2213	—	71	1	no	maximal, Sylow
62	C₇₁	#2213	1	1	yes	characteristic, complemented, Sylow
63	#509	—	71	2	no	—
64	#493	—	71	1	no	—
65	#443	—	71	1	no	—
66	#364	D₇₁	1	1	yes	characteristic, complemented
67	#361	—	71	2	no	—
68	#135	—	71	1	no	—
69	C₂×D₈	—	71	2	no	—
70	C₂×D₈	—	142	1	no	—
71	#131	—	142	1	no	—
72	#127	—	142	1	no	—
73	SD₃₂	—	142	2	no	—
74	D₁₆	—	142	2	no	—
75	M₃₂	C₂×D₇₁	1	2	yes	—
76	#108	—	71	1	no	—
77	#104	—	142	1	no	—
78	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
79	#41	—	142	1	no	—
80	C₂×D₄	—	284	2	no	—
81	C₂×D₄	—	142	2	no	—
82	Q₁₆	—	71	1	no	—
83	SD₁₆	—	142	1	no	—
84	D₈	—	284	1	no	—
85	D₈	—	284	1	no	—
86	D₈	—	71	1	no	—
87	D₈	—	142	2	no	—
88	D₈	—	142	2	no	—
89	M₁₆	—	284	1	no	—
90	M₁₆	—	142	1	no	—
91	C₂×C₈	—	284	1	no	—
92	C₂×C₈	D₂₈₄	1	1	yes	characteristic
93	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
94	C₄²	D₂₈₄	1	1	yes	characteristic
95	C₁₆	—	2	2	no	—
96	C₂³	—	284	2	no	—
97	Q₈	—	142	1	no	—
98	D₄	—	284	1	no	—
99	D₄	—	284	2	no	—
100	D₄	—	284	2	no	—
101	D₄	—	568	4	no	—
102	D₄	—	142	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	284	1	no	—
105	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
106	C₈	—	284	1	no	—
107	C₈	D₅₆₈	1	1	yes	characteristic
108	C₈	D₅₆₈	1	1	yes	characteristic
109	C₈	D₄×D₇₁	1	1	yes	characteristic
110	C₈	#11765866	1	1	yes	characteristic
111	C₂²	—	284	1	no	—
112	C₂²	—	568	4	no	—
113	C₂²	—	568	2	no	—
114	C₂²	#14556156	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	284	1	no	—
117	C₄	#14556103	1	1	yes	characteristic
118	C₄	C₂×D₅₆₈	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	568	1	no	—
121	C₂	—	1136	2	no	—
122	C₂	#17986195	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986437

C₂

yes

maximal, complemented

#17986376

C₂

yes

maximal, characteristic, complemented

#17986117

C₂

yes

maximal, characteristic, complemented

#17985813

C₂

yes

maximal, complemented

#364×C₇₁

C₂

yes

maximal, characteristic, complemented

#14556150

C₂²

yes

characteristic

C₂×D₅₆₈

—

C₂×D₅₆₈

C₂²

yes

—

#14556134

—

#14556125

—

#14556073

—

D₁₁₃₆

—

#14556064

C₂²

yes

characteristic

#14556056

—

M₃₂×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

#11765865

—

C₂×D₂₈₄

—

C₂×D₂₈₄

—

Dic₂₈₄

D₄

yes

characteristic, complemented

#11765854

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

—

D₅₆₈

D₄

yes

characteristic, complemented

#11765849

—

#11765848

—

C₈×D₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²⋊D₇₁

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

D₂₈₄

—

C₄×D₇₁

—

Dic₁₄₂

—

C₈×C₇₁

D₈

yes

characteristic

C₈×C₇₁

D₈

yes

characteristic

C₈×C₇₁

C₂×D₄

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

#121

yes

characteristic

C₄×C₇₁

C₂×D₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#459

yes

characteristic

D₇₁

—

D₇₁

—

#2213

—

maximal, Sylow

C₇₁

#2213

yes

characteristic, complemented, Sylow

—

—

—

D₇₁

yes

characteristic, complemented

#361

—

#135

—

C₂×D₈

—

C₂×D₈

—

142

—

#131

—

142

—

#127

—

142

—

SD₃₂

—

142

—

D₁₆

—

142

—

M₃₂

C₂×D₇₁

yes

—

#108

—

#104

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

#41

—

142

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

Q₁₆

—

SD₁₆

—

142

—

D₈

—

284

—

D₈

—

284

—

D₈

—

D₈

—

142

—

D₈

—

142

—

M₁₆

—

284

—

M₁₆

—

142

—

C₂×C₈

—

284

—

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄²

D₂₈₄

yes

characteristic

C₁₆

—

C₂³

—

284

—

Q₈

—

142

—

D₄

—

284

—

D₄

—

284

—

100

D₄

—

284

—

101

D₄

—

568

—

102

D₄

—

142

—

103

C₂×C₄

—

104

C₂×C₄

—

284

—

105

C₂×C₄

C₂×D₂₈₄

yes

characteristic

106

C₈

—

284

—

107

C₈

D₅₆₈

yes

characteristic

108

C₈

D₅₆₈

yes

characteristic

109

C₈

D₄×D₇₁

yes

characteristic

110

C₈

#11765866

yes

characteristic

111

C₂²

—

284

—

112

C₂²

—

568

—

113

C₂²

—

568

—

114

C₂²

#14556156

yes

characteristic

115

C₄

—

116

C₄

—

284

—

117

C₄

#14556103

yes

characteristic

118

C₄

C₂×D₅₆₈

yes

characteristic

119

C₂

—

120

C₂

—

568

—

121

C₂

—

1136

—

122

C₂

#17986195

yes

characteristic

	subgroup	G/H
center	C₂	#17986195
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#364×C₇₁	C₂
socle	C₂×C₇₁	#459
hypercenter	#364	D₇₁
hypocenter	C₇₁	#2213
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

subgroup

G/H

center

C₂

#17986195

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#364×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #364

N	H ≅ G/N	type
1. #17986437	C₂	semidirect
2. #17986376	C₂	semidirect
3. #17986117	C₂	semidirect
4. #17985813	C₂	semidirect
5. #364×C₇₁	C₂	semidirect
20. Dic₂₈₄	D₄	semidirect
26. D₅₆₈	D₄	semidirect
62. C₇₁	#2213	semidirect
66. #364	D₇₁	semidirect

H ≅ G/N

type

1. #17986437

C₂

semidirect

2. #17986376

C₂

semidirect

3. #17986117

C₂

semidirect

4. #17985813

C₂

semidirect

5. #364×C₇₁

C₂

semidirect

20. Dic₂₈₄

D₄

semidirect

26. D₅₆₈

D₄

semidirect

62. C₇₁

#2213

semidirect

66. #364

D₇₁

semidirect

Subgroups — #22590417

Conjugacy classes of subgroups

Complements