Group #22590554 — subgroups

Subgroups — #22590554

Order 9088.

134 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 45 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	#17986252	C₂	1	1	yes	maximal, characteristic
2	#17985823	C₂	1	2	yes	maximal, complemented
3	#17985811	C₂	1	2	yes	maximal, complemented
4	#17985781	C₂	1	1	yes	maximal, characteristic, complemented
5	#430×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
7	#14556102	—	2	1	no	—
8	#14556068	C₂²	1	2	yes	—
9	#14556060	C₂²	1	1	yes	characteristic
10	#14556050	—	2	1	no	—
11	C₂×M₁₆×C₇₁	—	2	1	no	—
12	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	#108×C₇₁	—	2	1	no	—
14	#108×C₇₁	C₂²	1	2	yes	—
15	C₂×D₂₈₄	—	2	1	no	—
16	C₂×C₂²⋊D₇₁	—	4	1	no	—
17	D₅₆₈	—	2	2	no	—
18	C₄⋊Dic₇₁	—	2	1	no	—
19	(C₂×D₇₁)⋊C₄	—	4	1	no	—
20	C₇₁⋊C₁₆	—	2	2	no	—
21	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
22	M₁₆×C₇₁	—	2	2	no	—
23	M₁₆×C₇₁	—	2	2	no	—
24	M₁₆×C₇₁	—	4	1	no	—
25	C₂×C₈×C₇₁	—	4	1	no	—
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
28	C₂³×C₇₁	SD₁₆	1	1	yes	characteristic
29	C₂²×D₇₁	—	4	1	no	—
30	C₂×C₄×C₇₁	—	2	1	no	—
31	C₂×C₄×C₇₁	—	2	1	no	—
32	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂²⋊D₇₁	—	8	2	no	—
35	C₂×Dic₇₁	—	4	1	no	—
36	D₂₈₄	—	4	1	no	—
37	D₂₈₄	—	4	1	no	—
38	C₈×C₇₁	—	2	1	no	—
39	C₈×C₇₁	—	4	2	no	—
40	C₈×C₇₁	#41	1	2	yes	—
41	C₂²×C₇₁	—	2	1	no	—
42	C₂²×C₇₁	—	4	1	no	—
43	C₂²×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
44	C₂×D₇₁	—	8	1	no	—
45	C₂×D₇₁	—	8	1	no	—
46	C₂×D₇₁	—	8	1	no	—
47	C₄×C₇₁	—	2	1	no	—
48	C₄×C₇₁	#123	1	1	yes	characteristic
49	C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₇₁	—	8	1	no	—
51	C₂×C₇₁	—	2	1	no	—
52	C₂×C₇₁	—	4	1	no	—
53	C₂×C₇₁	#481	1	1	yes	characteristic
54	D₇₁	—	16	1	no	—
55	#2238	—	71	1	no	maximal, Sylow
56	C₇₁	#2238	1	1	yes	characteristic, complemented, Sylow
57	#469	—	71	1	no	—
58	#430	D₇₁	1	1	yes	characteristic, complemented
59	#365	—	71	2	no	—
60	#361	—	71	2	no	—
61	#349	—	71	1	no	—
62	C₂×D₈	—	71	1	no	—
63	C₂×M₁₆	—	2	1	no	—
64	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
65	#121	—	142	1	no	—
66	M₃₂	—	71	2	no	—
67	#108	—	2	1	no	—
68	#108	C₂×D₇₁	1	2	yes	—
69	#106	—	71	1	no	—
70	#102	—	142	1	no	—
71	C₂×D₄	—	284	1	no	—
72	C₂×D₄	—	142	1	no	—
73	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
74	D₈	—	142	2	no	—
75	M₁₆	—	2	2	no	—
76	M₁₆	—	2	2	no	—
77	M₁₆	—	4	1	no	—
78	C₂×C₈	—	4	1	no	—
79	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
80	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
81	C₄⋊C₄	—	142	1	no	—
82	C₂²⋊C₄	—	284	1	no	—
83	C₁₆	—	142	2	no	—
84	C₂³	—	284	1	no	—
85	C₂³	#11765852	1	1	yes	characteristic
86	D₄	—	284	1	no	—
87	D₄	—	284	1	no	—
88	D₄	—	568	2	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	284	1	no	—
92	C₂×C₄	#11765852	1	1	yes	characteristic
93	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
94	C₈	—	2	1	no	—
95	C₈	—	4	2	no	—
96	C₈	#11765865	1	2	yes	—
97	C₂²	—	2	1	no	—
98	C₂²	—	4	1	no	—
99	C₂²	—	568	1	no	—
100	C₂²	—	568	1	no	—
101	C₂²	—	568	1	no	—
102	C₂²	C₂×#11765852	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	568	1	no	—
105	C₄	#14556111	1	1	yes	characteristic
106	C₄	#14556154	1	1	yes	characteristic
107	C₂	—	2	1	no	—
108	C₂	—	4	1	no	—
109	C₂	—	1136	1	no	—
110	C₂	#17986317	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986252

C₂

yes

maximal, characteristic

#17985823

C₂

yes

maximal, complemented

#17985811

C₂

yes

maximal, complemented

#17985781

C₂

yes

maximal, characteristic, complemented

#430×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×D₅₆₈

C₂²

yes

characteristic

#14556102

—

#14556068

C₂²

yes

—

#14556060

C₂²

yes

characteristic

#14556050

—

C₂×M₁₆×C₇₁

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#108×C₇₁

—

#108×C₇₁

C₂²

yes

—

C₂×D₂₈₄

—

C₂×C₂²⋊D₇₁

—

D₅₆₈

—

C₄⋊Dic₇₁

—

(C₂×D₇₁)⋊C₄

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₂³×C₇₁

SD₁₆

yes

characteristic

C₂²×D₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂²⋊D₇₁

—

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

#41

yes

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×SD₁₆

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#481

yes

characteristic

D₇₁

—

#2238

—

maximal, Sylow

C₇₁

#2238

yes

characteristic, complemented, Sylow

#469

—

#430

D₇₁

yes

characteristic, complemented

#365

—

#361

—

#349

—

C₂×D₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

#121

—

142

—

M₃₂

—

#108

—

#108

C₂×D₇₁

yes

—

#106

—

#102

—

142

—

C₂×D₄

—

284

—

C₂×D₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

D₈

—

142

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

284

—

C₁₆

—

142

—

C₂³

—

284

—

C₂³

#11765852

yes

characteristic

D₄

—

284

—

D₄

—

284

—

D₄

—

568

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

#11765852

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₈

—

C₈

—

C₈

#11765865

yes

—

C₂²

—

C₂²

—

C₂²

—

568

—

100

C₂²

—

568

—

101

C₂²

—

568

—

102

C₂²

C₂×#11765852

yes

characteristic

103

C₄

—

104

C₄

—

568

—

105

C₄

#14556111

yes

characteristic

106

C₄

#14556154

yes

characteristic

107

C₂

—

108

C₂

—

109

C₂

—

1136

—

110

C₂

#17986317

yes

characteristic

	subgroup	G/H
center	C₂	#17986317
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#430×C₇₁	C₂
socle	C₂×C₇₁	#481
hypercenter	#430	D₇₁
hypocenter	C₇₁	#2238
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #430

subgroup

G/H

center

C₂

#17986317

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#430×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #430

N	H ≅ G/N	type
2. #17985823	C₂	semidirect
3. #17985811	C₂	semidirect
4. #17985781	C₂	semidirect
5. #430×C₇₁	C₂	semidirect
56. C₇₁	#2238	semidirect
58. #430	D₇₁	semidirect

H ≅ G/N

type

2. #17985823

C₂

semidirect

3. #17985811

C₂

semidirect

4. #17985781

C₂

semidirect

5. #430×C₇₁

C₂

semidirect

56. C₇₁

#2238

semidirect

58. #430

D₇₁

semidirect

Subgroups — #22590554

Conjugacy classes of subgroups

Complements