Group #22588474 — subgroups

Subgroups — #22588474

Order 9088.

224 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 109 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	#17985869	C₂	1	1	yes	maximal, characteristic
2	#17985865	C₂	1	1	yes	maximal, characteristic
3	#17985762	C₂	1	4	yes	maximal
4	C₂×#97×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₇₁	—	2	2	no	—
6	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
8	C₇₁⋊#95	—	2	2	no	—
9	C₂²×C₈×C₇₁	C₂²	1	2	yes	—
10	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
11	#97×C₇₁	C₄	1	4	yes	complemented
12	C₂²×Dic₇₁	—	2	2	no	—
13	C₂²×Dic₇₁	—	2	2	no	—
14	C₄⋊Dic₇₁	—	2	4	no	—
15	C₄⋊Dic₇₁	—	2	4	no	—
16	C₄⋊Dic₇₁	—	4	4	no	—
17	C₂²×C₄×C₇₁	D₄	1	2	yes	—
18	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
19	C₂×C₈×C₇₁	—	2	2	no	—
20	C₂×C₈×C₇₁	—	2	2	no	—
21	C₂×C₈×C₇₁	C₂×C₄	1	4	yes	—
22	C₄²×C₇₁	C₂×C₄	1	2	yes	—
23	C₄²×C₇₁	Q₈	1	2	yes	—
24	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
25	C₂×C₄×C₇₁	—	2	2	no	—
26	C₂×C₄×C₇₁	—	2	2	no	—
27	C₂×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
28	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₇₁	#41	1	2	yes	—
31	C₂×C₄×C₇₁	#41	1	2	yes	—
32	C₂×Dic₇₁	—	4	2	no	—
33	C₂×Dic₇₁	—	4	2	no	—
34	C₂×Dic₇₁	—	4	4	no	—
35	C₂×Dic₇₁	—	4	2	no	—
36	C₂×Dic₇₁	—	4	2	no	—
37	C₈×C₇₁	—	2	4	no	—
38	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂²×C₇₁	#124	1	1	yes	characteristic
40	C₂²×C₇₁	#124	1	1	yes	characteristic
41	C₂²×C₇₁	AGL(1,ℤ/8)	1	2	yes	—
42	C₂²×C₇₁	#137	1	2	yes	—
43	C₄×C₇₁	—	2	4	no	—
44	C₄×C₇₁	#117	1	2	yes	—
45	C₄×C₇₁	#125	1	2	yes	—
46	Dic₇₁	—	8	2	no	—
47	Dic₇₁	—	8	2	no	—
48	C₂×C₇₁	#382	1	1	yes	characteristic
49	C₂×C₇₁	#428	1	2	yes	—
50	C₂×C₇₁	#489	1	2	yes	—
51	C₂×C₇₁	#490	1	1	yes	characteristic
52	C₂×C₇₁	#491	1	1	yes	characteristic
53	#1836	—	71	1	no	maximal, Sylow
54	C₇₁	#1836	1	1	yes	characteristic, complemented, Sylow
55	C₂×#97	D₇₁	1	1	yes	characteristic
56	#384	—	71	1	no	—
57	#382	—	71	1	no	—
58	#340	—	71	4	no	—
59	C₂²×C₈	C₂×D₇₁	1	2	yes	—
60	C₂×C₄⋊C₄	—	71	2	no	—
61	C₂×C₄⋊C₄	—	71	2	no	—
62	C₂×C₄⋊C₄	—	142	2	no	—
63	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
64	#97	Dic₇₁	1	4	yes	complemented
65	#95	—	142	2	no	—
66	C₂²×C₄	—	142	2	no	—
67	C₂²×C₄	—	142	2	no	—
68	C₂²×C₄	D₂₈₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
70	C₂×C₈	—	2	2	no	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×Dic₇₁	1	4	yes	—
73	C₄⋊C₄	—	284	4	no	—
74	C₄⋊C₄	—	142	4	no	—
75	C₄⋊C₄	—	142	4	no	—
76	C₄²	Dic₁₄₂	1	2	yes	—
77	C₄²	C₂×Dic₇₁	1	2	yes	—
78	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	284	2	no	—
82	C₂×C₄	—	284	2	no	—
83	C₂×C₄	—	284	2	no	—
84	C₂×C₄	—	284	4	no	—
85	C₂×C₄	—	284	2	no	—
86	C₂×C₄	C₄⋊Dic₇₁	1	4	yes	—
87	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
89	C₂×C₄	#11765865	1	2	yes	—
90	C₂×C₄	#11765865	1	2	yes	—
91	C₈	—	2	4	no	—
92	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
93	C₂²	#14556116	1	1	yes	characteristic
94	C₂²	#14556116	1	1	yes	characteristic
95	C₂²	#14556156	1	2	yes	—
96	C₂²	#14556161	1	2	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	568	2	no	—
99	C₄	—	568	2	no	—
100	C₄	#14556087	1	2	yes	—
101	C₄	#14556120	1	2	yes	—
102	C₂	#17985865	1	1	yes	characteristic
103	C₂	#17986041	1	2	yes	—
104	C₂	#17986359	1	2	yes	—
105	C₂	#17986364	1	1	yes	characteristic
106	C₂	#17986368	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985869

C₂

yes

maximal, characteristic

#17985865

C₂

yes

maximal, characteristic

#17985762

C₂

yes

maximal

C₂×#97×C₇₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₁

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

#97×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

Q₈

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

C₂²×C₇₁

yes

—

C₂²×C₇₁

#137

yes

—

C₄×C₇₁

—

C₄×C₇₁

#117

yes

—

C₄×C₇₁

#125

yes

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#382

yes

characteristic

C₂×C₇₁

#428

yes

—

C₂×C₇₁

#489

yes

—

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₂×#97

D₇₁

yes

—

—

—

C₂²×C₈

C₂×D₇₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

142

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#97

Dic₇₁

yes

complemented

#95

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

D₂₈₄

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

Dic₁₄₂

yes

—

C₄²

C₂×Dic₇₁

yes

—

C₂³

C₂×D₂₈₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

C₄⋊Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₂×C₄

#11765865

yes

—

C₂×C₄

#11765865

yes

—

C₈

—

C₂²

C₂×C₄⋊Dic₇₁

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

#14556161

yes

—

C₄

—

C₄

—

568

—

C₄

—

568

—

100

C₄

#14556087

yes

—

101

C₄

#14556120

yes

—

102

C₂

#17985865

yes

characteristic

103

C₂

#17986041

yes

—

104

C₂

#17986359

yes

—

105

C₂

#17986364

yes

characteristic

106

C₂

#17986368

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#97×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#97	D₇₁
hypocenter	C₇₁	#1836
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#97×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₂×#97

D₇₁

hypocenter

C₇₁

#1836

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
11. #97×C₇₁	C₄	semidirect
54. C₇₁	#1836	semidirect
64. #97	Dic₇₁	semidirect

H ≅ G/N

type

11. #97×C₇₁

C₄

semidirect

54. C₇₁

#1836

semidirect

64. #97

Dic₇₁

semidirect

Subgroups — #22588474

Conjugacy classes of subgroups

Complements