Group #22588476 — subgroups

Subgroups — #22588476

Order 9088.

188 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 81 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	#17986022	C₂	1	2	yes	maximal
2	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985764	C₂	1	1	yes	maximal, characteristic
4	#17985763	C₂	1	2	yes	maximal
5	#341×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	2	yes	—
7	C₂²×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
9	Dic₇₁⋊C₈	—	2	4	no	—
10	C₈×Dic₇₁	C₄	1	2	yes	—
11	C₈×Dic₇₁	C₄	1	2	yes	—
12	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
13	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₇₁	Q₈	1	2	yes	—
15	#11765849	—	2	2	no	—
16	#11765849	—	2	2	no	—
17	C₂×C₇₁⋊C₈	—	2	2	no	—
18	C₂×C₇₁⋊C₈	—	2	2	no	—
19	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
20	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
22	C₄×Dic₇₁	D₄	1	2	yes	—
23	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
24	M₁₆×C₇₁	—	2	2	no	—
25	M₁₆×C₇₁	—	2	2	no	—
26	C₂×C₈×C₇₁	—	2	2	no	—
27	C₂×C₈×C₇₁	—	2	2	no	—
28	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
29	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	#41	1	2	yes	—
33	C₂×C₄×C₇₁	#41	1	2	yes	—
34	C₂×Dic₇₁	—	2	4	no	—
35	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
36	C₂×Dic₇₁	C₄⋊C₄	1	2	yes	—
37	C₈×C₇₁	—	2	2	no	—
38	C₈×C₇₁	—	4	2	no	—
39	C₇₁⋊C₈	—	2	2	no	—
40	C₇₁⋊C₈	—	4	2	no	—
41	C₂²×C₇₁	—	2	2	no	—
42	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₇₁	#125	1	2	yes	—
44	C₄×C₇₁	#117	1	1	yes	characteristic
45	C₄×C₇₁	#117	1	1	yes	characteristic
46	C₄×C₇₁	#124	1	2	yes	—
47	Dic₇₁	—	2	2	no	—
48	Dic₇₁	#108	1	4	yes	—
49	C₂×C₇₁	—	2	2	no	—
50	C₂×C₇₁	#382	1	1	yes	characteristic
51	C₂×C₇₁	C₂×#108	1	2	yes	—
52	#1837	—	71	1	no	maximal, Sylow
53	C₇₁	#1837	1	1	yes	characteristic, complemented, Sylow
54	#423	—	71	2	no	—
55	C₂×C₄×C₈	—	71	1	no	—
56	#341	—	71	2	no	—
57	#341	—	71	1	no	—
58	#341	D₇₁	1	1	yes	characteristic
59	C₂×M₁₆	—	71	2	no	—
60	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
61	C₂²×C₈	—	71	1	no	—
62	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
63	C₂×C₄²	—	71	1	no	—
64	C₄⋊C₈	—	142	4	no	—
65	C₄×C₈	—	71	2	no	—
66	C₄×C₈	—	71	2	no	—
67	C₂²×C₄	—	71	2	no	—
68	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	142	2	no	—
72	M₁₆	—	142	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	71	2	no	—
76	C₂×C₈	—	142	2	no	—
77	C₂×C₈	—	142	2	no	—
78	C₂×C₈	C₄×D₇₁	1	2	yes	—
79	C₄²	—	71	1	no	—
80	C₄²	—	71	1	no	—
81	C₄²	—	71	2	no	—
82	C₂³	Q₈×D₇₁	1	1	yes	characteristic
83	C₂×C₄	—	71	2	no	—
84	C₂×C₄	—	71	2	no	—
85	C₂×C₄	—	142	4	no	—
86	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
87	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
88	C₂×C₄	#11765864	1	2	yes	—
89	C₂×C₄	#11765865	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	284	2	no	—
93	C₈	—	142	2	no	—
94	C₂²	—	2	2	no	—
95	C₂²	C₄⋊C₄×D₇₁	1	1	yes	characteristic
96	C₂²	#14556118	1	2	yes	—
97	C₄	—	71	4	no	—
98	C₄	—	142	2	no	—
99	C₄	#14556084	1	1	yes	characteristic
100	C₄	#14556085	1	1	yes	characteristic
101	C₄	#14556114	1	2	yes	—
102	C₂	—	2	2	no	—
103	C₂	#17985863	1	1	yes	characteristic
104	C₂	#108×D₇₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17986022

C₂

yes

maximal

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

#17985764

C₂

yes

maximal, characteristic

#17985763

C₂

yes

maximal

#341×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

C₂²×C₇₁⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

Dic₇₁⋊C₈

—

C₈×Dic₇₁

C₄

yes

—

C₈×Dic₇₁

C₄

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

Q₈

yes

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₂×Dic₇₁

C₄⋊C₄

yes

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#125

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

#108

yes

—

C₂×C₇₁

—

C₂×C₇₁

#382

yes

characteristic

C₂×C₇₁

C₂×#108

yes

—

#1837

—

maximal, Sylow

C₇₁

#1837

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

—

#341

—

#341

—

#341

D₇₁

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

142

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

C₄×D₇₁

yes

—

C₄²

—

C₄²

—

C₄²

—

C₂³

Q₈×D₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

D₄×D₇₁

yes

characteristic

C₂×C₄

#11765864

yes

—

C₂×C₄

#11765865

yes

—

C₈

—

C₈

—

C₈

—

284

—

C₈

—

142

—

C₂²

—

C₂²

C₄⋊C₄×D₇₁

yes

characteristic

C₂²

#14556118

yes

—

C₄

—

C₄

—

142

—

C₄

#14556084

yes

characteristic

100

C₄

#14556085

yes

characteristic

101

C₄

#14556114

yes

—

102

C₂

—

103

C₂

#17985863

yes

characteristic

104

C₂

#108×D₇₁

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#341×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	#341	D₇₁
hypocenter	C₇₁	#1837
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

subgroup

G/H

center

C₂×C₄

D₄×D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#341×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #341

N	H ≅ G/N	type
53. C₇₁	#1837	semidirect

H ≅ G/N

type

53. C₇₁

#1837

semidirect

Subgroups — #22588476

Conjugacy classes of subgroups

Complements