Group #22590624 — subgroups

Subgroups — #22590624

Order 9088.

120 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17986403	C₂	1	1	yes	maximal, characteristic
2	#17986400	C₂	1	1	yes	maximal, characteristic
3	#17985827	C₂	1	1	yes	maximal, characteristic
4	#17985826	C₂	1	1	yes	maximal, characteristic
5	#17985824	C₂	1	1	yes	maximal, characteristic
6	#17985774	C₂	1	1	yes	maximal, characteristic
7	#366×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556127	—	2	1	no	—
9	#14556119	—	2	1	no	—
10	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	#14556062	C₂²	1	1	yes	characteristic
12	#14556062	C₂²	1	1	yes	characteristic
13	#14556061	—	2	1	no	—
14	#14556059	—	2	1	no	—
15	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	#107×C₇₁	C₂²	1	1	yes	characteristic
18	#107×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₂	—	4	1	no	—
20	C₂×C₇₁⋊C₈	D₄	1	1	yes	characteristic
21	C₄⋊Dic₇₁	—	2	1	no	—
22	C₄⋊Dic₇₁	—	2	1	no	—
23	C₄⋊Dic₇₁	—	4	1	no	—
24	Dic₇₁⋊C₄	—	4	1	no	—
25	Dic₇₁⋊C₄	—	4	1	no	—
26	C₄×Dic₇₁	D₄	1	1	yes	characteristic
27	C₇₁⋊C₁₆	Q₈	1	2	yes	complemented
28	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₁	—	2	1	no	—
30	C₄⋊C₄×C₇₁	—	2	1	no	—
31	C₁₆×C₇₁	Q₈	1	2	yes	complemented
32	C₂×C₄×C₇₁	—	4	1	no	—
33	C₂×C₄×C₇₁	—	4	1	no	—
34	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₇₁	—	4	1	no	—
36	C₂×Dic₇₁	—	4	1	no	—
37	C₂×Dic₇₁	D₈	1	1	yes	characteristic
38	C₂×Dic₇₁	D₈	1	1	yes	characteristic
39	Dic₁₄₂	—	8	2	no	—
40	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
41	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
42	C₇₁⋊C₈	Q₁₆	1	2	yes	—
43	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
44	C₄×C₇₁	—	8	1	no	—
45	C₄×C₇₁	—	8	1	no	—
46	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
48	Dic₇₁	—	2	1	no	—
49	Dic₇₁	—	2	1	no	—
50	Dic₇₁	—	8	1	no	—
51	Dic₇₁	—	8	1	no	—
52	C₂×C₇₁	#500	1	1	yes	characteristic
53	C₂×C₇₁	#509	1	1	yes	characteristic
54	C₂×C₇₁	#510	1	1	yes	characteristic
55	#2255	—	71	1	no	maximal, Sylow
56	C₇₁	#2255	1	1	yes	characteristic, complemented, Sylow
57	#500	—	71	1	no	—
58	#499	—	71	1	no	—
59	#367	—	71	1	no	—
60	#367	—	71	1	no	—
61	#366	—	71	1	no	—
62	#366	D₇₁	1	1	yes	characteristic
63	#346	—	71	1	no	—
64	C₄⋊Q₈	—	142	1	no	—
65	#125	—	142	1	no	—
66	C₂×C₁₆	—	71	1	no	—
67	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
68	#107	—	71	1	no	—
69	#107	—	71	1	no	—
70	#107	—	142	1	no	—
71	#107	C₂×D₇₁	1	1	yes	characteristic
72	#107	C₂×D₇₁	1	1	yes	characteristic
73	#106	—	142	1	no	—
74	C₄×C₈	—	71	1	no	—
75	C₂×Q₈	—	284	1	no	—
76	C₂×C₈	—	71	1	no	—
77	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
78	C₄⋊C₄	—	2	1	no	—
79	C₄⋊C₄	—	2	1	no	—
80	C₄⋊C₄	—	284	1	no	—
81	C₄⋊C₄	—	284	1	no	—
82	C₄⋊C₄	—	284	1	no	—
83	C₄⋊C₄	—	142	1	no	—
84	C₄⋊C₄	—	142	1	no	—
85	C₄²	—	71	1	no	—
86	C₁₆	—	71	2	no	—
87	C₁₆	Dic₁₄₂	1	2	yes	complemented
88	Q₈	—	568	2	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	284	1	no	—
92	C₂×C₄	—	284	1	no	—
93	C₂×C₄	—	71	1	no	—
94	C₂×C₄	—	71	1	no	—
95	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
96	C₈	—	71	2	no	—
97	C₈	Q₈×D₇₁	1	1	yes	characteristic
98	C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
99	C₂²	D₈×D₇₁	1	1	yes	characteristic
100	C₄	—	8	1	no	—
101	C₄	—	8	1	no	—
102	C₄	—	568	1	no	—
103	C₄	—	568	1	no	—
104	C₄	—	142	1	no	—
105	C₄	—	142	1	no	—
106	C₄	#14556127	1	1	yes	characteristic
107	C₄	Q₁₆×D₇₁	1	1	yes	characteristic
108	C₂	#17986403	1	1	yes	characteristic
109	C₂	#17986433	1	1	yes	characteristic
110	C₂	#17986441	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986403

C₂

yes

maximal, characteristic

#17986400

C₂

yes

maximal, characteristic

#17985827

C₂

yes

maximal, characteristic

#17985826

C₂

yes

maximal, characteristic

#17985824

C₂

yes

maximal, characteristic

#17985774

C₂

yes

maximal, characteristic

#366×C₇₁

C₂

yes

maximal, characteristic

#14556127

—

#14556119

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556061

—

#14556059

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

#107×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×C₇₁⋊C₈

D₄

yes

characteristic

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

Q₈

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

D₈

yes

characteristic

C₂×Dic₇₁

D₈

yes

characteristic

Dic₁₄₂

—

C₈×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

C₂×Q₈

yes

characteristic

C₇₁⋊C₈

Q₁₆

yes

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₄×C₇₁

C₂×Q₁₆

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₇₁

yes

characteristic

#346

—

C₄⋊Q₈

—

142

—

#125

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

—

—

—

142

—

#107

C₂×D₇₁

yes

characteristic

#107

C₂×D₇₁

yes

characteristic

#106

—

142

—

C₄×C₈

—

C₂×Q₈

—

284

—

C₂×C₈

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

C₁₆

—

C₁₆

Dic₁₄₂

yes

complemented

Q₈

—

568

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₇₁

yes

characteristic

C₈

—

C₈

Q₈×D₇₁

yes

characteristic

C₈

C₂×Dic₁₄₂

yes

characteristic

C₂²

D₈×D₇₁

yes

characteristic

100

C₄

—

101

C₄

—

102

C₄

—

568

—

103

C₄

—

568

—

104

C₄

—

142

—

105

C₄

—

142

—

106

C₄

#14556127

yes

characteristic

107

C₄

Q₁₆×D₇₁

yes

characteristic

108

C₂

#17986403

yes

characteristic

109

C₂

#17986433

yes

characteristic

110

C₂

#17986441

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#366×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#366	D₇₁
hypocenter	C₇₁	#2255
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#366×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #366

N	H ≅ G/N	type
27. C₇₁⋊C₁₆	Q₈	semidirect
31. C₁₆×C₇₁	Q₈	semidirect
56. C₇₁	#2255	semidirect
87. C₁₆	Dic₁₄₂	semidirect

H ≅ G/N

type

27. C₇₁⋊C₁₆

Q₈

semidirect

31. C₁₆×C₇₁

Q₈

semidirect

56. C₇₁

#2255

semidirect

87. C₁₆

Dic₁₄₂

semidirect

Subgroups — #22590624

Conjugacy classes of subgroups

Complements