Group #22590490 — subgroups

Subgroups — #22590490

Order 9088.

130 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 51 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	Q₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#17985825	C₂	1	1	yes	maximal, characteristic
4	#17985824	C₂	1	1	yes	maximal, characteristic
5	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	#17985803	C₂	1	1	yes	maximal, characteristic
7	#358×C₇₁	C₂	1	1	yes	maximal, characteristic
8	#14556127	—	2	1	no	—
9	Q₈×Dic₇₁	—	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	#14556052	—	2	1	no	—
15	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×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	C₄×Dic₇₁	—	4	1	no	—
26	C₄×Dic₇₁	D₄	1	1	yes	characteristic
27	C₇₁⋊C₁₆	—	2	1	no	—
28	C₂×Q₈×C₇₁	—	2	1	no	—
29	Q₁₆×C₇₁	—	2	1	no	—
30	Q₁₆×C₇₁	Q₈	1	2	yes	complemented
31	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₇₁	—	2	1	no	—
33	C₁₆×C₇₁	—	2	1	no	—
34	Q₈×C₇₁	—	2	2	no	—
35	Q₈×C₇₁	—	4	1	no	—
36	C₂×C₄×C₇₁	—	4	1	no	—
37	C₂×C₄×C₇₁	—	4	1	no	—
38	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₇₁	—	4	1	no	—
40	C₂×Dic₇₁	—	4	1	no	—
41	C₂×Dic₇₁	D₈	1	1	yes	characteristic
42	C₂×Dic₇₁	D₈	1	1	yes	characteristic
43	Dic₁₄₂	—	8	2	no	—
44	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
45	C₈×C₇₁	#41	1	1	yes	characteristic
46	C₇₁⋊C₈	—	2	1	no	—
47	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
48	C₄×C₇₁	—	4	2	no	—
49	C₄×C₇₁	—	8	1	no	—
50	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
51	C₄×C₇₁	#137	1	1	yes	characteristic
52	Dic₇₁	—	2	1	no	—
53	Dic₇₁	—	8	1	no	—
54	Dic₇₁	—	8	1	no	—
55	Dic₇₁	Q₃₂	1	2	yes	—
56	C₂×C₇₁	#474	1	1	yes	characteristic
57	C₂×C₇₁	C₂×Q₃₂	1	1	yes	characteristic
58	C₂×C₇₁	#509	1	1	yes	characteristic
59	#2227	—	71	1	no	maximal, Sylow
60	C₇₁	#2227	1	1	yes	characteristic, complemented, Sylow
61	#500	—	71	1	no	—
62	C₄×Q₁₆	—	71	1	no	—
63	#366	—	71	1	no	—
64	#366	—	71	1	no	—
65	C₄⋊C₁₆	—	71	1	no	—
66	#358	—	71	1	no	—
67	#358	D₇₁	1	1	yes	characteristic
68	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
69	C₄⋊Q₈	—	142	1	no	—
70	C₄×Q₈	—	142	1	no	—
71	C₂×C₁₆	—	71	1	no	—
72	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
73	#107	—	71	1	no	—
74	#107	—	71	1	no	—
75	#107	—	142	1	no	—
76	#107	C₂×D₇₁	1	1	yes	characteristic
77	#103	—	142	1	no	—
78	C₄×C₈	—	71	1	no	—
79	C₂×Q₈	—	2	1	no	—
80	C₂×Q₈	—	284	1	no	—
81	Q₁₆	—	2	1	no	—
82	Q₁₆	Dic₁₄₂	1	2	yes	complemented
83	C₂×C₈	—	71	1	no	—
84	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
85	C₄⋊C₄	—	2	1	no	—
86	C₄⋊C₄	—	284	1	no	—
87	C₄⋊C₄	—	284	1	no	—
88	C₄⋊C₄	—	142	1	no	—
89	C₄⋊C₄	—	142	1	no	—
90	C₄²	—	284	1	no	—
91	C₄²	—	71	1	no	—
92	C₁₆	—	2	1	no	—
93	C₁₆	—	142	1	no	—
94	Q₈	—	2	2	no	—
95	Q₈	—	4	1	no	—
96	Q₈	—	568	2	no	—
97	C₂×C₄	—	4	1	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	284	1	no	—
100	C₂×C₄	—	284	1	no	—
101	C₂×C₄	—	71	1	no	—
102	C₂×C₄	—	71	1	no	—
103	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
104	C₈	—	142	1	no	—
105	C₈	C₂×Dic₁₄₂	1	1	yes	characteristic
106	C₈	#11765864	1	1	yes	characteristic
107	C₂²	D₈×D₇₁	1	1	yes	characteristic
108	C₄	—	4	2	no	—
109	C₄	—	8	1	no	—
110	C₄	—	568	1	no	—
111	C₄	—	568	1	no	—
112	C₄	—	71	2	no	—
113	C₄	—	142	1	no	—
114	C₄	#14556105	1	1	yes	characteristic
115	C₄	#14556157	1	1	yes	characteristic
116	C₂	#17986277	1	1	yes	characteristic
117	C₂	Q₃₂×D₇₁	1	1	yes	characteristic
118	C₂	#17986436	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986403

C₂

yes

maximal, characteristic

Q₁₆×Dic₇₁

C₂

yes

maximal, characteristic

#17985825

C₂

yes

maximal, characteristic

#17985824

C₂

yes

maximal, characteristic

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic

#17985803

C₂

yes

maximal, characteristic

#358×C₇₁

C₂

yes

maximal, characteristic

#14556127

—

Q₈×Dic₇₁

—

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556062

C₂²

yes

characteristic

#14556061

—

#14556052

—

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×Q₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×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

—

C₄×Dic₇₁

—

C₄×Dic₇₁

D₄

yes

characteristic

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

Q₁₆×C₇₁

—

Q₁₆×C₇₁

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

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

#41

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#137

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

Q₃₂

yes

—

C₂×C₇₁

#474

yes

characteristic

C₂×C₇₁

C₂×Q₃₂

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#500

—

C₄×Q₁₆

—

#366

—

#366

—

C₄⋊C₁₆

—

#358

—

#358

D₇₁

yes

characteristic

C₂×Q₁₆

C₂×D₇₁

yes

characteristic

C₄⋊Q₈

—

142

—

C₄×Q₈

—

142

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

—

—

—

142

—

#107

C₂×D₇₁

yes

characteristic

#103

—

142

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

284

—

Q₁₆

—

Q₁₆

Dic₁₄₂

yes

complemented

C₂×C₈

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

—

284

—

C₄²

—

C₁₆

—

C₁₆

—

142

—

Q₈

—

Q₈

—

Q₈

—

568

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

100

C₂×C₄

—

284

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

D₄×D₇₁

yes

characteristic

104

C₈

—

142

—

105

C₈

C₂×Dic₁₄₂

yes

characteristic

106

C₈

#11765864

yes

characteristic

107

C₂²

D₈×D₇₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

568

—

111

C₄

—

568

—

112

C₄

—

113

C₄

—

142

—

114

C₄

#14556105

yes

characteristic

115

C₄

#14556157

yes

characteristic

116

C₂

#17986277

yes

characteristic

117

C₂

Q₃₂×D₇₁

yes

characteristic

118

C₂

#17986436

yes

characteristic

	subgroup	G/H
center	C₂²	D₈×D₇₁
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#358×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#358	D₇₁
hypocenter	C₇₁	#2227
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #358

subgroup

G/H

center

C₂²

D₈×D₇₁

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#358×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 ◁ #358

N	H ≅ G/N	type
30. Q₁₆×C₇₁	Q₈	semidirect
60. C₇₁	#2227	semidirect
82. Q₁₆	Dic₁₄₂	semidirect

H ≅ G/N

type

30. Q₁₆×C₇₁

Q₈

semidirect

60. C₇₁

#2227

semidirect

82. Q₁₆

Dic₁₄₂

semidirect

Subgroups — #22590490

Conjugacy classes of subgroups

Complements