Group #22590247 — subgroups

Subgroups — #22590247

Order 9088.

126 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 69 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	C₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic
2	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	Dic₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#17985775	C₂	1	1	yes	maximal, characteristic
5	#17985774	C₂	1	1	yes	maximal, characteristic
6	C₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₄×Dic₁₄₂	C₄	1	1	yes	characteristic
9	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₇₁⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₄⋊(C₇₁⋊C₈)	C₄	1	1	yes	characteristic
12	Dic₇₁⋊C₈	C₄	1	1	yes	characteristic
13	Dic₇₁⋊C₈	C₄	1	1	yes	characteristic
14	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
15	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
17	C₂×C₁₆×C₇₁	C₂²	1	1	yes	characteristic
18	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×Dic₁₄₂	C₈	1	1	yes	characteristic
20	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
21	C₂×C₇₁⋊C₈	C₂×C₄	1	1	yes	characteristic
22	C₄⋊Dic₇₁	C₈	1	1	yes	characteristic
23	Dic₇₁⋊C₄	C₈	1	1	yes	characteristic
24	Dic₇₁⋊C₄	C₈	1	1	yes	characteristic
25	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
26	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
27	C₇₁⋊C₁₆	—	2	1	no	—
28	C₇₁⋊C₁₆	Q₈	1	2	yes	complemented
29	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
31	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
32	C₁₆×C₇₁	—	2	1	no	—
33	C₁₆×C₇₁	—	2	1	no	—
34	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
38	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
39	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
40	C₂×Dic₇₁	C₂×C₈	1	1	yes	characteristic
41	Dic₁₄₂	—	2	2	no	—
42	C₈×C₇₁	—	2	1	no	—
43	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₇₁	#41	1	1	yes	characteristic
45	C₇₁⋊C₈	—	2	1	no	—
46	C₇₁⋊C₈	—	2	1	no	—
47	C₂²×C₇₁	C₂²×C₈	1	1	yes	characteristic
48	C₄×C₇₁	—	2	1	no	—
49	C₄×C₇₁	—	2	1	no	—
50	C₄×C₇₁	C₄×Q₈	1	1	yes	characteristic
51	C₄×C₇₁	#131	1	1	yes	characteristic
52	Dic₇₁	—	2	1	no	—
53	Dic₇₁	—	2	1	no	—
54	Dic₇₁	—	2	1	no	—
55	Dic₇₁	M₃₂	1	2	yes	—
56	C₂×C₇₁	C₈×Q₈	1	1	yes	characteristic
57	C₂×C₇₁	C₂×M₃₂	1	1	yes	characteristic
58	C₂×C₇₁	#504	1	1	yes	characteristic
59	#2183	—	71	1	no	maximal, Sylow
60	C₇₁	#2183	1	1	yes	characteristic, complemented, Sylow
61	C₈×Q₈	—	71	1	no	—
62	C₄⋊C₁₆	—	71	1	no	—
63	C₄⋊C₁₆	—	71	1	no	—
64	C₄⋊C₁₆	D₇₁	1	1	yes	characteristic
65	#346	—	71	1	no	—
66	#346	—	71	1	no	—
67	C₄×C₁₆	—	71	1	no	—
68	C₄×Q₈	—	71	1	no	—
69	C₂×C₁₆	—	71	1	no	—
70	C₂×C₁₆	—	71	1	no	—
71	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
72	C₂×C₁₆	C₂×D₇₁	1	1	yes	characteristic
73	C₄⋊C₈	—	71	1	no	—
74	C₄⋊C₈	—	71	1	no	—
75	C₄⋊C₈	—	71	1	no	—
76	C₄×C₈	—	71	1	no	—
77	C₄×C₈	—	71	1	no	—
78	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
79	C₂×Q₈	—	71	1	no	—
80	C₂×C₈	—	71	1	no	—
81	C₂×C₈	—	71	1	no	—
82	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
83	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
84	C₄⋊C₄	—	71	1	no	—
85	C₄⋊C₄	—	71	1	no	—
86	C₄⋊C₄	—	71	1	no	—
87	C₄²	—	71	1	no	—
88	C₄²	—	71	1	no	—
89	C₄²	C₄×D₇₁	1	1	yes	characteristic
90	C₁₆	—	2	1	no	—
91	C₁₆	—	2	1	no	—
92	C₁₆	—	71	2	no	—
93	C₁₆	—	142	1	no	—
94	Q₈	—	142	2	no	—
95	C₂×C₄	—	71	1	no	—
96	C₂×C₄	—	71	1	no	—
97	C₂×C₄	—	71	1	no	—
98	C₂×C₄	—	71	1	no	—
99	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
100	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
101	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
102	C₈	—	2	1	no	—
103	C₈	—	142	1	no	—
104	C₈	—	142	1	no	—
105	C₈	Q₈×D₇₁	1	1	yes	characteristic
106	C₈	#11765864	1	1	yes	characteristic
107	C₂²	C₂×C₈×D₇₁	1	1	yes	characteristic
108	C₄	—	2	1	no	—
109	C₄	—	2	1	no	—
110	C₄	—	71	2	no	—
111	C₄	—	142	1	no	—
112	C₄	—	142	1	no	—
113	C₄	—	142	1	no	—
114	C₄	#14556093	1	1	yes	characteristic
115	C₄	#14556136	1	1	yes	characteristic
116	C₂	#17986123	1	1	yes	characteristic
117	C₂	M₃₂×D₇₁	1	1	yes	characteristic
118	C₂	#17986415	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₄₂

C₂

yes

maximal, characteristic

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic

Dic₇₁⋊C₁₆

C₂

yes

maximal, characteristic

#17985775

C₂

yes

maximal, characteristic

#17985774

C₂

yes

maximal, characteristic

C₁₆×Dic₇₁

C₂

yes

maximal, characteristic

C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₄×Dic₁₄₂

C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₂×C₇₁⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₇₁⋊C₈)

C₄

yes

characteristic

Dic₇₁⋊C₈

C₄

yes

characteristic

Dic₇₁⋊C₈

C₄

yes

characteristic

C₈×Dic₇₁

C₂²

yes

characteristic

C₈×Dic₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

C₈

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₂×C₇₁⋊C₈

C₂×C₄

yes

characteristic

C₄⋊Dic₇₁

C₈

yes

characteristic

Dic₇₁⋊C₄

C₈

yes

characteristic

Dic₇₁⋊C₄

C₈

yes

characteristic

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

Q₈

yes

complemented

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₁₆×C₇₁

—

C₁₆×C₇₁

—

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

C₂×Dic₇₁

C₂×C₈

yes

characteristic

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₇₁⋊C₈

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂²×C₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×Q₈

yes

characteristic

C₄×C₇₁

#131

yes

characteristic

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

M₃₂

yes

—

C₂×C₇₁

C₈×Q₈

yes

characteristic

C₂×C₇₁

C₂×M₃₂

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

D₇₁

yes

characteristic

#346

—

#346

—

C₄×C₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₇₁

yes

characteristic

C₂×C₁₆

C₂×D₇₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×D₇₁

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

142

—

Q₈

—

142

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₈×D₇₁

yes

characteristic

100

C₂×C₄

C₈×D₇₁

yes

characteristic

101

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

102

C₈

—

103

C₈

—

142

—

104

C₈

—

142

—

105

C₈

Q₈×D₇₁

yes

characteristic

106

C₈

#11765864

yes

characteristic

107

C₂²

C₂×C₈×D₇₁

yes

characteristic

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

142

—

112

C₄

—

142

—

113

C₄

—

142

—

114

C₄

#14556093

yes

characteristic

115

C₄

#14556136

yes

characteristic

116

C₂

#17986123

yes

characteristic

117

C₂

M₃₂×D₇₁

yes

characteristic

118

C₂

#17986415

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₇₁
commutator, socle	C₂²×C₇₁	C₂²×C₈
Fitting	C₄⋊C₁₆×C₇₁	C₂
hypercenter	C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#2183
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₇₁

commutator, socle

C₂²×C₇₁

C₂²×C₈

Fitting

C₄⋊C₁₆×C₇₁

C₂

hypercenter

C₄⋊C₁₆

D₇₁

C₇₁

G ▷ C₂²×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
28. C₇₁⋊C₁₆	Q₈	semidirect
60. C₇₁	#2183	semidirect

H ≅ G/N

type

28. C₇₁⋊C₁₆

Q₈

semidirect

60. C₇₁

#2183

semidirect

Subgroups — #22590247

Conjugacy classes of subgroups

Complements