Group #22586417 — subgroups

Subgroups — #22586417

Order 9088.

160 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 97 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	C₂²×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₇₁⋊C₈	C₄	1	1	yes	characteristic
5	C₂×C₄×Dic₇₁	C₄	1	1	yes	characteristic
6	C₂×C₇₁⋊C₁₆	—	2	1	no	—
7	C₂×C₇₁⋊C₁₆	—	2	1	no	—
8	C₂×C₇₁⋊C₁₆	C₄	1	2	yes	complemented
9	C₈×Dic₇₁	—	2	1	no	—
10	C₈×Dic₇₁	—	2	1	no	—
11	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×C₇₁	—	2	1	no	—
13	C₂×C₁₆×C₇₁	—	2	1	no	—
14	C₂×C₁₆×C₇₁	C₄	1	2	yes	complemented
15	C₂²×Dic₇₁	C₈	1	2	yes	—
16	C₂×C₇₁⋊C₈	—	2	1	no	—
17	C₂×C₇₁⋊C₈	—	2	1	no	—
18	C₂×C₇₁⋊C₈	C₈	1	2	yes	—
19	C₄×Dic₇₁	—	2	1	no	—
20	C₄×Dic₇₁	—	2	1	no	—
21	C₇₁⋊C₁₆	—	2	2	no	—
22	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
26	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₁	Q₈	1	1	yes	characteristic
29	C₁₆×C₇₁	—	2	2	no	—
30	C₂³×C₇₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
35	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
36	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
37	C₂×Dic₇₁	—	2	2	no	—
38	C₂×Dic₇₁	—	2	2	no	—
39	C₂×Dic₇₁	C₁₆	1	4	yes	complemented
40	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
42	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
44	C₇₁⋊C₈	—	2	2	no	—
45	C₂²×C₇₁	C₄×C₈	1	1	yes	characteristic
46	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
47	C₂²×C₇₁	C₄⋊C₈	1	1	yes	characteristic
48	C₂²×C₇₁	C₂×C₁₆	1	2	yes	—
49	C₂²×C₇₁	M₃₂	1	2	yes	—
50	C₄×C₇₁	#95	1	1	yes	characteristic
51	C₄×C₇₁	#97	1	1	yes	characteristic
52	C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
53	C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
54	Dic₇₁	—	2	4	no	—
55	C₂×C₇₁	#336	1	1	yes	characteristic
56	C₂×C₇₁	C₄×C₁₆	1	1	yes	characteristic
57	C₂×C₇₁	#346	1	1	yes	characteristic
58	C₂×C₇₁	C₂²⋊C₁₆	1	2	yes	—
59	C₂×C₇₁	C₄⋊C₁₆	1	2	yes	—
60	#1374	—	71	1	no	maximal, Sylow
61	C₇₁	#1374	1	1	yes	characteristic, complemented, Sylow
62	C₂²×C₁₆	—	71	1	no	—
63	C₂²×C₁₆	D₇₁	1	1	yes	characteristic
64	C₂×C₄×C₈	—	71	1	no	—
65	C₂²×C₈	—	71	1	no	—
66	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
67	C₂×C₄²	—	71	1	no	—
68	C₂×C₁₆	—	2	1	no	—
69	C₂×C₁₆	—	2	1	no	—
70	C₂×C₁₆	—	71	2	no	—
71	C₂×C₁₆	—	142	1	no	—
72	C₂×C₁₆	—	142	1	no	—
73	C₂×C₁₆	Dic₇₁	1	2	yes	complemented
74	C₄×C₈	—	142	1	no	—
75	C₄×C₈	—	142	1	no	—
76	C₂²×C₄	—	71	2	no	—
77	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
78	C₂×C₈	—	71	2	no	—
79	C₂×C₈	—	142	1	no	—
80	C₂×C₈	—	142	1	no	—
81	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
82	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
83	C₂×C₈	D₂₈₄	1	1	yes	characteristic
84	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
85	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
86	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
87	C₄²	—	142	1	no	—
88	C₄²	—	142	1	no	—
89	C₁₆	—	2	2	no	—
90	C₁₆	—	142	2	no	—
91	C₂³	C₈×D₇₁	1	1	yes	characteristic
92	C₂×C₄	—	71	4	no	—
93	C₂×C₄	—	142	2	no	—
94	C₂×C₄	—	142	2	no	—
95	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
96	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
97	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
98	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
99	C₂×C₄	#11765848	1	1	yes	characteristic
100	C₂×C₄	#11765848	1	1	yes	characteristic
101	C₈	—	142	2	no	—
102	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
103	C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
104	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
105	C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
106	C₂²	C₈×Dic₇₁	1	1	yes	characteristic
107	C₂²	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
108	C₂²	Dic₇₁⋊C₈	1	1	yes	characteristic
109	C₂²	C₁₆×D₇₁	1	2	yes	—
110	C₂²	#14556067	1	2	yes	—
111	C₄	—	142	4	no	—
112	C₄	C₇₁⋊#95	1	1	yes	characteristic
113	C₄	#14556038	1	1	yes	characteristic
114	C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
115	C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
116	C₂	#17985753	1	1	yes	characteristic
117	C₂	C₁₆×Dic₇₁	1	1	yes	characteristic
118	C₂	#17985774	1	1	yes	characteristic
119	C₂	(C₂×D₇₁)⋊C₁₆	1	2	yes	—
120	C₂	Dic₇₁⋊C₁₆	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic

C₂×C₈×Dic₇₁

C₂

yes

maximal, characteristic

C₂²×C₁₆×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₄

yes

characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₄

yes

complemented

C₈×Dic₇₁

—

C₈×Dic₇₁

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

—

C₂×C₁₆×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

C₈

yes

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₈

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

Q₈

yes

characteristic

C₁₆×C₇₁

—

C₂³×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

C₁₆

yes

complemented

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₂²⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₈×C₇₁

C₄⋊C₄

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

C₄×C₈

yes

characteristic

C₂²×C₇₁

C₂²⋊C₈

yes

characteristic

C₂²×C₇₁

C₄⋊C₈

yes

characteristic

C₂²×C₇₁

C₂×C₁₆

yes

—

C₂²×C₇₁

M₃₂

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₄×C₇₁

C₄⋊C₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

#336

yes

characteristic

C₂×C₇₁

C₄×C₁₆

yes

characteristic

C₂×C₇₁

#346

yes

characteristic

C₂×C₇₁

C₂²⋊C₁₆

yes

—

C₂×C₇₁

C₄⋊C₁₆

yes

—

#1374

—

maximal, Sylow

C₇₁

#1374

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

D₇₁

yes

characteristic

C₂×C₄×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

—

142

—

C₂×C₁₆

Dic₇₁

yes

complemented

C₄×C₈

—

142

—

C₄×C₈

—

142

—

C₂²×C₄

—

C₂²×C₄

C₄×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₂×C₈

Dic₁₄₂

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄²

—

142

—

C₄²

—

142

—

C₁₆

—

C₁₆

—

142

—

C₂³

C₈×D₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

characteristic

C₂×C₄

C₈×D₇₁

yes

characteristic

C₂×C₄

#11765848

yes

characteristic

100

C₂×C₄

#11765848

yes

characteristic

101

C₈

—

142

—

102

C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

103

C₈

C₂²⋊Dic₇₁

yes

characteristic

104

C₈

Dic₇₁⋊C₄

yes

characteristic

105

C₈

C₄⋊Dic₇₁

yes

characteristic

106

C₂²

C₈×Dic₇₁

yes

characteristic

107

C₂²

(C₂×D₇₁)⋊C₈

yes

characteristic

108

C₂²

Dic₇₁⋊C₈

yes

characteristic

109

C₂²

C₁₆×D₇₁

yes

—

110

C₂²

#14556067

yes

—

111

C₄

—

142

—

112

C₄

C₇₁⋊#95

yes

characteristic

113

C₄

#14556038

yes

characteristic

114

C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

115

C₄

Dic₇₁⋊C₈

yes

characteristic

116

C₂

#17985753

yes

characteristic

117

C₂

C₁₆×Dic₇₁

yes

characteristic

118

C₂

#17985774

yes

characteristic

119

C₂

(C₂×D₇₁)⋊C₁₆

yes

—

120

C₂

Dic₇₁⋊C₁₆

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂×D₇₁
commutator	C₂×C₇₁	C₄×C₁₆
Fitting	C₂²×C₁₆×C₇₁	C₂
socle	C₂³×C₇₁	C₂×C₈
hypercenter	C₂²×C₁₆	D₇₁
hypocenter	C₇₁	#1374
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂×D₇₁

commutator

C₂×C₇₁

C₄×C₁₆

Fitting

C₂²×C₁₆×C₇₁

C₂

socle

C₂³×C₇₁

C₂×C₈

hypercenter

C₂²×C₁₆

D₇₁

C₇₁

G ▷ C₂×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
8. C₂×C₇₁⋊C₁₆	C₄	semidirect
14. C₂×C₁₆×C₇₁	C₄	semidirect
39. C₂×Dic₇₁	C₁₆	semidirect
61. C₇₁	#1374	semidirect
73. C₂×C₁₆	Dic₇₁	semidirect

H ≅ G/N

type

8. C₂×C₇₁⋊C₁₆

C₄

semidirect

14. C₂×C₁₆×C₇₁

C₄

semidirect

39. C₂×Dic₇₁

C₁₆

semidirect

61. C₇₁

#1374

semidirect

73. C₂×C₁₆

Dic₇₁

semidirect

Subgroups — #22586417

Conjugacy classes of subgroups

Complements