Group #22586419 — subgroups

Subgroups — #22586419

Order 9088.

140 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₂×#14556068	C₂	1	1	yes	maximal, characteristic
2	C₂×#14556038	C₂	1	1	yes	maximal, characteristic
3	C₂×M₃₂×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	#14556068	—	2	1	no	—
7	#14556068	—	2	1	no	—
8	C₂×C₇₁⋊C₁₆	C₄	1	2	yes	complemented
9	#14556038	—	2	1	no	—
10	#14556038	—	2	1	no	—
11	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	M₃₂×C₇₁	—	2	1	no	—
13	M₃₂×C₇₁	—	2	1	no	—
14	C₂×C₁₆×C₇₁	C₄	1	2	yes	complemented
15	C₂²×Dic₇₁	—	2	1	no	—
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	C₈	1	2	yes	—
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	M₁₆	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	C₂×C₈	1	1	yes	characteristic
36	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
37	C₂×Dic₇₁	—	2	2	no	—
38	C₂×Dic₇₁	—	4	2	no	—
39	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
40	C₈×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
41	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
42	C₈×C₇₁	C₄⋊C₄	1	1	yes	characteristic
43	C₇₁⋊C₈	—	2	2	no	—
44	C₂²×C₇₁	—	2	2	no	—
45	C₂²×C₇₁	#97	1	1	yes	characteristic
46	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
47	C₂²×C₇₁	C₄⋊C₈	1	1	yes	characteristic
48	C₄×C₇₁	#95	1	1	yes	characteristic
49	C₄×C₇₁	C₄×C₈	1	1	yes	characteristic
50	C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
51	C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
52	Dic₇₁	—	4	2	no	—
53	C₂×C₇₁	—	2	1	no	—
54	C₂×C₇₁	—	2	1	no	—
55	C₂×C₇₁	#336	1	1	yes	characteristic
56	C₂×C₇₁	#347	1	2	yes	—
57	#1375	—	71	1	no	maximal, Sylow
58	C₇₁	#1375	1	1	yes	characteristic, complemented, Sylow
59	C₂×M₃₂	—	71	1	no	—
60	C₂×M₃₂	D₇₁	1	1	yes	characteristic
61	C₂×#97	—	71	1	no	—
62	C₂²×C₈	—	71	1	no	—
63	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
64	C₂×C₄²	—	71	1	no	—
65	M₃₂	—	2	1	no	—
66	M₃₂	—	2	1	no	—
67	M₃₂	—	142	1	no	—
68	M₃₂	—	142	1	no	—
69	C₂×C₁₆	—	71	2	no	—
70	C₂×C₁₆	Dic₇₁	1	2	yes	complemented
71	#97	—	142	1	no	—
72	#97	—	142	1	no	—
73	C₂²×C₄	—	142	1	no	—
74	C₂²×C₄	C₄×D₇₁	1	1	yes	characteristic
75	C₂×C₈	—	71	2	no	—
76	C₂×C₈	—	142	1	no	—
77	C₂×C₈	—	142	1	no	—
78	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
79	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
80	C₂×C₈	D₂₈₄	1	1	yes	characteristic
81	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
82	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
83	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
84	C₄²	—	71	2	no	—
85	C₄²	—	142	1	no	—
86	C₁₆	—	2	2	no	—
87	C₁₆	—	142	2	no	—
88	C₂³	#11765848	1	1	yes	characteristic
89	C₂×C₄	—	284	2	no	—
90	C₂×C₄	—	142	2	no	—
91	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
92	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
93	C₂×C₄	Dic₇₁⋊C₄	1	1	yes	characteristic
94	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
95	C₂×C₄	C₈×D₇₁	1	1	yes	characteristic
96	C₂×C₄	#11765848	1	1	yes	characteristic
97	C₈	—	142	2	no	—
98	C₈	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
99	C₈	C₂²⋊Dic₇₁	1	1	yes	characteristic
100	C₈	Dic₇₁⋊C₄	1	1	yes	characteristic
101	C₈	C₄⋊Dic₇₁	1	1	yes	characteristic
102	C₂²	—	2	2	no	—
103	C₂²	#14556038	1	1	yes	characteristic
104	C₂²	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
105	C₂²	Dic₇₁⋊C₈	1	1	yes	characteristic
106	C₄	—	284	2	no	—
107	C₄	C₇₁⋊#95	1	1	yes	characteristic
108	C₄	C₈×Dic₇₁	1	1	yes	characteristic
109	C₄	(C₂×D₇₁)⋊C₈	1	1	yes	characteristic
110	C₄	Dic₇₁⋊C₈	1	1	yes	characteristic
111	C₂	—	2	1	no	—
112	C₂	—	2	1	no	—
113	C₂	#17985753	1	1	yes	characteristic
114	C₂	#17985776	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556068

C₂

yes

maximal, characteristic

C₂×#14556038

C₂

yes

maximal, characteristic

C₂×M₃₂×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₄

yes

characteristic

C₂×C₄×Dic₇₁

C₄

yes

characteristic

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

C₄

yes

complemented

#14556038

—

#14556038

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

C₈

yes

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₈

yes

—

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

M₁₆

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

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

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

—

C₂²×C₇₁

#97

yes

characteristic

C₂²×C₇₁

C₂²⋊C₈

yes

characteristic

C₂²×C₇₁

C₄⋊C₈

yes

characteristic

C₄×C₇₁

#95

yes

characteristic

C₄×C₇₁

C₄×C₈

yes

characteristic

C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₄×C₇₁

C₄⋊C₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#336

yes

characteristic

C₂×C₇₁

#347

yes

—

#1375

—

maximal, Sylow

C₇₁

#1375

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

C₂×M₃₂

D₇₁

yes

characteristic

C₂×#97

—

C₂²×C₈

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄²

—

M₃₂

—

M₃₂

—

M₃₂

—

142

—

M₃₂

—

142

—

C₂×C₁₆

—

C₂×C₁₆

Dic₇₁

yes

complemented

#97

—

142

—

#97

—

142

—

C₂²×C₄

—

142

—

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

—

C₄²

—

142

—

C₁₆

—

C₁₆

—

142

—

C₂³

#11765848

yes

characteristic

C₂×C₄

—

284

—

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

C₈×D₇₁

yes

characteristic

C₂×C₄

#11765848

yes

characteristic

C₈

—

142

—

C₈

(C₂×D₇₁)⋊C₄

yes

characteristic

C₈

C₂²⋊Dic₇₁

yes

characteristic

100

C₈

Dic₇₁⋊C₄

yes

characteristic

101

C₈

C₄⋊Dic₇₁

yes

characteristic

102

C₂²

—

103

C₂²

#14556038

yes

characteristic

104

C₂²

(C₂×D₇₁)⋊C₈

yes

characteristic

105

C₂²

Dic₇₁⋊C₈

yes

characteristic

106

C₄

—

284

—

107

C₄

C₇₁⋊#95

yes

characteristic

108

C₄

C₈×Dic₇₁

yes

characteristic

109

C₄

(C₂×D₇₁)⋊C₈

yes

characteristic

110

C₄

Dic₇₁⋊C₈

yes

characteristic

111

C₂

—

112

C₂

—

113

C₂

#17985753

yes

characteristic

114

C₂

#17985776

yes

—

	subgroup	G/H
center	C₂×C₄	C₄×Dic₇₁
commutator	C₄×C₇₁	C₄×C₈
Frattini	C₂²×C₈	C₂×D₇₁
Fitting	C₂×M₃₂×C₇₁	C₂
socle	C₂²×C₇₁	#97
hypercenter	C₂×M₃₂	D₇₁
hypocenter	C₇₁	#1375
LCS	G ▷ C₄×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

subgroup

G/H

center

C₂×C₄

C₄×Dic₇₁

commutator

C₄×C₇₁

C₄×C₈

Frattini

C₂²×C₈

C₂×D₇₁

Fitting

C₂×M₃₂×C₇₁

C₂

socle

C₂²×C₇₁

#97

hypercenter

C₂×M₃₂

D₇₁

C₇₁

G ▷ C₄×C₇₁ ▷ C₇₁

C₁ ◁ C₂×C₄ ◁ C₂×M₃₂

N	H ≅ G/N	type
8. C₂×C₇₁⋊C₁₆	C₄	semidirect
14. C₂×C₁₆×C₇₁	C₄	semidirect
58. C₇₁	#1375	semidirect
70. C₂×C₁₆	Dic₇₁	semidirect

H ≅ G/N

type

8. C₂×C₇₁⋊C₁₆

C₄

semidirect

14. C₂×C₁₆×C₇₁

C₄

semidirect

58. C₇₁

#1375

semidirect

70. C₂×C₁₆

Dic₇₁

semidirect

Subgroups — #22586419

Conjugacy classes of subgroups

Complements