Group #22588378 — subgroups

Subgroups — #22588378

Order 9088.

292 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 117 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×#14556064	C₂	1	2	yes	maximal, complemented
2	#17985963	C₂	1	2	yes	maximal
3	#17985764	C₂	1	2	yes	maximal, complemented
4	C₂³×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#11765849	C₂²	1	4	yes	—
6	#14556064	—	2	4	no	—
7	C₂²⋊(C₇₁⋊C₈)	—	2	4	no	—
8	C₂³×C₄×C₇₁	C₂²	1	1	yes	characteristic
9	C₂²×C₈×C₇₁	—	2	4	no	—
10	C₂²×C₈×C₇₁	C₄	1	4	yes	—
11	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
13	#11765849	—	2	4	no	—
14	#11765849	—	2	4	no	—
15	C₂×C₇₁⋊C₈	—	2	4	no	—
16	C₂⁴×C₇₁	Q₈	1	1	yes	characteristic
17	C₂²×C₄×C₇₁	—	2	2	no	—
18	C₂²×C₄×C₇₁	—	2	2	no	—
19	C₂²×C₄×C₇₁	C₂×C₄	1	2	yes	—
20	C₂²×C₄×C₇₁	D₄	1	2	yes	—
21	C₂²×C₄×C₇₁	Q₈	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
23	C₂×C₈×C₇₁	—	2	2	no	—
24	C₂×C₈×C₇₁	—	2	8	no	—
25	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₇₁	D₄	1	4	yes	—
28	C₂³×C₇₁	—	2	4	no	—
29	C₂³×C₇₁	—	2	2	no	—
30	C₂³×C₇₁	C₄⋊C₄	1	2	yes	—
31	C₂³×C₇₁	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	—	2	2	no	—
33	C₂×C₄×C₇₁	—	2	2	no	—
34	C₂×C₄×C₇₁	—	2	2	no	—
35	C₂×C₄×C₇₁	—	2	4	no	—
36	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
37	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₁	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₇₁	#41	1	2	yes	—
41	C₈×C₇₁	—	2	2	no	—
42	C₈×C₇₁	C₂²⋊C₄	1	4	yes	—
43	C₇₁⋊C₈	—	4	4	no	—
44	C₂²×C₇₁	—	2	8	no	—
45	C₂²×C₇₁	—	2	2	no	—
46	C₂²×C₇₁	—	2	2	no	—
47	C₂²×C₇₁	—	2	2	no	—
48	C₂²×C₇₁	#108	1	4	yes	complemented
49	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂²×C₇₁	C₂²⋊Q₈	1	2	yes	—
51	C₄×C₇₁	—	2	2	no	—
52	C₄×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₄×C₇₁	#117	1	1	yes	characteristic
54	C₄×C₇₁	#121	1	2	yes	—
55	C₂×C₇₁	—	2	4	no	—
56	C₂×C₇₁	—	2	2	no	—
57	C₂×C₇₁	#380	1	1	yes	characteristic
58	C₂×C₇₁	C₂×#108	1	2	yes	—
59	#1810	—	71	1	no	maximal, Sylow
60	C₇₁	#1810	1	1	yes	characteristic, complemented, Sylow
61	C₂³×C₈	D₇₁	1	1	yes	characteristic
62	C₂×#108	—	71	2	no	—
63	#407	—	71	2	no	—
64	#341	—	71	2	no	—
65	C₂³×C₄	C₂×D₇₁	1	1	yes	characteristic
66	C₂×M₁₆	—	71	4	no	—
67	C₂²×C₈	—	2	4	no	—
68	C₂²×C₈	Dic₇₁	1	4	yes	—
69	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
70	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
71	#108	—	142	4	no	—
72	C₂²⋊C₈	—	142	4	no	—
73	C₂⁴	Dic₁₄₂	1	1	yes	characteristic
74	C₂²×C₄	—	2	2	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	Dic₁₄₂	1	1	yes	characteristic
77	C₂²×C₄	D₂₈₄	1	2	yes	—
78	C₂²×C₄	C₂×Dic₇₁	1	2	yes	—
79	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
80	M₁₆	—	142	4	no	—
81	M₁₆	—	142	4	no	—
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	8	no	—
84	C₂×C₈	—	142	4	no	—
85	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
86	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
87	C₂×C₈	C₂²⋊D₇₁	1	4	yes	—
88	C₂³	—	2	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	C₄⋊Dic₇₁	1	2	yes	—
91	C₂³	C₂×Dic₁₄₂	1	1	yes	characteristic
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	C₄⋊Dic₇₁	1	2	yes	—
97	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₇₁	1	2	yes	—
99	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
100	C₂×C₄	#11765865	1	2	yes	—
101	C₈	—	2	2	no	—
102	C₈	—	284	4	no	—
103	C₈	C₂²⋊Dic₇₁	1	4	yes	—
104	C₂²	—	2	8	no	—
105	C₂²	—	2	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	#14556064	1	4	yes	complemented
109	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
110	C₂²	#14556107	1	2	yes	—
111	C₄	—	2	2	no	—
112	C₄	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
113	C₄	#14556087	1	1	yes	characteristic
114	C₄	#14556102	1	2	yes	—
115	C₂	—	2	4	no	—
116	C₂	—	2	2	no	—
117	C₂	#17985859	1	1	yes	characteristic
118	C₂	C₂×#14556064	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556064

C₂

yes

maximal, complemented

#17985963

C₂

yes

maximal

#17985764

C₂

yes

maximal, complemented

C₂³×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

—

#14556064

—

C₂²⋊(C₇₁⋊C₈)

—

C₂³×C₄×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

—

C₂²×C₈×C₇₁

C₄

yes

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₂⁴×C₇₁

Q₈

yes

characteristic

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×C₄

yes

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

Q₈

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂³×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

C₄⋊C₄

yes

—

C₂³×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊C₄

yes

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#108

yes

complemented

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

C₂²⋊Q₈

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₂²⋊C₄

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

#380

yes

characteristic

C₂×C₇₁

C₂×#108

yes

—

#1810

—

maximal, Sylow

C₇₁

#1810

yes

characteristic, complemented, Sylow

C₂³×C₈

D₇₁

yes

—

—

—

C₂³×C₄

C₂×D₇₁

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

Dic₇₁

yes

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

#108

—

142

—

C₂²⋊C₈

—

142

—

C₂⁴

Dic₁₄₂

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

Dic₁₄₂

yes

characteristic

C₂²×C₄

D₂₈₄

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

142

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₂×C₈

C₂²⋊D₇₁

yes

—

C₂³

—

C₂³

—

C₂³

C₄⋊Dic₇₁

yes

—

C₂³

C₂×Dic₁₄₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

—

C₂×C₄

C₂×D₂₈₄

yes

characteristic

100

C₂×C₄

#11765865

yes

—

101

C₈

—

102

C₈

—

284

—

103

C₈

C₂²⋊Dic₇₁

yes

—

104

C₂²

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

#14556064

yes

complemented

109

C₂²

C₂×C₄⋊Dic₇₁

yes

characteristic

110

C₂²

#14556107

yes

—

111

C₄

—

112

C₄

C₂×C₂²⋊Dic₇₁

yes

characteristic

113

C₄

#14556087

yes

characteristic

114

C₄

#14556102

yes

—

115

C₂

—

116

C₂

—

117

C₂

#17985859

yes

characteristic

118

C₂

C₂×#14556064

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂³×C₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂×C₄⋊C₄
hypercenter	C₂³×C₈	D₇₁
hypocenter	C₇₁	#1810
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

subgroup

G/H

center

C₂×C₄

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂³×C₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂×C₄⋊C₄

hypercenter

C₂³×C₈

D₇₁

hypocenter

C₇₁

#1810

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈

N	H ≅ G/N	type
1. C₂×#14556064	C₂	semidirect
3. #17985764	C₂	semidirect
48. C₂²×C₇₁	#108	semidirect
60. C₇₁	#1810	semidirect
108. C₂²	#14556064	semidirect

H ≅ G/N

type

1. C₂×#14556064

C₂

semidirect

3. #17985764

C₂

semidirect

48. C₂²×C₇₁

#108

semidirect

60. C₇₁

#1810

semidirect

108. C₂²

#14556064

semidirect

Subgroups — #22588378

Conjugacy classes of subgroups

Complements