Group #86088 — subgroups

Subgroups — C₂×C₃⋊#564 #86088

Order 384.

896 conjugacy classes of subgroups; only the 469 normal subgroups are shown.

normal subgroups only

This group has 896 conjugacy classes of subgroups — too many to draw. Only its 469 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×C₃×#125	C₂	1	1	yes	maximal, characteristic
2	C₃⋊#564	C₂	1	16	yes	maximal, direct product factor
3	C₂×#890	C₂	1	4	yes	maximal
4	C₂×#889	C₂	1	4	yes	maximal
5	C₂×#889	C₂	1	4	yes	maximal
6	C₂×#870	C₂	1	2	yes	maximal
7	C₂×#564	—	3	1	no	maximal, Sylow
8	C₃×#125	C₂²	1	8	yes	—
9	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
15	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
16	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
17	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
18	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
19	C₂×C₄×Dic₃	C₂²	1	4	yes	—
20	#890	C₂²	1	32	yes	—
21	#889	C₂²	1	32	yes	—
22	#889	C₂²	1	32	yes	—
23	#870	C₂²	1	16	yes	—
24	#564	—	3	16	no	—
25	C₂×#125	—	3	4	no	—
26	C₂×#125	—	3	4	no	—
27	C₂×#125	—	3	4	no	—
28	C₂×#125	—	3	2	no	—
29	C₂×#125	D₃	1	1	yes	characteristic
30	C₂²×C₃×C₄	C₂³	1	4	yes	—
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂³	1	2	yes	—
33	C₂²×Dic₃	C₂³	1	4	yes	—
34	C₂²×Dic₃	C₂³	1	4	yes	—
35	C₃×C₄⋊C₄	C₂³	1	8	yes	—
36	C₃×C₄⋊C₄	C₂³	1	16	yes	—
37	C₃×C₄²	C₂³	1	4	yes	—
38	C₄⋊Dic₃	C₂³	1	8	yes	—
39	C₄⋊Dic₃	C₂³	1	8	yes	—
40	C₄⋊Dic₃	C₂³	1	16	yes	—
41	Dic₃⋊C₄	C₂³	1	16	yes	—
42	Dic₃⋊C₄	C₂³	1	16	yes	—
43	Dic₃⋊C₄	C₂³	1	16	yes	—
44	Dic₃⋊C₄	C₂³	1	16	yes	—
45	C₄×Dic₃	C₂³	1	16	yes	—
46	#125	—	3	32	no	—
47	#125	—	3	32	no	—
48	#125	—	3	32	no	—
49	#125	—	3	16	no	—
50	#125	C₂×D₃	1	8	yes	—
51	C₂×C₄⋊C₄	—	3	4	no	—
52	C₂×C₄⋊C₄	—	3	4	no	—
53	C₂×C₄⋊C₄	—	3	4	no	—
54	C₂×C₄⋊C₄	—	3	4	no	—
55	C₂×C₄⋊C₄	—	3	4	no	—
56	C₂×C₄⋊C₄	—	3	2	no	—
57	C₂×C₄⋊C₄	—	3	2	no	—
58	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
59	C₂×C₄⋊C₄	C₂×D₃	1	4	yes	—
60	C₂×C₄²	—	3	4	no	—
61	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
62	C₂³×C₃	C₂⁴	1	1	yes	characteristic
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	8	no	—
65	C₂×C₃×C₄	—	2	4	no	—
66	C₂×C₃×C₄	C₂⁴	1	8	yes	—
67	C₂×C₃×C₄	C₂⁴	1	2	yes	—
68	C₂×C₃×C₄	C₂⁴	1	4	yes	—
69	C₂×Dic₃	—	2	8	no	—
70	C₂×Dic₃	—	2	8	no	—
71	C₂×Dic₃	C₂⁴	1	8	yes	—
72	C₂×Dic₃	C₂⁴	1	8	yes	—
73	C₂²×C₄	—	3	4	no	—
74	C₂²×C₄	—	3	4	no	—
75	C₂²×C₄	C₂²×D₃	1	4	yes	—
76	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
77	C₂²×C₄	C₂²×D₃	1	2	yes	—
78	C₄⋊C₄	—	3	16	no	—
79	C₄⋊C₄	—	3	16	no	—
80	C₄⋊C₄	—	3	16	no	—
81	C₄⋊C₄	—	3	16	no	—
82	C₄⋊C₄	—	3	16	no	—
83	C₄⋊C₄	—	3	8	no	—
84	C₄⋊C₄	—	3	8	no	—
85	C₄⋊C₄	C₂²×D₃	1	8	yes	—
86	C₄⋊C₄	C₂²×D₃	1	16	yes	—
87	C₄²	—	3	16	no	—
88	C₄²	C₂²×D₃	1	4	yes	—
89	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
90	C₂²×C₃	2₋¹⁺⁴	1	4	yes	—
91	C₂²×C₃	C₂⁵	1	1	yes	characteristic
92	C₃×C₄	—	2	2	no	—
93	C₃×C₄	—	2	8	no	—
94	C₃×C₄	—	2	4	no	—
95	Dic₃	—	2	8	no	—
96	Dic₃	—	2	8	no	—
97	C₂³	C₂³×D₃	1	1	yes	characteristic
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	3	8	no	—
102	C₂×C₄	—	3	8	no	—
103	C₂×C₄	—	6	8	no	—
104	C₂×C₄	—	6	8	no	—
105	C₂×C₄	C₂³×D₃	1	8	yes	—
106	C₂×C₄	C₂³×D₃	1	2	yes	—
107	C₂×C₄	C₂³×D₃	1	4	yes	—
108	C₂×C₃	#564	1	4	yes	complemented
109	C₂×C₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×C₃	C₂×2₋¹⁺⁴	1	2	yes	—
111	C₂²	#1007	1	2	yes	—
112	C₂²	#1010	1	4	yes	—
113	C₂²	C₂⁴×D₃	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	—	2	8	no	—
116	C₄	—	2	4	no	—
117	C₄	—	6	8	no	—
118	C₄	—	6	8	no	—
119	C₃	C₂×#564	1	1	yes	characteristic, complemented, Sylow
120	C₂	C₃⋊#564	1	4	yes	direct product factor
121	C₂	C₂×#1007	1	1	yes	characteristic
122	C₂	C₂×#1010	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#125

C₂

yes

maximal, characteristic

C₃⋊#564

C₂

yes

maximal, direct product factor

C₂×#890

C₂

yes

maximal

C₂×#889

C₂

yes

maximal

C₂×#889

C₂

yes

maximal

C₂×#870

C₂

yes

—

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

#890

C₂²

yes

—

#889

C₂²

yes

—

#889

C₂²

yes

—

#870

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

C₂³

yes

—

C₃×C₄⋊C₄

C₂³

yes

—

C₃×C₄²

C₂³

yes

—

C₄⋊Dic₃

C₂³

yes

—

C₄⋊Dic₃

C₂³

yes

—

C₄⋊Dic₃

C₂³

yes

—

Dic₃⋊C₄

C₂³

yes

—

Dic₃⋊C₄

C₂³

yes

—

Dic₃⋊C₄

C₂³

yes

—

Dic₃⋊C₄

C₂³

yes

—

C₄×Dic₃

C₂³

yes

—

—

—

—

—

C₂×D₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂⁴

yes

—

C₂×C₃×C₄

C₂⁴

yes

—

C₂×C₃×C₄

C₂⁴

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂⁴

yes

—

C₂×Dic₃

C₂⁴

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃

yes

—

C₄⋊C₄

C₂²×D₃

yes

—

C₄²

—

C₄²

C₂²×D₃

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₂²×C₃

C₂⁵

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₂³

C₂³×D₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂³×D₃

yes

—

106

C₂×C₄

C₂³×D₃

yes

—

107

C₂×C₄

C₂³×D₃

yes

—

108

C₂×C₃

#564

yes

complemented

109

C₂×C₃

C₂×2₋¹⁺⁴

yes

characteristic

110

C₂×C₃

C₂×2₋¹⁺⁴

yes

—

111

C₂²

#1007

yes

—

112

C₂²

#1010

yes

—

113

C₂²

C₂⁴×D₃

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₃

C₂×#564

yes

characteristic, complemented, Sylow

120

C₂

C₃⋊#564

yes

direct product factor

121

C₂

C₂×#1007

yes

characteristic

122

C₂

C₂×#1010

yes

—

	subgroup	G/H
center	C₂³	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁵
Frattini	C₂²	C₂⁴×D₃
Fitting	C₂×C₃×#125	C₂
socle, prime radical	C₂³×C₃	C₂⁴
hypercenter	C₂×#125	D₃
hypocenter	C₃	C₂×#564
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂³×D₃

commutator, cosocle

C₂²×C₃

C₂⁵

Frattini

C₂²

C₂⁴×D₃

Fitting

C₂×C₃×#125

C₂

socle, prime radical

C₂³×C₃

C₂⁴

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
2. C₃⋊#564	C₂	direct (N × H)
108. C₂×C₃	#564	semidirect
119. C₃	C₂×#564	semidirect
120. C₂	C₃⋊#564	direct (N × H)

H ≅ G/N

type

2. C₃⋊#564

C₂

direct (N × H)

108. C₂×C₃

#564

semidirect

119. C₃

C₂×#564

semidirect

120. C₂

C₃⋊#564

direct (N × H)

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#125	2
C₃⋊#564	2
C₂×#890	2
C₂×#889	2
C₂×#870	2
C₂×#564	3

subgroup

index

C₂×C₃×#125

Subgroups — C₂×C₃⋊#564 #86088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×C3⋊#564 #86088

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃⋊#564 #86088