Group #78094 — subgroups

Subgroups — #78094

Order 384.

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

normal subgroups only

This group has 372 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	C₃×#399	C₂	1	1	yes	maximal, characteristic
2	#4982	C₂	1	6	yes	maximal, complemented
3	#4953	C₂	1	3	yes	maximal
4	#4946	C₂	1	3	yes	maximal
5	C₃⋊#401	C₂	1	2	yes	maximal, complemented
6	#2834	—	3	1	no	maximal, Sylow
7	C₂³×Dic₃	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄⋊C₄	C₂²	1	3	yes	—
9	C₂×C₃×C₂²⋊C₄	C₂²	1	3	yes	—
10	C₂×#217	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₃	C₂²	1	3	yes	—
12	C₂×Dic₃⋊C₄	C₂²	1	6	yes	—
13	C₃×#95	C₂²	1	1	yes	characteristic
14	C₃⋊#95	C₂²	1	6	yes	—
15	C₃⋊#95	C₂²	1	6	yes	—
16	C₃⋊#95	C₂²	1	3	yes	—
17	#401	—	3	2	no	—
18	#399	D₃	1	1	yes	characteristic
19	#398	—	3	6	no	—
20	#397	—	3	3	no	—
21	#385	—	3	3	no	—
22	C₂⁴×C₃	C₂³	1	1	yes	characteristic
23	C₂²×C₃×C₄	C₂³	1	3	yes	—
24	C₂²×C₃×C₄	C₂³	1	3	yes	—
25	C₂²×Dic₃	—	4	3	no	—
26	C₂²×Dic₃	C₂³	1	6	yes	—
27	C₂²×Dic₃	C₂³	1	2	yes	—
28	C₃×C₄⋊C₄	—	4	3	no	—
29	C₃×C₂²⋊C₄	—	4	3	no	—
30	#217	—	2	6	no	—
31	C₄⋊Dic₃	—	4	3	no	—
32	Dic₃⋊C₄	—	4	6	no	—
33	C₂³×C₄	—	3	1	no	—
34	C₂×C₄⋊C₄	—	3	6	no	—
35	C₂×C₄⋊C₄	—	3	3	no	—
36	C₂×C₄⋊C₄	C₂×D₃	1	3	yes	—
37	C₂×C₂²⋊C₄	—	3	3	no	—
38	C₂×C₂²⋊C₄	C₂×D₃	1	3	yes	—
39	#95	—	3	6	no	—
40	#95	—	3	3	no	—
41	#95	—	3	6	no	—
42	#95	C₂×D₃	1	1	yes	characteristic
43	C₂³×C₃	—	2	3	no	—
44	C₂³×C₃	—	2	3	no	—
45	C₂³×C₃	#41	1	2	yes	—
46	C₂³×C₃	C₂⁴	1	1	yes	characteristic
47	C₂×C₃×C₄	—	2	6	no	—
48	C₂×C₃×C₄	—	2	3	no	—
49	C₂×C₃×C₄	—	2	3	no	—
50	C₂×C₃×C₄	—	2	6	no	—
51	C₂×Dic₃	—	2	6	no	—
52	C₂×Dic₃	—	2	6	no	—
53	C₂×Dic₃	—	2	6	no	—
54	C₂×Dic₃	—	2	6	no	—
55	C₂×Dic₃	—	4	4	no	—
56	C₂⁴	C₂²×D₃	1	1	yes	characteristic
57	C₂²×C₄	—	3	6	no	—
58	C₂²×C₄	—	3	2	no	—
59	C₂²×C₄	—	12	3	no	—
60	C₂²×C₄	C₂²×D₃	1	3	yes	—
61	C₂²×C₄	C₂²×D₃	1	3	yes	—
62	C₄⋊C₄	—	4	3	no	—
63	C₄⋊C₄	—	12	6	no	—
64	C₄⋊C₄	—	12	3	no	—
65	C₂²⋊C₄	—	4	3	no	—
66	C₂²⋊C₄	—	6	6	no	—
67	C₂²×C₃	—	2	6	no	—
68	C₂²×C₃	—	4	1	no	—
69	C₂²×C₃	—	4	3	no	—
70	C₂²×C₃	C₂×#41	1	1	yes	characteristic
71	C₂²×C₃	H(5,𝔽₂)	1	3	yes	—
72	C₂²×C₃	2₋¹⁺⁴	1	3	yes	—
73	C₃×C₄	—	4	3	no	—
74	C₃×C₄	—	4	3	no	—
75	Dic₃	—	4	6	no	—
76	Dic₃	—	4	2	no	—
77	C₂³	—	2	3	no	—
78	C₂³	—	2	3	no	—
79	C₂³	#239	1	2	yes	—
80	C₂³	C₂³×D₃	1	1	yes	characteristic
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	6	no	—
84	C₂×C₄	—	2	6	no	—
85	C₂×C₄	—	6	6	no	—
86	C₂×C₄	—	6	6	no	—
87	C₂×C₄	—	6	6	no	—
88	C₂×C₄	—	6	6	no	—
89	C₂×C₄	—	12	4	no	—
90	C₂×C₃	—	4	2	no	—
91	C₂×C₃	#539	1	3	yes	—
92	C₂×C₃	#560	1	1	yes	characteristic
93	C₂×C₃	#563	1	3	yes	—
94	C₂²	—	2	6	no	—
95	C₂²	—	4	3	no	—
96	C₂²	—	4	1	no	—
97	C₂²	#1004	1	3	yes	—
98	C₂²	C₂×#239	1	1	yes	characteristic
99	C₂²	#1010	1	3	yes	—
100	C₄	—	4	3	no	—
101	C₄	—	4	3	no	—
102	C₄	—	12	6	no	—
103	C₄	—	12	2	no	—
104	C₃	#2834	1	1	yes	characteristic, complemented, Sylow
105	C₂	—	4	2	no	—
106	C₂	#5645	1	3	yes	—
107	C₂	#5651	1	3	yes	—
108	C₂	#5720	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#399

C₂

yes

maximal, characteristic

#4982

C₂

yes

maximal, complemented

#4953

C₂

yes

maximal

#4946

C₂

yes

maximal

C₃⋊#401

C₂

yes

maximal, complemented

#2834

—

maximal, Sylow

C₂³×Dic₃

C₂²

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₂²⋊C₄

C₂²

yes

—

C₂×#217

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₃×#95

C₂²

yes

characteristic

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

#401

—

#399

D₃

yes

—

—

—

C₂⁴×C₃

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

#217

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₂³×C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₂×D₃

yes

—

—

—

—

C₂×D₃

yes

characteristic

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

#41

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂⁴

C₂²×D₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×#41

yes

characteristic

C₂²×C₃

H(5,𝔽₂)

yes

—

C₂²×C₃

2₋¹⁺⁴

yes

—

C₃×C₄

—

C₃×C₄

—

Dic₃

—

Dic₃

—

C₂³

—

C₂³

—

C₂³

#239

yes

—

C₂³

C₂³×D₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₃

—

C₂×C₃

#539

yes

—

C₂×C₃

#560

yes

characteristic

C₂×C₃

#563

yes

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#1004

yes

—

C₂²

C₂×#239

yes

characteristic

C₂²

#1010

yes

—

100

C₄

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₃

#2834

yes

characteristic, complemented, Sylow

105

C₂

—

106

C₂

#5645

yes

—

107

C₂

#5651

yes

—

108

C₂

#5720

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#399	C₂
prime radical	C₂⁴×C₃	C₂³
hypercenter	#399	D₃
hypocenter	C₃	#2834
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #399

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, cosocle

C₂³×C₃

C₂⁴

Fitting

C₃×#399

C₂

prime radical

C₂⁴×C₃

C₂³

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #399

N	H ≅ G/N	type
2. #4982	C₂	semidirect
5. C₃⋊#401	C₂	semidirect
104. C₃	#2834	semidirect

H ≅ G/N

type

2. #4982

C₂

semidirect

5. C₃⋊#401

C₂

semidirect

104. C₃

#2834

semidirect

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

subgroup	index
C₃×#399	2
#4982	2
#4953	2
#4946	2
C₃⋊#401	2
#2834	3

subgroup

index

Subgroups — #78094

Conjugacy classes of subgroups

Complements

Maximal subgroups