Group #70526 — subgroups

Subgroups — #70526

Order 384.

158 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 61 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×#423	C₂	1	1	yes	maximal, characteristic
2	#5125	C₂	1	1	yes	maximal, characteristic
3	#4559	C₂	1	2	yes	maximal
4	#4557	C₂	1	1	yes	maximal, characteristic
5	#4523	C₂	1	2	yes	maximal
6	#2102	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	#926	C₂²	1	2	yes	—
11	C₂×#208	C₂²	1	2	yes	—
12	C₃×C₄⋊C₈	—	2	2	no	—
13	#819	—	2	2	no	—
14	#430	—	3	1	no	—
15	#423	D₃	1	1	yes	characteristic
16	#344	—	3	2	no	—
17	#339	—	3	2	no	—
18	H(3,ℤ/4)	—	3	1	no	—
19	C₂²×C₃×C₄	—	2	1	no	—
20	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
21	C₃×M₁₆	—	2	2	no	—
22	C₃×M₁₆	—	2	2	no	—
23	C₂×C₃×C₈	—	2	1	no	—
24	C₂×C₃×C₈	D₄	1	2	yes	—
25	C₃×C₄²	—	2	1	no	—
26	C₃×C₄²	Q₈	1	1	yes	characteristic
27	C₃×C₄²	Q₈	1	1	yes	characteristic
28	#217	—	2	2	no	—
29	C₄⋊Dic₃	—	2	2	no	—
30	C₄×Dic₃	—	2	2	no	—
31	#208	—	2	2	no	—
32	#208	—	2	2	no	—
33	C₂×C₃⋊C₈	—	2	2	no	—
34	C₂×M₁₆	—	3	2	no	—
35	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
36	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
37	#117	—	3	2	no	—
38	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
39	#108	—	6	2	no	—
40	C₄⋊C₈	—	2	2	no	—
41	C₂³×C₃	#41	1	1	yes	characteristic
42	C₂×C₃×C₄	—	2	1	no	—
43	C₂×C₃×C₄	—	2	1	no	—
44	C₂×C₃×C₄	—	4	2	no	—
45	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₃×C₄	#41	1	2	yes	—
48	C₂×C₃×C₄	#41	1	2	yes	—
49	C₂×Dic₃	—	4	2	no	—
50	C₃×C₈	—	2	2	no	—
51	C₃×C₈	—	4	1	no	—
52	C₃⋊C₈	—	4	2	no	—
53	C₂²×C₄	—	2	1	no	—
54	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
55	M₁₆	—	2	2	no	—
56	M₁₆	—	2	2	no	—
57	M₁₆	—	6	2	no	—
58	M₁₆	—	6	2	no	—
59	C₂×C₈	—	2	1	no	—
60	C₂×C₈	—	6	2	no	—
61	C₂×C₈	#67	1	2	yes	—
62	C₄⋊C₄	—	6	2	no	—
63	C₂²⋊C₄	—	6	2	no	—
64	C₄²	—	2	1	no	—
65	C₄²	—	6	2	no	—
66	C₄²	Dic₆	1	1	yes	characteristic
67	C₄²	Dic₆	1	1	yes	characteristic
68	C₂²×C₃	—	4	1	no	—
69	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
70	C₂²×C₃	#126	1	2	yes	—
71	C₃×C₄	—	2	2	no	—
72	C₃×C₄	—	4	1	no	—
73	C₃×C₄	#123	1	1	yes	characteristic
74	C₃×C₄	#124	1	1	yes	characteristic
75	C₃×C₄	#125	1	2	yes	—
76	Dic₃	—	8	2	no	—
77	C₂³	#235	1	1	yes	characteristic
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	2	1	no	—
80	C₂×C₄	—	4	2	no	—
81	C₂×C₄	—	12	2	no	—
82	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
83	C₂×C₄	#237	1	2	yes	—
84	C₂×C₄	#239	1	2	yes	—
85	C₂×C₄	C₂×#67	1	1	yes	characteristic
86	C₈	—	2	2	no	—
87	C₈	—	4	1	no	—
88	C₈	—	12	2	no	—
89	C₂×C₃	—	2	2	no	—
90	C₂×C₃	—	2	1	no	—
91	C₂×C₃	#400	1	1	yes	characteristic
92	C₂²	—	4	1	no	—
93	C₂²	#898	1	2	yes	—
94	C₂²	#924	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	4	1	no	—
97	C₄	—	24	2	no	—
98	C₄	#890	1	2	yes	—
99	C₄	#935	1	1	yes	characteristic
100	C₄	#947	1	1	yes	characteristic
101	C₃	#2102	1	1	yes	characteristic, complemented, Sylow
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#4986	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#423

C₂

yes

maximal, characteristic

#5125

C₂

yes

maximal, characteristic

#4559

C₂

yes

maximal

#4557

C₂

yes

maximal, characteristic

#4523

C₂

yes

maximal

#2102

—

maximal, Sylow

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

#926

C₂²

yes

—

C₂×#208

C₂²

yes

—

C₃×C₄⋊C₈

—

#819

—

#430

—

#423

D₃

yes

—

—

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

—

C₃×C₄²

Q₈

yes

characteristic

C₃×C₄²

Q₈

yes

characteristic

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

characteristic

C₂×M₁₆

C₂×D₃

yes

characteristic

#117

—

C₂×C₄²

C₂×D₃

yes

characteristic

#108

—

C₄⋊C₈

—

C₂³×C₃

#41

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#67

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

Dic₆

yes

characteristic

C₄²

Dic₆

yes

characteristic

C₂²×C₃

—

C₂²×C₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₃

#126

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

yes

C₃×C₄

yes

C₃×C₄

yes

—

Dic₃

—

C₂³

#235

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₆

yes

characteristic

C₂×C₄

#237

yes

—

C₂×C₄

#239

yes

—

C₂×C₄

C₂×#67

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

#400

yes

characteristic

C₂²

—

C₂²

#898

yes

—

C₂²

#924

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#890

yes

—

C₄

#935

yes

characteristic

100

C₄

#947

yes

characteristic

101

C₃

#2102

yes

characteristic, complemented, Sylow

102

C₂

—

103

C₂

—

104

C₂

#4986

yes

characteristic

	subgroup	G/H
center	C₄	#947
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#423	C₂
socle	C₂×C₃	#400
prime radical	C₂³×C₃	#41
hypercenter	#423	D₃
hypocenter	C₃	#2102
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

subgroup

G/H

center

C₄

#947

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #423

N	H ≅ G/N	type
101. C₃	#2102	semidirect

H ≅ G/N

type

101. C₃

#2102

semidirect

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

subgroup	index
C₃×#423	2
#5125	2
#4559	2
#4557	2
#4523	2
#2102	3

subgroup

index

Subgroups — #70526

Conjugacy classes of subgroups

Complements

Maximal subgroups