Group #70533 — subgroups

Subgroups — #70533

Order 384.

176 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 69 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×#424	C₂	1	1	yes	maximal, characteristic
2	C₂×#817	C₂	1	1	yes	maximal, characteristic
3	#4995	C₂	1	2	yes	maximal
4	#4554	C₂	1	2	yes	maximal
5	#4552	C₂	1	1	yes	maximal, characteristic
6	#2090	—	3	1	no	maximal, Sylow
7	C₂×C₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₃×#117	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×#208	C₂²	1	2	yes	—
12	C₃×C₄⋊C₈	—	2	2	no	—
13	#817	—	2	2	no	—
14	#808	—	2	2	no	—
15	#807	—	2	2	no	—
16	#428	—	3	2	no	—
17	C₂×#106	—	3	1	no	—
18	#424	D₃	1	1	yes	characteristic
19	#343	—	3	2	no	—
20	#341	—	3	1	no	—
21	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
22	C₂²×Dic₃	—	2	2	no	—
23	C₃×M₁₆	—	2	2	no	—
24	C₂×C₃×C₈	—	2	2	no	—
25	C₂×C₃×C₈	—	2	1	no	—
26	C₂×C₃×C₈	D₄	1	2	yes	—
27	C₃×C₄⋊C₄	D₄	1	2	yes	—
28	C₃×C₂²⋊C₄	—	2	1	no	—
29	C₃×C₄²	—	2	1	no	—
30	C₄⋊Dic₃	—	2	2	no	—
31	C₄⋊Dic₃	—	2	2	no	—
32	#208	—	2	2	no	—
33	#208	Q₈	1	4	yes	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂×M₁₆	—	3	2	no	—
36	C₂×M₁₆	C₂×D₃	1	1	yes	characteristic
37	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
38	#117	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄⋊C₄	—	3	2	no	—
40	#107	—	6	2	no	—
41	#106	—	6	2	no	—
42	#106	—	6	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₂³×C₃	#41	1	1	yes	characteristic
45	C₂×C₃×C₄	—	2	2	no	—
46	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
49	C₂×C₃×C₄	#41	1	2	yes	—
50	C₂×Dic₃	—	4	2	no	—
51	C₂×Dic₃	—	8	2	no	—
52	C₃×C₈	—	2	2	no	—
53	C₃×C₈	—	4	1	no	—
54	C₃⋊C₈	—	2	4	no	—
55	C₂²×C₄	—	6	2	no	—
56	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
57	M₁₆	—	2	2	no	—
58	M₁₆	—	3	4	no	—
59	M₁₆	—	6	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	6	2	no	—
63	C₂×C₈	#67	1	2	yes	—
64	C₄⋊C₄	—	6	2	no	—
65	C₄⋊C₄	—	6	2	no	—
66	C₄⋊C₄	D₁₂	1	2	yes	—
67	C₂²⋊C₄	—	2	1	no	—
68	C₄²	—	2	1	no	—
69	C₂²×C₃	—	2	2	no	—
70	C₂²×C₃	#121	1	1	yes	characteristic
71	C₂²×C₃	#125	1	2	yes	—
72	C₃×C₄	—	4	2	no	—
73	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
74	C₃×C₄	#123	1	1	yes	characteristic
75	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
76	Dic₃	—	8	2	no	—
77	C₂³	#235	1	1	yes	characteristic
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	12	2	no	—
80	C₂×C₄	—	24	2	no	—
81	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
82	C₂×C₄	#237	1	2	yes	—
83	C₂×C₄	D₃×Q₈	1	2	yes	—
84	C₂×C₄	C₂×#67	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	4	1	no	—
87	C₈	—	6	4	no	—
88	C₂×C₃	—	2	2	no	—
89	C₂×C₃	#398	1	1	yes	characteristic
90	C₂×C₃	#471	1	2	yes	—
91	C₂²	—	2	2	no	—
92	C₂²	#889	1	2	yes	—
93	C₂²	#930	1	1	yes	characteristic
94	C₄	—	4	2	no	—
95	C₄	—	24	2	no	—
96	C₄	#897	1	2	yes	—
97	C₄	#935	1	1	yes	characteristic
98	C₄	#944	1	1	yes	characteristic
99	C₃	#2090	1	1	yes	characteristic, complemented, Sylow
100	C₂	—	2	2	no	—
101	C₂	#4981	1	1	yes	characteristic
102	C₂	#5141	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#424

C₂

yes

maximal, characteristic

C₂×#817

C₂

yes

maximal, characteristic

#4995

C₂

yes

maximal

#4554

C₂

yes

maximal

#4552

C₂

yes

maximal, characteristic

#2090

—

maximal, Sylow

C₂×C₃×M₁₆

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

characteristic

C₃×#117

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×#208

C₂²

yes

—

C₃×C₄⋊C₈

—

—

—

—

—

—

D₃

yes

characteristic

#343

—

#341

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

D₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

#208

—

#208

Q₈

yes

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃

yes

characteristic

C₂²×C₈

C₂×D₃

yes

characteristic

#117

C₂×D₃

yes

characteristic

C₂×C₄⋊C₄

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₂³×C₃

#41

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#67

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₂

yes

—

C₂²⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

#121

yes

characteristic

C₂²×C₃

#125

yes

—

C₃×C₄

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₃×C₄

#123

yes

characteristic

C₃×C₄

C₄⋊Q₈

yes

characteristic

Dic₃

—

C₂³

#235

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

#237

yes

—

C₂×C₄

D₃×Q₈

yes

—

C₂×C₄

C₂×#67

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

#398

yes

characteristic

C₂×C₃

#471

yes

—

C₂²

—

C₂²

#889

yes

—

C₂²

#930

yes

characteristic

C₄

—

C₄

—

C₄

#897

yes

—

C₄

yes

C₄

yes

C₃

yes

characteristic, complemented, Sylow

100

C₂

—

101

C₂

#4981

yes

characteristic

102

C₂

#5141

yes

—

	subgroup	G/H
center	C₂²	#930
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#424	C₂
socle	C₂²×C₃	#121
prime radical	C₂³×C₃	#41
hypercenter	#424	D₃
hypocenter	C₃	#2090
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

subgroup

G/H

center

C₂²

#930

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #424

N	H ≅ G/N	type
99. C₃	#2090	semidirect

H ≅ G/N

type

99. C₃

#2090

semidirect

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

subgroup	index
C₃×#424	2
C₂×#817	2
#4995	2
#4554	2
#4552	2
#2090	3

subgroup

index

Subgroups — #70533

Conjugacy classes of subgroups

Complements

Maximal subgroups