Group #70573 — subgroups

Subgroups — #70573

Order 384.

176 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 67 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×#106	C₂	1	1	yes	maximal, characteristic
2	#5102	C₂	1	1	yes	maximal, characteristic
3	#4964	C₂	1	2	yes	maximal
4	#4552	C₂	1	1	yes	maximal, characteristic
5	#4530	C₂	1	2	yes	maximal
6	#2090	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
9	#926	C₂²	1	1	yes	characteristic
10	C₂×#208	C₂²	1	1	yes	characteristic
11	C₂×#208	C₂²	1	2	yes	—
12	C₃×#106	—	2	2	no	—
13	Dic₃⋊C₈	—	2	2	no	—
14	#808	—	2	2	no	—
15	#807	—	2	2	no	—
16	#428	—	3	2	no	—
17	C₂×#106	D₃	1	1	yes	characteristic
18	#424	—	3	1	no	—
19	#343	—	3	2	no	—
20	#341	—	3	1	no	—
21	C₂²×C₃×C₄	—	2	2	no	—
22	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂×C₃×C₈	—	2	2	no	—
24	C₂×C₃×C₈	D₄	1	2	yes	—
25	C₃×C₄⋊C₄	—	2	2	no	—
26	C₃×C₄⋊C₄	—	2	2	no	—
27	#217	—	2	1	no	—
28	C₄⋊Dic₃	D₄	1	2	yes	—
29	C₄×Dic₃	—	2	1	no	—
30	#208	—	2	2	no	—
31	#208	—	2	2	no	—
32	#208	Q₈	1	4	yes	—
33	C₂×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃⋊C₈	—	2	1	no	—
35	C₂×M₁₆	—	3	2	no	—
36	C₂×M₁₆	—	3	1	no	—
37	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
38	#117	—	3	1	no	—
39	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
40	#107	—	6	2	no	—
41	#106	—	2	2	no	—
42	#106	—	6	2	no	—
43	C₄⋊C₈	—	6	2	no	—
44	C₂³×C₃	#41	1	1	yes	characteristic
45	C₂×C₃×C₄	—	4	2	no	—
46	C₂×C₃×C₄	—	8	2	no	—
47	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
50	C₂×C₃×C₄	#41	1	2	yes	—
51	C₂×Dic₃	—	2	2	no	—
52	C₃×C₈	—	2	2	no	—
53	C₃⋊C₈	—	2	4	no	—
54	C₃⋊C₈	—	4	1	no	—
55	C₂²×C₄	—	2	2	no	—
56	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
57	M₁₆	—	3	4	no	—
58	M₁₆	—	6	2	no	—
59	M₁₆	—	6	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	6	2	no	—
62	C₂×C₈	—	6	1	no	—
63	C₂×C₈	#67	1	2	yes	—
64	C₄⋊C₄	—	2	2	no	—
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	3	2	no	—
67	C₂²⋊C₄	—	6	1	no	—
68	C₄²	—	6	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	—	8	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	—	4	2	no	—
77	C₂³	#237	1	1	yes	characteristic
78	C₂×C₄	—	4	2	no	—
79	C₂×C₄	—	6	2	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	#235	1	2	yes	—
82	C₂×C₄	D₃×D₄	1	1	yes	characteristic
83	C₂×C₄	D₃×Q₈	1	2	yes	—
84	C₂×C₄	C₂×#67	1	1	yes	characteristic
85	C₈	—	2	2	no	—
86	C₈	—	6	4	no	—
87	C₈	—	12	1	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	#938	1	1	yes	characteristic
94	C₄	—	8	2	no	—
95	C₄	—	12	2	no	—
96	C₄	#896	1	2	yes	—
97	C₄	#929	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	#4983	1	1	yes	characteristic
102	C₂	#5177	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#106

C₂

yes

maximal, characteristic

#5102

C₂

yes

maximal, characteristic

#4964

C₂

yes

maximal

#4552

C₂

yes

maximal, characteristic

#4530

C₂

yes

maximal

#2090

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²

yes

—

#926

C₂²

yes

characteristic

C₂×#208

C₂²

yes

characteristic

C₂×#208

C₂²

yes

—

C₃×#106

—

Dic₃⋊C₈

—

—

—

—

D₃

yes

—

—

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

#217

—

C₄⋊Dic₃

D₄

yes

—

C₄×Dic₃

—

#208

—

#208

—

#208

Q₈

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₂³×C₃

#41

yes

characteristic

C₂×C₃×C₄

—

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

—

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

—

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

#237

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#235

yes

—

C₂×C₄

D₃×D₄

yes

characteristic

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

#938

yes

characteristic

C₄

—

C₄

—

C₄

#896

yes

—

C₄

yes

C₄

yes

C₃

yes

characteristic, complemented, Sylow

100

C₂

—

101

C₂

#4983

yes

characteristic

102

C₂

#5177

yes

—

	subgroup	G/H
center	C₂²	#938
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#106	C₂
socle	C₂²×C₃	#121
prime radical	C₂³×C₃	#41
hypercenter	C₂×#106	D₃
hypocenter	C₃	#2090
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#938

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×#106

C₂

C₂²×C₃

C₂³×C₃

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

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
C₂×C₃×#106	2
#5102	2
#4964	2
#4552	2
#4530	2
#2090	3

subgroup

index

C₂×C₃×#106

Subgroups — #70573

Conjugacy classes of subgroups

Complements

Maximal subgroups