Group #67853 — subgroups

Subgroups — C₃×#1293 #67853

Order 384.

184 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 68 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×#407	C₂	1	2	yes	maximal
2	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#1293	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
4	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
5	C₂×C₃×M₁₆	—	2	2	no	—
6	C₂²×C₃×C₈	C₄	1	2	yes	—
7	C₃×C₂²⋊C₈	—	2	2	no	—
8	C₃×C₂²⋊C₈	—	2	2	no	—
9	C₃×C₂²⋊C₈	C₄	1	4	yes	—
10	#407	C₂×C₃	1	2	yes	—
11	C₂×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
12	C₂⁴×C₃	Q₈	1	1	yes	characteristic
13	C₂²×C₃×C₄	—	4	2	no	—
14	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
16	C₂²×C₃×C₄	D₄	1	2	yes	—
17	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
18	C₃×M₁₆	—	4	4	no	—
19	C₂×C₃×C₈	—	2	2	no	—
20	C₂×C₃×C₈	—	2	4	no	—
21	C₂×C₃×C₈	—	4	2	no	—
22	C₂³×C₄	C₂²×C₃	1	1	yes	characteristic
23	C₂×M₁₆	—	2	2	no	—
24	C₂²×C₈	C₃×C₄	1	2	yes	—
25	C₂²⋊C₈	—	2	2	no	—
26	C₂²⋊C₈	—	2	2	no	—
27	C₂²⋊C₈	C₃×C₄	1	4	yes	—
28	C₂³×C₃	—	2	2	no	—
29	C₂³×C₃	—	4	1	no	—
30	C₂³×C₃	—	4	1	no	—
31	C₂³×C₃	C₄⋊C₄	1	2	yes	—
32	C₂³×C₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₃×C₄	—	2	2	no	—
34	C₂×C₃×C₄	—	2	4	no	—
35	C₂×C₃×C₄	—	4	2	no	—
36	C₂×C₃×C₄	—	4	1	no	—
37	C₂×C₃×C₄	C₄²	1	1	yes	characteristic
38	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
39	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
40	C₃×C₈	—	4	4	no	—
41	C₃×C₈	—	4	2	no	—
42	C₂⁴	C₃×Q₈	1	1	yes	characteristic
43	C₂²×C₄	—	4	2	no	—
44	C₂²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
45	C₂²×C₄	C₂×C₃×C₄	1	2	yes	—
46	C₂²×C₄	C₃×D₄	1	2	yes	—
47	C₂²×C₄	C₃×D₄	1	1	yes	characteristic
48	M₁₆	—	4	4	no	—
49	C₂×C₈	—	2	2	no	—
50	C₂×C₈	—	2	4	no	—
51	C₂×C₈	—	4	2	no	—
52	C₂²×C₃	—	2	1	no	—
53	C₂²×C₃	—	2	1	no	—
54	C₂²×C₃	—	2	2	no	—
55	C₂²×C₃	—	2	2	no	—
56	C₂²×C₃	—	2	2	no	—
57	C₂²×C₃	—	4	4	no	—
58	C₂²×C₃	#95	1	1	yes	characteristic
59	C₂²×C₃	#108	1	2	yes	—
60	C₃×C₄	—	2	2	no	—
61	C₃×C₄	—	2	1	no	—
62	C₃×C₄	#99	1	2	yes	—
63	C₂³	—	2	2	no	—
64	C₂³	—	4	1	no	—
65	C₂³	—	4	1	no	—
66	C₂³	C₃×C₄⋊C₄	1	2	yes	—
67	C₂³	C₃×C₄⋊C₄	1	1	yes	characteristic
68	C₂×C₄	—	2	2	no	—
69	C₂×C₄	—	2	4	no	—
70	C₂×C₄	—	4	2	no	—
71	C₂×C₄	—	4	1	no	—
72	C₂×C₄	C₃×C₄²	1	1	yes	characteristic
73	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
74	C₂×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
75	C₈	—	4	4	no	—
76	C₈	—	4	2	no	—
77	C₂×C₃	—	2	2	no	—
78	C₂×C₃	—	4	2	no	—
79	C₂×C₃	#338	1	1	yes	characteristic
80	C₂×C₃	#341	1	1	yes	characteristic
81	C₂×C₃	#342	1	1	yes	characteristic
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	2	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	4	4	no	—
88	C₂²	C₃×#95	1	1	yes	characteristic
89	C₂²	C₃×#108	1	2	yes	—
90	C₄	—	2	2	no	—
91	C₄	—	2	1	no	—
92	C₄	C₃×#99	1	2	yes	—
93	C₃	#1293	1	1	yes	characteristic, direct product factor, Sylow
94	C₂	—	2	2	no	—
95	C₂	—	4	2	no	—
96	C₂	C₃×#338	1	1	yes	characteristic
97	C₂	C₃×#341	1	1	yes	characteristic
98	C₂	C₃×#342	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#407

C₂

yes

maximal

C₂×C₃×C₂²⋊C₈

C₂

yes

maximal, characteristic

#1293

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃×C₄

C₂²

yes

characteristic

C₂×C₃×M₁₆

—

C₂²×C₃×C₈

C₄

yes

—

C₃×C₂²⋊C₈

—

C₃×C₂²⋊C₈

—

C₃×C₂²⋊C₈

C₄

yes

—

#407

C₂×C₃

yes

—

C₂×C₂²⋊C₈

C₂×C₃

yes

characteristic

C₂⁴×C₃

Q₈

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

D₄

yes

characteristic

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂³×C₄

C₂²×C₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

C₃×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₃×C₄

yes

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₄⋊C₄

yes

—

C₂³×C₃

C₄⋊C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄²

yes

characteristic

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₂⁴

C₃×Q₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₂×C₃×C₄

yes

characteristic

C₂²×C₄

C₂×C₃×C₄

yes

—

C₂²×C₄

C₃×D₄

yes

—

C₂²×C₄

C₃×D₄

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

#95

yes

characteristic

C₂²×C₃

#108

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#99

yes

—

C₂³

—

C₂³

—

C₂³

—

C₂³

C₃×C₄⋊C₄

yes

—

C₂³

C₃×C₄⋊C₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₃×C₄²

yes

characteristic

C₂×C₄

C₃×C₂²⋊C₄

yes

—

C₂×C₄

C₃×C₂²⋊C₄

yes

characteristic

C₈

—

C₈

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₃×#95

yes

characteristic

C₂²

C₃×#108

yes

—

C₄

—

C₄

—

C₄

C₃×#99

yes

—

C₃

#1293

yes

characteristic, direct product factor, Sylow

C₂

—

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center, socle	C₂²×C₃	#95
commutator	C₂×C₄	C₃×C₄²
Frattini, cosocle	C₂³×C₄	C₂²×C₃
prime radical	C₂⁴×C₃	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

subgroup

G/H

center, socle

C₂²×C₃

#95

commutator

C₂×C₄

C₃×C₄²

Frattini, cosocle

C₂³×C₄

C₂²×C₃

prime radical

C₂⁴×C₃

Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃ ◁ C₂³×C₃×C₄ ◁ G

N	H ≅ G/N	type
3. #1293	C₃	direct (N × H)
93. C₃	#1293	direct (N × H)

H ≅ G/N

type

3. #1293

C₃

direct (N × H)

93. C₃

#1293

direct (N × H)

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

subgroup	index
C₃×#407	2
C₂×C₃×C₂²⋊C₈	2
#1293	3

subgroup

index

C₃×#407

C₂×C₃×C₂²⋊C₈

#1293

Subgroups — C₃×#1293 #67853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#1293 #67853

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1293 #67853