Group #72709 — subgroups

Subgroups — C₃×#1994 #72709

Order 384.

284 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 120 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×#533	C₂	1	1	yes	maximal, characteristic
2	C₃×#423	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
4	C₃×#339	C₂	1	4	yes	maximal
5	#1994	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
6	C₂×C₃×M₁₆	C₂²	1	2	yes	—
7	C₃×C₄⋊Q₈	C₄	1	2	yes	complemented
8	C₃×#125	C₄	1	2	yes	complemented
9	C₃×C₂²⋊Q₈	—	2	2	no	—
10	C₃×C₄×Q₈	—	2	2	no	—
11	C₃×#117	C₂²	1	2	yes	—
12	C₂×C₃×C₄²	—	2	2	no	—
13	C₂×C₃×C₄²	C₂²	1	2	yes	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₃×C₄⋊C₈	—	2	2	no	—
16	#533	C₂×C₃	1	1	yes	characteristic
17	#423	C₂×C₃	1	1	yes	characteristic
18	C₄³	C₂×C₃	1	1	yes	characteristic
19	#339	C₂×C₃	1	4	yes	—
20	C₂×C₃×Q₈	—	2	2	no	—
21	C₂²×C₃×C₄	—	2	2	no	—
22	C₂²×C₃×C₄	D₄	1	2	yes	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₃×M₁₆	—	2	4	no	—
25	C₂×C₃×C₈	—	2	2	no	—
26	C₃×C₄⋊C₄	—	2	4	no	—
27	C₃×C₄⋊C₄	—	2	2	no	—
28	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
29	C₃×C₂²⋊C₄	—	2	2	no	—
30	C₃×C₄²	—	2	8	no	—
31	C₃×C₄²	—	2	2	no	—
32	C₃×C₄²	—	2	2	no	—
33	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
34	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
35	C₃×C₄²	D₄	1	2	yes	—
36	C₃×C₄²	Q₈	1	4	yes	complemented
37	C₂×M₁₆	C₂²×C₃	1	2	yes	—
38	C₄⋊Q₈	C₃×C₄	1	2	yes	complemented
39	#125	C₃×C₄	1	2	yes	complemented
40	C₂²⋊Q₈	—	2	2	no	—
41	C₄×Q₈	—	2	2	no	—
42	#117	C₂²×C₃	1	2	yes	—
43	C₂×C₄²	—	2	2	no	—
44	C₂×C₄²	C₂²×C₃	1	2	yes	—
45	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
46	C₄⋊C₈	—	2	2	no	—
47	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
48	C₃×Q₈	—	4	4	no	—
49	C₂×C₃×C₄	—	2	8	no	—
50	C₂×C₃×C₄	—	2	4	no	—
51	C₂×C₃×C₄	—	2	4	no	—
52	C₂×C₃×C₄	—	2	4	no	—
53	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
54	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
58	C₂×C₃×C₄	#41	1	2	yes	—
59	C₃×C₈	—	4	2	no	—
60	C₂×Q₈	—	2	2	no	—
61	C₂²×C₄	—	2	2	no	—
62	C₂²×C₄	C₃×D₄	1	2	yes	—
63	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
64	M₁₆	—	2	4	no	—
65	C₂×C₈	—	2	2	no	—
66	C₄⋊C₄	—	2	4	no	—
67	C₄⋊C₄	—	2	2	no	—
68	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
69	C₂²⋊C₄	—	2	2	no	—
70	C₄²	—	2	8	no	—
71	C₄²	—	2	2	no	—
72	C₄²	—	2	2	no	—
73	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
74	C₄²	C₂×C₃×C₄	1	1	yes	characteristic
75	C₄²	C₃×D₄	1	2	yes	—
76	C₄²	C₃×Q₈	1	4	yes	complemented
77	C₂²×C₃	—	2	2	no	—
78	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
80	C₃×C₄	—	2	8	no	—
81	C₃×C₄	—	2	2	no	—
82	C₃×C₄	—	4	4	no	—
83	C₃×C₄	#104	1	4	yes	—
84	C₃×C₄	C₄×Q₈	1	2	yes	—
85	C₃×C₄	#124	1	1	yes	characteristic
86	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
87	C₂³	C₂×C₃×D₄	1	1	yes	characteristic
88	Q₈	—	4	4	no	—
89	C₂×C₄	—	2	8	no	—
90	C₂×C₄	—	2	4	no	—
91	C₂×C₄	—	2	4	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
94	C₂×C₄	C₃×C₂²⋊C₄	1	2	yes	—
95	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
96	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
97	C₂×C₄	C₂×C₃×Q₈	1	2	yes	—
98	C₂×C₄	C₃×#41	1	2	yes	—
99	C₈	—	4	2	no	—
100	C₂×C₃	—	2	2	no	—
101	C₂×C₃	#391	1	1	yes	characteristic
102	C₂×C₃	C₂×#104	1	2	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
105	C₂²	C₃×C₂²⋊Q₈	1	2	yes	—
106	C₄	—	2	8	no	—
107	C₄	—	2	2	no	—
108	C₄	—	4	4	no	—
109	C₄	C₃×#104	1	4	yes	—
110	C₄	C₃×C₄×Q₈	1	2	yes	—
111	C₄	C₃×#124	1	1	yes	characteristic
112	C₄	C₃×C₄⋊Q₈	1	1	yes	characteristic
113	C₃	#1994	1	1	yes	characteristic, direct product factor, Sylow
114	C₂	—	2	2	no	—
115	C₂	C₃×#391	1	1	yes	characteristic
116	C₂	C₂×C₃×#104	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#533

C₂

yes

maximal, characteristic

C₃×#423

C₂

yes

maximal, characteristic

C₃×C₄³

C₂

yes

maximal, characteristic

C₃×#339

C₂

yes

maximal

#1994

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃×M₁₆

C₂²

yes

—

C₃×C₄⋊Q₈

C₄

yes

complemented

C₃×#125

C₄

yes

complemented

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

C₂²

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×C₄⋊C₈

—

#533

C₂×C₃

yes

characteristic

#423

C₂×C₃

yes

characteristic

C₄³

C₂×C₃

yes

characteristic

#339

C₂×C₃

yes

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

—

C₃×C₄²

Q₈

yes

complemented

C₂×M₁₆

C₂²×C₃

yes

—

C₄⋊Q₈

C₃×C₄

yes

complemented

#125

C₃×C₄

yes

complemented

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₂²×C₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

C₄⋊C₈

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₃×C₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₃×D₄

yes

—

C₂²×C₄

C₂³×C₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₃×C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×C₃×C₄

yes

characteristic

C₄²

C₂×C₃×C₄

yes

characteristic

C₄²

C₃×D₄

yes

—

C₄²

C₃×Q₈

yes

complemented

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#104

yes

—

C₃×C₄

C₄×Q₈

yes

—

C₃×C₄

#124

yes

characteristic

C₃×C₄

C₄⋊Q₈

yes

characteristic

C₂³

C₂×C₃×D₄

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₃×C₂²⋊C₄

yes

—

C₂×C₄

C₃×C₂²⋊C₄

yes

—

C₂×C₄

C₂²×C₃×C₄

yes

characteristic

C₂×C₄

C₂×C₃×D₄

yes

characteristic

C₂×C₄

C₂×C₃×Q₈

yes

—

C₂×C₄

C₃×#41

yes

—

C₈

—

100

C₂×C₃

—

101

C₂×C₃

#391

yes

characteristic

102

C₂×C₃

C₂×#104

yes

—

103

C₂²

—

104

C₂²

C₂×C₃×C₂²⋊C₄

yes

characteristic

105

C₂²

C₃×C₂²⋊Q₈

yes

—

106

C₄

—

107

C₄

—

108

C₄

—

109

C₄

C₃×#104

yes

—

110

C₄

C₃×C₄×Q₈

yes

—

111

C₄

C₃×#124

yes

characteristic

112

C₄

C₃×C₄⋊Q₈

yes

characteristic

113

C₃

#1994

yes

characteristic, direct product factor, Sylow

114

C₂

—

115

C₂

C₃×#391

yes

characteristic

116

C₂

C₂×C₃×#104

yes

—

	subgroup	G/H
center	C₂×C₃×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₃	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

subgroup

G/H

center

C₂×C₃×C₄

C₂×D₄

commutator

C₂×C₄

C₂²×C₃×C₄

Frattini, cosocle

C₂²×C₄

C₂³×C₃

socle

C₂²×C₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₃

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ G

N	H ≅ G/N	type
5. #1994	C₃	direct (N × H)
7. C₃×C₄⋊Q₈	C₄	semidirect
8. C₃×#125	C₄	semidirect
36. C₃×C₄²	Q₈	semidirect
38. C₄⋊Q₈	C₃×C₄	semidirect
39. #125	C₃×C₄	semidirect
76. C₄²	C₃×Q₈	semidirect
113. C₃	#1994	direct (N × H)

H ≅ G/N

type

5. #1994

C₃

direct (N × H)

7. C₃×C₄⋊Q₈

C₄

semidirect

8. C₃×#125

C₄

semidirect

36. C₃×C₄²

Q₈

semidirect

38. C₄⋊Q₈

C₃×C₄

semidirect

39. #125

C₃×C₄

semidirect

76. C₄²

C₃×Q₈

semidirect

113. C₃

#1994

direct (N × H)

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

subgroup	index
C₃×#533	2
C₃×#423	2
C₃×C₄³	2
C₃×#339	2
#1994	3

subgroup

index

C₃×C₄³

Subgroups — C₃×#1994 #72709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#1994 #72709

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1994 #72709