Group #44480331 — subgroups

Subgroups — #1561×C₂₃₉ #44480331

Order 30592.

210 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 96 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	#533×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	#423×C₂₃₉	C₂	1	2	yes	maximal
3	#332×C₂₃₉	C₂	1	4	yes	maximal, complemented
4	C₂×M₁₆×C₂₃₉	—	2	2	no	—
5	C₄⋊Q₈×C₂₃₉	C₂²	1	2	yes	—
6	#125×C₂₃₉	C₄	1	2	yes	—
7	C₂²⋊Q₈×C₂₃₉	—	2	2	no	—
8	C₄×Q₈×C₂₃₉	—	2	2	no	—
9	#117×C₂₃₉	C₄	1	2	yes	—
10	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₂₃₉	—	2	2	no	—
12	C₄⋊C₈×C₂₃₉	C₂²	1	4	yes	—
13	C₂×Q₈×C₂₃₉	—	2	2	no	—
14	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃₉	D₄	1	2	yes	—
16	M₁₆×C₂₃₉	—	4	4	no	—
17	C₂×C₈×C₂₃₉	—	2	4	no	—
18	C₄⋊C₄×C₂₃₉	—	2	4	no	—
19	C₄⋊C₄×C₂₃₉	—	2	2	no	—
20	C₄⋊C₄×C₂₃₉	C₂×C₄	1	4	yes	—
21	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
22	C₄²×C₂₃₉	—	2	2	no	—
23	C₄²×C₂₃₉	C₂×C₄	1	1	yes	characteristic
24	C₄²×C₂₃₉	D₄	1	2	yes	—
25	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
26	C₂³×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₂₃₉	—	4	2	no	—
28	Q₈×C₂₃₉	—	4	2	no	—
29	C₂×C₄×C₂₃₉	—	2	2	no	—
30	C₂×C₄×C₂₃₉	—	2	2	no	—
31	C₂×C₄×C₂₃₉	—	2	2	no	—
32	C₂×C₄×C₂₃₉	—	2	4	no	—
33	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
37	C₈×C₂₃₉	—	4	4	no	—
38	C₂²×C₂₃₉	—	2	1	no	—
39	C₂²×C₂₃₉	—	2	2	no	—
40	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₂₃₉	—	2	1	no	—
42	C₄×C₂₃₉	—	2	2	no	—
43	C₄×C₂₃₉	—	4	4	no	—
44	C₄×C₂₃₉	#101	1	2	yes	—
45	C₄×C₂₃₉	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₂₃₉	#137	1	2	yes	—
47	C₂×C₂₃₉	—	4	1	no	—
48	C₂×C₂₃₉	C₂×#101	1	1	yes	characteristic
49	C₂×C₂₃₉	#417	1	2	yes	—
50	C₂₃₉	#1561	1	1	yes	characteristic, direct product factor, Sylow
51	#1561	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	#533	C₂×C₂₃₉	1	1	yes	characteristic
53	#423	C₂×C₂₃₉	1	2	yes	—
54	#332	C₂×C₂₃₉	1	4	yes	complemented
55	C₂×M₁₆	—	2	2	no	—
56	C₄⋊Q₈	C₂²×C₂₃₉	1	2	yes	—
57	#125	C₄×C₂₃₉	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	C₄×C₂₃₉	1	2	yes	—
61	C₂×C₄²	C₂²×C₂₃₉	1	1	yes	characteristic
62	C₄⋊C₈	—	2	2	no	—
63	C₄⋊C₈	C₂²×C₂₃₉	1	4	yes	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×C₄×C₂₃₉	1	1	yes	characteristic
66	C₂²×C₄	D₄×C₂₃₉	1	2	yes	—
67	M₁₆	—	4	4	no	—
68	C₂×C₈	—	2	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	C₂×C₄×C₂₃₉	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×C₄×C₂₃₉	1	1	yes	characteristic
75	C₄²	D₄×C₂₃₉	1	2	yes	—
76	C₄²	C₂³×C₂₃₉	1	1	yes	characteristic
77	C₂³	C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊C₄×C₂₃₉	1	2	yes	—
85	C₂×C₄	C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
86	C₂×C₄	C₂²×C₄×C₂₃₉	1	1	yes	characteristic
87	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
88	C₈	—	4	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#101×C₂₃₉	1	2	yes	—
96	C₄	AGL(1,ℤ/8)×C₂₃₉	1	2	yes	—
97	C₄	#137×C₂₃₉	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#101×C₂₃₉	1	1	yes	characteristic
100	C₂	#417×C₂₃₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#533×C₂₃₉

C₂

yes

maximal, characteristic

#423×C₂₃₉

C₂

yes

maximal

#332×C₂₃₉

C₂

yes

maximal, complemented

C₂×M₁₆×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

C₂²

yes

—

#125×C₂₃₉

C₄

yes

—

C₂²⋊Q₈×C₂₃₉

—

C₄×Q₈×C₂₃₉

—

#117×C₂₃₉

C₄

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₃₉

—

C₄⋊C₈×C₂₃₉

C₂²

yes

—

C₂×Q₈×C₂₃₉

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

D₄

yes

—

M₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

C₂×C₄

yes

characteristic

C₄²×C₂₃₉

D₄

yes

—

C₄²×C₂₃₉

C₂³

yes

characteristic

C₂³×C₂₃₉

C₂²⋊C₄

yes

characteristic

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#101

yes

—

C₄×C₂₃₉

AGL(1,ℤ/8)

yes

—

C₄×C₂₃₉

#137

yes

—

C₂×C₂₃₉

—

C₂×C₂₃₉

C₂×#101

yes

characteristic

C₂×C₂₃₉

#417

yes

—

C₂₃₉

#1561

yes

characteristic, direct product factor, Sylow

#1561

C₂₃₉

yes

maximal, characteristic, direct product factor, Sylow

#533

C₂×C₂₃₉

yes

characteristic

#423

C₂×C₂₃₉

yes

—

#332

C₂×C₂₃₉

yes

complemented

C₂×M₁₆

—

C₄⋊Q₈

C₂²×C₂₃₉

yes

—

#125

C₄×C₂₃₉

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

C₄×C₂₃₉

yes

—

C₂×C₄²

C₂²×C₂₃₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₃₉

yes

—

C₂×Q₈

—

C₂²×C₄

C₂×C₄×C₂₃₉

yes

characteristic

C₂²×C₄

D₄×C₂₃₉

yes

—

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₂₃₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×C₄×C₂₃₉

yes

characteristic

C₄²

D₄×C₂₃₉

yes

—

C₄²

C₂³×C₂₃₉

yes

characteristic

C₂³

C₂²⋊C₄×C₂₃₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊C₄×C₂₃₉

yes

—

C₂×C₄

C₂²⋊C₄×C₂₃₉

yes

characteristic

C₂×C₄

C₂²×C₄×C₂₃₉

yes

characteristic

C₂×C₄

C₂×D₄×C₂₃₉

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊C₄×C₂₃₉

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#101×C₂₃₉

yes

—

C₄

AGL(1,ℤ/8)×C₂₃₉

yes

—

C₄

#137×C₂₃₉

yes

—

C₂

—

C₂

C₂×#101×C₂₃₉

yes

characteristic

100

C₂

#417×C₂₃₉

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₂₃₉
Frattini, cosocle	C₄²	C₂³×C₂₃₉
prime radical	C₂³×C₂₃₉	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ C₂×C₄²×C₂₃₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃₉

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₂₃₉

Frattini, cosocle

C₄²

C₂³×C₂₃₉

prime radical

C₂³×C₂₃₉

C₂²⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃₉ ◁ C₂×C₄²×C₂₃₉ ◁ G

N	H ≅ G/N	type
3. #332×C₂₃₉	C₂	semidirect
50. C₂₃₉	#1561	direct (N × H)
51. #1561	C₂₃₉	direct (N × H)
54. #332	C₂×C₂₃₉	semidirect

H ≅ G/N

type

3. #332×C₂₃₉

C₂

semidirect

50. C₂₃₉

#1561

direct (N × H)

51. #1561

C₂₃₉

direct (N × H)

54. #332

C₂×C₂₃₉

semidirect

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

subgroup	index
#533×C₂₃₉	2
#423×C₂₃₉	2
#332×C₂₃₉	2
#1561	239

subgroup

index

#533×C₂₃₉

#423×C₂₃₉

#332×C₂₃₉

#1561

239

Subgroups — #1561×C₂₃₉ #44480331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1561×C239 #44480331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1561×C₂₃₉ #44480331