Group #44483686 — subgroups

Subgroups — #44483686

Order 30592.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#29505502	C₂	1	1	yes	maximal, characteristic
2	#29505064	C₂	1	2	yes	maximal
3	#29504521	C₂	1	1	yes	maximal, characteristic
4	#29504504	C₂	1	2	yes	maximal
5	#494×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	#21439959	—	2	2	no	—
7	#21439958	—	2	2	no	—
8	C₄×Dic₄₇₈	C₂²	1	2	yes	—
9	#21439925	—	2	2	no	—
10	#21439893	—	2	2	no	—
11	C₄⋊(C₂₃₉⋊C₈)	C₂²	1	2	yes	—
12	#21439883	—	2	2	no	—
13	C₂×Q₁₆×C₂₃₉	—	2	2	no	—
14	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₄₇₈	—	2	2	no	—
18	C₂×Dic₄₇₈	D₄	1	2	yes	—
19	C₂×C₂₃₉⋊C₈	—	2	2	no	—
20	C₄⋊Dic₂₃₉	D₄	1	2	yes	—
21	Dic₂₃₉⋊C₄	—	2	4	no	—
22	Dic₂₃₉⋊C₄	—	2	2	no	—
23	C₄×Dic₂₃₉	—	2	2	no	—
24	C₄×Dic₂₃₉	—	2	2	no	—
25	C₂×Q₈×C₂₃₉	—	2	1	no	—
26	C₂×Q₈×C₂₃₉	D₄	1	2	yes	—
27	Q₁₆×C₂₃₉	—	4	4	no	—
28	C₂×C₈×C₂₃₉	—	2	1	no	—
29	C₄⋊C₄×C₂₃₉	—	2	2	no	—
30	C₄⋊C₄×C₂₃₉	—	2	2	no	—
31	C₄²×C₂₃₉	C₂³	1	1	yes	characteristic
32	Dic₄₇₈	—	2	2	no	—
33	Dic₄₇₈	—	8	2	no	—
34	Dic₄₇₈	Q₁₆	1	4	yes	complemented
35	C₂×Dic₂₃₉	—	2	2	no	—
36	C₂×Dic₂₃₉	—	2	2	no	—
37	C₂₃₉⋊C₈	—	4	2	no	—
38	Q₈×C₂₃₉	—	4	2	no	—
39	Q₈×C₂₃₉	—	4	2	no	—
40	C₂×C₄×C₂₃₉	—	2	2	no	—
41	C₂×C₄×C₂₃₉	—	4	1	no	—
42	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
44	C₈×C₂₃₉	—	2	2	no	—
45	C₂²×C₂₃₉	#120	1	1	yes	characteristic
46	C₄×C₂₃₉	—	2	1	no	—
47	C₄×C₂₃₉	—	4	2	no	—
48	C₄×C₂₃₉	—	8	1	no	—
49	C₄×C₂₃₉	C₂×Q₁₆	1	2	yes	—
50	C₄×C₂₃₉	#137	1	2	yes	—
51	Dic₂₃₉	—	2	4	no	—
52	Dic₂₃₉	—	4	2	no	—
53	C₂×C₂₃₉	#451	1	2	yes	—
54	C₂×C₂₃₉	#453	1	1	yes	characteristic
55	C₂₃₉	#1633	1	1	yes	characteristic, complemented, Sylow
56	#1633	—	239	1	no	maximal, Sylow
57	Q₈²	—	239	1	no	—
58	#494	D₂₃₉	1	1	yes	characteristic
59	#477	—	239	2	no	—
60	#333	—	239	1	no	—
61	Q₈⋊C₈	—	239	2	no	—
62	C₂×Q₁₆	—	2	2	no	—
63	C₄⋊Q₈	—	478	2	no	—
64	C₄⋊Q₈	—	478	2	no	—
65	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
66	C₄⋊Q₈	C₂×D₂₃₉	1	1	yes	characteristic
67	C₄×Q₈	—	478	2	no	—
68	C₄×Q₈	—	239	2	no	—
69	#107	—	478	2	no	—
70	C₄⋊C₈	—	239	2	no	—
71	#103	—	478	2	no	—
72	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	478	2	no	—
75	C₂×Q₈	—	239	2	no	—
76	C₂×Q₈	C₂²⋊D₂₃₉	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	478	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	478	4	no	—
83	C₄⋊C₄	—	478	2	no	—
84	C₄⋊C₄	—	239	2	no	—
85	C₄²	—	478	2	no	—
86	C₄²	—	478	2	no	—
87	C₄²	C₂²×D₂₃₉	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	1912	2	no	—
91	Q₈	—	478	2	no	—
92	Q₈	—	239	4	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	478	2	no	—
96	C₂×C₄	—	478	2	no	—
97	C₂×C₄	D₄×D₂₃₉	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	956	2	no	—
101	C₂²	#21439928	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	956	2	no	—
106	C₄	—	478	4	no	—
107	C₄	Q₁₆×D₂₃₉	1	2	yes	—
108	C₄	#21439992	1	2	yes	—
109	C₂	#29504926	1	2	yes	—
110	C₂	#29504939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29505502

C₂

yes

maximal, characteristic

#29505064

C₂

yes

maximal

#29504521

C₂

yes

maximal, characteristic

#29504504

C₂

yes

maximal

#494×C₂₃₉

C₂

yes

maximal, characteristic

#21439959

—

#21439958

—

C₄×Dic₄₇₈

C₂²

yes

—

#21439925

—

#21439893

—

C₄⋊(C₂₃₉⋊C₈)

C₂²

yes

—

#21439883

—

C₂×Q₁₆×C₂₃₉

—

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄⋊Q₈×C₂₃₉

C₂²

yes

characteristic

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

D₄

yes

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

D₄

yes

—

Dic₂₃₉⋊C₄

—

Dic₂₃₉⋊C₄

—

C₄×Dic₂₃₉

—

C₄×Dic₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

D₄

yes

—

Q₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

C₂³

yes

characteristic

Dic₄₇₈

—

Dic₄₇₈

—

Dic₄₇₈

Q₁₆

yes

complemented

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

#120

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂×Q₁₆

yes

—

C₄×C₂₃₉

#137

yes

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

#451

yes

—

C₂×C₂₃₉

#453

yes

characteristic

C₂₃₉

#1633

yes

characteristic, complemented, Sylow

#1633

—

239

maximal, Sylow

Q₈²

—

239

—

#494

D₂₃₉

yes

characteristic

#477

—

239

—

#333

—

239

—

Q₈⋊C₈

—

239

—

C₂×Q₁₆

—

C₄⋊Q₈

—

478

—

C₄⋊Q₈

—

478

—

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

C₄⋊Q₈

C₂×D₂₃₉

yes

characteristic

C₄×Q₈

—

478

—

C₄×Q₈

—

239

—

#107

—

478

—

C₄⋊C₈

—

239

—

#103

—

478

—

C₄×C₈

C₂×D₂₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

478

—

C₂×Q₈

—

239

—

C₂×Q₈

C₂²⋊D₂₃₉

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

478

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

239

—

C₄²

—

478

—

C₄²

—

478

—

C₄²

C₂²×D₂₃₉

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

1912

—

Q₈

—

478

—

Q₈

—

239

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

478

—

C₂×C₄

—

478

—

C₂×C₄

D₄×D₂₃₉

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

C₈

—

100

C₈

—

956

—

101

C₂²

#21439928

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

956

—

106

C₄

—

478

—

107

C₄

Q₁₆×D₂₃₉

yes

—

108

C₄

#21439992

yes

—

109

C₂

#29504926

yes

—

110

C₂

#29504939

yes

characteristic

	subgroup	G/H
center	C₂²	#21439928
commutator, cosocle	C₄²×C₂₃₉	C₂³
Frattini	C₄²	C₂²×D₂₃₉
Fitting	#494×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	#120
hypercenter	#494	D₂₃₉
hypocenter	C₂₃₉	#1633
LCS	G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#21439928

commutator, cosocle

C₄²×C₂₃₉

C₂³

Frattini

C₄²

C₂²×D₂₃₉

Fitting

#494×C₂₃₉

C₂

C₂²×C₂₃₉

D₂₃₉

C₂₃₉

G ▷ C₄²×C₂₃₉ ▷ C₂²×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
34. Dic₄₇₈	Q₁₆	semidirect
55. C₂₃₉	#1633	semidirect

H ≅ G/N

type

34. Dic₄₇₈

Q₁₆

semidirect

55. C₂₃₉

#1633

semidirect

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

subgroup	index
#29505502	2
#29505064	2
#29504521	2
#29504504	2
#494×C₂₃₉	2
#1633	239

subgroup

index

#494×C₂₃₉

239

Subgroups — #44483686

Conjugacy classes of subgroups

Complements

Maximal subgroups