Group #44484455 — subgroups

Subgroups — #44484455

Order 30592.

220 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 113 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×#21439895	C₂	1	2	yes	maximal
2	C₄×#16838532	C₂	1	1	yes	maximal, characteristic
3	C₂×#21439871	C₂	1	1	yes	maximal, characteristic
4	#29504536	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838532	C₂²	1	2	yes	—
7	C₂²×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
9	#21439895	C₄	1	8	yes	complemented
10	#21439871	—	2	1	no	—
11	#21439871	C₄	1	4	yes	—
12	C₄×C₂₃₉⋊C₈	—	2	1	no	—
13	C₂×M₁₆×C₂₃₉	C₂²	1	2	yes	—
14	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
15	#97×C₂₃₉	—	2	1	no	—
16	C₄×C₈×C₂₃₉	—	2	1	no	—
17	#16838532	—	2	2	no	—
18	#16838532	C₂×C₄	1	4	yes	—
19	C₂×C₂₃₉⋊C₈	—	2	2	no	—
20	C₂×C₂₃₉⋊C₈	—	2	2	no	—
21	C₂×C₂₃₉⋊C₈	C₂×C₄	1	2	yes	—
22	C₂×C₂₃₉⋊C₈	C₂×C₄	1	2	yes	—
23	C₂×C₂₃₉⋊C₈	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₃₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₃₉	—	2	2	no	—
27	M₁₆×C₂₃₉	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₂₃₉	—	2	2	no	—
29	C₄²×C₂₃₉	C₂×C₄	1	2	yes	—
30	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
31	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
32	C₂₃₉⋊C₈	—	2	2	no	—
33	C₂₃₉⋊C₈	—	2	4	no	—
34	C₂₃₉⋊C₈	C₄²	1	4	yes	—
35	C₂³×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
36	C₂×C₄×C₂₃₉	—	4	2	no	—
37	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃₉	C₂²×C₄	1	2	yes	—
41	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃₉	#41	1	2	yes	—
43	C₈×C₂₃₉	—	2	4	no	—
44	C₂²×C₂₃₉	—	2	2	no	—
45	C₂²×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₃₉	C₄×Q₈	1	2	yes	—
47	C₄×C₂₃₉	—	2	2	no	—
48	C₄×C₂₃₉	—	2	2	no	—
49	C₄×C₂₃₉	C₂×C₄²	1	1	yes	characteristic
50	C₄×C₂₃₉	#117	1	1	yes	characteristic
51	C₄×C₂₃₉	C₄×D₄	1	2	yes	—
52	C₂×C₂₃₉	—	2	2	no	—
53	C₂×C₂₃₉	C₄×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₂₃₉	#430	1	2	yes	—
55	C₂₃₉	#1778	1	1	yes	characteristic, complemented, Sylow
56	#1778	—	239	1	no	maximal, Sylow
57	C₂×#108	—	239	2	no	—
58	C₄×M₁₆	—	239	1	no	—
59	C₄×M₁₆	D₂₃₉	1	1	yes	characteristic
60	C₂×#97	—	239	1	no	—
61	#341	—	239	2	no	—
62	C₂×M₁₆	—	239	2	no	—
63	C₂×M₁₆	C₂×D₂₃₉	1	2	yes	—
64	C₂²×C₈	—	239	1	no	—
65	C₂²×C₈	—	239	1	no	—
66	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
67	#108	—	239	8	no	—
68	#97	—	2	1	no	—
69	#97	—	478	1	no	—
70	#97	—	239	4	no	—
71	C₄×C₈	—	2	1	no	—
72	C₄×C₈	—	478	1	no	—
73	C₂²×C₄	Dic₄₇₈	1	2	yes	—
74	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
75	M₁₆	—	2	2	no	—
76	M₁₆	—	478	2	no	—
77	M₁₆	—	239	4	no	—
78	M₁₆	C₄×D₂₃₉	1	4	yes	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	478	2	no	—
81	C₂×C₈	—	478	2	no	—
82	C₂×C₈	—	239	2	no	—
83	C₂×C₈	—	239	2	no	—
84	C₂×C₈	—	239	4	no	—
85	C₄²	C₄×D₂₃₉	1	2	yes	—
86	C₄²	C₂²⋊D₂₃₉	1	1	yes	characteristic
87	C₄²	C₂²⋊D₂₃₉	1	1	yes	characteristic
88	C₂³	C₂×Dic₄₇₈	1	1	yes	characteristic
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	Dic₂₃₉⋊C₄	1	2	yes	—
91	C₂×C₄	Dic₂₃₉⋊C₄	1	2	yes	—
92	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
93	C₂×C₄	C₂×C₄×D₂₃₉	1	2	yes	—
94	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
95	C₂×C₄	#16838548	1	2	yes	—
96	C₈	—	2	4	no	—
97	C₈	—	478	2	no	—
98	C₈	—	478	4	no	—
99	C₈	—	239	4	no	—
100	C₂²	—	2	2	no	—
101	C₂²	C₂×Dic₂₃₉⋊C₄	1	1	yes	characteristic
102	C₂²	C₄×Dic₄₇₈	1	2	yes	—
103	C₄	—	2	2	no	—
104	C₄	—	2	2	no	—
105	C₄	C₄²×D₂₃₉	1	1	yes	characteristic
106	C₄	#21439917	1	1	yes	characteristic
107	C₄	C₄×C₂²⋊D₂₃₉	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₄×Dic₂₃₉⋊C₄	1	1	yes	characteristic
110	C₂	#29504820	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439895

C₂

yes

maximal

C₄×#16838532

C₂

yes

maximal, characteristic

C₂×#21439871

C₂

yes

maximal, characteristic

#29504536

C₂

yes

maximal

C₄×M₁₆×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838532

C₂²

yes

—

C₂²×C₂₃₉⋊C₈

C₂²

yes

characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

characteristic

#21439895

C₄

yes

complemented

#21439871

—

#21439871

C₄

yes

—

C₄×C₂₃₉⋊C₈

—

C₂×M₁₆×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

#97×C₂₃₉

—

C₄×C₈×C₂₃₉

—

#16838532

—

#16838532

C₂×C₄

yes

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₂₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₂₃₉⋊C₈

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

Q₈

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

M₁₆×C₂₃₉

—

M₁₆×C₂₃₉

C₂×C₄

yes

—

C₂×C₈×C₂₃₉

—

C₄²×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

D₄

yes

characteristic

C₄²×C₂₃₉

D₄

yes

characteristic

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

C₄²

yes

—

C₂³×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂²×C₄

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃₉

C₄×Q₈

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂×C₄²

yes

characteristic

C₄×C₂₃₉

#117

yes

characteristic

C₄×C₂₃₉

C₄×D₄

yes

—

C₂×C₂₃₉

—

C₂×C₂₃₉

C₄×C₄⋊C₄

yes

characteristic

C₂×C₂₃₉

#430

yes

—

C₂₃₉

#1778

yes

characteristic, complemented, Sylow

#1778

—

239

maximal, Sylow

C₂×#108

—

239

—

C₄×M₁₆

—

239

—

C₄×M₁₆

D₂₃₉

yes

characteristic

C₂×#97

—

239

—

#341

—

239

—

C₂×M₁₆

—

239

—

C₂×M₁₆

C₂×D₂₃₉

yes

—

C₂²×C₈

—

239

—

C₂²×C₈

—

239

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

#108

—

239

—

#97

—

#97

—

478

—

#97

—

239

—

C₄×C₈

—

C₄×C₈

—

478

—

C₂²×C₄

Dic₄₇₈

yes

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

M₁₆

—

M₁₆

—

478

—

M₁₆

—

239

—

M₁₆

C₄×D₂₃₉

yes

—

C₂×C₈

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

239

—

C₂×C₈

—

239

—

C₂×C₈

—

239

—

C₄²

C₄×D₂₃₉

yes

—

C₄²

C₂²⋊D₂₃₉

yes

characteristic

C₄²

C₂²⋊D₂₃₉

yes

characteristic

C₂³

C₂×Dic₄₇₈

yes

characteristic

C₂×C₄

—

C₂×C₄

Dic₂₃₉⋊C₄

yes

—

C₂×C₄

Dic₂₃₉⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₃₉

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃₉

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

C₂×C₄

#16838548

yes

—

C₈

—

C₈

—

478

—

C₈

—

478

—

C₈

—

239

—

100

C₂²

—

101

C₂²

C₂×Dic₂₃₉⋊C₄

yes

characteristic

102

C₂²

C₄×Dic₄₇₈

yes

—

103

C₄

—

104

C₄

—

105

C₄

C₄²×D₂₃₉

yes

characteristic

106

C₄

#21439917

yes

characteristic

107

C₄

C₄×C₂²⋊D₂₃₉

yes

—

108

C₂

—

109

C₂

C₄×Dic₂₃₉⋊C₄

yes

characteristic

110

C₂

#29504820

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃₉
commutator	C₄×C₂₃₉	C₂×C₄²
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₄×M₁₆×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃₉	C₂×Q₈
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₄×M₁₆	D₂₃₉
hypocenter	C₂₃₉	#1778
LCS	G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃₉

commutator

C₄×C₂₃₉

C₂×C₄²

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₄×M₁₆×C₂₃₉

C₂

socle

C₂²×C₂₃₉

C₂×C₄⋊C₄

prime radical

C₂³×C₂₃₉

C₂×Q₈

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₄×M₁₆

D₂₃₉

hypocenter

C₂₃₉

#1778

LCS

G ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
9. #21439895	C₄	semidirect
55. C₂₃₉	#1778	semidirect

H ≅ G/N

type

9. #21439895

C₄

semidirect

55. C₂₃₉

#1778

semidirect

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

subgroup	index
C₂×#21439895	2
C₄×#16838532	2
C₂×#21439871	2
#29504536	2
C₄×M₁₆×C₂₃₉	2
#1778	239

subgroup

index

C₄×M₁₆×C₂₃₉

239

Subgroups — #44484455

Conjugacy classes of subgroups

Complements

Maximal subgroups