Group #44485528 — subgroups

Subgroups — #44485528

Order 30592.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 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	#29505329	C₂	1	1	yes	maximal, characteristic
2	#29504618	C₂	1	1	yes	maximal, characteristic
3	#29504533	C₂	1	4	yes	maximal
4	#423×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	#21439960	C₄	1	2	yes	complemented
6	#21439952	C₄	1	2	yes	complemented
7	#21439939	—	2	2	no	—
8	C₄×Dic₄₇₈	—	2	2	no	—
9	#21439919	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₃₉	C₂²	1	2	yes	—
11	C₂₃₉⋊#95	—	2	2	no	—
12	C₂×M₁₆×C₂₃₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₃₉	—	2	2	no	—
15	C₂×Dic₄₇₈	—	2	2	no	—
16	C₂²×Dic₂₃₉	—	2	2	no	—
17	C₄⋊Dic₂₃₉	—	2	2	no	—
18	C₄⋊Dic₂₃₉	C₂×C₄	1	4	yes	—
19	Dic₂₃₉⋊C₄	—	2	4	no	—
20	C₂²⋊Dic₂₃₉	—	2	2	no	—
21	C₄×Dic₂₃₉	—	2	2	no	—
22	C₄×Dic₂₃₉	—	2	2	no	—
23	C₄×Dic₂₃₉	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₂₃₉	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₃₉	—	2	4	no	—
27	C₂×C₈×C₂₃₉	—	2	2	no	—
28	C₄²×C₂₃₉	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₃₉	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₃₉	D₄	1	2	yes	—
31	Dic₄₇₈	—	4	4	no	—
32	C₂×Dic₂₃₉	—	2	4	no	—
33	C₂×Dic₂₃₉	—	2	4	no	—
34	C₂×Dic₂₃₉	—	4	4	no	—
35	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃₉	—	4	2	no	—
37	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₂₃₉	#41	1	2	yes	—
43	C₈×C₂₃₉	—	4	2	no	—
44	C₂²×C₂₃₉	—	2	2	no	—
45	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₃₉	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₂₃₉	—	2	2	no	—
48	C₄×C₂₃₉	—	2	2	no	—
49	C₄×C₂₃₉	C₄×Q₈	1	2	yes	—
50	C₄×C₂₃₉	#124	1	1	yes	characteristic
51	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
52	Dic₂₃₉	—	4	4	no	—
53	Dic₂₃₉	—	4	4	no	—
54	C₂×C₂₃₉	—	2	2	no	—
55	C₂×C₂₃₉	#391	1	1	yes	characteristic
56	C₂×C₂₃₉	#421	1	2	yes	—
57	C₂₃₉	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	239	1	no	maximal, Sylow
59	#533	—	239	1	no	—
60	#423	D₂₃₉	1	1	yes	characteristic
61	#376	—	239	1	no	—
62	#339	—	239	4	no	—
63	C₂×M₁₆	C₂×D₂₃₉	1	2	yes	—
64	C₄⋊Q₈	—	239	2	no	—
65	#125	—	239	2	no	—
66	C₂²⋊Q₈	—	478	2	no	—
67	C₄×Q₈	—	478	2	no	—
68	#117	—	239	2	no	—
69	C₂×C₄²	—	239	2	no	—
70	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	#95	—	478	2	no	—
73	C₂×Q₈	—	478	2	no	—
74	C₂²×C₄	—	478	2	no	—
75	C₂²×C₄	D₉₅₆	1	2	yes	—
76	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	478	4	no	—
80	C₄⋊C₄	—	478	2	no	—
81	C₄⋊C₄	—	239	4	no	—
82	C₂²⋊C₄	—	478	2	no	—
83	C₄²	—	478	2	no	—
84	C₄²	—	478	2	no	—
85	C₄²	—	239	4	no	—
86	C₄²	C₄×D₂₃₉	1	1	yes	characteristic
87	C₄²	C₄×D₂₃₉	1	1	yes	characteristic
88	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
89	C₂³	C₂×D₉₅₆	1	1	yes	characteristic
90	Q₈	—	956	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	956	4	no	—
93	C₂×C₄	—	478	4	no	—
94	C₂×C₄	—	478	4	no	—
95	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	2	yes	—
96	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₂₃₉	1	2	yes	—
100	C₂×C₄	#16838547	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
104	C₂²	#21439940	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	956	4	no	—
108	C₄	—	956	4	no	—
109	C₄	#21439925	1	2	yes	—
110	C₄	#21439947	1	1	yes	characteristic
111	C₄	#21439958	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#29504677	1	1	yes	characteristic
114	C₂	#29504788	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#29505329

C₂

yes

maximal, characteristic

#29504618

C₂

yes

maximal, characteristic

#29504533

C₂

yes

maximal

#423×C₂₃₉

C₂

yes

maximal, characteristic

#21439960

C₄

yes

complemented

#21439952

C₄

yes

complemented

#21439939

—

C₄×Dic₄₇₈

—

#21439919

C₂²

yes

—

C₂×C₄×Dic₂₃₉

C₂²

yes

—

C₂₃₉⋊#95

—

C₂×M₁₆×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₃₉

—

C₂×Dic₄₇₈

—

C₂²×Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

C₂×C₄

yes

—

Dic₂₃₉⋊C₄

—

C₂²⋊Dic₂₃₉

—

C₄×Dic₂₃₉

—

C₄×Dic₂₃₉

—

C₄×Dic₂₃₉

Q₈

yes

complemented

C₂²×C₄×C₂₃₉

D₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

M₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄²×C₂₃₉

C₂×C₄

yes

characteristic

C₄²×C₂₃₉

C₂×C₄

yes

characteristic

C₄²×C₂₃₉

D₄

yes

—

Dic₄₇₈

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

—

C₂×C₄×C₂₃₉

#41

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃₉

C₂²⋊Q₈

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₄×Q₈

yes

—

C₄×C₂₃₉

#124

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#391

yes

characteristic

C₂×C₂₃₉

#421

yes

—

C₂₃₉

#1995

yes

characteristic, complemented, Sylow

#1995

—

239

maximal, Sylow

#533

—

239

—

#423

D₂₃₉

yes

characteristic

#376

—

239

—

#339

—

239

—

C₂×M₁₆

C₂×D₂₃₉

yes

—

C₄⋊Q₈

—

239

—

#125

—

239

—

C₂²⋊Q₈

—

478

—

C₄×Q₈

—

478

—

#117

—

239

—

C₂×C₄²

—

239

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₄⋊C₈

—

#95

—

478

—

C₂×Q₈

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

D₉₅₆

yes

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

239

—

C₂²⋊C₄

—

478

—

C₄²

—

478

—

C₄²

—

478

—

C₄²

—

239

—

C₄²

C₄×D₂₃₉

yes

characteristic

C₄²

C₄×D₂₃₉

yes

characteristic

C₄²

C₂²⋊D₂₃₉

yes

—

C₂³

C₂×D₉₅₆

yes

characteristic

Q₈

—

956

—

C₂×C₄

—

C₂×C₄

—

956

—

C₂×C₄

—

478

—

C₂×C₄

—

478

—

C₂×C₄

(C₂×D₂₃₉)⋊C₄

yes

—

C₂×C₄

(C₂×D₂₃₉)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₃₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

C₂×C₄

Q₈×D₂₃₉

yes

—

100

C₂×C₄

#16838547

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂×(C₂×D₂₃₉)⋊C₄

yes

characteristic

104

C₂²

#21439940

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

956

—

108

C₄

—

956

—

109

C₄

#21439925

yes

—

110

C₄

#21439947

yes

characteristic

111

C₄

#21439958

yes

characteristic

112

C₂

—

113

C₂

#29504677

yes

characteristic

114

C₂

#29504788

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₃₉
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	#423×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₃₉	C₂×D₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	#423	D₂₃₉
hypocenter	C₂₃₉	#1995
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₃₉

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

#423×C₂₃₉

C₂

socle

C₂²×C₂₃₉

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₃₉

C₂×D₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

D₂₃₉

C₂₃₉

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #21439960	C₄	semidirect
6. #21439952	C₄	semidirect
23. C₄×Dic₂₃₉	Q₈	semidirect
57. C₂₃₉	#1995	semidirect

H ≅ G/N

type

5. #21439960

C₄

semidirect

6. #21439952

C₄

semidirect

23. C₄×Dic₂₃₉

Q₈

semidirect

57. C₂₃₉

#1995

semidirect

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

subgroup	index
#29505329	2
#29504618	2
#29504533	2
#423×C₂₃₉	2
#1995	239

subgroup

index

#423×C₂₃₉

239

Subgroups — #44485528

Conjugacy classes of subgroups

Complements

Maximal subgroups