Group #44485297 — subgroups

Subgroups — #44485297

Order 30592.

208 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 208 conjugacy classes of subgroups — too many to draw. Only its 97 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×#21439896	C₂	1	1	yes	maximal, characteristic
2	#29504799	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₂₃₉	C₂	1	1	yes	maximal, characteristic
4	#29504531	C₂	1	2	yes	maximal
5	#427×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838532	C₂²	1	2	yes	—
7	C₂²×C₂₃₉⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₂₃₉	C₂²	1	1	yes	characteristic
9	#21439896	—	2	2	no	—
10	C₄⋊(C₂₃₉⋊C₈)	—	2	4	no	—
11	C₈×Dic₂₃₉	—	2	2	no	—
12	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
13	#117×C₂₃₉	C₂²	1	2	yes	—
14	#107×C₂₃₉	C₄	1	2	yes	complemented
15	#106×C₂₃₉	C₄	1	2	yes	complemented
16	C₂²×Dic₂₃₉	—	2	1	no	—
17	#16838532	—	2	2	no	—
18	#16838532	—	2	2	no	—
19	C₂×C₂₃₉⋊C₈	—	2	1	no	—
20	C₂×C₂₃₉⋊C₈	—	2	2	no	—
21	C₂×C₂₃₉⋊C₈	—	2	1	no	—
22	C₂×C₂₃₉⋊C₈	D₄	1	2	yes	—
23	C₄×Dic₂₃₉	—	2	1	no	—
24	C₄×Dic₂₃₉	Q₈	1	2	yes	—
25	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₂₃₉	C₂×C₄	1	1	yes	characteristic
28	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
29	C₂×C₈×C₂₃₉	Q₈	1	2	yes	—
30	C₄⋊C₄×C₂₃₉	C₂×C₄	1	4	yes	—
31	C₂²⋊C₄×C₂₃₉	—	2	2	no	—
32	C₄²×C₂₃₉	—	2	2	no	—
33	C₂×Dic₂₃₉	—	2	4	no	—
34	C₂×Dic₂₃₉	—	2	2	no	—
35	C₂₃₉⋊C₈	—	2	2	no	—
36	C₂₃₉⋊C₈	—	4	2	no	—
37	C₂³×C₂₃₉	#41	1	1	yes	characteristic
38	C₂×C₄×C₂₃₉	—	2	4	no	—
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×D₄	1	1	yes	characteristic
42	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
45	C₈×C₂₃₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₃₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₃₉	—	2	2	no	—
48	C₂²×C₂₃₉	C₄×D₄	1	1	yes	characteristic
49	C₂²×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₃₉	#125	1	1	yes	characteristic
51	C₄×C₂₃₉	—	4	4	no	—
52	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₄×C₂₃₉	C₄×Q₈	1	1	yes	characteristic
54	C₄×C₂₃₉	#121	1	1	yes	characteristic
55	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
56	Dic₂₃₉	—	2	4	no	—
57	C₂×C₂₃₉	—	2	1	no	—
58	C₂×C₂₃₉	—	2	1	no	—
59	C₂×C₂₃₉	#389	1	1	yes	characteristic
60	C₂×C₂₃₉	#443	1	2	yes	—
61	C₂₃₉	#1945	1	1	yes	characteristic, complemented, Sylow
62	#1945	—	239	1	no	maximal, Sylow
63	C₂×#108	—	239	1	no	—
64	#427	D₂₃₉	1	1	yes	characteristic
65	#424	—	239	2	no	—
66	C₂×C₄×C₈	—	239	1	no	—
67	#339	—	239	2	no	—
68	C₂×M₁₆	—	239	2	no	—
69	C₂²×C₈	—	239	1	no	—
70	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
71	#117	C₂×D₂₃₉	1	2	yes	—
72	C₂×C₄²	—	239	1	no	—
73	#108	—	478	2	no	—
74	#107	Dic₂₃₉	1	2	yes	complemented
75	#106	Dic₂₃₉	1	2	yes	complemented
76	C₄⋊C₈	—	478	4	no	—
77	C₄×C₈	—	478	2	no	—
78	C₂²×C₄	—	478	1	no	—
79	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
80	M₁₆	—	478	2	no	—
81	M₁₆	—	478	2	no	—
82	C₂×C₈	—	478	2	no	—
83	C₂×C₈	—	478	1	no	—
84	C₂×C₈	—	478	1	no	—
85	C₂×C₈	—	239	2	no	—
86	C₂×C₈	C₂×Dic₂₃₉	1	1	yes	characteristic
87	C₂×C₈	C₂×Dic₂₃₉	1	1	yes	characteristic
88	C₂×C₈	D₉₅₆	1	2	yes	—
89	C₂×C₈	Dic₄₇₈	1	2	yes	—
90	C₄⋊C₄	C₂×Dic₂₃₉	1	4	yes	—
91	C₂²⋊C₄	—	2	2	no	—
92	C₄²	—	2	2	no	—
93	C₄²	—	478	1	no	—
94	C₄²	—	239	2	no	—
95	C₂³	#16838547	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	478	4	no	—
98	C₂×C₄	—	478	2	no	—
99	C₂×C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
100	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
102	C₂×C₄	Q₈×D₂₃₉	1	1	yes	characteristic
103	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
104	C₂×C₄	#16838549	1	1	yes	characteristic
105	C₈	—	956	2	no	—
106	C₈	—	478	2	no	—
107	C₈	C₄⋊Dic₂₃₉	1	2	yes	—
108	C₈	C₄⋊Dic₂₃₉	1	2	yes	—
109	C₂²	—	2	2	no	—
110	C₂²	D₄×Dic₂₃₉	1	1	yes	characteristic
111	C₂²	#21439940	1	1	yes	characteristic
112	C₂²	#21439951	1	1	yes	characteristic
113	C₄	—	4	4	no	—
114	C₄	—	478	4	no	—
115	C₄	C₂×C₄⋊Dic₂₃₉	1	1	yes	characteristic
116	C₄	Q₈×Dic₂₃₉	1	1	yes	characteristic
117	C₄	#21439935	1	1	yes	characteristic
118	C₄	#21439959	1	1	yes	characteristic
119	C₂	—	2	1	no	—
120	C₂	—	2	1	no	—
121	C₂	#29504671	1	1	yes	characteristic
122	C₂	#29504889	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439896

C₂

yes

maximal, characteristic

#29504799

C₂

yes

maximal

C₂×C₈×Dic₂₃₉

C₂

yes

maximal, characteristic

#29504531

C₂

yes

maximal

#427×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838532

C₂²

yes

—

C₂²×C₂₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₂₃₉

C₂²

yes

characteristic

#21439896

—

C₄⋊(C₂₃₉⋊C₈)

—

C₈×Dic₂₃₉

—

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

#117×C₂₃₉

C₂²

yes

—

#107×C₂₃₉

C₄

yes

complemented

#106×C₂₃₉

C₄

yes

complemented

C₂²×Dic₂₃₉

—

#16838532

—

#16838532

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

D₄

yes

—

C₄×Dic₂₃₉

—

C₄×Dic₂₃₉

Q₈

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃₉

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

Q₈

yes

—

C₄⋊C₄×C₂₃₉

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃₉

—

C₄²×C₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

#41

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

characteristic

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₄×D₄

yes

characteristic

C₂²×C₂₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃₉

#125

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃₉

C₄×Q₈

yes

characteristic

C₄×C₂₃₉

#121

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#389

yes

characteristic

C₂×C₂₃₉

#443

yes

—

C₂₃₉

#1945

yes

characteristic, complemented, Sylow

#1945

—

239

maximal, Sylow

C₂×#108

—

239

—

#427

D₂₃₉

yes

characteristic

#424

—

239

—

C₂×C₄×C₈

—

239

—

#339

—

239

—

C₂×M₁₆

—

239

—

C₂²×C₈

—

239

—

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

#117

C₂×D₂₃₉

yes

—

C₂×C₄²

—

239

—

#108

—

478

—

#107

Dic₂₃₉

yes

complemented

#106

Dic₂₃₉

yes

complemented

C₄⋊C₈

—

478

—

C₄×C₈

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

M₁₆

—

478

—

M₁₆

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

—

239

—

C₂×C₈

C₂×Dic₂₃₉

yes

characteristic

C₂×C₈

C₂×Dic₂₃₉

yes

characteristic

C₂×C₈

D₉₅₆

yes

—

C₂×C₈

Dic₄₇₈

yes

—

C₄⋊C₄

C₂×Dic₂₃₉

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

478

—

C₄²

—

239

—

C₂³

#16838547

yes

characteristic

C₂×C₄

—

C₂×C₄

—

478

—

C₂×C₄

—

478

—

C₂×C₄

C₂²×Dic₂₃₉

yes

characteristic

100

C₂×C₄

D₄×D₂₃₉

yes

characteristic

101

C₂×C₄

C₂×D₉₅₆

yes

characteristic

102

C₂×C₄

Q₈×D₂₃₉

yes

characteristic

103

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

104

C₂×C₄

#16838549

yes

characteristic

105

C₈

—

956

—

106

C₈

—

478

—

107

C₈

C₄⋊Dic₂₃₉

yes

—

108

C₈

C₄⋊Dic₂₃₉

yes

—

109

C₂²

—

110

C₂²

D₄×Dic₂₃₉

yes

characteristic

111

C₂²

#21439940

yes

characteristic

112

C₂²

#21439951

yes

characteristic

113

C₄

—

114

C₄

—

478

—

115

C₄

C₂×C₄⋊Dic₂₃₉

yes

characteristic

116

C₄

Q₈×Dic₂₃₉

yes

characteristic

117

C₄

#21439935

yes

characteristic

118

C₄

#21439959

yes

characteristic

119

C₂

—

120

C₂

—

121

C₂

#29504671

yes

characteristic

122

C₂

#29504889

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃₉
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	#427×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₄×D₄
prime radical	C₂³×C₂₃₉	#41
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	#427	D₂₃₉
hypocenter	C₂₃₉	#1945
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

subgroup

G/H

center

C₂×C₄

D₄×D₂₃₉

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

#427×C₂₃₉

C₂

socle

C₂²×C₂₃₉

C₄×D₄

prime radical

C₂³×C₂₃₉

#41

cosocle

C₂²×C₄×C₂₃₉

C₂³

D₂₃₉

C₂₃₉

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #427

N	H ≅ G/N	type
14. #107×C₂₃₉	C₄	semidirect
15. #106×C₂₃₉	C₄	semidirect
61. C₂₃₉	#1945	semidirect
74. #107	Dic₂₃₉	semidirect
75. #106	Dic₂₃₉	semidirect

H ≅ G/N

type

14. #107×C₂₃₉

C₄

semidirect

15. #106×C₂₃₉

C₄

semidirect

61. C₂₃₉

#1945

semidirect

74. #107

Dic₂₃₉

semidirect

75. #106

Dic₂₃₉

semidirect

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

subgroup	index
C₂×#21439896	2
#29504799	2
C₂×C₈×Dic₂₃₉	2
#29504531	2
#427×C₂₃₉	2
#1945	239

subgroup

index

C₂×C₈×Dic₂₃₉

#427×C₂₃₉

239

Subgroups — #44485297

Conjugacy classes of subgroups

Complements

Maximal subgroups