Group #44485339 — subgroups

Subgroups — #44485339

Order 30592.

184 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 93 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	#29504821	C₂	1	2	yes	maximal
2	C₂×#21439896	C₂	1	1	yes	maximal, characteristic
3	#29504727	C₂	1	1	yes	maximal, characteristic
4	#29504544	C₂	1	2	yes	maximal
5	C₂×#108×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#16838532	C₂²	1	2	yes	—
7	C₂×#16838532	C₂²	1	1	yes	characteristic
8	#21439919	C₂²	1	1	yes	characteristic
9	#21439896	—	2	4	no	—
10	#21439896	—	2	1	no	—
11	#21439896	—	2	1	no	—
12	#21439870	—	2	1	no	—
13	C₈×Dic₂₃₉	—	2	1	no	—
14	C₂×M₁₆×C₂₃₉	C₂²	1	2	yes	—
15	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
16	#108×C₂₃₉	C₄	1	4	yes	complemented
17	#16838532	—	2	2	no	—
18	#16838532	—	2	2	no	—
19	#16838532	—	2	1	no	—
20	#16838532	D₄	1	2	yes	—
21	C₂×C₂₃₉⋊C₈	—	2	1	no	—
22	C₂×C₂₃₉⋊C₈	—	2	2	no	—
23	C₄⋊Dic₂₃₉	—	2	1	no	—
24	C₂²⋊Dic₂₃₉	Q₈	1	2	yes	—
25	C₄×Dic₂₃₉	—	2	1	no	—
26	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₃₉	—	2	2	no	—
28	M₁₆×C₂₃₉	C₂×C₄	1	4	yes	—
29	C₂×C₈×C₂₃₉	—	2	2	no	—
30	C₂×C₈×C₂₃₉	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
33	C₂×C₈×C₂₃₉	Q₈	1	2	yes	—
34	C₂×Dic₂₃₉	—	2	2	no	—
35	C₂₃₉⋊C₈	—	2	2	no	—
36	C₂₃₉⋊C₈	—	4	2	no	—
37	C₂³×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
43	C₂×C₄×C₂₃₉	#41	1	1	yes	characteristic
44	C₈×C₂₃₉	—	2	4	no	—
45	C₈×C₂₃₉	C₄⋊C₄	1	2	yes	—
46	C₈×C₂₃₉	C₄⋊C₄	1	2	yes	—
47	C₂²×C₂₃₉	—	4	1	no	—
48	C₂²×C₂₃₉	C₄×Q₈	1	1	yes	characteristic
49	C₂²×C₂₃₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
52	C₄×C₂₃₉	C₄×D₄	1	1	yes	characteristic
53	C₄×C₂₃₉	#121	1	1	yes	characteristic
54	C₄×C₂₃₉	#125	1	1	yes	characteristic
55	Dic₂₃₉	—	4	2	no	—
56	C₂×C₂₃₉	—	2	1	no	—
57	C₂×C₂₃₉	—	2	1	no	—
58	C₂×C₂₃₉	—	2	1	no	—
59	C₂×C₂₃₉	#389	1	1	yes	characteristic
60	C₂₃₉	#1953	1	1	yes	characteristic, complemented, Sylow
61	#1953	—	239	1	no	maximal, Sylow
62	#430	—	239	2	no	—
63	C₂×#108	—	239	1	no	—
64	C₂×#108	D₂₃₉	1	1	yes	characteristic
65	#405	—	239	1	no	—
66	#344	—	239	2	no	—
67	C₂×M₁₆	—	239	2	no	—
68	C₂×M₁₆	—	239	1	no	—
69	C₂×M₁₆	C₂×D₂₃₉	1	2	yes	—
70	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
71	#117	—	239	1	no	—
72	#108	—	478	4	no	—
73	#108	—	478	1	no	—
74	#108	—	478	1	no	—
75	#108	Dic₂₃₉	1	4	yes	complemented
76	#97	—	478	1	no	—
77	C₄×C₈	—	478	1	no	—
78	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
79	M₁₆	—	2	2	no	—
80	M₁₆	—	478	2	no	—
81	M₁₆	—	478	2	no	—
82	M₁₆	—	478	1	no	—
83	M₁₆	—	239	2	no	—
84	M₁₆	C₂×Dic₂₃₉	1	4	yes	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	478	2	no	—
87	C₂×C₈	—	478	1	no	—
88	C₂×C₈	C₂×Dic₂₃₉	1	2	yes	—
89	C₂×C₈	D₉₅₆	1	1	yes	characteristic
90	C₂×C₈	D₉₅₆	1	1	yes	characteristic
91	C₂×C₈	Dic₄₇₈	1	2	yes	—
92	C₄⋊C₄	—	478	1	no	—
93	C₂²⋊C₄	—	239	2	no	—
94	C₄²	—	478	1	no	—
95	C₂³	Q₈×D₂₃₉	1	1	yes	characteristic
96	C₂×C₄	—	478	2	no	—
97	C₂×C₄	C₂²×Dic₂₃₉	1	1	yes	characteristic
98	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
99	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
101	C₂×C₄	#16838547	1	1	yes	characteristic
102	C₂×C₄	#16838549	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₈	—	956	2	no	—
105	C₈	—	478	2	no	—
106	C₈	C₄⋊Dic₂₃₉	1	2	yes	—
107	C₈	C₄⋊Dic₂₃₉	1	2	yes	—
108	C₂²	—	4	1	no	—
109	C₂²	Q₈×Dic₂₃₉	1	1	yes	characteristic
110	C₂²	#21439940	1	1	yes	characteristic
111	C₂²	#21439959	1	1	yes	characteristic
112	C₄	—	956	2	no	—
113	C₄	C₂×C₄⋊Dic₂₃₉	1	1	yes	characteristic
114	C₄	D₄×Dic₂₃₉	1	1	yes	characteristic
115	C₄	#21439935	1	1	yes	characteristic
116	C₄	#21439951	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	2	1	no	—
119	C₂	—	2	1	no	—
120	C₂	#29504671	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#29504821

C₂

yes

maximal

C₂×#21439896

C₂

yes

maximal, characteristic

#29504727

C₂

yes

maximal, characteristic

#29504544

C₂

yes

maximal

C₂×#108×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838532

C₂²

yes

—

C₂×#16838532

C₂²

yes

characteristic

#21439919

C₂²

yes

—

—

—

—

C₈×Dic₂₃₉

—

C₂×M₁₆×C₂₃₉

C₂²

yes

—

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

#108×C₂₃₉

C₄

yes

—

—

—

D₄

yes

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₄⋊Dic₂₃₉

—

C₂²⋊Dic₂₃₉

Q₈

yes

—

C₄×Dic₂₃₉

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

M₁₆×C₂₃₉

—

M₁₆×C₂₃₉

C₂×C₄

yes

—

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

C₂×C₄

yes

—

C₂×C₈×C₂₃₉

D₄

yes

characteristic

C₂×C₈×C₂₃₉

D₄

yes

characteristic

C₂×C₈×C₂₃₉

Q₈

yes

—

C₂×Dic₂₃₉

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂×Q₈

yes

characteristic

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

C₂×C₄×C₂₃₉

yes

C₂×C₄×C₂₃₉

yes

C₈×C₂₃₉

—

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₄×Q₈

yes

characteristic

C₂²×C₂₃₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₄×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃₉

C₄×D₄

yes

C₄×C₂₃₉

yes

C₄×C₂₃₉

yes

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#389

yes

characteristic

C₂₃₉

#1953

yes

characteristic, complemented, Sylow

#1953

—

239

maximal, Sylow

#430

—

239

—

C₂×#108

—

239

—

C₂×#108

D₂₃₉

yes

characteristic

#405

—

239

—

#344

—

239

—

C₂×M₁₆

—

239

—

C₂×M₁₆

—

239

—

C₂×M₁₆

C₂×D₂₃₉

yes

—

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

#117

—

239

—

#108

—

478

—

#108

—

478

—

#108

—

478

—

#108

Dic₂₃₉

yes

complemented

#97

—

478

—

C₄×C₈

—

478

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

M₁₆

—

M₁₆

—

478

—

M₁₆

—

478

—

M₁₆

—

478

—

M₁₆

—

239

—

M₁₆

C₂×Dic₂₃₉

yes

—

C₂×C₈

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₂×C₈

C₂×Dic₂₃₉

yes

—

C₂×C₈

D₉₅₆

yes

characteristic

C₂×C₈

D₉₅₆

yes

characteristic

C₂×C₈

Dic₄₇₈

yes

—

C₄⋊C₄

—

478

—

C₂²⋊C₄

—

239

—

C₄²

—

478

—

C₂³

Q₈×D₂₃₉

yes

characteristic

C₂×C₄

—

478

—

C₂×C₄

C₂²×Dic₂₃₉

yes

characteristic

C₂×C₄

D₄×D₂₃₉

yes

characteristic

C₂×C₄

C₂×D₉₅₆

yes

characteristic

100

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

101

C₂×C₄

#16838547

yes

characteristic

102

C₂×C₄

#16838549

yes

characteristic

103

C₈

—

104

C₈

—

956

—

105

C₈

—

478

—

106

C₈

C₄⋊Dic₂₃₉

yes

—

107

C₈

C₄⋊Dic₂₃₉

yes

—

108

C₂²

—

109

C₂²

Q₈×Dic₂₃₉

yes

characteristic

110

C₂²

#21439940

yes

characteristic

111

C₂²

#21439959

yes

characteristic

112

C₄

—

956

—

113

C₄

C₂×C₄⋊Dic₂₃₉

yes

characteristic

114

C₄

D₄×Dic₂₃₉

yes

characteristic

115

C₄

#21439935

yes

characteristic

116

C₄

#21439951

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

120

C₂

#29504671

yes

characteristic

	subgroup	G/H
center	C₄	#21439935
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂×#108×C₂₃₉	C₂
socle	C₂×C₂₃₉	#389
prime radical	C₂³×C₂₃₉	C₂×Q₈
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×#108	D₂₃₉
hypocenter	C₂₃₉	#1953
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₄

#21439935

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₂×#108×C₂₃₉

C₂

socle

C₂×C₂₃₉

#389

prime radical

C₂³×C₂₃₉

C₂×Q₈

cosocle

C₂²×C₄×C₂₃₉

C₂³

D₂₃₉

C₂₃₉

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
16. #108×C₂₃₉	C₄	semidirect
60. C₂₃₉	#1953	semidirect
75. #108	Dic₂₃₉	semidirect

H ≅ G/N

type

16. #108×C₂₃₉

C₄

semidirect

60. C₂₃₉

#1953

semidirect

75. #108

Dic₂₃₉

semidirect

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

subgroup	index
#29504821	2
C₂×#21439896	2
#29504727	2
#29504544	2
C₂×#108×C₂₃₉	2
#1953	239

subgroup

index

C₂×#108×C₂₃₉

239

Subgroups — #44485339

Conjugacy classes of subgroups

Complements

Maximal subgroups