Group #44480554 — subgroups

Subgroups — #1784×C₂₃₉ #44480554

Order 30592.

564 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 144 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	C₂³×Q₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#101×C₂₃₉	C₂	1	4	yes	maximal, complemented
3	#407×C₂₃₉	C₂	1	2	yes	maximal
4	C₂²×Q₈×C₂₃₉	—	2	8	no	—
5	C₂²×Q₈×C₂₃₉	—	2	4	no	—
6	C₂²×Q₈×C₂₃₉	C₄	1	2	yes	—
7	C₂²×Q₈×C₂₃₉	C₂²	1	2	yes	—
8	C₂³×C₄×C₂₃₉	C₄	1	2	yes	—
9	C₂³×C₄×C₂₃₉	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃₉	C₂²	1	4	yes	—
11	#101×C₂₃₉	—	2	8	no	—
12	C₂²⋊C₈×C₂₃₉	—	2	4	no	—
13	C₂⁴×C₂₃₉	C₂×C₄	1	1	yes	characteristic
14	C₂×Q₈×C₂₃₉	—	2	32	no	—
15	C₂×Q₈×C₂₃₉	—	2	8	no	—
16	C₂×Q₈×C₂₃₉	—	2	4	no	—
17	C₂×Q₈×C₂₃₉	—	2	4	no	—
18	C₂×Q₈×C₂₃₉	—	2	4	no	—
19	C₂×Q₈×C₂₃₉	D₄	1	8	yes	—
20	C₂²×C₄×C₂₃₉	—	2	4	no	—
21	C₂²×C₄×C₂₃₉	—	2	8	no	—
22	C₂²×C₄×C₂₃₉	—	2	4	no	—
23	C₂²×C₄×C₂₃₉	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃₉	D₄	1	4	yes	—
26	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₂₃₉	—	2	8	no	—
28	C₂×C₈×C₂₃₉	—	2	4	no	—
29	C₂³×C₂₃₉	—	2	4	no	—
30	C₂³×C₂₃₉	—	2	2	no	—
31	C₂³×C₂₃₉	C₂²⋊C₄	1	2	yes	—
32	C₂³×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
33	Q₈×C₂₃₉	—	2	16	no	—
34	Q₈×C₂₃₉	—	2	16	no	—
35	C₂×C₄×C₂₃₉	—	2	16	no	—
36	C₂×C₄×C₂₃₉	—	2	4	no	—
37	C₂×C₄×C₂₃₉	—	2	2	no	—
38	C₂×C₄×C₂₃₉	—	2	4	no	—
39	C₂×C₄×C₂₃₉	—	2	8	no	—
40	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	8	yes	—
41	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₂₃₉	C₂×D₄	1	4	yes	—
43	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
44	C₈×C₂₃₉	—	4	4	no	—
45	C₂²×C₂₃₉	—	2	8	no	—
46	C₂²×C₂₃₉	—	2	2	no	—
47	C₂²×C₂₃₉	—	2	2	no	—
48	C₂²×C₂₃₉	—	2	2	no	—
49	C₂²×C₂₃₉	#101	1	4	yes	complemented
50	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	2	yes	—
51	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₄×C₂₃₉	—	2	8	no	—
53	C₄×C₂₃₉	—	2	2	no	—
54	C₄×C₂₃₉	#120	1	4	yes	—
55	C₂×C₂₃₉	—	2	4	no	—
56	C₂×C₂₃₉	—	2	2	no	—
57	C₂×C₂₃₉	#379	1	1	yes	characteristic
58	C₂×C₂₃₉	C₂×#101	1	2	yes	—
59	C₂₃₉	#1784	1	1	yes	characteristic, direct product factor, Sylow
60	#1784	C₂₃₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×Q₈	C₂×C₂₃₉	1	1	yes	characteristic
62	C₂×#101	C₂×C₂₃₉	1	4	yes	complemented
63	#407	C₂×C₂₃₉	1	2	yes	—
64	C₂²×Q₈	—	2	8	no	—
65	C₂²×Q₈	—	2	4	no	—
66	C₂²×Q₈	C₄×C₂₃₉	1	2	yes	—
67	C₂²×Q₈	C₂²×C₂₃₉	1	2	yes	—
68	C₂³×C₄	C₄×C₂₃₉	1	2	yes	—
69	C₂³×C₄	C₂²×C₂₃₉	1	1	yes	characteristic
70	C₂×M₁₆	C₂²×C₂₃₉	1	4	yes	—
71	#101	—	2	8	no	—
72	C₂²⋊C₈	—	2	4	no	—
73	C₂⁴	C₂×C₄×C₂₃₉	1	1	yes	characteristic
74	C₂×Q₈	—	2	32	no	—
75	C₂×Q₈	—	2	8	no	—
76	C₂×Q₈	—	2	4	no	—
77	C₂×Q₈	—	2	4	no	—
78	C₂×Q₈	—	2	4	no	—
79	C₂×Q₈	D₄×C₂₃₉	1	8	yes	—
80	C₂²×C₄	—	2	4	no	—
81	C₂²×C₄	—	2	8	no	—
82	C₂²×C₄	—	2	4	no	—
83	C₂²×C₄	C₂×C₄×C₂₃₉	1	4	yes	—
84	C₂²×C₄	C₂×C₄×C₂₃₉	1	1	yes	characteristic
85	C₂²×C₄	D₄×C₂₃₉	1	4	yes	—
86	C₂²×C₄	C₂³×C₂₃₉	1	1	yes	characteristic
87	M₁₆	—	2	8	no	—
88	C₂×C₈	—	2	4	no	—
89	C₂³	—	2	4	no	—
90	C₂³	—	2	2	no	—
91	C₂³	C₂²⋊C₄×C₂₃₉	1	2	yes	—
92	C₂³	C₂²×C₄×C₂₃₉	1	1	yes	characteristic
93	Q₈	—	2	16	no	—
94	Q₈	—	2	16	no	—
95	C₂×C₄	—	2	16	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	8	no	—
100	C₂×C₄	C₂²⋊C₄×C₂₃₉	1	8	yes	—
101	C₂×C₄	C₂²⋊C₄×C₂₃₉	1	2	yes	—
102	C₂×C₄	C₂×D₄×C₂₃₉	1	2	yes	—
103	C₂×C₄	C₂×D₄×C₂₃₉	1	4	yes	—
104	C₈	—	4	4	no	—
105	C₂²	—	2	8	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	2	no	—
108	C₂²	—	2	2	no	—
109	C₂²	#101×C₂₃₉	1	4	yes	complemented
110	C₂²	C₂×C₂²⋊C₄×C₂₃₉	1	2	yes	—
111	C₂²	C₂×C₂²⋊C₄×C₂₃₉	1	1	yes	characteristic
112	C₄	—	2	8	no	—
113	C₄	—	2	2	no	—
114	C₄	#120×C₂₃₉	1	4	yes	—
115	C₂	—	2	4	no	—
116	C₂	—	2	2	no	—
117	C₂	#379×C₂₃₉	1	1	yes	characteristic
118	C₂	C₂×#101×C₂₃₉	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Q₈×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#101×C₂₃₉

C₂

yes

maximal, complemented

#407×C₂₃₉

C₂

yes

maximal

C₂²×Q₈×C₂₃₉

—

C₂²×Q₈×C₂₃₉

—

C₂²×Q₈×C₂₃₉

C₄

yes

—

C₂²×Q₈×C₂₃₉

C₂²

yes

—

C₂³×C₄×C₂₃₉

C₄

yes

—

C₂³×C₄×C₂₃₉

C₂²

yes

characteristic

C₂×M₁₆×C₂₃₉

C₂²

yes

—

#101×C₂₃₉

—

C₂²⋊C₈×C₂₃₉

—

C₂⁴×C₂₃₉

C₂×C₄

yes

characteristic

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

—

C₂×Q₈×C₂₃₉

D₄

yes

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

D₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

M₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

—

C₂³×C₂₃₉

C₂²⋊C₄

yes

—

C₂³×C₂₃₉

C₂²×C₄

yes

characteristic

Q₈×C₂₃₉

—

Q₈×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

#101

yes

complemented

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

—

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#120

yes

—

C₂×C₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#379

yes

characteristic

C₂×C₂₃₉

C₂×#101

yes

—

C₂₃₉

#1784

yes

characteristic, direct product factor, Sylow

#1784

C₂₃₉

yes

maximal, characteristic, direct product factor, Sylow

C₂³×Q₈

C₂×C₂₃₉

yes

characteristic

C₂×#101

C₂×C₂₃₉

yes

complemented

#407

C₂×C₂₃₉

yes

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₄×C₂₃₉

yes

—

C₂²×Q₈

C₂²×C₂₃₉

yes

—

C₂³×C₄

C₄×C₂₃₉

yes

—

C₂³×C₄

C₂²×C₂₃₉

yes

characteristic

C₂×M₁₆

C₂²×C₂₃₉

yes

—

#101

—

C₂²⋊C₈

—

C₂⁴

C₂×C₄×C₂₃₉

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

D₄×C₂₃₉

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₃₉

yes

—

C₂²×C₄

C₂×C₄×C₂₃₉

yes

characteristic

C₂²×C₄

D₄×C₂₃₉

yes

—

C₂²×C₄

C₂³×C₂₃₉

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

C₂²⋊C₄×C₂₃₉

yes

—

C₂³

C₂²×C₄×C₂₃₉

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊C₄×C₂₃₉

yes

—

101

C₂×C₄

C₂²⋊C₄×C₂₃₉

yes

—

102

C₂×C₄

C₂×D₄×C₂₃₉

yes

—

103

C₂×C₄

C₂×D₄×C₂₃₉

yes

—

104

C₈

—

105

C₂²

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

#101×C₂₃₉

yes

complemented

110

C₂²

C₂×C₂²⋊C₄×C₂₃₉

yes

—

111

C₂²

C₂×C₂²⋊C₄×C₂₃₉

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

#120×C₂₃₉

yes

—

115

C₂

—

116

C₂

—

117

C₂

#379×C₂₃₉

yes

characteristic

118

C₂

C₂×#101×C₂₃₉

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃₉	C₂×C₂²⋊C₄
commutator	C₂³	C₂²×C₄×C₂₃₉
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃₉
prime radical	C₂⁴×C₂₃₉	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃₉ ◁ C₂³×C₄×C₂₃₉ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃₉

C₂×C₂²⋊C₄

commutator

C₂³

C₂²×C₄×C₂₃₉

Frattini, cosocle

C₂²×C₄

C₂³×C₂₃₉

prime radical

C₂⁴×C₂₃₉

C₂×C₄

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃₉ ◁ C₂³×C₄×C₂₃₉ ◁ G

N	H ≅ G/N	type
2. C₂×#101×C₂₃₉	C₂	semidirect
49. C₂²×C₂₃₉	#101	semidirect
59. C₂₃₉	#1784	direct (N × H)
60. #1784	C₂₃₉	direct (N × H)
62. C₂×#101	C₂×C₂₃₉	semidirect
109. C₂²	#101×C₂₃₉	semidirect

H ≅ G/N

type

2. C₂×#101×C₂₃₉

C₂

semidirect

49. C₂²×C₂₃₉

#101

semidirect

59. C₂₃₉

#1784

direct (N × H)

60. #1784

C₂₃₉

direct (N × H)

62. C₂×#101

C₂×C₂₃₉

semidirect

109. C₂²

#101×C₂₃₉

semidirect

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

subgroup	index
C₂³×Q₈×C₂₃₉	2
C₂×#101×C₂₃₉	2
#407×C₂₃₉	2
#1784	239

subgroup

index

C₂³×Q₈×C₂₃₉

C₂×#101×C₂₃₉

#407×C₂₃₉

#1784

239

Subgroups — #1784×C₂₃₉ #44480554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1784×C239 #44480554

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1784×C₂₃₉ #44480554