Group #44484721 — subgroups

Subgroups — #44484721

Order 30592.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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	C₂×#21439895	C₂	1	2	yes	maximal
3	#29504795	C₂	1	2	yes	maximal
4	C₂×#21439870	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×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	#21439895	—	2	4	no	—
11	Dic₂₃₉⋊C₈	—	2	4	no	—
12	#21439870	C₄	1	4	yes	—
13	C₂×M₁₆×C₂₃₉	C₂²	1	2	yes	—
14	C₂²×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
15	#108×C₂₃₉	—	2	2	no	—
16	C₂²×Dic₂₃₉	Q₈	1	2	yes	—
17	#16838532	—	2	2	no	—
18	#16838532	—	2	2	no	—
19	C₂×C₂₃₉⋊C₈	—	2	2	no	—
20	C₂×C₂₃₉⋊C₈	C₂×C₄	1	2	yes	—
21	C₂×C₂₃₉⋊C₈	D₄	1	2	yes	—
22	C₂×C₂₃₉⋊C₈	Q₈	1	2	yes	—
23	C₄×Dic₂₃₉	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₃₉	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₃₉	—	2	2	no	—
27	M₁₆×C₂₃₉	—	2	2	no	—
28	C₂×C₈×C₂₃₉	—	2	2	no	—
29	C₂×C₈×C₂₃₉	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₂₃₉	D₄	1	2	yes	—
31	C₂×C₈×C₂₃₉	Q₈	1	2	yes	—
32	C₂×Dic₂₃₉	—	4	2	no	—
33	C₂×Dic₂₃₉	C₄⋊C₄	1	4	yes	—
34	C₂₃₉⋊C₈	—	4	2	no	—
35	C₂₃₉⋊C₈	C₄⋊C₄	1	4	yes	—
36	C₂³×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃₉	C₂×D₄	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	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₂₃₉	—	4	2	no	—
44	C₈×C₂₃₉	C₄⋊C₄	1	4	yes	—
45	C₂²×C₂₃₉	—	2	2	no	—
46	C₂²×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₂₃₉	#127	1	1	yes	characteristic
52	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₂₃₉	—	2	4	no	—
54	C₂×C₂₃₉	—	2	1	no	—
55	C₂×C₂₃₉	—	2	1	no	—
56	C₂×C₂₃₉	#384	1	1	yes	characteristic
57	C₂×C₂₃₉	#430	1	2	yes	—
58	C₂₃₉	#1849	1	1	yes	characteristic, complemented, Sylow
59	#1849	—	239	1	no	maximal, Sylow
60	C₂×#108	—	239	2	no	—
61	C₂×#108	—	239	1	no	—
62	C₂×#108	D₂₃₉	1	1	yes	characteristic
63	#423	—	239	2	no	—
64	C₂×#97	—	239	1	no	—
65	C₂×M₁₆	—	239	2	no	—
66	C₂×M₁₆	C₂×D₂₃₉	1	2	yes	—
67	C₂²×C₈	—	239	1	no	—
68	C₂²×C₈	C₂×D₂₃₉	1	1	yes	characteristic
69	C₂×C₄²	—	239	1	no	—
70	#108	—	2	2	no	—
71	#108	—	478	4	no	—
72	#108	—	478	2	no	—
73	C₄⋊C₈	—	478	4	no	—
74	#97	—	239	4	no	—
75	C₂²×C₄	—	239	2	no	—
76	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	478	2	no	—
80	M₁₆	—	478	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	478	2	no	—
83	C₂×C₈	—	239	2	no	—
84	C₂×C₈	—	239	2	no	—
85	C₂×C₈	—	239	2	no	—
86	C₂×C₈	C₄×D₂₃₉	1	2	yes	—
87	C₂×C₈	C₂²⋊D₂₃₉	1	2	yes	—
88	C₂×C₈	Dic₄₇₈	1	2	yes	—
89	C₄²	—	239	2	no	—
90	C₄²	—	239	2	no	—
91	C₂³	Q₈×D₂₃₉	1	1	yes	characteristic
92	C₂×C₄	—	956	2	no	—
93	C₂×C₄	—	239	4	no	—
94	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
95	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
96	C₂×C₄	D₄×D₂₃₉	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₂₃₉	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	956	2	no	—
102	C₈	—	239	4	no	—
103	C₈	Dic₂₃₉⋊C₄	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₂₃₉	1	1	yes	characteristic
106	C₂²	#21439958	1	1	yes	characteristic
107	C₂²	#21439959	1	1	yes	characteristic
108	C₄	—	478	4	no	—
109	C₄	C₄⋊C₄×D₂₃₉	1	1	yes	characteristic
110	C₄	C₂×Dic₂₃₉⋊C₄	1	1	yes	characteristic
111	C₄	#21439956	1	1	yes	characteristic
112	C₄	#21439959	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#29504641	1	1	yes	characteristic
116	C₂	#29504819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439896

C₂

yes

maximal, characteristic

C₂×#21439895

C₂

yes

maximal

#29504795

C₂

yes

maximal

C₂×#21439870

C₂

yes

maximal, characteristic

C₂×#108×C₂₃₉

C₂

yes

maximal, characteristic

C₂×#16838532

C₂²

yes

—

C₂²×C₂₃₉⋊C₈

C₂²

yes

characteristic

C₂×C₄×Dic₂₃₉

C₂²

yes

characteristic

#21439896

—

#21439895

—

Dic₂₃₉⋊C₈

—

#21439870

C₄

yes

—

C₂×M₁₆×C₂₃₉

C₂²

yes

—

C₂²×C₈×C₂₃₉

C₂²

yes

characteristic

#108×C₂₃₉

—

C₂²×Dic₂₃₉

Q₈

yes

—

#16838532

—

#16838532

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

C₂×C₄

yes

—

C₂×C₂₃₉⋊C₈

D₄

yes

—

C₂×C₂₃₉⋊C₈

Q₈

yes

—

C₄×Dic₂₃₉

C₂×C₄

yes

—

C₄×Dic₂₃₉

D₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

M₁₆×C₂₃₉

—

M₁₆×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

C₂×C₄

yes

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₂×C₈×C₂₃₉

Q₈

yes

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

C₄⋊C₄

yes

—

C₂₃₉⋊C₈

—

C₂₃₉⋊C₈

C₄⋊C₄

yes

—

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×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₈×C₂₃₉

—

C₈×C₂₃₉

C₄⋊C₄

yes

—

C₂²×C₂₃₉

—

C₂²×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₄×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃₉

#127

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

Dic₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

—

C₂×C₂₃₉

#384

yes

characteristic

C₂×C₂₃₉

#430

yes

—

C₂₃₉

#1849

yes

characteristic, complemented, Sylow

#1849

—

239

maximal, Sylow

C₂×#108

—

239

—

C₂×#108

—

239

—

C₂×#108

D₂₃₉

yes

characteristic

#423

—

239

—

C₂×#97

—

239

—

C₂×M₁₆

—

239

—

C₂×M₁₆

C₂×D₂₃₉

yes

—

C₂²×C₈

—

239

—

C₂²×C₈

C₂×D₂₃₉

yes

characteristic

C₂×C₄²

—

239

—

#108

—

#108

—

478

—

#108

—

478

—

C₄⋊C₈

—

478

—

#97

—

239

—

C₂²×C₄

—

239

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

478

—

M₁₆

—

478

—

C₂×C₈

—

C₂×C₈

—

478

—

C₂×C₈

—

239

—

C₂×C₈

—

239

—

C₂×C₈

—

239

—

C₂×C₈

C₄×D₂₃₉

yes

—

C₂×C₈

C₂²⋊D₂₃₉

yes

—

C₂×C₈

Dic₄₇₈

yes

—

C₄²

—

239

—

C₄²

—

239

—

C₂³

Q₈×D₂₃₉

yes

characteristic

C₂×C₄

—

956

—

C₂×C₄

—

239

—

C₂×C₄

C₂×C₄×D₂₃₉

yes

characteristic

C₂×C₄

D₄×D₂₃₉

yes

characteristic

C₂×C₄

D₄×D₂₃₉

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

C₂×C₄

Q₈×D₂₃₉

yes

characteristic

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

100

C₈

—

101

C₈

—

956

—

102

C₈

—

239

—

103

C₈

Dic₂₃₉⋊C₄

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₂₃₉

yes

characteristic

106

C₂²

#21439958

yes

characteristic

107

C₂²

#21439959

yes

characteristic

108

C₄

—

478

—

109

C₄

C₄⋊C₄×D₂₃₉

yes

characteristic

110

C₄

C₂×Dic₂₃₉⋊C₄

yes

characteristic

111

C₄

#21439956

yes

characteristic

112

C₄

#21439959

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#29504641

yes

characteristic

116

C₂

#29504819

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃₉
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂×#108×C₂₃₉	C₂
socle	C₂²×C₂₃₉	C₂×C₄⋊C₄
prime radical	C₂³×C₂₃₉	C₂×Q₈
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×#108	D₂₃₉
hypocenter	C₂₃₉	#1849
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₂₃₉

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₂×#108×C₂₃₉

C₂

socle

C₂²×C₂₃₉

C₂×C₄⋊C₄

prime radical

C₂³×C₂₃₉

C₂×Q₈

cosocle

C₂²×C₄×C₂₃₉

C₂³

D₂₃₉

C₂₃₉

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
58. C₂₃₉	#1849	semidirect

H ≅ G/N

type

58. C₂₃₉

#1849

semidirect

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

subgroup	index
C₂×#21439896	2
C₂×#21439895	2
#29504795	2
C₂×#21439870	2
C₂×#108×C₂₃₉	2
#1849	239

subgroup

index

C₂×#108×C₂₃₉

239

Subgroups — #44484721

Conjugacy classes of subgroups

Complements

Maximal subgroups