Group #44484665 — subgroups

Subgroups — #44484665

Order 30592.

260 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 149 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×C₄⋊(C₂₃₉⋊C₈)	C₂	1	2	yes	maximal
2	#29504527	C₂	1	4	yes	maximal
3	C₄³×C₂₃₉	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₂₃₉⋊C₈	C₂²	1	4	yes	—
5	C₄⋊(C₂₃₉⋊C₈)	—	2	4	no	—
6	C₂×C₄²×C₂₃₉	C₄	1	2	yes	—
7	C₂×C₄²×C₂₃₉	C₄	1	1	yes	characteristic
8	C₂×C₄²×C₂₃₉	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₂₃₉	C₂²	1	2	yes	—
10	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
11	C₂×C₂₃₉⋊C₈	—	2	4	no	—
12	C₂×C₂₃₉⋊C₈	—	2	8	no	—
13	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
15	C₂²×C₄×C₂₃₉	C₂×C₄	1	2	yes	—
16	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
18	C₄²×C₂₃₉	—	2	4	no	—
19	C₄²×C₂₃₉	—	2	2	no	—
20	C₄²×C₂₃₉	—	2	4	no	—
21	C₄²×C₂₃₉	C₈	1	4	yes	complemented
22	C₄²×C₂₃₉	D₄	1	2	yes	—
23	C₄²×C₂₃₉	Q₈	1	2	yes	—
24	C₂₃₉⋊C₈	—	4	4	no	—
25	C₂³×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
26	C₂×C₄×C₂₃₉	—	2	2	no	—
27	C₂×C₄×C₂₃₉	—	2	2	no	—
28	C₂×C₄×C₂₃₉	—	2	2	no	—
29	C₂×C₄×C₂₃₉	—	2	2	no	—
30	C₂×C₄×C₂₃₉	—	2	2	no	—
31	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
32	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
33	C₂×C₄×C₂₃₉	C₂×C₈	1	4	yes	—
34	C₂×C₄×C₂₃₉	C₂×C₈	1	2	yes	—
35	C₂×C₄×C₂₃₉	M₁₆	1	4	yes	—
36	C₂×C₄×C₂₃₉	M₁₆	1	2	yes	—
37	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₄×C₂₃₉	#41	1	4	yes	—
40	C₂²×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₂₃₉	#117	1	2	yes	—
42	C₂²×C₂₃₉	C₂²×C₈	1	1	yes	characteristic
43	C₂²×C₂₃₉	C₂×M₁₆	1	1	yes	characteristic
44	C₂²×C₂₃₉	C₂×M₁₆	1	2	yes	—
45	C₄×C₂₃₉	—	2	2	no	—
46	C₄×C₂₃₉	—	2	4	no	—
47	C₄×C₂₃₉	—	2	4	no	—
48	C₄×C₂₃₉	C₄⋊C₈	1	4	yes	—
49	C₄×C₂₃₉	#124	1	2	yes	—
50	C₄×C₂₃₉	#125	1	2	yes	—
51	C₂×C₂₃₉	#382	1	1	yes	characteristic
52	C₂×C₂₃₉	C₂×C₄⋊C₈	1	1	yes	characteristic
53	C₂×C₂₃₉	#423	1	1	yes	characteristic
54	C₂×C₂₃₉	#431	1	2	yes	—
55	C₂×C₂₃₉	#432	1	2	yes	—
56	C₂₃₉	#1831	1	1	yes	characteristic, complemented, Sylow
57	#1831	—	239	1	no	maximal, Sylow
58	C₂×C₄⋊C₈	—	239	2	no	—
59	C₄³	D₂₃₉	1	1	yes	characteristic
60	#336	—	239	4	no	—
61	C₂²×C₈	—	239	4	no	—
62	C₂×C₄²	Dic₂₃₉	1	2	yes	—
63	C₂×C₄²	Dic₂₃₉	1	1	yes	characteristic
64	C₂×C₄²	Dic₂₃₉	1	1	yes	characteristic
65	C₂×C₄²	C₂×D₂₃₉	1	2	yes	—
66	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
67	C₄⋊C₈	—	478	4	no	—
68	C₂²×C₄	C₂×Dic₂₃₉	1	1	yes	characteristic
69	C₂²×C₄	C₂×Dic₂₃₉	1	2	yes	—
70	C₂²×C₄	C₂×Dic₂₃₉	1	2	yes	—
71	C₂²×C₄	C₂×Dic₂₃₉	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
73	C₂×C₈	—	478	4	no	—
74	C₂×C₈	—	478	8	no	—
75	C₄²	—	2	4	no	—
76	C₄²	—	2	2	no	—
77	C₄²	—	2	4	no	—
78	C₄²	C₂₃₉⋊C₈	1	4	yes	complemented
79	C₄²	D₉₅₆	1	2	yes	—
80	C₄²	Dic₄₇₈	1	2	yes	—
81	C₂³	C₂²×Dic₂₃₉	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	2	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	C₄⋊Dic₂₃₉	1	2	yes	—
88	C₂×C₄	C₄⋊Dic₂₃₉	1	2	yes	—
89	C₂×C₄	C₂×C₂₃₉⋊C₈	1	4	yes	—
90	C₂×C₄	C₂×C₂₃₉⋊C₈	1	2	yes	—
91	C₂×C₄	#16838532	1	4	yes	—
92	C₂×C₄	#16838532	1	2	yes	—
93	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
94	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
95	C₂×C₄	#16838548	1	4	yes	—
96	C₈	—	956	4	no	—
97	C₂²	C₂×C₄⋊Dic₂₃₉	1	1	yes	characteristic
98	C₂²	#21439919	1	2	yes	—
99	C₂²	C₂²×C₂₃₉⋊C₈	1	1	yes	characteristic
100	C₂²	C₂×#16838532	1	1	yes	characteristic
101	C₂²	C₂×#16838532	1	2	yes	—
102	C₄	—	2	2	no	—
103	C₄	—	2	4	no	—
104	C₄	—	2	4	no	—
105	C₄	C₄⋊(C₂₃₉⋊C₈)	1	4	yes	—
106	C₄	#21439948	1	2	yes	—
107	C₄	#21439952	1	2	yes	—
108	C₂	#29504638	1	1	yes	characteristic
109	C₂	C₂×C₄⋊(C₂₃₉⋊C₈)	1	1	yes	characteristic
110	C₂	#29504796	1	1	yes	characteristic
111	C₂	#29504825	1	2	yes	—
112	C₂	#29504829	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃₉⋊C₈)

C₂

yes

maximal

#29504527

C₂

yes

maximal

C₄³×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

C₄⋊(C₂₃₉⋊C₈)

—

C₂×C₄²×C₂₃₉

C₄

yes

—

C₂×C₄²×C₂₃₉

C₄

yes

characteristic

C₂×C₄²×C₂₃₉

C₄

yes

characteristic

C₂×C₄²×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₄²×C₂₃₉

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

—

C₄²×C₂₃₉

C₈

yes

complemented

C₄²×C₂₃₉

D₄

yes

—

C₄²×C₂₃₉

Q₈

yes

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂×C₈

yes

—

C₂×C₄×C₂₃₉

C₂×C₈

yes

—

C₂×C₄×C₂₃₉

M₁₆

yes

—

C₂×C₄×C₂₃₉

M₁₆

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

—

C₂²×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃₉

#117

yes

—

C₂²×C₂₃₉

C₂²×C₈

yes

characteristic

C₂²×C₂₃₉

C₂×M₁₆

yes

characteristic

C₂²×C₂₃₉

C₂×M₁₆

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₄⋊C₈

yes

—

C₄×C₂₃₉

#124

yes

—

C₄×C₂₃₉

#125

yes

—

C₂×C₂₃₉

#382

yes

characteristic

C₂×C₂₃₉

C₂×C₄⋊C₈

yes

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

—

C₂×C₂₃₉

#432

yes

—

C₂₃₉

#1831

yes

characteristic, complemented, Sylow

#1831

—

239

maximal, Sylow

C₂×C₄⋊C₈

—

239

—

C₄³

D₂₃₉

yes

characteristic

#336

—

239

—

C₂²×C₈

—

239

—

C₂×C₄²

Dic₂₃₉

yes

—

C₂×C₄²

Dic₂₃₉

yes

characteristic

C₂×C₄²

Dic₂₃₉

yes

characteristic

C₂×C₄²

C₂×D₂₃₉

yes

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₄⋊C₈

—

478

—

C₂²×C₄

C₂×Dic₂₃₉

yes

characteristic

C₂²×C₄

C₂×Dic₂₃₉

yes

—

C₂²×C₄

C₂×Dic₂₃₉

yes

—

C₂²×C₄

C₂×Dic₂₃₉

yes

characteristic

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂₃₉⋊C₈

yes

complemented

C₄²

D₉₅₆

yes

—

C₄²

Dic₄₇₈

yes

—

C₂³

C₂²×Dic₂₃₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₃₉

yes

—

C₂×C₄

C₄⋊Dic₂₃₉

yes

—

C₂×C₄

C₂×C₂₃₉⋊C₈

yes

—

C₂×C₄

C₂×C₂₃₉⋊C₈

yes

—

C₂×C₄

#16838532

yes

—

C₂×C₄

#16838532

yes

—

C₂×C₄

C₂×D₉₅₆

yes

characteristic

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

C₂×C₄

#16838548

yes

—

C₈

—

956

—

C₂²

C₂×C₄⋊Dic₂₃₉

yes

characteristic

C₂²

#21439919

yes

—

C₂²

C₂²×C₂₃₉⋊C₈

yes

characteristic

100

C₂²

C₂×#16838532

yes

characteristic

101

C₂²

C₂×#16838532

yes

—

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

C₄⋊(C₂₃₉⋊C₈)

yes

—

106

C₄

#21439948

yes

—

107

C₄

#21439952

yes

—

108

C₂

#29504638

yes

characteristic

109

C₂

C₂×C₄⋊(C₂₃₉⋊C₈)

yes

characteristic

110

C₂

#29504796

yes

characteristic

111

C₂

#29504825

yes

—

112

C₂

#29504829

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃₉
commutator	C₂²×C₂₃₉	C₂²×C₈
Fitting	C₄³×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂²×C₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₄³	D₂₃₉
hypocenter	C₂₃₉	#1831
LCS	G ▷ C₂²×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃₉

commutator

C₂²×C₂₃₉

C₂²×C₈

Fitting

C₄³×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂²×C₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₄³

D₂₃₉

C₂₃₉

G ▷ C₂²×C₂₃₉ ▷ C₂₃₉

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
21. C₄²×C₂₃₉	C₈	semidirect
56. C₂₃₉	#1831	semidirect
78. C₄²	C₂₃₉⋊C₈	semidirect

H ≅ G/N

type

21. C₄²×C₂₃₉

C₈

semidirect

56. C₂₃₉

#1831

semidirect

78. C₄²

C₂₃₉⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃₉⋊C₈)	2
#29504527	2
C₄³×C₂₃₉	2
#1831	239

subgroup

index

C₂×C₄⋊(C₂₃₉⋊C₈)

#29504527

C₄³×C₂₃₉

#1831

239

Subgroups — #44484665

Conjugacy classes of subgroups

Complements

Maximal subgroups