Group #44486599 — subgroups

Subgroups — #44486599

Order 30592.

154 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 154 conjugacy classes of subgroups — too many to draw. Only its 55 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×Dic₁₉₁₂	C₂	1	2	yes	maximal
2	#29505152	C₂	1	1	yes	maximal, characteristic
3	C₄×Dic₉₅₆	C₂	1	1	yes	maximal, characteristic
4	#29504577	C₂	1	2	yes	maximal
5	C₄⋊C₁₆×C₂₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₉₅₆	—	2	1	no	—
7	C₂×Dic₉₅₆	C₂²	1	2	yes	—
8	C₂×Dic₉₅₆	C₂²	1	1	yes	characteristic
9	#21439960	—	2	1	no	—
10	C₄×Dic₄₇₈	—	2	1	no	—
11	Dic₁₉₁₂	—	2	4	no	—
12	#21439894	C₂²	1	1	yes	characteristic
13	#21439885	—	2	1	no	—
14	C₂×C₁₆×C₂₃₉	C₂²	1	2	yes	—
15	C₄×C₈×C₂₃₉	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₇₈	—	2	1	no	—
17	C₂×Dic₄₇₈	—	2	2	no	—
18	Dic₉₅₆	—	2	2	no	—
19	Dic₉₅₆	—	2	1	no	—
20	Dic₉₅₆	—	2	2	no	—
21	Dic₉₅₆	—	4	2	no	—
22	Dic₉₅₆	D₄	1	2	yes	—
23	C₄⋊Dic₂₃₉	—	2	1	no	—
24	C₄⋊Dic₂₃₉	—	4	1	no	—
25	C₄⋊Dic₂₃₉	—	4	1	no	—
26	Dic₂₃₉⋊C₄	—	4	1	no	—
27	C₄×Dic₂₃₉	—	4	1	no	—
28	C₂×C₈×C₂₃₉	D₄	1	1	yes	characteristic
29	C₂×C₈×C₂₃₉	C₂³	1	1	yes	characteristic
30	C₄²×C₂₃₉	D₄	1	1	yes	characteristic
31	C₁₆×C₂₃₉	—	2	2	no	—
32	Dic₄₇₈	—	2	2	no	—
33	Dic₄₇₈	—	4	1	no	—
34	Dic₄₇₈	—	4	2	no	—
35	Dic₄₇₈	—	4	2	no	—
36	C₂×Dic₂₃₉	—	4	1	no	—
37	C₂×Dic₂₃₉	—	4	1	no	—
38	C₂×Dic₂₃₉	—	4	2	no	—
39	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
40	C₂×C₄×C₂₃₉	D₈	1	1	yes	characteristic
41	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
42	C₈×C₂₃₉	—	2	1	no	—
43	C₈×C₂₃₉	C₂×D₄	1	1	yes	characteristic
44	C₈×C₂₃₉	#41	1	1	yes	characteristic
45	C₂²×C₂₃₉	C₂×D₈	1	1	yes	characteristic
46	C₄×C₂₃₉	—	2	1	no	—
47	C₄×C₂₃₉	Q₃₂	1	2	yes	complemented
48	C₄×C₂₃₉	#121	1	1	yes	characteristic
49	C₄×C₂₃₉	AGL(1,ℤ/8)	1	1	yes	characteristic
50	Dic₂₃₉	—	4	2	no	—
51	Dic₂₃₉	—	8	2	no	—
52	Dic₂₃₉	—	8	1	no	—
53	C₂×C₂₃₉	#459	1	1	yes	characteristic
54	C₂×C₂₃₉	C₂×Q₃₂	1	1	yes	characteristic
55	C₂×C₂₃₉	#510	1	1	yes	characteristic
56	C₂₃₉	#2209	1	1	yes	characteristic, complemented, Sylow
57	#2209	—	239	1	no	maximal, Sylow
58	C₂×Q₃₂	—	239	2	no	—
59	#494	—	239	1	no	—
60	C₄×Q₁₆	—	239	1	no	—
61	C₄⋊C₁₆	D₂₃₉	1	1	yes	characteristic
62	#358	—	239	2	no	—
63	C₂×Q₁₆	—	478	1	no	—
64	C₂×Q₁₆	—	239	2	no	—
65	C₂×Q₁₆	—	239	1	no	—
66	C₄⋊Q₈	—	478	1	no	—
67	C₄×Q₈	—	478	1	no	—
68	Q₃₂	—	478	4	no	—
69	C₂×C₁₆	C₂×D₂₃₉	1	2	yes	—
70	#107	—	239	1	no	—
71	#103	—	478	1	no	—
72	C₄×C₈	C₂×D₂₃₉	1	1	yes	characteristic
73	C₂×Q₈	—	478	2	no	—
74	C₂×Q₈	—	478	1	no	—
75	Q₁₆	—	956	2	no	—
76	Q₁₆	—	478	2	no	—
77	Q₁₆	—	478	2	no	—
78	Q₁₆	—	478	1	no	—
79	Q₁₆	—	239	2	no	—
80	C₂×C₈	D₉₅₆	1	1	yes	characteristic
81	C₂×C₈	C₂²×D₂₃₉	1	1	yes	characteristic
82	C₄⋊C₄	—	956	1	no	—
83	C₄⋊C₄	—	956	1	no	—
84	C₄⋊C₄	—	956	1	no	—
85	C₄⋊C₄	—	478	1	no	—
86	C₄²	—	956	1	no	—
87	C₄²	D₉₅₆	1	1	yes	characteristic
88	C₁₆	—	2	2	no	—
89	Q₈	—	956	2	no	—
90	Q₈	—	956	2	no	—
91	Q₈	—	956	1	no	—
92	Q₈	—	478	2	no	—
93	C₂×C₄	—	956	1	no	—
94	C₂×C₄	—	956	2	no	—
95	C₂×C₄	—	956	1	no	—
96	C₂×C₄	D₁₉₁₂	1	1	yes	characteristic
97	C₂×C₄	D₁₉₁₂	1	1	yes	characteristic
98	C₂×C₄	C₂×D₉₅₆	1	1	yes	characteristic
99	C₈	—	2	1	no	—
100	C₈	D₄×D₂₃₉	1	1	yes	characteristic
101	C₈	#16838549	1	1	yes	characteristic
102	C₂²	C₂×D₁₉₁₂	1	1	yes	characteristic
103	C₄	—	2	1	no	—
104	C₄	—	1912	1	no	—
105	C₄	—	1912	2	no	—
106	C₄	—	956	2	no	—
107	C₄	Dic₁₉₁₂	1	2	yes	complemented
108	C₄	#21439935	1	1	yes	characteristic
109	C₄	#21439988	1	1	yes	characteristic
110	C₂	#29504968	1	1	yes	characteristic
111	C₂	C₂×Dic₁₉₁₂	1	1	yes	characteristic
112	C₂	#29505215	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₉₁₂

C₂

yes

maximal

#29505152

C₂

yes

maximal, characteristic

C₄×Dic₉₅₆

C₂

yes

maximal, characteristic

#29504577

C₂

yes

maximal

C₄⋊C₁₆×C₂₃₉

C₂

yes

maximal, characteristic

C₂×Dic₉₅₆

—

C₂×Dic₉₅₆

C₂²

yes

—

C₂×Dic₉₅₆

C₂²

yes

characteristic

#21439960

—

C₄×Dic₄₇₈

—

Dic₁₉₁₂

—

#21439894

C₂²

yes

characteristic

#21439885

—

C₂×C₁₆×C₂₃₉

C₂²

yes

—

C₄×C₈×C₂₃₉

C₂²

yes

characteristic

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

—

Dic₉₅₆

—

Dic₉₅₆

—

Dic₉₅₆

—

Dic₉₅₆

—

Dic₉₅₆

D₄

yes

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

Dic₂₃₉⋊C₄

—

C₄×Dic₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

characteristic

C₂×C₈×C₂₃₉

C₂³

yes

characteristic

C₄²×C₂₃₉

D₄

yes

characteristic

C₁₆×C₂₃₉

—

Dic₄₇₈

—

Dic₄₇₈

—

Dic₄₇₈

—

Dic₄₇₈

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×C₄×C₂₃₉

D₈

yes

characteristic

C₂×C₄×C₂₃₉

D₈

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₈×C₂₃₉

—

C₈×C₂₃₉

C₂×D₄

yes

characteristic

C₈×C₂₃₉

#41

yes

characteristic

C₂²×C₂₃₉

C₂×D₈

yes

characteristic

C₄×C₂₃₉

—

C₄×C₂₃₉

Q₃₂

yes

C₄×C₂₃₉

yes

C₄×C₂₃₉

yes

Dic₂₃₉

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

#459

yes

characteristic

C₂×C₂₃₉

C₂×Q₃₂

yes

C₂×C₂₃₉

yes

C₂₃₉

yes

characteristic, complemented, Sylow

#2209

—

239

maximal, Sylow

C₂×Q₃₂

—

239

—

#494

—

239

—

C₄×Q₁₆

—

239

—

C₄⋊C₁₆

D₂₃₉

yes

characteristic

#358

—

239

—

C₂×Q₁₆

—

478

—

C₂×Q₁₆

—

239

—

C₂×Q₁₆

—

239

—

C₄⋊Q₈

—

478

—

C₄×Q₈

—

478

—

Q₃₂

—

478

—

C₂×C₁₆

C₂×D₂₃₉

yes

—

#107

—

239

—

#103

—

478

—

C₄×C₈

C₂×D₂₃₉

yes

characteristic

C₂×Q₈

—

478

—

C₂×Q₈

—

478

—

Q₁₆

—

956

—

Q₁₆

—

478

—

Q₁₆

—

478

—

Q₁₆

—

478

—

Q₁₆

—

239

—

C₂×C₈

D₉₅₆

yes

characteristic

C₂×C₈

C₂²×D₂₃₉

yes

characteristic

C₄⋊C₄

—

956

—

C₄⋊C₄

—

956

—

C₄⋊C₄

—

956

—

C₄⋊C₄

—

478

—

C₄²

—

956

—

C₄²

D₉₅₆

yes

characteristic

C₁₆

—

Q₈

—

956

—

Q₈

—

956

—

Q₈

—

956

—

Q₈

—

478

—

C₂×C₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

D₁₉₁₂

yes

characteristic

C₂×C₄

D₁₉₁₂

yes

characteristic

C₂×C₄

C₂×D₉₅₆

yes

characteristic

C₈

—

100

C₈

D₄×D₂₃₉

yes

characteristic

101

C₈

#16838549

yes

characteristic

102

C₂²

C₂×D₁₉₁₂

yes

characteristic

103

C₄

—

104

C₄

—

1912

—

105

C₄

—

1912

—

106

C₄

—

956

—

107

C₄

Dic₁₉₁₂

yes

complemented

108

C₄

#21439935

yes

characteristic

109

C₄

#21439988

yes

characteristic

110

C₂

#29504968

yes

characteristic

111

C₂

C₂×Dic₁₉₁₂

yes

characteristic

112

C₂

#29505215

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₉₁₂
commutator, cosocle	C₂×C₈×C₂₃₉	C₂³
Frattini	C₂×C₈	C₂²×D₂₃₉
Fitting	C₄⋊C₁₆×C₂₃₉	C₂
socle, prime radical	C₂²×C₂₃₉	C₂×D₈
hypercenter	C₄⋊C₁₆	D₂₃₉
hypocenter	C₂₃₉	#2209
LCS	G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₁₉₁₂

commutator, cosocle

C₂×C₈×C₂₃₉

C₂³

Frattini

C₂×C₈

C₂²×D₂₃₉

Fitting

C₄⋊C₁₆×C₂₃₉

C₂

socle, prime radical

C₂²×C₂₃₉

C₂×D₈

hypercenter

C₄⋊C₁₆

D₂₃₉

hypocenter

C₂₃₉

#2209

LCS

G ▷ C₂×C₈×C₂₃₉ ▷ C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
47. C₄×C₂₃₉	Q₃₂	semidirect
56. C₂₃₉	#2209	semidirect
107. C₄	Dic₁₉₁₂	semidirect

H ≅ G/N

type

47. C₄×C₂₃₉

Q₃₂

semidirect

56. C₂₃₉

#2209

semidirect

107. C₄

Dic₁₉₁₂

semidirect

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

subgroup	index
C₂×Dic₁₉₁₂	2
#29505152	2
C₄×Dic₉₅₆	2
#29504577	2
C₄⋊C₁₆×C₂₃₉	2
#2209	239

subgroup

index

C₂×Dic₁₉₁₂

#29505152

C₄×Dic₉₅₆

#29504577

C₄⋊C₁₆×C₂₃₉

#2209

239

Subgroups — #44486599

Conjugacy classes of subgroups

Complements

Maximal subgroups