Group #44484701 — subgroups

Subgroups — #44484701

Order 30592.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	1	yes	maximal, characteristic
2	C₂×Dic₂₃₉⋊C₈	C₂	1	4	yes	maximal
3	#29504618	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₂₃₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×Dic₂₃₉	C₂²	1	2	yes	—
7	C₄⋊(C₂₃₉⋊C₈)	—	2	2	no	—
8	Dic₂₃₉⋊C₈	—	2	8	no	—
9	C₂₃₉⋊#95	C₄	1	2	yes	—
10	C₂₃₉⋊#95	C₄	1	2	yes	—
11	C₂²×C₈×C₂₃₉	C₂²	1	2	yes	—
12	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₂₃₉	—	2	2	no	—
14	C₂²×Dic₂₃₉	C₂×C₄	1	4	yes	—
15	C₂×C₂₃₉⋊C₈	—	2	4	no	—
16	C₂×C₂₃₉⋊C₈	—	2	2	no	—
17	C₄×Dic₂₃₉	D₄	1	4	yes	—
18	C₄×Dic₂₃₉	Q₈	1	4	yes	—
19	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₂₃₉	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₃₉	—	2	2	no	—
23	C₂×C₈×C₂₃₉	—	2	4	no	—
24	C₄²×C₂₃₉	D₄	1	2	yes	—
25	C₄²×C₂₃₉	Q₈	1	2	yes	—
26	C₂×Dic₂₃₉	—	2	4	no	—
27	C₂×Dic₂₃₉	—	2	4	no	—
28	C₂×Dic₂₃₉	C₄⋊C₄	1	8	yes	—
29	C₂₃₉⋊C₈	—	4	2	no	—
30	C₂³×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃₉	—	2	2	no	—
32	C₂×C₄×C₂₃₉	—	2	2	no	—
33	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₂₃₉	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₂₃₉	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₂₃₉	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₂₃₉	—	4	2	no	—
40	C₂²×C₂₃₉	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₂₃₉	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₂₃₉	#131	1	2	yes	—
43	C₂²×C₂₃₉	#131	1	2	yes	—
44	C₄×C₂₃₉	—	2	2	no	—
45	C₄×C₂₃₉	—	2	2	no	—
46	C₄×C₂₃₉	#127	1	1	yes	characteristic
47	C₄×C₂₃₉	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₂₃₉	C₄⋊Q₈	1	2	yes	—
49	Dic₂₃₉	—	2	8	no	—
50	C₂×C₂₃₉	#384	1	1	yes	characteristic
51	C₂×C₂₃₉	#424	1	4	yes	—
52	C₂×C₂₃₉	#424	1	1	yes	characteristic
53	C₂×C₂₃₉	#424	1	1	yes	characteristic
54	C₂₃₉	#1843	1	1	yes	characteristic, complemented, Sylow
55	#1843	—	239	1	no	maximal, Sylow
56	C₂×C₄⋊C₈	—	239	4	no	—
57	C₂×C₄⋊C₈	—	239	1	no	—
58	C₂×C₄⋊C₈	D₂₃₉	1	1	yes	characteristic
59	#376	—	239	1	no	—
60	C₂²×C₈	—	239	2	no	—
61	C₂²×C₈	C₂×D₂₃₉	1	2	yes	—
62	C₂×C₄²	—	239	2	no	—
63	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	478	8	no	—
66	C₄⋊C₈	—	478	2	no	—
67	#95	—	239	2	no	—
68	#95	—	239	2	no	—
69	C₂²×C₄	—	239	4	no	—
70	C₂²×C₄	C₄×D₂₃₉	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₂₃₉	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	478	4	no	—
76	C₂×C₈	—	478	2	no	—
77	C₄²	—	239	4	no	—
78	C₄²	—	239	4	no	—
79	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
80	C₄²	Dic₄₇₈	1	2	yes	—
81	C₂³	C₂×C₄×D₂₃₉	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	478	4	no	—
85	C₂×C₄	—	478	4	no	—
86	C₂×C₄	—	239	8	no	—
87	C₂×C₄	Dic₂₃₉⋊C₄	1	2	yes	—
88	C₂×C₄	Dic₂₃₉⋊C₄	1	2	yes	—
89	C₂×C₄	D₄×D₂₃₉	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₂₃₉	1	2	yes	—
92	C₂×C₄	C₂×Dic₄₇₈	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	956	2	no	—
95	C₂²	C₄⋊C₄×D₂₃₉	1	2	yes	—
96	C₂²	C₂×Dic₂₃₉⋊C₄	1	1	yes	characteristic
97	C₂²	#21439966	1	2	yes	—
98	C₂²	#21439968	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	478	8	no	—
102	C₄	#21439956	1	1	yes	characteristic
103	C₄	#21439958	1	1	yes	characteristic
104	C₄	#21439959	1	2	yes	—
105	C₂	#29504641	1	1	yes	characteristic
106	C₂	#29504797	1	4	yes	—
107	C₂	#29504798	1	1	yes	characteristic
108	C₂	#29504800	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃₉⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₂₃₉⋊C₈

C₂

yes

maximal

#29504618

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×C₂₃₉⋊C₈

C₂²

yes

—

C₂×C₄×Dic₂₃₉

C₂²

yes

—

C₄⋊(C₂₃₉⋊C₈)

—

Dic₂₃₉⋊C₈

—

C₂₃₉⋊#95

C₄

yes

—

C₂₃₉⋊#95

C₄

yes

—

C₂²×C₈×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

C₄⋊C₈×C₂₃₉

—

C₂²×Dic₂₃₉

C₂×C₄

yes

—

C₂×C₂₃₉⋊C₈

—

C₂×C₂₃₉⋊C₈

—

C₄×Dic₂₃₉

D₄

yes

—

C₄×Dic₂₃₉

Q₈

yes

—

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₄²×C₂₃₉

D₄

yes

—

C₄²×C₂₃₉

Q₈

yes

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

C₄⋊C₄

yes

—

C₂₃₉⋊C₈

—

C₂³×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₄⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×Q₈

yes

—

C₂×C₄×C₂₃₉

C₂×Q₈

yes

characteristic

C₈×C₂₃₉

—

C₂²×C₂₃₉

C₂×C₄⋊C₄

yes

—

C₂²×C₂₃₉

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃₉

#131

yes

—

C₂²×C₂₃₉

#131

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

—

C₄×C₂₃₉

#127

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

characteristic

C₄×C₂₃₉

C₄⋊Q₈

yes

—

Dic₂₃₉

—

C₂×C₂₃₉

#384

yes

characteristic

C₂×C₂₃₉

#424

yes

—

C₂×C₂₃₉

yes

C₂×C₂₃₉

yes

C₂₃₉

yes

characteristic, complemented, Sylow

#1843

—

239

maximal, Sylow

C₂×C₄⋊C₈

—

239

—

C₂×C₄⋊C₈

—

239

—

C₂×C₄⋊C₈

D₂₃₉

yes

characteristic

#376

—

239

—

C₂²×C₈

—

239

—

C₂²×C₈

C₂×D₂₃₉

yes

—

C₂×C₄²

—

239

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

478

—

C₄⋊C₈

—

478

—

#95

—

239

—

#95

—

239

—

C₂²×C₄

—

239

—

C₂²×C₄

C₄×D₂₃₉

yes

characteristic

C₂²×C₄

C₄×D₂₃₉

yes

characteristic

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

478

—

C₂×C₈

—

478

—

C₄²

—

239

—

C₄²

—

239

—

C₄²

C₂²⋊D₂₃₉

yes

—

C₄²

Dic₄₇₈

yes

—

C₂³

C₂×C₄×D₂₃₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

478

—

C₂×C₄

—

478

—

C₂×C₄

—

239

—

C₂×C₄

Dic₂₃₉⋊C₄

yes

—

C₂×C₄

Dic₂₃₉⋊C₄

yes

—

C₂×C₄

D₄×D₂₃₉

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

C₂×C₄

Q₈×D₂₃₉

yes

—

C₂×C₄

C₂×Dic₄₇₈

yes

characteristic

C₈

—

C₈

—

956

—

C₂²

C₄⋊C₄×D₂₃₉

yes

—

C₂²

C₂×Dic₂₃₉⋊C₄

yes

characteristic

C₂²

#21439966

yes

—

C₂²

#21439968

yes

—

C₄

—

100

C₄

—

101

C₄

—

478

—

102

C₄

#21439956

yes

characteristic

103

C₄

#21439958

yes

characteristic

104

C₄

#21439959

yes

—

105

C₂

#29504641

yes

characteristic

106

C₂

#29504797

yes

—

107

C₂

#29504798

yes

characteristic

108

C₂

#29504800

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃₉
commutator, socle, prime radical	C₂³×C₂₃₉	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₂₃₉	C₂
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₃₉
hypocenter	C₂₃₉	#1843
LCS	G ▷ C₂³×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃₉

commutator, socle, prime radical

C₂³×C₂₃₉

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₂₃₉

C₂

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₃₉

C₂₃₉

G ▷ C₂³×C₂₃₉ ▷ C₂₃₉

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
54. C₂₃₉	#1843	semidirect

H ≅ G/N

type

54. C₂₃₉

#1843

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
C₂×Dic₂₃₉⋊C₈	2
#29504618	2
C₂×C₄⋊C₈×C₂₃₉	2
#1843	239

subgroup

index

C₂×C₄⋊(C₂₃₉⋊C₈)

C₂×Dic₂₃₉⋊C₈

#29504618

C₂×C₄⋊C₈×C₂₃₉

#1843

239

Subgroups — #44484701

Conjugacy classes of subgroups

Complements

Maximal subgroups