Group #44485368 — subgroups

Subgroups — #44485368

Order 30592.

308 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 109 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×#21439960	C₂	1	1	yes	maximal, characteristic
2	C₂×#21439885	C₂	1	4	yes	maximal
3	#29504638	C₂	1	1	yes	maximal, characteristic
4	C₂×#97×C₂₃₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₇₈	C₂²	1	2	yes	—
6	#21439960	—	2	2	no	—
7	#21439960	C₄	1	4	yes	complemented
8	C₂×C₄⋊Dic₂₃₉	—	2	2	no	—
9	C₂×C₄⋊Dic₂₃₉	C₂²	1	2	yes	—
10	#21439885	—	2	8	no	—
11	C₂₃₉⋊#95	—	2	2	no	—
12	C₂²×C₈×C₂₃₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₃₉	C₂²	1	1	yes	characteristic
14	#97×C₂₃₉	—	2	2	no	—
15	C₂×Dic₄₇₈	—	2	2	no	—
16	C₂×Dic₄₇₈	—	2	4	no	—
17	C₂×Dic₄₇₈	—	2	4	no	—
18	C₂×Dic₄₇₈	C₂×C₄	1	4	yes	—
19	C₂²×Dic₂₃₉	—	2	2	no	—
20	C₂²×Dic₂₃₉	—	2	2	no	—
21	C₄⋊Dic₂₃₉	—	2	8	no	—
22	C₄⋊Dic₂₃₉	—	2	4	no	—
23	C₂²×C₄×C₂₃₉	D₄	1	2	yes	—
24	C₂²×C₄×C₂₃₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₂₃₉	—	2	2	no	—
26	C₂×C₈×C₂₃₉	—	2	2	no	—
27	C₂×C₈×C₂₃₉	D₄	1	4	yes	—
28	C₄²×C₂₃₉	C₂×C₄	1	2	yes	—
29	C₄²×C₂₃₉	D₄	1	2	yes	—
30	Dic₄₇₈	—	2	8	no	—
31	Dic₄₇₈	—	4	4	no	—
32	C₂×Dic₂₃₉	—	2	4	no	—
33	C₂×Dic₂₃₉	—	4	2	no	—
34	C₂×Dic₂₃₉	—	4	2	no	—
35	C₂×Dic₂₃₉	—	4	2	no	—
36	C₂×Dic₂₃₉	—	4	4	no	—
37	C₂³×C₂₃₉	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₂₃₉	—	2	4	no	—
39	C₂×C₄×C₂₃₉	C₂²⋊C₄	1	4	yes	—
40	C₂×C₄×C₂₃₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₂₃₉	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₂₃₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₂₃₉	#41	1	2	yes	—
44	C₈×C₂₃₉	—	2	4	no	—
45	C₂²×C₂₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₃₉	#124	1	1	yes	characteristic
47	C₂²×C₂₃₉	#127	1	1	yes	characteristic
48	C₂²×C₂₃₉	#137	1	4	yes	—
49	C₄×C₂₃₉	—	2	4	no	—
50	C₄×C₂₃₉	C₄×D₄	1	2	yes	—
51	C₄×C₂₃₉	#121	1	2	yes	—
52	Dic₂₃₉	—	4	4	no	—
53	Dic₂₃₉	—	8	2	no	—
54	C₂×C₂₃₉	#390	1	1	yes	characteristic
55	C₂×C₂₃₉	#419	1	2	yes	—
56	C₂×C₂₃₉	#491	1	2	yes	—
57	C₂×C₂₃₉	#497	1	2	yes	—
58	C₂₃₉	#1960	1	1	yes	characteristic, complemented, Sylow
59	#1960	—	239	1	no	maximal, Sylow
60	C₂×C₄⋊Q₈	—	239	1	no	—
61	C₂×#103	—	239	4	no	—
62	C₂×#97	D₂₃₉	1	1	yes	characteristic
63	#382	—	239	1	no	—
64	C₂²×Q₈	—	239	2	no	—
65	C₂²×C₈	C₂×D₂₃₉	1	2	yes	—
66	C₄⋊Q₈	—	478	2	no	—
67	C₄⋊Q₈	—	239	4	no	—
68	C₂×C₄⋊C₄	—	478	2	no	—
69	C₂×C₄⋊C₄	—	239	2	no	—
70	C₂×C₄²	C₂×D₂₃₉	1	1	yes	characteristic
71	#103	—	478	8	no	—
72	#97	—	2	2	no	—
73	#95	—	478	2	no	—
74	C₂×Q₈	—	478	4	no	—
75	C₂×Q₈	—	478	4	no	—
76	C₂×Q₈	—	478	2	no	—
77	C₂×Q₈	—	239	4	no	—
78	C₂²×C₄	—	478	2	no	—
79	C₂²×C₄	—	478	2	no	—
80	C₂²×C₄	D₉₅₆	1	2	yes	—
81	C₂²×C₄	C₂²×D₂₃₉	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	D₉₅₆	1	4	yes	—
85	C₄⋊C₄	—	478	4	no	—
86	C₄⋊C₄	—	478	8	no	—
87	C₄²	C₄×D₂₃₉	1	2	yes	—
88	C₄²	C₂²⋊D₂₃₉	1	2	yes	—
89	C₂³	C₂×D₉₅₆	1	1	yes	characteristic
90	Q₈	—	956	4	no	—
91	Q₈	—	478	8	no	—
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	956	2	no	—
94	C₂×C₄	—	956	4	no	—
95	C₂×C₄	—	956	2	no	—
96	C₂×C₄	—	956	2	no	—
97	C₂×C₄	—	478	4	no	—
98	C₂×C₄	(C₂×D₂₃₉)⋊C₄	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₂₃₉	1	1	yes	characteristic
100	C₂×C₄	C₂×C₂²⋊D₂₃₉	1	1	yes	characteristic
101	C₂×C₄	C₂×D₉₅₆	1	2	yes	—
102	C₂×C₄	#16838548	1	2	yes	—
103	C₈	—	2	4	no	—
104	C₂²	C₂×(C₂×D₂₃₉)⋊C₄	1	1	yes	characteristic
105	C₂²	#21439948	1	1	yes	characteristic
106	C₂²	#21439957	1	1	yes	characteristic
107	C₂²	#21439993	1	4	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	1912	2	no	—
110	C₄	—	956	4	no	—
111	C₄	C₄×D₉₅₆	1	2	yes	—
112	C₄	#21439934	1	2	yes	—
113	C₂	#29504674	1	1	yes	characteristic
114	C₂	#29504782	1	2	yes	—
115	C₂	#29505141	1	2	yes	—
116	C₂	#29505166	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#21439960

C₂

yes

maximal, characteristic

C₂×#21439885

C₂

yes

maximal

#29504638

C₂

yes

maximal, characteristic

C₂×#97×C₂₃₉

C₂

yes

maximal, characteristic

C₂²×Dic₄₇₈

C₂²

yes

—

#21439960

—

#21439960

C₄

yes

complemented

C₂×C₄⋊Dic₂₃₉

—

C₂×C₄⋊Dic₂₃₉

C₂²

yes

—

#21439885

—

C₂₃₉⋊#95

—

C₂²×C₈×C₂₃₉

C₂²

yes

—

C₂×C₄²×C₂₃₉

C₂²

yes

characteristic

#97×C₂₃₉

—

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

—

C₂×Dic₄₇₈

C₂×C₄

yes

—

C₂²×Dic₂₃₉

—

C₂²×Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₄⋊Dic₂₃₉

—

C₂²×C₄×C₂₃₉

D₄

yes

—

C₂²×C₄×C₂₃₉

C₂³

yes

characteristic

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

—

C₂×C₈×C₂₃₉

D₄

yes

—

C₄²×C₂₃₉

C₂×C₄

yes

—

C₄²×C₂₃₉

D₄

yes

—

Dic₄₇₈

—

Dic₄₇₈

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂×Dic₂₃₉

—

C₂³×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

—

C₂×C₄×C₂₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃₉

C₂×D₄

yes

—

C₂×C₄×C₂₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃₉

#41

yes

—

C₈×C₂₃₉

—

C₂²×C₂₃₉

C₂×C₂²⋊C₄

yes

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

yes

C₂²×C₂₃₉

yes

—

C₄×C₂₃₉

—

C₄×C₂₃₉

C₄×D₄

yes

—

C₄×C₂₃₉

#121

yes

—

Dic₂₃₉

—

Dic₂₃₉

—

C₂×C₂₃₉

#390

yes

characteristic

C₂×C₂₃₉

#419

yes

—

C₂×C₂₃₉

#491

yes

—

C₂×C₂₃₉

#497

yes

—

C₂₃₉

#1960

yes

characteristic, complemented, Sylow

#1960

—

239

maximal, Sylow

C₂×C₄⋊Q₈

—

239

—

C₂×#103

—

239

—

C₂×#97

D₂₃₉

yes

characteristic

#382

—

239

—

C₂²×Q₈

—

239

—

C₂²×C₈

C₂×D₂₃₉

yes

—

C₄⋊Q₈

—

478

—

C₄⋊Q₈

—

239

—

C₂×C₄⋊C₄

—

478

—

C₂×C₄⋊C₄

—

239

—

C₂×C₄²

C₂×D₂₃₉

yes

characteristic

#103

—

478

—

#97

—

#95

—

478

—

C₂×Q₈

—

478

—

C₂×Q₈

—

478

—

C₂×Q₈

—

478

—

C₂×Q₈

—

239

—

C₂²×C₄

—

478

—

C₂²×C₄

—

478

—

C₂²×C₄

D₉₅₆

yes

—

C₂²×C₄

C₂²×D₂₃₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₉₅₆

yes

—

C₄⋊C₄

—

478

—

C₄⋊C₄

—

478

—

C₄²

C₄×D₂₃₉

yes

—

C₄²

C₂²⋊D₂₃₉

yes

—

C₂³

C₂×D₉₅₆

yes

characteristic

Q₈

—

956

—

Q₈

—

478

—

C₂×C₄

—

C₂×C₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

—

956

—

C₂×C₄

—

478

—

C₂×C₄

(C₂×D₂₃₉)⋊C₄

yes

—

C₂×C₄

C₂×C₄×D₂₃₉

yes

characteristic

100

C₂×C₄

C₂×C₂²⋊D₂₃₉

yes

characteristic

101

C₂×C₄

C₂×D₉₅₆

yes

—

102

C₂×C₄

#16838548

yes

—

103

C₈

—

104

C₂²

C₂×(C₂×D₂₃₉)⋊C₄

yes

characteristic

105

C₂²

#21439948

yes

characteristic

106

C₂²

#21439957

yes

characteristic

107

C₂²

#21439993

yes

—

108

C₄

—

109

C₄

—

1912

—

110

C₄

—

956

—

111

C₄

C₄×D₉₅₆

yes

—

112

C₄

#21439934

yes

—

113

C₂

#29504674

yes

characteristic

114

C₂

#29504782

yes

—

115

C₂

#29505141

yes

—

116

C₂

#29505166

yes

—

	subgroup	G/H
center	C₂³	C₂×D₉₅₆
commutator	C₂×C₄×C₂₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃₉
Fitting	C₂×#97×C₂₃₉	C₂
socle, prime radical	C₂³×C₂₃₉	C₂×D₄
cosocle	C₂²×C₄×C₂₃₉	C₂³
hypercenter	C₂×#97	D₂₃₉
hypocenter	C₂₃₉	#1960
LCS	G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₉₅₆

commutator

C₂×C₄×C₂₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃₉

Fitting

C₂×#97×C₂₃₉

C₂

socle, prime radical

C₂³×C₂₃₉

C₂×D₄

cosocle

C₂²×C₄×C₂₃₉

C₂³

hypercenter

C₂×#97

D₂₃₉

hypocenter

C₂₃₉

#1960

LCS

G ▷ C₂×C₄×C₂₃₉ ▷ C₂×C₂₃₉ ▷ C₂₃₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
7. #21439960	C₄	semidirect
58. C₂₃₉	#1960	semidirect

H ≅ G/N

type

7. #21439960

C₄

semidirect

58. C₂₃₉

#1960

semidirect

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

subgroup	index
C₂×#21439960	2
C₂×#21439885	2
#29504638	2
C₂×#97×C₂₃₉	2
#1960	239

subgroup

index

C₂×#97×C₂₃₉

239

Subgroups — #44485368

Conjugacy classes of subgroups

Complements

Maximal subgroups