Group #44508425 — subgroups

Subgroups — #44508425

Order 30848.

244 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 89 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	#29508274	C₂	1	1	yes	maximal, characteristic
2	#29507563	C₂	1	1	yes	maximal, characteristic
3	#29507478	C₂	1	4	yes	maximal
4	#423×C₂₄₁	C₂	1	1	yes	maximal, characteristic
5	#21440642	C₄	1	2	yes	complemented
6	#21440634	C₄	1	2	yes	complemented
7	#21440621	—	2	2	no	—
8	C₄×Dic₄₈₂	—	2	2	no	—
9	#21440601	C₂²	1	2	yes	—
10	C₂×C₄×Dic₂₄₁	C₂²	1	2	yes	—
11	#21440549	—	2	2	no	—
12	C₂×M₁₆×C₂₄₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₂₄₁	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₂₄₁	—	2	2	no	—
15	C₂×Dic₄₈₂	—	2	2	no	—
16	C₂²×Dic₂₄₁	—	2	2	no	—
17	C₄⋊Dic₂₄₁	—	2	2	no	—
18	C₄⋊Dic₂₄₁	C₂×C₄	1	4	yes	—
19	#16838749	—	2	4	no	—
20	C₂²⋊Dic₂₄₁	—	2	2	no	—
21	C₄×Dic₂₄₁	—	2	2	no	—
22	C₄×Dic₂₄₁	—	2	2	no	—
23	C₄×Dic₂₄₁	Q₈	1	4	yes	complemented
24	C₂²×C₄×C₂₄₁	D₄	1	2	yes	—
25	C₂²×C₄×C₂₄₁	C₂³	1	1	yes	characteristic
26	M₁₆×C₂₄₁	—	2	4	no	—
27	C₂×C₈×C₂₄₁	—	2	2	no	—
28	C₄²×C₂₄₁	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₂₄₁	C₂×C₄	1	1	yes	characteristic
30	C₄²×C₂₄₁	D₄	1	2	yes	—
31	Dic₄₈₂	—	4	4	no	—
32	C₂×Dic₂₄₁	—	2	4	no	—
33	C₂×Dic₂₄₁	—	2	4	no	—
34	C₂×Dic₂₄₁	—	4	4	no	—
35	C₂³×C₂₄₁	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₄₁	—	4	2	no	—
37	C₂×C₄×C₂₄₁	C₂²⋊C₄	1	2	yes	—
38	C₂×C₄×C₂₄₁	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₂₄₁	C₂×Q₈	1	2	yes	—
42	C₂×C₄×C₂₄₁	#41	1	2	yes	—
43	C₈×C₂₄₁	—	4	2	no	—
44	C₂²×C₂₄₁	—	2	2	no	—
45	C₂²×C₂₄₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
46	C₂²×C₂₄₁	C₂²⋊Q₈	1	2	yes	—
47	C₄×C₂₄₁	—	2	2	no	—
48	C₄×C₂₄₁	—	2	2	no	—
49	C₄×C₂₄₁	C₄×Q₈	1	2	yes	—
50	C₄×C₂₄₁	#124	1	1	yes	characteristic
51	C₄×C₂₄₁	C₄⋊Q₈	1	1	yes	characteristic
52	Dic₂₄₁	—	4	4	no	—
53	Dic₂₄₁	—	4	4	no	—
54	C₂×C₂₄₁	—	2	2	no	—
55	C₂×C₂₄₁	#391	1	1	yes	characteristic
56	C₂×C₂₄₁	#421	1	2	yes	—
57	C₂₄₁	#1995	1	1	yes	characteristic, complemented, Sylow
58	#1995	—	241	1	no	maximal, Sylow
59	#533	—	241	1	no	—
60	#423	D₂₄₁	1	1	yes	characteristic
61	#376	—	241	1	no	—
62	#339	—	241	4	no	—
63	C₂×M₁₆	C₂×D₂₄₁	1	2	yes	—
64	C₄⋊Q₈	—	241	2	no	—
65	#125	—	241	2	no	—
66	C₂²⋊Q₈	—	482	2	no	—
67	C₄×Q₈	—	482	2	no	—
68	#117	—	241	2	no	—
69	C₂×C₄²	—	241	2	no	—
70	C₂×C₄²	C₂×D₂₄₁	1	1	yes	characteristic
71	C₄⋊C₈	—	2	2	no	—
72	#95	—	482	2	no	—
73	C₂×Q₈	—	482	2	no	—
74	C₂²×C₄	—	482	2	no	—
75	C₂²×C₄	D₉₆₄	1	2	yes	—
76	C₂²×C₄	C₂²×D₂₄₁	1	1	yes	characteristic
77	M₁₆	—	2	4	no	—
78	C₂×C₈	—	2	2	no	—
79	C₄⋊C₄	—	482	4	no	—
80	C₄⋊C₄	—	482	2	no	—
81	C₄⋊C₄	—	241	4	no	—
82	C₂²⋊C₄	—	482	2	no	—
83	C₄²	—	482	2	no	—
84	C₄²	—	482	2	no	—
85	C₄²	—	241	4	no	—
86	C₄²	C₄×D₂₄₁	1	1	yes	characteristic
87	C₄²	C₄×D₂₄₁	1	1	yes	characteristic
88	C₄²	C₂²⋊D₂₄₁	1	2	yes	—
89	C₂³	C₂×D₉₆₄	1	1	yes	characteristic
90	Q₈	—	964	4	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	964	4	no	—
93	C₂×C₄	—	482	4	no	—
94	C₂×C₄	—	482	4	no	—
95	C₂×C₄	#16838747	1	2	yes	—
96	C₂×C₄	#16838747	1	2	yes	—
97	C₂×C₄	C₂×C₄×D₂₄₁	1	1	yes	characteristic
98	C₂×C₄	C₂×C₂²⋊D₂₄₁	1	1	yes	characteristic
99	C₂×C₄	Q₈×D₂₄₁	1	2	yes	—
100	C₂×C₄	#16838769	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂²	—	2	2	no	—
103	C₂²	C₂×#16838747	1	1	yes	characteristic
104	C₂²	#21440622	1	2	yes	—
105	C₄	—	2	2	no	—
106	C₄	—	2	2	no	—
107	C₄	—	964	4	no	—
108	C₄	—	964	4	no	—
109	C₄	#21440607	1	2	yes	—
110	C₄	#21440629	1	1	yes	characteristic
111	C₄	#21440640	1	1	yes	characteristic
112	C₂	—	2	2	no	—
113	C₂	#29507622	1	1	yes	characteristic
114	C₂	#29507733	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#29508274

C₂

yes

maximal, characteristic

#29507563

C₂

yes

maximal, characteristic

#29507478

C₂

yes

maximal

#423×C₂₄₁

C₂

yes

maximal, characteristic

#21440642

C₄

yes

complemented

#21440634

C₄

yes

complemented

#21440621

—

C₄×Dic₄₈₂

—

#21440601

C₂²

yes

—

C₂×C₄×Dic₂₄₁

C₂²

yes

—

#21440549

—

C₂×M₁₆×C₂₄₁

C₂²

yes

—

C₂×C₄²×C₂₄₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₄₁

—

C₂×Dic₄₈₂

—

C₂²×Dic₂₄₁

—

C₄⋊Dic₂₄₁

—

C₄⋊Dic₂₄₁

C₂×C₄

yes

—

#16838749

—

C₂²⋊Dic₂₄₁

—

C₄×Dic₂₄₁

—

C₄×Dic₂₄₁

—

C₄×Dic₂₄₁

Q₈

yes

complemented

C₂²×C₄×C₂₄₁

D₄

yes

—

C₂²×C₄×C₂₄₁

C₂³

yes

characteristic

M₁₆×C₂₄₁

—

C₂×C₈×C₂₄₁

—

C₄²×C₂₄₁

C₂×C₄

yes

characteristic

C₄²×C₂₄₁

C₂×C₄

yes

characteristic

C₄²×C₂₄₁

D₄

yes

—

Dic₄₈₂

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

C₂×Dic₂₄₁

—

C₂³×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₂²⋊C₄

yes

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×Q₈

yes

—

C₂×C₄×C₂₄₁

#41

yes

—

C₈×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₄₁

C₂²⋊Q₈

yes

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

C₄×Q₈

yes

—

C₄×C₂₄₁

#124

yes

characteristic

C₄×C₂₄₁

C₄⋊Q₈

yes

characteristic

Dic₂₄₁

—

Dic₂₄₁

—

C₂×C₂₄₁

—

C₂×C₂₄₁

#391

yes

characteristic

C₂×C₂₄₁

#421

yes

—

C₂₄₁

#1995

yes

characteristic, complemented, Sylow

#1995

—

241

maximal, Sylow

#533

—

241

—

#423

D₂₄₁

yes

characteristic

#376

—

241

—

#339

—

241

—

C₂×M₁₆

C₂×D₂₄₁

yes

—

C₄⋊Q₈

—

241

—

#125

—

241

—

C₂²⋊Q₈

—

482

—

C₄×Q₈

—

482

—

#117

—

241

—

C₂×C₄²

—

241

—

C₂×C₄²

C₂×D₂₄₁

yes

characteristic

C₄⋊C₈

—

#95

—

482

—

C₂×Q₈

—

482

—

C₂²×C₄

—

482

—

C₂²×C₄

D₉₆₄

yes

—

C₂²×C₄

C₂²×D₂₄₁

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

482

—

C₄⋊C₄

—

482

—

C₄⋊C₄

—

241

—

C₂²⋊C₄

—

482

—

C₄²

—

482

—

C₄²

—

482

—

C₄²

—

241

—

C₄²

C₄×D₂₄₁

yes

characteristic

C₄²

C₄×D₂₄₁

yes

characteristic

C₄²

C₂²⋊D₂₄₁

yes

—

C₂³

C₂×D₉₆₄

yes

characteristic

Q₈

—

964

—

C₂×C₄

—

C₂×C₄

—

964

—

C₂×C₄

—

482

—

C₂×C₄

—

482

—

C₂×C₄

#16838747

yes

—

C₂×C₄

#16838747

yes

—

C₂×C₄

C₂×C₄×D₂₄₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₄₁

yes

characteristic

C₂×C₄

Q₈×D₂₄₁

yes

—

100

C₂×C₄

#16838769

yes

—

101

C₈

—

102

C₂²

—

103

C₂²

C₂×#16838747

yes

characteristic

104

C₂²

#21440622

yes

—

105

C₄

—

106

C₄

—

107

C₄

—

964

—

108

C₄

—

964

—

109

C₄

#21440607

yes

—

110

C₄

#21440629

yes

characteristic

111

C₄

#21440640

yes

characteristic

112

C₂

—

113

C₂

#29507622

yes

characteristic

114

C₂

#29507733

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊D₂₄₁
commutator	C₂×C₄×C₂₄₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₄₁
Fitting	#423×C₂₄₁	C₂
socle	C₂²×C₂₄₁	C₂×C₂²⋊C₄
prime radical	C₂³×C₂₄₁	C₂×D₄
cosocle	C₂²×C₄×C₂₄₁	C₂³
hypercenter	#423	D₂₄₁
hypocenter	C₂₄₁	#1995
LCS	G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊D₂₄₁

commutator

C₂×C₄×C₂₄₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₄₁

Fitting

#423×C₂₄₁

C₂

socle

C₂²×C₂₄₁

C₂×C₂²⋊C₄

prime radical

C₂³×C₂₄₁

C₂×D₄

cosocle

C₂²×C₄×C₂₄₁

C₂³

D₂₄₁

C₂₄₁

G ▷ C₂×C₄×C₂₄₁ ▷ C₂×C₂₄₁ ▷ C₂₄₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #423

N	H ≅ G/N	type
5. #21440642	C₄	semidirect
6. #21440634	C₄	semidirect
23. C₄×Dic₂₄₁	Q₈	semidirect
57. C₂₄₁	#1995	semidirect

H ≅ G/N

type

5. #21440642

C₄

semidirect

6. #21440634

C₄

semidirect

23. C₄×Dic₂₄₁

Q₈

semidirect

57. C₂₄₁

#1995

semidirect

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

subgroup	index
#29508274	2
#29507563	2
#29507478	2
#423×C₂₄₁	2
#1995	241

subgroup

index

#423×C₂₄₁

241

Subgroups — #44508425

Conjugacy classes of subgroups

Complements

Maximal subgroups