Group #49169289 — subgroups

Subgroups — C₂×#34036531 #49169289

Order 37504.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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×#22627616	C₂	1	2	yes	maximal
2	#34036531	C₂	1	8	yes	maximal, direct product factor
3	C₂×#22627558	C₂	1	2	yes	maximal
4	C₂×#22627556	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	#22627616	—	2	4	no	—
7	#22627616	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₂₉₃	C₂²	1	4	yes	—
9	C₂×#17989781	—	2	4	no	—
10	#22627558	C₂²	1	8	yes	—
11	#22627556	C₂²	1	8	yes	—
12	C₂²×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₂₉₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₂₉₃	C₂²	1	4	yes	—
16	C₂²×Dic₂₉₃	—	2	4	no	—
17	C₄⋊Dic₂₉₃	—	2	4	no	—
18	C₄⋊Dic₂₉₃	C₂³	1	8	yes	—
19	#17989781	—	2	16	no	—
20	C₂²×C₄×C₂₉₃	D₄	1	2	yes	—
21	C₂²×C₄×C₂₉₃	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₂₉₃	Q₈	1	4	yes	—
23	C₂×C₈×C₂₉₃	Q₈	1	4	yes	—
24	C₂×C₈×C₂₉₃	C₂³	1	2	yes	—
25	C₂×C₈×C₂₉₃	C₂³	1	2	yes	—
26	C₄²×C₂₉₃	D₄	1	2	yes	—
27	C₄²×C₂₉₃	C₂³	1	2	yes	—
28	C₂×Dic₂₉₃	—	2	8	no	—
29	C₂×Dic₂₉₃	—	4	8	no	—
30	C₂³×C₂₉₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₉₃	—	2	4	no	—
32	C₂×C₄×C₂₉₃	C₂×D₄	1	4	yes	—
33	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₂₉₃	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₂₉₃	C₂×Q₈	1	2	yes	—
36	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
37	C₈×C₂₉₃	C₂×Q₈	1	4	yes	—
38	C₈×C₂₉₃	C₂×Q₈	1	4	yes	—
39	C₂²×C₂₉₃	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₂₉₃	#135	1	4	yes	—
41	C₂²×C₂₉₃	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₂₉₃	—	2	4	no	—
43	C₄×C₂₉₃	C₄⋊Q₈	1	2	yes	—
44	C₄×C₂₉₃	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₂₉₃	C₂²×Q₈	1	1	yes	characteristic
46	Dic₂₉₃	—	4	8	no	—
47	C₂×C₂₉₃	#499	1	4	yes	complemented
48	C₂×C₂₉₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₂₉₃	C₂×#135	1	2	yes	—
50	C₂₉₃	C₂×#499	1	1	yes	characteristic, complemented, Sylow
51	C₂×#499	—	293	1	no	maximal, Sylow
52	C₂×#125	—	293	2	no	—
53	#499	—	293	8	no	—
54	C₂×#107	—	293	2	no	—
55	C₂×#106	—	293	2	no	—
56	C₂×C₄×C₈	D₂₉₃	1	1	yes	characteristic
57	C₂²×C₈	C₂×D₂₉₃	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₂₉₃	1	1	yes	characteristic
59	#125	—	293	8	no	—
60	#125	—	586	4	no	—
61	C₂×C₄⋊C₄	—	293	4	no	—
62	C₂×C₄⋊C₄	—	586	4	no	—
63	C₂×C₄²	C₂×D₂₉₃	1	1	yes	characteristic
64	#107	—	293	8	no	—
65	#106	—	293	8	no	—
66	C₄×C₈	C₂×D₂₉₃	1	4	yes	—
67	C₂²×C₄	—	586	4	no	—
68	C₂²×C₄	D₁₁₇₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₉₃	1	1	yes	characteristic
70	C₂×C₈	Dic₅₈₆	1	4	yes	—
71	C₂×C₈	Dic₅₈₆	1	4	yes	—
72	C₂×C₈	C₂²×D₂₉₃	1	2	yes	—
73	C₂×C₈	C₂²×D₂₉₃	1	2	yes	—
74	C₄⋊C₄	—	293	8	no	—
75	C₄⋊C₄	—	586	4	no	—
76	C₄⋊C₄	—	586	16	no	—
77	C₄²	D₁₁₇₂	1	2	yes	—
78	C₄²	C₂²×D₂₉₃	1	2	yes	—
79	C₂³	C₂×D₁₁₇₂	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	586	8	no	—
82	C₂×C₄	—	1172	8	no	—
83	C₂×C₄	C₂×D₁₁₇₂	1	4	yes	—
84	C₂×C₄	C₂×D₁₁₇₂	1	1	yes	characteristic
85	C₂×C₄	C₂×Dic₅₈₆	1	2	yes	—
86	C₂×C₄	C₂×Dic₅₈₆	1	2	yes	—
87	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
88	C₈	C₂×Dic₅₈₆	1	4	yes	—
89	C₈	C₂×Dic₅₈₆	1	4	yes	—
90	C₂²	#22627624	1	2	yes	—
91	C₂²	#22627646	1	4	yes	—
92	C₂²	C₂²×D₁₁₇₂	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	1172	8	no	—
95	C₄	#22627624	1	2	yes	—
96	C₄	C₂²×Dic₅₈₆	1	1	yes	characteristic
97	C₄	C₂²×Dic₅₈₆	1	1	yes	characteristic
98	C₂	#34036531	1	4	yes	direct product factor
99	C₂	C₂×#22627624	1	1	yes	characteristic
100	C₂	C₂×#22627646	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#22627616

C₂

yes

maximal

#34036531

C₂

yes

maximal, direct product factor

C₂×#22627558

C₂

yes

maximal

C₂×#22627556

C₂

yes

maximal

C₂×C₄×C₈×C₂₉₃

C₂

yes

maximal, characteristic

#22627616

—

#22627616

C₂²

yes

—

C₂×C₄⋊Dic₂₉₃

C₂²

yes

—

C₂×#17989781

—

#22627558

C₂²

yes

—

#22627556

C₂²

yes

—

C₂²×C₈×C₂₉₃

C₂²

yes

characteristic

C₂²×C₈×C₂₉₃

C₂²

yes

characteristic

C₂×C₄²×C₂₉₃

C₂²

yes

characteristic

C₄×C₈×C₂₉₃

C₂²

yes

—

C₂²×Dic₂₉₃

—

C₄⋊Dic₂₉₃

—

C₄⋊Dic₂₉₃

C₂³

yes

—

#17989781

—

C₂²×C₄×C₂₉₃

D₄

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

characteristic

C₂×C₈×C₂₉₃

Q₈

yes

—

C₂×C₈×C₂₉₃

Q₈

yes

—

C₂×C₈×C₂₉₃

C₂³

yes

—

C₂×C₈×C₂₉₃

C₂³

yes

—

C₄²×C₂₉₃

D₄

yes

—

C₄²×C₂₉₃

C₂³

yes

—

C₂×Dic₂₉₃

—

C₂×Dic₂₉₃

—

C₂³×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₉₃

C₂×Q₈

yes

—

C₂×C₄×C₂₉₃

C₂×Q₈

yes

—

C₂×C₄×C₂₉₃

C₂⁴

yes

characteristic

C₈×C₂₉₃

C₂×Q₈

yes

—

C₈×C₂₉₃

C₂×Q₈

yes

—

C₂²×C₂₉₃

C₄⋊Q₈

yes

—

C₂²×C₂₉₃

#135

yes

—

C₂²×C₂₉₃

C₂²×D₄

yes

characteristic

C₄×C₂₉₃

—

C₄×C₂₉₃

C₄⋊Q₈

yes

—

C₄×C₂₉₃

C₂²×Q₈

yes

characteristic

C₄×C₂₉₃

C₂²×Q₈

yes

characteristic

Dic₂₉₃

—

C₂×C₂₉₃

#499

yes

complemented

C₂×C₂₉₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₉₃

C₂×#135

yes

—

C₂₉₃

C₂×#499

yes

characteristic, complemented, Sylow

C₂×#499

—

293

maximal, Sylow

C₂×#125

—

293

—

#499

—

293

—

C₂×#107

—

293

—

C₂×#106

—

293

—

C₂×C₄×C₈

D₂₉₃

yes

characteristic

C₂²×C₈

C₂×D₂₉₃

yes

characteristic

C₂²×C₈

C₂×D₂₉₃

yes

characteristic

#125

—

293

—

#125

—

586

—

C₂×C₄⋊C₄

—

293

—

C₂×C₄⋊C₄

—

586

—

C₂×C₄²

C₂×D₂₉₃

yes

characteristic

#107

—

293

—

#106

—

293

—

C₄×C₈

C₂×D₂₉₃

yes

—

C₂²×C₄

—

586

—

C₂²×C₄

D₁₁₇₂

yes

—

C₂²×C₄

C₂²×D₂₉₃

yes

characteristic

C₂×C₈

Dic₅₈₆

yes

—

C₂×C₈

Dic₅₈₆

yes

—

C₂×C₈

C₂²×D₂₉₃

yes

—

C₂×C₈

C₂²×D₂₉₃

yes

—

C₄⋊C₄

—

293

—

C₄⋊C₄

—

586

—

C₄⋊C₄

—

586

—

C₄²

D₁₁₇₂

yes

—

C₄²

C₂²×D₂₉₃

yes

—

C₂³

C₂×D₁₁₇₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

586

—

C₂×C₄

—

1172

—

C₂×C₄

C₂×D₁₁₇₂

yes

—

C₂×C₄

C₂×D₁₁₇₂

yes

characteristic

C₂×C₄

C₂×Dic₅₈₆

yes

—

C₂×C₄

C₂×Dic₅₈₆

yes

—

C₂×C₄

C₂³×D₂₉₃

yes

characteristic

C₈

C₂×Dic₅₈₆

yes

—

C₈

C₂×Dic₅₈₆

yes

—

C₂²

#22627624

yes

—

C₂²

#22627646

yes

—

C₂²

C₂²×D₁₁₇₂

yes

characteristic

C₄

—

C₄

—

1172

—

C₄

#22627624

yes

—

C₄

C₂²×Dic₅₈₆

yes

characteristic

C₄

C₂²×Dic₅₈₆

yes

characteristic

C₂

#34036531

yes

direct product factor

C₂

C₂×#22627624

yes

characteristic

100

C₂

C₂×#22627646

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₁₇₂
commutator, cosocle	C₂×C₄×C₂₉₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₉₃
Fitting	C₂×C₄×C₈×C₂₉₃	C₂
socle, prime radical	C₂³×C₂₉₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₂₉₃
hypocenter	C₂₉₃	C₂×#499
LCS	G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₁₇₂

commutator, cosocle

C₂×C₄×C₂₉₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₉₃

Fitting

C₂×C₄×C₈×C₂₉₃

C₂

socle, prime radical

C₂³×C₂₉₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₂₉₃

hypocenter

C₂₉₃

C₂×#499

LCS

G ▷ C₂×C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #34036531	C₂	direct (N × H)
47. C₂×C₂₉₃	#499	semidirect
50. C₂₉₃	C₂×#499	semidirect
98. C₂	#34036531	direct (N × H)

H ≅ G/N

type

2. #34036531

C₂

direct (N × H)

47. C₂×C₂₉₃

#499

semidirect

50. C₂₉₃

C₂×#499

semidirect

98. C₂

#34036531

direct (N × H)

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

subgroup	index
C₂×#22627616	2
#34036531	2
C₂×#22627558	2
C₂×#22627556	2
C₂×C₄×C₈×C₂₉₃	2
C₂×#499	293

subgroup

index

C₂×C₄×C₈×C₂₉₃

293

Subgroups — C₂×#34036531 #49169289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#34036531 #49169289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#34036531 #49169289