Group #49158414 — subgroups

Subgroups — #49158414

Order 37504.

220 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 65 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	#34036506	C₂	1	1	yes	maximal, characteristic
2	#34036503	C₂	1	2	yes	maximal, complemented
3	#34036477	C₂	1	2	yes	maximal, complemented
4	#34035850	C₂	1	1	yes	maximal, characteristic, complemented
5	#498×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#17989789	—	2	4	no	—
7	C₂×D₂₃₄₄	—	2	2	no	—
8	#22627621	C₂²	1	1	yes	characteristic
9	#22627621	C₂²	1	1	yes	characteristic
10	#22627544	—	2	4	no	—
11	C₄×#14556873	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₂₉₃	C₂²	1	2	yes	—
13	#106×C₂₉₃	—	2	2	no	—
14	C₄×C₈×C₂₉₃	C₂²	1	1	yes	characteristic
15	C₂×D₁₁₇₂	—	2	1	no	—
16	C₂×D₁₁₇₂	—	2	1	no	—
17	C₂×D₁₁₇₂	—	2	2	no	—
18	C₂×D₁₁₇₂	—	2	2	no	—
19	#17989789	—	4	8	no	—
20	D₂₃₄₄	—	4	4	no	—
21	C₂×#14556873	D₄	1	4	yes	—
22	C₂×Q₈×C₂₉₃	—	2	2	no	—
23	C₂×C₈×C₂₉₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₂₉₃	—	2	2	no	—
25	C₄⋊C₄×C₂₉₃	—	2	2	no	—
26	C₄²×C₂₉₃	C₂³	1	1	yes	characteristic
27	C₂²×D₂₉₃	—	4	1	no	—
28	C₂²×D₂₉₃	—	4	1	no	—
29	D₁₁₇₂	—	4	2	no	—
30	D₁₁₇₂	—	4	2	no	—
31	D₁₁₇₂	—	4	4	no	—
32	D₁₁₇₂	—	4	4	no	—
33	#14556873	—	2	4	no	—
34	Q₈×C₂₉₃	—	4	4	no	—
35	C₂×C₄×C₂₉₃	—	4	2	no	—
36	C₂×C₄×C₂₉₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₉₃	C₂×D₄	1	1	yes	characteristic
38	C₈×C₂₉₃	SD₁₆	1	4	yes	complemented
39	C₂²×C₂₉₃	#127	1	1	yes	characteristic
40	C₂×D₂₉₃	—	8	1	no	—
41	C₂×D₂₉₃	—	8	1	no	—
42	C₂×D₂₉₃	—	8	2	no	—
43	C₂×D₂₉₃	—	8	2	no	—
44	C₄×C₂₉₃	—	8	2	no	—
45	C₄×C₂₉₃	C₂×SD₁₆	1	2	yes	—
46	C₄×C₂₉₃	AGL(1,ℤ/8)	1	4	yes	—
47	C₂×C₂₉₃	#492	1	1	yes	characteristic
48	C₂×C₂₉₃	#496	1	2	yes	—
49	D₂₉₃	—	16	1	no	—
50	D₂₉₃	—	16	1	no	—
51	C₂₉₃	#1715	1	1	yes	characteristic, complemented, Sylow
52	#1715	—	293	1	no	maximal, Sylow
53	#498	D₂₉₃	1	1	yes	characteristic, complemented
54	#493	—	293	1	no	—
55	#492	—	293	2	no	—
56	#486	—	293	2	no	—
57	#322	—	293	1	no	—
58	C₂×SD₁₆	—	586	4	no	—
59	C₂×D₈	—	586	2	no	—
60	C₄⋊Q₈	C₂×D₂₉₃	1	2	yes	—
61	#127	—	293	1	no	—
62	#127	—	293	1	no	—
63	#106	—	2	2	no	—
64	#102	—	586	4	no	—
65	C₄×C₈	—	293	2	no	—
66	C₄×C₈	C₂×D₂₉₃	1	1	yes	characteristic
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	586	2	no	—
69	C₂×D₄	—	586	2	no	—
70	C₂×D₄	—	586	1	no	—
71	C₂×D₄	—	586	1	no	—
72	SD₁₆	—	1172	8	no	—
73	D₈	—	1172	4	no	—
74	C₂×C₈	—	293	4	no	—
75	C₂×C₈	C₂²⋊D₂₉₃	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂²×D₂₉₃	1	1	yes	characteristic
79	C₂³	—	1172	1	no	—
80	C₂³	—	1172	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	1172	2	no	—
83	D₄	—	1172	2	no	—
84	D₄	—	1172	4	no	—
85	D₄	—	1172	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	D₄×D₂₉₃	1	2	yes	—
88	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	1	yes	characteristic
89	C₈	—	586	4	no	—
90	C₈	#17989789	1	4	yes	complemented
91	C₂²	—	2344	1	no	—
92	C₂²	—	2344	1	no	—
93	C₂²	—	2344	2	no	—
94	C₂²	—	2344	2	no	—
95	C₂²	#22627620	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×#17989789	1	2	yes	—
98	C₄	#22627651	1	4	yes	—
99	C₂	—	4688	1	no	—
100	C₂	—	4688	1	no	—
101	C₂	#34036503	1	1	yes	characteristic
102	C₂	#34036515	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#34036506

C₂

yes

maximal, characteristic

#34036503

C₂

yes

maximal, complemented

#34036477

C₂

yes

maximal, complemented

#34035850

C₂

yes

maximal, characteristic, complemented

#498×C₂₉₃

C₂

yes

maximal, characteristic, complemented

C₂×#17989789

—

C₂×D₂₃₄₄

—

#22627621

C₂²

yes

characteristic

#22627621

C₂²

yes

characteristic

#22627544

—

C₄×#14556873

C₂²

yes

—

C₄⋊Q₈×C₂₉₃

C₂²

yes

—

#106×C₂₉₃

—

C₄×C₈×C₂₉₃

C₂²

yes

characteristic

C₂×D₁₁₇₂

—

C₂×D₁₁₇₂

—

C₂×D₁₁₇₂

—

C₂×D₁₁₇₂

—

#17989789

—

D₂₃₄₄

—

C₂×#14556873

D₄

yes

—

C₂×Q₈×C₂₉₃

—

C₂×C₈×C₂₉₃

D₄

yes

—

C₄⋊C₄×C₂₉₃

—

C₄⋊C₄×C₂₉₃

—

C₄²×C₂₉₃

C₂³

yes

characteristic

C₂²×D₂₉₃

—

C₂²×D₂₉₃

—

D₁₁₇₂

—

D₁₁₇₂

—

D₁₁₇₂

—

D₁₁₇₂

—

#14556873

—

Q₈×C₂₉₃

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

characteristic

C₈×C₂₉₃

SD₁₆

yes

complemented

C₂²×C₂₉₃

#127

yes

characteristic

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₂×D₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×SD₁₆

yes

—

C₄×C₂₉₃

AGL(1,ℤ/8)

yes

—

C₂×C₂₉₃

#492

yes

characteristic

C₂×C₂₉₃

#496

yes

—

D₂₉₃

—

D₂₉₃

—

C₂₉₃

#1715

yes

characteristic, complemented, Sylow

#1715

—

293

maximal, Sylow

#498

D₂₉₃

yes

characteristic, complemented

#493

—

293

—

#492

—

293

—

#486

—

293

—

#322

—

293

—

C₂×SD₁₆

—

586

—

C₂×D₈

—

586

—

C₄⋊Q₈

C₂×D₂₉₃

yes

—

#127

—

293

—

#127

—

293

—

#106

—

#102

—

586

—

C₄×C₈

—

293

—

C₄×C₈

C₂×D₂₉₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

586

—

C₂×D₄

—

586

—

C₂×D₄

—

586

—

C₂×D₄

—

586

—

SD₁₆

—

1172

—

D₈

—

1172

—

C₂×C₈

—

293

—

C₂×C₈

C₂²⋊D₂₉₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×D₂₉₃

yes

characteristic

C₂³

—

1172

—

C₂³

—

1172

—

Q₈

—

D₄

—

1172

—

D₄

—

1172

—

D₄

—

1172

—

D₄

—

1172

—

C₂×C₄

—

C₂×C₄

D₄×D₂₉₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

characteristic

C₈

—

586

—

C₈

#17989789

yes

complemented

C₂²

—

2344

—

C₂²

—

2344

—

C₂²

—

2344

—

C₂²

—

2344

—

C₂²

#22627620

yes

characteristic

C₄

—

C₄

C₂×#17989789

yes

—

C₄

#22627651

yes

—

C₂

—

4688

—

100

C₂

—

4688

—

101

C₂

#34036503

yes

characteristic

102

C₂

#34036515

yes

—

	subgroup	G/H
center	C₂²	#22627620
commutator, cosocle	C₄²×C₂₉₃	C₂³
Frattini	C₄²	C₂²×D₂₉₃
Fitting	#498×C₂₉₃	C₂
socle	C₂²×C₂₉₃	#127
prime radical	#34036506	C₂
hypercenter	#498	D₂₉₃
hypocenter	C₂₉₃	#1715
LCS	G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #498

subgroup

G/H

center

C₂²

#22627620

commutator, cosocle

C₄²×C₂₉₃

C₂³

Frattini

C₄²

C₂²×D₂₉₃

Fitting

#498×C₂₉₃

C₂

C₂²×C₂₉₃

C₂

D₂₉₃

C₂₉₃

G ▷ C₄²×C₂₉₃ ▷ C₂²×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #498

N	H ≅ G/N	type
2. #34036503	C₂	semidirect
3. #34036477	C₂	semidirect
4. #34035850	C₂	semidirect
5. #498×C₂₉₃	C₂	semidirect
38. C₈×C₂₉₃	SD₁₆	semidirect
51. C₂₉₃	#1715	semidirect
53. #498	D₂₉₃	semidirect
90. C₈	#17989789	semidirect

H ≅ G/N

type

2. #34036503

C₂

semidirect

3. #34036477

C₂

semidirect

4. #34035850

C₂

semidirect

5. #498×C₂₉₃

C₂

semidirect

38. C₈×C₂₉₃

SD₁₆

semidirect

51. C₂₉₃

#1715

semidirect

53. #498

D₂₉₃

semidirect

90. C₈

#17989789

semidirect

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

subgroup	index
#34036506	2
#34036503	2
#34036477	2
#34035850	2
#498×C₂₉₃	2
#1715	293

subgroup

index

#498×C₂₉₃

293

Subgroups — #49158414

Conjugacy classes of subgroups

Complements

Maximal subgroups