Group #49167394 — subgroups

Subgroups — #49167394

Order 37504.

440 conjugacy classes of subgroups; only the 249 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 249 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	#34036896	C₂	1	1	yes	maximal, characteristic
2	C₂×#22627631	C₂	1	1	yes	maximal, characteristic
3	#34036177	C₂	1	6	yes	maximal, complemented
4	C₂×#22627559	C₂	1	6	yes	maximal, complemented
5	#568×C₂₉₃	C₂	1	1	yes	maximal, characteristic
6	#22627631	—	2	4	no	—
7	#22627631	C₄	1	8	yes	—
8	C₂×#17989786	C₂²	1	6	yes	—
9	C₂×#17989786	C₂²	1	3	yes	—
10	C₂²×#14556873	C₂²	1	3	yes	—
11	#22627559	C₂²	1	4	yes	—
12	#22627559	C₂²	1	12	yes	—
13	C₂×#41×C₂₉₃	C₂²	1	1	yes	characteristic
14	#131×C₂₉₃	—	2	4	no	—
15	C₂×M₁₆×C₂₉₃	C₂²	1	6	yes	—
16	#17989786	—	2	6	no	—
17	#17989786	C₂×C₄	1	12	yes	—
18	#17989786	C₂³	1	4	yes	—
19	C₂×#14556873	—	2	6	no	—
20	C₂×#14556873	C₂×C₄	1	12	yes	—
21	C₂×#14556873	C₂³	1	1	yes	characteristic
22	C₂×#14556873	C₂³	1	3	yes	—
23	#41×C₂₉₃	C₂×C₄	1	4	yes	—
24	#41×C₂₉₃	D₄	1	4	yes	—
25	C₂×Q₈×C₂₉₃	Q₈	1	1	yes	characteristic
26	C₂×D₄×C₂₉₃	Q₈	1	3	yes	—
27	C₂²×C₄×C₂₉₃	C₂³	1	3	yes	—
28	M₁₆×C₂₉₃	—	2	6	no	—
29	M₁₆×C₂₉₃	C₂³	1	4	yes	—
30	C₂×C₈×C₂₉₃	—	2	6	no	—
31	#14556873	—	2	4	no	—
32	#14556873	C₂²×C₄	1	2	yes	—
33	#14556873	C₂²×C₄	1	6	yes	—
34	C₂³×C₂₉₃	C₂×Q₈	1	3	yes	—
35	Q₈×C₂₉₃	C₄⋊C₄	1	4	yes	—
36	D₄×C₂₉₃	C₄⋊C₄	1	12	yes	—
37	C₂×C₄×C₂₉₃	C₂²×C₄	1	6	yes	—
38	C₂×C₄×C₂₉₃	C₂×D₄	1	6	yes	—
39	C₂×C₄×C₂₉₃	C₂×Q₈	1	3	yes	—
40	C₂×C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
41	C₈×C₂₉₃	—	2	4	no	—
42	C₂²×C₂₉₃	—	4	3	no	—
43	C₂²×C₂₉₃	C₂×C₄⋊C₄	1	6	yes	—
44	C₂²×C₂₉₃	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₂₉₃	C₂×C₄⋊C₄	1	6	yes	—
46	C₄×C₂₉₃	C₂³×C₄	1	1	yes	characteristic
47	C₄×C₂₉₃	C₂²×D₄	1	1	yes	characteristic
48	C₂×C₂₉₃	—	2	1	no	—
49	C₂×C₂₉₃	—	2	6	no	—
50	C₂×C₂₉₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
51	C₂₉₃	#2916	1	1	yes	characteristic, complemented, Sylow
52	#2916	—	293	1	no	maximal, Sylow
53	#568	—	293	1	no	—
54	#568	D₂₉₃	1	1	yes	characteristic
55	C₂×#131	—	293	1	no	—
56	#430	—	293	6	no	—
57	C₂×#108	—	293	6	no	—
58	C₂×#41	C₂×D₂₉₃	1	1	yes	characteristic
59	#131	—	2	4	no	—
60	#131	—	293	8	no	—
61	#131	—	586	4	no	—
62	C₂×M₁₆	—	293	3	no	—
63	C₂×M₁₆	—	293	6	no	—
64	C₂×M₁₆	C₂×D₂₉₃	1	6	yes	—
65	C₂²×C₈	—	293	3	no	—
66	#108	—	293	4	no	—
67	#108	—	293	12	no	—
68	#41	C₄×D₂₉₃	1	4	yes	—
69	#41	C₂²⋊D₂₉₃	1	4	yes	—
70	C₂×Q₈	Dic₅₈₆	1	1	yes	characteristic
71	C₂×D₄	Dic₅₈₆	1	3	yes	—
72	C₂²×C₄	C₂²×D₂₉₃	1	3	yes	—
73	M₁₆	—	2	6	no	—
74	M₁₆	—	293	12	no	—
75	M₁₆	—	293	4	no	—
76	M₁₆	—	586	6	no	—
77	M₁₆	C₂²×D₂₉₃	1	4	yes	—
78	C₂×C₈	—	2	6	no	—
79	C₂×C₈	—	293	12	no	—
80	C₂×C₈	—	293	1	no	—
81	C₂×C₈	—	293	3	no	—
82	C₂×C₈	—	586	6	no	—
83	C₂³	C₂×Dic₅₈₆	1	3	yes	—
84	Q₈	#17989781	1	4	yes	—
85	D₄	#17989781	1	12	yes	—
86	C₂×C₄	C₂×C₄×D₂₉₃	1	6	yes	—
87	C₂×C₄	C₂×C₂²⋊D₂₉₃	1	6	yes	—
88	C₂×C₄	C₂×Dic₅₈₆	1	3	yes	—
89	C₂×C₄	C₂³×D₂₉₃	1	1	yes	characteristic
90	C₈	—	2	4	no	—
91	C₈	—	293	2	no	—
92	C₈	—	293	6	no	—
93	C₈	—	586	4	no	—
94	C₂²	—	4	3	no	—
95	C₂²	C₂×#17989781	1	6	yes	—
96	C₂²	C₂²×Dic₅₈₆	1	1	yes	characteristic
97	C₄	C₂×#17989781	1	6	yes	—
98	C₄	C₂²×C₄×D₂₉₃	1	1	yes	characteristic
99	C₄	C₂²×C₂²⋊D₂₉₃	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	2	6	no	—
102	C₂	C₂²×#17989781	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#34036896

C₂

yes

maximal, characteristic

C₂×#22627631

C₂

yes

maximal, characteristic

#34036177

C₂

yes

maximal, complemented

C₂×#22627559

C₂

yes

maximal, complemented

#568×C₂₉₃

C₂

yes

maximal, characteristic

#22627631

—

#22627631

C₄

yes

—

C₂×#17989786

C₂²

yes

—

C₂×#17989786

C₂²

yes

—

C₂²×#14556873

C₂²

yes

—

#22627559

C₂²

yes

—

#22627559

C₂²

yes

—

C₂×#41×C₂₉₃

C₂²

yes

characteristic

#131×C₂₉₃

—

C₂×M₁₆×C₂₉₃

C₂²

yes

—

#17989786

—

#17989786

C₂×C₄

yes

—

#17989786

C₂³

yes

—

C₂×#14556873

—

C₂×#14556873

C₂×C₄

yes

—

C₂×#14556873

C₂³

yes

characteristic

C₂×#14556873

C₂³

yes

—

#41×C₂₉₃

C₂×C₄

yes

—

#41×C₂₉₃

D₄

yes

—

C₂×Q₈×C₂₉₃

Q₈

yes

characteristic

C₂×D₄×C₂₉₃

Q₈

yes

—

C₂²×C₄×C₂₉₃

C₂³

yes

—

M₁₆×C₂₉₃

—

M₁₆×C₂₉₃

C₂³

yes

—

C₂×C₈×C₂₉₃

—

#14556873

—

#14556873

C₂²×C₄

yes

—

#14556873

C₂²×C₄

yes

—

C₂³×C₂₉₃

C₂×Q₈

yes

—

Q₈×C₂₉₃

C₄⋊C₄

yes

—

D₄×C₂₉₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

—

C₂×C₄×C₂₉₃

C₂×D₄

yes

—

C₂×C₄×C₂₉₃

C₂×Q₈

yes

—

C₂×C₄×C₂₉₃

C₂⁴

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₂×C₄⋊C₄

yes

—

C₂²×C₂₉₃

C₂²×Q₈

yes

characteristic

C₄×C₂₉₃

C₂×C₄⋊C₄

yes

—

C₄×C₂₉₃

C₂³×C₄

yes

characteristic

C₄×C₂₉₃

C₂²×D₄

yes

characteristic

C₂×C₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

C₂²×C₄⋊C₄

yes

characteristic

C₂₉₃

#2916

yes

characteristic, complemented, Sylow

#2916

—

293

maximal, Sylow

#568

—

293

—

#568

D₂₉₃

yes

characteristic

C₂×#131

—

293

—

#430

—

293

—

C₂×#108

—

293

—

C₂×#41

C₂×D₂₉₃

yes

characteristic

#131

—

#131

—

293

—

#131

—

586

—

C₂×M₁₆

—

293

—

C₂×M₁₆

—

293

—

C₂×M₁₆

C₂×D₂₉₃

yes

—

C₂²×C₈

—

293

—

#108

—

293

—

#108

—

293

—

#41

C₄×D₂₉₃

yes

—

#41

C₂²⋊D₂₉₃

yes

—

C₂×Q₈

Dic₅₈₆

yes

characteristic

C₂×D₄

Dic₅₈₆

yes

—

C₂²×C₄

C₂²×D₂₉₃

yes

—

M₁₆

—

M₁₆

—

293

—

M₁₆

—

293

—

M₁₆

—

586

—

M₁₆

C₂²×D₂₉₃

yes

—

C₂×C₈

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

586

—

C₂³

C₂×Dic₅₈₆

yes

—

Q₈

#17989781

yes

—

D₄

#17989781

yes

—

C₂×C₄

C₂×C₄×D₂₉₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₉₃

yes

—

C₂×C₄

C₂×Dic₅₈₆

yes

—

C₂×C₄

C₂³×D₂₉₃

yes

characteristic

C₈

—

C₈

—

293

—

C₈

—

293

—

C₈

—

586

—

C₂²

—

C₂²

C₂×#17989781

yes

—

C₂²

C₂²×Dic₅₈₆

yes

characteristic

C₄

C₂×#17989781

yes

—

C₄

C₂²×C₄×D₂₉₃

yes

characteristic

C₄

C₂²×C₂²⋊D₂₉₃

yes

characteristic

100

C₂

—

101

C₂

—

102

C₂

C₂²×#17989781

yes

characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₂₉₃
commutator	C₄×C₂₉₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₉₃
Fitting	#568×C₂₉₃	C₂
socle	C₂×C₂₉₃	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₂₉₃	C₂²
cosocle	C₂×C₄×C₂₉₃	C₂⁴
hypercenter	#568	D₂₉₃
hypocenter	C₂₉₃	#2916
LCS	G ▷ C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

subgroup

G/H

center

C₄

C₂²×C₂²⋊D₂₉₃

commutator

C₄×C₂₉₃

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₂₉₃

Fitting

#568×C₂₉₃

C₂

socle

C₂×C₂₉₃

C₂²×C₄⋊C₄

prime radical

C₂×#41×C₂₉₃

C₂²

cosocle

C₂×C₄×C₂₉₃

C₂⁴

D₂₉₃

C₂₉₃

G ▷ C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₄ ◁ C₂×#41 ◁ #568

N	H ≅ G/N	type
3. #34036177	C₂	semidirect
4. C₂×#22627559	C₂	semidirect
51. C₂₉₃	#2916	semidirect

H ≅ G/N

type

3. #34036177

C₂

semidirect

4. C₂×#22627559

C₂

semidirect

51. C₂₉₃

#2916

semidirect

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

subgroup	index
#34036896	2
C₂×#22627631	2
#34036177	2
C₂×#22627559	2
#568×C₂₉₃	2
#2916	293

subgroup

index

#568×C₂₉₃

293

Subgroups — #49167394

Conjugacy classes of subgroups

Complements

Maximal subgroups