Group #49174501 — subgroups

Subgroups — #49174501

Order 37504.

272 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 98 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	#34037088	C₂	1	1	yes	maximal, characteristic
2	C₄²×#11924078	C₂	1	1	yes	maximal, characteristic
3	(C₂×#11924078)⋊C₈	C₂	1	4	yes	maximal
4	#97×D₂₉₃	C₂	1	1	yes	maximal, characteristic
5	C₂×D₂₉₃⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×#11924078	C₄	1	4	yes	—
7	C₂×C₄×#11924078	C₂²	1	2	yes	—
8	#22627679	C₄	1	4	yes	complemented
9	C₂×C₈×D₂₉₃	C₂²	1	2	yes	—
10	C₄²×D₂₉₃	C₂²	1	1	yes	characteristic
11	#22627535	C₄	1	1	yes	characteristic, complemented
12	#22627534	C₄	1	2	yes	complemented
13	#97×C₂₉₃	C₄	1	1	yes	characteristic, complemented
14	C₂²×#11924078	C₂×C₄	1	4	yes	—
15	D₂₉₃⋊C₈	—	2	4	no	—
16	C₂×#14556880	C₂×C₄	1	4	yes	—
17	C₄×#11924078	—	2	8	no	—
18	C₄×#11924078	—	2	4	no	—
19	C₂×C₄×D₂₉₃	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₂₉₃	C₂³	1	1	yes	characteristic
21	C₂×#14556873	C₂×C₄	1	2	yes	—
22	C₈×D₂₉₃	—	2	4	no	—
23	C₄×Dic₂₉₃	C₂×C₄	1	2	yes	—
24	C₄×Dic₂₉₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₈×C₂₉₃	C₂×C₄	1	2	yes	—
26	C₄²×C₂₉₃	C₂×C₄	1	1	yes	characteristic
27	C₂×#11924078	—	2	4	no	—
28	C₂×#11924078	—	2	4	no	—
29	C₂×#11924078	M₁₆	1	8	yes	—
30	#14556880	—	2	4	no	—
31	C₂²×D₂₉₃	C₂²×C₄	1	1	yes	characteristic
32	C₂×Dic₂₉₃	C₄²	1	2	yes	—
33	C₂×Dic₂₉₃	C₂²×C₄	1	1	yes	characteristic
34	C₄×D₂₉₃	—	2	4	no	—
35	C₄×D₂₉₃	#41	1	4	yes	—
36	#14556873	—	2	2	no	—
37	C₂×C₄×C₂₉₃	C₄²	1	2	yes	—
38	C₂×C₄×C₂₉₃	C₂²×C₄	1	1	yes	characteristic
39	C₈×C₂₉₃	—	2	2	no	—
40	C₂²×C₂₉₃	C₂×C₄²	1	1	yes	characteristic
41	C₂×D₂₉₃	#117	1	2	yes	—
42	C₂×D₂₉₃	C₂×M₁₆	1	4	yes	—
43	#11924078	—	2	8	no	—
44	C₄×C₂₉₃	—	2	2	no	—
45	C₄×C₂₉₃	#117	1	2	yes	—
46	Dic₂₉₃	—	2	2	no	—
47	Dic₂₉₃	#117	1	2	yes	—
48	C₂×C₂₉₃	#376	1	1	yes	characteristic
49	C₂×C₂₉₃	C₄×M₁₆	1	2	yes	—
50	D₂₉₃	#432	1	4	yes	—
51	C₂₉₃	#1745	1	1	yes	characteristic, complemented, Sylow
52	#1745	—	293	1	no	maximal, Sylow
53	C₂×#97	—	293	1	no	—
54	C₂×#97	—	293	1	no	—
55	C₄³	—	293	1	no	—
56	#336	—	293	4	no	—
57	C₂²×C₈	—	293	2	no	—
58	C₂²×C₈	—	293	2	no	—
59	C₂×C₄²	—	293	1	no	—
60	C₂×C₄²	—	293	2	no	—
61	C₂×C₄²	—	293	4	no	—
62	#97	—	293	4	no	—
63	#97	—	293	2	no	—
64	#97	—	293	1	no	—
65	#97	#11924078	1	1	yes	characteristic, complemented
66	C₂²×C₄	—	293	1	no	—
67	C₂²×C₄	—	293	4	no	—
68	C₂²×C₄	—	293	2	no	—
69	C₂×C₈	—	293	4	no	—
70	C₂×C₈	—	293	2	no	—
71	C₂×C₈	—	586	4	no	—
72	C₂×C₈	—	586	4	no	—
73	C₂×C₈	C₂×#11924078	1	2	yes	—
74	C₄²	—	293	2	no	—
75	C₄²	—	293	1	no	—
76	C₄²	—	586	4	no	—
77	C₄²	—	586	8	no	—
78	C₄²	C₂×#11924078	1	1	yes	characteristic
79	C₂³	—	293	1	no	—
80	C₂×C₄	—	293	1	no	—
81	C₂×C₄	—	293	4	no	—
82	C₂×C₄	—	293	8	no	—
83	C₂×C₄	—	293	2	no	—
84	C₂×C₄	—	586	4	no	—
85	C₂×C₄	—	586	4	no	—
86	C₂×C₄	—	586	4	no	—
87	C₂×C₄	C₄×#11924078	1	2	yes	—
88	C₂×C₄	C₂²×#11924078	1	1	yes	characteristic
89	C₈	—	2	2	no	—
90	C₈	—	586	4	no	—
91	C₈	—	586	2	no	—
92	C₂²	—	293	2	no	—
93	C₂²	—	293	4	no	—
94	C₂²	C₂×C₄×#11924078	1	1	yes	characteristic
95	C₄	—	2	2	no	—
96	C₄	—	293	2	no	—
97	C₄	—	586	8	no	—
98	C₄	—	586	2	no	—
99	C₄	#22627703	1	2	yes	—
100	C₂	—	293	4	no	—
101	C₂	#34037065	1	1	yes	characteristic
102	C₂	M₁₆×#11924078	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#34037088

C₂

yes

maximal, characteristic

C₄²×#11924078

C₂

yes

maximal, characteristic

(C₂×#11924078)⋊C₈

C₂

yes

maximal

#97×D₂₉₃

C₂

yes

maximal, characteristic

C₂×D₂₉₃⋊C₈

C₂²

yes

—

C₂×C₄×#11924078

C₄

yes

—

C₂×C₄×#11924078

C₂²

yes

—

#22627679

C₄

yes

complemented

C₂×C₈×D₂₉₃

C₂²

yes

—

C₄²×D₂₉₃

C₂²

yes

characteristic

#22627535

C₄

yes

characteristic, complemented

#22627534

C₄

yes

complemented

#97×C₂₉₃

C₄

yes

characteristic, complemented

C₂²×#11924078

C₂×C₄

yes

—

D₂₉₃⋊C₈

—

C₂×#14556880

C₂×C₄

yes

—

C₄×#11924078

—

C₄×#11924078

—

C₂×C₄×D₂₉₃

C₂×C₄

yes

—

C₂×C₄×D₂₉₃

C₂³

yes

characteristic

C₂×#14556873

C₂×C₄

yes

—

C₈×D₂₉₃

—

C₄×Dic₂₉₃

C₂×C₄

yes

—

C₄×Dic₂₉₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₉₃

C₂×C₄

yes

—

C₄²×C₂₉₃

C₂×C₄

yes

—

—

M₁₆

yes

—

#14556880

—

C₂²×D₂₉₃

C₂²×C₄

yes

characteristic

C₂×Dic₂₉₃

C₄²

yes

—

C₂×Dic₂₉₃

C₂²×C₄

yes

characteristic

C₄×D₂₉₃

—

C₄×D₂₉₃

#41

yes

—

#14556873

—

C₂×C₄×C₂₉₃

C₄²

yes

—

C₂×C₄×C₂₉₃

C₂²×C₄

yes

characteristic

C₈×C₂₉₃

—

C₂²×C₂₉₃

C₂×C₄²

yes

characteristic

C₂×D₂₉₃

#117

yes

—

C₂×D₂₉₃

C₂×M₁₆

yes

—

#11924078

—

C₄×C₂₉₃

—

C₄×C₂₉₃

#117

yes

—

Dic₂₉₃

—

Dic₂₉₃

#117

yes

—

C₂×C₂₉₃

#376

yes

characteristic

C₂×C₂₉₃

C₄×M₁₆

yes

—

D₂₉₃

#432

yes

—

C₂₉₃

#1745

yes

characteristic, complemented, Sylow

#1745

—

293

maximal, Sylow

C₂×#97

—

293

—

C₂×#97

—

293

—

C₄³

—

293

—

#336

—

293

—

C₂²×C₈

—

293

—

C₂²×C₈

—

293

—

C₂×C₄²

—

293

—

C₂×C₄²

—

293

—

C₂×C₄²

—

293

—

#97

—

293

—

#97

—

293

—

#97

—

293

—

#97

#11924078

yes

characteristic, complemented

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂²×C₄

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

293

—

C₂×C₈

—

586

—

C₂×C₈

—

586

—

C₂×C₈

C₂×#11924078

yes

—

C₄²

—

293

—

C₄²

—

293

—

C₄²

—

586

—

C₄²

—

586

—

C₄²

C₂×#11924078

yes

characteristic

C₂³

—

293

—

C₂×C₄

—

293

—

C₂×C₄

—

293

—

C₂×C₄

—

293

—

C₂×C₄

—

293

—

C₂×C₄

—

586

—

C₂×C₄

—

586

—

C₂×C₄

—

586

—

C₂×C₄

C₄×#11924078

yes

—

C₂×C₄

C₂²×#11924078

yes

characteristic

C₈

—

C₈

—

586

—

C₈

—

586

—

C₂²

—

293

—

C₂²

—

293

—

C₂²

C₂×C₄×#11924078

yes

characteristic

C₄

—

C₄

—

293

—

C₄

—

586

—

C₄

—

586

—

C₄

#22627703

yes

—

100

C₂

—

293

—

101

C₂

#34037065

yes

characteristic

102

C₂

M₁₆×#11924078

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×#11924078
commutator, socle	C₂²×C₂₉₃	C₂×C₄²
Fitting	#97×C₂₉₃	C₄
prime radical	C₂²×D₂₉₃	C₂²×C₄
cosocle	C₂×C₄×D₂₉₃	C₂³
hypercenter	#97	#11924078
hypocenter	C₂₉₃	#1745
LCS	G ▷ C₂²×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂×C₄ ◁ #97

subgroup

G/H

center, Frattini

C₂×C₄

C₂²×#11924078

commutator, socle

C₂²×C₂₉₃

C₂×C₄²

Fitting

#97×C₂₉₃

C₄

prime radical

C₂²×D₂₉₃

C₂²×C₄

cosocle

C₂×C₄×D₂₉₃

C₂³

C₂₉₃

G ▷ C₂²×C₂₉₃ ▷ C₂₉₃

C₁ ◁ C₂×C₄ ◁ #97

N	H ≅ G/N	type
8. #22627679	C₄	semidirect
11. #22627535	C₄	semidirect
12. #22627534	C₄	semidirect
13. #97×C₂₉₃	C₄	semidirect
51. C₂₉₃	#1745	semidirect
65. #97	#11924078	semidirect

H ≅ G/N

type

8. #22627679

C₄

semidirect

11. #22627535

C₄

semidirect

12. #22627534

C₄

semidirect

13. #97×C₂₉₃

C₄

semidirect

51. C₂₉₃

#1745

semidirect

65. #97

#11924078

semidirect

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

subgroup	index
#34037088	2
C₄²×#11924078	2
(C₂×#11924078)⋊C₈	2
#97×D₂₉₃	2
#1745	293

subgroup

index

#97×D₂₉₃

293

Subgroups — #49174501

Conjugacy classes of subgroups

Complements

Maximal subgroups