Group #34036940 — subgroups

Subgroups — #34036940

Order 18752.

258 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 107 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	#22627671	C₂	1	1	yes	maximal, characteristic, complemented
2	#22627654	C₂	1	3	yes	maximal, complemented
3	#22627650	C₂	1	3	yes	maximal, complemented
4	#22627644	C₂	1	3	yes	maximal, complemented
5	C₂×#17989790	C₂	1	3	yes	maximal, complemented
6	#22627631	C₂	1	1	yes	maximal, characteristic, complemented
7	H(5,𝔽₂)×C₂₉₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#17989802	C₂²	1	3	yes	complemented
9	#17989801	—	2	3	no	—
10	C₂×Dic₅₈₆	C₂²	1	3	yes	complemented
11	Q₈×D₂₉₃	—	2	1	no	—
12	#17989792	C₂²	1	3	yes	complemented
13	#17989790	C₂²	1	3	yes	complemented
14	#17989790	C₂²	1	6	yes	complemented
15	#17989789	C₂²	1	1	yes	characteristic, complemented
16	#17989787	C₂²	1	3	yes	complemented
17	#17989786	C₂²	1	3	yes	complemented
18	C₂×#14556873	C₂²	1	3	yes	complemented
19	#41×C₂₉₃	—	2	1	no	—
20	#41×C₂₉₃	C₂²	1	3	yes	—
21	#41×C₂₉₃	C₂²	1	1	yes	characteristic
22	C₂×D₄×C₂₉₃	—	2	3	no	—
23	C₂×D₄×C₂₉₃	C₂²	1	3	yes	—
24	Dic₅₈₆	—	2	3	no	—
25	Dic₅₈₆	C₂³	1	3	yes	—
26	D₁₁₇₂	C₂³	1	1	yes	characteristic
27	C₂²⋊D₂₉₃	—	2	3	no	—
28	C₂×Dic₂₉₃	—	2	3	no	—
29	C₄×D₂₉₃	—	2	3	no	—
30	#14556873	C₂³	1	1	yes	characteristic
31	#14556873	C₂³	1	3	yes	—
32	C₂³×C₂₉₃	—	2	3	no	—
33	Q₈×C₂₉₃	D₄	1	1	yes	characteristic
34	Q₈×C₂₉₃	C₂³	1	1	yes	characteristic
35	D₄×C₂₉₃	—	2	3	no	—
36	D₄×C₂₉₃	—	2	3	no	—
37	D₄×C₂₉₃	D₄	1	3	yes	—
38	D₄×C₂₉₃	C₂³	1	3	yes	—
39	C₂×C₄×C₂₉₃	—	2	3	no	—
40	C₂×C₄×C₂₉₃	C₂³	1	3	yes	—
41	C₂²×C₂₉₃	—	2	3	no	—
42	C₂²×C₂₉₃	—	8	3	no	—
43	C₂²×C₂₉₃	C₂×D₄	1	3	yes	—
44	C₂×D₂₉₃	—	2	1	no	—
45	C₄×C₂₉₃	—	2	1	no	—
46	C₄×C₂₉₃	C₂×D₄	1	3	yes	—
47	C₄×C₂₉₃	C₂⁴	1	1	yes	characteristic
48	Dic₂₉₃	—	2	3	no	—
49	C₂×C₂₉₃	—	2	3	no	—
50	C₂×C₂₉₃	—	4	3	no	—
51	C₂×C₂₉₃	C₂²×D₄	1	1	yes	characteristic
52	D₂₉₃	—	4	1	no	—
53	C₂₉₃	#577	1	1	yes	characteristic, complemented, Sylow
54	#577	—	293	1	no	maximal, Sylow
55	2₋¹⁺⁴	—	293	1	no	—
56	H(5,𝔽₂)	D₂₉₃	1	1	yes	characteristic, complemented
57	#137	—	293	3	no	—
58	AGL(1,ℤ/8)	—	293	3	no	—
59	#135	—	293	3	no	—
60	C₂×SD₁₆	—	293	3	no	—
61	#131	—	293	1	no	—
62	#41	—	2	1	no	—
63	#41	—	293	3	no	—
64	#41	—	586	3	no	—
65	#41	C₂×D₂₉₃	1	3	yes	—
66	#41	C₂×D₂₉₃	1	1	yes	characteristic
67	C₂×Q₈	—	293	3	no	—
68	C₂×Q₈	—	586	1	no	—
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂×D₂₉₃	1	3	yes	—
71	Q₁₆	—	293	3	no	—
72	SD₁₆	—	293	1	no	—
73	SD₁₆	—	293	3	no	—
74	SD₁₆	—	293	6	no	—
75	D₈	—	293	3	no	—
76	M₁₆	—	293	3	no	—
77	C₂×C₈	—	293	3	no	—
78	C₂³	—	2	3	no	—
79	Q₈	—	293	3	no	—
80	Q₈	—	586	3	no	—
81	Q₈	C₂²⋊D₂₉₃	1	1	yes	characteristic
82	Q₈	C₂²×D₂₉₃	1	1	yes	characteristic
83	D₄	—	2	3	no	—
84	D₄	—	2	3	no	—
85	D₄	—	293	1	no	—
86	D₄	—	586	3	no	—
87	D₄	C₂²⋊D₂₉₃	1	3	yes	—
88	D₄	C₂²×D₂₉₃	1	3	yes	—
89	C₂×C₄	—	2	3	no	—
90	C₂×C₄	—	586	3	no	—
91	C₂×C₄	—	586	3	no	—
92	C₂×C₄	C₂²×D₂₉₃	1	3	yes	—
93	C₈	—	293	1	no	—
94	C₈	—	293	3	no	—
95	C₂²	—	2	3	no	—
96	C₂²	—	8	3	no	—
97	C₂²	—	586	1	no	—
98	C₂²	C₂×C₂²⋊D₂₉₃	1	3	yes	—
99	C₄	—	2	1	no	—
100	C₄	—	586	3	no	—
101	C₄	C₂×C₂²⋊D₂₉₃	1	3	yes	—
102	C₄	C₂³×D₂₉₃	1	1	yes	characteristic
103	C₂	—	2	3	no	—
104	C₂	—	4	3	no	—
105	C₂	—	1172	1	no	—
106	C₂	C₂²×C₂²⋊D₂₉₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22627671

C₂

yes

maximal, characteristic, complemented

#22627654

C₂

yes

maximal, complemented

#22627650

C₂

yes

maximal, complemented

#22627644

C₂

yes

maximal, complemented

C₂×#17989790

C₂

yes

maximal, complemented

#22627631

C₂

yes

maximal, characteristic, complemented

H(5,𝔽₂)×C₂₉₃

C₂

yes

maximal, characteristic, complemented

#17989802

C₂²

yes

complemented

#17989801

—

C₂×Dic₅₈₆

C₂²

yes

complemented

Q₈×D₂₉₃

—

#17989792

C₂²

yes

complemented

#17989790

C₂²

yes

complemented

#17989790

C₂²

yes

complemented

#17989789

C₂²

yes

characteristic, complemented

#17989787

C₂²

yes

complemented

#17989786

C₂²

yes

complemented

C₂×#14556873

C₂²

yes

complemented

#41×C₂₉₃

—

#41×C₂₉₃

C₂²

yes

—

#41×C₂₉₃

C₂²

yes

characteristic

C₂×D₄×C₂₉₃

—

C₂×D₄×C₂₉₃

C₂²

yes

—

Dic₅₈₆

—

Dic₅₈₆

C₂³

yes

—

D₁₁₇₂

C₂³

yes

characteristic

C₂²⋊D₂₉₃

—

C₂×Dic₂₉₃

—

C₄×D₂₉₃

—

#14556873

C₂³

yes

characteristic

#14556873

C₂³

yes

—

C₂³×C₂₉₃

—

Q₈×C₂₉₃

D₄

yes

characteristic

Q₈×C₂₉₃

C₂³

yes

characteristic

D₄×C₂₉₃

—

D₄×C₂₉₃

—

D₄×C₂₉₃

D₄

yes

—

D₄×C₂₉₃

C₂³

yes

—

C₂×C₄×C₂₉₃

—

C₂×C₄×C₂₉₃

C₂³

yes

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

—

C₂²×C₂₉₃

C₂×D₄

yes

—

C₂×D₂₉₃

—

C₄×C₂₉₃

—

C₄×C₂₉₃

C₂×D₄

yes

—

C₄×C₂₉₃

C₂⁴

yes

characteristic

Dic₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

—

C₂×C₂₉₃

C₂²×D₄

yes

characteristic

D₂₉₃

—

C₂₉₃

#577

yes

characteristic, complemented, Sylow

#577

—

293

maximal, Sylow

2₋¹⁺⁴

—

293

—

H(5,𝔽₂)

D₂₉₃

yes

characteristic, complemented

#137

—

293

—

AGL(1,ℤ/8)

—

293

—

#135

—

293

—

C₂×SD₁₆

—

293

—

#131

—

293

—

#41

—

#41

—

293

—

#41

—

586

—

#41

C₂×D₂₉₃

yes

—

#41

C₂×D₂₉₃

yes

characteristic

C₂×Q₈

—

293

—

C₂×Q₈

—

586

—

C₂×D₄

—

C₂×D₄

C₂×D₂₉₃

yes

—

Q₁₆

—

293

—

SD₁₆

—

293

—

SD₁₆

—

293

—

SD₁₆

—

293

—

D₈

—

293

—

M₁₆

—

293

—

C₂×C₈

—

293

—

C₂³

—

Q₈

—

293

—

Q₈

—

586

—

Q₈

C₂²⋊D₂₉₃

yes

characteristic

Q₈

C₂²×D₂₉₃

yes

characteristic

D₄

—

D₄

—

D₄

—

293

—

D₄

—

586

—

D₄

C₂²⋊D₂₉₃

yes

—

D₄

C₂²×D₂₉₃

yes

—

C₂×C₄

—

C₂×C₄

—

586

—

C₂×C₄

—

586

—

C₂×C₄

C₂²×D₂₉₃

yes

—

C₈

—

293

—

C₈

—

293

—

C₂²

—

C₂²

—

C₂²

—

586

—

C₂²

C₂×C₂²⋊D₂₉₃

yes

—

C₄

—

100

C₄

—

586

—

101

C₄

C₂×C₂²⋊D₂₉₃

yes

—

102

C₄

C₂³×D₂₉₃

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

1172

—

106

C₂

C₂²×C₂²⋊D₂₉₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×C₂²⋊D₂₉₃
commutator, cosocle	C₄×C₂₉₃	C₂⁴
Frattini	C₄	C₂³×D₂₉₃
Fitting	H(5,𝔽₂)×C₂₉₃	C₂
socle	C₂×C₂₉₃	C₂²×D₄
hypercenter	H(5,𝔽₂)	D₂₉₃
hypocenter	C₂₉₃	#577
LCS	G ▷ C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

subgroup

G/H

center

C₂

C₂²×C₂²⋊D₂₉₃

commutator, cosocle

C₄×C₂₉₃

C₂⁴

Frattini

C₄

C₂³×D₂₉₃

Fitting

H(5,𝔽₂)×C₂₉₃

C₂

socle

C₂×C₂₉₃

C₂²×D₄

D₂₉₃

C₂₉₃

G ▷ C₄×C₂₉₃ ▷ C₂×C₂₉₃ ▷ C₂₉₃

UCS

C₁ ◁ C₂ ◁ #41 ◁ H(5,𝔽₂)

N	H ≅ G/N	type
1. #22627671	C₂	semidirect
2. #22627654	C₂	semidirect
3. #22627650	C₂	semidirect
4. #22627644	C₂	semidirect
5. C₂×#17989790	C₂	semidirect
6. #22627631	C₂	semidirect
7. H(5,𝔽₂)×C₂₉₃	C₂	semidirect
8. #17989802	C₂²	semidirect
10. C₂×Dic₅₈₆	C₂²	semidirect
12. #17989792	C₂²	semidirect
13. #17989790	C₂²	semidirect
14. #17989790	C₂²	semidirect
15. #17989789	C₂²	semidirect
16. #17989787	C₂²	semidirect
17. #17989786	C₂²	semidirect
18. C₂×#14556873	C₂²	semidirect
53. C₂₉₃	#577	semidirect
56. H(5,𝔽₂)	D₂₉₃	semidirect

H ≅ G/N

type

1. #22627671

C₂

semidirect

2. #22627654

C₂

semidirect

3. #22627650

C₂

semidirect

4. #22627644

C₂

semidirect

5. C₂×#17989790

C₂

semidirect

6. #22627631

C₂

semidirect

7. H(5,𝔽₂)×C₂₉₃

C₂

semidirect

8. #17989802

C₂²

semidirect

10. C₂×Dic₅₈₆

C₂²

semidirect

12. #17989792

C₂²

semidirect

13. #17989790

C₂²

semidirect

14. #17989790

C₂²

semidirect

15. #17989789

C₂²

semidirect

16. #17989787

C₂²

semidirect

17. #17989786

C₂²

semidirect

18. C₂×#14556873

C₂²

semidirect

53. C₂₉₃

#577

semidirect

56. H(5,𝔽₂)

D₂₉₃

semidirect

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

subgroup	index
#22627671	2
#22627654	2
#22627650	2
#22627644	2
C₂×#17989790	2
#22627631	2
H(5,𝔽₂)×C₂₉₃	2
#577	293

subgroup

index

H(5,𝔽₂)×C₂₉₃

293

Subgroups — #34036940

Conjugacy classes of subgroups

Complements

Maximal subgroups