Subgroups — #26013685

Order 11168.

166 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19150234	C₂	1	2	yes	maximal, complemented
2	#19150233	C₂	1	3	yes	maximal, complemented
3	C₂×D₁₃₉₆	C₂	1	3	yes	maximal, complemented
4	D₄×D₃₄₉	C₂	1	6	yes	maximal, complemented
5	#41×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×D₃₄₉	C₂²	1	6	yes	complemented
7	Dic₆₉₈	C₂²	1	1	yes	characteristic, complemented
8	D₁₃₉₆	C₂²	1	6	yes	complemented
9	D₁₃₉₆	C₂²	1	3	yes	complemented
10	C₂²⋊D₃₄₉	C₂²	1	6	yes	complemented
11	C₄×D₃₄₉	C₂²	1	6	yes	complemented
12	Q₈×C₃₄₉	C₂²	1	1	yes	characteristic, complemented
13	D₄×C₃₄₉	C₂²	1	3	yes	complemented
14	C₂×C₄×C₃₄₉	C₂²	1	3	yes	complemented
15	C₂²×C₃₄₉	C₂³	1	3	yes	—
16	C₂×D₃₄₉	—	4	6	no	—
17	C₂×D₃₄₉	C₂³	1	6	yes	—
18	C₄×C₃₄₉	C₂³	1	1	yes	characteristic
19	C₄×C₃₄₉	C₂³	1	3	yes	—
20	Dic₃₄₉	C₂³	1	2	yes	—
21	C₂×C₃₄₉	—	2	3	no	—
22	C₂×C₃₄₉	C₂⁴	1	1	yes	characteristic
23	D₃₄₉	—	2	6	no	—
24	C₃₄₉	H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
25	H(5,𝔽₂)	—	349	1	no	maximal, Sylow
26	#41	—	349	2	no	—
27	#41	—	349	3	no	—
28	#41	D₃₄₉	1	1	yes	characteristic, complemented
29	C₂×D₄	—	349	6	no	—
30	C₂×D₄	—	349	3	no	—
31	C₂³	—	349	6	no	—
32	Q₈	—	349	1	no	—
33	Q₈	C₂×D₃₄₉	1	1	yes	characteristic, complemented
34	D₄	—	349	6	no	—
35	D₄	—	349	6	no	—
36	D₄	—	349	3	no	—
37	D₄	C₂×D₃₄₉	1	3	yes	complemented
38	C₂×C₄	—	349	6	no	—
39	C₂×C₄	C₂×D₃₄₉	1	3	yes	complemented
40	C₂²	—	349	6	no	—
41	C₂²	—	1396	6	no	—
42	C₂²	C₂²×D₃₄₉	1	3	yes	—
43	C₄	—	349	2	no	—
44	C₄	C₂²×D₃₄₉	1	3	yes	—
45	C₄	C₂²×D₃₄₉	1	1	yes	characteristic
46	C₂	—	2	3	no	—
47	C₂	—	698	6	no	—
48	C₂	C₂³×D₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂³×D₃₄₉
commutator, socle, cosocle	C₂×C₃₄₉	C₂⁴
Fitting	#41×C₃₄₉	C₂
hypercenter	#41	D₃₄₉
hypocenter	C₃₄₉	H(5,𝔽₂)
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ #41

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #19150234	C₂	semidirect
2. #19150233	C₂	semidirect
3. C₂×D₁₃₉₆	C₂	semidirect
4. D₄×D₃₄₉	C₂	semidirect
5. #41×C₃₄₉	C₂	semidirect
6. C₂²×D₃₄₉	C₂²	semidirect
7. Dic₆₉₈	C₂²	semidirect
8. D₁₃₉₆	C₂²	semidirect
9. D₁₃₉₆	C₂²	semidirect
10. C₂²⋊D₃₄₉	C₂²	semidirect
11. C₄×D₃₄₉	C₂²	semidirect
12. Q₈×C₃₄₉	C₂²	semidirect
13. D₄×C₃₄₉	C₂²	semidirect
14. C₂×C₄×C₃₄₉	C₂²	semidirect
24. C₃₄₉	H(5,𝔽₂)	semidirect
28. #41	D₃₄₉	semidirect
33. Q₈	C₂×D₃₄₉	semidirect
37. D₄	C₂×D₃₄₉	semidirect
39. C₂×C₄	C₂×D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#19150234	2
#19150233	2
C₂×D₁₃₉₆	2
D₄×D₃₄₉	2
#41×C₃₄₉	2
H(5,𝔽₂)	349