Subgroups — #26013718

Order 11168.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄⋊#13063703	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#13063703	C₂	1	2	yes	maximal
3	C₄×#13063703	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×D₃₄₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#13063703	C₂²	1	4	yes	—
6	C₂²×D₃₄₉	C₂²	1	1	yes	characteristic
7	C₂×Dic₃₄₉	C₄	1	1	yes	characteristic, complemented
8	C₄×D₃₄₉	C₄	1	1	yes	characteristic, complemented
9	C₄×D₃₄₉	C₄	1	1	yes	characteristic, complemented
10	C₄×D₃₄₉	C₂²	1	1	yes	characteristic
11	C₄×D₃₄₉	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₃₄₉	C₄	1	1	yes	characteristic, complemented
13	C₂²×C₃₄₉	C₂×C₄	1	1	yes	characteristic
14	C₂×D₃₄₉	—	2	2	no	—
15	C₂×D₃₄₉	C₂×C₄	1	1	yes	characteristic
16	C₂×D₃₄₉	C₂³	1	1	yes	characteristic
17	#13063703	—	2	4	no	—
18	C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
20	Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
21	Dic₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₂×C₃₄₉	—	2	1	no	—
23	C₂×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
24	D₃₄₉	—	2	1	no	—
25	D₃₄₉	#41	1	2	yes	—
26	C₃₄₉	#117	1	1	yes	characteristic, complemented, Sylow
27	#117	—	349	1	no	maximal, Sylow
28	C₂²×C₄	—	349	1	no	—
29	C₄⋊C₄	—	349	2	no	—
30	C₂²⋊C₄	—	349	2	no	—
31	C₄²	—	349	2	no	—
32	C₂³	—	349	1	no	—
33	C₂×C₄	—	349	1	no	—
34	C₂×C₄	—	349	1	no	—
35	C₂×C₄	—	349	1	no	—
36	C₂×C₄	—	349	1	no	—
37	C₂×C₄	—	349	1	no	—
38	C₂×C₄	—	349	4	no	—
39	C₂×C₄	#13063703	1	1	yes	characteristic, complemented
40	C₂²	—	349	1	no	—
41	C₂²	—	349	1	no	—
42	C₂²	—	698	2	no	—
43	C₂²	C₂×#13063703	1	1	yes	characteristic
44	C₄	—	349	1	no	—
45	C₄	—	349	1	no	—
46	C₄	—	698	4	no	—
47	C₄	C₂×#13063703	1	1	yes	characteristic
48	C₄	C₂×#13063703	1	1	yes	characteristic
49	C₂	—	2	1	no	—
50	C₂	—	349	2	no	—
51	C₂	—	698	1	no	—
52	C₂	C₂²×#13063703	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×#13063703
commutator, socle	C₂×C₃₄₉	C₂²×C₄
Frattini	C₂	C₂²×#13063703
Fitting	C₂×C₄×C₃₄₉	C₄
prime radical	C₂²×D₃₄₉	C₂²
cosocle	C₂×D₃₄₉	C₂³
hypercenter	C₂×C₄	#13063703
hypocenter	C₃₄₉	#117
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄

Complements

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

N	H ≅ G/N	type
1. C₄⋊#13063703	C₂	semidirect
3. C₄×#13063703	C₂	semidirect
7. C₂×Dic₃₄₉	C₄	semidirect
8. C₄×D₃₄₉	C₄	semidirect
9. C₄×D₃₄₉	C₄	semidirect
12. C₂×C₄×C₃₄₉	C₄	semidirect
26. C₃₄₉	#117	semidirect
39. C₂×C₄	#13063703	semidirect

Maximal subgroups

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

subgroup	index
C₄⋊#13063703	2
C₂²⋊#13063703	2
C₄×#13063703	2
C₂×C₄×D₃₄₉	2
#117	349