Subgroups — #36374501

Order 22336.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013627	C₂	1	1	yes	maximal, characteristic
2	#26013562	C₂	1	2	yes	maximal, complemented
3	#26013559	C₂	1	2	yes	maximal
4	C₄×#14584095	C₂	1	1	yes	maximal, characteristic, complemented
5	#125×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₆₉₈	C₂²	1	1	yes	characteristic
7	C₂×D₁₃₉₆	C₂²	1	1	yes	characteristic
8	C₂×#14584095	C₂²	1	2	yes	—
9	#19150210	—	2	2	no	—
10	C₄⋊C₄×C₃₄₉	—	2	2	no	—
11	C₄⋊C₄×C₃₄₉	C₂²	1	2	yes	—
12	C₄²×C₃₄₉	C₂²	1	1	yes	characteristic
13	C₂²×D₃₄₉	—	2	1	no	—
14	Dic₆₉₈	—	2	2	no	—
15	D₁₃₉₆	—	2	2	no	—
16	C₂×Dic₃₄₉	—	2	1	no	—
17	#14584095	—	2	2	no	—
18	C₂×C₄×C₃₄₉	—	2	2	no	—
19	C₂×C₄×C₃₄₉	D₄	1	2	yes	—
20	C₂×C₄×C₃₄₉	C₂³	1	1	yes	characteristic
21	C₂²×C₃₄₉	C₂×D₄	1	1	yes	characteristic
22	C₂×D₃₄₉	—	4	1	no	—
23	C₂×D₃₄₉	—	4	2	no	—
24	C₄×C₃₄₉	—	2	2	no	—
25	C₄×C₃₄₉	—	4	2	no	—
26	C₄×C₃₄₉	#41	1	2	yes	—
27	Dic₃₄₉	—	4	1	no	—
28	C₂×C₃₄₉	#124	1	1	yes	characteristic
29	C₂×C₃₄₉	#135	1	2	yes	—
30	D₃₄₉	—	8	1	no	—
31	C₃₄₉	#488	1	1	yes	characteristic, complemented, Sylow
32	#488	—	349	1	no	maximal, Sylow
33	#125	D₃₄₉	1	1	yes	characteristic, complemented
34	#124	—	349	1	no	—
35	#103	—	349	2	no	—
36	#102	—	349	2	no	—
37	C₄×C₈	—	349	1	no	—
38	C₂×Q₈	—	349	1	no	—
39	C₂×D₄	—	349	1	no	—
40	C₂×C₈	—	349	2	no	—
41	C₄⋊C₄	—	2	2	no	—
42	C₄⋊C₄	C₂×D₃₄₉	1	2	yes	—
43	C₂²⋊C₄	—	698	2	no	—
44	C₄²	C₂×D₃₄₉	1	1	yes	characteristic
45	C₂³	—	698	1	no	—
46	Q₈	—	698	2	no	—
47	D₄	—	698	2	no	—
48	C₂×C₄	—	2	2	no	—
49	C₂×C₄	—	698	1	no	—
50	C₂×C₄	C₂²⋊D₃₄₉	1	2	yes	—
51	C₂×C₄	C₂²×D₃₄₉	1	1	yes	characteristic
52	C₈	—	698	2	no	—
53	C₂²	—	1396	1	no	—
54	C₂²	—	1396	2	no	—
55	C₂²	C₂×C₂²⋊D₃₄₉	1	1	yes	characteristic
56	C₄	—	2	2	no	—
57	C₄	—	4	2	no	—
58	C₄	—	1396	1	no	—
59	C₄	#19150234	1	2	yes	—
60	C₂	—	2792	1	no	—
61	C₂	#26013628	1	1	yes	characteristic
62	C₂	#26013659	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₄₉
commutator, cosocle	C₂×C₄×C₃₄₉	C₂³
Frattini	C₂×C₄	C₂²×D₃₄₉
Fitting	#125×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×D₄
prime radical	C₂×D₁₃₉₆	C₂²
hypercenter	#125	D₃₄₉
hypocenter	C₃₄₉	#488
LCS	G ▷ C₂×C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #125

Complements

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

N	H ≅ G/N	type
2. #26013562	C₂	semidirect
4. C₄×#14584095	C₂	semidirect
5. #125×C₃₄₉	C₂	semidirect
31. C₃₄₉	#488	semidirect
33. #125	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013627	2
#26013562	2
#26013559	2
C₄×#14584095	2
#125×C₃₄₉	2
#488	349