Subgroups — #36373925

Order 22336.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19150216	C₂	1	1	yes	maximal, characteristic
2	#26013579	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₃₄₉	C₄	1	1	yes	characteristic
5	#19150216	—	2	1	no	—
6	#19150216	—	2	1	no	—
7	C₂×#14584095	C₄	1	1	yes	characteristic
8	#19150208	—	2	1	no	—
9	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
10	C₁₆×C₃₄₉	—	2	1	no	—
11	C₂²×D₃₄₉	C₈	1	1	yes	characteristic
12	C₂×Dic₃₄₉	C₈	1	1	yes	characteristic
13	C₄×D₃₄₉	—	2	1	no	—
14	C₄×D₃₄₉	—	2	1	no	—
15	#14584095	—	2	1	no	—
16	C₂×C₄×C₃₄₉	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₄₉	D₄	1	1	yes	characteristic
18	C₈×C₃₄₉	D₄	1	1	yes	characteristic
19	C₂²×C₃₄₉	C₂×C₈	1	1	yes	characteristic
20	C₂×D₃₄₉	—	2	1	no	—
21	C₂×D₃₄₉	—	4	1	no	—
22	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
23	C₄×C₃₄₉	M₁₆	1	1	yes	characteristic
24	Dic₃₄₉	—	2	1	no	—
25	C₂×C₃₄₉	—	2	1	no	—
26	C₂×C₃₄₉	C₂²⋊C₈	1	1	yes	characteristic
27	D₃₄₉	—	4	1	no	—
28	C₃₄₉	#349	1	1	yes	characteristic, complemented, Sylow
29	#349	—	349	1	no	maximal, Sylow
30	C₂×M₁₆	—	349	1	no	—
31	M₃₂	—	349	1	no	—
32	M₃₂	D₃₄₉	1	1	yes	characteristic, complemented
33	C₂²×C₄	—	349	1	no	—
34	M₁₆	—	698	1	no	—
35	M₁₆	—	698	1	no	—
36	C₂×C₈	—	349	1	no	—
37	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
38	C₁₆	—	2	1	no	—
39	C₁₆	—	698	1	no	—
40	C₂³	—	349	1	no	—
41	C₂×C₄	—	349	1	no	—
42	C₂×C₄	—	698	1	no	—
43	C₂×C₄	—	698	1	no	—
44	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
45	C₈	—	698	1	no	—
46	C₈	C₂²⋊D₃₄₉	1	1	yes	characteristic
47	C₈	D₁₃₉₆	1	1	yes	characteristic
48	C₂²	—	1396	1	no	—
49	C₂²	—	698	1	no	—
50	C₂²	C₈×D₃₄₉	1	1	yes	characteristic
51	C₄	—	698	1	no	—
52	C₄	#19150210	1	1	yes	characteristic
53	C₄	#19150216	1	1	yes	characteristic
54	C₂	—	2	1	no	—
55	C₂	—	1396	1	no	—
56	C₂	#26013551	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19150210
commutator	C₂²×C₃₄₉	C₂×C₈
Frattini	C₂×C₈	C₂×D₃₄₉
Fitting	M₃₂×C₃₄₉	C₂
socle	C₂×C₃₄₉	C₂²⋊C₈
prime radical	C₂²×D₃₄₉	C₈
cosocle	C₂×C₈×C₃₄₉	C₂²
hypercenter	M₃₂	D₃₄₉
hypocenter	C₃₄₉	#349
LCS	G ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. #26013579	C₂	semidirect
3. M₃₂×C₃₄₉	C₂	semidirect
28. C₃₄₉	#349	semidirect
32. M₃₂	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19150216	2
#26013579	2
M₃₂×C₃₄₉	2
#349	349