Subgroups — #36373923

Order 22336.

66 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#19150208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×D₃₄₉	C₄	1	1	yes	characteristic
5	C₂×#14584095	C₄	1	1	yes	characteristic
6	C₈×D₃₄₉	—	2	1	no	—
7	C₈×D₃₄₉	—	2	1	no	—
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	—	2	1	no	—
22	C₂×D₃₄₉	C₁₆	1	2	yes	complemented
23	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
24	C₄×C₃₄₉	M₁₆	1	1	yes	characteristic
25	Dic₃₄₉	—	2	1	no	—
26	C₂×C₃₄₉	C₂²⋊C₈	1	1	yes	characteristic
27	C₂×C₃₄₉	C₂×C₁₆	1	1	yes	characteristic
28	C₂×C₃₄₉	M₃₂	1	1	yes	characteristic
29	D₃₄₉	—	2	2	no	—
30	C₃₄₉	C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
31	C₂²⋊C₁₆	—	349	1	no	maximal, Sylow
32	C₂²×C₈	—	349	1	no	—
33	C₂×C₁₆	—	349	1	no	—
34	C₂×C₁₆	D₃₄₉	1	1	yes	characteristic, complemented
35	C₂²×C₄	—	349	1	no	—
36	C₂×C₈	—	349	1	no	—
37	C₂×C₈	—	698	1	no	—
38	C₂×C₈	—	698	1	no	—
39	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
40	C₁₆	—	2	1	no	—
41	C₁₆	—	698	1	no	—
42	C₂³	—	349	1	no	—
43	C₂×C₄	—	349	1	no	—
44	C₂×C₄	—	698	1	no	—
45	C₂×C₄	—	698	1	no	—
46	C₂×C₄	C₄×D₃₄₉	1	1	yes	characteristic
47	C₈	—	698	1	no	—
48	C₈	C₂²⋊D₃₄₉	1	1	yes	characteristic
49	C₈	D₁₃₉₆	1	1	yes	characteristic
50	C₂²	—	349	2	no	—
51	C₂²	—	698	1	no	—
52	C₂²	—	698	1	no	—
53	C₂²	C₈×D₃₄₉	1	1	yes	characteristic
54	C₄	—	698	1	no	—
55	C₄	#19150210	1	1	yes	characteristic
56	C₄	#19150216	1	1	yes	characteristic
57	C₂	—	698	2	no	—
58	C₂	#26013551	1	1	yes	characteristic
59	C₂	C₁₆×D₃₄₉	1	1	yes	characteristic
60	C₂	#26013578	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃₄₉
commutator	C₂×C₃₄₉	C₂×C₁₆
Fitting	C₂×C₁₆×C₃₄₉	C₂
socle	C₂²×C₃₄₉	C₂×C₈
prime radical	C₂²×D₃₄₉	C₈
cosocle	C₂×C₈×C₃₄₉	C₂²
hypercenter	C₂×C₁₆	D₃₄₉
hypocenter	C₃₄₉	C₂²⋊C₁₆
LCS	G ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#19150208	C₂	semidirect
3. C₂×C₁₆×C₃₄₉	C₂	semidirect
22. C₂×D₃₄₉	C₁₆	semidirect
30. C₃₄₉	C₂²⋊C₁₆	semidirect
34. C₂×C₁₆	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₈×D₃₄₉	2
C₂×#19150208	2
C₂×C₁₆×C₃₄₉	2
C₂²⋊C₁₆	349