Subgroups — #36373950

Order 22336.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013661	C₂	1	1	yes	maximal, characteristic
2	C₂×#19150208	C₂	1	1	yes	maximal, characteristic, complemented
3	#108×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	#19150233	—	2	1	no	—
5	Dic₁₃₉₆	C₄	1	1	yes	characteristic
6	#19150222	—	2	1	no	—
7	D₂₇₉₂	C₄	1	1	yes	characteristic
8	#19150208	—	2	1	no	—
9	M₁₆×C₃₄₉	—	2	1	no	—
10	C₂×C₈×C₃₄₉	C₂²	1	1	yes	characteristic
11	Dic₆₉₈	—	2	1	no	—
12	D₁₃₉₆	—	2	1	no	—
13	C₂²⋊D₃₄₉	—	4	1	no	—
14	C₄×D₃₄₉	—	4	1	no	—
15	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
16	C₈×C₃₄₉	—	4	1	no	—
17	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
18	C₈×C₃₄₉	D₄	1	1	yes	characteristic
19	C₂²×C₃₄₉	SD₁₆	1	1	yes	characteristic
20	C₂×D₃₄₉	—	4	1	no	—
21	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₄₉	D₈	1	1	yes	characteristic
23	Dic₃₄₉	—	4	1	no	—
24	C₂×C₃₄₉	—	2	1	no	—
25	C₂×C₃₄₉	#102	1	1	yes	characteristic
26	D₃₄₉	—	8	1	no	—
27	C₃₄₉	#359	1	1	yes	characteristic, complemented, Sylow
28	#359	—	349	1	no	maximal, Sylow
29	#135	—	349	1	no	—
30	C₂×C₁₆	—	349	1	no	—
31	#108	D₃₄₉	1	1	yes	characteristic, complemented
32	#41	—	698	1	no	—
33	Q₁₆	—	349	1	no	—
34	SD₁₆	—	698	1	no	—
35	D₈	—	349	1	no	—
36	M₁₆	—	2	1	no	—
37	C₂×C₈	C₂×D₃₄₉	1	1	yes	characteristic
38	C₁₆	—	698	1	no	—
39	Q₈	—	698	1	no	—
40	D₄	—	1396	1	no	—
41	D₄	—	698	1	no	—
42	C₂×C₄	—	1396	1	no	—
43	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
44	C₈	—	4	1	no	—
45	C₈	C₄×D₃₄₉	1	1	yes	characteristic
46	C₈	D₁₃₉₆	1	1	yes	characteristic
47	C₂²	—	1396	1	no	—
48	C₂²	#19150220	1	1	yes	characteristic
49	C₄	—	1396	1	no	—
50	C₄	#19150210	1	1	yes	characteristic
51	C₄	#19150218	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	2792	1	no	—
54	C₂	#26013559	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19150218
commutator	C₈×C₃₄₉	C₂×C₄
Frattini	C₂×C₈	C₂×D₃₄₉
Fitting	#108×C₃₄₉	C₂
socle	C₂×C₃₄₉	#102
prime radical	#26013661	C₂
cosocle	C₂×C₈×C₃₄₉	C₂²
hypercenter	#108	D₃₄₉
hypocenter	C₃₄₉	#359
LCS	G ▷ C₈×C₃₄₉ ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #108

Complements

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

N	H ≅ G/N	type
2. C₂×#19150208	C₂	semidirect
3. #108×C₃₄₉	C₂	semidirect
27. C₃₄₉	#359	semidirect
31. #108	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013661	2
C₂×#19150208	2
#108×C₃₄₉	2
#359	349