Subgroups — #36375062

Order 22336.

62 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013714	C₂	1	1	yes	maximal, characteristic, complemented
2	#26013709	C₂	1	1	yes	maximal, characteristic, complemented
3	D₈×D₃₄₉	C₂	1	1	yes	maximal, characteristic
4	#19150243	—	2	1	no	—
5	#19150236	—	2	1	no	—
6	D₄×D₃₄₉	—	2	1	no	—
7	D₂₇₉₂	C₄	1	1	yes	characteristic
8	#19150218	—	2	1	no	—
9	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
10	D₈×C₃₄₉	C₄	1	1	yes	characteristic
11	#14584102	—	4	1	no	—
12	C₂²×D₃₄₉	—	4	1	no	—
13	D₁₃₉₆	—	2	1	no	—
14	C₂²⋊D₃₄₉	—	4	1	no	—
15	C₄×D₃₄₉	D₄	1	1	yes	characteristic
16	#14584095	D₄	1	1	yes	characteristic
17	D₄×C₃₄₉	—	2	1	no	—
18	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃₄₉	—	4	1	no	—
20	C₂×D₃₄₉	—	4	1	no	—
21	C₂×D₃₄₉	—	8	2	no	—
22	C₂×D₃₄₉	SD₁₆	1	1	yes	characteristic
23	C₄×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
24	Dic₃₄₉	D₈	1	1	yes	characteristic
25	C₂×C₃₄₉	—	8	1	no	—
26	C₂×C₃₄₉	#102	1	1	yes	characteristic
27	D₃₄₉	—	2	1	no	—
28	D₃₄₉	—	8	1	no	—
29	C₃₄₉	#361	1	1	yes	characteristic, complemented, Sylow
30	#361	—	349	1	no	maximal, Sylow
31	C₂×D₈	—	349	1	no	—
32	M₃₂	—	349	1	no	—
33	#108	—	349	1	no	—
34	C₂×D₄	—	698	1	no	—
35	D₈	—	349	1	no	—
36	D₈	—	698	1	no	—
37	D₈	#13063703	1	1	yes	characteristic
38	M₁₆	—	698	1	no	—
39	C₂×C₈	—	349	1	no	—
40	C₁₆	—	698	1	no	—
41	C₂³	—	1396	1	no	—
42	D₄	—	2	1	no	—
43	D₄	—	1396	1	no	—
44	D₄	—	698	1	no	—
45	C₂×C₄	—	349	1	no	—
46	C₈	—	349	1	no	—
47	C₈	—	1396	1	no	—
48	C₈	C₂×#13063703	1	1	yes	characteristic
49	C₂²	—	4	1	no	—
50	C₂²	—	349	1	no	—
51	C₂²	—	1396	1	no	—
52	C₂²	—	2792	2	no	—
53	C₄	—	349	1	no	—
54	C₄	C₂²⋊#13063703	1	1	yes	characteristic
55	C₂	—	8	1	no	—
56	C₂	—	698	1	no	—
57	C₂	—	2792	1	no	—
58	C₂	#26013701	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013701
commutator	C₈×C₃₄₉	C₂×C₄
Frattini	C₈	C₂×#13063703
Fitting	D₈×C₃₄₉	C₄
socle	C₂×C₃₄₉	#102
prime radical	D₈×D₃₄₉	C₂
cosocle	C₈×D₃₄₉	C₂²
hypercenter	D₈	#13063703
hypocenter	C₃₄₉	#361
LCS	G ▷ C₈×C₃₄₉ ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #26013714	C₂	semidirect
2. #26013709	C₂	semidirect
29. C₃₄₉	#361	semidirect

Maximal subgroups

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

subgroup	index
#26013714	2
#26013709	2
D₈×D₃₄₉	2
#361	349