Subgroups — #36374575

Order 22336.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013662	C₂	1	1	yes	maximal, characteristic, complemented
2	#26013658	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013585	C₂	1	1	yes	maximal, characteristic, complemented
4	#26013584	C₂	1	1	yes	maximal, characteristic, complemented
5	#26013580	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×D₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
7	SD₃₂×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
8	#19150234	—	2	1	no	—
9	#19150232	—	2	1	no	—
10	Dic₁₃₉₆	C₂²	1	1	yes	characteristic
11	#19150221	—	2	1	no	—
12	#19150220	—	2	1	no	—
13	D₂₇₉₂	C₂²	1	1	yes	characteristic
14	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
15	#19150208	C₂²	1	1	yes	characteristic
16	Q₁₆×C₃₄₉	C₂²	1	1	yes	characteristic
17	D₈×C₃₄₉	C₂²	1	1	yes	characteristic
18	C₁₆×C₃₄₉	C₂²	1	1	yes	characteristic
19	Dic₆₉₈	—	2	1	no	—
20	D₁₃₉₆	—	2	1	no	—
21	D₁₃₉₆	—	4	1	no	—
22	C₂²⋊D₃₄₉	—	4	1	no	—
23	C₂×Dic₃₄₉	—	4	1	no	—
24	C₄×D₃₄₉	—	4	1	no	—
25	C₄×D₃₄₉	D₄	1	1	yes	characteristic
26	#14584095	D₄	1	1	yes	characteristic
27	Q₈×C₃₄₉	—	2	1	no	—
28	D₄×C₃₄₉	—	2	1	no	—
29	C₈×C₃₄₉	C₂³	1	1	yes	characteristic
30	C₂²×C₃₄₉	—	4	1	no	—
31	C₂×D₃₄₉	—	4	1	no	—
32	C₂×D₃₄₉	D₈	1	1	yes	characteristic
33	C₄×C₃₄₉	—	4	1	no	—
34	C₄×C₃₄₉	C₂×D₄	1	1	yes	characteristic
35	Dic₃₄₉	—	4	1	no	—
36	Dic₃₄₉	D₈	1	1	yes	characteristic
37	C₂×C₃₄₉	—	8	1	no	—
38	C₂×C₃₄₉	C₂×D₈	1	1	yes	characteristic
39	D₃₄₉	—	2	1	no	—
40	D₃₄₉	—	8	1	no	—
41	C₃₄₉	#508	1	1	yes	characteristic, complemented, Sylow
42	#508	—	349	1	no	maximal, Sylow
43	#135	—	349	1	no	—
44	#135	—	349	1	no	—
45	Q₃₂	—	349	1	no	—
46	SD₃₂	—	349	1	no	—
47	SD₃₂	D₃₄₉	1	1	yes	characteristic, complemented
48	D₁₆	—	349	1	no	—
49	C₂×C₁₆	—	349	1	no	—
50	#41	—	698	1	no	—
51	#41	—	698	1	no	—
52	Q₁₆	—	349	1	no	—
53	Q₁₆	C₂×D₃₄₉	1	1	yes	characteristic
54	SD₁₆	—	698	1	no	—
55	SD₁₆	—	698	1	no	—
56	D₈	—	349	1	no	—
57	D₈	C₂×D₃₄₉	1	1	yes	characteristic
58	C₂×C₈	—	349	1	no	—
59	C₁₆	—	349	1	no	—
60	C₁₆	C₂×D₃₄₉	1	1	yes	characteristic
61	Q₈	—	2	1	no	—
62	Q₈	—	698	1	no	—
63	D₄	—	2	1	no	—
64	D₄	—	1396	1	no	—
65	D₄	—	1396	1	no	—
66	D₄	—	698	1	no	—
67	C₂×C₄	—	349	1	no	—
68	C₂×C₄	—	1396	1	no	—
69	C₂×C₄	—	1396	1	no	—
70	C₈	—	349	1	no	—
71	C₈	C₂²×D₃₄₉	1	1	yes	characteristic
72	C₂²	—	4	1	no	—
73	C₂²	—	349	1	no	—
74	C₂²	—	1396	1	no	—
75	C₄	—	4	1	no	—
76	C₄	—	349	1	no	—
77	C₄	—	1396	1	no	—
78	C₄	D₄×D₃₄₉	1	1	yes	characteristic
79	C₂	—	8	1	no	—
80	C₂	—	698	1	no	—
81	C₂	—	2792	1	no	—
82	C₂	D₈×D₃₄₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₃₄₉
commutator, cosocle	C₈×C₃₄₉	C₂³
Frattini	C₈	C₂²×D₃₄₉
Fitting	SD₃₂×C₃₄₉	C₂
socle	C₂×C₃₄₉	C₂×D₈
hypercenter	SD₃₂	D₃₄₉
hypocenter	C₃₄₉	#508
LCS	G ▷ C₈×C₃₄₉ ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ SD₃₂

Complements

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

N	H ≅ G/N	type
1. #26013662	C₂	semidirect
2. #26013658	C₂	semidirect
3. #26013585	C₂	semidirect
4. #26013584	C₂	semidirect
5. #26013580	C₂	semidirect
6. C₁₆×D₃₄₉	C₂	semidirect
7. SD₃₂×C₃₄₉	C₂	semidirect
41. C₃₄₉	#508	semidirect
47. SD₃₂	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013662	2
#26013658	2
#26013585	2
#26013584	2
#26013580	2
C₁₆×D₃₄₉	2
SD₃₂×C₃₄₉	2
#508	349