Subgroups — #36374961

Order 22336.

248 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013686	C₂	1	2	yes	maximal, complemented
2	#26013672	C₂	1	6	yes	maximal, complemented
3	#26013661	C₂	1	3	yes	maximal, complemented
4	C₂×Dic₁₃₉₆	C₂	1	3	yes	maximal, complemented
5	#131×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19150233	C₂²	1	6	yes	—
7	#19150232	—	2	6	no	—
8	C₂×Dic₆₉₈	C₂²	1	6	yes	—
9	Q₈×D₃₄₉	—	2	2	no	—
10	Dic₁₃₉₆	C₂²	1	3	yes	—
11	Dic₁₃₉₆	C₂²	1	6	yes	—
12	#19150222	C₂²	1	6	yes	—
13	D₂₇₉₂	C₂²	1	1	yes	characteristic
14	#41×C₃₄₉	C₂²	1	1	yes	characteristic
15	M₁₆×C₃₄₉	C₂²	1	3	yes	—
16	C₂×C₈×C₃₄₉	C₂²	1	3	yes	—
17	Dic₆₉₈	—	2	6	no	—
18	Dic₆₉₈	C₂³	1	6	yes	—
19	D₁₃₉₆	C₂³	1	2	yes	—
20	C₂²⋊D₃₄₉	—	2	6	no	—
21	C₂×Dic₃₄₉	—	2	6	no	—
22	C₄×D₃₄₉	—	2	6	no	—
23	Q₈×C₃₄₉	D₄	1	1	yes	characteristic
24	D₄×C₃₄₉	D₄	1	3	yes	—
25	C₂×C₄×C₃₄₉	C₂³	1	3	yes	—
26	C₈×C₃₄₉	C₂³	1	1	yes	characteristic
27	C₈×C₃₄₉	C₂³	1	3	yes	—
28	C₂²×C₃₄₉	C₂×D₄	1	3	yes	—
29	C₂×D₃₄₉	—	2	2	no	—
30	C₄×C₃₄₉	C₂×D₄	1	3	yes	—
31	C₄×C₃₄₉	C₂⁴	1	1	yes	characteristic
32	Dic₃₄₉	—	2	6	no	—
33	C₂×C₃₄₉	—	2	3	no	—
34	C₂×C₃₄₉	C₂²×D₄	1	1	yes	characteristic
35	D₃₄₉	—	4	2	no	—
36	C₃₄₉	#578	1	1	yes	characteristic, complemented, Sylow
37	#578	—	349	1	no	maximal, Sylow
38	2₋¹⁺⁴	—	349	2	no	—
39	#137	—	349	6	no	—
40	#135	—	349	3	no	—
41	C₂×Q₁₆	—	349	3	no	—
42	#131	D₃₄₉	1	1	yes	characteristic, complemented
43	#41	—	349	6	no	—
44	#41	—	698	6	no	—
45	#41	C₂×D₃₄₉	1	1	yes	characteristic
46	C₂×Q₈	—	349	6	no	—
47	C₂×Q₈	—	698	2	no	—
48	Q₁₆	—	349	3	no	—
49	Q₁₆	—	349	6	no	—
50	SD₁₆	—	349	6	no	—
51	D₈	—	349	1	no	—
52	M₁₆	C₂×D₃₄₉	1	3	yes	—
53	C₂×C₈	C₂×D₃₄₉	1	3	yes	—
54	Q₈	—	349	6	no	—
55	Q₈	—	698	6	no	—
56	Q₈	D₁₃₉₆	1	1	yes	characteristic
57	D₄	—	349	2	no	—
58	D₄	—	698	6	no	—
59	D₄	D₁₃₉₆	1	3	yes	—
60	C₂×C₄	—	698	6	no	—
61	C₂×C₄	—	698	6	no	—
62	C₂×C₄	C₂²×D₃₄₉	1	3	yes	—
63	C₈	C₂²×D₃₄₉	1	1	yes	characteristic
64	C₈	C₂²×D₃₄₉	1	3	yes	—
65	C₂²	—	698	2	no	—
66	C₂²	C₂×D₁₃₉₆	1	3	yes	—
67	C₄	—	698	6	no	—
68	C₄	C₂×D₁₃₉₆	1	3	yes	—
69	C₄	C₂³×D₃₄₉	1	1	yes	characteristic
70	C₂	—	2	3	no	—
71	C₂	—	1396	2	no	—
72	C₂	C₂²×D₁₃₉₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₁₃₉₆
commutator, cosocle	C₄×C₃₄₉	C₂⁴
Frattini	C₄	C₂³×D₃₄₉
Fitting	#131×C₃₄₉	C₂
socle	C₂×C₃₄₉	C₂²×D₄
hypercenter	#131	D₃₄₉
hypocenter	C₃₄₉	#578
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131

Complements

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

N	H ≅ G/N	type
1. #26013686	C₂	semidirect
2. #26013672	C₂	semidirect
3. #26013661	C₂	semidirect
4. C₂×Dic₁₃₉₆	C₂	semidirect
5. #131×C₃₄₉	C₂	semidirect
36. C₃₄₉	#578	semidirect
42. #131	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013686	2
#26013672	2
#26013661	2
C₂×Dic₁₃₉₆	2
#131×C₃₄₉	2
#578	349