Subgroups — #36375190

Order 22336.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013727	C₂	1	2	yes	maximal, complemented
2	#26013709	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013704	C₂	1	2	yes	maximal, complemented
4	#26013693	C₂	1	1	yes	maximal, characteristic, complemented
5	#26013662	C₂	1	1	yes	maximal, characteristic
6	#19150243	—	2	2	no	—
7	#19150243	C₂²	1	2	yes	—
8	D₃₄₉⋊C₈	—	2	2	no	—
9	D₃₄₉⋊C₈	C₂²	1	1	yes	characteristic
10	C₄×#13063703	C₂²	1	1	yes	characteristic
11	#19150234	C₂²	1	2	yes	—
12	#19150220	C₄	1	2	yes	complemented
13	D₂₇₉₂	C₄	1	1	yes	characteristic, complemented
14	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
15	Q₁₆×C₃₄₉	C₄	1	1	yes	characteristic, complemented
16	C₂×#13063703	—	2	1	no	—
17	#14584102	—	2	2	no	—
18	#14584102	D₄	1	2	yes	—
19	D₁₃₉₆	—	2	2	no	—
20	D₁₃₉₆	C₂×C₄	1	2	yes	—
21	C₄×D₃₄₉	—	2	2	no	—
22	C₄×D₃₄₉	C₂³	1	1	yes	characteristic
23	#14584095	C₂×C₄	1	1	yes	characteristic
24	Q₈×C₃₄₉	C₂×C₄	1	2	yes	—
25	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₂×D₃₄₉	—	2	2	no	—
27	C₂×D₃₄₉	#41	1	1	yes	characteristic
28	#13063703	—	4	1	no	—
29	C₄×C₃₄₉	—	2	2	no	—
30	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
31	Dic₃₄₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃₄₉	C₄×D₄	1	1	yes	characteristic
33	D₃₄₉	—	2	1	no	—
34	D₃₄₉	—	4	2	no	—
35	C₃₄₉	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	349	1	no	maximal, Sylow
37	#135	—	349	1	no	—
38	#131	—	349	2	no	—
39	#108	—	349	1	no	—
40	#104	—	349	2	no	—
41	#97	—	349	1	no	—
42	#41	—	349	2	no	—
43	Q₁₆	#13063703	1	1	yes	characteristic, complemented
44	SD₁₆	—	349	2	no	—
45	D₈	—	349	1	no	—
46	M₁₆	—	349	2	no	—
47	M₁₆	—	698	2	no	—
48	C₂×C₈	—	349	1	no	—
49	C₂×C₈	—	349	1	no	—
50	C₂×C₈	—	698	2	no	—
51	C₄²	—	349	1	no	—
52	Q₈	C₂×#13063703	1	2	yes	—
53	D₄	—	349	2	no	—
54	D₄	—	698	2	no	—
55	C₂×C₄	—	349	1	no	—
56	C₂×C₄	—	698	2	no	—
57	C₂×C₄	—	698	1	no	—
58	C₈	—	349	2	no	—
59	C₈	—	349	1	no	—
60	C₈	—	698	2	no	—
61	C₈	C₂×#13063703	1	1	yes	characteristic
62	C₂²	—	349	1	no	—
63	C₂²	—	698	2	no	—
64	C₄	—	2	2	no	—
65	C₄	—	349	1	no	—
66	C₄	—	1396	1	no	—
67	C₄	C₂²×#13063703	1	1	yes	characteristic
68	C₂	—	1396	2	no	—
69	C₂	—	698	1	no	—
70	C₂	D₄×#13063703	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×#13063703
commutator	C₄×C₃₄₉	C₂²×C₄
Frattini	C₄	C₂²×#13063703
Fitting	Q₁₆×C₃₄₉	C₄
socle	C₂×C₃₄₉	C₄×D₄
prime radical	#26013662	C₂
cosocle	C₄×D₃₄₉	C₂³
hypercenter	Q₁₆	#13063703
hypocenter	C₃₄₉	#444
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
1. #26013727	C₂	semidirect
2. #26013709	C₂	semidirect
3. #26013704	C₂	semidirect
4. #26013693	C₂	semidirect
12. #19150220	C₄	semidirect
13. D₂₇₉₂	C₄	semidirect
15. Q₁₆×C₃₄₉	C₄	semidirect
35. C₃₄₉	#444	semidirect
43. Q₁₆	#13063703	semidirect

Maximal subgroups

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

subgroup	index
#26013727	2
#26013709	2
#26013704	2
#26013693	2
#26013662	2
#444	349