Subgroups — #36375191

Order 22336.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013726	C₂	1	2	yes	maximal, complemented
2	#26013709	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013703	C₂	1	2	yes	maximal, complemented
4	#26013693	C₂	1	1	yes	maximal, characteristic, complemented
5	#26013658	C₂	1	1	yes	maximal, characteristic
6	#19150243	C₂²	1	2	yes	—
7	#19150242	—	2	2	no	—
8	D₃₄₉⋊C₈	C₂²	1	1	yes	characteristic
9	C₂×#14584102	—	2	2	no	—
10	C₄×#13063703	C₂²	1	1	yes	characteristic
11	#19150232	C₂²	1	2	yes	—
12	Dic₁₃₉₆	C₄	1	1	yes	characteristic, complemented
13	#19150221	C₄	1	2	yes	complemented
14	C₈×D₃₄₉	C₂²	1	1	yes	characteristic
15	D₈×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	Dic₆₉₈	C₂×C₄	1	2	yes	—
20	C₂²⋊D₃₄₉	—	2	2	no	—
21	C₂×Dic₃₄₉	—	2	2	no	—
22	C₄×D₃₄₉	C₂³	1	1	yes	characteristic
23	#14584095	C₂×C₄	1	1	yes	characteristic
24	D₄×C₃₄₉	C₂×C₄	1	2	yes	—
25	C₈×C₃₄₉	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₄₉	—	2	2	no	—
27	C₂×D₃₄₉	#41	1	1	yes	characteristic
28	#13063703	—	4	1	no	—
29	C₄×C₃₄₉	C₂²×C₄	1	1	yes	characteristic
30	Dic₃₄₉	—	2	2	no	—
31	Dic₃₄₉	C₂×D₄	1	1	yes	characteristic
32	C₂×C₃₄₉	—	4	2	no	—
33	C₂×C₃₄₉	C₄×D₄	1	1	yes	characteristic
34	D₃₄₉	—	2	1	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	—	349	1	no	—
44	SD₁₆	—	349	2	no	—
45	D₈	#13063703	1	1	yes	characteristic, complemented
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	—	349	2	no	—
53	D₄	—	698	2	no	—
54	D₄	C₂×#13063703	1	2	yes	—
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	—	2	2	no	—
63	C₂²	—	349	1	no	—
64	C₄	—	349	1	no	—
65	C₄	—	1396	1	no	—
66	C₄	—	698	2	no	—
67	C₄	C₂²×#13063703	1	1	yes	characteristic
68	C₂	—	4	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	D₈×C₃₄₉	C₄
socle	C₂×C₃₄₉	C₄×D₄
prime radical	#26013658	C₂
cosocle	C₄×D₃₄₉	C₂³
hypercenter	D₈	#13063703
hypocenter	C₃₄₉	#444
LCS	G ▷ C₄×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ D₈

Complements

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

N	H ≅ G/N	type
1. #26013726	C₂	semidirect
2. #26013709	C₂	semidirect
3. #26013703	C₂	semidirect
4. #26013693	C₂	semidirect
12. Dic₁₃₉₆	C₄	semidirect
13. #19150221	C₄	semidirect
15. D₈×C₃₄₉	C₄	semidirect
35. C₃₄₉	#444	semidirect
45. D₈	#13063703	semidirect

Maximal subgroups

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

subgroup	index
#26013726	2
#26013709	2
#26013703	2
#26013693	2
#26013658	2
#444	349