Subgroups — #36373932

Order 22336.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013607	C₂	1	1	yes	maximal, characteristic
2	#26013554	C₂	1	1	yes	maximal, characteristic, complemented
3	#100×C₃₄₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂³×D₃₄₉	C₄	1	1	yes	characteristic, complemented
5	C₂×C₂²⋊D₃₄₉	—	2	1	no	—
6	C₂²⋊Dic₃₄₉	—	2	1	no	—
7	C₂²⋊Dic₃₄₉	C₄	1	1	yes	characteristic, complemented
8	#19150210	—	2	1	no	—
9	C₂×D₄×C₃₄₉	C₂²	1	1	yes	characteristic
10	M₁₆×C₃₄₉	—	2	1	no	—
11	C₂²×D₃₄₉	—	2	1	no	—
12	C₂²×D₃₄₉	—	4	2	no	—
13	C₂²×D₃₄₉	—	4	1	no	—
14	C₂²⋊D₃₄₉	—	4	2	no	—
15	C₂×Dic₃₄₉	—	2	1	no	—
16	C₂×Dic₃₄₉	—	4	1	no	—
17	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
18	C₂³×C₃₄₉	D₄	1	1	yes	characteristic
19	D₄×C₃₄₉	—	4	1	no	—
20	C₂×C₄×C₃₄₉	D₄	1	1	yes	characteristic
21	C₈×C₃₄₉	—	4	1	no	—
22	C₂²×C₃₄₉	—	2	1	no	—
23	C₂²×C₃₄₉	—	2	1	no	—
24	C₂²×C₃₄₉	—	4	1	no	—
25	C₂²×C₃₄₉	—	4	1	no	—
26	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
27	C₂×D₃₄₉	—	2	2	no	—
28	C₂×D₃₄₉	—	4	1	no	—
29	C₂×D₃₄₉	—	4	2	no	—
30	C₂×D₃₄₉	—	4	1	no	—
31	C₂×D₃₄₉	—	4	2	no	—
32	C₄×C₃₄₉	—	2	1	no	—
33	Dic₃₄₉	—	4	1	no	—
34	Dic₃₄₉	—	8	1	no	—
35	C₂×C₃₄₉	—	2	1	no	—
36	C₂×C₃₄₉	—	4	1	no	—
37	C₂×C₃₄₉	—	4	1	no	—
38	C₂×C₃₄₉	#99	1	1	yes	characteristic
39	D₃₄₉	—	4	2	no	—
40	C₃₄₉	AΣL(1,𝔽₁₆)	1	1	yes	characteristic, complemented, Sylow
41	AΣL(1,𝔽₁₆)	—	349	1	no	maximal, Sylow
42	#120	—	349	1	no	—
43	#100	D₃₄₉	1	1	yes	characteristic, complemented
44	#99	—	349	1	no	—
45	C₂⁴	—	349	1	no	—
46	C₂×D₄	—	698	1	no	—
47	C₂×D₄	C₂×D₃₄₉	1	1	yes	characteristic
48	M₁₆	—	2	1	no	—
49	C₂²⋊C₄	—	349	1	no	—
50	C₂²⋊C₄	—	698	1	no	—
51	C₂²⋊C₄	—	698	1	no	—
52	C₂³	—	1396	1	no	—
53	C₂³	—	1396	2	no	—
54	C₂³	—	698	1	no	—
55	C₂³	C₄×D₃₄₉	1	1	yes	characteristic
56	C₂³	D₁₃₉₆	1	1	yes	characteristic
57	D₄	—	4	1	no	—
58	D₄	—	1396	2	no	—
59	C₂×C₄	—	1396	1	no	—
60	C₂×C₄	—	698	1	no	—
61	C₂×C₄	C₂²⋊D₃₄₉	1	1	yes	characteristic
62	C₈	—	4	1	no	—
63	C₂²	—	2	1	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	1396	1	no	—
68	C₂²	—	1396	2	no	—
69	C₂²	—	1396	2	no	—
70	C₂²	—	1396	1	no	—
71	C₂²	—	698	2	no	—
72	C₂²	#19150210	1	1	yes	characteristic
73	C₄	—	2	1	no	—
74	C₄	—	1396	1	no	—
75	C₄	—	2792	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	1396	2	no	—
80	C₂	#26013553	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013553
commutator	C₂³×C₃₄₉	C₂×C₄
Frattini	C₂×D₄	C₂×D₃₄₉
Fitting	#100×C₃₄₉	C₂
socle	C₂×C₃₄₉	#99
prime radical	#26013607	C₂
cosocle	C₂×D₄×C₃₄₉	C₂²
hypercenter	#100	D₃₄₉
hypocenter	C₃₄₉	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₃₄₉ ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×D₄ ◁ #100

Complements

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

N	H ≅ G/N	type
2. #26013554	C₂	semidirect
3. #100×C₃₄₉	C₂	semidirect
4. C₂³×D₃₄₉	C₄	semidirect
7. C₂²⋊Dic₃₄₉	C₄	semidirect
40. C₃₄₉	AΣL(1,𝔽₁₆)	semidirect
43. #100	D₃₄₉	semidirect

Maximal subgroups

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

subgroup	index
#26013607	2
#26013554	2
#100×C₃₄₉	2
AΣL(1,𝔽₁₆)	349