Subgroups — #36375044

Order 22336.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26013699	C₂	1	1	yes	maximal, characteristic, complemented
2	#26013697	C₂	1	1	yes	maximal, characteristic, complemented
3	#26013608	C₂	1	1	yes	maximal, characteristic
4	#19150242	—	2	1	no	—
5	C₂²⋊#13063703	—	2	1	no	—
6	C₂×C₂²⋊D₃₄₉	—	2	1	no	—
7	C₂×C₂²⋊D₃₄₉	C₂²	1	1	yes	characteristic
8	C₂²⋊Dic₃₄₉	—	2	1	no	—
9	C₂²⋊Dic₃₄₉	C₄	1	1	yes	characteristic, complemented
10	C₂⁴×C₃₄₉	C₄	1	1	yes	characteristic, complemented
11	C₂×#13063703	—	4	1	no	—
12	#14584102	—	4	1	no	—
13	C₂²×D₃₄₉	D₄	1	1	yes	characteristic
14	C₂²⋊D₃₄₉	—	4	1	no	—
15	C₂²⋊D₃₄₉	—	4	2	no	—
16	C₂×Dic₃₄₉	—	2	1	no	—
17	C₂×Dic₃₄₉	D₄	1	1	yes	characteristic
18	C₂³×C₃₄₉	—	2	1	no	—
19	C₂³×C₃₄₉	—	4	2	no	—
20	C₂³×C₃₄₉	—	4	1	no	—
21	C₂³×C₃₄₉	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃₄₉	—	2	2	no	—
23	C₂²×C₃₄₉	—	2	1	no	—
24	C₂²×C₃₄₉	—	4	1	no	—
25	C₂²×C₃₄₉	—	4	1	no	—
26	C₂²×C₃₄₉	—	4	2	no	—
27	C₂²×C₃₄₉	—	4	2	no	—
28	C₂²×C₃₄₉	—	4	1	no	—
29	C₂²×C₃₄₉	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×D₃₄₉	—	2	1	no	—
31	C₂×D₃₄₉	—	4	1	no	—
32	#13063703	—	8	1	no	—
33	Dic₃₄₉	—	2	1	no	—
34	Dic₃₄₉	—	4	1	no	—
35	C₂×C₃₄₉	—	2	1	no	—
36	C₂×C₃₄₉	—	4	2	no	—
37	C₂×C₃₄₉	—	4	1	no	—
38	C₂×C₃₄₉	#99	1	1	yes	characteristic
39	D₃₄₉	—	4	1	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	—	349	1	no	—
44	#99	—	349	1	no	—
45	C₂⁴	#13063703	1	1	yes	characteristic, complemented
46	C₂×D₄	—	349	1	no	—
47	C₂×D₄	—	698	1	no	—
48	M₁₆	—	698	1	no	—
49	C₂²⋊C₄	—	349	1	no	—
50	C₂²⋊C₄	—	698	1	no	—
51	C₂²⋊C₄	—	698	1	no	—
52	C₂³	—	2	1	no	—
53	C₂³	—	4	2	no	—
54	C₂³	—	4	1	no	—
55	C₂³	—	349	1	no	—
56	C₂³	C₂×#13063703	1	1	yes	characteristic
57	D₄	—	1396	2	no	—
58	D₄	—	1396	1	no	—
59	C₂×C₄	—	349	1	no	—
60	C₂×C₄	—	1396	1	no	—
61	C₂×C₄	—	698	1	no	—
62	C₈	—	1396	1	no	—
63	C₂²	—	2	2	no	—
64	C₂²	—	2	1	no	—
65	C₂²	—	4	1	no	—
66	C₂²	—	4	1	no	—
67	C₂²	—	4	2	no	—
68	C₂²	—	4	2	no	—
69	C₂²	—	4	1	no	—
70	C₂²	—	1396	1	no	—
71	C₂²	—	698	1	no	—
72	C₂²	C₂²⋊#13063703	1	1	yes	characteristic
73	C₄	—	1396	1	no	—
74	C₄	—	698	1	no	—
75	C₄	—	2792	1	no	—
76	C₂	—	2	1	no	—
77	C₂	—	4	2	no	—
78	C₂	—	4	1	no	—
79	C₂	—	1396	1	no	—
80	C₂	#26013697	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#26013697
commutator	C₂³×C₃₄₉	C₂×C₄
Frattini	C₂³	C₂×#13063703
Fitting	C₂⁴×C₃₄₉	C₄
socle	C₂×C₃₄₉	#99
prime radical	#26013608	C₂
cosocle	C₂×C₂²⋊D₃₄₉	C₂²
hypercenter	C₂⁴	#13063703
hypocenter	C₃₄₉	AΣL(1,𝔽₁₆)
LCS	G ▷ C₂³×C₃₄₉ ▷ C₂²×C₃₄₉ ▷ C₂×C₃₄₉ ▷ C₃₄₉
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂³ ◁ C₂⁴

Complements

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

N	H ≅ G/N	type
1. #26013699	C₂	semidirect
2. #26013697	C₂	semidirect
9. C₂²⋊Dic₃₄₉	C₄	semidirect
10. C₂⁴×C₃₄₉	C₄	semidirect
40. C₃₄₉	AΣL(1,𝔽₁₆)	semidirect
45. C₂⁴	#13063703	semidirect

Maximal subgroups

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

subgroup	index
#26013699	2
#26013697	2
#26013608	2
AΣL(1,𝔽₁₆)	349