Subgroups — #34046966

Order 19072.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750030	C₂	1	2	yes	maximal
2	#23749451	C₂	1	4	yes	maximal
3	C₄×C₁₆×C₁₄₉	C₂	1	1	yes	maximal, characteristic
4	#18011911	—	2	2	no	—
5	#18011845	—	2	2	no	—
6	#18011845	C₂²	1	4	yes	—
7	C₂×C₁₆×C₁₄₉	C₂²	1	2	yes	—
8	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₉₈	—	4	2	no	—
10	C₄⋊Dic₁₄₉	—	2	4	no	—
11	C₄⋊Dic₁₄₉	—	4	2	no	—
12	C₂×C₈×C₁₄₉	D₄	1	1	yes	characteristic
13	C₂×C₈×C₁₄₉	C₂³	1	1	yes	characteristic
14	C₄²×C₁₄₉	D₄	1	1	yes	characteristic
15	C₁₆×C₁₄₉	Q₈	1	4	yes	complemented
16	Dic₂₉₈	—	8	4	no	—
17	C₂×Dic₁₄₉	—	4	4	no	—
18	C₂×C₄×C₁₄₉	D₈	1	1	yes	characteristic
19	C₂×C₄×C₁₄₉	D₈	1	1	yes	characteristic
20	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
21	C₈×C₁₄₉	Q₁₆	1	2	yes	—
22	C₈×C₁₄₉	C₂×Q₈	1	2	yes	—
23	C₂²×C₁₄₉	C₂×D₈	1	1	yes	characteristic
24	C₄×C₁₄₉	D₁₆	1	2	yes	—
25	C₄×C₁₄₉	Q₃₂	1	2	yes	—
26	C₄×C₁₄₉	C₄⋊Q₈	1	1	yes	characteristic
27	C₄×C₁₄₉	C₂×Q₁₆	1	1	yes	characteristic
28	Dic₁₄₉	—	8	4	no	—
29	C₂×C₁₄₉	#500	1	1	yes	characteristic
30	C₂×C₁₄₉	C₂×D₁₆	1	1	yes	characteristic
31	C₂×C₁₄₉	C₂×Q₃₂	1	1	yes	characteristic
32	C₁₄₉	#2252	1	1	yes	characteristic, complemented, Sylow
33	#2252	—	149	1	no	maximal, Sylow
34	#500	—	149	2	no	—
35	#366	—	149	4	no	—
36	C₄×C₁₆	D₁₄₉	1	1	yes	characteristic
37	C₄⋊Q₈	—	298	2	no	—
38	C₂×C₁₆	C₂×D₁₄₉	1	2	yes	—
39	#107	—	298	2	no	—
40	#107	—	149	4	no	—
41	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
42	C₂×Q₈	—	596	2	no	—
43	C₂×C₈	D₅₉₆	1	1	yes	characteristic
44	C₂×C₈	C₂²×D₁₄₉	1	1	yes	characteristic
45	C₄⋊C₄	—	298	4	no	—
46	C₄⋊C₄	—	596	2	no	—
47	C₄²	D₅₉₆	1	1	yes	characteristic
48	C₁₆	Dic₂₉₈	1	4	yes	complemented
49	Q₈	—	1192	4	no	—
50	C₂×C₄	—	596	4	no	—
51	C₂×C₄	D₁₁₉₂	1	1	yes	characteristic
52	C₂×C₄	D₁₁₉₂	1	1	yes	characteristic
53	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
54	C₈	Dic₅₉₆	1	2	yes	—
55	C₈	C₂×Dic₂₉₈	1	2	yes	—
56	C₂²	C₂×D₁₁₉₂	1	1	yes	characteristic
57	C₄	—	1192	4	no	—
58	C₄	D₂₃₈₄	1	2	yes	—
59	C₄	Dic₁₁₉₂	1	2	yes	—
60	C₄	#18011911	1	1	yes	characteristic
61	C₄	C₂×Dic₅₉₆	1	1	yes	characteristic
62	C₂	#23750030	1	1	yes	characteristic
63	C₂	C₂×D₂₃₈₄	1	1	yes	characteristic
64	C₂	C₂×Dic₁₁₉₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₁₉₂
commutator, cosocle	C₂×C₈×C₁₄₉	C₂³
Frattini	C₂×C₈	C₂²×D₁₄₉
Fitting	C₄×C₁₆×C₁₄₉	C₂
socle, prime radical	C₂²×C₁₄₉	C₂×D₈
hypercenter	C₄×C₁₆	D₁₄₉
hypocenter	C₁₄₉	#2252
LCS	G ▷ C₂×C₈×C₁₄₉ ▷ C₄×C₁₄₉ ▷ C₂×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₁₄₉	Q₈	semidirect
32. C₁₄₉	#2252	semidirect
48. C₁₆	Dic₂₉₈	semidirect

Maximal subgroups

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

subgroup	index
#23750030	2
#23749451	2
C₄×C₁₆×C₁₄₉	2
#2252	149