Subgroups — #34044301

Order 19072.

168 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23750030	C₂	1	2	yes	maximal
2	#23749976	C₂	1	4	yes	maximal
3	C₈²×C₁₄₉	C₂	1	1	yes	maximal, characteristic
4	#18011911	C₂²	1	4	yes	—
5	#18011845	—	2	4	no	—
6	#18011836	—	2	8	no	—
7	C₄×C₈×C₁₄₉	C₂²	1	2	yes	—
8	C₄×C₈×C₁₄₉	C₂²	1	1	yes	characteristic
9	C₂×Dic₂₉₈	—	2	4	no	—
10	C₄⋊Dic₁₄₉	—	2	4	no	—
11	C₄⋊Dic₁₄₉	—	2	4	no	—
12	C₂×C₈×C₁₄₉	D₄	1	4	yes	—
13	C₂×C₈×C₁₄₉	D₄	1	2	yes	—
14	C₄²×C₁₄₉	C₂³	1	1	yes	characteristic
15	Dic₂₉₈	—	4	8	no	—
16	C₂×Dic₁₄₉	—	4	4	no	—
17	C₂×C₄×C₁₄₉	C₂×D₄	1	1	yes	characteristic
18	C₂×C₄×C₁₄₉	C₂×D₄	1	2	yes	—
19	C₈×C₁₄₉	—	2	4	no	—
20	C₈×C₁₄₉	Q₁₆	1	4	yes	—
21	C₂²×C₁₄₉	#127	1	1	yes	characteristic
22	C₄×C₁₄₉	C₂×Q₁₆	1	2	yes	—
23	C₄×C₁₄₉	#135	1	4	yes	—
24	Dic₁₄₉	—	8	4	no	—
25	C₂×C₁₄₉	#494	1	1	yes	characteristic
26	C₂×C₁₄₉	#495	1	2	yes	—
27	C₁₄₉	#1710	1	1	yes	characteristic, complemented, Sylow
28	#1710	—	149	1	no	maximal, Sylow
29	#500	—	149	2	no	—
30	#487	—	149	4	no	—
31	C₈²	D₁₄₉	1	1	yes	characteristic
32	C₄⋊Q₈	—	149	4	no	—
33	#107	—	298	4	no	—
34	#103	—	298	8	no	—
35	C₄×C₈	C₂×D₁₄₉	1	2	yes	—
36	C₄×C₈	C₂×D₁₄₉	1	1	yes	characteristic
37	C₂×Q₈	—	298	4	no	—
38	C₂×C₈	D₅₉₆	1	4	yes	—
39	C₂×C₈	D₅₉₆	1	2	yes	—
40	C₄⋊C₄	—	298	4	no	—
41	C₄⋊C₄	—	298	4	no	—
42	C₄²	C₂²×D₁₄₉	1	1	yes	characteristic
43	Q₈	—	596	8	no	—
44	C₂×C₄	—	596	4	no	—
45	C₂×C₄	C₂×D₅₉₆	1	1	yes	characteristic
46	C₂×C₄	C₂×D₅₉₆	1	2	yes	—
47	C₈	—	2	4	no	—
48	C₈	Dic₅₉₆	1	4	yes	—
49	C₂²	#18011908	1	1	yes	characteristic
50	C₄	—	1192	4	no	—
51	C₄	C₂×Dic₅₉₆	1	2	yes	—
52	C₄	#18011933	1	4	yes	—
53	C₂	#23750005	1	1	yes	characteristic
54	C₂	#23750009	1	2	yes	—

	subgroup	G/H
center	C₂²	#18011908
commutator, cosocle	C₄²×C₁₄₉	C₂³
Frattini	C₄²	C₂²×D₁₄₉
Fitting	C₈²×C₁₄₉	C₂
socle, prime radical	C₂²×C₁₄₉	#127
hypercenter	C₈²	D₁₄₉
hypocenter	C₁₄₉	#1710
LCS	G ▷ C₄²×C₁₄₉ ▷ C₂²×C₁₄₉ ▷ C₁₄₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈²

Complements

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

N	H ≅ G/N	type
27. C₁₄₉	#1710	semidirect

Maximal subgroups

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

subgroup	index
#23750030	2
#23749976	2
C₈²×C₁₄₉	2
#1710	149