Subgroups — #34068412

Order 19328.

182 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23752578	C₂	1	3	yes	maximal
2	#23751954	C₂	1	3	yes	maximal, complemented
3	#564×C₁₅₁	C₂	1	1	yes	maximal, characteristic, complemented
4	#18012736	C₂²	1	1	yes	characteristic
5	#18012659	—	2	6	no	—
6	#18012650	C₂²	1	3	yes	—
7	#125×C₁₅₁	—	2	6	no	—
8	#125×C₁₅₁	C₂²	1	3	yes	—
9	C₂×D₆₀₄	—	2	3	no	—
10	C₂×C₁₅₁⋊C₈	—	2	3	no	—
11	C₄⋊C₄×C₁₅₁	—	2	6	no	—
12	C₄⋊C₄×C₁₅₁	—	2	6	no	—
13	C₄⋊C₄×C₁₅₁	D₄	1	6	yes	—
14	C₄²×C₁₅₁	—	2	2	no	—
15	C₄²×C₁₅₁	C₂³	1	1	yes	characteristic
16	C₂²×D₁₅₁	—	4	1	no	—
17	D₆₀₄	—	4	6	no	—
18	C₁₅₁⋊C₈	—	4	3	no	—
19	C₂×C₄×C₁₅₁	—	2	6	no	—
20	C₂×C₄×C₁₅₁	C₂×D₄	1	3	yes	—
21	C₂²×C₁₅₁	#120	1	1	yes	characteristic
22	C₂×D₁₅₁	—	8	3	no	—
23	C₄×C₁₅₁	—	2	3	no	—
24	C₄×C₁₅₁	—	4	6	no	—
25	C₂×C₁₅₁	#454	1	3	yes	—
26	D₁₅₁	—	16	1	no	—
27	C₁₅₁	#1658	1	1	yes	characteristic, complemented, Sylow
28	#1658	—	151	1	no	maximal, Sylow
29	#564	D₁₅₁	1	1	yes	characteristic, complemented
30	#490	—	151	3	no	—
31	#330	—	151	3	no	—
32	#127	—	151	1	no	—
33	#125	—	2	6	no	—
34	#125	C₂×D₁₅₁	1	3	yes	—
35	#102	—	302	6	no	—
36	#97	—	151	3	no	—
37	C₂×D₄	—	302	3	no	—
38	C₂×C₈	—	302	3	no	—
39	C₄⋊C₄	—	2	6	no	—
40	C₄⋊C₄	—	2	6	no	—
41	C₄⋊C₄	C₂²⋊D₁₅₁	1	6	yes	—
42	C₄²	—	2	2	no	—
43	C₄²	C₂²×D₁₅₁	1	1	yes	characteristic
44	C₂³	—	604	1	no	—
45	D₄	—	604	6	no	—
46	C₂×C₄	—	2	6	no	—
47	C₂×C₄	C₂×C₂²⋊D₁₅₁	1	3	yes	—
48	C₈	—	604	3	no	—
49	C₂²	—	1208	3	no	—
50	C₂²	#18012708	1	1	yes	characteristic
51	C₄	—	2	3	no	—
52	C₄	—	4	6	no	—
53	C₂	—	2416	1	no	—
54	C₂	#23752385	1	3	yes	—

	subgroup	G/H
center	C₂²	#18012708
commutator, cosocle	C₄²×C₁₅₁	C₂³
Frattini	C₄²	C₂²×D₁₅₁
Fitting	#564×C₁₅₁	C₂
socle	C₂²×C₁₅₁	#120
prime radical	#18012736	C₂²
hypercenter	#564	D₁₅₁
hypocenter	C₁₅₁	#1658
LCS	G ▷ C₄²×C₁₅₁ ▷ C₂²×C₁₅₁ ▷ C₁₅₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #564

Complements

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

N	H ≅ G/N	type
2. #23751954	C₂	semidirect
3. #564×C₁₅₁	C₂	semidirect
27. C₁₅₁	#1658	semidirect
29. #564	D₁₅₁	semidirect

Maximal subgroups

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

subgroup	index
#23752578	2
#23751954	2
#564×C₁₅₁	2
#1658	151