Subgroups — #11701658

Order 768.

2038 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#3573	C₂	1	1	yes	maximal, characteristic
2	#83780	C₂	1	15	yes	maximal, complemented
3	C₂×#5140	C₂	1	15	yes	maximal, complemented
4	#62624	—	3	1	no	maximal, Sylow
5	C₃×#585	C₄	1	1	yes	characteristic
6	C₃×#568	C₄	1	15	yes	—
7	C₂×C₃×#131	C₂²	1	15	yes	—
8	#5140	C₂²	1	20	yes	complemented
9	#5140	C₂²	1	60	yes	complemented
10	C₂×#812	C₂²	1	45	yes	complemented
11	C₂²×C₃⋊C₁₆	C₂²	1	15	yes	complemented
12	#3573	D₃	1	1	yes	characteristic
13	#3407	—	3	15	no	—
14	C₂×#504	—	3	15	no	—
15	C₃×2₋¹⁺⁴	C₈	1	6	yes	—
16	C₃×H(5,𝔽₂)	C₈	1	10	yes	—
17	C₂×C₃×#41	C₂×C₄	1	15	yes	—
18	C₃×#131	C₂×C₄	1	60	yes	—
19	C₃×#131	C₂³	1	20	yes	—
20	C₂×C₃×M₁₆	C₂×C₄	1	45	yes	—
21	C₂²×C₃×C₈	C₂³	1	15	yes	—
22	#812	C₂³	1	60	yes	—
23	C₂×C₃⋊C₁₆	C₂³	1	15	yes	—
24	C₂×C₃⋊C₁₆	C₂³	1	45	yes	—
25	#585	Dic₃	1	1	yes	characteristic
26	#568	Dic₃	1	15	yes	—
27	C₂×#131	C₂×D₃	1	15	yes	—
28	#504	—	3	20	no	—
29	#504	—	3	60	no	—
30	C₂×M₃₂	—	3	45	no	—
31	C₂²×C₁₆	—	3	15	no	—
32	C₃×#41	C₂×C₈	1	60	yes	—
33	C₃×#41	C₂²×C₄	1	20	yes	—
34	C₂×C₃×Q₈	C₂×C₈	1	15	yes	—
35	C₂×C₃×D₄	C₂×C₈	1	45	yes	—
36	C₂²×C₃×C₄	C₂²×C₄	1	15	yes	—
37	C₃×M₁₆	C₂²×C₄	1	60	yes	—
38	C₂×C₃×C₈	C₂²×C₄	1	45	yes	—
39	C₂×C₃×C₈	C₂⁴	1	15	yes	—
40	C₃⋊C₁₆	C₂⁴	1	1	yes	characteristic
41	C₃⋊C₁₆	C₂⁴	1	15	yes	—
42	2₋¹⁺⁴	C₃⋊C₈	1	6	yes	—
43	H(5,𝔽₂)	C₃⋊C₈	1	10	yes	—
44	C₂×#41	C₂×Dic₃	1	15	yes	—
45	#131	C₂×Dic₃	1	60	yes	—
46	#131	C₂²×D₃	1	20	yes	—
47	C₂×M₁₆	C₂×Dic₃	1	45	yes	—
48	C₂²×C₈	C₂²×D₃	1	15	yes	—
49	M₃₂	—	3	60	no	—
50	C₂×C₁₆	—	3	15	no	—
51	C₂×C₁₆	—	3	45	no	—
52	C₂³×C₃	C₂²×C₈	1	15	yes	—
53	C₃×Q₈	C₂²×C₈	1	20	yes	—
54	C₃×D₄	C₂²×C₈	1	60	yes	—
55	C₂×C₃×C₄	C₂²×C₈	1	45	yes	—
56	C₂×C₃×C₄	C₂³×C₄	1	15	yes	—
57	C₃×C₈	C₂³×C₄	1	15	yes	—
58	C₃×C₈	C₂⁵	1	1	yes	characteristic
59	#41	C₂×C₃⋊C₈	1	60	yes	—
60	#41	C₂²×Dic₃	1	20	yes	—
61	C₂×Q₈	C₂×C₃⋊C₈	1	15	yes	—
62	C₂×D₄	C₂×C₃⋊C₈	1	45	yes	—
63	C₂²×C₄	C₂²×Dic₃	1	15	yes	—
64	M₁₆	C₂²×Dic₃	1	60	yes	—
65	C₂×C₈	C₂²×Dic₃	1	45	yes	—
66	C₂×C₈	C₂³×D₃	1	15	yes	—
67	C₁₆	—	3	1	no	—
68	C₁₆	—	3	15	no	—
69	C₂²×C₃	—	4	15	no	—
70	C₂²×C₃	C₂³×C₈	1	15	yes	—
71	C₃×C₄	C₂³×C₈	1	15	yes	—
72	C₃×C₄	C₂⁴×C₄	1	1	yes	characteristic
73	C₂³	C₂²×C₃⋊C₈	1	15	yes	—
74	Q₈	C₂²×C₃⋊C₈	1	20	yes	—
75	D₄	C₂²×C₃⋊C₈	1	60	yes	—
76	C₂×C₄	C₂²×C₃⋊C₈	1	45	yes	—
77	C₂×C₄	C₂³×Dic₃	1	15	yes	—
78	C₈	C₂³×Dic₃	1	15	yes	—
79	C₈	C₂⁴×D₃	1	1	yes	characteristic
80	C₂×C₃	—	2	15	no	—
81	C₂×C₃	C₂⁴×C₈	1	1	yes	characteristic
82	C₂²	—	4	15	no	—
83	C₂²	C₂³×C₃⋊C₈	1	15	yes	—
84	C₄	C₂³×C₃⋊C₈	1	15	yes	—
85	C₄	C₂⁴×Dic₃	1	1	yes	characteristic
86	C₃	#62624	1	1	yes	characteristic, complemented, Sylow
87	C₂	—	2	15	no	—
88	C₂	C₂⁴×C₃⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₈	C₂⁴×D₃
commutator, socle	C₂×C₃	C₂⁴×C₈
Fitting	C₃×#3573	C₂
prime radical	C₃×#585	C₄
cosocle	C₃×C₈	C₂⁵
hypercenter	#3573	D₃
hypocenter	C₃	#62624
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₈ ◁ #3573

Complements

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

N	H ≅ G/N	type
2. #83780	C₂	semidirect
3. C₂×#5140	C₂	semidirect
8. #5140	C₂²	semidirect
9. #5140	C₂²	semidirect
10. C₂×#812	C₂²	semidirect
11. C₂²×C₃⋊C₁₆	C₂²	semidirect
86. C₃	#62624	semidirect

Maximal subgroups

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

subgroup	index
C₃×#3573	2
#83780	2
C₂×#5140	2
#62624	3