Subgroups — #10709924

Order 768.

262 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69044	C₂	1	4	yes	maximal
2	#68065	C₂	1	2	yes	maximal
3	C₃×C₈×C₁₆	C₂	1	1	yes	maximal, characteristic
4	#12521	—	3	1	no	maximal, Sylow
5	#4699	—	2	2	no	—
6	#4682	—	2	2	no	—
7	#4682	C₂²	1	4	yes	—
8	C₃×C₄×C₁₆	C₂²	1	2	yes	—
9	C₃×C₈²	C₂²	1	1	yes	characteristic
10	#4504	—	2	8	no	—
11	#2252	—	3	4	no	—
12	#1713	—	3	2	no	—
13	C₈×C₁₆	D₃	1	1	yes	characteristic
14	C₂×Dic₁₂	—	4	4	no	—
15	#869	—	2	4	no	—
16	C₂×C₃×C₁₆	D₄	1	4	yes	—
17	C₃×C₄×C₈	D₄	1	1	yes	characteristic
18	C₃×C₄×C₈	D₄	1	1	yes	characteristic
19	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
20	#818	—	2	8	no	—
21	#818	—	4	4	no	—
22	#500	—	3	4	no	—
23	#500	—	6	2	no	—
24	#494	—	6	2	no	—
25	#366	—	6	8	no	—
26	C₄×C₁₆	C₂×D₃	1	2	yes	—
27	C₈²	C₂×D₃	1	1	yes	characteristic
28	C₂×Dic₆	—	4	4	no	—
29	C₂×C₃×C₈	D₈	1	2	yes	—
30	C₂×C₃×C₈	D₈	1	2	yes	—
31	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
32	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
33	C₄⋊Dic₃	—	4	4	no	—
34	C₄⋊Dic₃	—	4	4	no	—
35	Dic₁₂	—	8	8	no	—
36	C₃×C₁₆	Q₁₆	1	8	yes	complemented
37	C₂×Q₁₆	—	12	4	no	—
38	C₄⋊Q₈	—	6	4	no	—
39	C₂×C₁₆	D₁₂	1	4	yes	—
40	#107	—	6	8	no	—
41	#107	—	12	4	no	—
42	C₄×C₈	D₁₂	1	1	yes	characteristic
43	C₄×C₈	D₁₂	1	1	yes	characteristic
44	C₄×C₈	C₂²×D₃	1	1	yes	characteristic
45	C₂×C₃×C₄	#127	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
47	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₃	—	8	4	no	—
49	Dic₆	—	8	8	no	—
50	C₃×C₈	D₁₆	1	4	yes	—
51	C₃×C₈	Q₃₂	1	4	yes	—
52	C₃×C₈	C₂×Q₁₆	1	4	yes	—
53	C₂×Q₈	—	12	4	no	—
54	Q₁₆	—	24	8	no	—
55	C₂×C₈	D₂₄	1	2	yes	—
56	C₂×C₈	D₂₄	1	2	yes	—
57	C₂×C₈	C₂×D₁₂	1	2	yes	—
58	C₄⋊C₄	—	12	4	no	—
59	C₄⋊C₄	—	12	4	no	—
60	C₄²	C₂×D₁₂	1	1	yes	characteristic
61	C₁₆	Dic₁₂	1	8	yes	complemented
62	C₂²×C₃	#493	1	1	yes	characteristic
63	C₃×C₄	#494	1	2	yes	—
64	C₃×C₄	C₂×D₁₆	1	2	yes	—
65	C₃×C₄	C₂×Q₃₂	1	2	yes	—
66	Dic₃	—	16	4	no	—
67	Q₈	—	24	8	no	—
68	C₂×C₄	—	24	4	no	—
69	C₂×C₄	#874	1	1	yes	characteristic
70	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
71	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
72	C₈	D₄₈	1	4	yes	—
73	C₈	Dic₂₄	1	4	yes	—
74	C₈	C₂×Dic₁₂	1	4	yes	—
75	C₂×C₃	#1713	1	1	yes	characteristic
76	C₂×C₃	#2246	1	1	yes	characteristic
77	C₂×C₃	#2247	1	1	yes	characteristic
78	C₂²	#4695	1	1	yes	characteristic
79	C₄	—	48	4	no	—
80	C₄	#4699	1	2	yes	—
81	C₄	C₂×D₄₈	1	2	yes	—
82	C₄	C₂×Dic₂₄	1	2	yes	—
83	C₃	#12521	1	1	yes	characteristic, complemented, Sylow
84	C₂	#68065	1	1	yes	characteristic
85	C₂	#69060	1	1	yes	characteristic
86	C₂	#69066	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#4695
commutator, cosocle	C₃×C₄×C₈	C₂³
Frattini	C₄×C₈	C₂²×D₃
Fitting	C₃×C₈×C₁₆	C₂
socle, prime radical	C₂²×C₃	#493
hypercenter	C₈×C₁₆	D₃
hypocenter	C₃	#12521
LCS	G ▷ C₃×C₄×C₈ ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ C₈×C₁₆

Complements

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

N	H ≅ G/N	type
36. C₃×C₁₆	Q₁₆	semidirect
61. C₁₆	Dic₁₂	semidirect
83. C₃	#12521	semidirect

Maximal subgroups

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

subgroup	index
#69044	2
#68065	2
C₃×C₈×C₁₆	2
#12521	3