Subgroups — #10683072

Order 768.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69369	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₆₄	C₂	1	1	yes	maximal, characteristic, complemented
3	#67593	C₂	1	1	yes	maximal, characteristic, complemented
4	#7539	—	3	1	no	maximal, Sylow
5	#4904	—	2	1	no	—
6	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
7	#4506	—	2	1	no	—
8	Dic₄₈	C₄	1	1	yes	characteristic
9	#4449	—	2	1	no	—
10	D₉₆	C₄	1	1	yes	characteristic
11	C₃×C₆₄	—	2	1	no	—
12	#2262	—	3	1	no	—
13	C₂×C₆₄	D₃	1	1	yes	characteristic, complemented
14	#1425	—	3	1	no	—
15	#904	—	4	1	no	—
16	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
17	#819	—	4	1	no	—
18	Dic₂₄	—	2	1	no	—
19	#800	—	4	1	no	—
20	D₄₈	—	2	1	no	—
21	C₃×C₃₂	C₂×C₄	1	1	yes	characteristic
22	C₃×C₃₂	D₄	1	1	yes	characteristic
23	#508	—	6	1	no	—
24	Q₆₄	—	3	1	no	—
25	SD₆₄	—	6	1	no	—
26	D₃₂	—	3	1	no	—
27	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
28	#368	—	6	1	no	—
29	C₆₄	—	2	1	no	—
30	#235	—	8	1	no	—
31	C₂×C₃×C₈	D₈	1	1	yes	characteristic
32	#208	—	8	1	no	—
33	Dic₁₂	—	4	1	no	—
34	D₂₄	—	4	1	no	—
35	#204	—	8	1	no	—
36	C₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
37	C₃×C₁₆	SD₁₆	1	1	yes	characteristic
38	#135	—	12	1	no	—
39	Q₃₂	—	6	1	no	—
40	SD₃₂	—	12	1	no	—
41	D₁₆	—	6	1	no	—
42	C₂×C₁₆	D₁₂	1	1	yes	characteristic
43	#108	—	12	1	no	—
44	C₃₂	C₄×D₃	1	1	yes	characteristic
45	C₃₂	#67	1	1	yes	characteristic
46	C₂×C₃×C₄	D₁₆	1	1	yes	characteristic
47	#67	—	16	1	no	—
48	D₁₂	—	8	1	no	—
49	C₄×D₃	—	16	1	no	—
50	Dic₆	—	8	1	no	—
51	C₃×C₈	#102	1	1	yes	characteristic
52	C₃×C₈	SD₃₂	1	1	yes	characteristic
53	C₃⋊C₈	—	16	1	no	—
54	#41	—	24	1	no	—
55	Q₁₆	—	12	1	no	—
56	SD₁₆	—	24	1	no	—
57	D₈	—	12	1	no	—
58	M₁₆	—	24	1	no	—
59	C₂×C₈	D₂₄	1	1	yes	characteristic
60	C₁₆	#204	1	1	yes	characteristic
61	C₁₆	(C₂×D₃)⋊C₄	1	1	yes	characteristic
62	C₂²×C₃	SD₆₄	1	1	yes	characteristic
63	C₂×D₃	—	16	1	no	—
64	C₃×C₄	#357	1	1	yes	characteristic
65	C₃×C₄	D₃₂	1	1	yes	characteristic
66	Dic₃	—	16	1	no	—
67	Q₈	—	24	1	no	—
68	D₄	—	24	1	no	—
69	D₄	—	48	1	no	—
70	C₂×C₄	—	48	1	no	—
71	C₂×C₄	D₄₈	1	1	yes	characteristic
72	C₈	—	48	1	no	—
73	C₈	#800	1	1	yes	characteristic
74	C₈	#821	1	1	yes	characteristic
75	C₂×C₃	—	2	1	no	—
76	C₂×C₃	#1415	1	1	yes	characteristic
77	D₃	—	32	1	no	—
78	C₂²	—	48	1	no	—
79	C₂²	#4449	1	1	yes	characteristic
80	C₄	—	48	1	no	—
81	C₄	D₉₆	1	1	yes	characteristic
82	C₄	#4509	1	1	yes	characteristic
83	C₃	#7539	1	1	yes	characteristic, complemented, Sylow
84	C₂	—	2	1	no	—
85	C₂	—	96	1	no	—
86	C₂	#67596	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₉₆
commutator	C₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₃
Fitting	C₂×C₃×C₆₄	C₂
socle	C₂×C₃	#1415
prime radical	#69369	C₂
cosocle	C₂×C₃×C₃₂	C₂²
hypercenter	C₂×C₆₄	D₃
hypocenter	C₃	#7539
LCS	G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₆₄	C₂	semidirect
3. #67593	C₂	semidirect
13. C₂×C₆₄	D₃	semidirect
83. C₃	#7539	semidirect

Maximal subgroups

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

subgroup	index
#69369	2
C₂×C₃×C₆₄	2
#67593	2
#7539	3