Subgroups — #10683077

Order 768.

94 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×D₃₂	C₂	1	1	yes	maximal, characteristic
2	#67593	C₂	1	1	yes	maximal, characteristic, complemented
3	#67587	C₂	1	1	yes	maximal, characteristic, complemented
4	#7541	—	3	1	no	maximal, Sylow
5	C₂×C₃×D₁₆	—	2	1	no	—
6	C₃×D₃₂	—	2	1	no	—
7	C₃×D₃₂	C₄	1	2	yes	—
8	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
9	#4506	—	2	1	no	—
10	C₃⋊C₆₄	—	2	1	no	—
11	C₂×D₃₂	D₃	1	1	yes	characteristic
12	M₁₂₈	—	3	1	no	—
13	#1425	—	3	1	no	—
14	C₂×C₃×D₈	—	4	1	no	—
15	C₃×D₁₆	—	2	2	no	—
16	C₃×D₁₆	—	4	1	no	—
17	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
18	#819	—	4	1	no	—
19	C₃×C₃₂	C₂×C₄	1	1	yes	characteristic
20	C₃×C₃₂	D₄	1	1	yes	characteristic
21	C₂×D₁₆	—	2	1	no	—
22	D₃₂	—	2	1	no	—
23	D₃₂	Dic₃	1	2	yes	—
24	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
25	#368	—	6	1	no	—
26	C₆₄	—	6	1	no	—
27	C₂×C₃×D₄	—	8	1	no	—
28	C₃×D₈	—	4	2	no	—
29	C₃×D₈	—	8	1	no	—
30	C₂×C₃×C₈	D₈	1	1	yes	characteristic
31	#208	—	8	1	no	—
32	C₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
33	C₃×C₁₆	SD₁₆	1	1	yes	characteristic
34	C₂×D₈	—	4	1	no	—
35	D₁₆	—	2	2	no	—
36	D₁₆	—	4	1	no	—
37	C₂×C₁₆	#67	1	1	yes	characteristic
38	#108	—	12	1	no	—
39	C₃₂	C₂×Dic₃	1	1	yes	characteristic
40	C₃₂	#67	1	1	yes	characteristic
41	C₂³×C₃	—	16	1	no	—
42	C₃×D₄	—	8	2	no	—
43	C₃×D₄	—	16	1	no	—
44	C₂×C₃×C₄	D₁₆	1	1	yes	characteristic
45	C₃×C₈	#102	1	1	yes	characteristic
46	C₃×C₈	SD₃₂	1	1	yes	characteristic
47	C₃⋊C₈	—	16	1	no	—
48	C₂×D₄	—	8	1	no	—
49	D₈	—	4	2	no	—
50	D₈	—	8	1	no	—
51	M₁₆	—	24	1	no	—
52	C₂×C₈	#213	1	1	yes	characteristic
53	C₁₆	#214	1	1	yes	characteristic
54	C₁₆	#217	1	1	yes	characteristic
55	C₂²×C₃	—	16	2	no	—
56	C₂²×C₃	—	32	2	no	—
57	C₂²×C₃	SD₆₄	1	1	yes	characteristic
58	C₃×C₄	#357	1	1	yes	characteristic
59	C₃×C₄	D₃₂	1	1	yes	characteristic
60	C₂³	—	16	1	no	—
61	D₄	—	8	2	no	—
62	D₄	—	16	1	no	—
63	C₂×C₄	#826	1	1	yes	characteristic
64	C₈	—	48	1	no	—
65	C₈	#827	1	1	yes	characteristic
66	C₈	#832	1	1	yes	characteristic
67	C₂×C₃	—	2	1	no	—
68	C₂×C₃	—	32	2	no	—
69	C₂×C₃	#1415	1	1	yes	characteristic
70	C₂²	—	16	2	no	—
71	C₂²	—	32	2	no	—
72	C₂²	#4520	1	1	yes	characteristic
73	C₄	#4519	1	1	yes	characteristic
74	C₄	#4562	1	1	yes	characteristic
75	C₃	#7541	1	1	yes	characteristic, complemented, Sylow
76	C₂	—	2	1	no	—
77	C₂	—	32	2	no	—
78	C₂	#67824	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#67824
commutator	C₃×C₃₂	C₂×C₄
Frattini	C₂×C₃₂	C₂×D₃
Fitting, prime radical	C₂×C₃×D₃₂	C₂
socle	C₂×C₃	#1415
cosocle	C₂×C₃×C₃₂	C₂²
hypercenter	C₂×D₃₂	D₃
hypocenter	C₃	#7541
LCS	G ▷ C₃×C₃₂ ▷ C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×D₃₂

Complements

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

N	H ≅ G/N	type
2. #67593	C₂	semidirect
3. #67587	C₂	semidirect
75. C₃	#7541	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃×D₃₂	2
#67593	2
#67587	2
#7541	3