Subgroups — #10716211

Order 768.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69045	C₂	1	2	yes	maximal
2	C₃×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#67592	C₂	1	2	yes	maximal
4	#67591	C₂	1	2	yes	maximal
5	#13732	—	3	1	no	maximal, Sylow
6	#4683	—	2	2	no	—
7	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
9	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
10	#4505	—	2	2	no	—
11	#4504	C₂²	1	4	yes	—
12	#2253	—	3	2	no	—
13	#1424	—	3	2	no	—
14	#1423	—	3	2	no	—
15	C₄×C₃₂	D₃	1	1	yes	characteristic
16	#870	—	4	2	no	—
17	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
18	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
19	C₃×C₄×C₈	D₄	1	1	yes	characteristic
20	#818	—	2	4	no	—
21	#817	—	4	2	no	—
22	C₃×C₃₂	Q₈	1	2	yes	—
23	C₃×C₃₂	Q₈	1	2	yes	—
24	#499	—	6	2	no	—
25	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
26	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
27	#367	—	6	2	no	—
28	#366	—	3	4	no	—
29	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
30	C₂×C₃×C₈	D₈	1	1	yes	characteristic
31	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
32	C₃×C₄²	D₈	1	1	yes	characteristic
33	C₄⋊Dic₃	—	4	4	no	—
34	Dic₃⋊C₄	—	8	4	no	—
35	C₃×C₁₆	Q₁₆	1	2	yes	—
36	C₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
37	C₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
38	#125	—	12	2	no	—
39	C₂×C₁₆	D₁₂	1	1	yes	characteristic
40	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
41	#107	—	6	4	no	—
42	#106	—	12	2	no	—
43	C₄×C₈	D₁₂	1	1	yes	characteristic
44	C₃₂	Dic₆	1	2	yes	—
45	C₃₂	Dic₆	1	2	yes	—
46	C₂×C₃×C₄	D₁₆	1	2	yes	—
47	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
48	C₂×Dic₃	—	8	4	no	—
49	C₃×C₈	Q₃₂	1	2	yes	—
50	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
51	C₃×C₈	C₂×Q₁₆	1	1	yes	characteristic
52	C₂×C₈	D₂₄	1	1	yes	characteristic
53	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
54	C₄⋊C₄	—	12	4	no	—
55	C₄⋊C₄	—	24	4	no	—
56	C₄²	D₂₄	1	1	yes	characteristic
57	C₁₆	Dic₁₂	1	2	yes	—
58	C₁₆	C₂×Dic₆	1	1	yes	characteristic
59	C₁₆	C₂×Dic₆	1	1	yes	characteristic
60	C₂²×C₃	C₂×D₁₆	1	1	yes	characteristic
61	C₃×C₄	—	2	2	no	—
62	C₃×C₄	#500	1	1	yes	characteristic
63	C₃×C₄	C₂×Q₃₂	1	1	yes	characteristic
64	Dic₃	—	16	4	no	—
65	C₂×C₄	—	24	4	no	—
66	C₂×C₄	D₄₈	1	2	yes	—
67	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
68	C₈	Dic₂₄	1	2	yes	—
69	C₈	#869	1	1	yes	characteristic
70	C₈	C₂×Dic₁₂	1	1	yes	characteristic
71	C₂×C₃	#2252	1	1	yes	characteristic
72	C₂×C₃	#2262	1	2	yes	—
73	C₂²	C₂×D₄₈	1	1	yes	characteristic
74	C₄	—	2	2	no	—
75	C₄	—	48	4	no	—
76	C₄	#4682	1	1	yes	characteristic
77	C₄	C₂×Dic₂₄	1	1	yes	characteristic
78	C₃	#13732	1	1	yes	characteristic, complemented, Sylow
79	C₂	#69044	1	1	yes	characteristic
80	C₂	#69369	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄₈
commutator, cosocle	C₂×C₃×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₃
Fitting	C₃×C₄×C₃₂	C₂
socle, prime radical	C₂²×C₃	C₂×D₁₆
hypercenter	C₄×C₃₂	D₃
hypocenter	C₃	#13732
LCS	G ▷ C₂×C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
78. C₃	#13732	semidirect

Maximal subgroups

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

subgroup	index
#69045	2
C₃×C₄×C₃₂	2
#67592	2
#67591	2
#13732	3