Subgroups — #10716214

Order 768.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69044	C₂	1	2	yes	maximal
2	C₃×C₄×C₃₂	C₂	1	1	yes	maximal, characteristic
3	#67592	C₂	1	4	yes	maximal
4	#13733	—	3	1	no	maximal, Sylow
5	#4682	—	2	2	no	—
6	C₂×C₃×C₃₂	C₂²	1	2	yes	—
7	C₃×C₄×C₁₆	C₂²	1	1	yes	characteristic
8	#4504	—	2	2	no	—
9	#4504	C₂²	1	4	yes	—
10	#2252	—	3	2	no	—
11	#1424	—	3	4	no	—
12	C₄×C₃₂	D₃	1	1	yes	characteristic
13	#869	—	4	2	no	—
14	C₂×C₃×C₁₆	D₄	1	1	yes	characteristic
15	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
16	C₃×C₄×C₈	D₄	1	1	yes	characteristic
17	#818	—	2	4	no	—
18	#818	—	4	2	no	—
19	C₃×C₃₂	Q₈	1	4	yes	complemented
20	#500	—	6	2	no	—
21	C₂×C₃₂	C₂×D₃	1	2	yes	—
22	#366	—	3	4	no	—
23	#366	—	6	2	no	—
24	C₄×C₁₆	C₂×D₃	1	1	yes	characteristic
25	C₂×Dic₆	—	8	2	no	—
26	C₂×C₃×C₈	D₈	1	1	yes	characteristic
27	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
28	C₃×C₄²	D₈	1	1	yes	characteristic
29	C₄⋊Dic₃	—	4	4	no	—
30	C₄⋊Dic₃	—	8	2	no	—
31	C₃×C₁₆	Q₁₆	1	2	yes	—
32	C₃×C₁₆	C₂×Q₈	1	2	yes	—
33	C₄⋊Q₈	—	12	2	no	—
34	C₂×C₁₆	D₁₂	1	1	yes	characteristic
35	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
36	#107	—	6	4	no	—
37	#107	—	12	2	no	—
38	C₄×C₈	D₁₂	1	1	yes	characteristic
39	C₃₂	Dic₆	1	4	yes	complemented
40	C₂×C₃×C₄	D₁₆	1	2	yes	—
41	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
42	C₂×Dic₃	—	8	4	no	—
43	Dic₆	—	16	4	no	—
44	C₃×C₈	Q₃₂	1	2	yes	—
45	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
46	C₃×C₈	C₂×Q₁₆	1	1	yes	characteristic
47	C₂×Q₈	—	24	2	no	—
48	C₂×C₈	D₂₄	1	1	yes	characteristic
49	C₂×C₈	C₂×D₁₂	1	1	yes	characteristic
50	C₄⋊C₄	—	12	4	no	—
51	C₄⋊C₄	—	24	2	no	—
52	C₄²	D₂₄	1	1	yes	characteristic
53	C₁₆	Dic₁₂	1	2	yes	—
54	C₁₆	C₂×Dic₆	1	2	yes	—
55	C₂²×C₃	C₂×D₁₆	1	1	yes	characteristic
56	C₃×C₄	SD₆₄	1	4	yes	—
57	C₃×C₄	#500	1	1	yes	characteristic
58	C₃×C₄	C₂×Q₃₂	1	1	yes	characteristic
59	Dic₃	—	16	4	no	—
60	Q₈	—	48	4	no	—
61	C₂×C₄	—	24	4	no	—
62	C₂×C₄	D₄₈	1	2	yes	—
63	C₂×C₄	C₂×D₂₄	1	1	yes	characteristic
64	C₈	Dic₂₄	1	2	yes	—
65	C₈	#869	1	1	yes	characteristic
66	C₈	C₂×Dic₁₂	1	1	yes	characteristic
67	C₂×C₃	#2252	1	1	yes	characteristic
68	C₂×C₃	C₂×SD₆₄	1	2	yes	—
69	C₂²	C₂×D₄₈	1	1	yes	characteristic
70	C₄	—	48	4	no	—
71	C₄	#4449	1	4	yes	—
72	C₄	#4682	1	1	yes	characteristic
73	C₄	C₂×Dic₂₄	1	1	yes	characteristic
74	C₃	#13733	1	1	yes	characteristic, complemented, Sylow
75	C₂	#69044	1	1	yes	characteristic
76	C₂	C₂×#4449	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	#13733
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
19. C₃×C₃₂	Q₈	semidirect
39. C₃₂	Dic₆	semidirect
74. C₃	#13733	semidirect

Maximal subgroups

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

subgroup	index
#69044	2
C₃×C₄×C₃₂	2
#67592	2
#13733	3