Subgroups — #10716213

Order 768.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#69320	C₂	1	2	yes	maximal
2	C₃×#1424	C₂	1	1	yes	maximal, characteristic
3	#67592	C₂	1	1	yes	maximal, characteristic
4	Dic₃×C₃₂	C₂	1	1	yes	maximal, characteristic
5	#67576	C₂	1	2	yes	maximal
6	#13733	—	3	1	no	maximal, Sylow
7	#4874	—	2	2	no	—
8	C₂×C₃×C₃₂	C₂²	1	1	yes	characteristic
9	C₃×#366	C₂²	1	2	yes	—
10	#4504	C₂²	1	2	yes	—
11	Dic₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₂×C₃⋊C₃₂	C₂²	1	1	yes	characteristic
13	#4489	—	2	2	no	—
14	#2252	—	3	2	no	—
15	#1424	—	3	1	no	—
16	#1424	—	3	2	no	—
17	#1424	D₃	1	1	yes	characteristic
18	C₄×C₃₂	—	3	1	no	—
19	#888	—	4	2	no	—
20	C₂×C₃×C₁₆	C₂³	1	1	yes	characteristic
21	C₃×#107	—	2	2	no	—
22	#818	—	2	2	no	—
23	C₈×Dic₃	D₄	1	1	yes	characteristic
24	C₂×C₃⋊C₁₆	D₄	1	1	yes	characteristic
25	#807	—	4	2	no	—
26	C₃×C₃₂	Q₈	1	2	yes	complemented
27	C₃⋊C₃₂	Q₈	1	2	yes	complemented
28	#500	—	6	2	no	—
29	C₂×C₃₂	—	3	1	no	—
30	C₂×C₃₂	C₂×D₃	1	1	yes	characteristic
31	#366	—	3	2	no	—
32	#366	—	6	2	no	—
33	#366	C₂×D₃	1	2	yes	—
34	C₄×C₁₆	—	3	1	no	—
35	C₂×Dic₆	—	8	2	no	—
36	C₂×C₃×C₈	C₂×D₄	1	1	yes	characteristic
37	C₃×C₄⋊C₄	—	4	2	no	—
38	C₄⋊Dic₃	—	4	2	no	—
39	Dic₃⋊C₄	—	8	2	no	—
40	C₄×Dic₃	D₈	1	1	yes	characteristic
41	C₂×C₃⋊C₈	D₈	1	1	yes	characteristic
42	C₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
43	C₃×C₁₆	C₂×Q₈	1	1	yes	characteristic
44	C₃⋊C₁₆	Q₁₆	1	2	yes	—
45	C₄⋊Q₈	—	12	2	no	—
46	C₂×C₁₆	—	3	1	no	—
47	C₂×C₁₆	C₂²×D₃	1	1	yes	characteristic
48	#107	—	2	2	no	—
49	#107	—	6	2	no	—
50	#107	—	12	2	no	—
51	C₄×C₈	—	3	1	no	—
52	C₃₂	—	3	2	no	—
53	C₃₂	Dic₆	1	2	yes	complemented
54	C₂×C₃×C₄	—	8	2	no	—
55	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
56	C₂×Dic₃	—	8	2	no	—
57	C₂×Dic₃	D₁₆	1	2	yes	—
58	Dic₆	—	16	4	no	—
59	C₃×C₈	C₄⋊Q₈	1	1	yes	characteristic
60	C₃×C₈	C₂×Q₁₆	1	1	yes	characteristic
61	C₃⋊C₈	Q₃₂	1	2	yes	—
62	C₂×Q₈	—	24	2	no	—
63	C₂×C₈	—	3	1	no	—
64	C₂×C₈	D₃×D₄	1	1	yes	characteristic
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	—	12	2	no	—
67	C₄⋊C₄	—	24	2	no	—
68	C₄²	—	3	1	no	—
69	C₁₆	—	3	2	no	—
70	C₁₆	C₂×Dic₆	1	1	yes	characteristic
71	C₁₆	D₃×Q₈	1	1	yes	characteristic
72	C₂²×C₃	C₂×D₁₆	1	1	yes	characteristic
73	C₃×C₄	—	16	2	no	—
74	C₃×C₄	#500	1	1	yes	characteristic
75	C₃×C₄	C₂×Q₃₂	1	1	yes	characteristic
76	Dic₃	—	16	2	no	—
77	Dic₃	SD₆₄	1	4	yes	—
78	Q₈	—	48	4	no	—
79	C₂×C₄	—	3	2	no	—
80	C₂×C₄	—	8	2	no	—
81	C₂×C₄	—	24	2	no	—
82	C₂×C₄	D₃×D₈	1	1	yes	characteristic
83	C₈	—	3	2	no	—
84	C₈	#888	1	1	yes	characteristic
85	C₈	D₃×Q₁₆	1	1	yes	characteristic
86	C₂×C₃	#2252	1	1	yes	characteristic
87	C₂×C₃	C₂×SD₆₄	1	2	yes	—
88	C₂²	D₃×D₁₆	1	1	yes	characteristic
89	C₄	—	3	4	no	—
90	C₄	—	16	2	no	—
91	C₄	—	48	2	no	—
92	C₄	#4874	1	1	yes	characteristic
93	C₄	D₃×Q₃₂	1	1	yes	characteristic
94	C₃	#13733	1	1	yes	characteristic, complemented, Sylow
95	C₂	#69320	1	1	yes	characteristic
96	C₂	D₃×SD₆₄	1	2	yes	—

	subgroup	G/H
center	C₂²	D₃×D₁₆
commutator, cosocle	C₂×C₃×C₁₆	C₂³
Frattini	C₂×C₁₆	C₂²×D₃
Fitting	C₃×#1424	C₂
socle, prime radical	C₂²×C₃	C₂×D₁₆
hypercenter	#1424	D₃
hypocenter	C₃	#13733
LCS	G ▷ C₂×C₃×C₁₆ ▷ C₃×C₈ ▷ C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1424

Complements

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

N	H ≅ G/N	type
26. C₃×C₃₂	Q₈	semidirect
27. C₃⋊C₃₂	Q₈	semidirect
53. C₃₂	Dic₆	semidirect
94. C₃	#13733	semidirect

Maximal subgroups

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

subgroup	index
#69320	2
C₃×#1424	2
#67592	2
Dic₃×C₃₂	2
#67576	2
#13733	3