Subgroups — #10683049

Order 768.

146 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#1709	C₂	1	1	yes	maximal, characteristic
2	#67795	C₂	1	2	yes	maximal, complemented
3	#7531	—	3	1	no	maximal, Sylow
4	C₃×#494	C₂²	1	1	yes	characteristic
5	C₃×#493	C₄	1	1	yes	characteristic
6	C₃×#486	—	2	2	no	—
7	C₃×C₈²	C₄	1	1	yes	characteristic
8	#4548	—	2	2	no	—
9	#1709	D₃	1	1	yes	characteristic
10	#1411	—	3	2	no	—
11	C₂×C₃×Q₁₆	—	4	1	no	—
12	C₂×C₃×D₈	—	4	1	no	—
13	C₃×C₄⋊Q₈	D₄	1	2	yes	—
14	C₃×#127	—	2	1	no	—
15	C₃×#102	—	4	4	no	—
16	C₃×C₄×C₈	—	2	1	no	—
17	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
18	#836	—	4	2	no	—
19	#494	C₂×D₃	1	1	yes	characteristic
20	#493	Dic₃	1	1	yes	characteristic
21	#486	—	2	2	no	—
22	#356	—	6	2	no	—
23	C₈²	Dic₃	1	1	yes	characteristic
24	C₂×C₃×Q₈	—	2	2	no	—
25	C₂×C₃×D₄	—	4	2	no	—
26	C₂×C₃×D₄	—	4	1	no	—
27	C₃×Q₁₆	—	8	2	no	—
28	C₃×D₈	—	8	2	no	—
29	C₂×C₃×C₈	—	2	1	no	—
30	C₂×C₃×C₈	—	2	2	no	—
31	C₃×C₄⋊C₄	—	4	2	no	—
32	C₃×C₄²	C₂²⋊C₄	1	1	yes	characteristic
33	#208	—	8	2	no	—
34	C₂×Q₁₆	—	4	1	no	—
35	C₂×D₈	—	4	1	no	—
36	C₄⋊Q₈	#67	1	2	yes	—
37	#127	—	2	1	no	—
38	#102	—	4	4	no	—
39	#101	—	12	2	no	—
40	C₄×C₈	—	2	1	no	—
41	C₄×C₈	C₂×Dic₃	1	1	yes	characteristic
42	C₂³×C₃	—	8	1	no	—
43	C₃×Q₈	—	8	2	no	—
44	C₃×D₄	—	8	2	no	—
45	C₃×D₄	—	8	4	no	—
46	C₂×C₃×C₄	—	2	1	no	—
47	C₂×C₃×C₄	—	4	2	no	—
48	C₂×C₃×C₄	#99	1	1	yes	characteristic
49	C₃×C₈	—	2	2	no	—
50	C₃×C₈	—	4	2	no	—
51	C₃⋊C₈	—	16	2	no	—
52	C₂×Q₈	—	2	2	no	—
53	C₂×D₄	—	4	2	no	—
54	C₂×D₄	—	4	1	no	—
55	Q₁₆	—	8	2	no	—
56	D₈	—	8	2	no	—
57	M₁₆	—	24	2	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	2	no	—
60	C₄⋊C₄	—	4	2	no	—
61	C₄²	#217	1	1	yes	characteristic
62	C₂²×C₃	—	16	2	no	—
63	C₂²×C₃	—	16	1	no	—
64	C₂²×C₃	#353	1	1	yes	characteristic
65	C₃×C₄	—	2	2	no	—
66	C₃×C₄	—	2	1	no	—
67	C₃×C₄	—	8	2	no	—
68	C₂³	—	8	1	no	—
69	Q₈	—	8	2	no	—
70	D₄	—	8	2	no	—
71	D₄	—	8	4	no	—
72	C₂×C₄	—	2	1	no	—
73	C₂×C₄	—	4	2	no	—
74	C₂×C₄	#834	1	1	yes	characteristic
75	C₈	—	2	2	no	—
76	C₈	—	4	2	no	—
77	C₈	—	48	2	no	—
78	C₂×C₃	—	2	1	no	—
79	C₂×C₃	—	32	1	no	—
80	C₂×C₃	#1408	1	1	yes	characteristic
81	C₂²	—	16	2	no	—
82	C₂²	—	16	1	no	—
83	C₂²	#4545	1	1	yes	characteristic
84	C₄	—	2	2	no	—
85	C₄	—	2	1	no	—
86	C₄	—	8	2	no	—
87	C₃	#7531	1	1	yes	characteristic, complemented, Sylow
88	C₂	—	2	1	no	—
89	C₂	—	32	1	no	—
90	C₂	#67792	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#67792
commutator	C₃×C₄×C₈	C₂×C₄
Frattini	#494	C₂×D₃
Fitting	C₃×#1709	C₂
socle	C₂×C₃	#1408
prime radical	C₃×#493	C₄
cosocle	C₃×#494	C₂²
hypercenter	#1709	D₃
hypocenter	C₃	#7531
LCS	G ▷ C₃×C₄×C₈ ▷ C₃×C₄² ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ #1709

Complements

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

N	H ≅ G/N	type
2. #67795	C₂	semidirect
87. C₃	#7531	semidirect

Maximal subgroups

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

subgroup	index
C₃×#1709	2
#67795	2
#7531	3