Subgroups — #13133555

Order 1701.

156 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₇⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×#4428	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#4427	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×ΓL(1,𝔽₈)	C₃	1	1	yes	maximal, characteristic, complemented
5	C₇×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
6	C₃²⋊(C₇⋊C₉)	C₃	1	2	yes	maximal, complemented
7	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	2	yes	maximal, complemented
8	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	2	yes	maximal, complemented
9	(C₃×ΓL(1,𝔽₈))⋊C₉	C₃	1	2	yes	maximal, complemented
10	#6893	—	7	1	no	maximal, Sylow
11	C₃³×C₇	C₃²	1	1	yes	characteristic
12	C₃²×ΓL(1,𝔽₈)	C₃²	1	2	yes	—
13	C₃²×ΓL(1,𝔽₈)	C₃²	1	1	yes	characteristic
14	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
15	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
16	C₃×C₇×C₉	C₃²	1	1	yes	characteristic
17	C₇⋊H(3,𝔽₃)	—	3	3	no	—
18	C₃×C₇⋊C₉	C₃²	1	2	yes	—
19	C₃×C₇⋊C₉	C₃²	1	2	yes	—
20	C₃×C₇⋊C₉	C₃²	1	2	yes	—
21	#4428	—	3	3	no	—
22	#4427	—	3	3	no	—
23	C₉×ΓL(1,𝔽₈)	—	3	3	no	—
24	C₃×3₋¹⁺²	—	7	1	no	—
25	C₃×3₋¹⁺²	—	7	1	no	—
26	C₃×H(3,𝔽₃)	—	7	1	no	—
27	C₃²×C₉	—	7	1	no	—
28	C₃²⋊C₉	—	7	2	no	—
29	C₃²⋊C₉	—	7	2	no	—
30	C₃²⋊C₉	—	7	2	no	—
31	C₃²⋊C₉	—	7	2	no	—
32	C₃²⋊C₉	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
33	C₃²×C₇	—	3	1	no	—
34	C₃²×C₇	—	3	1	no	—
35	C₃²×C₇	—	3	2	no	—
36	C₃²×C₇	C₃³	1	1	yes	characteristic
37	C₃×ΓL(1,𝔽₈)	—	3	1	no	—
38	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
39	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
40	C₃×ΓL(1,𝔽₈)	—	3	2	no	—
41	C₃×ΓL(1,𝔽₈)	—	3	4	no	—
42	C₃×ΓL(1,𝔽₈)	H(3,𝔽₃)	1	3	yes	—
43	C₇×C₉	—	3	1	no	—
44	C₇×C₉	—	3	1	no	—
45	C₇×C₉	—	3	1	no	—
46	C₇⋊C₉	—	3	2	no	—
47	C₇⋊C₉	—	3	2	no	—
48	C₇⋊C₉	—	3	2	no	—
49	C₃³	—	7	1	no	—
50	C₃³	—	7	2	no	—
51	C₃³	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
52	3₋¹⁺²	—	21	3	no	—
53	3₋¹⁺²	—	21	3	no	—
54	H(3,𝔽₃)	—	21	3	no	—
55	C₃×C₉	—	7	2	no	—
56	C₃×C₉	—	7	2	no	—
57	C₃×C₉	—	7	2	no	—
58	C₃×C₉	—	21	3	no	—
59	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
60	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
61	C₃×C₉	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic
62	C₃×C₇	—	9	1	no	—
63	C₃×C₇	C₃×H(3,𝔽₃)	1	1	yes	characteristic
64	C₃×C₇	#737	1	1	yes	characteristic
65	C₃×C₇	#737	1	2	yes	—
66	ΓL(1,𝔽₈)	—	3	3	no	—
67	ΓL(1,𝔽₈)	—	9	2	no	—
68	C₃²	—	3	1	no	—
69	C₃²	—	3	1	no	—
70	C₃²	—	3	2	no	—
71	C₃²	—	7	3	no	—
72	C₃²	—	21	1	no	—
73	C₃²	—	21	2	no	—
74	C₃²	—	21	2	no	—
75	C₃²	—	21	2	no	—
76	C₃²	—	21	4	no	—
77	C₃²	C₃²×ΓL(1,𝔽₈)	1	1	yes	characteristic
78	C₉	—	3	1	no	—
79	C₉	—	3	1	no	—
80	C₉	—	3	1	no	—
81	C₉	—	21	2	no	—
82	C₉	—	21	2	no	—
83	C₉	—	21	2	no	—
84	C₇	#6893	1	1	yes	characteristic, complemented, Sylow
85	C₃	—	9	1	no	—
86	C₃	—	21	3	no	—
87	C₃	—	63	2	no	—
88	C₃	#10588061	1	1	yes	characteristic
89	C₃	ΓL(1,𝔽₈)×H(3,𝔽₃)	1	1	yes	characteristic
90	C₃	#10588065	1	2	yes	—

	subgroup	G/H
center, Frattini	C₃²	C₃²×ΓL(1,𝔽₈)
commutator, socle, cosocle	C₃²×C₇	C₃³
Fitting	C₇×C₃²⋊C₉	C₃
prime radical	C₃×C₇⋊H(3,𝔽₃)	C₃
hypercenter	C₃²⋊C₉	ΓL(1,𝔽₈)
hypocenter	C₇	#6893
LCS	G ▷ C₃²×C₇ ▷ C₇
UCS	C₁ ◁ C₃² ◁ C₃²⋊C₉

Complements

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

N	H ≅ G/N	type
2. C₃×#4428	C₃	semidirect
3. C₃×#4427	C₃	semidirect
4. C₃×C₉×ΓL(1,𝔽₈)	C₃	semidirect
5. C₇×C₃²⋊C₉	C₃	semidirect
6. C₃²⋊(C₇⋊C₉)	C₃	semidirect
7. (C₃×ΓL(1,𝔽₈))⋊C₉	C₃	semidirect
8. (C₃×ΓL(1,𝔽₈))⋊C₉	C₃	semidirect
9. (C₃×ΓL(1,𝔽₈))⋊C₉	C₃	semidirect
32. C₃²⋊C₉	ΓL(1,𝔽₈)	semidirect
84. C₇	#6893	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₇⋊H(3,𝔽₃)	3
C₃×#4428	3
C₃×#4427	3
C₃×C₉×ΓL(1,𝔽₈)	3
C₇×C₃²⋊C₉	3
C₃²⋊(C₇⋊C₉)	3
(C₃×ΓL(1,𝔽₈))⋊C₉	3
#6893	7