Subgroups — #44527677

Order 31122.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29509868	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	#23780627	C₃	1	1	yes	maximal, characteristic, complemented
3	#23780625	C₃	1	1	yes	maximal, characteristic, complemented
4	#23780623	C₃	1	1	yes	maximal, characteristic, complemented
5	#23780620	C₃	1	1	yes	maximal, characteristic, complemented
6	C₁₉×#63271	C₂×C₃	1	1	yes	characteristic, complemented
7	#19104440	C₂×C₃	1	1	yes	characteristic, complemented
8	C₁₃×#87705	C₂×C₃	1	1	yes	characteristic, complemented
9	C₇×#10625203	C₂×C₃	1	1	yes	characteristic, complemented
10	(C₁₃⋊C₃)⋊(D₁₉⋊C₃)	—	7	1	no	maximal, Hall
11	D₁₇₂₉	C₃²	1	1	yes	characteristic, complemented, Hall
12	ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	—	13	1	no	maximal, Hall
13	C₁₃⋊C₃×C₁₉⋊C₃	—	7	1	no	Hall
14	C₇×C₁₃×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
15	ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	—	19	1	no	maximal, Hall
16	#13093367	—	7	1	no	—
17	#13093366	—	7	1	no	—
18	(C₁₃⋊C₃)⋊D₁₉	—	7	1	no	—
19	(C₁₉⋊C₃)⋊D₁₃	—	7	1	no	—
20	ΓL(1,𝔽₈)×C₁₉⋊C₃	—	13	1	no	Hall
21	ΓL(1,𝔽₈)×C₁₃⋊C₃	—	19	1	no	Hall
22	#11716328	—	13	1	no	—
23	#11716327	—	13	1	no	—
24	ΓL(1,𝔽₈)⋊D₁₉	—	13	1	no	—
25	(C₁₉⋊C₃)⋊D₇	—	13	1	no	—
26	#10625203	AGL(1,𝔽₇)	1	1	yes	characteristic, complemented
27	#10625202	—	7	1	no	—
28	C₁₃×C₁₉⋊C₃	—	7	1	no	—
29	C₁₉×C₁₃⋊C₃	—	7	1	no	—
30	#10587723	—	19	1	no	—
31	#10587722	—	19	1	no	—
32	ΓL(1,𝔽₈)⋊D₁₃	—	19	1	no	—
33	(C₁₃⋊C₃)⋊D₇	—	19	1	no	—
34	D₂₄₇	—	7	1	no	Hall
35	#87705	D₁₃⋊C₃	1	1	yes	characteristic, complemented
36	#87704	—	13	1	no	—
37	C₇×C₁₉⋊C₃	—	13	1	no	—
38	C₁₉×ΓL(1,𝔽₈)	—	13	1	no	—
39	C₃×D₁₉⋊C₃	—	91	1	no	Hall
40	#63271	D₁₉⋊C₃	1	1	yes	characteristic, complemented
41	#63270	—	19	1	no	—
42	C₇×C₁₃⋊C₃	—	19	1	no	—
43	C₁₃×ΓL(1,𝔽₈)	—	19	1	no	—
44	D₁₃₃	—	13	1	no	Hall
45	C₁₃×C₁₉	C₃×AGL(1,𝔽₇)	1	1	yes	characteristic, complemented, Hall
46	C₃×D₁₃⋊C₃	—	133	1	no	Hall
47	D₉₁	—	19	1	no	Hall
48	C₃×C₁₉⋊C₃	—	91	1	no	Hall
49	C₇×C₁₉	C₃×D₁₃⋊C₃	1	1	yes	characteristic, complemented, Hall
50	C₃×AGL(1,𝔽₇)	—	247	1	no	Hall
51	C₃×C₁₃⋊C₃	—	133	1	no	Hall
52	C₃×D₁₉	—	91	1	no	—
53	D₁₉⋊C₃	—	91	1	no	—
54	D₁₉⋊C₃	—	91	1	no	—
55	D₁₉⋊C₃	—	91	1	no	—
56	C₇×C₁₃	C₃×D₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
57	C₃×D₁₃	—	133	1	no	—
58	D₁₃⋊C₃	—	133	1	no	—
59	D₁₃⋊C₃	—	133	1	no	—
60	D₁₃⋊C₃	—	133	1	no	—
61	C₃×ΓL(1,𝔽₈)	—	247	1	no	Hall
62	C₃×C₁₉	—	91	1	no	—
63	C₁₉⋊C₃	—	7	1	no	—
64	C₁₉⋊C₃	—	13	1	no	—
65	C₁₉⋊C₃	—	91	1	no	—
66	C₃×D₇	—	247	1	no	—
67	AGL(1,𝔽₇)	—	247	1	no	—
68	AGL(1,𝔽₇)	—	247	1	no	—
69	AGL(1,𝔽₇)	—	247	1	no	—
70	C₃×C₁₃	—	133	1	no	—
71	C₁₃⋊C₃	—	7	1	no	—
72	C₁₃⋊C₃	—	19	1	no	—
73	C₁₃⋊C₃	—	133	1	no	—
74	D₁₉	—	91	1	no	Hall
75	D₁₃	—	133	1	no	Hall
76	C₃×C₇	—	247	1	no	—
77	ΓL(1,𝔽₈)	—	13	1	no	—
78	ΓL(1,𝔽₈)	—	19	1	no	—
79	ΓL(1,𝔽₈)	—	247	1	no	—
80	C₁₉	ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	1	1	yes	characteristic, complemented, Sylow
81	C₂×C₃²	—	1729	1	no	Hall
82	D₇	—	247	1	no	Hall
83	C₁₃	ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
84	C₃²	—	1729	1	no	Sylow
85	C₇	(C₁₃⋊C₃)⋊(D₁₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
86	C₂×C₃	—	1729	1	no	—
87	C₂×C₃	—	1729	1	no	—
88	C₂×C₃	—	1729	1	no	—
89	C₂×C₃	—	1729	1	no	—
90	C₃	—	91	1	no	—
91	C₃	—	133	1	no	—
92	C₃	—	1729	1	no	—
93	C₃	—	247	1	no	—
94	C₂	—	1729	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇×C₁₃×C₁₉	C₂×C₃²
LCS	G ▷ C₇×C₁₃×C₁₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #29509868	C₂	semidirect
2. #23780627	C₃	semidirect
3. #23780625	C₃	semidirect
4. #23780623	C₃	semidirect
5. #23780620	C₃	semidirect
6. C₁₉×#63271	C₂×C₃	semidirect
7. #19104440	C₂×C₃	semidirect
8. C₁₃×#87705	C₂×C₃	semidirect
9. C₇×#10625203	C₂×C₃	semidirect
11. D₁₇₂₉	C₃²	semidirect
14. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
26. #10625203	AGL(1,𝔽₇)	semidirect
35. #87705	D₁₃⋊C₃	semidirect
40. #63271	D₁₉⋊C₃	semidirect
45. C₁₃×C₁₉	C₃×AGL(1,𝔽₇)	semidirect
49. C₇×C₁₉	C₃×D₁₃⋊C₃	semidirect
56. C₇×C₁₃	C₃×D₁₉⋊C₃	semidirect
80. C₁₉	ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	semidirect
83. C₁₃	ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	semidirect
85. C₇	(C₁₃⋊C₃)⋊(D₁₉⋊C₃)	semidirect

Maximal subgroups

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

subgroup	index
#29509868	2
#23780627	3
#23780625	3
#23780623	3
#23780620	3
(C₁₃⋊C₃)⋊(D₁₉⋊C₃)	7
ΓL(1,𝔽₈)⋊(D₁₉⋊C₃)	13
ΓL(1,𝔽₈)⋊(D₁₃⋊C₃)	19