Subgroups — #44527744

Order 31122.

96 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#29509866	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	#23780576	C₃	1	1	yes	maximal, characteristic, complemented
3	#23780572	C₃	1	1	yes	maximal, characteristic, complemented
4	#23780571	C₃	1	1	yes	maximal, characteristic, complemented
5	C₇×#13093366	C₃	1	1	yes	maximal, characteristic, complemented
6	#19104443	C₂×C₃	1	1	yes	characteristic, complemented
7	C₁₉×#63270	C₂×C₃	1	1	yes	characteristic, complemented
8	C₁₃×#87705	C₂×C₃	1	1	yes	characteristic, complemented
9	C₇×#10625202	C₂×C₃	1	1	yes	characteristic, complemented
10	(C₁₃⋊C₃)⋊(D₁₉⋊C₃)	—	7	1	no	maximal, Hall
11	C₇×D₂₄₇	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	ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
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	#11716322	—	13	1	no	—
23	#11716321	—	13	1	no	—
24	ΓL(1,𝔽₈)×D₁₉	—	13	1	no	—
25	C₇×D₁₉⋊C₃	—	13	1	no	—
26	#10625203	—	7	1	no	—
27	#10625202	C₂×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented
28	C₁₃×C₁₉⋊C₃	—	7	1	no	—
29	C₁₉×C₁₃⋊C₃	—	7	1	no	—
30	#10587717	—	19	1	no	—
31	#10587716	—	19	1	no	—
32	ΓL(1,𝔽₈)×D₁₃	—	19	1	no	—
33	C₇×D₁₃⋊C₃	—	19	1	no	—
34	D₂₄₇	C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, 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	—	19	1	no	—
41	#63270	D₁₉⋊C₃	1	1	yes	characteristic, complemented
42	C₇×C₁₃⋊C₃	—	19	1	no	—
43	C₁₃×ΓL(1,𝔽₈)	—	19	1	no	—
44	C₇×D₁₉	—	13	1	no	Hall
45	C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	1	yes	characteristic, complemented, Hall
46	C₃×D₁₃⋊C₃	—	133	1	no	Hall
47	C₇×D₁₃	—	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	C₂×C₃×ΓL(1,𝔽₈)	—	247	1	no	Hall
51	C₃×C₁₃⋊C₃	—	133	1	no	Hall
52	C₃×D₁₉	—	91	1	no	—
53	D₁₉⋊C₃	—	13	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	—	19	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	C₂×C₃×C₇	—	247	1	no	—
67	C₂×ΓL(1,𝔽₈)	—	247	1	no	—
68	C₂×ΓL(1,𝔽₈)	—	247	1	no	—
69	C₂×ΓL(1,𝔽₈)	—	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	—	13	1	no	Hall
75	D₁₃	—	19	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	C₂×C₇	—	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	—	247	1	no	—
90	C₃	—	91	1	no	—
91	C₃	—	133	1	no	—
92	C₃	—	1729	1	no	—
93	C₃	—	247	1	no	—
94	C₂	—	247	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. #29509866	C₂	semidirect
2. #23780576	C₃	semidirect
3. #23780572	C₃	semidirect
4. #23780571	C₃	semidirect
5. C₇×#13093366	C₃	semidirect
6. #19104443	C₂×C₃	semidirect
7. C₁₉×#63270	C₂×C₃	semidirect
8. C₁₃×#87705	C₂×C₃	semidirect
9. C₇×#10625202	C₂×C₃	semidirect
11. C₇×D₂₄₇	C₃²	semidirect
14. C₇×C₁₃×C₁₉	C₂×C₃²	semidirect
17. #13093366	ΓL(1,𝔽₈)	semidirect
27. #10625202	C₂×ΓL(1,𝔽₈)	semidirect
34. D₂₄₇	C₃×ΓL(1,𝔽₈)	semidirect
35. #87705	D₁₃⋊C₃	semidirect
41. #63270	D₁₉⋊C₃	semidirect
45. C₁₃×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	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
#29509866	2
#23780576	3
#23780572	3
#23780571	3
C₇×#13093366	3
(C₁₃⋊C₃)⋊(D₁₉⋊C₃)	7
ΓL(1,𝔽₈)×D₁₉⋊C₃	13
ΓL(1,𝔽₈)×D₁₃⋊C₃	19