Subgroups — #30660713

Order 16758.

112 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22552727	C₂	1	1	yes	maximal, characteristic, complemented, Hall
2	D₁₉×#3862	C₃	1	1	yes	maximal, characteristic, complemented
3	#19150312	C₃	1	1	yes	maximal, characteristic, complemented
4	C₇×#11716321	C₃	1	2	yes	maximal, complemented
5	C₁₉×#3862	C₂×C₃	1	1	yes	characteristic, complemented
6	#14584116	C₂×C₃	1	1	yes	characteristic, complemented
7	C₇×#87704	C₂×C₃	1	2	yes	complemented
8	ΓL(1,𝔽₈)×D₁₉⋊C₃	—	7	2	no	maximal
9	C₇²×D₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
10	ΓL(1,𝔽₈)×C₁₉⋊C₃	—	7	2	no	—
11	C₇²×C₁₉	C₂×C₃²	1	1	yes	characteristic, complemented, Hall
12	C₂×ΓL(1,𝔽₈)²	—	19	1	no	maximal, Hall
13	#11716322	—	7	2	no	—
14	#11716322	—	21	1	no	—
15	#11716322	—	21	1	no	—
16	#11716321	ΓL(1,𝔽₈)	1	2	yes	complemented
17	ΓL(1,𝔽₈)×D₁₉	—	7	2	no	—
18	C₇×D₁₉⋊C₃	—	7	2	no	—
19	ΓL(1,𝔽₈)²	—	19	1	no	Hall
20	#87705	—	7	2	no	—
21	#87705	—	21	1	no	—
22	#87705	—	21	1	no	—
23	#87704	C₂×ΓL(1,𝔽₈)	1	2	yes	complemented
24	C₇×C₁₉⋊C₃	—	7	2	no	—
25	C₁₉×ΓL(1,𝔽₈)	—	7	2	no	—
26	C₃×D₁₉⋊C₃	—	49	1	no	Hall
27	C₂×#3862	—	57	1	no	—
28	C₂×#3861	—	19	1	no	—
29	C₂×C₇×ΓL(1,𝔽₈)	—	19	2	no	—
30	C₇×D₁₉	—	3	1	no	—
31	C₇×D₁₉	—	3	1	no	—
32	C₇×D₁₉	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
33	C₃×C₁₉⋊C₃	—	49	1	no	Hall
34	#3862	D₁₉⋊C₃	1	1	yes	characteristic, complemented
35	#3861	—	19	1	no	—
36	C₇×ΓL(1,𝔽₈)	—	19	2	no	—
37	C₇×C₁₉	—	3	1	no	—
38	C₇×C₁₉	—	3	1	no	—
39	C₇×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
40	C₂×C₃×ΓL(1,𝔽₈)	—	133	2	no	—
41	C₃×D₁₉	—	49	1	no	—
42	D₁₉⋊C₃	—	7	2	no	—
43	D₁₉⋊C₃	—	49	1	no	—
44	C₂×C₇²	—	57	1	no	Hall
45	C₃×ΓL(1,𝔽₈)	—	133	2	no	—
46	C₃×C₁₉	—	49	1	no	—
47	C₁₉⋊C₃	—	7	2	no	—
48	C₁₉⋊C₃	—	49	1	no	—
49	C₇²	C₃×D₁₉⋊C₃	1	1	yes	characteristic, complemented, Sylow
50	C₂×C₃×C₇	—	133	2	no	—
51	C₂×ΓL(1,𝔽₈)	—	19	2	no	—
52	C₂×ΓL(1,𝔽₈)	—	133	2	no	—
53	C₂×ΓL(1,𝔽₈)	—	399	1	no	—
54	C₂×ΓL(1,𝔽₈)	—	399	1	no	—
55	C₂×ΓL(1,𝔽₈)	—	399	2	no	—
56	D₁₉	ΓL(1,𝔽₈)²	1	1	yes	characteristic, complemented, Hall
57	C₃×C₇	—	133	2	no	—
58	ΓL(1,𝔽₈)	—	7	2	no	—
59	ΓL(1,𝔽₈)	—	19	2	no	—
60	ΓL(1,𝔽₈)	—	133	2	no	—
61	ΓL(1,𝔽₈)	—	399	1	no	—
62	ΓL(1,𝔽₈)	—	399	1	no	—
63	C₁₉	C₂×ΓL(1,𝔽₈)²	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₃²	—	931	1	no	Hall
65	C₂×C₇	—	57	1	no	—
66	C₂×C₇	—	57	2	no	—
67	C₂×C₇	—	171	1	no	—
68	C₃²	—	931	1	no	Sylow
69	C₇	—	3	1	no	—
70	C₇	—	3	1	no	—
71	C₇	ΓL(1,𝔽₈)×D₁₉⋊C₃	1	2	yes	complemented
72	C₂×C₃	—	133	2	no	—
73	C₂×C₃	—	931	1	no	—
74	C₂×C₃	—	2793	1	no	—
75	C₃	—	49	1	no	—
76	C₃	—	133	2	no	—
77	C₃	—	931	1	no	—
78	C₂	—	57	1	no	Sylow

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₇²×C₁₉	C₂×C₃²
LCS	G ▷ C₇²×C₁₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. #22552727	C₂	semidirect
2. D₁₉×#3862	C₃	semidirect
3. #19150312	C₃	semidirect
4. C₇×#11716321	C₃	semidirect
5. C₁₉×#3862	C₂×C₃	semidirect
6. #14584116	C₂×C₃	semidirect
7. C₇×#87704	C₂×C₃	semidirect
9. C₇²×D₁₉	C₃²	semidirect
11. C₇²×C₁₉	C₂×C₃²	semidirect
16. #11716321	ΓL(1,𝔽₈)	semidirect
23. #87704	C₂×ΓL(1,𝔽₈)	semidirect
32. C₇×D₁₉	C₃×ΓL(1,𝔽₈)	semidirect
34. #3862	D₁₉⋊C₃	semidirect
39. C₇×C₁₉	C₂×C₃×ΓL(1,𝔽₈)	semidirect
49. C₇²	C₃×D₁₉⋊C₃	semidirect
56. D₁₉	ΓL(1,𝔽₈)²	semidirect
63. C₁₉	C₂×ΓL(1,𝔽₈)²	semidirect
71. C₇	ΓL(1,𝔽₈)×D₁₉⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
#22552727	2
D₁₉×#3862	3
#19150312	3
C₇×#11716321	3
ΓL(1,𝔽₈)×D₁₉⋊C₃	7
C₂×ΓL(1,𝔽₈)²	19