Subgroups — #45692513

Order 33516.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₉×#3862	C₃	1	1	yes	maximal, characteristic, complemented
2	C₂²⋊#14584116	C₃	1	1	yes	maximal, characteristic, complemented
3	C₇×C₂²⋊#87704	C₃	1	2	yes	maximal, complemented
4	#22552727	—	4	1	no	maximal, Hall
5	C₂×C₁₉×#3862	—	3	1	no	—
6	ΓL(1,𝔽₈)×C₂²⋊(C₁₉⋊C₃)	—	7	2	no	maximal
7	C₂²×C₇²×C₁₉	C₃²	1	1	yes	characteristic, complemented, Hall
8	C₁₉×#3862	A₄	1	1	yes	characteristic, complemented
9	#14584116	—	4	1	no	—
10	C₇×#87704	—	4	2	no	—
11	C₂×C₇²×C₁₉	—	3	1	no	—
12	#3862⋊A₄	—	19	1	no	maximal, Hall
13	C₂²⋊#87705	—	7	2	no	—
14	C₂²⋊#87705	—	21	2	no	—
15	C₂²⋊#87704	ΓL(1,𝔽₈)	1	2	yes	complemented
16	C₇×C₂²⋊(C₁₉⋊C₃)	—	7	2	no	—
17	C₂²×C₁₉×ΓL(1,𝔽₈)	—	7	2	no	—
18	ΓL(1,𝔽₈)×C₁₉⋊C₃	—	28	2	no	—
19	C₇²×C₁₉	C₃×A₄	1	1	yes	characteristic, complemented, Hall
20	C₂×C₁₉×ΓL(1,𝔽₈)	—	21	2	no	—
21	C₃×C₂²⋊(C₁₉⋊C₃)	—	49	1	no	Hall
22	C₂²⋊#3861	—	19	1	no	—
23	C₇×C₂²⋊ΓL(1,𝔽₈)	—	19	2	no	—
24	C₂²×#3862	C₁₉⋊C₃	1	1	yes	characteristic, complemented
25	C₂²×C₇×C₁₉	—	3	2	no	—
26	C₂²×C₇×C₁₉	C₃×ΓL(1,𝔽₈)	1	2	yes	complemented
27	ΓL(1,𝔽₈)²	—	76	1	no	Hall
28	#87705	—	28	2	no	—
29	#87705	—	84	2	no	—
30	#87704	—	4	2	no	—
31	C₇×C₁₉⋊C₃	—	28	2	no	—
32	C₁₉×ΓL(1,𝔽₈)	—	7	2	no	—
33	C₂×#3862	—	3	1	no	—
34	C₂×C₇×C₁₉	—	3	2	no	—
35	C₂×C₇×C₁₉	—	9	2	no	—
36	A₄×ΓL(1,𝔽₈)	—	133	2	no	—
37	C₂²×C₃×C₁₉	—	49	1	no	—
38	C₂²⋊(C₁₉⋊C₃)	—	7	2	no	—
39	C₂²⋊(C₁₉⋊C₃)	—	49	1	no	—
40	C₂²×C₇²	C₃×C₁₉⋊C₃	1	1	yes	characteristic, complemented, Hall
41	C₃×C₁₉⋊C₃	—	196	1	no	Hall
42	#3862	C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, complemented
43	#3861	—	76	1	no	—
44	C₇×ΓL(1,𝔽₈)	—	76	2	no	—
45	C₇×C₁₉	—	3	2	no	—
46	C₇×C₁₉	A₄×ΓL(1,𝔽₈)	1	2	yes	complemented
47	C₂×C₃×C₁₉	—	147	1	no	—
48	C₂×C₇²	—	3	1	no	—
49	C₂²⋊ΓL(1,𝔽₈)	—	19	2	no	—
50	C₂²⋊ΓL(1,𝔽₈)	—	133	2	no	—
51	C₂²⋊ΓL(1,𝔽₈)	—	399	2	no	—
52	C₇×A₄	—	133	2	no	—
53	C₂²×ΓL(1,𝔽₈)	—	7	2	no	—
54	C₂²×C₁₉	ΓL(1,𝔽₈)²	1	1	yes	characteristic, complemented, Hall
55	C₃×ΓL(1,𝔽₈)	—	532	2	no	—
56	C₃×C₁₉	—	49	1	no	—
57	C₁₉⋊C₃	—	28	2	no	—
58	C₁₉⋊C₃	—	196	1	no	—
59	C₇²	C₃×C₂²⋊(C₁₉⋊C₃)	1	1	yes	characteristic, complemented, Sylow
60	C₂×ΓL(1,𝔽₈)	—	21	2	no	—
61	C₂×C₁₉	—	3	1	no	—
62	C₃×A₄	—	931	1	no	Hall
63	C₂²×C₇	—	3	2	no	—
64	C₂²×C₇	ΓL(1,𝔽₈)×C₁₉⋊C₃	1	2	yes	complemented
65	C₃×C₇	—	532	2	no	—
66	ΓL(1,𝔽₈)	—	7	2	no	—
67	ΓL(1,𝔽₈)	—	532	2	no	—
68	ΓL(1,𝔽₈)	—	1596	2	no	—
69	ΓL(1,𝔽₈)	—	76	2	no	—
70	C₁₉	#3862⋊A₄	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₇	—	3	2	no	—
72	C₂×C₇	—	9	2	no	—
73	C₂²×C₃	—	49	1	no	—
74	A₄	—	133	2	no	—
75	A₄	—	931	1	no	—
76	C₃²	—	3724	1	no	Sylow
77	C₇	—	3	2	no	—
78	C₇	ΓL(1,𝔽₈)×C₂²⋊(C₁₉⋊C₃)	1	2	yes	complemented
79	C₂×C₃	—	147	1	no	—
80	C₂²	#22552727	1	1	yes	characteristic, complemented, Sylow
81	C₃	—	532	2	no	—
82	C₃	—	49	1	no	—
83	C₃	—	3724	1	no	—
84	C₂	—	3	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₂²×C₇²×C₁₉	C₃²
LCS	G ▷ C₂²×C₇²×C₁₉
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₉×#3862	C₃	semidirect
2. C₂²⋊#14584116	C₃	semidirect
3. C₇×C₂²⋊#87704	C₃	semidirect
7. C₂²×C₇²×C₁₉	C₃²	semidirect
8. C₁₉×#3862	A₄	semidirect
15. C₂²⋊#87704	ΓL(1,𝔽₈)	semidirect
19. C₇²×C₁₉	C₃×A₄	semidirect
24. C₂²×#3862	C₁₉⋊C₃	semidirect
26. C₂²×C₇×C₁₉	C₃×ΓL(1,𝔽₈)	semidirect
40. C₂²×C₇²	C₃×C₁₉⋊C₃	semidirect
42. #3862	C₂²⋊(C₁₉⋊C₃)	semidirect
46. C₇×C₁₉	A₄×ΓL(1,𝔽₈)	semidirect
54. C₂²×C₁₉	ΓL(1,𝔽₈)²	semidirect
59. C₇²	C₃×C₂²⋊(C₁₉⋊C₃)	semidirect
64. C₂²×C₇	ΓL(1,𝔽₈)×C₁₉⋊C₃	semidirect
70. C₁₉	#3862⋊A₄	semidirect
78. C₇	ΓL(1,𝔽₈)×C₂²⋊(C₁₉⋊C₃)	semidirect
80. C₂²	#22552727	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₁₉×#3862	3
C₂²⋊#14584116	3
C₇×C₂²⋊#87704	3
#22552727	4
ΓL(1,𝔽₈)×C₂²⋊(C₁₉⋊C₃)	7
#3862⋊A₄	19