Subgroups — #28413334

Order 14700.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅⋊(C₅⋊#64425)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×C₅⋊#64418	C₂	1	2	yes	maximal, complemented
3	D₃₅²	—	3	1	no	maximal, Hall
4	C₅²×#3862	C₂²	1	1	yes	characteristic, complemented, Hall
5	#64418⋊D₅	—	5	2	no	maximal
6	C₅⋊(C₅⋊(C₇⋊D₇))	D₃	1	1	yes	characteristic, complemented
7	C₅×C₇×D₃₅	—	3	2	no	—
8	C₅⋊#64425	—	5	2	no	—
9	C₅⋊#64425	—	10	2	no	—
10	C₅⋊#64418	D₅	1	2	yes	complemented
11	C₅×#64418	—	5	2	no	—
12	C₅²×C₇²	C₂×D₃	1	1	yes	characteristic, complemented, Hall
13	C₅⋊(C₅⋊AGL(1,𝔽₇))	—	14	1	no	—
14	D₇×D₃₅	—	15	2	no	—
15	C₅×#3862	—	2	2	no	—
16	C₅×#3862	C₂×D₅	1	2	yes	complemented
17	D₅×D₃₅	—	21	2	no	—
18	#64418⋊C₂	—	25	1	no	Hall
19	C₅²×ΓL(1,𝔽₈)	—	14	1	no	—
20	C₅⋊(C₇⋊D₇)	—	5	2	no	—
21	C₅⋊(C₇⋊D₇)	—	10	2	no	—
22	C₇×D₃₅	—	3	2	no	—
23	C₅×C₇×D₇	—	15	2	no	—
24	C₅⋊D₃₅	—	14	1	no	—
25	C₅⋊D₃₅	—	21	2	no	—
26	C₅×D₃₅	—	3	2	no	—
27	C₅×C₇×D₅	—	21	2	no	—
28	D₁₅⋊D₅	—	49	1	no	maximal, Hall
29	#64425	—	25	1	no	—
30	#64418	—	5	2	no	—
31	C₅×C₇²	—	2	2	no	—
32	C₅×C₇²	D₃×D₅	1	2	yes	complemented
33	C₅⋊AGL(1,𝔽₇)	—	70	4	no	—
34	C₅⋊AGL(1,𝔽₇)	—	70	2	no	—
35	D₇²	—	75	1	no	Hall
36	C₅²×C₇	—	2	1	no	—
37	C₅²×C₇	—	3	2	no	—
38	C₅×D₁₅	—	49	2	no	—
39	C₃×C₅⋊D₅	—	49	1	no	—
40	#3862	D₅²	1	1	yes	characteristic, complemented, Hall
41	C₂×D₃₅	—	105	2	no	—
42	D₅×D₇	—	105	2	no	—
43	C₅×ΓL(1,𝔽₈)	—	14	2	no	—
44	C₅×ΓL(1,𝔽₈)	—	14	4	no	—
45	D₅²	—	147	1	no	Hall
46	C₇⋊D₇	—	25	1	no	—
47	C₇×D₇	—	15	2	no	—
48	C₃×C₅²	—	49	1	no	Hall
49	C₂×C₅×C₇	—	105	2	no	—
50	D₃₅	—	3	2	no	—
51	D₃₅	—	70	4	no	—
52	D₃₅	—	70	2	no	—
53	D₃₅	—	105	2	no	—
54	D₃₅	—	105	2	no	—
55	D₃₅	—	210	4	no	—
56	C₅×D₇	—	15	2	no	—
57	C₇×D₅	—	21	2	no	—
58	D₃×D₅	—	245	2	no	—
59	C₅⋊D₅	—	49	1	no	—
60	C₅×D₅	—	21	2	no	—
61	C₇²	D₁₅⋊D₅	1	1	yes	characteristic, complemented, Sylow
62	AGL(1,𝔽₇)	—	350	1	no	—
63	C₅×C₇	—	2	4	no	—
64	C₅×C₇	—	2	2	no	—
65	C₅×C₇	—	3	2	no	—
66	C₅×C₇	—	3	2	no	—
67	C₅×C₇	—	6	4	no	—
68	D₁₅	—	49	2	no	—
69	C₃×D₅	—	490	2	no	—
70	C₃×D₅	—	245	2	no	—
71	C₅×D₃	—	245	2	no	—
72	C₂×D₇	—	525	2	no	—
73	C₅²	#64418⋊C₂	1	1	yes	characteristic, complemented, Sylow
74	ΓL(1,𝔽₈)	—	14	1	no	—
75	C₂×D₅	—	735	2	no	—
76	C₃×C₅	—	49	2	no	—
77	C₃×C₅	—	98	2	no	—
78	C₂×C₇	—	105	2	no	—
79	D₇	—	525	2	no	—
80	D₇	—	15	2	no	—
81	D₇	—	350	1	no	—
82	C₂×D₃	—	1225	1	no	Hall
83	C₂×C₅	—	105	2	no	—
84	D₅	—	21	2	no	—
85	D₅	—	490	2	no	—
86	D₅	—	245	2	no	—
87	C₇	—	2	1	no	—
88	C₇	—	3	2	no	—
89	C₂×C₃	—	1225	1	no	—
90	D₃	—	245	2	no	—
91	C₅	—	2	2	no	—
92	C₅	#64418⋊D₅	1	2	yes	complemented
93	C₂²	—	3675	1	no	Sylow
94	C₃	—	49	1	no	Sylow
95	C₂	—	105	2	no	—
96	C₂	—	1225	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, cosocle, hypocenter	C₅²×#3862	C₂²
Fitting, socle	C₅²×C₇²	C₂×D₃
LCS	G ▷ C₅²×#3862
UCS	C₁

Complements

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

N	H ≅ G/N	type
1. C₅⋊(C₅⋊#64425)	C₂	semidirect
2. C₅×C₅⋊#64418	C₂	semidirect
4. C₅²×#3862	C₂²	semidirect
6. C₅⋊(C₅⋊(C₇⋊D₇))	D₃	semidirect
10. C₅⋊#64418	D₅	semidirect
12. C₅²×C₇²	C₂×D₃	semidirect
16. C₅×#3862	C₂×D₅	semidirect
32. C₅×C₇²	D₃×D₅	semidirect
40. #3862	D₅²	semidirect
61. C₇²	D₁₅⋊D₅	semidirect
73. C₅²	#64418⋊C₂	semidirect
92. C₅	#64418⋊D₅	semidirect

Maximal subgroups

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

subgroup	index
C₅⋊(C₅⋊#64425)	2
C₅×C₅⋊#64418	2
D₃₅²	3
#64418⋊D₅	5
D₁₅⋊D₅	49