Subgroups — C₅×#41×Dic₅ #13106975

Order 1600.

784 conjugacy classes of subgroups.

The subgroup lattice has 164 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅²×#41	C₂	1	1	yes	maximal, characteristic
2	C₅×Q₈×Dic₅	C₂	1	2	yes	maximal, complemented
3	C₅×D₄×Dic₅	C₂	1	6	yes	maximal, complemented
4	C₅×#4072	C₂	1	3	yes	maximal, complemented
5	C₂×C₄×C₅×Dic₅	C₂	1	3	yes	maximal, complemented
6	C₅²×#41	C₄	1	8	yes	complemented
7	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₅²×D₄	C₂²	1	3	yes	—
9	C₂²×C₄×C₅²	C₂²	1	3	yes	—
10	C₂²×C₅×Dic₅	C₂²	1	6	yes	—
11	C₅×C₂²⋊Dic₅	C₂²	1	6	yes	—
12	C₅×C₄⋊Dic₅	C₂²	1	6	yes	—
13	C₄×C₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₄×C₅×Dic₅	C₂²	1	3	yes	—
15	C₄×C₅×Dic₅	C₂²	1	6	yes	—
16	C₄×C₅×#41	—	5	1	no	maximal
17	#41×Dic₅	C₅	1	1	yes	maximal, characteristic, direct product factor
18	C₂³×C₅²	C₂³	1	3	yes	—
19	C₅²×Q₈	C₂×C₄	1	4	yes	complemented
20	C₅²×D₄	C₂×C₄	1	12	yes	complemented
21	C₂×C₄×C₅²	C₂×C₄	1	12	yes	complemented
22	C₂×C₄×C₅²	C₂³	1	3	yes	—
23	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
24	C₂×C₅×Dic₅	—	2	12	no	—
25	C₂×C₅×Dic₅	C₂³	1	6	yes	—
26	C₂×C₅×Dic₅	C₂³	1	2	yes	—
27	C₂×C₅×#41	—	2	2	no	—
28	C₂×C₅×#41	D₅	1	1	yes	characteristic
29	C₂×C₅×#41	C₂×C₅	1	1	yes	characteristic
30	C₄×C₅×Q₈	—	5	2	no	—
31	C₄×C₅×D₄	—	5	6	no	—
32	C₅×#117	—	5	3	no	—
33	C₂×C₄²×C₅	—	5	3	no	—
34	Q₈×Dic₅	C₂×C₅	1	2	yes	complemented
35	D₄×Dic₅	C₂×C₅	1	6	yes	complemented
36	#4072	C₂×C₅	1	3	yes	complemented
37	C₂×C₄×Dic₅	C₂×C₅	1	3	yes	complemented
38	C₂²×C₅²	—	2	6	no	—
39	C₂²×C₅²	C₂²×C₄	1	6	yes	—
40	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
41	C₄×C₅²	C₂²×C₄	1	2	yes	—
42	C₄×C₅²	C₂²×C₄	1	6	yes	—
43	C₅×Dic₅	—	2	6	no	—
44	C₅×Dic₅	#41	1	4	yes	direct product factor
45	C₅×#41	—	2	16	no	—
46	C₅×#41	Dic₅	1	8	yes	direct product factor
47	C₅×#41	C₄×C₅	1	8	yes	complemented
48	C₂×C₅×Q₈	—	2	2	no	—
49	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic
50	C₂×C₅×Q₈	C₂²×C₅	1	1	yes	characteristic
51	C₂×C₅×D₄	—	2	6	no	—
52	C₂×C₅×D₄	C₂×D₅	1	3	yes	—
53	C₂×C₅×D₄	C₂²×C₅	1	3	yes	—
54	C₂²×C₄×C₅	—	2	6	no	—
55	C₂²×C₄×C₅	—	5	6	no	—
56	C₂²×C₄×C₅	C₂×D₅	1	3	yes	—
57	C₂²×C₄×C₅	C₂²×C₅	1	3	yes	—
58	C₂²×Dic₅	C₂²×C₅	1	6	yes	—
59	C₅×C₄⋊C₄	—	5	6	no	—
60	C₅×C₂²⋊C₄	—	5	6	no	—
61	C₄²×C₅	—	5	6	no	—
62	C₄²×C₅	—	5	1	no	—
63	C₄²×C₅	—	5	3	no	—
64	C₂²⋊Dic₅	C₂²×C₅	1	6	yes	—
65	C₄⋊Dic₅	C₂²×C₅	1	6	yes	—
66	C₄×Dic₅	C₂²×C₅	1	1	yes	characteristic
67	C₄×Dic₅	C₂²×C₅	1	3	yes	—
68	C₄×Dic₅	C₂²×C₅	1	6	yes	—
69	C₄×#41	—	5	1	no	Sylow
70	C₂×C₅²	—	2	6	no	—
71	C₂×C₅²	C₂³×C₄	1	1	yes	characteristic
72	C₂×C₅²	C₂×#41	1	2	yes	—
73	C₂³×C₅	—	2	6	no	—
74	C₂³×C₅	C₂²×D₅	1	3	yes	—
75	C₂³×C₅	C₂³×C₅	1	3	yes	—
76	C₅×Q₈	—	2	8	no	—
77	C₅×Q₈	C₂×Dic₅	1	4	yes	complemented
78	C₅×Q₈	C₂×C₄×C₅	1	4	yes	complemented
79	C₅×D₄	—	2	24	no	—
80	C₅×D₄	C₂×Dic₅	1	12	yes	complemented
81	C₅×D₄	C₂×C₄×C₅	1	12	yes	complemented
82	C₂×C₄×C₅	—	2	24	no	—
83	C₂×C₄×C₅	—	2	2	no	—
84	C₂×C₄×C₅	—	2	6	no	—
85	C₂×C₄×C₅	—	5	2	no	—
86	C₂×C₄×C₅	—	5	6	no	—
87	C₂×C₄×C₅	—	10	12	no	—
88	C₂×C₄×C₅	C₂×Dic₅	1	12	yes	complemented
89	C₂×C₄×C₅	C₂×C₄×C₅	1	12	yes	complemented
90	C₂×C₄×C₅	C₂²×D₅	1	3	yes	—
91	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
92	C₂×C₄×C₅	C₂³×C₅	1	1	yes	characteristic
93	C₂×C₄×C₅	C₂³×C₅	1	3	yes	—
94	C₂×Dic₅	—	2	12	no	—
95	C₂×Dic₅	C₂³×C₅	1	6	yes	—
96	C₂×Dic₅	C₂³×C₅	1	2	yes	—
97	C₂×#41	C₅×D₅	1	1	yes	characteristic
98	C₄×Q₈	—	5	2	no	—
99	C₄×D₄	—	5	6	no	—
100	#117	—	5	3	no	—
101	C₂×C₄²	—	5	3	no	—
102	C₅²	C₄×#41	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₅	—	2	6	no	—
104	C₂²×C₅	—	2	6	no	—
105	C₂²×C₅	—	2	12	no	—
106	C₂²×C₅	—	2	2	no	—
107	C₂²×C₅	—	4	12	no	—
108	C₂²×C₅	C₂²×Dic₅	1	6	yes	—
109	C₂²×C₅	C₂²×C₄×C₅	1	6	yes	—
110	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
111	C₂²×C₅	C₂⁴×C₅	1	1	yes	characteristic
112	C₄×C₅	—	2	4	no	—
113	C₄×C₅	—	2	12	no	—
114	C₄×C₅	—	5	4	no	—
115	C₄×C₅	—	10	6	no	—
116	C₄×C₅	C₂²×Dic₅	1	2	yes	—
117	C₄×C₅	C₂²×Dic₅	1	6	yes	—
118	C₄×C₅	C₂²×C₄×C₅	1	2	yes	—
119	C₄×C₅	C₂²×C₄×C₅	1	6	yes	—
120	Dic₅	—	2	6	no	—
121	Dic₅	C₅×#41	1	4	yes	direct product factor
122	#41	C₅×Dic₅	1	8	yes	direct product factor
123	C₂×Q₈	C₂×C₅×D₅	1	1	yes	characteristic
124	C₂×D₄	C₂×C₅×D₅	1	3	yes	—
125	C₂²×C₄	—	5	6	no	—
126	C₂²×C₄	C₂×C₅×D₅	1	3	yes	—
127	C₄⋊C₄	—	5	6	no	—
128	C₂²⋊C₄	—	5	6	no	—
129	C₄²	—	5	1	no	—
130	C₄²	—	5	3	no	—
131	C₄²	—	5	6	no	—
132	C₂×C₅	—	2	6	no	—
133	C₂×C₅	—	2	6	no	—
134	C₂×C₅	—	2	4	no	—
135	C₂×C₅	—	2	2	no	—
136	C₂×C₅	—	4	12	no	—
137	C₂×C₅	D₅×#41	1	2	yes	—
138	C₂×C₅	C₂³×Dic₅	1	1	yes	characteristic
139	C₂×C₅	C₂³×C₄×C₅	1	1	yes	characteristic
140	C₂×C₅	C₂×C₅×#41	1	2	yes	—
141	C₂³	C₂²×C₅×D₅	1	3	yes	—
142	Q₈	C₂×C₅×Dic₅	1	4	yes	complemented
143	D₄	C₂×C₅×Dic₅	1	12	yes	complemented
144	C₂×C₄	—	5	2	no	—
145	C₂×C₄	—	5	6	no	—
146	C₂×C₄	—	10	12	no	—
147	C₂×C₄	C₂×C₅×Dic₅	1	12	yes	complemented
148	C₂×C₄	C₂²×C₅×D₅	1	1	yes	characteristic
149	C₂×C₄	C₂²×C₅×D₅	1	3	yes	—
150	C₅	—	2	2	no	—
151	C₅	#41×Dic₅	1	1	yes	characteristic, direct product factor
152	C₅	C₄×C₅×#41	1	1	yes	characteristic, complemented
153	C₂²	—	2	6	no	—
154	C₂²	C₂²×C₅×Dic₅	1	6	yes	—
155	C₂²	C₂³×C₅×D₅	1	1	yes	characteristic
156	C₄	—	5	4	no	—
157	C₄	—	10	6	no	—
158	C₄	C₂²×C₅×Dic₅	1	2	yes	—
159	C₄	C₂²×C₅×Dic₅	1	6	yes	—
160	C₂	—	2	6	no	—
161	C₂	C₅×D₅×#41	1	2	yes	—
162	C₂	C₂³×C₅×Dic₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₅
commutator	C₂×C₅	C₂³×C₄×C₅
Frattini	C₂²	C₂³×C₅×D₅
Fitting, prime radical	C₂×C₅²×#41	C₂
socle	C₂²×C₅²	C₂⁴
cosocle	C₂²×C₅	C₂⁴×C₅
hypercenter	C₂×C₅×#41	D₅
hypocenter	C₅	C₄×C₅×#41
LCS	G ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#41

Complements

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

N	H ≅ G/N	type
2. C₅×Q₈×Dic₅	C₂	semidirect
3. C₅×D₄×Dic₅	C₂	semidirect
4. C₅×#4072	C₂	semidirect
5. C₂×C₄×C₅×Dic₅	C₂	semidirect
6. C₅²×#41	C₄	semidirect
17. #41×Dic₅	C₅	direct (N × H)
19. C₅²×Q₈	C₂×C₄	semidirect
20. C₅²×D₄	C₂×C₄	semidirect
21. C₂×C₄×C₅²	C₂×C₄	semidirect
34. Q₈×Dic₅	C₂×C₅	semidirect
35. D₄×Dic₅	C₂×C₅	semidirect
36. #4072	C₂×C₅	semidirect
37. C₂×C₄×Dic₅	C₂×C₅	semidirect
44. C₅×Dic₅	#41	direct (N × H)
46. C₅×#41	Dic₅	direct (N × H)
47. C₅×#41	C₄×C₅	semidirect
77. C₅×Q₈	C₂×Dic₅	semidirect
78. C₅×Q₈	C₂×C₄×C₅	semidirect
80. C₅×D₄	C₂×Dic₅	semidirect
81. C₅×D₄	C₂×C₄×C₅	semidirect
88. C₂×C₄×C₅	C₂×Dic₅	semidirect
89. C₂×C₄×C₅	C₂×C₄×C₅	semidirect
102. C₅²	C₄×#41	semidirect
121. Dic₅	C₅×#41	direct (N × H)
122. #41	C₅×Dic₅	direct (N × H)
142. Q₈	C₂×C₅×Dic₅	semidirect
143. D₄	C₂×C₅×Dic₅	semidirect
147. C₂×C₄	C₂×C₅×Dic₅	semidirect
151. C₅	#41×Dic₅	direct (N × H)
152. C₅	C₄×C₅×#41	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅²×#41	2
C₅×Q₈×Dic₅	2
C₅×D₄×Dic₅	2
C₅×#4072	2
C₂×C₄×C₅×Dic₅	2
C₄×C₅×#41	5
#41×Dic₅	5

Subgroups — C5×#41×Dic5 #13106975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#41×Dic₅ #13106975