Subgroups — C₅×#2484 #10616079

Order 640.

512 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#117	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₅×#400	C₂	1	2	yes	maximal, complemented
4	C₅×#398	C₂	1	2	yes	maximal, complemented
5	C₅×#387	C₂	1	4	yes	maximal, complemented
6	C₅×#385	C₂	1	2	yes	maximal
7	C₅×#381	C₂	1	1	yes	maximal, characteristic
8	C₅×#381	C₂	1	2	yes	maximal
9	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₅×#117	—	2	4	no	—
13	C₂×C₅×C₄⋊C₄	—	2	4	no	—
14	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
15	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₄²×C₅	C₂²	1	2	yes	—
22	C₅×#95	C₂²	1	2	yes	—
23	C₅×#95	C₂²	1	4	yes	—
24	C₅×#95	C₂²	1	4	yes	—
25	C₅×#95	C₂²	1	2	yes	—
26	C₅×#95	C₂²	1	4	yes	—
27	#2484	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
28	C₂⁴×C₅	C₂³	1	1	yes	characteristic
29	C₂²×C₄×C₅	—	2	2	no	—
30	C₂²×C₄×C₅	—	2	4	no	—
31	C₂²×C₄×C₅	—	2	4	no	—
32	C₂²×C₄×C₅	—	2	2	no	—
33	C₂²×C₄×C₅	—	2	4	no	—
34	C₂²×C₄×C₅	D₄	1	4	yes	—
35	C₂²×C₄×C₅	C₂³	1	2	yes	—
36	C₂²×C₄×C₅	C₂³	1	2	yes	—
37	C₂²×C₄×C₅	C₂³	1	4	yes	—
38	C₂²×C₄×C₅	C₂³	1	2	yes	—
39	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
40	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
41	C₂²×C₄×C₅	C₂³	1	2	yes	—
42	C₅×C₄⋊C₄	—	2	4	no	—
43	C₅×C₄⋊C₄	—	2	4	no	—
44	C₅×C₄⋊C₄	—	2	2	no	—
45	C₅×C₄⋊C₄	—	2	2	no	—
46	C₅×C₄⋊C₄	—	2	8	no	—
47	C₅×C₂²⋊C₄	—	2	4	no	—
48	C₅×C₂²⋊C₄	—	2	4	no	—
49	C₄²×C₅	—	2	4	no	—
50	C₂×#117	C₂×C₅	1	1	yes	characteristic
51	C₂²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic
52	#400	C₂×C₅	1	2	yes	complemented
53	#398	C₂×C₅	1	2	yes	complemented
54	#387	C₂×C₅	1	4	yes	complemented
55	#385	C₂×C₅	1	2	yes	—
56	#381	C₂×C₅	1	1	yes	characteristic
57	#381	C₂×C₅	1	2	yes	—
58	C₂³×C₅	—	2	2	no	—
59	C₂³×C₅	—	2	2	no	—
60	C₂³×C₅	C₂×D₄	1	2	yes	—
61	C₂³×C₅	#41	1	4	yes	—
62	C₂³×C₅	C₂⁴	1	1	yes	characteristic
63	C₂×C₄×C₅	—	2	8	no	—
64	C₂×C₄×C₅	—	2	8	no	—
65	C₂×C₄×C₅	—	2	2	no	—
66	C₂×C₄×C₅	—	2	2	no	—
67	C₂×C₄×C₅	—	2	2	no	—
68	C₂×C₄×C₅	—	2	2	no	—
69	C₂×C₄×C₅	—	2	2	no	—
70	C₂×C₄×C₅	—	2	2	no	—
71	C₂×C₄×C₅	—	2	2	no	—
72	C₂×C₄×C₅	—	2	4	no	—
73	C₂×C₄×C₅	—	2	4	no	—
74	C₂×C₄×C₅	—	2	4	no	—
75	C₂×C₄×C₅	—	2	4	no	—
76	C₂×C₄×C₅	—	2	4	no	—
77	C₂×C₄×C₅	—	2	4	no	—
78	C₂×C₄×C₅	—	4	4	no	—
79	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₅	#41	1	4	yes	—
82	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
83	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
84	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
85	#117	—	2	4	no	—
86	C₂×C₄⋊C₄	—	2	4	no	—
87	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
88	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
89	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
90	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
91	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
92	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
93	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
94	C₂×C₄²	C₂²×C₅	1	2	yes	—
95	#95	C₂²×C₅	1	2	yes	—
96	#95	C₂²×C₅	1	4	yes	—
97	#95	C₂²×C₅	1	4	yes	—
98	#95	C₂²×C₅	1	2	yes	—
99	#95	C₂²×C₅	1	4	yes	—
100	C₂²×C₅	—	2	2	no	—
101	C₂²×C₅	—	2	2	no	—
102	C₂²×C₅	—	2	8	no	—
103	C₂²×C₅	#123	1	4	yes	—
104	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
105	C₂²×C₅	C₂×#41	1	2	yes	—
106	C₂²×C₅	C₂×#41	1	2	yes	—
107	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
108	C₄×C₅	—	2	2	no	—
109	C₄×C₅	—	2	2	no	—
110	C₄×C₅	—	2	4	no	—
111	C₄×C₅	—	4	2	no	—
112	C₄×C₅	—	4	2	no	—
113	C₄×C₅	—	4	2	no	—
114	C₄×C₅	—	4	4	no	—
115	C₂⁴	C₂³×C₅	1	1	yes	characteristic
116	C₂²×C₄	—	2	2	no	—
117	C₂²×C₄	—	2	4	no	—
118	C₂²×C₄	—	2	4	no	—
119	C₂²×C₄	—	2	2	no	—
120	C₂²×C₄	—	2	4	no	—
121	C₂²×C₄	C₅×D₄	1	4	yes	—
122	C₂²×C₄	C₂³×C₅	1	4	yes	—
123	C₂²×C₄	C₂³×C₅	1	2	yes	—
124	C₂²×C₄	C₂³×C₅	1	2	yes	—
125	C₂²×C₄	C₂³×C₅	1	2	yes	—
126	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
127	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
128	C₂²×C₄	C₂³×C₅	1	2	yes	—
129	C₄⋊C₄	—	2	4	no	—
130	C₄⋊C₄	—	2	4	no	—
131	C₄⋊C₄	—	2	2	no	—
132	C₄⋊C₄	—	2	2	no	—
133	C₄⋊C₄	—	2	8	no	—
134	C₂²⋊C₄	—	2	4	no	—
135	C₂²⋊C₄	—	2	4	no	—
136	C₄²	—	2	4	no	—
137	C₂×C₅	—	2	4	no	—
138	C₂×C₅	C₂×#123	1	1	yes	characteristic
139	C₂×C₅	#525	1	1	yes	characteristic
140	C₂×C₅	#536	1	1	yes	characteristic
141	C₂×C₅	#552	1	4	yes	—
142	C₂³	—	2	2	no	—
143	C₂³	—	2	2	no	—
144	C₂³	C₂×C₅×D₄	1	2	yes	—
145	C₂³	C₅×#41	1	4	yes	—
146	C₂³	C₂⁴×C₅	1	1	yes	characteristic
147	C₂×C₄	—	2	8	no	—
148	C₂×C₄	—	2	8	no	—
149	C₂×C₄	—	2	2	no	—
150	C₂×C₄	—	2	2	no	—
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	2	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	2	2	no	—
156	C₂×C₄	—	2	4	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	4	no	—
159	C₂×C₄	—	2	4	no	—
160	C₂×C₄	—	2	4	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	4	4	no	—
163	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
164	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
165	C₂×C₄	C₅×#41	1	4	yes	—
166	C₅	#2484	1	1	yes	characteristic, direct product factor, Sylow
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	8	no	—
170	C₂²	C₅×#123	1	4	yes	—
171	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
172	C₂²	C₂×C₅×#41	1	2	yes	—
173	C₂²	C₂×C₅×#41	1	2	yes	—
174	C₂²	C₅×2₋¹⁺⁴	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	2	2	no	—
177	C₄	—	2	4	no	—
178	C₄	—	4	2	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	—	4	4	no	—
182	C₂	—	2	4	no	—
183	C₂	C₂×C₅×#123	1	1	yes	characteristic
184	C₂	C₅×#525	1	1	yes	characteristic
185	C₂	C₅×#536	1	1	yes	characteristic
186	C₂	C₅×#552	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂⁴×C₅	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₅×#400	C₂	semidirect
4. C₅×#398	C₂	semidirect
5. C₅×#387	C₂	semidirect
27. #2484	C₅	direct (N × H)
52. #400	C₂×C₅	semidirect
53. #398	C₂×C₅	semidirect
54. #387	C₂×C₅	semidirect
166. C₅	#2484	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#117	2
C₂²×C₅×C₄⋊C₄	2
C₅×#400	2
C₅×#398	2
C₅×#387	2
C₅×#385	2
C₅×#381	2
#2484	5

Subgroups — C5×#2484 #10616079

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2484 #10616079