Subgroups — C₅×#2499 #10616094

Order 640.

484 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#117	C₂	1	1	yes	maximal, characteristic
3	C₅×#397	C₂	1	4	yes	maximal
4	C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#387	C₂	1	4	yes	maximal, complemented
7	C₅×#384	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#382	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
10	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
12	C₅×C₂²⋊Q₈	—	2	4	no	—
13	C₅×#117	—	2	4	no	—
14	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
17	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
19	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
23	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₂×C₄²×C₅	C₂²	1	2	yes	—
25	C₅×#95	C₂²	1	2	yes	—
26	C₅×#95	C₂²	1	4	yes	—
27	C₅×#95	C₂²	1	4	yes	—
28	C₅×#95	C₂²	1	4	yes	—
29	#2499	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂⁴×C₅	C₂³	1	1	yes	characteristic
31	C₂×C₅×Q₈	—	2	2	no	—
32	C₂×C₅×Q₈	—	2	2	no	—
33	C₂×C₅×Q₈	—	2	2	no	—
34	C₂²×C₄×C₅	—	2	2	no	—
35	C₂²×C₄×C₅	—	2	2	no	—
36	C₂²×C₄×C₅	D₄	1	4	yes	—
37	C₂²×C₄×C₅	C₂³	1	2	yes	—
38	C₂²×C₄×C₅	C₂³	1	2	yes	—
39	C₂²×C₄×C₅	C₂³	1	2	yes	—
40	C₂²×C₄×C₅	C₂³	1	2	yes	—
41	C₂²×C₄×C₅	C₂³	1	2	yes	—
42	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₅	C₂³	1	2	yes	—
44	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
45	C₅×C₄⋊C₄	—	2	2	no	—
46	C₅×C₄⋊C₄	—	2	2	no	—
47	C₅×C₄⋊C₄	—	2	2	no	—
48	C₅×C₄⋊C₄	—	2	4	no	—
49	C₅×C₄⋊C₄	—	2	8	no	—
50	C₅×C₂²⋊C₄	—	2	4	no	—
51	C₅×C₂²⋊C₄	—	2	4	no	—
52	C₅×C₂²⋊C₄	—	2	4	no	—
53	C₄²×C₅	—	2	4	no	—
54	C₄²×C₅	—	2	2	no	—
55	C₄²×C₅	—	2	2	no	—
56	C₂×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic
57	C₂×#117	C₂×C₅	1	1	yes	characteristic
58	#397	C₂×C₅	1	4	yes	—
59	#391	C₂×C₅	1	1	yes	characteristic, complemented
60	#391	C₂×C₅	1	1	yes	characteristic, complemented
61	#387	C₂×C₅	1	4	yes	complemented
62	#384	C₂×C₅	1	1	yes	characteristic, complemented
63	#382	C₂×C₅	1	1	yes	characteristic, complemented
64	C₄×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
65	C₂³×C₅	—	2	2	no	—
66	C₂³×C₅	—	2	1	no	—
67	C₂³×C₅	—	2	1	no	—
68	C₂³×C₅	—	2	2	no	—
69	C₂³×C₅	C₂×D₄	1	2	yes	—
70	C₂³×C₅	C₂⁴	1	1	yes	characteristic
71	C₅×Q₈	—	4	4	no	—
72	C₂×C₄×C₅	—	2	2	no	—
73	C₂×C₄×C₅	—	2	1	no	—
74	C₂×C₄×C₅	—	2	1	no	—
75	C₂×C₄×C₅	—	2	2	no	—
76	C₂×C₄×C₅	—	2	8	no	—
77	C₂×C₄×C₅	—	2	2	no	—
78	C₂×C₄×C₅	—	2	2	no	—
79	C₂×C₄×C₅	—	2	2	no	—
80	C₂×C₄×C₅	—	2	2	no	—
81	C₂×C₄×C₅	—	2	2	no	—
82	C₂×C₄×C₅	—	2	2	no	—
83	C₂×C₄×C₅	—	2	4	no	—
84	C₂×C₄×C₅	—	2	4	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	4	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
91	C₂×C₄×C₅	#41	1	4	yes	—
92	C₂×C₄×C₅	#41	1	2	yes	—
93	C₂×C₄×C₅	#41	1	2	yes	—
94	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
95	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
96	C₂²⋊Q₈	—	2	4	no	—
97	#117	—	2	4	no	—
98	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
99	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
100	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
101	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
102	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
103	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
104	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
105	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
106	C₂×C₄²	C₂²×C₅	1	2	yes	—
107	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
108	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
109	#95	C₂²×C₅	1	2	yes	—
110	#95	C₂²×C₅	1	4	yes	—
111	#95	C₂²×C₅	1	4	yes	—
112	#95	C₂²×C₅	1	4	yes	—
113	C₂²×C₅	—	2	2	no	—
114	C₂²×C₅	—	2	2	no	—
115	C₂²×C₅	—	2	2	no	—
116	C₂²×C₅	—	4	2	no	—
117	C₂²×C₅	—	4	2	no	—
118	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
119	C₂²×C₅	C₂×#41	1	1	yes	characteristic
120	C₂²×C₅	C₂×#41	1	1	yes	characteristic
121	C₂²×C₅	C₂×#41	1	2	yes	—
122	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
123	C₄×C₅	—	2	2	no	—
124	C₄×C₅	—	2	4	no	—
125	C₄×C₅	—	4	2	no	—
126	C₄×C₅	—	4	2	no	—
127	C₄×C₅	—	4	2	no	—
128	C₄×C₅	—	4	2	no	—
129	C₄×C₅	—	4	2	no	—
130	C₄×C₅	#123	1	4	yes	—
131	C₂⁴	C₂³×C₅	1	1	yes	characteristic
132	C₂×Q₈	—	2	2	no	—
133	C₂×Q₈	—	2	2	no	—
134	C₂×Q₈	—	2	2	no	—
135	C₂²×C₄	—	2	2	no	—
136	C₂²×C₄	—	2	2	no	—
137	C₂²×C₄	C₅×D₄	1	4	yes	—
138	C₂²×C₄	C₂³×C₅	1	2	yes	—
139	C₂²×C₄	C₂³×C₅	1	2	yes	—
140	C₂²×C₄	C₂³×C₅	1	2	yes	—
141	C₂²×C₄	C₂³×C₅	1	2	yes	—
142	C₂²×C₄	C₂³×C₅	1	2	yes	—
143	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₅	1	2	yes	—
145	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
146	C₄⋊C₄	—	2	2	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	2	no	—
149	C₄⋊C₄	—	2	4	no	—
150	C₄⋊C₄	—	2	8	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₂²⋊C₄	—	2	4	no	—
154	C₄²	—	2	4	no	—
155	C₄²	—	2	2	no	—
156	C₄²	—	2	2	no	—
157	C₂×C₅	—	4	2	no	—
158	C₂×C₅	C₂×#123	1	1	yes	characteristic
159	C₂×C₅	#532	1	1	yes	characteristic
160	C₂×C₅	#536	1	1	yes	characteristic
161	C₂×C₅	#552	1	2	yes	—
162	C₂×C₅	#556	1	2	yes	—
163	C₂³	—	2	2	no	—
164	C₂³	—	2	1	no	—
165	C₂³	—	2	1	no	—
166	C₂³	—	2	2	no	—
167	C₂³	C₂×C₅×D₄	1	2	yes	—
168	C₂³	C₂⁴×C₅	1	1	yes	characteristic
169	Q₈	—	4	4	no	—
170	C₂×C₄	—	2	2	no	—
171	C₂×C₄	—	2	1	no	—
172	C₂×C₄	—	2	1	no	—
173	C₂×C₄	—	2	2	no	—
174	C₂×C₄	—	2	8	no	—
175	C₂×C₄	—	2	2	no	—
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	4	no	—
182	C₂×C₄	—	2	4	no	—
183	C₂×C₄	—	2	2	no	—
184	C₂×C₄	—	2	2	no	—
185	C₂×C₄	—	2	4	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
188	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
189	C₂×C₄	C₅×#41	1	4	yes	—
190	C₂×C₄	C₅×#41	1	2	yes	—
191	C₂×C₄	C₅×#41	1	2	yes	—
192	C₅	#2499	1	1	yes	characteristic, direct product factor, Sylow
193	C₂²	—	2	2	no	—
194	C₂²	—	2	2	no	—
195	C₂²	—	2	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
199	C₂²	C₂×C₅×#41	1	1	yes	characteristic
200	C₂²	C₂×C₅×#41	1	1	yes	characteristic
201	C₂²	C₂×C₅×#41	1	2	yes	—
202	C₂²	C₅×2₋¹⁺⁴	1	2	yes	—
203	C₄	—	2	2	no	—
204	C₄	—	2	4	no	—
205	C₄	—	4	2	no	—
206	C₄	—	4	2	no	—
207	C₄	—	4	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	C₅×#123	1	4	yes	—
211	C₂	—	4	2	no	—
212	C₂	C₂×C₅×#123	1	1	yes	characteristic
213	C₂	C₅×#532	1	1	yes	characteristic
214	C₂	C₅×#536	1	1	yes	characteristic
215	C₂	C₅×#552	1	2	yes	—
216	C₂	C₅×#556	1	2	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
4. C₅×#391	C₂	semidirect
5. C₅×#391	C₂	semidirect
6. C₅×#387	C₂	semidirect
7. C₅×#384	C₂	semidirect
8. C₅×#382	C₂	semidirect
29. #2499	C₅	direct (N × H)
59. #391	C₂×C₅	semidirect
60. #391	C₂×C₅	semidirect
61. #387	C₂×C₅	semidirect
62. #384	C₂×C₅	semidirect
63. #382	C₂×C₅	semidirect
192. C₅	#2499	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×C₂²⋊Q₈	2
C₂×C₅×#117	2
C₅×#397	2
C₅×#391	2
C₅×#387	2
C₅×#384	2
C₅×#382	2
C₄×C₅×C₂²⋊C₄	2
#2499	5

Subgroups — C5×#2499 #10616094

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2499 #10616094