Subgroups — C₅×#2545 #10616140

Order 640.

576 conjugacy classes of subgroups.

The subgroup lattice has 232 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#124	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#124	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#396	C₂	1	4	yes	maximal, complemented
6	C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#390	C₂	1	1	yes	maximal, characteristic
8	C₅×#388	C₂	1	4	yes	maximal, complemented
9	C₄×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
11	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₅×C₄⋊Q₈	—	2	4	no	—
14	C₅×#124	—	2	4	no	—
15	C₅×#124	—	2	4	no	—
16	C₄×C₅×Q₈	—	2	4	no	—
17	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
19	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
23	C₂×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
24	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
26	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂²	1	2	yes	—
28	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
29	C₅×#95	C₂²	1	2	yes	—
30	C₅×#95	C₂²	1	4	yes	—
31	#2545	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₅	C₂³	1	2	yes	—
33	C₂×C₅×Q₈	—	2	2	no	—
34	C₂×C₅×Q₈	—	2	2	no	—
35	C₂×C₅×Q₈	—	2	2	no	—
36	C₂×C₅×Q₈	—	2	2	no	—
37	C₂×C₅×Q₈	—	2	2	no	—
38	C₂×C₅×Q₈	D₄	1	4	yes	—
39	C₂×C₅×D₄	—	2	2	no	—
40	C₂×C₅×D₄	—	2	2	no	—
41	C₂×C₅×D₄	—	2	2	no	—
42	C₂²×C₄×C₅	C₂³	1	2	yes	—
43	C₂²×C₄×C₅	C₂³	1	2	yes	—
44	C₂²×C₄×C₅	C₂³	1	2	yes	—
45	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅	C₂³	1	2	yes	—
48	C₂²×C₄×C₅	C₂³	1	2	yes	—
49	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₅×C₄⋊C₄	—	2	4	no	—
51	C₅×C₄⋊C₄	—	2	2	no	—
52	C₅×C₄⋊C₄	—	2	8	no	—
53	C₅×C₄⋊C₄	—	2	4	no	—
54	C₅×C₂²⋊C₄	—	2	8	no	—
55	C₅×C₂²⋊C₄	—	2	4	no	—
56	C₅×C₂²⋊C₄	—	2	4	no	—
57	C₅×C₂²⋊C₄	—	2	8	no	—
58	C₄²×C₅	—	2	4	no	—
59	C₄²×C₅	—	2	2	no	—
60	C₄²×C₅	—	2	2	no	—
61	C₄²×C₅	D₄	1	4	yes	—
62	C₂×C₄⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
63	C₂×#124	C₂×C₅	1	1	yes	characteristic
64	C₂×#124	C₂×C₅	1	1	yes	characteristic
65	C₂×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
66	#396	C₂×C₅	1	4	yes	complemented
67	#391	C₂×C₅	1	1	yes	characteristic, complemented
68	#390	C₂×C₅	1	1	yes	characteristic
69	#388	C₂×C₅	1	4	yes	complemented
70	C₄×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂³×C₅	—	2	2	no	—
72	C₂³×C₅	—	2	2	no	—
73	C₂³×C₅	—	2	4	no	—
74	C₂³×C₅	—	2	2	no	—
75	C₂³×C₅	—	2	4	no	—
76	C₂³×C₅	C₂⁴	1	1	yes	characteristic
77	C₅×Q₈	—	2	8	no	—
78	C₅×Q₈	—	4	4	no	—
79	C₅×D₄	—	4	4	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	2	no	—
84	C₂×C₄×C₅	—	2	2	no	—
85	C₂×C₄×C₅	—	2	2	no	—
86	C₂×C₄×C₅	—	2	2	no	—
87	C₂×C₄×C₅	—	2	2	no	—
88	C₂×C₄×C₅	—	2	2	no	—
89	C₂×C₄×C₅	—	2	2	no	—
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	2	no	—
93	C₂×C₄×C₅	—	2	2	no	—
94	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
96	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
97	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
98	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₂×C₄×C₅	#41	1	4	yes	—
100	C₂×C₄×C₅	#41	1	2	yes	—
101	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
102	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
103	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
104	C₄⋊Q₈	—	2	4	no	—
105	#124	—	2	4	no	—
106	#124	—	2	4	no	—
107	C₄×Q₈	—	2	4	no	—
108	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
109	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
110	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
111	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
112	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
113	C₂×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
114	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
115	C₂×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
116	C₂×C₄²	C₂²×C₅	1	2	yes	—
117	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
119	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
120	#95	C₂²×C₅	1	2	yes	—
121	#95	C₂²×C₅	1	4	yes	—
122	C₂²×C₅	—	4	2	no	—
123	C₂²×C₅	—	4	2	no	—
124	C₂²×C₅	—	4	2	no	—
125	C₂²×C₅	—	4	4	no	—
126	C₂²×C₅	—	4	4	no	—
127	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
128	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
129	C₂²×C₅	C₂×#41	1	1	yes	characteristic
130	C₂²×C₅	C₂×#41	1	2	yes	—
131	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
132	C₄×C₅	—	2	4	no	—
133	C₄×C₅	—	2	2	no	—
134	C₄×C₅	—	2	2	no	—
135	C₄×C₅	—	2	4	no	—
136	C₄×C₅	—	4	2	no	—
137	C₄×C₅	—	4	2	no	—
138	C₄×C₅	—	4	2	no	—
139	C₄×C₅	#121	1	4	yes	—
140	C₂⁴	C₂³×C₅	1	2	yes	—
141	C₂×Q₈	—	2	2	no	—
142	C₂×Q₈	—	2	2	no	—
143	C₂×Q₈	—	2	2	no	—
144	C₂×Q₈	—	2	2	no	—
145	C₂×Q₈	—	2	2	no	—
146	C₂×Q₈	C₅×D₄	1	4	yes	—
147	C₂×D₄	—	2	2	no	—
148	C₂×D₄	—	2	2	no	—
149	C₂×D₄	—	2	2	no	—
150	C₂²×C₄	C₂³×C₅	1	2	yes	—
151	C₂²×C₄	C₂³×C₅	1	2	yes	—
152	C₂²×C₄	C₂³×C₅	1	2	yes	—
153	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
154	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
155	C₂²×C₄	C₂³×C₅	1	2	yes	—
156	C₂²×C₄	C₂³×C₅	1	2	yes	—
157	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
158	C₄⋊C₄	—	2	4	no	—
159	C₄⋊C₄	—	2	2	no	—
160	C₄⋊C₄	—	2	8	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₂²⋊C₄	—	2	8	no	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₂²⋊C₄	—	2	8	no	—
166	C₄²	—	2	4	no	—
167	C₄²	—	2	2	no	—
168	C₄²	—	2	2	no	—
169	C₄²	C₅×D₄	1	4	yes	—
170	C₂×C₅	—	8	2	no	—
171	C₂×C₅	C₂×#121	1	1	yes	characteristic
172	C₂×C₅	#532	1	1	yes	characteristic
173	C₂×C₅	#536	1	1	yes	characteristic
174	C₂×C₅	#548	1	2	yes	—
175	C₂×C₅	#556	1	2	yes	—
176	C₂³	—	2	2	no	—
177	C₂³	—	2	2	no	—
178	C₂³	—	2	4	no	—
179	C₂³	—	2	2	no	—
180	C₂³	—	2	4	no	—
181	C₂³	C₂⁴×C₅	1	1	yes	characteristic
182	Q₈	—	2	8	no	—
183	Q₈	—	4	4	no	—
184	D₄	—	4	4	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	2	2	no	—
188	C₂×C₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	4	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	2	2	no	—
199	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
200	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
201	C₂×C₄	C₂×C₅×D₄	1	4	yes	—
202	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
203	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
204	C₂×C₄	C₅×#41	1	4	yes	—
205	C₂×C₄	C₅×#41	1	2	yes	—
206	C₅	#2545	1	1	yes	characteristic, direct product factor, Sylow
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	—	4	4	no	—
211	C₂²	—	4	4	no	—
212	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
213	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
214	C₂²	C₂×C₅×#41	1	1	yes	characteristic
215	C₂²	C₂×C₅×#41	1	2	yes	—
216	C₂²	C₅×2₋¹⁺⁴	1	2	yes	—
217	C₄	—	2	4	no	—
218	C₄	—	2	2	no	—
219	C₄	—	2	2	no	—
220	C₄	—	2	4	no	—
221	C₄	—	4	2	no	—
222	C₄	—	4	2	no	—
223	C₄	—	4	2	no	—
224	C₄	C₅×#121	1	4	yes	—
225	C₂	—	8	2	no	—
226	C₂	C₂×C₅×#121	1	1	yes	characteristic
227	C₂	C₅×#532	1	1	yes	characteristic
228	C₂	C₅×#536	1	1	yes	characteristic
229	C₂	C₅×#548	1	2	yes	—
230	C₂	C₅×#556	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₅
prime radical	C₂²×C₅×D₄	C₂²
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
1. C₂×C₅×C₄⋊Q₈	C₂	semidirect
4. C₂×C₄×C₅×Q₈	C₂	semidirect
5. C₅×#396	C₂	semidirect
6. C₅×#391	C₂	semidirect
8. C₅×#388	C₂	semidirect
9. C₄×C₅×C₄⋊C₄	C₂	semidirect
31. #2545	C₅	direct (N × H)
62. C₂×C₄⋊Q₈	C₂×C₅	semidirect
65. C₂×C₄×Q₈	C₂×C₅	semidirect
66. #396	C₂×C₅	semidirect
67. #391	C₂×C₅	semidirect
69. #388	C₂×C₅	semidirect
70. C₄×C₄⋊C₄	C₂×C₅	semidirect
206. C₅	#2545	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×C₄⋊Q₈	2
C₂×C₅×#124	2
C₂×C₄×C₅×Q₈	2
C₅×#396	2
C₅×#391	2
C₅×#390	2
C₅×#388	2
C₄×C₅×C₄⋊C₄	2
#2545	5

Subgroups — C5×#2545 #10616140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2545 #10616140