Subgroups — C₅×#2409 #10616004

Order 640.

840 conjugacy classes of subgroups.

The subgroup lattice has 254 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×C₂²⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
4	C₅×#395	C₂	1	2	yes	maximal, complemented
5	C₅×#393	C₂	1	2	yes	maximal, complemented
6	C₅×#391	C₂	1	2	yes	maximal, complemented
7	C₅×#386	C₂	1	2	yes	maximal
8	C₅×#385	C₂	1	2	yes	maximal, complemented
9	C₅×#380	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×Q₈	—	2	8	no	—
11	C₂²×C₅×Q₈	—	2	4	no	—
12	C₂²×C₅×Q₈	C₂²	1	2	yes	—
13	C₂²×C₅×Q₈	C₂²	1	2	yes	—
14	C₂²×C₅×D₄	C₂²	1	1	yes	characteristic
15	C₂³×C₄×C₅	C₂²	1	2	yes	—
16	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₅	C₂²	1	1	yes	characteristic
18	C₅×C₂²⋊Q₈	—	2	8	no	—
19	C₄×C₅×D₄	—	2	4	no	—
20	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₂×C₅×C₄⋊C₄	C₂²	1	4	yes	—
22	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
26	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
28	C₅×#95	C₂²	1	2	yes	—
29	C₅×#95	C₂²	1	4	yes	—
30	C₅×#95	C₂²	1	4	yes	—
31	#2409	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂⁴×C₅	C₂³	1	1	yes	characteristic
33	C₂⁴×C₅	C₂³	1	1	yes	characteristic
34	C₂×C₅×Q₈	—	2	32	no	—
35	C₂×C₅×Q₈	—	2	8	no	—
36	C₂×C₅×Q₈	—	2	4	no	—
37	C₂×C₅×Q₈	—	2	4	no	—
38	C₂×C₅×Q₈	—	2	4	no	—
39	C₂×C₅×Q₈	D₄	1	8	yes	complemented
40	C₂×C₅×D₄	—	2	2	no	—
41	C₂×C₅×D₄	—	2	2	no	—
42	C₂×C₅×D₄	Q₈	1	4	yes	complemented
43	C₂²×C₄×C₅	—	2	4	no	—
44	C₂²×C₄×C₅	—	2	8	no	—
45	C₂²×C₄×C₅	—	2	4	no	—
46	C₂²×C₄×C₅	—	2	2	no	—
47	C₂²×C₄×C₅	—	2	4	no	—
48	C₂²×C₄×C₅	D₄	1	4	yes	—
49	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
50	C₂²×C₄×C₅	C₂³	1	4	yes	—
51	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
52	C₂²×C₄×C₅	C₂³	1	4	yes	—
53	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
56	C₅×C₄⋊C₄	—	2	2	no	—
57	C₅×C₄⋊C₄	—	2	4	no	—
58	C₅×C₄⋊C₄	—	2	8	no	—
59	C₅×C₂²⋊C₄	—	2	2	no	—
60	C₅×C₂²⋊C₄	—	2	4	no	—
61	C₅×C₂²⋊C₄	—	2	2	no	—
62	C₅×C₂²⋊C₄	—	2	8	no	—
63	C₄²×C₅	—	2	2	no	—
64	C₂³×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
65	C₂×C₂²⋊Q₈	C₂×C₅	1	2	yes	complemented
66	C₂×C₄×D₄	C₂×C₅	1	1	yes	characteristic
67	#395	C₂×C₅	1	2	yes	complemented
68	#393	C₂×C₅	1	2	yes	complemented
69	#391	C₂×C₅	1	2	yes	complemented
70	#386	C₂×C₅	1	2	yes	—
71	#385	C₂×C₅	1	2	yes	complemented
72	#380	C₂×C₅	1	1	yes	characteristic, complemented
73	C₂³×C₅	—	2	2	no	—
74	C₂³×C₅	—	2	2	no	—
75	C₂³×C₅	—	2	2	no	—
76	C₂³×C₅	—	2	1	no	—
77	C₂³×C₅	—	2	1	no	—
78	C₂³×C₅	—	2	2	no	—
79	C₂³×C₅	C₂×D₄	1	2	yes	—
80	C₂³×C₅	C₂×Q₈	1	2	yes	—
81	C₂³×C₅	C₂×Q₈	1	2	yes	complemented
82	C₂³×C₅	#41	1	2	yes	—
83	C₂³×C₅	C₂⁴	1	1	yes	characteristic
84	C₅×Q₈	—	2	16	no	—
85	C₅×Q₈	—	2	16	no	—
86	C₅×D₄	—	2	8	no	—
87	C₂×C₄×C₅	—	2	16	no	—
88	C₂×C₄×C₅	—	2	1	no	—
89	C₂×C₄×C₅	—	2	4	no	—
90	C₂×C₄×C₅	—	2	4	no	—
91	C₂×C₄×C₅	—	2	2	no	—
92	C₂×C₄×C₅	—	2	1	no	—
93	C₂×C₄×C₅	—	2	8	no	—
94	C₂×C₄×C₅	—	2	1	no	—
95	C₂×C₄×C₅	—	2	4	no	—
96	C₂×C₄×C₅	—	2	1	no	—
97	C₂×C₄×C₅	—	2	2	no	—
98	C₂×C₄×C₅	—	2	4	no	—
99	C₂×C₄×C₅	—	2	2	no	—
100	C₂×C₄×C₅	—	2	4	no	—
101	C₂×C₄×C₅	—	4	4	no	—
102	C₂×C₄×C₅	C₂×D₄	1	8	yes	—
103	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₅	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₅	C₂×D₄	1	4	yes	—
106	C₂×C₄×C₅	C₂×Q₈	1	2	yes	complemented
107	C₂²×Q₈	—	2	8	no	—
108	C₂²×Q₈	—	2	4	no	—
109	C₂²×Q₈	C₂²×C₅	1	2	yes	—
110	C₂²×Q₈	C₂²×C₅	1	2	yes	—
111	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
112	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
113	C₂³×C₄	C₂²×C₅	1	2	yes	—
114	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
115	C₂²⋊Q₈	—	2	8	no	—
116	C₄×D₄	—	2	4	no	—
117	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
118	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
119	C₂×C₄⋊C₄	C₂²×C₅	1	4	yes	—
120	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
123	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
124	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
125	#95	C₂²×C₅	1	2	yes	—
126	#95	C₂²×C₅	1	4	yes	—
127	#95	C₂²×C₅	1	4	yes	—
128	C₂²×C₅	—	2	2	no	—
129	C₂²×C₅	—	2	2	no	—
130	C₂²×C₅	—	2	4	no	—
131	C₂²×C₅	—	2	4	no	—
132	C₂²×C₅	—	2	2	no	—
133	C₂²×C₅	—	2	4	no	—
134	C₂²×C₅	—	4	1	no	—
135	C₂²×C₅	—	4	1	no	—
136	C₂²×C₅	—	4	2	no	—
137	C₂²×C₅	C₂²⋊Q₈	1	4	yes	complemented
138	C₂²×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₂²×C₅	C₂²×D₄	1	2	yes	—
140	C₂²×C₅	C₂²×Q₈	1	1	yes	characteristic
141	C₂²×C₅	C₂×#41	1	1	yes	characteristic
142	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
143	C₄×C₅	—	2	8	no	—
144	C₄×C₅	—	2	2	no	—
145	C₄×C₅	—	2	2	no	—
146	C₄×C₅	—	4	1	no	—
147	C₄×C₅	—	4	1	no	—
148	C₄×C₅	—	4	4	no	—
149	C₄×C₅	#120	1	4	yes	—
150	C₂⁴	C₂³×C₅	1	1	yes	characteristic
151	C₂⁴	C₂³×C₅	1	1	yes	characteristic
152	C₂×Q₈	—	2	32	no	—
153	C₂×Q₈	—	2	8	no	—
154	C₂×Q₈	—	2	4	no	—
155	C₂×Q₈	—	2	4	no	—
156	C₂×Q₈	—	2	4	no	—
157	C₂×Q₈	C₅×D₄	1	8	yes	complemented
158	C₂×D₄	—	2	2	no	—
159	C₂×D₄	—	2	2	no	—
160	C₂×D₄	C₅×Q₈	1	4	yes	complemented
161	C₂²×C₄	—	2	4	no	—
162	C₂²×C₄	—	2	8	no	—
163	C₂²×C₄	—	2	4	no	—
164	C₂²×C₄	—	2	2	no	—
165	C₂²×C₄	—	2	4	no	—
166	C₂²×C₄	C₅×D₄	1	4	yes	—
167	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
168	C₂²×C₄	C₂³×C₅	1	4	yes	—
169	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
170	C₂²×C₄	C₂³×C₅	1	4	yes	—
171	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
172	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
173	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
174	C₄⋊C₄	—	2	2	no	—
175	C₄⋊C₄	—	2	4	no	—
176	C₄⋊C₄	—	2	8	no	—
177	C₂²⋊C₄	—	2	2	no	—
178	C₂²⋊C₄	—	2	4	no	—
179	C₂²⋊C₄	—	2	2	no	—
180	C₂²⋊C₄	—	2	8	no	—
181	C₄²	—	2	2	no	—
182	C₂×C₅	—	2	4	no	—
183	C₂×C₅	—	4	2	no	—
184	C₂×C₅	C₂×#120	1	1	yes	characteristic
185	C₂×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
186	C₂×C₅	#536	1	1	yes	characteristic
187	C₂×C₅	#548	1	2	yes	—
188	C₂×C₅	D₄×Q₈	1	2	yes	—
189	C₂³	—	2	2	no	—
190	C₂³	—	2	2	no	—
191	C₂³	—	2	2	no	—
192	C₂³	—	2	1	no	—
193	C₂³	—	2	1	no	—
194	C₂³	—	2	2	no	—
195	C₂³	C₂×C₅×D₄	1	2	yes	—
196	C₂³	C₂×C₅×Q₈	1	2	yes	—
197	C₂³	C₂×C₅×Q₈	1	2	yes	complemented
198	C₂³	C₅×#41	1	2	yes	—
199	C₂³	C₂⁴×C₅	1	1	yes	characteristic
200	Q₈	—	2	16	no	—
201	Q₈	—	2	16	no	—
202	D₄	—	2	8	no	—
203	C₂×C₄	—	2	16	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	8	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	4	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	2	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	4	no	—
217	C₂×C₄	—	4	4	no	—
218	C₂×C₄	C₂×C₅×D₄	1	8	yes	—
219	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
220	C₂×C₄	C₂×C₅×D₄	1	2	yes	—
221	C₂×C₄	C₂×C₅×D₄	1	4	yes	—
222	C₂×C₄	C₂×C₅×Q₈	1	2	yes	complemented
223	C₅	#2409	1	1	yes	characteristic, direct product factor, Sylow
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	2	4	no	—
227	C₂²	—	2	4	no	—
228	C₂²	—	2	2	no	—
229	C₂²	—	2	4	no	—
230	C₂²	—	4	1	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	4	2	no	—
233	C₂²	C₅×C₂²⋊Q₈	1	4	yes	complemented
234	C₂²	C₂²×C₅×D₄	1	1	yes	characteristic
235	C₂²	C₂²×C₅×D₄	1	2	yes	—
236	C₂²	C₂²×C₅×Q₈	1	1	yes	characteristic
237	C₂²	C₂×C₅×#41	1	1	yes	characteristic
238	C₂²	C₅×2₋¹⁺⁴	1	2	yes	—
239	C₄	—	2	8	no	—
240	C₄	—	2	2	no	—
241	C₄	—	2	2	no	—
242	C₄	—	4	1	no	—
243	C₄	—	4	1	no	—
244	C₄	—	4	4	no	—
245	C₄	C₅×#120	1	4	yes	—
246	C₂	—	2	4	no	—
247	C₂	—	4	2	no	—
248	C₂	C₂×C₅×#120	1	1	yes	characteristic
249	C₂	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
250	C₂	C₅×#536	1	1	yes	characteristic
251	C₂	C₅×#548	1	2	yes	—
252	C₂	C₅×D₄×Q₈	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. C₂³×C₅×Q₈	C₂	semidirect
2. C₂×C₅×C₂²⋊Q₈	C₂	semidirect
4. C₅×#395	C₂	semidirect
5. C₅×#393	C₂	semidirect
6. C₅×#391	C₂	semidirect
8. C₅×#385	C₂	semidirect
9. C₅×#380	C₂	semidirect
31. #2409	C₅	direct (N × H)
39. C₂×C₅×Q₈	D₄	semidirect
42. C₂×C₅×D₄	Q₈	semidirect
64. C₂³×Q₈	C₂×C₅	semidirect
65. C₂×C₂²⋊Q₈	C₂×C₅	semidirect
67. #395	C₂×C₅	semidirect
68. #393	C₂×C₅	semidirect
69. #391	C₂×C₅	semidirect
71. #385	C₂×C₅	semidirect
72. #380	C₂×C₅	semidirect
81. C₂³×C₅	C₂×Q₈	semidirect
106. C₂×C₄×C₅	C₂×Q₈	semidirect
137. C₂²×C₅	C₂²⋊Q₈	semidirect
157. C₂×Q₈	C₅×D₄	semidirect
160. C₂×D₄	C₅×Q₈	semidirect
197. C₂³	C₂×C₅×Q₈	semidirect
222. C₂×C₄	C₂×C₅×Q₈	semidirect
223. C₅	#2409	direct (N × H)
233. C₂²	C₅×C₂²⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₅×Q₈	2
C₂×C₅×C₂²⋊Q₈	2
C₂×C₄×C₅×D₄	2
C₅×#395	2
C₅×#393	2
C₅×#391	2
C₅×#386	2
C₅×#385	2
C₅×#380	2
#2409	5

Subgroups — C5×#2409 #10616004

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2409 #10616004