Subgroups — C₂×C₅×#552 #10619255

Order 640.

1124 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#552	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₅×#126	C₂	1	2	yes	maximal, complemented
3	C₂×C₅×#125	C₂	1	2	yes	maximal, complemented
4	C₂×C₅×#125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₅×#123	C₂	1	2	yes	maximal
6	C₂×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×C₅×Q₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄×C₅×D₄	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#117	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₅×#117	C₂	1	2	yes	maximal, complemented
12	C₂²×C₅×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
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×#126	C₂²	1	16	yes	complemented
19	C₅×#125	C₂²	1	16	yes	complemented
20	C₅×#125	C₂²	1	8	yes	complemented
21	C₅×#123	C₂²	1	16	yes	—
22	C₅×C₂²⋊Q₈	C₂²	1	8	yes	complemented
23	C₅×C₂²⋊Q₈	C₂²	1	8	yes	complemented
24	C₄×C₅×Q₈	C₂²	1	8	yes	complemented
25	C₄×C₅×D₄	C₂²	1	8	yes	—
26	C₅×#117	C₂²	1	8	yes	complemented
27	C₅×#117	C₂²	1	16	yes	complemented
28	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
31	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
32	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
33	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
34	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
35	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
36	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
37	C₂×C₅×C₄⋊C₄	C₂²	1	2	yes	—
38	C₂×C₅×C₄⋊C₄	C₂²	1	8	yes	complemented
39	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
40	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
41	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
42	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
43	C₂×C₅×C₂²⋊C₄	C₂²	1	2	yes	—
44	C₂×C₄²×C₅	C₂²	1	2	yes	—
45	C₂×C₄²×C₅	C₂²	1	2	yes	—
46	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
47	C₂×#552	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
48	C₂⁴×C₅	C₂³	1	1	yes	characteristic
49	C₂⁴×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₅×Q₈	—	2	4	no	—
51	C₂×C₅×Q₈	C₂³	1	4	yes	—
52	C₂×C₅×D₄	—	2	4	no	—
53	C₂×C₅×D₄	C₂³	1	4	yes	—
54	C₂²×C₄×C₅	—	2	8	no	—
55	C₂²×C₄×C₅	—	2	4	no	—
56	C₂²×C₄×C₅	—	2	4	no	—
57	C₂²×C₄×C₅	C₂³	1	8	yes	—
58	C₂²×C₄×C₅	C₂³	1	4	yes	—
59	C₂²×C₄×C₅	C₂³	1	4	yes	—
60	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂³	1	2	yes	—
63	C₂²×C₄×C₅	C₂³	1	2	yes	—
64	C₂²×C₄×C₅	C₂³	1	2	yes	—
65	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂³	1	2	yes	—
68	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₅×C₄⋊C₄	C₂³	1	4	yes	—
70	C₅×C₄⋊C₄	C₂³	1	4	yes	—
71	C₅×C₄⋊C₄	C₂³	1	8	yes	—
72	C₅×C₄⋊C₄	C₂³	1	8	yes	—
73	C₅×C₄⋊C₄	C₂³	1	4	yes	—
74	C₅×C₄⋊C₄	C₂³	1	8	yes	—
75	C₅×C₄⋊C₄	C₂³	1	4	yes	—
76	C₅×C₄⋊C₄	C₂³	1	8	yes	—
77	C₅×C₄⋊C₄	C₂³	1	8	yes	—
78	C₅×C₄⋊C₄	C₂³	1	8	yes	—
79	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
80	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
81	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
82	C₅×C₂²⋊C₄	C₂³	1	4	yes	—
83	C₅×C₂²⋊C₄	C₂³	1	8	yes	—
84	C₄²×C₅	C₂³	1	8	yes	—
85	C₄²×C₅	C₂³	1	8	yes	—
86	C₄²×C₅	C₂³	1	4	yes	—
87	#552	C₂×C₅	1	16	yes	direct product factor
88	C₂×#126	C₂×C₅	1	2	yes	complemented
89	C₂×#125	C₂×C₅	1	2	yes	complemented
90	C₂×#125	C₂×C₅	1	1	yes	characteristic, complemented
91	C₂×#123	C₂×C₅	1	2	yes	—
92	C₂×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
93	C₂×C₂²⋊Q₈	C₂×C₅	1	1	yes	characteristic, complemented
94	C₂×C₄×Q₈	C₂×C₅	1	1	yes	characteristic, complemented
95	C₂×C₄×D₄	C₂×C₅	1	1	yes	characteristic
96	C₂×#117	C₂×C₅	1	1	yes	characteristic, complemented
97	C₂×#117	C₂×C₅	1	2	yes	complemented
98	C₂²×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
99	C₂³×C₅	—	2	2	no	—
100	C₂³×C₅	—	2	2	no	—
101	C₂³×C₅	—	2	2	no	—
102	C₂³×C₅	—	2	2	no	—
103	C₂³×C₅	—	2	2	no	—
104	C₂³×C₅	#41	1	4	yes	—
105	C₂³×C₅	C₂⁴	1	2	yes	—
106	C₂³×C₅	C₂⁴	1	2	yes	—
107	C₂³×C₅	C₂⁴	1	1	yes	characteristic
108	C₅×Q₈	—	2	8	no	—
109	C₅×D₄	—	2	8	no	—
110	C₂×C₄×C₅	—	2	16	no	—
111	C₂×C₄×C₅	—	2	4	no	—
112	C₂×C₄×C₅	—	2	4	no	—
113	C₂×C₄×C₅	—	2	2	no	—
114	C₂×C₄×C₅	—	2	2	no	—
115	C₂×C₄×C₅	—	2	2	no	—
116	C₂×C₄×C₅	—	2	4	no	—
117	C₂×C₄×C₅	—	2	2	no	—
118	C₂×C₄×C₅	—	2	4	no	—
119	C₂×C₄×C₅	—	2	8	no	—
120	C₂×C₄×C₅	—	2	8	no	—
121	C₂×C₄×C₅	#41	1	4	yes	—
122	C₂×C₄×C₅	C₂⁴	1	2	yes	—
123	C₂×C₄×C₅	C₂⁴	1	2	yes	—
124	C₂×C₄×C₅	C₂⁴	1	4	yes	—
125	C₂×C₄×C₅	C₂⁴	1	4	yes	—
126	C₂×C₄×C₅	C₂⁴	1	4	yes	—
127	C₂×C₄×C₅	C₂⁴	1	2	yes	—
128	C₂×C₄×C₅	C₂⁴	1	2	yes	—
129	C₂×C₄×C₅	C₂⁴	1	4	yes	—
130	C₂×C₄×C₅	C₂⁴	1	2	yes	—
131	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
132	C₂²×D₄	C₂²×C₅	1	1	yes	characteristic
133	C₂³×C₄	C₂²×C₅	1	2	yes	—
134	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
135	C₂³×C₄	C₂²×C₅	1	1	yes	characteristic
136	#126	C₂²×C₅	1	16	yes	complemented
137	#125	C₂²×C₅	1	16	yes	complemented
138	#125	C₂²×C₅	1	8	yes	complemented
139	#123	C₂²×C₅	1	16	yes	—
140	C₂²⋊Q₈	C₂²×C₅	1	8	yes	complemented
141	C₂²⋊Q₈	C₂²×C₅	1	8	yes	complemented
142	C₄×Q₈	C₂²×C₅	1	8	yes	complemented
143	C₄×D₄	C₂²×C₅	1	8	yes	—
144	#117	C₂²×C₅	1	8	yes	complemented
145	#117	C₂²×C₅	1	16	yes	complemented
146	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
147	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
148	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
149	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
150	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
151	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
154	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
155	C₂×C₄⋊C₄	C₂²×C₅	1	2	yes	—
156	C₂×C₄⋊C₄	C₂²×C₅	1	8	yes	complemented
157	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
158	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
159	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
160	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
161	C₂×C₂²⋊C₄	C₂²×C₅	1	2	yes	—
162	C₂×C₄²	C₂²×C₅	1	2	yes	—
163	C₂×C₄²	C₂²×C₅	1	2	yes	—
164	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
165	C₂²×C₅	—	2	8	no	—
166	C₂²×C₅	—	2	2	no	—
167	C₂²×C₅	—	2	2	no	—
168	C₂²×C₅	—	2	2	no	—
169	C₂²×C₅	—	2	2	no	—
170	C₂²×C₅	—	2	2	no	—
171	C₂²×C₅	—	4	4	no	—
172	C₂²×C₅	C₂×#41	1	4	yes	—
173	C₂²×C₅	C₂×#41	1	2	yes	—
174	C₂²×C₅	C₂×#41	1	2	yes	—
175	C₂²×C₅	2₋¹⁺⁴	1	2	yes	—
176	C₂²×C₅	C₂⁵	1	1	yes	characteristic
177	C₄×C₅	—	2	4	no	—
178	C₄×C₅	—	2	4	no	—
179	C₄×C₅	—	2	2	no	—
180	C₄×C₅	—	2	2	no	—
181	C₄×C₅	—	2	2	no	—
182	C₄×C₅	—	2	4	no	—
183	C₄×C₅	—	2	2	no	—
184	C₄×C₅	—	2	4	no	—
185	C₄×C₅	C₂×#41	1	4	yes	—
186	C₂⁴	C₂³×C₅	1	1	yes	characteristic
187	C₂⁴	C₂³×C₅	1	1	yes	characteristic
188	C₂×Q₈	—	2	4	no	—
189	C₂×Q₈	C₂³×C₅	1	4	yes	—
190	C₂×D₄	—	2	4	no	—
191	C₂×D₄	C₂³×C₅	1	4	yes	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
196	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
197	C₂²×C₄	C₂³×C₅	1	2	yes	—
198	C₂²×C₄	C₂³×C₅	1	2	yes	—
199	C₂²×C₄	C₂³×C₅	1	2	yes	—
200	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
202	C₂²×C₄	C₂³×C₅	1	2	yes	—
203	C₂²×C₄	C₂³×C₅	1	8	yes	—
204	C₂²×C₄	C₂³×C₅	1	4	yes	—
205	C₂²×C₄	C₂³×C₅	1	4	yes	—
206	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
207	C₄⋊C₄	C₂³×C₅	1	4	yes	—
208	C₄⋊C₄	C₂³×C₅	1	4	yes	—
209	C₄⋊C₄	C₂³×C₅	1	8	yes	—
210	C₄⋊C₄	C₂³×C₅	1	8	yes	—
211	C₄⋊C₄	C₂³×C₅	1	4	yes	—
212	C₄⋊C₄	C₂³×C₅	1	8	yes	—
213	C₄⋊C₄	C₂³×C₅	1	4	yes	—
214	C₄⋊C₄	C₂³×C₅	1	8	yes	—
215	C₄⋊C₄	C₂³×C₅	1	8	yes	—
216	C₄⋊C₄	C₂³×C₅	1	8	yes	—
217	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
218	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
219	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
220	C₂²⋊C₄	C₂³×C₅	1	4	yes	—
221	C₂²⋊C₄	C₂³×C₅	1	8	yes	—
222	C₄²	C₂³×C₅	1	8	yes	—
223	C₄²	C₂³×C₅	1	8	yes	—
224	C₄²	C₂³×C₅	1	4	yes	—
225	C₂×C₅	—	2	4	no	—
226	C₂×C₅	—	4	2	no	—
227	C₂×C₅	#552	1	4	yes	direct product factor
228	C₂×C₅	C₂²×#41	1	1	yes	characteristic
229	C₂×C₅	C₂²×#41	1	1	yes	characteristic
230	C₂×C₅	C₂×2₋¹⁺⁴	1	1	yes	characteristic
231	C₂³	—	2	2	no	—
232	C₂³	—	2	2	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	2	no	—
235	C₂³	—	2	2	no	—
236	C₂³	C₅×#41	1	4	yes	—
237	C₂³	C₂⁴×C₅	1	2	yes	—
238	C₂³	C₂⁴×C₅	1	2	yes	—
239	C₂³	C₂⁴×C₅	1	1	yes	characteristic
240	Q₈	—	2	8	no	—
241	D₄	—	2	8	no	—
242	C₂×C₄	—	2	16	no	—
243	C₂×C₄	—	2	4	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	2	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	2	2	no	—
250	C₂×C₄	—	2	4	no	—
251	C₂×C₄	—	2	8	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	C₅×#41	1	4	yes	—
254	C₂×C₄	C₂⁴×C₅	1	2	yes	—
255	C₂×C₄	C₂⁴×C₅	1	2	yes	—
256	C₂×C₄	C₂⁴×C₅	1	4	yes	—
257	C₂×C₄	C₂⁴×C₅	1	4	yes	—
258	C₂×C₄	C₂⁴×C₅	1	4	yes	—
259	C₂×C₄	C₂⁴×C₅	1	2	yes	—
260	C₂×C₄	C₂⁴×C₅	1	2	yes	—
261	C₂×C₄	C₂⁴×C₅	1	4	yes	—
262	C₂×C₄	C₂⁴×C₅	1	2	yes	—
263	C₅	C₂×#552	1	1	yes	characteristic, direct product factor, Sylow
264	C₂²	—	2	8	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	2	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	4	no	—
271	C₂²	C₂×C₅×#41	1	4	yes	—
272	C₂²	C₂×C₅×#41	1	2	yes	—
273	C₂²	C₂×C₅×#41	1	2	yes	—
274	C₂²	C₅×2₋¹⁺⁴	1	2	yes	—
275	C₂²	C₂⁵×C₅	1	1	yes	characteristic
276	C₄	—	2	4	no	—
277	C₄	—	2	4	no	—
278	C₄	—	2	2	no	—
279	C₄	—	2	2	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	4	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	4	no	—
284	C₄	C₂×C₅×#41	1	4	yes	—
285	C₂	—	2	4	no	—
286	C₂	—	4	2	no	—
287	C₂	C₅×#552	1	4	yes	direct product factor
288	C₂	C₂²×C₅×#41	1	1	yes	characteristic
289	C₂	C₂²×C₅×#41	1	1	yes	characteristic
290	C₂	C₂×C₅×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁵×C₅
prime radical	C₂²×C₅×D₄	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#552	C₂	direct (N × H)
2. C₂×C₅×#126	C₂	semidirect
3. C₂×C₅×#125	C₂	semidirect
4. C₂×C₅×#125	C₂	semidirect
6. C₂×C₅×C₂²⋊Q₈	C₂	semidirect
7. C₂×C₅×C₂²⋊Q₈	C₂	semidirect
8. C₂×C₄×C₅×Q₈	C₂	semidirect
10. C₂×C₅×#117	C₂	semidirect
11. C₂×C₅×#117	C₂	semidirect
12. C₂²×C₅×C₄⋊C₄	C₂	semidirect
18. C₅×#126	C₂²	semidirect
19. C₅×#125	C₂²	semidirect
20. C₅×#125	C₂²	semidirect
22. C₅×C₂²⋊Q₈	C₂²	semidirect
23. C₅×C₂²⋊Q₈	C₂²	semidirect
24. C₄×C₅×Q₈	C₂²	semidirect
26. C₅×#117	C₂²	semidirect
27. C₅×#117	C₂²	semidirect
38. C₂×C₅×C₄⋊C₄	C₂²	semidirect
47. C₂×#552	C₅	direct (N × H)
87. #552	C₂×C₅	direct (N × H)
88. C₂×#126	C₂×C₅	semidirect
89. C₂×#125	C₂×C₅	semidirect
90. C₂×#125	C₂×C₅	semidirect
92. C₂×C₂²⋊Q₈	C₂×C₅	semidirect
93. C₂×C₂²⋊Q₈	C₂×C₅	semidirect
94. C₂×C₄×Q₈	C₂×C₅	semidirect
96. C₂×#117	C₂×C₅	semidirect
97. C₂×#117	C₂×C₅	semidirect
98. C₂²×C₄⋊C₄	C₂×C₅	semidirect
136. #126	C₂²×C₅	semidirect
137. #125	C₂²×C₅	semidirect
138. #125	C₂²×C₅	semidirect
140. C₂²⋊Q₈	C₂²×C₅	semidirect
141. C₂²⋊Q₈	C₂²×C₅	semidirect
142. C₄×Q₈	C₂²×C₅	semidirect
144. #117	C₂²×C₅	semidirect
145. #117	C₂²×C₅	semidirect
156. C₂×C₄⋊C₄	C₂²×C₅	semidirect
227. C₂×C₅	#552	direct (N × H)
263. C₅	C₂×#552	direct (N × H)
287. C₂	C₅×#552	direct (N × H)

Maximal subgroups

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

subgroup	index
C₅×#552	2
C₂×C₅×#126	2
C₂×C₅×#125	2
C₂×C₅×#123	2
C₂×C₅×C₂²⋊Q₈	2
C₂×C₄×C₅×Q₈	2
C₂×C₄×C₅×D₄	2
C₂×C₅×#117	2
C₂²×C₅×C₄⋊C₄	2
C₂×#552	5

Subgroups — C2×C5×#552 #10619255

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#552 #10619255