Subgroups — C₂²×C₅×#360 #11956106

Order 1280.

1214 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#135	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₅×#106	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#360	C₂	1	12	yes	maximal, direct product factor
5	C₂²×C₅×#41	—	2	1	no	—
6	C₂×C₅×#135	—	2	3	no	—
7	C₂×C₅×#135	—	2	3	no	—
8	C₂×C₅×#135	C₄	1	6	yes	complemented
9	C₂×C₅×#135	C₂²	1	6	yes	—
10	C₂²×C₅×Q₁₆	C₄	1	1	yes	characteristic, complemented
11	C₂²×C₅×SD₁₆	—	2	1	no	—
12	C₂²×C₅×D₈	C₄	1	1	yes	characteristic, complemented
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₄⋊C₄	—	2	1	no	—
15	C₂×C₅×M₃₂	—	2	3	no	—
16	C₂×C₅×M₃₂	C₂²	1	6	yes	complemented
17	C₂²×C₅×C₁₆	—	2	1	no	—
18	C₂×C₅×#106	—	2	3	no	—
19	C₂×C₅×#106	C₂²	1	6	yes	complemented
20	C₅×#360	C₂²	1	16	yes	direct product factor
21	C₂²×#360	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
22	C₂×C₅×#41	—	2	6	no	—
23	C₂×C₅×#41	—	2	6	no	—
24	C₂×C₅×#41	—	4	3	no	—
25	C₂×C₅×#41	—	4	3	no	—
26	C₂²×C₅×Q₈	—	2	1	no	—
27	C₂²×C₅×D₄	—	2	1	no	—
28	C₂²×C₅×D₄	—	4	1	no	—
29	C₂³×C₄×C₅	—	4	1	no	—
30	C₂³×C₄×C₅	—	4	1	no	—
31	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
32	C₅×#135	—	2	12	no	—
33	C₅×#135	—	2	6	no	—
34	C₅×#135	—	2	6	no	—
35	C₅×#135	C₂×C₄	1	12	yes	complemented
36	C₅×#135	C₂³	1	4	yes	—
37	C₂×C₅×Q₁₆	—	2	3	no	—
38	C₂×C₅×Q₁₆	C₂×C₄	1	6	yes	complemented
39	C₂×C₅×SD₁₆	—	2	6	no	—
40	C₂×C₅×SD₁₆	—	2	6	no	—
41	C₂×C₅×D₈	—	2	3	no	—
42	C₂×C₅×D₈	C₂×C₄	1	6	yes	complemented
43	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₅×C₈	C₂×C₄	1	3	yes	—
45	C₂²×C₅×C₈	D₄	1	3	yes	—
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	D₄	1	3	yes	—
48	C₂²×C₅×C₈	C₂³	1	3	yes	—
49	C₂×C₅×C₄⋊C₄	—	2	6	no	—
50	C₂×C₅×C₄⋊C₄	—	4	3	no	—
51	C₅×M₃₂	—	2	6	no	—
52	C₅×M₃₂	C₂³	1	4	yes	complemented
53	C₂×C₅×C₁₆	—	2	6	no	—
54	C₅×#106	—	2	6	no	—
55	C₅×#106	C₂³	1	4	yes	complemented
56	C₂²×#135	C₂×C₅	1	1	yes	characteristic
57	C₂²×M₃₂	C₂×C₅	1	1	yes	characteristic, complemented
58	C₂²×#106	C₂×C₅	1	1	yes	characteristic, complemented
59	C₂×#360	C₂×C₅	1	12	yes	direct product factor
60	C₂⁴×C₅	—	4	1	no	—
61	C₂⁴×C₅	D₈	1	1	yes	characteristic
62	C₅×#41	—	2	12	no	—
63	C₅×#41	—	2	4	no	—
64	C₅×#41	—	4	12	no	—
65	C₅×#41	—	4	6	no	—
66	C₅×#41	—	4	6	no	—
67	C₂×C₅×Q₈	—	2	6	no	—
68	C₂×C₅×Q₈	—	4	3	no	—
69	C₂×C₅×D₄	—	2	6	no	—
70	C₂×C₅×D₄	—	4	6	no	—
71	C₂×C₅×D₄	—	4	6	no	—
72	C₂×C₅×D₄	—	4	3	no	—
73	C₂²×C₄×C₅	—	4	1	no	—
74	C₂²×C₄×C₅	—	4	6	no	—
75	C₂²×C₄×C₅	—	4	6	no	—
76	C₂²×C₄×C₅	—	4	6	no	—
77	C₂²×C₄×C₅	—	8	4	no	—
78	C₂²×C₄×C₅	C₂²⋊C₄	1	3	yes	—
79	C₂²×C₄×C₅	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	D₈	1	3	yes	—
81	C₂²×C₄×C₅	SD₁₆	1	3	yes	—
82	C₂²×C₄×C₅	SD₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	3	yes	—
84	C₅×Q₁₆	—	2	6	no	—
85	C₅×Q₁₆	C₂²×C₄	1	4	yes	complemented
86	C₅×SD₁₆	—	2	12	no	—
87	C₅×SD₁₆	—	2	4	no	—
88	C₅×D₈	—	2	6	no	—
89	C₅×D₈	C₂²×C₄	1	4	yes	complemented
90	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
91	C₂×C₅×C₈	C₂²⋊C₄	1	3	yes	—
92	C₂×C₅×C₈	C₂²⋊C₄	1	6	yes	—
93	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	3	yes	—
95	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
96	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	3	yes	—
98	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	—	4	6	no	—
101	C₅×C₁₆	—	2	4	no	—
102	C₂²×#41	—	2	1	no	—
103	C₂×#135	—	2	3	no	—
104	C₂×#135	—	2	3	no	—
105	C₂×#135	C₄×C₅	1	6	yes	complemented
106	C₂×#135	C₂²×C₅	1	6	yes	—
107	C₂²×Q₁₆	C₄×C₅	1	1	yes	characteristic, complemented
108	C₂²×SD₁₆	—	2	1	no	—
109	C₂²×D₈	C₄×C₅	1	1	yes	characteristic, complemented
110	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂²×C₄⋊C₄	—	2	1	no	—
112	C₂×M₃₂	—	2	3	no	—
113	C₂×M₃₂	C₂²×C₅	1	6	yes	complemented
114	C₂²×C₁₆	—	2	1	no	—
115	C₂×#106	—	2	3	no	—
116	C₂×#106	C₂²×C₅	1	6	yes	complemented
117	#360	C₂²×C₅	1	16	yes	direct product factor
118	C₂³×C₅	—	2	4	no	—
119	C₂³×C₅	—	4	6	no	—
120	C₂³×C₅	—	8	4	no	—
121	C₂³×C₅	#102	1	3	yes	—
122	C₂³×C₅	#102	1	1	yes	characteristic
123	C₂³×C₅	C₂×D₈	1	3	yes	—
124	C₅×Q₈	—	2	4	no	—
125	C₅×Q₈	—	4	6	no	—
126	C₅×D₄	—	2	4	no	—
127	C₅×D₄	—	4	4	no	—
128	C₅×D₄	—	4	12	no	—
129	C₅×D₄	—	4	6	no	—
130	C₂×C₄×C₅	—	4	6	no	—
131	C₂×C₄×C₅	—	4	4	no	—
132	C₂×C₄×C₅	—	4	12	no	—
133	C₂×C₄×C₅	—	4	4	no	—
134	C₂×C₄×C₅	—	8	12	no	—
135	C₂×C₄×C₅	#102	1	3	yes	—
136	C₂×C₄×C₅	#102	1	3	yes	—
137	C₂×C₄×C₅	#102	1	6	yes	—
138	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
139	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	3	yes	—
140	C₂×C₄×C₅	C₂×D₈	1	3	yes	—
141	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
142	C₂×C₄×C₅	C₂×SD₁₆	1	3	yes	—
143	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
144	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
145	C₅×C₈	C₂×C₂²⋊C₄	1	3	yes	—
146	C₅×C₈	C₂³×C₄	1	1	yes	characteristic
147	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
148	C₂×#41	—	2	6	no	—
149	C₂×#41	—	2	6	no	—
150	C₂×#41	—	4	3	no	—
151	C₂×#41	—	4	3	no	—
152	C₂²×Q₈	—	2	1	no	—
153	C₂²×D₄	—	2	1	no	—
154	C₂²×D₄	—	4	1	no	—
155	C₂³×C₄	—	4	1	no	—
156	C₂³×C₄	—	4	1	no	—
157	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
158	#135	—	2	12	no	—
159	#135	—	2	6	no	—
160	#135	—	2	6	no	—
161	#135	C₂×C₄×C₅	1	12	yes	complemented
162	#135	C₂³×C₅	1	4	yes	—
163	C₂×Q₁₆	—	2	3	no	—
164	C₂×Q₁₆	C₂×C₄×C₅	1	6	yes	complemented
165	C₂×SD₁₆	—	2	6	no	—
166	C₂×SD₁₆	—	2	6	no	—
167	C₂×D₈	—	2	3	no	—
168	C₂×D₈	C₂×C₄×C₅	1	6	yes	complemented
169	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
170	C₂²×C₈	C₂×C₄×C₅	1	3	yes	—
171	C₂²×C₈	C₅×D₄	1	3	yes	—
172	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
173	C₂²×C₈	C₅×D₄	1	3	yes	—
174	C₂²×C₈	C₂³×C₅	1	3	yes	—
175	C₂×C₄⋊C₄	—	2	6	no	—
176	C₂×C₄⋊C₄	—	4	3	no	—
177	M₃₂	—	2	6	no	—
178	M₃₂	C₂³×C₅	1	4	yes	complemented
179	C₂×C₁₆	—	2	6	no	—
180	#106	—	2	6	no	—
181	#106	C₂³×C₅	1	4	yes	complemented
182	C₂²×C₅	—	2	6	no	—
183	C₂²×C₅	—	2	6	no	—
184	C₂²×C₅	—	4	4	no	—
185	C₂²×C₅	—	8	12	no	—
186	C₂²×C₅	#360	1	4	yes	direct product factor
187	C₂²×C₅	C₂×#102	1	3	yes	—
188	C₂²×C₅	C₂×#102	1	3	yes	—
189	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
190	C₄×C₅	—	4	4	no	—
191	C₄×C₅	—	8	4	no	—
192	C₄×C₅	C₂×#102	1	3	yes	—
193	C₄×C₅	C₂×#102	1	3	yes	—
194	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
195	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
196	C₂⁴	—	4	1	no	—
197	C₂⁴	C₅×D₈	1	1	yes	characteristic
198	#41	—	2	12	no	—
199	#41	—	2	4	no	—
200	#41	—	4	6	no	—
201	#41	—	4	6	no	—
202	#41	—	4	12	no	—
203	C₂×Q₈	—	2	6	no	—
204	C₂×Q₈	—	4	3	no	—
205	C₂×D₄	—	2	6	no	—
206	C₂×D₄	—	4	6	no	—
207	C₂×D₄	—	4	6	no	—
208	C₂×D₄	—	4	3	no	—
209	C₂²×C₄	—	4	1	no	—
210	C₂²×C₄	—	4	6	no	—
211	C₂²×C₄	—	4	6	no	—
212	C₂²×C₄	—	4	6	no	—
213	C₂²×C₄	—	8	4	no	—
214	C₂²×C₄	C₅×C₂²⋊C₄	1	1	yes	characteristic
215	C₂²×C₄	C₅×C₂²⋊C₄	1	3	yes	—
216	C₂²×C₄	C₅×D₈	1	3	yes	—
217	C₂²×C₄	C₅×SD₁₆	1	1	yes	characteristic
218	C₂²×C₄	C₅×SD₁₆	1	3	yes	—
219	C₂²×C₄	C₂×C₅×D₄	1	3	yes	—
220	Q₁₆	—	2	6	no	—
221	Q₁₆	C₂²×C₄×C₅	1	4	yes	complemented
222	SD₁₆	—	2	12	no	—
223	SD₁₆	—	2	4	no	—
224	D₈	—	2	6	no	—
225	D₈	C₂²×C₄×C₅	1	4	yes	complemented
226	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
227	C₂×C₈	C₅×C₂²⋊C₄	1	3	yes	—
228	C₂×C₈	C₅×C₂²⋊C₄	1	6	yes	—
229	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
230	C₂×C₈	C₂²×C₄×C₅	1	3	yes	—
231	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
232	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
233	C₂×C₈	C₂×C₅×D₄	1	3	yes	—
234	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
235	C₄⋊C₄	—	2	4	no	—
236	C₄⋊C₄	—	4	6	no	—
237	C₁₆	—	2	4	no	—
238	C₂×C₅	—	2	1	no	—
239	C₂×C₅	—	2	3	no	—
240	C₂×C₅	—	8	4	no	—
241	C₂×C₅	C₂×#360	1	6	yes	direct product factor
242	C₂×C₅	C₂²×#102	1	1	yes	characteristic
243	C₂³	—	2	4	no	—
244	C₂³	—	4	6	no	—
245	C₂³	—	8	4	no	—
246	C₂³	C₅×#102	1	3	yes	—
247	C₂³	C₅×#102	1	1	yes	characteristic
248	C₂³	C₂×C₅×D₈	1	3	yes	—
249	Q₈	—	2	4	no	—
250	Q₈	—	4	6	no	—
251	D₄	—	2	4	no	—
252	D₄	—	4	12	no	—
253	D₄	—	4	4	no	—
254	D₄	—	4	6	no	—
255	C₂×C₄	—	4	6	no	—
256	C₂×C₄	—	4	12	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	8	12	no	—
260	C₂×C₄	C₅×#102	1	3	yes	—
261	C₂×C₄	C₅×#102	1	3	yes	—
262	C₂×C₄	C₅×#102	1	6	yes	—
263	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
264	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	3	yes	—
265	C₂×C₄	C₂×C₅×D₈	1	3	yes	—
266	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
267	C₂×C₄	C₂×C₅×SD₁₆	1	3	yes	—
268	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
269	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
270	C₈	C₂×C₅×C₂²⋊C₄	1	3	yes	—
271	C₈	C₂³×C₄×C₅	1	1	yes	characteristic
272	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
273	C₅	C₂²×#360	1	1	yes	characteristic, direct product factor, Sylow
274	C₂²	—	2	6	no	—
275	C₂²	—	2	6	no	—
276	C₂²	—	4	4	no	—
277	C₂²	—	8	12	no	—
278	C₂²	C₅×#360	1	4	yes	direct product factor
279	C₂²	C₂×C₅×#102	1	3	yes	—
280	C₂²	C₂×C₅×#102	1	3	yes	—
281	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
282	C₄	—	4	4	no	—
283	C₄	—	8	4	no	—
284	C₄	C₂×C₅×#102	1	3	yes	—
285	C₄	C₂×C₅×#102	1	3	yes	—
286	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
287	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
288	C₂	—	2	1	no	—
289	C₂	—	2	3	no	—
290	C₂	—	8	4	no	—
291	C₂	C₂×C₅×#360	1	6	yes	direct product factor
292	C₂	C₂²×C₅×#102	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	#102
commutator	C₈	C₂³×C₄×C₅
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂²×C₅×#135	C₂
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₅×M₃₂	C₂	semidirect
3. C₂²×C₅×#106	C₂	semidirect
4. C₂×C₅×#360	C₂	direct (N × H)
8. C₂×C₅×#135	C₄	semidirect
10. C₂²×C₅×Q₁₆	C₄	semidirect
12. C₂²×C₅×D₈	C₄	semidirect
16. C₂×C₅×M₃₂	C₂²	semidirect
19. C₂×C₅×#106	C₂²	semidirect
20. C₅×#360	C₂²	direct (N × H)
21. C₂²×#360	C₅	direct (N × H)
35. C₅×#135	C₂×C₄	semidirect
38. C₂×C₅×Q₁₆	C₂×C₄	semidirect
42. C₂×C₅×D₈	C₂×C₄	semidirect
52. C₅×M₃₂	C₂³	semidirect
55. C₅×#106	C₂³	semidirect
57. C₂²×M₃₂	C₂×C₅	semidirect
58. C₂²×#106	C₂×C₅	semidirect
59. C₂×#360	C₂×C₅	direct (N × H)
85. C₅×Q₁₆	C₂²×C₄	semidirect
89. C₅×D₈	C₂²×C₄	semidirect
105. C₂×#135	C₄×C₅	semidirect
107. C₂²×Q₁₆	C₄×C₅	semidirect
109. C₂²×D₈	C₄×C₅	semidirect
113. C₂×M₃₂	C₂²×C₅	semidirect
116. C₂×#106	C₂²×C₅	semidirect
117. #360	C₂²×C₅	direct (N × H)
161. #135	C₂×C₄×C₅	semidirect
164. C₂×Q₁₆	C₂×C₄×C₅	semidirect
168. C₂×D₈	C₂×C₄×C₅	semidirect
178. M₃₂	C₂³×C₅	semidirect
181. #106	C₂³×C₅	semidirect
186. C₂²×C₅	#360	direct (N × H)
221. Q₁₆	C₂²×C₄×C₅	semidirect
225. D₈	C₂²×C₄×C₅	semidirect
241. C₂×C₅	C₂×#360	direct (N × H)
273. C₅	C₂²×#360	direct (N × H)
278. C₂²	C₅×#360	direct (N × H)
291. C₂	C₂×C₅×#360	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#135	2
C₂²×C₅×M₃₂	2
C₂²×C₅×#106	2
C₂×C₅×#360	2
C₂²×#360	5

Subgroups — C22×C5×#360 #11956106

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₅×#360 #11956106