Subgroups — C₅×#33432 #11956152

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×C₅×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2929	C₂	1	1	yes	maximal, characteristic
4	C₅×#2168	C₂	1	4	yes	maximal, complemented
5	C₅×D₄×C₁₆	C₂	1	4	yes	maximal, complemented
6	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2113	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2112	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2108	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#131	C₂²	1	1	yes	characteristic
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#525	C₄	1	1	yes	characteristic
13	C₅×#504	—	2	4	no	—
14	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₅×M₃₂	C₂²	1	2	yes	complemented
17	C₂²×C₅×C₁₆	—	2	4	no	—
18	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
20	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
21	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
23	C₅×#435	C₄	1	4	yes	—
24	C₅×C₈×D₄	C₂²	1	4	yes	—
25	C₅×#424	C₄	1	1	yes	characteristic
26	C₅×#408	C₄	1	1	yes	characteristic
27	C₅×#407	C₄	1	1	yes	characteristic
28	C₅×#405	C₂²	1	1	yes	characteristic
29	C₅×C₄⋊C₁₆	C₂²	1	4	yes	complemented
30	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	—
31	C₅×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
32	C₅×#346	C₂²	1	2	yes	complemented
33	C₄×C₅×C₁₆	C₂²	1	2	yes	complemented
34	#33432	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
37	C₅×#131	—	2	2	no	—
38	C₅×#131	—	2	2	no	—
39	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
41	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
42	C₂²×C₅×C₈	—	2	2	no	—
43	C₂²×C₅×C₈	—	2	2	no	—
44	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
46	C₂²×C₅×C₈	C₂³	1	2	yes	—
47	C₂²×C₅×C₈	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×#123	C₈	1	2	yes	—
50	C₅×C₂²⋊Q₈	C₈	1	2	yes	—
51	C₅×#121	C₈	1	2	yes	—
52	C₅×#120	C₈	1	2	yes	—
53	C₄×C₅×D₄	C₂×C₄	1	4	yes	—
54	C₅×#117	C₂×C₄	1	1	yes	characteristic
55	C₅×M₃₂	—	2	2	no	—
56	C₅×M₃₂	—	2	2	no	—
57	C₅×M₃₂	—	2	4	no	—
58	C₂×C₅×C₁₆	—	2	2	no	—
59	C₂×C₅×C₁₆	—	2	4	no	—
60	C₂×C₅×C₁₆	—	2	4	no	—
61	C₂×C₅×C₁₆	—	2	4	no	—
62	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
63	C₂×C₅×C₁₆	C₂³	1	2	yes	—
64	C₂×C₅×C₁₆	C₂³	1	2	yes	—
65	C₂×C₅×C₁₆	C₂³	1	2	yes	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
68	C₅×C₄⋊C₈	C₂×C₄	1	4	yes	—
69	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
70	C₅×C₂²⋊C₈	C₂×C₄	1	4	yes	—
71	C₅×#97	C₂×C₄	1	2	yes	—
72	C₄×C₅×C₈	C₂³	1	2	yes	—
73	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
74	C₂³×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
75	#2929	C₂×C₅	1	1	yes	characteristic
76	#2168	C₂×C₅	1	4	yes	complemented
77	D₄×C₁₆	C₂×C₅	1	4	yes	complemented
78	#2151	C₂×C₅	1	1	yes	characteristic, complemented
79	#2113	C₂×C₅	1	1	yes	characteristic, complemented
80	#2112	C₂×C₅	1	1	yes	characteristic, complemented
81	#2108	C₂×C₅	1	1	yes	characteristic, complemented
82	C₂⁴×C₅	C₂×C₈	1	1	yes	characteristic
83	C₅×#41	—	2	2	no	—
84	C₅×#41	—	2	2	no	—
85	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
86	C₂×C₅×D₄	C₂×C₈	1	4	yes	—
87	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
88	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₄×C₅	—	2	2	no	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	C₂×C₈	1	2	yes	—
92	C₂²×C₄×C₅	C₂×C₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
94	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
95	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
96	C₅×M₁₆	—	2	2	no	—
97	C₅×M₁₆	—	2	2	no	—
98	C₅×M₁₆	—	2	4	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	2	no	—
105	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
107	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
108	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
110	C₂×C₅×C₈	#41	1	2	yes	—
111	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
112	C₅×C₄⋊C₄	C₂×C₈	1	2	yes	—
113	C₅×C₄⋊C₄	C₂×C₈	1	4	yes	—
114	C₅×C₂²⋊C₄	C₂×C₈	1	2	yes	—
115	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
116	C₅×C₂²⋊C₄	C₂×C₈	1	4	yes	—
117	C₄²×C₅	C₂²×C₄	1	2	yes	—
118	C₅×C₁₆	—	2	2	no	—
119	C₅×C₁₆	—	2	2	no	—
120	C₅×C₁₆	—	2	2	no	—
121	C₅×C₁₆	C₂×D₄	1	2	yes	—
122	C₅×C₁₆	C₂×D₄	1	2	yes	—
123	C₂×#131	C₂²×C₅	1	1	yes	characteristic
124	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
125	#525	C₄×C₅	1	1	yes	characteristic
126	#504	—	2	4	no	—
127	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
128	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
129	C₂×M₃₂	C₂²×C₅	1	2	yes	complemented
130	C₂²×C₁₆	—	2	4	no	—
131	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
132	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
133	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
134	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
135	C₂²×C₁₆	C₂²×C₅	1	2	yes	complemented
136	#435	C₄×C₅	1	4	yes	—
137	C₈×D₄	C₂²×C₅	1	4	yes	—
138	#424	C₄×C₅	1	1	yes	characteristic
139	#408	C₄×C₅	1	1	yes	characteristic
140	#407	C₄×C₅	1	1	yes	characteristic
141	#405	C₂²×C₅	1	1	yes	characteristic
142	C₄⋊C₁₆	C₂²×C₅	1	4	yes	complemented
143	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	—
144	C₂²⋊C₁₆	C₂²×C₅	1	4	yes	complemented
145	#346	C₂²×C₅	1	2	yes	complemented
146	C₄×C₁₆	C₂²×C₅	1	2	yes	complemented
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	—	2	4	no	—
149	C₂³×C₅	C₂²×C₈	1	2	yes	—
150	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
151	C₂³×C₅	C₂²×C₈	1	2	yes	—
152	C₅×Q₈	—	2	2	no	—
153	C₅×D₄	—	2	4	no	—
154	C₅×D₄	—	2	8	no	—
155	C₅×D₄	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	4	no	—
162	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
163	C₂×C₄×C₅	C₂²×C₈	1	4	yes	—
164	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
165	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
166	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
167	C₂×C₄×C₅	#131	1	2	yes	—
168	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
169	C₅×C₈	—	2	2	no	—
170	C₅×C₈	—	2	2	no	—
171	C₅×C₈	—	2	2	no	—
172	C₅×C₈	C₄×D₄	1	2	yes	—
173	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
174	C₅×C₈	C₂×#41	1	1	yes	characteristic
175	C₂×#41	C₂×C₄×C₅	1	1	yes	characteristic
176	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
177	#131	—	2	2	no	—
178	#131	—	2	2	no	—
179	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
180	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
181	C₂×M₁₆	C₂×C₄×C₅	1	1	yes	characteristic
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
185	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
186	C₂²×C₈	C₂³×C₅	1	2	yes	—
187	C₂²×C₈	C₂³×C₅	1	2	yes	—
188	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
189	#123	C₅×C₈	1	2	yes	—
190	C₂²⋊Q₈	C₅×C₈	1	2	yes	—
191	#121	C₅×C₈	1	2	yes	—
192	#120	C₅×C₈	1	2	yes	—
193	C₄×D₄	C₂×C₄×C₅	1	4	yes	—
194	#117	C₂×C₄×C₅	1	1	yes	characteristic
195	M₃₂	—	2	2	no	—
196	M₃₂	—	2	2	no	—
197	M₃₂	—	2	4	no	—
198	C₂×C₁₆	—	2	2	no	—
199	C₂×C₁₆	—	2	4	no	—
200	C₂×C₁₆	—	2	4	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	C₅×D₄	1	4	yes	complemented
203	C₂×C₁₆	C₂³×C₅	1	2	yes	—
204	C₂×C₁₆	C₂³×C₅	1	2	yes	—
205	C₂×C₁₆	C₂³×C₅	1	2	yes	—
206	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
207	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
208	C₄⋊C₈	C₂×C₄×C₅	1	4	yes	—
209	C₂²⋊C₈	C₂×C₄×C₅	1	4	yes	—
210	C₂²⋊C₈	C₂×C₄×C₅	1	4	yes	—
211	#97	C₂×C₄×C₅	1	2	yes	—
212	C₄×C₈	C₂³×C₅	1	2	yes	—
213	C₂²×C₅	—	2	8	no	—
214	C₂²×C₅	—	2	2	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	4	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	C₈×D₄	1	2	yes	—
220	C₂²×C₅	#504	1	4	yes	complemented
221	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	2	2	no	—
224	C₄×C₅	—	2	2	no	—
225	C₄×C₅	C₈×D₄	1	2	yes	—
226	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
227	C₄×C₅	C₂×#131	1	1	yes	characteristic
228	C₂⁴	C₂×C₅×C₈	1	1	yes	characteristic
229	#41	—	2	2	no	—
230	#41	—	2	2	no	—
231	C₂×Q₈	C₂×C₅×C₈	1	1	yes	characteristic
232	C₂×D₄	C₂×C₅×C₈	1	4	yes	—
233	C₂×D₄	C₂×C₅×C₈	1	2	yes	—
234	C₂×D₄	C₂×C₅×C₈	1	1	yes	characteristic
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	C₂×C₅×C₈	1	2	yes	—
238	C₂²×C₄	C₂×C₅×C₈	1	1	yes	characteristic
239	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
240	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
241	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
242	M₁₆	—	2	2	no	—
243	M₁₆	—	2	2	no	—
244	M₁₆	—	2	4	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
252	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
253	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
254	C₂×C₈	C₂²×C₄×C₅	1	1	yes	characteristic
255	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
256	C₂×C₈	C₅×#41	1	2	yes	—
257	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
258	C₄⋊C₄	C₂×C₅×C₈	1	2	yes	—
259	C₄⋊C₄	C₂×C₅×C₈	1	4	yes	—
260	C₂²⋊C₄	C₂×C₅×C₈	1	2	yes	—
261	C₂²⋊C₄	C₂×C₅×C₈	1	4	yes	—
262	C₂²⋊C₄	C₂×C₅×C₈	1	4	yes	—
263	C₄²	C₂²×C₄×C₅	1	2	yes	—
264	C₁₆	—	2	2	no	—
265	C₁₆	—	2	2	no	—
266	C₁₆	—	2	2	no	—
267	C₁₆	C₂×C₅×D₄	1	2	yes	—
268	C₁₆	C₂×C₅×D₄	1	2	yes	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	2	4	no	—
271	C₂×C₅	—	4	2	no	—
272	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
273	C₂×C₅	C₂×#504	1	2	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	4	no	—
276	C₂³	C₂²×C₅×C₈	1	2	yes	—
277	C₂³	C₂²×C₅×C₈	1	1	yes	characteristic
278	C₂³	C₂²×C₅×C₈	1	2	yes	—
279	Q₈	—	2	2	no	—
280	D₄	—	2	4	no	—
281	D₄	—	2	8	no	—
282	D₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
290	C₂×C₄	C₂²×C₅×C₈	1	4	yes	—
291	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
292	C₂×C₄	C₂²×C₅×C₈	1	2	yes	—
293	C₂×C₄	C₂²×C₅×C₈	1	1	yes	characteristic
294	C₂×C₄	C₅×#131	1	2	yes	—
295	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	2	no	—
298	C₈	—	2	2	no	—
299	C₈	C₄×C₅×D₄	1	2	yes	—
300	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
301	C₈	C₂×C₅×#41	1	1	yes	characteristic
302	C₅	#33432	1	1	yes	characteristic, direct product factor, Sylow
303	C₂²	—	2	8	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	4	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₅×C₈×D₄	1	2	yes	—
310	C₂²	C₅×#504	1	4	yes	complemented
311	C₂²	C₂³×C₅×C₈	1	1	yes	characteristic
312	C₄	—	2	4	no	—
313	C₄	—	2	2	no	—
314	C₄	—	2	2	no	—
315	C₄	C₅×C₈×D₄	1	2	yes	—
316	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
317	C₄	C₂×C₅×#131	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	2	4	no	—
320	C₂	—	4	2	no	—
321	C₂	C₂×C₅×C₈×D₄	1	1	yes	characteristic
322	C₂	C₂×C₅×#504	1	2	yes	—

	subgroup	G/H
center	C₂×C₅×C₁₆	C₂³
commutator	C₂²	C₂³×C₅×C₈
Frattini, cosocle	C₂×C₈	C₂⁴×C₅
socle	C₂²×C₅	C₂³×C₈
prime radical	C₅×#525	C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₅×C₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#504	C₂	semidirect
2. C₂³×C₅×C₁₆	C₂	semidirect
4. C₅×#2168	C₂	semidirect
5. C₅×D₄×C₁₆	C₂	semidirect
6. C₅×#2151	C₂	semidirect
7. C₅×#2113	C₂	semidirect
8. C₅×#2112	C₂	semidirect
9. C₅×#2108	C₂	semidirect
16. C₂×C₅×M₃₂	C₂²	semidirect
22. C₂²×C₅×C₁₆	C₂²	semidirect
29. C₅×C₄⋊C₁₆	C₂²	semidirect
31. C₅×C₂²⋊C₁₆	C₂²	semidirect
32. C₅×#346	C₂²	semidirect
33. C₄×C₅×C₁₆	C₂²	semidirect
34. #33432	C₅	direct (N × H)
62. C₂×C₅×C₁₆	D₄	semidirect
73. C₂×#504	C₂×C₅	semidirect
74. C₂³×C₁₆	C₂×C₅	semidirect
76. #2168	C₂×C₅	semidirect
77. D₄×C₁₆	C₂×C₅	semidirect
78. #2151	C₂×C₅	semidirect
79. #2113	C₂×C₅	semidirect
80. #2112	C₂×C₅	semidirect
81. #2108	C₂×C₅	semidirect
129. C₂×M₃₂	C₂²×C₅	semidirect
135. C₂²×C₁₆	C₂²×C₅	semidirect
142. C₄⋊C₁₆	C₂²×C₅	semidirect
144. C₂²⋊C₁₆	C₂²×C₅	semidirect
145. #346	C₂²×C₅	semidirect
146. C₄×C₁₆	C₂²×C₅	semidirect
202. C₂×C₁₆	C₅×D₄	semidirect
220. C₂²×C₅	#504	semidirect
302. C₅	#33432	direct (N × H)
310. C₂²	C₅×#504	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#504	2
C₂³×C₅×C₁₆	2
C₅×#2929	2
C₅×#2168	2
C₅×D₄×C₁₆	2
C₅×#2151	2
C₅×#2113	2
C₅×#2112	2
C₅×#2108	2
#33432	5

Subgroups — C5×#33432 #11956152

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33432 #11956152