Subgroups — C₅×#33609 #11956329

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#3058	C₂	1	1	yes	maximal, characteristic
5	C₅×#2220	C₂	1	4	yes	maximal, complemented
6	C₅×#2217	C₂	1	4	yes	maximal, complemented
7	C₅×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₅×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×#131	C₂²	1	1	yes	characteristic
13	C₅×#536	—	2	1	no	—
14	C₅×#535	—	2	1	no	—
15	C₂×C₅×SD₃₂	—	2	4	no	—
16	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
18	C₅×#504	—	2	4	no	—
19	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×C₅×M₃₂	C₂²	1	2	yes	complemented
21	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₅×C₁₆	C₂²	1	2	yes	complemented
23	C₅×#470	—	2	2	no	—
24	C₅×#469	—	2	2	no	—
25	C₅×#467	C₂²	1	4	yes	complemented
26	C₅×#466	C₂²	1	4	yes	—
27	C₅×#427	C₂²	1	1	yes	characteristic
28	C₅×#427	C₂²	1	1	yes	characteristic
29	C₅×#419	—	2	1	no	—
30	C₅×#418	—	2	1	no	—
31	C₅×#367	C₂²	1	2	yes	complemented
32	C₅×#366	C₂²	1	2	yes	complemented
33	C₅×#358	C₂²	1	4	yes	complemented
34	C₅×#357	C₂²	1	4	yes	—
35	#33609	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	D₄	1	1	yes	characteristic
37	C₂²×C₅×Q₈	—	2	1	no	—
38	C₂²×C₅×D₄	—	2	1	no	—
39	C₂×C₅×Q₁₆	—	2	2	no	—
40	C₂×C₅×Q₁₆	—	2	1	no	—
41	C₂×C₅×Q₁₆	—	2	2	no	—
42	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₅×D₈	—	2	2	no	—
44	C₂×C₅×D₈	—	2	1	no	—
45	C₂×C₅×D₈	—	2	2	no	—
46	C₂×C₅×D₈	C₂³	1	2	yes	—
47	C₅×#131	—	2	2	no	—
48	C₅×#131	—	2	2	no	—
49	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
50	C₂×C₅×M₁₆	D₄	1	2	yes	—
51	C₂²×C₅×C₈	C₂³	1	2	yes	—
52	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
53	C₅×C₄⋊Q₈	—	4	1	no	—
54	C₅×#127	—	4	1	no	—
55	C₅×#124	—	4	1	no	—
56	C₅×#124	—	4	1	no	—
57	C₅×#123	—	4	2	no	—
58	C₅×C₂²⋊Q₈	—	2	4	no	—
59	C₅×#121	—	2	4	no	—
60	C₅×#120	—	4	2	no	—
61	C₅×#117	—	2	1	no	—
62	C₅×#117	—	2	1	no	—
63	C₅×SD₃₂	—	2	4	no	—
64	C₅×SD₃₂	—	2	4	no	—
65	C₅×M₃₂	—	2	2	no	—
66	C₅×M₃₂	—	2	4	no	—
67	C₂×C₅×C₁₆	—	2	4	no	—
68	C₂×C₅×C₁₆	D₄	1	4	yes	complemented
69	C₂×C₅×C₁₆	C₂³	1	2	yes	—
70	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
71	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
72	C₅×#107	C₂³	1	2	yes	—
73	C₅×#107	C₂³	1	2	yes	—
74	C₅×#106	—	2	1	no	—
75	C₅×#106	—	2	1	no	—
76	C₅×#103	—	2	4	no	—
77	C₅×#102	—	2	4	no	—
78	C₂²×SD₃₂	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂×#504	C₂×C₅	1	1	yes	characteristic, complemented
80	#3059	C₂×C₅	1	1	yes	characteristic, complemented
81	#3058	C₂×C₅	1	1	yes	characteristic
82	#2220	C₂×C₅	1	4	yes	complemented
83	#2217	C₂×C₅	1	4	yes	complemented
84	#2158	C₂×C₅	1	1	yes	characteristic, complemented
85	#2141	C₂×C₅	1	1	yes	characteristic, complemented
86	#2140	C₂×C₅	1	1	yes	characteristic, complemented
87	C₂⁴×C₅	—	4	1	no	—
88	C₅×#41	—	2	2	no	—
89	C₅×#41	—	2	2	no	—
90	C₂×C₅×Q₈	—	2	2	no	—
91	C₂×C₅×Q₈	—	4	1	no	—
92	C₂×C₅×Q₈	—	4	2	no	—
93	C₂×C₅×Q₈	—	4	2	no	—
94	C₂×C₅×Q₈	D₈	1	1	yes	characteristic
95	C₂×C₅×D₄	—	2	2	no	—
96	C₂×C₅×D₄	—	4	4	no	—
97	C₂×C₅×D₄	—	4	2	no	—
98	C₂×C₅×D₄	—	4	1	no	—
99	C₂×C₅×D₄	—	4	2	no	—
100	C₂×C₅×D₄	D₈	1	2	yes	—
101	C₂×C₅×D₄	D₈	1	1	yes	characteristic
102	C₂²×C₄×C₅	—	4	1	no	—
103	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
105	C₅×Q₁₆	—	2	2	no	—
106	C₅×Q₁₆	—	2	2	no	—
107	C₅×Q₁₆	—	4	2	no	—
108	C₅×D₈	—	2	2	no	—
109	C₅×D₈	—	2	2	no	—
110	C₅×D₈	—	4	2	no	—
111	C₅×M₁₆	—	2	2	no	—
112	C₅×M₁₆	—	2	4	no	—
113	C₂×C₅×C₈	—	2	2	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
116	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
117	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
118	C₂×C₅×C₈	#41	1	2	yes	—
119	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
120	C₅×C₄⋊C₄	—	2	2	no	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	—	4	4	no	—
123	C₅×C₂²⋊C₄	—	4	1	no	—
124	C₅×C₂²⋊C₄	—	4	1	no	—
125	C₅×C₂²⋊C₄	—	4	4	no	—
126	C₅×C₂²⋊C₄	—	4	4	no	—
127	C₄²×C₅	—	4	1	no	—
128	C₄²×C₅	—	4	1	no	—
129	C₅×C₁₆	—	2	2	no	—
130	C₅×C₁₆	C₂×D₄	1	2	yes	—
131	C₅×C₁₆	C₂×D₄	1	2	yes	—
132	C₂²×Q₁₆	C₂²×C₅	1	1	yes	characteristic
133	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
134	C₂×#131	C₂²×C₅	1	1	yes	characteristic
135	#536	—	2	1	no	—
136	#535	—	2	1	no	—
137	C₂×SD₃₂	—	2	4	no	—
138	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
139	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
140	#504	—	2	4	no	—
141	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
142	C₂×M₃₂	C₂²×C₅	1	2	yes	complemented
143	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
144	C₂²×C₁₆	C₂²×C₅	1	2	yes	complemented
145	#470	—	2	2	no	—
146	#469	—	2	2	no	—
147	#467	C₂²×C₅	1	4	yes	complemented
148	#466	C₂²×C₅	1	4	yes	—
149	#427	C₂²×C₅	1	1	yes	characteristic
150	#427	C₂²×C₅	1	1	yes	characteristic
151	#419	—	2	1	no	—
152	#418	—	2	1	no	—
153	#367	C₂²×C₅	1	2	yes	complemented
154	#366	C₂²×C₅	1	2	yes	complemented
155	#358	C₂²×C₅	1	4	yes	complemented
156	#357	C₂²×C₅	1	4	yes	—
157	C₂³×C₅	—	4	2	no	—
158	C₂³×C₅	—	8	2	no	—
159	C₂³×C₅	—	16	2	no	—
160	C₂³×C₅	C₂×D₈	1	2	yes	—
161	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
162	C₅×Q₈	—	2	2	no	—
163	C₅×Q₈	—	4	2	no	—
164	C₅×Q₈	—	4	2	no	—
165	C₅×Q₈	—	8	2	no	—
166	C₅×D₄	—	2	4	no	—
167	C₅×D₄	—	2	2	no	—
168	C₅×D₄	—	4	2	no	—
169	C₅×D₄	—	4	2	no	—
170	C₅×D₄	—	8	2	no	—
171	C₅×D₄	—	16	4	no	—
172	C₂×C₄×C₅	—	2	2	no	—
173	C₂×C₄×C₅	—	2	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	4	2	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	8	2	no	—
178	C₂×C₄×C₅	#121	1	2	yes	—
179	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
180	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₅	#135	1	2	yes	—
182	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
183	C₅×C₈	—	2	2	no	—
184	C₅×C₈	#121	1	2	yes	—
185	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
186	C₅×C₈	C₂×#41	1	1	yes	characteristic
187	C₂×#41	C₅×D₄	1	1	yes	characteristic
188	C₂²×Q₈	—	2	1	no	—
189	C₂²×D₄	—	2	1	no	—
190	C₂×Q₁₆	—	2	2	no	—
191	C₂×Q₁₆	—	2	1	no	—
192	C₂×Q₁₆	—	2	2	no	—
193	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
194	C₂×D₈	—	2	2	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	—	2	2	no	—
197	C₂×D₈	C₂³×C₅	1	2	yes	—
198	#131	—	2	2	no	—
199	#131	—	2	2	no	—
200	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
201	C₂×M₁₆	C₅×D₄	1	2	yes	—
202	C₂²×C₈	C₂³×C₅	1	2	yes	—
203	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
204	C₄⋊Q₈	—	4	1	no	—
205	#127	—	4	1	no	—
206	#124	—	4	1	no	—
207	#124	—	4	1	no	—
208	#123	—	4	2	no	—
209	C₂²⋊Q₈	—	2	4	no	—
210	#121	—	2	4	no	—
211	#120	—	4	2	no	—
212	#117	—	2	1	no	—
213	#117	—	2	1	no	—
214	SD₃₂	—	2	4	no	—
215	SD₃₂	—	2	4	no	—
216	M₃₂	—	2	2	no	—
217	M₃₂	—	2	4	no	—
218	C₂×C₁₆	—	2	4	no	—
219	C₂×C₁₆	C₅×D₄	1	4	yes	complemented
220	C₂×C₁₆	C₂³×C₅	1	2	yes	—
221	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
222	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
223	#107	C₂³×C₅	1	2	yes	—
224	#107	C₂³×C₅	1	2	yes	—
225	#106	—	2	1	no	—
226	#106	—	2	1	no	—
227	#103	—	2	4	no	—
228	#102	—	2	4	no	—
229	C₂²×C₅	—	2	2	no	—
230	C₂²×C₅	—	2	4	no	—
231	C₂²×C₅	—	2	2	no	—
232	C₂²×C₅	—	8	4	no	—
233	C₂²×C₅	—	8	2	no	—
234	C₂²×C₅	—	16	4	no	—
235	C₂²×C₅	#466	1	2	yes	—
236	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
237	C₄×C₅	—	2	2	no	—
238	C₄×C₅	—	8	2	no	—
239	C₄×C₅	—	8	2	no	—
240	C₄×C₅	—	8	2	no	—
241	C₄×C₅	#466	1	2	yes	—
242	C₄×C₅	C₂×#121	1	1	yes	characteristic
243	C₄×C₅	C₂×#135	1	1	yes	characteristic
244	C₂⁴	—	4	1	no	—
245	#41	—	2	2	no	—
246	#41	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	—	4	2	no	—
250	C₂×Q₈	—	4	2	no	—
251	C₂×Q₈	C₅×D₈	1	1	yes	characteristic
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	4	4	no	—
254	C₂×D₄	—	4	2	no	—
255	C₂×D₄	—	4	1	no	—
256	C₂×D₄	—	4	2	no	—
257	C₂×D₄	C₅×D₈	1	2	yes	—
258	C₂×D₄	C₅×D₈	1	1	yes	characteristic
259	C₂²×C₄	—	4	1	no	—
260	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
261	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	4	2	no	—
265	D₈	—	2	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	4	2	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	2	4	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
273	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
274	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
275	C₂×C₈	C₅×#41	1	2	yes	—
276	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	4	4	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₂²⋊C₄	—	4	4	no	—
283	C₂²⋊C₄	—	4	4	no	—
284	C₄²	—	4	1	no	—
285	C₄²	—	4	1	no	—
286	C₁₆	—	2	2	no	—
287	C₁₆	C₂×C₅×D₄	1	2	yes	—
288	C₁₆	C₂×C₅×D₄	1	2	yes	—
289	C₂×C₅	—	2	2	no	—
290	C₂×C₅	—	4	2	no	—
291	C₂×C₅	—	16	2	no	—
292	C₂×C₅	C₂×#466	1	1	yes	characteristic
293	C₂×C₅	#3416	1	2	yes	—
294	C₂³	—	4	2	no	—
295	C₂³	—	8	2	no	—
296	C₂³	—	16	2	no	—
297	C₂³	C₂×C₅×D₈	1	2	yes	—
298	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	Q₈	—	4	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	8	2	no	—
303	D₄	—	2	4	no	—
304	D₄	—	2	2	no	—
305	D₄	—	4	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	8	2	no	—
308	D₄	—	16	4	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	C₅×#121	1	2	yes	—
316	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
317	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
318	C₂×C₄	C₅×#135	1	2	yes	—
319	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	C₅×#121	1	2	yes	—
322	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
323	C₈	C₂×C₅×#41	1	1	yes	characteristic
324	C₅	#33609	1	1	yes	characteristic, direct product factor, Sylow
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	4	no	—
330	C₂²	—	16	4	no	—
331	C₂²	C₅×#466	1	2	yes	—
332	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	C₅×#466	1	2	yes	—
338	C₄	C₂×C₅×#121	1	1	yes	characteristic
339	C₄	C₂×C₅×#135	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₂×C₅×#466	1	1	yes	characteristic
344	C₂	C₅×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₅×#41 ◁ C₂×C₅×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×SD₃₂	C₂	semidirect
2. C₂×C₅×#504	C₂	semidirect
3. C₅×#3059	C₂	semidirect
5. C₅×#2220	C₂	semidirect
6. C₅×#2217	C₂	semidirect
7. C₅×#2158	C₂	semidirect
8. C₅×#2141	C₂	semidirect
9. C₅×#2140	C₂	semidirect
20. C₂×C₅×M₃₂	C₂²	semidirect
22. C₂²×C₅×C₁₆	C₂²	semidirect
25. C₅×#467	C₂²	semidirect
31. C₅×#367	C₂²	semidirect
32. C₅×#366	C₂²	semidirect
33. C₅×#358	C₂²	semidirect
35. #33609	C₅	direct (N × H)
68. C₂×C₅×C₁₆	D₄	semidirect
78. C₂²×SD₃₂	C₂×C₅	semidirect
79. C₂×#504	C₂×C₅	semidirect
80. #3059	C₂×C₅	semidirect
82. #2220	C₂×C₅	semidirect
83. #2217	C₂×C₅	semidirect
84. #2158	C₂×C₅	semidirect
85. #2141	C₂×C₅	semidirect
86. #2140	C₂×C₅	semidirect
142. C₂×M₃₂	C₂²×C₅	semidirect
144. C₂²×C₁₆	C₂²×C₅	semidirect
147. #467	C₂²×C₅	semidirect
153. #367	C₂²×C₅	semidirect
154. #366	C₂²×C₅	semidirect
155. #358	C₂²×C₅	semidirect
219. C₂×C₁₆	C₅×D₄	semidirect
324. C₅	#33609	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×SD₃₂	2
C₂×C₅×#504	2
C₅×#3059	2
C₅×#3058	2
C₅×#2220	2
C₅×#2217	2
C₅×#2158	2
C₅×#2141	2
C₅×#2140	2
#33609	5

Subgroups — C5×#33609 #11956329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33609 #11956329