Subgroups — C₅×#33839 #11956559

Order 1280.

712 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#509	C₂	1	2	yes	maximal, complemented
2	C₅×#3280	C₂	1	2	yes	maximal, complemented
3	C₅×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2220	C₂	1	2	yes	maximal, complemented
5	C₅×#2210	C₂	1	2	yes	maximal, complemented
6	C₅×#2189	C₂	1	2	yes	maximal, complemented
7	C₅×#2184	C₂	1	2	yes	maximal, complemented
8	C₄×C₅×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2169	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#135	C₂²	1	2	yes	—
11	C₂²×C₅×D₈	C₂²	1	2	yes	complemented
12	C₅×#546	—	2	2	no	—
13	C₅×#509	—	2	8	no	—
14	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₅×SD₃₂	C₂²	1	2	yes	complemented
16	C₂×C₅×SD₃₂	C₂²	1	2	yes	complemented
17	C₂×C₅×D₁₆	C₂²	1	2	yes	—
18	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
19	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₅×M₃₂	C₂²	1	2	yes	complemented
21	C₅×#495	C₂²	1	2	yes	complemented
22	C₅×#467	C₂²	1	2	yes	complemented
23	C₅×#466	C₂²	1	2	yes	—
24	C₅×#463	—	2	2	no	—
25	C₅×#450	—	2	2	no	—
26	C₅×#449	—	2	2	no	—
27	C₄×C₅×D₈	C₂²	1	2	yes	complemented
28	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
29	C₅×#366	C₂²	1	1	yes	characteristic, complemented
30	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic, complemented
31	C₅×#358	C₂²	1	2	yes	complemented
32	C₅×#357	C₂²	1	2	yes	complemented
33	C₅×#357	C₂²	1	2	yes	complemented
34	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	complemented
35	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic, complemented
36	#33839	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	C₂×C₅×#41	—	2	2	no	—
38	C₂²×C₅×D₄	—	2	2	no	—
39	C₅×#135	—	2	4	no	—
40	C₅×#135	—	4	2	no	—
41	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₅×SD₁₆	—	2	2	no	—
43	C₂×C₅×SD₁₆	—	2	2	no	—
44	C₂×C₅×D₈	—	2	4	no	—
45	C₂×C₅×D₈	—	2	2	no	—
46	C₂×C₅×D₈	—	4	2	no	—
47	C₂×C₅×D₈	C₂³	1	2	yes	—
48	C₂×C₅×D₈	C₂³	1	2	yes	—
49	C₂²×C₅×C₈	C₂³	1	2	yes	—
50	C₅×#124	—	2	2	no	—
51	C₅×#123	—	4	2	no	—
52	C₅×C₂²⋊Q₈	—	2	2	no	—
53	C₅×#121	—	2	2	no	—
54	C₅×#121	—	4	2	no	—
55	C₅×#120	—	4	2	no	—
56	C₄×C₅×D₄	—	2	2	no	—
57	C₄×C₅×D₄	D₄	1	1	yes	characteristic
58	C₂×C₅×C₂²⋊C₄	—	4	2	no	—
59	C₅×Q₃₂	—	4	1	no	—
60	C₅×SD₃₂	—	2	4	no	—
61	C₅×SD₃₂	—	2	4	no	—
62	C₅×D₁₆	—	2	4	no	—
63	C₅×D₁₆	—	4	1	no	—
64	C₅×D₁₆	D₄	1	4	yes	complemented
65	C₅×M₃₂	—	2	4	no	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₁₆	C₂³	1	2	yes	—
68	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
69	C₅×#107	C₂³	1	2	yes	—
70	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₅×#103	—	2	2	no	—
72	C₅×#102	—	2	2	no	—
73	C₅×#102	—	2	2	no	—
74	C₅×C₂²⋊C₈	D₄	1	2	yes	—
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₂×#509	C₂×C₅	1	2	yes	complemented
77	#3280	C₂×C₅	1	2	yes	complemented
78	#2249	C₂×C₅	1	1	yes	characteristic, complemented
79	#2220	C₂×C₅	1	2	yes	complemented
80	#2210	C₂×C₅	1	2	yes	complemented
81	#2189	C₂×C₅	1	2	yes	complemented
82	#2184	C₂×C₅	1	2	yes	complemented
83	C₄×D₁₆	C₂×C₅	1	1	yes	characteristic, complemented
84	#2169	C₂×C₅	1	1	yes	characteristic, complemented
85	C₂⁴×C₅	—	4	2	no	—
86	C₅×#41	—	4	4	no	—
87	C₅×#41	—	8	2	no	—
88	C₂×C₅×Q₈	—	2	2	no	—
89	C₂×C₅×D₄	—	2	2	no	—
90	C₂×C₅×D₄	—	2	2	no	—
91	C₂×C₅×D₄	—	4	4	no	—
92	C₂×C₅×D₄	—	4	2	no	—
93	C₂×C₅×D₄	—	4	2	no	—
94	C₂×C₅×D₄	—	4	2	no	—
95	C₂×C₅×D₄	—	8	2	no	—
96	C₂×C₅×D₄	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	—	4	2	no	—
98	C₂²×C₄×C₅	—	4	2	no	—
99	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
100	C₅×Q₁₆	—	2	2	no	—
101	C₅×Q₁₆	—	2	2	no	—
102	C₅×SD₁₆	—	4	4	no	—
103	C₅×SD₁₆	—	8	2	no	—
104	C₅×D₈	—	2	2	no	—
105	C₅×D₈	—	2	2	no	—
106	C₅×D₈	—	2	2	no	—
107	C₅×D₈	—	4	4	no	—
108	C₅×D₈	C₂×D₄	1	4	yes	—
109	C₂×C₅×C₈	—	2	2	no	—
110	C₂×C₅×C₈	—	2	2	no	—
111	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
112	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	2	no	—
115	C₅×C₄⋊C₄	—	4	2	no	—
116	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
117	C₅×C₂²⋊C₄	—	4	2	no	—
118	C₅×C₂²⋊C₄	—	4	2	no	—
119	C₅×C₂²⋊C₄	—	4	2	no	—
120	C₅×C₂²⋊C₄	—	4	2	no	—
121	C₅×C₂²⋊C₄	—	4	2	no	—
122	C₅×C₂²⋊C₄	D₈	1	2	yes	—
123	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
124	C₅×C₁₆	—	2	2	no	—
125	C₅×C₁₆	—	2	1	no	—
126	C₅×C₁₆	C₂×D₄	1	2	yes	—
127	C₂×#135	C₂²×C₅	1	2	yes	—
128	C₂²×D₈	C₂²×C₅	1	2	yes	complemented
129	#546	—	2	2	no	—
130	#509	—	2	8	no	—
131	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
132	C₂×SD₃₂	C₂²×C₅	1	2	yes	complemented
133	C₂×SD₃₂	C₂²×C₅	1	2	yes	complemented
134	C₂×D₁₆	C₂²×C₅	1	2	yes	—
135	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
136	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic
137	C₂×M₃₂	C₂²×C₅	1	2	yes	complemented
138	#495	C₂²×C₅	1	2	yes	complemented
139	#467	C₂²×C₅	1	2	yes	complemented
140	#466	C₂²×C₅	1	2	yes	—
141	#463	—	2	2	no	—
142	#450	—	2	2	no	—
143	#449	—	2	2	no	—
144	C₄×D₈	C₂²×C₅	1	2	yes	complemented
145	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
146	#366	C₂²×C₅	1	1	yes	characteristic, complemented
147	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
148	#358	C₂²×C₅	1	2	yes	complemented
149	#357	C₂²×C₅	1	2	yes	complemented
150	#357	C₂²×C₅	1	2	yes	complemented
151	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	complemented
152	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
153	C₂³×C₅	—	4	2	no	—
154	C₂³×C₅	—	4	2	no	—
155	C₂³×C₅	—	8	4	no	—
156	C₂³×C₅	—	8	4	no	—
157	C₂³×C₅	—	16	2	no	—
158	C₂³×C₅	C₂×D₈	1	2	yes	—
159	C₅×Q₈	—	4	2	no	—
160	C₅×Q₈	—	4	2	no	—
161	C₅×D₄	—	2	4	no	—
162	C₅×D₄	—	4	1	no	—
163	C₅×D₄	—	4	2	no	—
164	C₅×D₄	—	4	2	no	—
165	C₅×D₄	—	4	2	no	—
166	C₅×D₄	—	8	4	no	—
167	C₅×D₄	—	8	4	no	—
168	C₅×D₄	—	16	2	no	—
169	C₂×C₄×C₅	—	2	2	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	4	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	—	8	4	no	—
174	C₂×C₄×C₅	—	16	2	no	—
175	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
178	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
179	C₅×C₈	—	2	2	no	—
180	C₅×C₈	—	2	1	no	—
181	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
182	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
183	C₂×#41	—	2	2	no	—
184	C₂²×D₄	—	2	2	no	—
185	#135	—	2	4	no	—
186	#135	—	4	2	no	—
187	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
188	C₂×SD₁₆	—	2	2	no	—
189	C₂×SD₁₆	—	2	2	no	—
190	C₂×D₈	—	2	4	no	—
191	C₂×D₈	—	2	2	no	—
192	C₂×D₈	—	4	2	no	—
193	C₂×D₈	C₂³×C₅	1	2	yes	—
194	C₂×D₈	C₂³×C₅	1	2	yes	—
195	C₂²×C₈	C₂³×C₅	1	2	yes	—
196	#124	—	2	2	no	—
197	#123	—	4	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	#121	—	2	2	no	—
200	#121	—	4	2	no	—
201	#120	—	4	2	no	—
202	C₄×D₄	—	2	2	no	—
203	C₄×D₄	C₅×D₄	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	—	4	2	no	—
205	Q₃₂	—	4	1	no	—
206	SD₃₂	—	2	4	no	—
207	SD₃₂	—	2	4	no	—
208	D₁₆	—	2	4	no	—
209	D₁₆	—	4	1	no	—
210	D₁₆	C₅×D₄	1	4	yes	complemented
211	M₃₂	—	2	4	no	—
212	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
213	C₂×C₁₆	C₂³×C₅	1	2	yes	—
214	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
215	#107	C₂³×C₅	1	2	yes	—
216	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
217	#103	—	2	2	no	—
218	#102	—	2	2	no	—
219	#102	—	2	2	no	—
220	C₂²⋊C₈	C₅×D₄	1	2	yes	—
221	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
222	C₂²×C₅	—	2	2	no	—
223	C₂²×C₅	—	2	2	no	—
224	C₂²×C₅	—	2	2	no	—
225	C₂²×C₅	—	4	4	no	—
226	C₂²×C₅	—	8	2	no	—
227	C₂²×C₅	—	8	2	no	—
228	C₂²×C₅	—	8	2	no	—
229	C₂²×C₅	—	8	2	no	—
230	C₂²×C₅	—	8	2	no	—
231	C₂²×C₅	—	16	8	no	—
232	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
233	C₄×C₅	—	2	1	no	—
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	8	2	no	—
237	C₄×C₅	—	8	2	no	—
238	C₄×C₅	D₄²	1	1	yes	characteristic
239	C₄×C₅	#576	1	1	yes	characteristic
240	C₂⁴	—	4	2	no	—
241	#41	—	4	4	no	—
242	#41	—	8	2	no	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×D₄	—	2	2	no	—
245	C₂×D₄	—	2	2	no	—
246	C₂×D₄	—	4	4	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	8	2	no	—
251	C₂×D₄	C₅×D₈	1	1	yes	characteristic
252	C₂²×C₄	—	4	2	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
255	Q₁₆	—	2	2	no	—
256	Q₁₆	—	2	2	no	—
257	SD₁₆	—	4	4	no	—
258	SD₁₆	—	8	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	4	4	no	—
263	D₈	C₂×C₅×D₄	1	4	yes	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
267	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
268	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	C₅×D₈	1	2	yes	—
278	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
279	C₁₆	—	2	2	no	—
280	C₁₆	—	2	1	no	—
281	C₁₆	C₂×C₅×D₄	1	2	yes	—
282	C₂×C₅	—	4	2	no	—
283	C₂×C₅	—	8	4	no	—
284	C₂×C₅	—	16	2	no	—
285	C₂×C₅	D₄×D₈	1	1	yes	characteristic
286	C₂×C₅	#3414	1	1	yes	characteristic
287	C₂×C₅	#3415	1	1	yes	characteristic
288	C₂³	—	4	2	no	—
289	C₂³	—	4	2	no	—
290	C₂³	—	8	4	no	—
291	C₂³	—	8	4	no	—
292	C₂³	—	16	2	no	—
293	C₂³	C₂×C₅×D₈	1	2	yes	—
294	Q₈	—	4	2	no	—
295	Q₈	—	4	2	no	—
296	D₄	—	2	4	no	—
297	D₄	—	4	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	4	no	—
302	D₄	—	8	4	no	—
303	D₄	—	16	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	8	4	no	—
309	C₂×C₄	—	16	2	no	—
310	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
311	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
312	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
313	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
314	C₈	—	2	2	no	—
315	C₈	—	2	1	no	—
316	C₈	C₂²×C₅×D₄	1	1	yes	characteristic
317	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
318	C₅	#33839	1	1	yes	characteristic, direct product factor, Sylow
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	4	4	no	—
323	C₂²	—	8	2	no	—
324	C₂²	—	8	2	no	—
325	C₂²	—	8	2	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	16	8	no	—
329	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	2	no	—
335	C₄	C₅×D₄²	1	1	yes	characteristic
336	C₄	C₅×#576	1	1	yes	characteristic
337	C₂	—	4	2	no	—
338	C₂	—	8	4	no	—
339	C₂	—	16	2	no	—
340	C₂	C₅×D₄×D₈	1	1	yes	characteristic
341	C₂	C₅×#3414	1	1	yes	characteristic
342	C₂	C₅×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄×C₅×D₄ ◁ C₅×C₈×D₄ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#509	C₂	semidirect
2. C₅×#3280	C₂	semidirect
3. C₅×#2249	C₂	semidirect
4. C₅×#2220	C₂	semidirect
5. C₅×#2210	C₂	semidirect
6. C₅×#2189	C₂	semidirect
7. C₅×#2184	C₂	semidirect
8. C₄×C₅×D₁₆	C₂	semidirect
9. C₅×#2169	C₂	semidirect
11. C₂²×C₅×D₈	C₂²	semidirect
14. C₂×C₅×Q₃₂	C₂²	semidirect
15. C₂×C₅×SD₃₂	C₂²	semidirect
16. C₂×C₅×SD₃₂	C₂²	semidirect
20. C₂×C₅×M₃₂	C₂²	semidirect
21. C₅×#495	C₂²	semidirect
22. C₅×#467	C₂²	semidirect
27. C₄×C₅×D₈	C₂²	semidirect
29. C₅×#366	C₂²	semidirect
30. C₅×C₄⋊C₁₆	C₂²	semidirect
31. C₅×#358	C₂²	semidirect
32. C₅×#357	C₂²	semidirect
33. C₅×#357	C₂²	semidirect
34. C₅×C₂²⋊C₁₆	C₂²	semidirect
35. C₄×C₅×C₁₆	C₂²	semidirect
36. #33839	C₅	direct (N × H)
64. C₅×D₁₆	D₄	semidirect
76. C₂×#509	C₂×C₅	semidirect
77. #3280	C₂×C₅	semidirect
78. #2249	C₂×C₅	semidirect
79. #2220	C₂×C₅	semidirect
80. #2210	C₂×C₅	semidirect
81. #2189	C₂×C₅	semidirect
82. #2184	C₂×C₅	semidirect
83. C₄×D₁₆	C₂×C₅	semidirect
84. #2169	C₂×C₅	semidirect
128. C₂²×D₈	C₂²×C₅	semidirect
131. C₂×Q₃₂	C₂²×C₅	semidirect
132. C₂×SD₃₂	C₂²×C₅	semidirect
133. C₂×SD₃₂	C₂²×C₅	semidirect
137. C₂×M₃₂	C₂²×C₅	semidirect
138. #495	C₂²×C₅	semidirect
139. #467	C₂²×C₅	semidirect
144. C₄×D₈	C₂²×C₅	semidirect
146. #366	C₂²×C₅	semidirect
147. C₄⋊C₁₆	C₂²×C₅	semidirect
148. #358	C₂²×C₅	semidirect
149. #357	C₂²×C₅	semidirect
150. #357	C₂²×C₅	semidirect
151. C₂²⋊C₁₆	C₂²×C₅	semidirect
152. C₄×C₁₆	C₂²×C₅	semidirect
210. D₁₆	C₅×D₄	semidirect
318. C₅	#33839	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₅×#509	2
C₅×#3280	2
C₅×#2249	2
C₅×#2220	2
C₅×#2210	2
C₅×#2189	2
C₅×#2184	2
C₄×C₅×D₁₆	2
C₅×#2169	2
#33839	5

Subgroups — C5×#33839 #11956559

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33839 #11956559