Subgroups — C₅×#33776 #11956496

Order 1280.

592 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3147	C₂	1	1	yes	maximal, characteristic
3	C₅×#2248	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#2241	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2240	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2211	C₂	1	4	yes	maximal, complemented
8	C₅×#2189	C₂	1	4	yes	maximal, complemented
9	C₅×#2162	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#135	—	2	1	no	—
11	C₂×C₅×#135	C₂²	1	2	yes	—
12	C₅×#533	—	2	1	no	—
13	C₅×#532	—	2	1	no	—
14	C₂×C₅×Q₃₂	C₂²	1	4	yes	—
15	C₂×C₅×SD₃₂	C₂²	1	4	yes	—
16	C₅×#500	C₂²	1	1	yes	characteristic
17	C₅×#495	—	2	1	no	—
18	C₅×#495	C₂²	1	2	yes	—
19	C₅×#494	C₂²	1	1	yes	characteristic
20	C₅×#494	C₂²	1	1	yes	characteristic
21	C₅×#493	C₂²	1	1	yes	characteristic
22	C₅×#492	—	2	1	no	—
23	C₅×#487	—	2	1	no	—
24	C₅×#467	—	2	1	no	—
25	C₅×#467	C₂²	1	2	yes	—
26	C₄×C₅×Q₁₆	—	2	1	no	—
27	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
28	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₅×#358	C₂²	1	4	yes	—
31	C₅×#357	C₂²	1	4	yes	—
32	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
33	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
35	#33776	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	2	2	no	—
37	C₅×#135	—	4	4	no	—
38	C₅×#135	—	4	4	no	—
39	C₂×C₅×Q₁₆	—	2	1	no	—
40	C₂×C₅×Q₁₆	—	2	1	no	—
41	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₅×SD₁₆	—	2	2	no	—
44	C₂×C₅×SD₁₆	—	2	2	no	—
45	C₂×C₅×D₈	—	2	1	no	—
46	C₂×C₅×D₈	C₂³	1	2	yes	—
47	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×C₄⋊Q₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	1	no	—
51	C₅×C₄⋊Q₈	—	2	1	no	—
52	C₅×#127	—	2	1	no	—
53	C₅×#125	—	4	1	no	—
54	C₅×#124	—	2	2	no	—
55	C₅×C₂²⋊Q₈	—	2	2	no	—
56	C₅×C₂²⋊Q₈	—	4	1	no	—
57	C₅×#121	—	4	1	no	—
58	C₅×#121	—	4	1	no	—
59	C₄×C₅×Q₈	—	2	2	no	—
60	C₄×C₅×Q₈	—	4	1	no	—
61	C₄×C₅×D₄	—	4	1	no	—
62	C₄×C₅×D₄	—	4	1	no	—
63	C₅×#117	—	4	1	no	—
64	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
65	C₅×Q₃₂	—	4	4	no	—
66	C₅×SD₃₂	—	4	4	no	—
67	C₂×C₅×C₁₆	C₂³	1	2	yes	—
68	C₂×C₅×C₁₆	C₂³	1	2	yes	—
69	C₅×#107	—	2	1	no	—
70	C₅×#107	C₂³	1	2	yes	—
71	C₅×#103	—	2	2	no	—
72	C₅×#103	—	2	2	no	—
73	C₄×C₅×C₈	D₄	1	1	yes	characteristic
74	C₄×C₅×C₈	D₄	1	1	yes	characteristic
75	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
76	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
77	#3170	C₂×C₅	1	1	yes	characteristic, complemented
78	#3147	C₂×C₅	1	1	yes	characteristic
79	#2248	C₂×C₅	1	1	yes	characteristic, complemented
80	#2247	C₂×C₅	1	1	yes	characteristic, complemented
81	#2241	C₂×C₅	1	1	yes	characteristic, complemented
82	#2240	C₂×C₅	1	1	yes	characteristic, complemented
83	#2211	C₂×C₅	1	4	yes	complemented
84	#2189	C₂×C₅	1	4	yes	complemented
85	#2162	C₂×C₅	1	1	yes	characteristic, complemented
86	C₅×#41	—	4	4	no	—
87	C₅×#41	—	4	4	no	—
88	C₂×C₅×Q₈	—	2	2	no	—
89	C₂×C₅×Q₈	—	2	2	no	—
90	C₂×C₅×Q₈	—	4	1	no	—
91	C₂×C₅×D₄	—	2	2	no	—
92	C₂×C₅×D₄	—	4	2	no	—
93	C₂×C₅×D₄	—	4	1	no	—
94	C₂×C₅×D₄	—	4	1	no	—
95	C₂²×C₄×C₅	—	4	2	no	—
96	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
98	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
99	C₅×Q₁₆	—	2	4	no	—
100	C₅×Q₁₆	—	2	4	no	—
101	C₅×Q₁₆	—	4	2	no	—
102	C₅×Q₁₆	—	4	2	no	—
103	C₅×SD₁₆	—	4	4	no	—
104	C₅×SD₁₆	—	4	4	no	—
105	C₅×D₈	—	2	4	no	—
106	C₅×D₈	—	4	2	no	—
107	C₂×C₅×C₈	—	2	2	no	—
108	C₂×C₅×C₈	—	2	2	no	—
109	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
113	C₅×C₄⋊C₄	—	2	2	no	—
114	C₅×C₄⋊C₄	—	4	1	no	—
115	C₅×C₄⋊C₄	—	4	1	no	—
116	C₅×C₄⋊C₄	—	4	1	no	—
117	C₅×C₄⋊C₄	—	4	1	no	—
118	C₅×C₄⋊C₄	—	4	2	no	—
119	C₅×C₄⋊C₄	—	4	2	no	—
120	C₅×C₄⋊C₄	—	4	1	no	—
121	C₅×C₂²⋊C₄	—	4	2	no	—
122	C₅×C₂²⋊C₄	—	4	2	no	—
123	C₅×C₂²⋊C₄	—	4	2	no	—
124	C₄²×C₅	—	4	2	no	—
125	C₄²×C₅	D₈	1	1	yes	characteristic
126	C₄²×C₅	D₈	1	1	yes	characteristic
127	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
129	C₅×C₁₆	—	2	2	no	—
130	C₅×C₁₆	—	2	2	no	—
131	C₂×#135	—	2	1	no	—
132	C₂×#135	C₂²×C₅	1	2	yes	—
133	#533	—	2	1	no	—
134	#532	—	2	1	no	—
135	C₂×Q₃₂	C₂²×C₅	1	4	yes	—
136	C₂×SD₃₂	C₂²×C₅	1	4	yes	—
137	#500	C₂²×C₅	1	1	yes	characteristic
138	#495	—	2	1	no	—
139	#495	C₂²×C₅	1	2	yes	—
140	#494	C₂²×C₅	1	1	yes	characteristic
141	#494	C₂²×C₅	1	1	yes	characteristic
142	#493	C₂²×C₅	1	1	yes	characteristic
143	#492	—	2	1	no	—
144	#487	—	2	1	no	—
145	#467	—	2	1	no	—
146	#467	C₂²×C₅	1	2	yes	—
147	C₄×Q₁₆	—	2	1	no	—
148	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
149	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
150	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
151	#358	C₂²×C₅	1	4	yes	—
152	#357	C₂²×C₅	1	4	yes	—
153	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
154	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
155	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
156	C₂³×C₅	—	4	2	no	—
157	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
158	C₅×Q₈	—	4	2	no	—
159	C₅×Q₈	—	4	2	no	—
160	C₅×Q₈	—	4	2	no	—
161	C₅×Q₈	—	4	2	no	—
162	C₅×Q₈	—	8	2	no	—
163	C₅×D₄	—	4	2	no	—
164	C₅×D₄	—	4	2	no	—
165	C₅×D₄	—	8	4	no	—
166	C₅×D₄	—	8	2	no	—
167	C₅×D₄	—	8	2	no	—
168	C₂×C₄×C₅	—	2	1	no	—
169	C₂×C₄×C₅	—	2	1	no	—
170	C₂×C₄×C₅	—	2	2	no	—
171	C₂×C₄×C₅	—	2	2	no	—
172	C₂×C₄×C₅	—	4	2	no	—
173	C₂×C₄×C₅	—	4	2	no	—
174	C₂×C₄×C₅	—	4	2	no	—
175	C₂×C₄×C₅	—	8	4	no	—
176	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
182	C₅×C₈	—	2	1	no	—
183	C₅×C₈	—	2	1	no	—
184	C₅×C₈	AGL(1,ℤ/8)	1	2	yes	—
185	C₅×C₈	H(5,𝔽₂)	1	2	yes	—
186	C₂×#41	—	2	2	no	—
187	#135	—	4	4	no	—
188	#135	—	4	4	no	—
189	C₂×Q₁₆	—	2	1	no	—
190	C₂×Q₁₆	—	2	1	no	—
191	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
192	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
193	C₂×SD₁₆	—	2	2	no	—
194	C₂×SD₁₆	—	2	2	no	—
195	C₂×D₈	—	2	1	no	—
196	C₂×D₈	C₂³×C₅	1	2	yes	—
197	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
198	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
199	C₄⋊Q₈	—	2	1	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	#127	—	2	1	no	—
203	#125	—	4	1	no	—
204	#124	—	2	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	—	4	1	no	—
207	#121	—	4	1	no	—
208	#121	—	4	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	4	1	no	—
211	C₄×D₄	—	4	1	no	—
212	C₄×D₄	—	4	1	no	—
213	#117	—	4	1	no	—
214	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
215	Q₃₂	—	4	4	no	—
216	SD₃₂	—	4	4	no	—
217	C₂×C₁₆	C₂³×C₅	1	2	yes	—
218	C₂×C₁₆	C₂³×C₅	1	2	yes	—
219	#107	—	2	1	no	—
220	#107	C₂³×C₅	1	2	yes	—
221	#103	—	2	2	no	—
222	#103	—	2	2	no	—
223	C₄×C₈	C₅×D₄	1	1	yes	characteristic
224	C₄×C₈	C₅×D₄	1	1	yes	characteristic
225	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
226	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	8	2	no	—
231	C₂²×C₅	—	8	2	no	—
232	C₂²×C₅	—	8	2	no	—
233	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
234	C₄×C₅	—	2	1	no	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
237	C₄×C₅	—	8	2	no	—
238	C₄×C₅	—	8	2	no	—
239	C₄×C₅	—	8	2	no	—
240	C₄×C₅	#508	1	2	yes	—
241	C₄×C₅	#510	1	2	yes	—
242	C₄×C₅	#535	1	1	yes	characteristic
243	C₄×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
244	#41	—	4	4	no	—
245	#41	—	4	4	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂²×C₄	—	4	2	no	—
254	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
255	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
256	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
257	Q₁₆	—	2	4	no	—
258	Q₁₆	—	2	4	no	—
259	Q₁₆	—	4	2	no	—
260	Q₁₆	—	4	2	no	—
261	SD₁₆	—	4	4	no	—
262	SD₁₆	—	4	4	no	—
263	D₈	—	2	4	no	—
264	D₈	—	4	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
269	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
270	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	4	2	no	—
283	C₄²	C₅×D₈	1	1	yes	characteristic
284	C₄²	C₅×D₈	1	1	yes	characteristic
285	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
286	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
287	C₁₆	—	2	2	no	—
288	C₁₆	—	2	2	no	—
289	C₂×C₅	—	4	1	no	—
290	C₂×C₅	—	16	2	no	—
291	C₂×C₅	#3205	1	1	yes	characteristic
292	C₂×C₅	C₂×#508	1	1	yes	characteristic
293	C₂×C₅	C₂×#510	1	1	yes	characteristic
294	C₂³	—	4	2	no	—
295	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
296	Q₈	—	4	2	no	—
297	Q₈	—	4	2	no	—
298	Q₈	—	4	2	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	8	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	8	4	no	—
304	D₄	—	8	2	no	—
305	D₄	—	8	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	8	4	no	—
314	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
316	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
317	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
320	C₈	—	2	1	no	—
321	C₈	—	2	1	no	—
322	C₈	C₅×AGL(1,ℤ/8)	1	2	yes	—
323	C₈	C₅×H(5,𝔽₂)	1	2	yes	—
324	C₅	#33776	1	1	yes	characteristic, direct product factor, Sylow
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	C₅×#508	1	2	yes	—
339	C₄	C₅×#510	1	2	yes	—
340	C₄	C₅×#535	1	1	yes	characteristic
341	C₄	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
342	C₂	—	4	1	no	—
343	C₂	—	16	2	no	—
344	C₂	C₅×#3205	1	1	yes	characteristic
345	C₂	C₂×C₅×#508	1	1	yes	characteristic
346	C₂	C₂×C₅×#510	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3147	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₂×C₄²×C₅ ◁ C₂×C₄×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3170	C₂	semidirect
3. C₅×#2248	C₂	semidirect
4. C₅×#2247	C₂	semidirect
5. C₅×#2241	C₂	semidirect
6. C₅×#2240	C₂	semidirect
7. C₅×#2211	C₂	semidirect
8. C₅×#2189	C₂	semidirect
9. C₅×#2162	C₂	semidirect
35. #33776	C₅	direct (N × H)
77. #3170	C₂×C₅	semidirect
79. #2248	C₂×C₅	semidirect
80. #2247	C₂×C₅	semidirect
81. #2241	C₂×C₅	semidirect
82. #2240	C₂×C₅	semidirect
83. #2211	C₂×C₅	semidirect
84. #2189	C₂×C₅	semidirect
85. #2162	C₂×C₅	semidirect
324. C₅	#33776	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×#3170	2
C₅×#3147	2
C₅×#2248	2
C₅×#2247	2
C₅×#2241	2
C₅×#2240	2
C₅×#2211	2
C₅×#2189	2
C₅×#2162	2
#33776	5

Subgroups — C5×#33776 #11956496

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33776 #11956496