Subgroups — C₅×#19930 #11942650

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3385	C₂	1	2	yes	maximal
2	C₅×Q₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#3338	C₂	1	1	yes	maximal, characteristic
4	C₅×#1713	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1696	C₂	1	2	yes	maximal, complemented
6	C₅×#1676	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#1675	C₂	1	1	yes	maximal, characteristic
8	C₅×#1633	C₂	1	2	yes	maximal, complemented
9	C₅×#1632	C₂	1	2	yes	maximal
10	C₅×#1591	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×C₈×SD₁₆	C₂	1	1	yes	maximal, characteristic
12	C₅×Q₈²	C₂²	1	2	yes	—
13	C₅×#554	C₂²	1	2	yes	—
14	C₅×#501	—	2	2	no	—
15	C₅×#500	—	2	1	no	—
16	C₅×#500	C₂²	1	1	yes	characteristic
17	C₅×#500	C₂²	1	1	yes	characteristic
18	C₅×#500	C₂²	1	2	yes	—
19	C₅×#494	C₂²	1	2	yes	—
20	C₅×#484	—	2	2	no	—
21	C₅×#479	—	2	2	no	—
22	C₅×#477	—	2	2	no	—
23	C₅×#477	C₂²	1	2	yes	—
24	C₅×#477	C₂²	1	2	yes	—
25	C₅×#477	C₂²	1	2	yes	—
26	C₅×#476	C₂²	1	2	yes	—
27	C₅×#474	—	2	2	no	—
28	C₅×#467	—	2	1	no	—
29	C₅×#446	C₂²	1	2	yes	—
30	C₅×C₈×Q₈	C₂²	1	1	yes	characteristic
31	C₅×#440	—	2	2	no	—
32	C₄×C₅×Q₁₆	—	2	1	no	—
33	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
34	C₄×C₅×SD₁₆	C₂²	1	1	yes	characteristic
35	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
36	C₂×C₅×#107	—	2	1	no	—
37	C₅×#335	C₂²	1	1	yes	characteristic
38	C₅×#334	C₂²	1	1	yes	characteristic
39	C₅×#333	C₂²	1	2	yes	—
40	C₅×#333	C₂²	1	2	yes	—
41	C₅×Q₈⋊C₈	C₂²	1	1	yes	characteristic
42	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
43	C₅×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₅×C₈²	C₂²	1	1	yes	characteristic
45	#19930	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
46	C₂×C₅×Q₁₆	—	4	2	no	—
47	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
48	C₂×C₅×SD₁₆	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	—	2	1	no	—
50	C₅×C₄⋊Q₈	—	2	2	no	—
51	C₅×C₄⋊Q₈	—	2	2	no	—
52	C₅×C₄⋊Q₈	—	4	2	no	—
53	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
55	C₅×#125	—	2	2	no	—
56	C₅×C₂²⋊Q₈	—	2	2	no	—
57	C₅×C₂²⋊Q₈	—	4	2	no	—
58	C₄×C₅×Q₈	—	2	2	no	—
59	C₄×C₅×Q₈	—	2	2	no	—
60	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
61	C₄×C₅×Q₈	C₂³	1	2	yes	—
62	C₄×C₅×D₄	—	2	2	no	—
63	C₄×C₅×D₄	C₂³	1	1	yes	characteristic
64	C₂×C₅×C₄⋊C₄	—	2	2	no	—
65	C₅×#107	—	2	1	no	—
66	C₅×#107	—	2	1	no	—
67	C₅×#107	—	2	2	no	—
68	C₅×#107	—	2	2	no	—
69	C₅×#107	—	2	1	no	—
70	C₅×#107	—	2	1	no	—
71	C₅×#107	—	2	2	no	—
72	C₅×#107	—	2	2	no	—
73	C₅×#107	—	2	2	no	—
74	C₅×#107	D₄	1	1	yes	characteristic
75	C₅×#106	—	2	2	no	—
76	C₅×#106	D₄	1	1	yes	characteristic
77	C₅×C₄⋊C₈	—	2	1	no	—
78	C₅×C₄⋊C₈	—	2	2	no	—
79	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
80	C₅×C₄⋊C₈	C₂³	1	1	yes	characteristic
81	C₅×C₄⋊C₈	C₂³	1	2	yes	—
82	C₅×#103	—	2	2	no	—
83	C₅×#103	—	2	2	no	—
84	C₅×#103	—	2	2	no	—
85	C₅×#103	D₄	1	1	yes	characteristic
86	C₅×#103	D₄	1	2	yes	—
87	C₅×#102	—	2	2	no	—
88	C₅×#102	D₄	1	1	yes	characteristic
89	C₅×C₂²⋊C₈	—	2	1	no	—
90	C₅×#97	—	2	2	no	—
91	C₄×C₅×C₈	—	2	1	no	—
92	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
93	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
94	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
95	#3385	C₂×C₅	1	2	yes	—
96	Q₈×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
97	#3338	C₂×C₅	1	1	yes	characteristic
98	#1713	C₂×C₅	1	1	yes	characteristic, complemented
99	#1696	C₂×C₅	1	2	yes	complemented
100	#1676	C₂×C₅	1	1	yes	characteristic, complemented
101	#1675	C₂×C₅	1	1	yes	characteristic
102	#1633	C₂×C₅	1	2	yes	complemented
103	#1632	C₂×C₅	1	2	yes	—
104	#1591	C₂×C₅	1	1	yes	characteristic, complemented
105	C₈×SD₁₆	C₂×C₅	1	1	yes	characteristic
106	C₂×C₅×Q₈	—	2	2	no	—
107	C₂×C₅×Q₈	—	2	2	no	—
108	C₂×C₅×Q₈	—	4	2	no	—
109	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
110	C₂×C₅×Q₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₅×D₄	C₂×D₄	1	1	yes	characteristic
112	C₂²×C₄×C₅	—	2	1	no	—
113	C₂²×C₄×C₅	—	4	2	no	—
114	C₅×Q₁₆	—	2	2	no	—
115	C₅×Q₁₆	—	8	4	no	—
116	C₅×SD₁₆	Q₁₆	1	4	yes	complemented
117	C₂×C₅×C₈	—	2	1	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	1	no	—
120	C₂×C₅×C₈	—	2	2	no	—
121	C₂×C₅×C₈	—	4	2	no	—
122	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
123	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
125	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	—	2	1	no	—
128	C₅×C₄⋊C₄	—	2	2	no	—
129	C₅×C₄⋊C₄	—	2	2	no	—
130	C₅×C₄⋊C₄	—	2	2	no	—
131	C₅×C₄⋊C₄	—	2	2	no	—
132	C₅×C₄⋊C₄	—	4	2	no	—
133	C₅×C₄⋊C₄	—	4	2	no	—
134	C₅×C₄⋊C₄	—	4	2	no	—
135	C₅×C₄⋊C₄	—	4	2	no	—
136	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
137	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
138	C₅×C₄⋊C₄	C₂×D₄	1	1	yes	characteristic
139	C₅×C₂²⋊C₄	—	2	1	no	—
140	C₅×C₂²⋊C₄	—	4	2	no	—
141	C₄²×C₅	—	2	2	no	—
142	C₄²×C₅	—	2	1	no	—
143	C₄²×C₅	—	4	2	no	—
144	C₄²×C₅	C₂⁴	1	1	yes	characteristic
145	Q₈²	C₂²×C₅	1	2	yes	—
146	#554	C₂²×C₅	1	2	yes	—
147	#501	—	2	2	no	—
148	#500	—	2	1	no	—
149	#500	C₂²×C₅	1	1	yes	characteristic
150	#500	C₂²×C₅	1	1	yes	characteristic
151	#500	C₂²×C₅	1	2	yes	—
152	#494	C₂²×C₅	1	2	yes	—
153	#484	—	2	2	no	—
154	#479	—	2	2	no	—
155	#477	—	2	2	no	—
156	#477	C₂²×C₅	1	2	yes	—
157	#477	C₂²×C₅	1	2	yes	—
158	#477	C₂²×C₅	1	2	yes	—
159	#476	C₂²×C₅	1	2	yes	—
160	#474	—	2	2	no	—
161	#467	—	2	1	no	—
162	#446	C₂²×C₅	1	2	yes	—
163	C₈×Q₈	C₂²×C₅	1	1	yes	characteristic
164	#440	—	2	2	no	—
165	C₄×Q₁₆	—	2	1	no	—
166	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
167	C₄×SD₁₆	C₂²×C₅	1	1	yes	characteristic
168	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
169	C₂×#107	—	2	1	no	—
170	#335	C₂²×C₅	1	1	yes	characteristic
171	#334	C₂²×C₅	1	1	yes	characteristic
172	#333	C₂²×C₅	1	2	yes	—
173	#333	C₂²×C₅	1	2	yes	—
174	Q₈⋊C₈	C₂²×C₅	1	1	yes	characteristic
175	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
176	D₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
177	C₈²	C₂²×C₅	1	1	yes	characteristic
178	C₂³×C₅	—	2	1	no	—
179	C₅×Q₈	—	2	1	no	—
180	C₅×Q₈	—	2	2	no	—
181	C₅×Q₈	—	2	2	no	—
182	C₅×Q₈	—	8	2	no	—
183	C₅×Q₈	—	8	2	no	—
184	C₅×Q₈	—	16	2	no	—
185	C₅×Q₈	C₂×Q₁₆	1	2	yes	—
186	C₅×D₄	—	2	1	no	—
187	C₅×D₄	C₂×Q₁₆	1	2	yes	—
188	C₂×C₄×C₅	—	2	1	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	—	2	1	no	—
192	C₂×C₄×C₅	—	2	1	no	—
193	C₂×C₄×C₅	—	4	1	no	—
194	C₂×C₄×C₅	—	4	1	no	—
195	C₂×C₄×C₅	—	4	2	no	—
196	C₂×C₄×C₅	—	4	2	no	—
197	C₂×C₄×C₅	—	16	2	no	—
198	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
200	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
201	C₅×C₈	—	2	2	no	—
202	C₅×C₈	—	2	1	no	—
203	C₅×C₈	—	2	1	no	—
204	C₅×C₈	—	4	1	no	—
205	C₅×C₈	—	4	1	no	—
206	C₅×C₈	—	4	2	no	—
207	C₅×C₈	C₂×Q₁₆	1	2	yes	—
208	C₅×C₈	#137	1	2	yes	—
209	C₂×Q₁₆	—	4	2	no	—
210	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
211	C₂×SD₁₆	C₅×D₄	1	1	yes	characteristic
212	C₂²×C₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	2	no	—
214	C₄⋊Q₈	—	2	2	no	—
215	C₄⋊Q₈	—	4	2	no	—
216	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
217	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
218	#125	—	2	2	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	C₂²⋊Q₈	—	4	2	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	C₂³×C₅	1	2	yes	—
224	C₄×Q₈	C₂³×C₅	1	1	yes	characteristic
225	C₄×D₄	—	2	2	no	—
226	C₄×D₄	C₂³×C₅	1	1	yes	characteristic
227	C₂×C₄⋊C₄	—	2	2	no	—
228	#107	—	2	1	no	—
229	#107	—	2	1	no	—
230	#107	—	2	2	no	—
231	#107	—	2	2	no	—
232	#107	—	2	1	no	—
233	#107	—	2	1	no	—
234	#107	—	2	2	no	—
235	#107	—	2	2	no	—
236	#107	—	2	2	no	—
237	#107	C₅×D₄	1	1	yes	characteristic
238	#106	—	2	2	no	—
239	#106	C₅×D₄	1	1	yes	characteristic
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	2	no	—
242	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
243	C₄⋊C₈	C₂³×C₅	1	1	yes	characteristic
244	C₄⋊C₈	C₂³×C₅	1	2	yes	—
245	#103	—	2	2	no	—
246	#103	—	2	2	no	—
247	#103	—	2	2	no	—
248	#103	C₅×D₄	1	1	yes	characteristic
249	#103	C₅×D₄	1	2	yes	—
250	#102	—	2	2	no	—
251	#102	C₅×D₄	1	1	yes	characteristic
252	C₂²⋊C₈	—	2	1	no	—
253	#97	—	2	2	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
256	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
257	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
258	C₂²×C₅	—	2	2	no	—
259	C₂²×C₅	—	4	1	no	—
260	C₂²×C₅	—	4	1	no	—
261	C₂²×C₅	D₄²	1	1	yes	characteristic
262	C₄×C₅	—	2	2	no	—
263	C₄×C₅	—	2	1	no	—
264	C₄×C₅	—	4	1	no	—
265	C₄×C₅	—	4	2	no	—
266	C₄×C₅	—	4	2	no	—
267	C₄×C₅	—	4	1	no	—
268	C₄×C₅	—	8	2	no	—
269	C₄×C₅	—	8	2	no	—
270	C₄×C₅	C₂²×Q₁₆	1	1	yes	characteristic
271	C₄×C₅	C₂×#137	1	1	yes	characteristic
272	C₄×C₅	#578	1	1	yes	characteristic
273	C₄×C₅	#578	1	1	yes	characteristic
274	C₂×Q₈	—	2	2	no	—
275	C₂×Q₈	—	2	2	no	—
276	C₂×Q₈	—	4	2	no	—
277	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
278	C₂×Q₈	C₂×C₅×D₄	1	1	yes	characteristic
279	C₂×D₄	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂²×C₄	—	2	1	no	—
281	C₂²×C₄	—	4	2	no	—
282	Q₁₆	—	2	2	no	—
283	Q₁₆	—	8	4	no	—
284	SD₁₆	C₅×Q₁₆	1	4	yes	complemented
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	4	2	no	—
290	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
291	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
292	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
293	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	2	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	4	2	no	—
303	C₄⋊C₄	—	4	2	no	—
304	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
305	C₄⋊C₄	C₂×C₅×D₄	1	1	yes	characteristic
306	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
307	C₂²⋊C₄	—	2	1	no	—
308	C₂²⋊C₄	—	4	2	no	—
309	C₄²	—	2	1	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	4	2	no	—
312	C₄²	C₂⁴×C₅	1	1	yes	characteristic
313	C₂×C₅	—	4	2	no	—
314	C₂×C₅	#3276	1	1	yes	characteristic
315	C₂×C₅	D₄×Q₁₆	1	1	yes	characteristic
316	C₂×C₅	#3292	1	1	yes	characteristic
317	C₂³	—	2	1	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	2	no	—
320	Q₈	—	2	2	no	—
321	Q₈	—	8	2	no	—
322	Q₈	—	8	2	no	—
323	Q₈	—	16	2	no	—
324	Q₈	C₂×C₅×Q₁₆	1	2	yes	—
325	D₄	—	2	1	no	—
326	D₄	C₂×C₅×Q₁₆	1	2	yes	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	2	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	2	no	—
335	C₂×C₄	—	4	2	no	—
336	C₂×C₄	—	16	2	no	—
337	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
338	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
339	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	4	1	no	—
344	C₈	—	4	1	no	—
345	C₈	—	4	2	no	—
346	C₈	C₂×C₅×Q₁₆	1	2	yes	—
347	C₈	C₅×#137	1	2	yes	—
348	C₅	#19930	1	1	yes	characteristic, direct product factor, Sylow
349	C₂²	—	2	2	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	C₅×D₄²	1	1	yes	characteristic
353	C₄	—	2	2	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	4	2	no	—
358	C₄	—	4	1	no	—
359	C₄	—	8	2	no	—
360	C₄	—	8	2	no	—
361	C₄	C₂²×C₅×Q₁₆	1	1	yes	characteristic
362	C₄	C₂×C₅×#137	1	1	yes	characteristic
363	C₄	C₅×#578	1	1	yes	characteristic
364	C₄	C₅×#578	1	1	yes	characteristic
365	C₂	—	4	2	no	—
366	C₂	C₅×#3276	1	1	yes	characteristic
367	C₂	C₅×D₄×Q₁₆	1	1	yes	characteristic
368	C₂	C₅×#3292	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
prime radical	C₂×C₅×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×Q₈×Q₁₆	C₂	semidirect
4. C₅×#1713	C₂	semidirect
5. C₅×#1696	C₂	semidirect
6. C₅×#1676	C₂	semidirect
8. C₅×#1633	C₂	semidirect
10. C₅×#1591	C₂	semidirect
45. #19930	C₅	direct (N × H)
96. Q₈×Q₁₆	C₂×C₅	semidirect
98. #1713	C₂×C₅	semidirect
99. #1696	C₂×C₅	semidirect
100. #1676	C₂×C₅	semidirect
102. #1633	C₂×C₅	semidirect
104. #1591	C₂×C₅	semidirect
116. C₅×SD₁₆	Q₁₆	semidirect
284. SD₁₆	C₅×Q₁₆	semidirect
348. C₅	#19930	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×#3385	2
C₅×Q₈×Q₁₆	2
C₅×#3338	2
C₅×#1713	2
C₅×#1696	2
C₅×#1676	2
C₅×#1675	2
C₅×#1633	2
C₅×#1632	2
C₅×#1591	2
C₅×C₈×SD₁₆	2
#19930	5

Subgroups — C5×#19930 #11942650

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19930 #11942650