Subgroups — C₅×#19888 #11942608

Order 1280.

710 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3358	C₂	1	2	yes	maximal, complemented
2	C₅×#3278	C₂	1	2	yes	maximal, complemented
3	C₅×#1706	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#1685	C₂	1	2	yes	maximal, complemented
5	C₅×#1682	C₂	1	2	yes	maximal, complemented
6	C₅×#1649	C₂	1	2	yes	maximal, complemented
7	C₅×#1621	C₂	1	2	yes	maximal, complemented
8	C₅×#1589	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#137	—	2	2	no	—
11	C₅×#550	C₂²	1	2	yes	—
12	C₅×D₄×Q₈	C₂²	1	2	yes	—
13	C₅×#500	C₂²	1	1	yes	characteristic
14	C₅×#493	C₂²	1	1	yes	characteristic
15	C₅×#492	C₂²	1	2	yes	—
16	C₅×#486	—	2	2	no	—
17	C₅×#486	C₂²	1	2	yes	—
18	C₅×#474	—	2	2	no	—
19	C₅×#474	C₂²	1	2	yes	—
20	C₅×#468	—	2	2	no	—
21	C₅×#462	C₂²	1	2	yes	—
22	C₅×#460	C₂²	1	2	yes	—
23	C₅×#459	—	2	2	no	—
24	C₅×#459	C₂²	1	2	yes	—
25	C₅×#452	—	2	2	no	—
26	C₅×#448	—	2	2	no	—
27	C₅×C₈×Q₈	C₂²	1	2	yes	—
28	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₅×D₈	—	2	2	no	—
30	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
31	C₅×#436	C₂²	1	2	yes	—
32	C₅×#335	C₂²	1	1	yes	characteristic
33	C₅×#335	C₂²	1	1	yes	characteristic
34	C₅×#332	C₂²	1	2	yes	—
35	C₅×#331	C₂²	1	2	yes	—
36	C₅×Q₈⋊C₈	C₂²	1	2	yes	—
37	C₅×D₄⋊C₈	C₂²	1	2	yes	—
38	C₅×C₈²	C₂²	1	1	yes	characteristic
39	#19888	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂×C₅×#41	—	2	2	no	—
41	C₂²×C₅×Q₈	—	2	2	no	—
42	C₅×#137	—	4	8	no	—
43	C₂×C₅×Q₁₆	—	2	2	no	—
44	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic
45	C₂×C₅×SD₁₆	—	2	2	no	—
46	C₂×C₅×SD₁₆	—	2	2	no	—
47	C₂×C₅×SD₁₆	—	2	2	no	—
48	C₂×C₅×D₈	—	2	2	no	—
49	C₂×C₅×D₈	—	4	1	no	—
50	C₂×C₅×D₈	D₄	1	1	yes	characteristic
51	C₂×C₅×M₁₆	—	2	2	no	—
52	C₅×C₄⋊Q₈	—	2	2	no	—
53	C₅×C₄⋊Q₈	C₂³	1	2	yes	—
54	C₅×#127	—	2	2	no	—
55	C₅×#127	C₂³	1	2	yes	—
56	C₅×C₂²⋊Q₈	—	2	2	no	—
57	C₅×C₂²⋊Q₈	—	4	2	no	—
58	C₅×#121	—	2	2	no	—
59	C₅×#121	—	4	2	no	—
60	C₄×C₅×Q₈	C₂³	1	2	yes	—
61	C₄×C₅×D₄	—	2	2	no	—
62	C₄×C₅×D₄	—	2	2	no	—
63	C₄×C₅×D₄	C₂³	1	2	yes	—
64	C₅×#107	—	2	2	no	—
65	C₅×#107	—	4	1	no	—
66	C₅×#107	D₄	1	1	yes	characteristic
67	C₅×#107	D₄	1	1	yes	characteristic
68	C₅×C₄⋊C₈	—	2	2	no	—
69	C₅×C₄⋊C₈	C₂³	1	2	yes	—
70	C₅×C₄⋊C₈	C₂³	1	2	yes	—
71	C₅×#103	—	2	2	no	—
72	C₅×#103	D₄	1	2	yes	—
73	C₅×#102	—	2	2	no	—
74	C₅×#102	—	2	2	no	—
75	C₅×#102	—	2	2	no	—
76	C₅×#102	—	2	2	no	—
77	C₅×#102	—	2	2	no	—
78	C₅×#102	D₄	1	2	yes	—
79	C₅×C₂²⋊C₈	—	2	2	no	—
80	C₄×C₅×C₈	—	2	2	no	—
81	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
83	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
84	#3358	C₂×C₅	1	2	yes	complemented
85	#3278	C₂×C₅	1	2	yes	complemented
86	#1706	C₂×C₅	1	1	yes	characteristic, complemented
87	#1685	C₂×C₅	1	2	yes	complemented
88	#1682	C₂×C₅	1	2	yes	complemented
89	#1649	C₂×C₅	1	2	yes	complemented
90	#1621	C₂×C₅	1	2	yes	complemented
91	#1589	C₂×C₅	1	1	yes	characteristic, complemented
92	C₈×Q₁₆	C₂×C₅	1	1	yes	characteristic, complemented
93	C₅×#41	—	4	4	no	—
94	C₅×#41	—	8	2	no	—
95	C₂×C₅×Q₈	—	2	2	no	—
96	C₂×C₅×Q₈	—	4	2	no	—
97	C₂×C₅×Q₈	—	4	4	no	—
98	C₂×C₅×Q₈	—	8	2	no	—
99	C₂×C₅×Q₈	C₂×D₄	1	2	yes	—
100	C₂×C₅×D₄	—	2	2	no	—
101	C₂×C₅×D₄	—	2	2	no	—
102	C₂×C₅×D₄	—	2	2	no	—
103	C₂×C₅×D₄	—	4	2	no	—
104	C₂×C₅×D₄	—	4	2	no	—
105	C₂×C₅×D₄	C₂×D₄	1	2	yes	—
106	C₂²×C₄×C₅	—	2	2	no	—
107	C₂²×C₄×C₅	—	4	2	no	—
108	C₂²×C₄×C₅	—	4	2	no	—
109	C₅×Q₁₆	—	4	4	no	—
110	C₅×Q₁₆	D₈	1	4	yes	complemented
111	C₅×SD₁₆	—	4	4	no	—
112	C₅×SD₁₆	—	4	4	no	—
113	C₅×SD₁₆	—	8	2	no	—
114	C₅×D₈	—	4	1	no	—
115	C₅×D₈	—	4	4	no	—
116	C₅×D₈	—	8	2	no	—
117	C₅×M₁₆	—	4	4	no	—
118	C₂×C₅×C₈	—	2	1	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₂×C₅×C₈	—	2	2	no	—
121	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
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	C₅×C₄⋊C₄	—	2	2	no	—
126	C₅×C₄⋊C₄	—	2	2	no	—
127	C₅×C₄⋊C₄	—	2	2	no	—
128	C₅×C₄⋊C₄	—	4	2	no	—
129	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
130	C₅×C₄⋊C₄	C₂×D₄	1	2	yes	—
131	C₅×C₂²⋊C₄	—	2	2	no	—
132	C₅×C₂²⋊C₄	—	4	2	no	—
133	C₅×C₂²⋊C₄	—	4	2	no	—
134	C₄²×C₅	—	2	2	no	—
135	C₄²×C₅	C₂⁴	1	1	yes	characteristic
136	C₂×#137	—	2	2	no	—
137	#550	C₂²×C₅	1	2	yes	—
138	D₄×Q₈	C₂²×C₅	1	2	yes	—
139	#500	C₂²×C₅	1	1	yes	characteristic
140	#493	C₂²×C₅	1	1	yes	characteristic
141	#492	C₂²×C₅	1	2	yes	—
142	#486	—	2	2	no	—
143	#486	C₂²×C₅	1	2	yes	—
144	#474	—	2	2	no	—
145	#474	C₂²×C₅	1	2	yes	—
146	#468	—	2	2	no	—
147	#462	C₂²×C₅	1	2	yes	—
148	#460	C₂²×C₅	1	2	yes	—
149	#459	—	2	2	no	—
150	#459	C₂²×C₅	1	2	yes	—
151	#452	—	2	2	no	—
152	#448	—	2	2	no	—
153	C₈×Q₈	C₂²×C₅	1	2	yes	—
154	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
155	C₄×D₈	—	2	2	no	—
156	C₄×D₈	C₂²×C₅	1	1	yes	characteristic
157	#436	C₂²×C₅	1	2	yes	—
158	#335	C₂²×C₅	1	1	yes	characteristic
159	#335	C₂²×C₅	1	1	yes	characteristic
160	#332	C₂²×C₅	1	2	yes	—
161	#331	C₂²×C₅	1	2	yes	—
162	Q₈⋊C₈	C₂²×C₅	1	2	yes	—
163	D₄⋊C₈	C₂²×C₅	1	2	yes	—
164	C₈²	C₂²×C₅	1	1	yes	characteristic
165	C₂³×C₅	—	2	2	no	—
166	C₂³×C₅	—	4	2	no	—
167	C₅×Q₈	—	2	2	no	—
168	C₅×Q₈	—	4	2	no	—
169	C₅×Q₈	—	4	2	no	—
170	C₅×Q₈	—	8	4	no	—
171	C₅×Q₈	C₂×D₈	1	4	yes	—
172	C₅×D₄	—	2	2	no	—
173	C₅×D₄	—	2	2	no	—
174	C₅×D₄	—	4	2	no	—
175	C₅×D₄	—	4	2	no	—
176	C₅×D₄	—	4	2	no	—
177	C₅×D₄	—	4	2	no	—
178	C₅×D₄	—	8	4	no	—
179	C₅×D₄	—	8	2	no	—
180	C₅×D₄	—	16	2	no	—
181	C₂×C₄×C₅	—	2	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	4	2	no	—
185	C₂×C₄×C₅	—	4	2	no	—
186	C₂×C₄×C₅	—	4	2	no	—
187	C₂×C₄×C₅	—	8	4	no	—
188	C₂×C₄×C₅	—	8	4	no	—
189	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
191	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
192	C₅×C₈	—	2	2	no	—
193	C₅×C₈	—	2	1	no	—
194	C₅×C₈	—	2	1	no	—
195	C₅×C₈	—	2	1	no	—
196	C₅×C₈	—	4	2	no	—
197	C₅×C₈	—	4	2	no	—
198	C₅×C₈	C₂×D₈	1	2	yes	—
199	C₂×#41	—	2	2	no	—
200	C₂²×Q₈	—	2	2	no	—
201	#137	—	4	8	no	—
202	C₂×Q₁₆	—	2	2	no	—
203	C₂×Q₁₆	C₅×D₄	1	1	yes	characteristic
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×SD₁₆	—	2	2	no	—
207	C₂×D₈	—	2	2	no	—
208	C₂×D₈	—	4	1	no	—
209	C₂×D₈	C₅×D₄	1	1	yes	characteristic
210	C₂×M₁₆	—	2	2	no	—
211	C₄⋊Q₈	—	2	2	no	—
212	C₄⋊Q₈	C₂³×C₅	1	2	yes	—
213	#127	—	2	2	no	—
214	#127	C₂³×C₅	1	2	yes	—
215	C₂²⋊Q₈	—	2	2	no	—
216	C₂²⋊Q₈	—	4	2	no	—
217	#121	—	2	2	no	—
218	#121	—	4	2	no	—
219	C₄×Q₈	C₂³×C₅	1	2	yes	—
220	C₄×D₄	—	2	2	no	—
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	C₂³×C₅	1	2	yes	—
223	#107	—	2	2	no	—
224	#107	—	4	1	no	—
225	#107	C₅×D₄	1	1	yes	characteristic
226	#107	C₅×D₄	1	1	yes	characteristic
227	C₄⋊C₈	—	2	2	no	—
228	C₄⋊C₈	C₂³×C₅	1	2	yes	—
229	C₄⋊C₈	C₂³×C₅	1	2	yes	—
230	#103	—	2	2	no	—
231	#103	C₅×D₄	1	2	yes	—
232	#102	—	2	2	no	—
233	#102	—	2	2	no	—
234	#102	—	2	2	no	—
235	#102	—	2	2	no	—
236	#102	—	2	2	no	—
237	#102	C₅×D₄	1	2	yes	—
238	C₂²⋊C₈	—	2	2	no	—
239	C₄×C₈	—	2	2	no	—
240	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
242	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
243	C₂²×C₅	—	4	2	no	—
244	C₂²×C₅	—	4	2	no	—
245	C₂²×C₅	—	4	2	no	—
246	C₂²×C₅	—	8	2	no	—
247	C₂²×C₅	—	8	2	no	—
248	C₂²×C₅	—	8	2	no	—
249	C₂²×C₅	D₄²	1	1	yes	characteristic
250	C₄×C₅	—	2	4	no	—
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	4	2	no	—
253	C₄×C₅	—	4	2	no	—
254	C₄×C₅	—	8	2	no	—
255	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
256	C₄×C₅	#575	1	1	yes	characteristic
257	C₄×C₅	#576	1	1	yes	characteristic
258	C₄×C₅	#578	1	1	yes	characteristic
259	#41	—	4	4	no	—
260	#41	—	8	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	4	2	no	—
263	C₂×Q₈	—	4	4	no	—
264	C₂×Q₈	—	8	2	no	—
265	C₂×Q₈	C₂×C₅×D₄	1	2	yes	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	4	2	no	—
270	C₂×D₄	—	4	2	no	—
271	C₂×D₄	C₂×C₅×D₄	1	2	yes	—
272	C₂²×C₄	—	2	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	—	4	2	no	—
275	Q₁₆	—	4	4	no	—
276	Q₁₆	C₅×D₈	1	4	yes	complemented
277	SD₁₆	—	4	4	no	—
278	SD₁₆	—	4	4	no	—
279	SD₁₆	—	8	2	no	—
280	D₈	—	4	1	no	—
281	D₈	—	4	4	no	—
282	D₈	—	8	2	no	—
283	M₁₆	—	4	4	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
288	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
289	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
290	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	4	2	no	—
295	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
296	C₄⋊C₄	C₂×C₅×D₄	1	2	yes	—
297	C₂²⋊C₄	—	2	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	C₂⁴×C₅	1	1	yes	characteristic
302	C₂×C₅	—	8	2	no	—
303	C₂×C₅	—	16	2	no	—
304	C₂×C₅	#3277	1	1	yes	characteristic
305	C₂×C₅	D₄×D₈	1	1	yes	characteristic
306	C₂×C₅	#3291	1	1	yes	characteristic
307	C₂³	—	2	2	no	—
308	C₂³	—	4	2	no	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	4	no	—
313	Q₈	C₂×C₅×D₈	1	4	yes	—
314	D₄	—	2	2	no	—
315	D₄	—	2	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	2	no	—
320	D₄	—	8	4	no	—
321	D₄	—	8	2	no	—
322	D₄	—	16	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	8	4	no	—
330	C₂×C₄	—	8	4	no	—
331	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
333	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
334	C₈	—	2	2	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	C₂×C₅×D₈	1	2	yes	—
341	C₅	#19888	1	1	yes	characteristic, direct product factor, Sylow
342	C₂²	—	4	2	no	—
343	C₂²	—	4	2	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	—	8	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	C₅×D₄²	1	1	yes	characteristic
349	C₄	—	2	4	no	—
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	8	2	no	—
354	C₄	C₂²×C₅×D₈	1	1	yes	characteristic
355	C₄	C₅×#575	1	1	yes	characteristic
356	C₄	C₅×#576	1	1	yes	characteristic
357	C₄	C₅×#578	1	1	yes	characteristic
358	C₂	—	8	2	no	—
359	C₂	—	16	2	no	—
360	C₂	C₅×#3277	1	1	yes	characteristic
361	C₂	C₅×D₄×D₈	1	1	yes	characteristic
362	C₂	C₅×#3291	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₅
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
1. C₅×#3358	C₂	semidirect
2. C₅×#3278	C₂	semidirect
3. C₅×#1706	C₂	semidirect
4. C₅×#1685	C₂	semidirect
5. C₅×#1682	C₂	semidirect
6. C₅×#1649	C₂	semidirect
7. C₅×#1621	C₂	semidirect
8. C₅×#1589	C₂	semidirect
9. C₅×C₈×Q₁₆	C₂	semidirect
39. #19888	C₅	direct (N × H)
84. #3358	C₂×C₅	semidirect
85. #3278	C₂×C₅	semidirect
86. #1706	C₂×C₅	semidirect
87. #1685	C₂×C₅	semidirect
88. #1682	C₂×C₅	semidirect
89. #1649	C₂×C₅	semidirect
90. #1621	C₂×C₅	semidirect
91. #1589	C₂×C₅	semidirect
92. C₈×Q₁₆	C₂×C₅	semidirect
110. C₅×Q₁₆	D₈	semidirect
276. Q₁₆	C₅×D₈	semidirect
341. C₅	#19888	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×#3358	2
C₅×#3278	2
C₅×#1706	2
C₅×#1685	2
C₅×#1682	2
C₅×#1649	2
C₅×#1621	2
C₅×#1589	2
C₅×C₈×Q₁₆	2
#19888	5

Subgroups — C5×#19888 #11942608

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#19888 #11942608