Subgroups — C₅×#33871 #11956591

Order 1280.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3342	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3287	C₂	1	1	yes	maximal, characteristic
3	C₅×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2234	C₂	1	2	yes	maximal, complemented
5	C₅×#2225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2217	C₂	1	2	yes	maximal, complemented
7	C₅×#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×D₄×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅×#2138	C₂	1	2	yes	maximal, complemented
12	C₂×C₅×#135	C₂²	1	2	yes	—
13	C₅×#556	—	2	1	no	—
14	C₅×#550	—	2	1	no	—
15	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₅×SD₃₂	C₂²	1	2	yes	—
17	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
18	C₅×#500	C₂²	1	1	yes	characteristic
19	C₅×#493	C₂²	1	1	yes	characteristic
20	C₅×#485	—	2	2	no	—
21	C₅×#475	—	2	1	no	—
22	C₅×#467	C₂²	1	2	yes	—
23	C₅×#466	C₂²	1	2	yes	—
24	C₅×#460	—	2	1	no	—
25	C₅×#449	—	2	2	no	—
26	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₅×Q₁₆	C₂²	1	1	yes	characteristic
28	C₅×C₈×D₄	C₂²	1	1	yes	characteristic
29	C₅×#427	C₂²	1	2	yes	—
30	C₅×#367	C₂²	1	2	yes	—
31	C₅×#366	C₂²	1	1	yes	characteristic
32	C₅×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₅×#358	C₂²	1	1	yes	characteristic
34	C₅×#358	C₂²	1	2	yes	—
35	C₅×#358	C₂²	1	1	yes	characteristic
36	C₅×#357	C₂²	1	2	yes	—
37	C₅×#357	C₂²	1	2	yes	—
38	C₅×#357	C₂²	1	2	yes	—
39	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₅×C₁₆	C₂²	1	1	yes	characteristic
41	#33871	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅×#41	—	2	2	no	—
43	C₅×#135	—	2	4	no	—
44	C₅×#135	—	4	2	no	—
45	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
47	C₂×C₅×SD₁₆	—	2	2	no	—
48	C₂×C₅×D₈	—	2	1	no	—
49	C₂×C₅×D₈	C₂³	1	2	yes	—
50	C₂²×C₅×C₈	C₂³	1	2	yes	—
51	C₅×C₄⋊Q₈	—	2	1	no	—
52	C₅×#127	—	2	1	no	—
53	C₅×#127	—	4	1	no	—
54	C₅×#126	—	4	2	no	—
55	C₅×#124	—	4	1	no	—
56	C₅×C₂²⋊Q₈	—	2	2	no	—
57	C₅×#121	—	2	2	no	—
58	C₅×#121	—	4	2	no	—
59	C₄×C₅×Q₈	—	2	1	no	—
60	C₄×C₅×Q₈	—	2	1	no	—
61	C₄×C₅×Q₈	—	4	1	no	—
62	C₄×C₅×D₄	—	4	1	no	—
63	C₄×C₅×D₄	D₄	1	1	yes	characteristic
64	C₅×#117	—	2	2	no	—
65	C₅×Q₃₂	—	2	2	no	—
66	C₅×SD₃₂	—	4	2	no	—
67	C₂×C₅×C₁₆	—	4	2	no	—
68	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₅×C₁₆	C₂³	1	2	yes	—
71	C₅×#107	—	2	1	no	—
72	C₅×#107	C₂³	1	1	yes	characteristic
73	C₅×#107	C₂³	1	2	yes	—
74	C₅×#107	C₂³	1	1	yes	characteristic
75	C₅×#106	—	2	2	no	—
76	C₅×C₄⋊C₈	D₄	1	1	yes	characteristic
77	C₅×#103	—	2	1	no	—
78	C₅×#103	—	2	1	no	—
79	C₅×#103	—	2	2	no	—
80	C₅×#102	—	2	2	no	—
81	C₅×C₂²⋊C₈	D₄	1	2	yes	—
82	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
83	#3342	C₂×C₅	1	1	yes	characteristic, complemented
84	#3287	C₂×C₅	1	1	yes	characteristic
85	#2240	C₂×C₅	1	1	yes	characteristic, complemented
86	#2234	C₂×C₅	1	2	yes	complemented
87	#2225	C₂×C₅	1	1	yes	characteristic, complemented
88	#2217	C₂×C₅	1	2	yes	complemented
89	#2207	C₂×C₅	1	1	yes	characteristic, complemented
90	#2189	C₂×C₅	1	2	yes	complemented
91	C₄×Q₃₂	C₂×C₅	1	1	yes	characteristic, complemented
92	D₄×C₁₆	C₂×C₅	1	1	yes	characteristic, complemented
93	#2138	C₂×C₅	1	2	yes	complemented
94	C₅×#41	—	4	4	no	—
95	C₅×#41	—	8	2	no	—
96	C₂×C₅×Q₈	—	2	1	no	—
97	C₂×C₅×Q₈	—	2	1	no	—
98	C₂×C₅×Q₈	—	4	1	no	—
99	C₂×C₅×D₄	—	2	2	no	—
100	C₂×C₅×D₄	—	4	2	no	—
101	C₂×C₅×D₄	—	4	2	no	—
102	C₂×C₅×D₄	—	4	1	no	—
103	C₂×C₅×D₄	—	4	1	no	—
104	C₂×C₅×D₄	D₈	1	1	yes	characteristic
105	C₂²×C₄×C₅	—	4	2	no	—
106	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
107	C₅×Q₁₆	—	2	1	no	—
108	C₅×Q₁₆	—	2	1	no	—
109	C₅×Q₁₆	—	2	1	no	—
110	C₅×Q₁₆	#41	1	2	yes	—
111	C₅×SD₁₆	—	4	4	no	—
112	C₅×D₈	—	2	2	no	—
113	C₅×D₈	—	2	2	no	—
114	C₅×D₈	—	8	1	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
118	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
119	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
120	C₅×C₄⋊C₄	—	2	1	no	—
121	C₅×C₄⋊C₄	—	2	2	no	—
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	4	1	no	—
124	C₅×C₄⋊C₄	—	4	1	no	—
125	C₅×C₄⋊C₄	—	4	2	no	—
126	C₅×C₄⋊C₄	—	4	1	no	—
127	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
128	C₅×C₂²⋊C₄	—	4	2	no	—
129	C₅×C₂²⋊C₄	—	4	2	no	—
130	C₅×C₂²⋊C₄	—	4	2	no	—
131	C₅×C₂²⋊C₄	D₈	1	2	yes	—
132	C₄²×C₅	—	4	2	no	—
133	C₄²×C₅	—	4	1	no	—
134	C₄²×C₅	—	4	1	no	—
135	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
136	C₅×C₁₆	—	2	1	no	—
137	C₅×C₁₆	—	2	1	no	—
138	C₅×C₁₆	—	2	2	no	—
139	C₂×#135	C₂²×C₅	1	2	yes	—
140	#556	—	2	1	no	—
141	#550	—	2	1	no	—
142	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic
143	C₂×SD₃₂	C₂²×C₅	1	2	yes	—
144	C₂²×C₁₆	C₂²×C₅	1	2	yes	—
145	#500	C₂²×C₅	1	1	yes	characteristic
146	#493	C₂²×C₅	1	1	yes	characteristic
147	#485	—	2	2	no	—
148	#475	—	2	1	no	—
149	#467	C₂²×C₅	1	2	yes	—
150	#466	C₂²×C₅	1	2	yes	—
151	#460	—	2	1	no	—
152	#449	—	2	2	no	—
153	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
154	C₄×Q₁₆	C₂²×C₅	1	1	yes	characteristic
155	C₈×D₄	C₂²×C₅	1	1	yes	characteristic
156	#427	C₂²×C₅	1	2	yes	—
157	#367	C₂²×C₅	1	2	yes	—
158	#366	C₂²×C₅	1	1	yes	characteristic
159	C₄⋊C₁₆	C₂²×C₅	1	1	yes	characteristic
160	#358	C₂²×C₅	1	1	yes	characteristic
161	#358	C₂²×C₅	1	2	yes	—
162	#358	C₂²×C₅	1	1	yes	characteristic
163	#357	C₂²×C₅	1	2	yes	—
164	#357	C₂²×C₅	1	2	yes	—
165	#357	C₂²×C₅	1	2	yes	—
166	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
167	C₄×C₁₆	C₂²×C₅	1	1	yes	characteristic
168	C₂³×C₅	—	4	2	no	—
169	C₂³×C₅	C₂×D₈	1	2	yes	—
170	C₅×Q₈	—	2	2	no	—
171	C₅×Q₈	—	4	1	no	—
172	C₅×Q₈	—	4	1	no	—
173	C₅×Q₈	—	4	1	no	—
174	C₅×Q₈	—	8	2	no	—
175	C₅×D₄	—	2	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	1	no	—
181	C₅×D₄	—	16	2	no	—
182	C₂×C₄×C₅	—	2	2	no	—
183	C₂×C₄×C₅	—	2	2	no	—
184	C₂×C₄×C₅	—	4	1	no	—
185	C₂×C₄×C₅	—	4	1	no	—
186	C₂×C₄×C₅	—	4	2	no	—
187	C₂×C₄×C₅	—	4	1	no	—
188	C₂×C₄×C₅	—	4	1	no	—
189	C₂×C₄×C₅	—	8	4	no	—
190	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
191	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
192	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
193	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
194	C₅×C₈	—	2	2	no	—
195	C₅×C₈	—	2	1	no	—
196	C₅×C₈	C₂×#41	1	1	yes	characteristic
197	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
198	C₂×#41	—	2	2	no	—
199	#135	—	2	4	no	—
200	#135	—	4	2	no	—
201	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
202	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
203	C₂×SD₁₆	—	2	2	no	—
204	C₂×D₈	—	2	1	no	—
205	C₂×D₈	C₂³×C₅	1	2	yes	—
206	C₂²×C₈	C₂³×C₅	1	2	yes	—
207	C₄⋊Q₈	—	2	1	no	—
208	#127	—	2	1	no	—
209	#127	—	4	1	no	—
210	#126	—	4	2	no	—
211	#124	—	4	1	no	—
212	C₂²⋊Q₈	—	2	2	no	—
213	#121	—	2	2	no	—
214	#121	—	4	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	1	no	—
217	C₄×Q₈	—	4	1	no	—
218	C₄×D₄	—	4	1	no	—
219	C₄×D₄	C₅×D₄	1	1	yes	characteristic
220	#117	—	2	2	no	—
221	Q₃₂	—	2	2	no	—
222	SD₃₂	—	4	2	no	—
223	C₂×C₁₆	—	4	2	no	—
224	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
226	C₂×C₁₆	C₂³×C₅	1	2	yes	—
227	#107	—	2	1	no	—
228	#107	C₂³×C₅	1	1	yes	characteristic
229	#107	C₂³×C₅	1	2	yes	—
230	#107	C₂³×C₅	1	1	yes	characteristic
231	#106	—	2	2	no	—
232	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
233	#103	—	2	1	no	—
234	#103	—	2	1	no	—
235	#103	—	2	2	no	—
236	#102	—	2	2	no	—
237	C₂²⋊C₈	C₅×D₄	1	2	yes	—
238	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
239	C₂²×C₅	—	2	2	no	—
240	C₂²×C₅	—	2	2	no	—
241	C₂²×C₅	—	2	2	no	—
242	C₂²×C₅	—	8	2	no	—
243	C₂²×C₅	—	8	2	no	—
244	C₂²×C₅	—	8	2	no	—
245	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
246	C₄×C₅	—	2	1	no	—
247	C₄×C₅	—	2	2	no	—
248	C₄×C₅	—	4	2	no	—
249	C₄×C₅	—	8	2	no	—
250	C₄×C₅	—	8	1	no	—
251	C₄×C₅	—	8	1	no	—
252	C₄×C₅	—	8	1	no	—
253	C₄×C₅	#508	1	2	yes	—
254	C₄×C₅	#546	1	1	yes	characteristic
255	C₄×C₅	#577	1	1	yes	characteristic
256	#41	—	4	4	no	—
257	#41	—	8	2	no	—
258	C₂×Q₈	—	2	1	no	—
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	4	1	no	—
261	C₂×D₄	—	2	2	no	—
262	C₂×D₄	—	4	2	no	—
263	C₂×D₄	—	4	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂×D₄	C₅×D₈	1	1	yes	characteristic
267	C₂²×C₄	—	4	2	no	—
268	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	C₅×#41	1	2	yes	—
273	SD₁₆	—	4	4	no	—
274	D₈	—	2	2	no	—
275	D₈	—	2	2	no	—
276	D₈	—	8	1	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
280	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
281	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	2	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	C₅×D₈	1	2	yes	—
294	C₄²	—	4	2	no	—
295	C₄²	—	4	1	no	—
296	C₄²	—	4	1	no	—
297	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	1	no	—
300	C₁₆	—	2	2	no	—
301	C₂×C₅	—	4	2	no	—
302	C₂×C₅	—	16	2	no	—
303	C₂×C₅	#3294	1	1	yes	characteristic
304	C₂×C₅	C₂×#508	1	1	yes	characteristic
305	C₂×C₅	#3416	1	1	yes	characteristic
306	C₂³	—	4	2	no	—
307	C₂³	C₂×C₅×D₈	1	2	yes	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	8	2	no	—
313	D₄	—	2	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	8	4	no	—
317	D₄	—	8	2	no	—
318	D₄	—	16	1	no	—
319	D₄	—	16	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	8	4	no	—
328	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
329	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
330	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
331	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
332	C₈	—	2	2	no	—
333	C₈	—	2	1	no	—
334	C₈	C₂×C₅×#41	1	1	yes	characteristic
335	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
336	C₅	#33871	1	1	yes	characteristic, direct product factor, Sylow
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	C₅×#508	1	2	yes	—
352	C₄	C₅×#546	1	1	yes	characteristic
353	C₄	C₅×#577	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	16	2	no	—
356	C₂	C₅×#3294	1	1	yes	characteristic
357	C₂	C₂×C₅×#508	1	1	yes	characteristic
358	C₂	C₅×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₅×#3287	C₂
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. C₅×#3342	C₂	semidirect
3. C₅×#2240	C₂	semidirect
4. C₅×#2234	C₂	semidirect
5. C₅×#2225	C₂	semidirect
6. C₅×#2217	C₂	semidirect
7. C₅×#2207	C₂	semidirect
8. C₅×#2189	C₂	semidirect
9. C₄×C₅×Q₃₂	C₂	semidirect
10. C₅×D₄×C₁₆	C₂	semidirect
11. C₅×#2138	C₂	semidirect
41. #33871	C₅	direct (N × H)
83. #3342	C₂×C₅	semidirect
85. #2240	C₂×C₅	semidirect
86. #2234	C₂×C₅	semidirect
87. #2225	C₂×C₅	semidirect
88. #2217	C₂×C₅	semidirect
89. #2207	C₂×C₅	semidirect
90. #2189	C₂×C₅	semidirect
91. C₄×Q₃₂	C₂×C₅	semidirect
92. D₄×C₁₆	C₂×C₅	semidirect
93. #2138	C₂×C₅	semidirect
336. C₅	#33871	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×#3342	2
C₅×#3287	2
C₅×#2240	2
C₅×#2234	2
C₅×#2225	2
C₅×#2217	2
C₅×#2207	2
C₅×#2189	2
C₄×C₅×Q₃₂	2
C₅×D₄×C₁₆	2
C₅×#2138	2
#33871	5

Subgroups — C5×#33871 #11956591

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33871 #11956591