Subgroups — C₅×#33872 #11956592

Order 1280.

568 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×#3341	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#3287	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#2234	C₂	1	2	yes	maximal, complemented
5	C₅×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2215	C₂	1	2	yes	maximal, complemented
7	C₅×#2206	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2186	C₂	1	2	yes	maximal, complemented
9	C₄×C₅×SD₃₂	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×#555	—	2	1	no	—
14	C₅×#550	—	2	1	no	—
15	C₂×C₅×SD₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₅×D₁₆	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×#482	—	2	2	no	—
21	C₅×#476	—	2	1	no	—
22	C₅×#466	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×D₈	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	1	yes	characteristic
31	C₅×#366	C₂²	1	2	yes	—
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×#357	C₂²	1	2	yes	—
36	C₅×#357	C₂²	1	2	yes	—
37	C₅×#357	C₂²	1	1	yes	characteristic
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	#33872	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×SD₁₆	—	2	2	no	—
47	C₂×C₅×D₈	—	2	1	no	—
48	C₂×C₅×D₈	C₂³	1	2	yes	—
49	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
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×#124	—	4	2	no	—
55	C₅×#121	—	2	2	no	—
56	C₅×#121	—	2	2	no	—
57	C₅×#121	—	4	1	no	—
58	C₅×#121	—	4	2	no	—
59	C₄×C₅×Q₈	—	2	1	no	—
60	C₄×C₅×D₄	—	2	1	no	—
61	C₄×C₅×D₄	—	4	1	no	—
62	C₄×C₅×D₄	D₄	1	1	yes	characteristic
63	C₅×#117	—	2	2	no	—
64	C₅×#117	—	4	1	no	—
65	C₅×SD₃₂	—	2	2	no	—
66	C₅×D₁₆	—	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×#102	—	2	1	no	—
79	C₅×#102	—	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	#3341	C₂×C₅	1	1	yes	characteristic, complemented
84	#3287	C₂×C₅	1	1	yes	characteristic, complemented
85	#2240	C₂×C₅	1	1	yes	characteristic, complemented
86	#2234	C₂×C₅	1	2	yes	complemented
87	#2227	C₂×C₅	1	1	yes	characteristic, complemented
88	#2215	C₂×C₅	1	2	yes	complemented
89	#2206	C₂×C₅	1	1	yes	characteristic, complemented
90	#2186	C₂×C₅	1	2	yes	complemented
91	C₄×SD₃₂	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	—	4	1	no	—
98	C₂×C₅×D₄	—	2	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	2	no	—
103	C₂×C₅×D₄	—	4	1	no	—
104	C₂×C₅×D₄	—	4	1	no	—
105	C₂×C₅×D₄	D₈	1	1	yes	characteristic
106	C₂²×C₄×C₅	—	4	1	no	—
107	C₂²×C₄×C₅	—	4	2	no	—
108	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
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	1	no	—
114	C₅×D₈	—	2	2	no	—
115	C₅×D₈	—	2	1	no	—
116	C₅×D₈	—	8	1	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
120	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
121	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
122	C₅×C₄⋊C₄	—	2	1	no	—
123	C₅×C₄⋊C₄	—	2	2	no	—
124	C₅×C₄⋊C₄	—	2	1	no	—
125	C₅×C₄⋊C₄	—	4	1	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	1	no	—
131	C₅×C₂²⋊C₄	—	4	2	no	—
132	C₅×C₂²⋊C₄	D₈	1	2	yes	—
133	C₄²×C₅	—	4	2	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	#555	—	2	1	no	—
141	#550	—	2	1	no	—
142	C₂×SD₃₂	C₂²×C₅	1	1	yes	characteristic
143	C₂×D₁₆	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	#482	—	2	2	no	—
148	#476	—	2	1	no	—
149	#466	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×D₈	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	1	yes	characteristic
158	#366	C₂²×C₅	1	2	yes	—
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	#357	C₂²×C₅	1	2	yes	—
163	#357	C₂²×C₅	1	2	yes	—
164	#357	C₂²×C₅	1	1	yes	characteristic
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	1	no	—
169	C₂³×C₅	—	4	2	no	—
170	C₂³×C₅	C₂×D₈	1	2	yes	—
171	C₅×Q₈	—	2	2	no	—
172	C₅×Q₈	—	4	1	no	—
173	C₅×Q₈	—	16	1	no	—
174	C₅×D₄	—	2	2	no	—
175	C₅×D₄	—	4	2	no	—
176	C₅×D₄	—	4	1	no	—
177	C₅×D₄	—	4	2	no	—
178	C₅×D₄	—	4	1	no	—
179	C₅×D₄	—	8	4	no	—
180	C₅×D₄	—	8	2	no	—
181	C₅×D₄	—	16	1	no	—
182	C₅×D₄	—	16	2	no	—
183	C₅×D₄	—	16	2	no	—
184	C₂×C₄×C₅	—	2	2	no	—
185	C₂×C₄×C₅	—	2	2	no	—
186	C₂×C₄×C₅	—	4	1	no	—
187	C₂×C₄×C₅	—	4	1	no	—
188	C₂×C₄×C₅	—	4	1	no	—
189	C₂×C₄×C₅	—	4	2	no	—
190	C₂×C₄×C₅	—	8	4	no	—
191	C₂×C₄×C₅	—	8	2	no	—
192	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
193	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
194	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
195	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
196	C₅×C₈	—	2	2	no	—
197	C₅×C₈	—	2	1	no	—
198	C₅×C₈	C₂×#41	1	1	yes	characteristic
199	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂×#41	—	2	2	no	—
201	#135	—	2	4	no	—
202	#135	—	4	2	no	—
203	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	C₂³×C₅	1	2	yes	—
207	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₅	1	2	yes	—
209	C₄⋊Q₈	—	2	1	no	—
210	#127	—	2	1	no	—
211	#127	—	4	1	no	—
212	#124	—	4	2	no	—
213	#121	—	2	2	no	—
214	#121	—	2	2	no	—
215	#121	—	4	1	no	—
216	#121	—	4	2	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	4	1	no	—
220	C₄×D₄	C₅×D₄	1	1	yes	characteristic
221	#117	—	2	2	no	—
222	#117	—	4	1	no	—
223	SD₃₂	—	2	2	no	—
224	D₁₆	—	4	2	no	—
225	C₂×C₁₆	—	4	2	no	—
226	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
227	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
228	C₂×C₁₆	C₂³×C₅	1	2	yes	—
229	#107	—	2	1	no	—
230	#107	C₂³×C₅	1	2	yes	—
231	#107	C₂³×C₅	1	1	yes	characteristic
232	#107	C₂³×C₅	1	1	yes	characteristic
233	#106	—	2	2	no	—
234	C₄⋊C₈	C₅×D₄	1	1	yes	characteristic
235	#103	—	2	1	no	—
236	#102	—	2	1	no	—
237	#102	—	2	2	no	—
238	#102	—	2	2	no	—
239	C₂²⋊C₈	C₅×D₄	1	2	yes	—
240	C₄×C₈	C₂³×C₅	1	1	yes	characteristic
241	C₂²×C₅	—	2	2	no	—
242	C₂²×C₅	—	2	2	no	—
243	C₂²×C₅	—	2	2	no	—
244	C₂²×C₅	—	8	1	no	—
245	C₂²×C₅	—	8	2	no	—
246	C₂²×C₅	—	8	2	no	—
247	C₂²×C₅	—	8	1	no	—
248	C₂²×C₅	—	8	2	no	—
249	C₂²×C₅	—	8	1	no	—
250	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
251	C₄×C₅	—	2	1	no	—
252	C₄×C₅	—	2	2	no	—
253	C₄×C₅	—	4	2	no	—
254	C₄×C₅	—	8	1	no	—
255	C₄×C₅	—	8	1	no	—
256	C₄×C₅	—	8	2	no	—
257	C₄×C₅	#508	1	2	yes	—
258	C₄×C₅	#546	1	1	yes	characteristic
259	C₄×C₅	#577	1	1	yes	characteristic
260	#41	—	4	4	no	—
261	#41	—	8	2	no	—
262	C₂×Q₈	—	2	1	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	4	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	C₅×D₈	1	1	yes	characteristic
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	C₅×#41	1	2	yes	—
277	SD₁₆	—	4	4	no	—
278	D₈	—	2	1	no	—
279	D₈	—	2	2	no	—
280	D₈	—	2	1	no	—
281	D₈	—	2	2	no	—
282	D₈	—	8	1	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
286	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
287	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	1	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	C₅×D₈	1	2	yes	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×C₅×D₄	1	1	yes	characteristic
302	C₁₆	—	2	1	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	2	no	—
305	C₂×C₅	—	4	2	no	—
306	C₂×C₅	—	16	1	no	—
307	C₂×C₅	—	16	2	no	—
308	C₂×C₅	#3294	1	1	yes	characteristic
309	C₂×C₅	C₂×#508	1	1	yes	characteristic
310	C₂×C₅	#3415	1	1	yes	characteristic
311	C₂³	—	4	1	no	—
312	C₂³	—	4	2	no	—
313	C₂³	C₂×C₅×D₈	1	2	yes	—
314	Q₈	—	2	2	no	—
315	Q₈	—	4	1	no	—
316	Q₈	—	16	1	no	—
317	D₄	—	2	2	no	—
318	D₄	—	4	2	no	—
319	D₄	—	4	1	no	—
320	D₄	—	4	2	no	—
321	D₄	—	4	1	no	—
322	D₄	—	8	4	no	—
323	D₄	—	8	2	no	—
324	D₄	—	16	1	no	—
325	D₄	—	16	2	no	—
326	D₄	—	16	2	no	—
327	C₂×C₄	—	2	2	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	2	no	—
333	C₂×C₄	—	8	4	no	—
334	C₂×C₄	—	8	2	no	—
335	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
336	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
337	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
338	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
339	C₈	—	2	2	no	—
340	C₈	—	2	1	no	—
341	C₈	C₂×C₅×#41	1	1	yes	characteristic
342	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
343	C₅	#33872	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²	—	2	2	no	—
345	C₂²	—	2	2	no	—
346	C₂²	—	2	2	no	—
347	C₂²	—	8	2	no	—
348	C₂²	—	8	1	no	—
349	C₂²	—	8	1	no	—
350	C₂²	—	8	2	no	—
351	C₂²	—	8	2	no	—
352	C₂²	—	8	1	no	—
353	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
354	C₄	—	2	2	no	—
355	C₄	—	2	1	no	—
356	C₄	—	4	2	no	—
357	C₄	—	8	1	no	—
358	C₄	—	8	1	no	—
359	C₄	—	8	2	no	—
360	C₄	C₅×#508	1	2	yes	—
361	C₄	C₅×#546	1	1	yes	characteristic
362	C₄	C₅×#577	1	1	yes	characteristic
363	C₂	—	4	2	no	—
364	C₂	—	16	2	no	—
365	C₂	—	16	1	no	—
366	C₂	C₅×#3294	1	1	yes	characteristic
367	C₂	C₂×C₅×#508	1	1	yes	characteristic
368	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. C₅×#3341	C₂	semidirect
2. C₅×#3287	C₂	semidirect
3. C₅×#2240	C₂	semidirect
4. C₅×#2234	C₂	semidirect
5. C₅×#2227	C₂	semidirect
6. C₅×#2215	C₂	semidirect
7. C₅×#2206	C₂	semidirect
8. C₅×#2186	C₂	semidirect
9. C₄×C₅×SD₃₂	C₂	semidirect
10. C₅×D₄×C₁₆	C₂	semidirect
11. C₅×#2138	C₂	semidirect
41. #33872	C₅	direct (N × H)
83. #3341	C₂×C₅	semidirect
84. #3287	C₂×C₅	semidirect
85. #2240	C₂×C₅	semidirect
86. #2234	C₂×C₅	semidirect
87. #2227	C₂×C₅	semidirect
88. #2215	C₂×C₅	semidirect
89. #2206	C₂×C₅	semidirect
90. #2186	C₂×C₅	semidirect
91. C₄×SD₃₂	C₂×C₅	semidirect
92. D₄×C₁₆	C₂×C₅	semidirect
93. #2138	C₂×C₅	semidirect
343. C₅	#33872	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×#3341	2
C₅×#3287	2
C₅×#2240	2
C₅×#2234	2
C₅×#2227	2
C₅×#2215	2
C₅×#2206	2
C₅×#2186	2
C₄×C₅×SD₃₂	2
C₅×D₄×C₁₆	2
C₅×#2138	2
#33872	5

Subgroups — C5×#33872 #11956592

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33872 #11956592