Subgroups — C₅×#33633 #11956353

Order 1280.

534 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×#3164	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2951	C₂	1	1	yes	maximal, characteristic
3	C₅×#2229	C₂	1	2	yes	maximal, complemented
4	C₅×#2228	C₂	1	2	yes	maximal, complemented
5	C₅×#2227	C₂	1	2	yes	maximal, complemented
6	C₅×#2224	C₂	1	2	yes	maximal, complemented
7	C₅×#2159	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₅×#366	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#2151	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#2139	C₂	1	1	yes	maximal, characteristic
11	C₅×#2138	C₂	1	1	yes	maximal, characteristic
12	C₂×C₅×#135	C₂²	1	1	yes	characteristic
13	C₅×#544	—	2	1	no	—
14	C₅×#520	—	2	1	no	—
15	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
16	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
17	C₅×#501	—	2	2	no	—
18	C₅×#500	C₂²	1	2	yes	—
19	C₅×#499	C₂²	1	2	yes	—
20	C₅×#440	—	2	2	no	—
21	C₄×C₅×Q₁₆	C₂²	1	2	yes	—
22	C₄×C₅×D₈	C₂²	1	2	yes	—
23	C₅×#428	—	2	1	no	—
24	C₅×#427	C₂²	1	1	yes	characteristic
25	C₂×C₅×#107	C₂²	1	1	yes	characteristic
26	C₂×C₅×#107	C₂²	1	1	yes	characteristic
27	C₅×#417	—	2	1	no	—
28	C₅×#405	C₂²	1	1	yes	characteristic
29	C₅×#367	C₂²	1	2	yes	—
30	C₅×#366	C₂²	1	2	yes	—
31	C₅×#366	C₂²	1	2	yes	—
32	C₅×#366	C₂²	1	2	yes	—
33	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
34	C₅×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₅×#358	C₂²	1	1	yes	characteristic
36	C₅×#358	C₂²	1	1	yes	characteristic
37	C₅×#358	C₂²	1	2	yes	—
38	C₅×#357	C₂²	1	1	yes	characteristic
39	C₅×#357	C₂²	1	1	yes	characteristic
40	C₅×#357	C₂²	1	2	yes	—
41	#33633	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	C₂×C₅×#41	—	2	1	no	—
43	C₅×#135	—	2	2	no	—
44	C₅×#135	Q₈	1	4	yes	complemented
45	C₂×C₅×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₅×SD₁₆	—	2	1	no	—
47	C₂×C₅×D₈	C₂³	1	1	yes	characteristic
48	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
49	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
50	C₅×C₄⋊Q₈	—	2	2	no	—
51	C₅×C₄⋊Q₈	—	4	1	no	—
52	C₅×#125	—	2	2	no	—
53	C₅×#125	—	4	1	no	—
54	C₅×C₂²⋊Q₈	—	4	2	no	—
55	C₄×C₅×Q₈	—	2	2	no	—
56	C₄×C₅×D₄	—	2	2	no	—
57	C₄×C₅×D₄	—	4	2	no	—
58	C₅×#117	—	2	1	no	—
59	C₅×#117	—	4	1	no	—
60	C₅×#117	D₄	1	1	yes	characteristic
61	C₂×C₅×C₄⋊C₄	—	2	1	no	—
62	C₂×C₅×C₄⋊C₄	—	2	1	no	—
63	C₂×C₅×C₄⋊C₄	—	4	1	no	—
64	C₅×M₃₂	—	2	2	no	—
65	C₂×C₅×C₁₆	—	2	2	no	—
66	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
67	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₅×C₁₆	C₂³	1	2	yes	—
69	C₅×#107	—	2	1	no	—
70	C₅×#107	—	2	1	no	—
71	C₅×#107	—	2	2	no	—
72	C₅×#107	C₂³	1	2	yes	—
73	C₅×#107	C₂³	1	2	yes	—
74	C₅×#107	C₂³	1	1	yes	characteristic
75	C₅×#107	C₂³	1	1	yes	characteristic
76	C₅×#106	—	2	1	no	—
77	C₅×#106	—	2	2	no	—
78	C₅×#103	—	2	2	no	—
79	C₅×#102	—	2	2	no	—
80	C₅×#97	D₄	1	2	yes	—
81	C₄×C₅×C₈	C₂³	1	2	yes	—
82	#3164	C₂×C₅	1	1	yes	characteristic, complemented
83	#2951	C₂×C₅	1	1	yes	characteristic
84	#2229	C₂×C₅	1	2	yes	complemented
85	#2228	C₂×C₅	1	2	yes	complemented
86	#2227	C₂×C₅	1	2	yes	complemented
87	#2224	C₂×C₅	1	2	yes	complemented
88	#2159	C₂×C₅	1	1	yes	characteristic, complemented
89	C₂×#366	C₂×C₅	1	1	yes	characteristic, complemented
90	#2151	C₂×C₅	1	1	yes	characteristic, complemented
91	#2139	C₂×C₅	1	1	yes	characteristic
92	#2138	C₂×C₅	1	1	yes	characteristic
93	C₅×#41	—	2	4	no	—
94	C₅×#41	—	4	2	no	—
95	C₂×C₅×Q₈	—	2	1	no	—
96	C₂×C₅×Q₈	—	4	2	no	—
97	C₂×C₅×D₄	—	2	1	no	—
98	C₂×C₅×D₄	—	4	1	no	—
99	C₂²×C₄×C₅	—	4	1	no	—
100	C₂²×C₄×C₅	—	4	1	no	—
101	C₂²×C₄×C₅	—	4	1	no	—
102	C₂²×C₄×C₅	—	4	2	no	—
103	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₅×Q₁₆	—	2	1	no	—
105	C₅×Q₁₆	C₂×Q₈	1	2	yes	—
106	C₅×SD₁₆	—	2	4	no	—
107	C₅×D₈	—	2	1	no	—
108	C₅×D₈	C₂×Q₈	1	2	yes	—
109	C₅×M₁₆	—	2	2	no	—
110	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
111	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₅×C₈	C₂×Q₈	1	2	yes	—
113	C₂×C₅×C₈	#41	1	2	yes	—
114	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	2	no	—
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	—	2	1	no	—
119	C₅×C₄⋊C₄	—	4	1	no	—
120	C₅×C₄⋊C₄	—	4	1	no	—
121	C₅×C₄⋊C₄	—	4	2	no	—
122	C₅×C₄⋊C₄	—	4	2	no	—
123	C₅×C₄⋊C₄	—	4	2	no	—
124	C₅×C₄⋊C₄	—	4	2	no	—
125	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
126	C₅×C₄⋊C₄	D₈	1	1	yes	characteristic
127	C₅×C₂²⋊C₄	—	4	1	no	—
128	C₅×C₂²⋊C₄	—	4	2	no	—
129	C₅×C₂²⋊C₄	D₈	1	2	yes	—
130	C₄²×C₅	—	4	1	no	—
131	C₄²×C₅	—	4	2	no	—
132	C₄²×C₅	C₂×D₄	1	2	yes	—
133	C₅×C₁₆	—	2	1	no	—
134	C₅×C₁₆	—	2	1	no	—
135	C₅×C₁₆	—	2	2	no	—
136	C₂×#135	C₂²×C₅	1	1	yes	characteristic
137	#544	—	2	1	no	—
138	#520	—	2	1	no	—
139	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
140	C₂²×C₁₆	C₂²×C₅	1	1	yes	characteristic
141	#501	—	2	2	no	—
142	#500	C₂²×C₅	1	2	yes	—
143	#499	C₂²×C₅	1	2	yes	—
144	#440	—	2	2	no	—
145	C₄×Q₁₆	C₂²×C₅	1	2	yes	—
146	C₄×D₈	C₂²×C₅	1	2	yes	—
147	#428	—	2	1	no	—
148	#427	C₂²×C₅	1	1	yes	characteristic
149	C₂×#107	C₂²×C₅	1	1	yes	characteristic
150	C₂×#107	C₂²×C₅	1	1	yes	characteristic
151	#417	—	2	1	no	—
152	#405	C₂²×C₅	1	1	yes	characteristic
153	#367	C₂²×C₅	1	2	yes	—
154	#366	C₂²×C₅	1	2	yes	—
155	#366	C₂²×C₅	1	2	yes	—
156	#366	C₂²×C₅	1	2	yes	—
157	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
158	C₄⋊C₁₆	C₂²×C₅	1	2	yes	—
159	#358	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	1	yes	characteristic
163	#357	C₂²×C₅	1	1	yes	characteristic
164	#357	C₂²×C₅	1	2	yes	—
165	C₂³×C₅	—	4	1	no	—
166	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
167	C₅×Q₈	—	2	2	no	—
168	C₅×Q₈	—	4	1	no	—
169	C₅×Q₈	—	16	2	no	—
170	C₅×D₄	—	2	2	no	—
171	C₅×D₄	—	4	4	no	—
172	C₅×D₄	—	4	1	no	—
173	C₂×C₄×C₅	—	4	1	no	—
174	C₂×C₄×C₅	—	4	4	no	—
175	C₂×C₄×C₅	—	4	1	no	—
176	C₂×C₄×C₅	—	4	2	no	—
177	C₂×C₄×C₅	—	4	1	no	—
178	C₂×C₄×C₅	—	4	2	no	—
179	C₂×C₄×C₅	—	16	2	no	—
180	C₂×C₄×C₅	—	16	1	no	—
181	C₂×C₄×C₅	—	16	1	no	—
182	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
183	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
184	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
185	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
186	C₂×C₄×C₅	#137	1	2	yes	—
187	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
188	C₅×C₈	—	2	2	no	—
189	C₅×C₈	C₂²⋊Q₈	1	2	yes	—
190	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
191	C₅×C₈	C₂×#41	1	1	yes	characteristic
192	C₂×#41	—	2	1	no	—
193	#135	—	2	2	no	—
194	#135	C₅×Q₈	1	4	yes	complemented
195	C₂×Q₁₆	C₂³×C₅	1	1	yes	characteristic
196	C₂×SD₁₆	—	2	1	no	—
197	C₂×D₈	C₂³×C₅	1	1	yes	characteristic
198	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	4	1	no	—
202	#125	—	2	2	no	—
203	#125	—	4	1	no	—
204	C₂²⋊Q₈	—	4	2	no	—
205	C₄×Q₈	—	2	2	no	—
206	C₄×D₄	—	2	2	no	—
207	C₄×D₄	—	4	2	no	—
208	#117	—	2	1	no	—
209	#117	—	4	1	no	—
210	#117	C₅×D₄	1	1	yes	characteristic
211	C₂×C₄⋊C₄	—	2	1	no	—
212	C₂×C₄⋊C₄	—	2	1	no	—
213	C₂×C₄⋊C₄	—	4	1	no	—
214	M₃₂	—	2	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
217	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
218	C₂×C₁₆	C₂³×C₅	1	2	yes	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	—	2	2	no	—
222	#107	C₂³×C₅	1	2	yes	—
223	#107	C₂³×C₅	1	1	yes	characteristic
224	#107	C₂³×C₅	1	1	yes	characteristic
225	#107	C₂³×C₅	1	2	yes	—
226	#106	—	2	1	no	—
227	#106	—	2	2	no	—
228	#103	—	2	2	no	—
229	#102	—	2	2	no	—
230	#97	C₅×D₄	1	2	yes	—
231	C₄×C₈	C₂³×C₅	1	2	yes	—
232	C₂²×C₅	—	2	1	no	—
233	C₂²×C₅	—	2	1	no	—
234	C₂²×C₅	—	4	2	no	—
235	C₂²×C₅	—	8	1	no	—
236	C₂²×C₅	—	8	1	no	—
237	C₂²×C₅	#474	1	2	yes	—
238	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
239	C₄×C₅	—	2	2	no	—
240	C₄×C₅	—	2	2	no	—
241	C₄×C₅	—	4	2	no	—
242	C₄×C₅	—	8	2	no	—
243	C₄×C₅	—	8	2	no	—
244	C₄×C₅	—	8	1	no	—
245	C₄×C₅	—	8	1	no	—
246	C₄×C₅	#474	1	2	yes	—
247	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
248	C₄×C₅	C₂×#137	1	1	yes	characteristic
249	#41	—	2	4	no	—
250	#41	—	4	2	no	—
251	C₂×Q₈	—	2	1	no	—
252	C₂×Q₈	—	4	2	no	—
253	C₂×D₄	—	2	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	—	4	1	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	—	4	2	no	—
259	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
260	Q₁₆	—	2	1	no	—
261	Q₁₆	C₂×C₅×Q₈	1	2	yes	—
262	SD₁₆	—	2	4	no	—
263	D₈	—	2	1	no	—
264	D₈	C₂×C₅×Q₈	1	2	yes	—
265	M₁₆	—	2	2	no	—
266	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
267	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
268	C₂×C₈	C₂×C₅×Q₈	1	2	yes	—
269	C₂×C₈	C₅×#41	1	2	yes	—
270	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	2	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
282	C₄⋊C₄	C₅×D₈	1	1	yes	characteristic
283	C₂²⋊C₄	—	4	1	no	—
284	C₂²⋊C₄	—	4	2	no	—
285	C₂²⋊C₄	C₅×D₈	1	2	yes	—
286	C₄²	—	4	1	no	—
287	C₄²	—	4	2	no	—
288	C₄²	C₂×C₅×D₄	1	2	yes	—
289	C₁₆	—	2	1	no	—
290	C₁₆	—	2	1	no	—
291	C₁₆	—	2	2	no	—
292	C₂×C₅	—	2	2	no	—
293	C₂×C₅	—	8	2	no	—
294	C₂×C₅	C₂×#474	1	1	yes	characteristic
295	C₂×C₅	#3415	1	1	yes	characteristic
296	C₂×C₅	#3417	1	1	yes	characteristic
297	C₂³	—	4	1	no	—
298	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
299	Q₈	—	2	2	no	—
300	Q₈	—	4	1	no	—
301	Q₈	—	16	2	no	—
302	D₄	—	2	2	no	—
303	D₄	—	4	4	no	—
304	D₄	—	4	1	no	—
305	C₂×C₄	—	4	1	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	2	no	—
311	C₂×C₄	—	16	2	no	—
312	C₂×C₄	—	16	1	no	—
313	C₂×C₄	—	16	1	no	—
314	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
315	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
316	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
317	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
318	C₂×C₄	C₅×#137	1	2	yes	—
319	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	C₅×C₂²⋊Q₈	1	2	yes	—
322	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
323	C₈	C₂×C₅×#41	1	1	yes	characteristic
324	C₅	#33633	1	1	yes	characteristic, direct product factor, Sylow
325	C₂²	—	2	1	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	4	2	no	—
328	C₂²	—	8	1	no	—
329	C₂²	—	8	1	no	—
330	C₂²	C₅×#474	1	2	yes	—
331	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	C₅×#474	1	2	yes	—
340	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
341	C₄	C₂×C₅×#137	1	1	yes	characteristic
342	C₂	—	2	2	no	—
343	C₂	—	8	2	no	—
344	C₂	C₂×C₅×#474	1	1	yes	characteristic
345	C₂	C₅×#3415	1	1	yes	characteristic
346	C₂	C₅×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₅	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅ ◁ C₅×#117 ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3164	C₂	semidirect
3. C₅×#2229	C₂	semidirect
4. C₅×#2228	C₂	semidirect
5. C₅×#2227	C₂	semidirect
6. C₅×#2224	C₂	semidirect
7. C₅×#2159	C₂	semidirect
8. C₂×C₅×#366	C₂	semidirect
9. C₅×#2151	C₂	semidirect
41. #33633	C₅	direct (N × H)
44. C₅×#135	Q₈	semidirect
82. #3164	C₂×C₅	semidirect
84. #2229	C₂×C₅	semidirect
85. #2228	C₂×C₅	semidirect
86. #2227	C₂×C₅	semidirect
87. #2224	C₂×C₅	semidirect
88. #2159	C₂×C₅	semidirect
89. C₂×#366	C₂×C₅	semidirect
90. #2151	C₂×C₅	semidirect
194. #135	C₅×Q₈	semidirect
324. C₅	#33633	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×#3164	2
C₅×#2951	2
C₅×#2229	2
C₅×#2228	2
C₅×#2227	2
C₅×#2224	2
C₅×#2159	2
C₂×C₅×#366	2
C₅×#2151	2
C₅×#2139	2
C₅×#2138	2
#33633	5

Subgroups — C5×#33633 #11956353

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33633 #11956353