Subgroups — C₅×#33638 #11956358

Order 1280.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#2957	C₂	1	1	yes	maximal, characteristic
3	C₅×#2231	C₂	1	4	yes	maximal, complemented
4	C₅×#2230	C₂	1	4	yes	maximal, complemented
5	C₅×#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#365	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#2145	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅×#360	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×#135	C₂²	1	1	yes	characteristic
11	C₅×#568	—	2	1	no	—
12	C₅×#544	—	2	1	no	—
13	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
15	C₅×#501	—	2	2	no	—
16	C₅×#499	C₂²	1	2	yes	complemented
17	C₅×#498	C₂²	1	2	yes	complemented
18	C₅×#444	—	2	2	no	—
19	C₅×#443	C₂²	1	4	yes	—
20	C₂×C₅×#108	C₂²	1	1	yes	characteristic
21	C₅×#428	—	2	1	no	—
22	C₅×#427	C₂²	1	1	yes	characteristic
23	C₂×C₅×#106	C₂²	1	1	yes	characteristic
24	C₅×#421	—	2	1	no	—
25	C₅×#405	C₂²	1	1	yes	characteristic
26	C₅×#365	C₂²	1	4	yes	complemented
27	C₅×#365	C₂²	1	2	yes	complemented
28	C₅×#365	C₂²	1	2	yes	complemented
29	C₅×#364	C₂²	1	4	yes	complemented
30	C₅×#360	C₂²	1	4	yes	—
31	C₅×#360	C₂²	1	2	yes	—
32	C₅×#360	C₂²	1	2	yes	—
33	#33638	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂×C₅×#41	—	2	1	no	—
35	C₅×#135	—	2	1	no	—
36	C₅×#135	—	2	1	no	—
37	C₅×#135	Q₈	1	2	yes	complemented
38	C₅×#135	C₂³	1	2	yes	—
39	C₂×C₅×Q₁₆	Q₈	1	1	yes	characteristic, complemented
40	C₂×C₅×SD₁₆	—	2	1	no	—
41	C₂×C₅×D₈	Q₈	1	1	yes	characteristic, complemented
42	C₅×#131	—	2	4	no	—
43	C₅×#131	—	4	2	no	—
44	C₂×C₅×M₁₆	—	2	1	no	—
45	C₂×C₅×M₁₆	—	4	2	no	—
46	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
48	C₅×C₄⋊Q₈	—	2	2	no	—
49	C₅×C₄⋊Q₈	—	4	1	no	—
50	C₅×#125	—	2	2	no	—
51	C₅×#125	—	4	1	no	—
52	C₅×C₂²⋊Q₈	—	4	2	no	—
53	C₅×#117	—	2	1	no	—
54	C₅×#117	D₄	1	1	yes	characteristic
55	C₂×C₅×C₄⋊C₄	—	2	1	no	—
56	C₅×M₃₂	—	2	1	no	—
57	C₅×M₃₂	—	2	1	no	—
58	C₅×M₃₂	C₂³	1	2	yes	—
59	C₅×M₃₂	C₂³	1	2	yes	—
60	C₂×C₅×C₁₆	—	2	1	no	—
61	C₂×C₅×C₁₆	—	2	1	no	—
62	C₅×#108	—	2	1	no	—
63	C₅×#108	C₂³	1	2	yes	—
64	C₅×#107	—	2	1	no	—
65	C₅×#107	—	2	2	no	—
66	C₅×#106	—	2	1	no	—
67	C₅×#106	—	2	2	no	—
68	C₅×#106	C₂³	1	2	yes	—
69	C₅×#106	C₂³	1	1	yes	characteristic
70	C₅×#106	C₂³	1	1	yes	characteristic
71	C₅×#104	—	2	4	no	—
72	C₅×#97	D₄	1	2	yes	—
73	C₄×C₅×C₈	C₂³	1	2	yes	—
74	#3163	C₂×C₅	1	1	yes	characteristic, complemented
75	#2957	C₂×C₅	1	1	yes	characteristic
76	#2231	C₂×C₅	1	4	yes	complemented
77	#2230	C₂×C₅	1	4	yes	complemented
78	#2155	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂×#365	C₂×C₅	1	1	yes	characteristic, complemented
80	#2153	C₂×C₅	1	1	yes	characteristic, complemented
81	#2145	C₂×C₅	1	1	yes	characteristic
82	C₂×#360	C₂×C₅	1	1	yes	characteristic
83	C₅×#41	—	2	2	no	—
84	C₅×#41	—	2	2	no	—
85	C₅×#41	—	4	1	no	—
86	C₅×#41	—	4	1	no	—
87	C₂×C₅×Q₈	—	2	1	no	—
88	C₂×C₅×Q₈	—	4	2	no	—
89	C₂×C₅×D₄	—	2	1	no	—
90	C₂×C₅×D₄	—	4	1	no	—
91	C₂²×C₄×C₅	—	4	1	no	—
92	C₂²×C₄×C₅	—	4	1	no	—
93	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
94	C₅×Q₁₆	—	2	1	no	—
95	C₅×Q₁₆	C₂×Q₈	1	2	yes	—
96	C₅×SD₁₆	—	2	2	no	—
97	C₅×SD₁₆	—	2	2	no	—
98	C₅×D₈	—	2	1	no	—
99	C₅×D₈	C₂×Q₈	1	2	yes	—
100	C₅×M₁₆	—	2	2	no	—
101	C₅×M₁₆	—	4	1	no	—
102	C₅×M₁₆	—	4	4	no	—
103	C₅×M₁₆	SD₁₆	1	2	yes	—
104	C₅×M₁₆	SD₁₆	1	2	yes	—
105	C₂×C₅×C₈	—	4	1	no	—
106	C₂×C₅×C₈	—	4	4	no	—
107	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
108	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₅×C₈	#41	1	1	yes	characteristic
112	C₂×C₅×C₈	#41	1	1	yes	characteristic
113	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
114	C₅×C₄⋊C₄	—	2	1	no	—
115	C₅×C₄⋊C₄	—	2	2	no	—
116	C₅×C₄⋊C₄	—	2	1	no	—
117	C₅×C₄⋊C₄	—	2	1	no	—
118	C₅×C₄⋊C₄	—	4	2	no	—
119	C₅×C₄⋊C₄	—	4	2	no	—
120	C₅×C₄⋊C₄	—	4	2	no	—
121	C₅×C₄⋊C₄	—	4	1	no	—
122	C₅×C₂²⋊C₄	—	2	1	no	—
123	C₅×C₂²⋊C₄	—	4	1	no	—
124	C₄²×C₅	—	4	1	no	—
125	C₄²×C₅	C₂×D₄	1	2	yes	—
126	C₅×C₁₆	—	2	2	no	—
127	C₅×C₁₆	—	2	2	no	—
128	C₂×#135	C₂²×C₅	1	1	yes	characteristic
129	#568	—	2	1	no	—
130	#544	—	2	1	no	—
131	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
132	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
133	#501	—	2	2	no	—
134	#499	C₂²×C₅	1	2	yes	complemented
135	#498	C₂²×C₅	1	2	yes	complemented
136	#444	—	2	2	no	—
137	#443	C₂²×C₅	1	4	yes	—
138	C₂×#108	C₂²×C₅	1	1	yes	characteristic
139	#428	—	2	1	no	—
140	#427	C₂²×C₅	1	1	yes	characteristic
141	C₂×#106	C₂²×C₅	1	1	yes	characteristic
142	#421	—	2	1	no	—
143	#405	C₂²×C₅	1	1	yes	characteristic
144	#365	C₂²×C₅	1	4	yes	complemented
145	#365	C₂²×C₅	1	2	yes	complemented
146	#365	C₂²×C₅	1	2	yes	complemented
147	#364	C₂²×C₅	1	4	yes	complemented
148	#360	C₂²×C₅	1	4	yes	—
149	#360	C₂²×C₅	1	2	yes	—
150	#360	C₂²×C₅	1	2	yes	—
151	C₂³×C₅	—	4	1	no	—
152	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
153	C₅×Q₈	—	2	2	no	—
154	C₅×Q₈	—	4	1	no	—
155	C₅×Q₈	—	16	2	no	—
156	C₅×D₄	—	2	2	no	—
157	C₅×D₄	—	4	2	no	—
158	C₅×D₄	—	4	2	no	—
159	C₅×D₄	—	4	1	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	4	1	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	1	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	16	1	no	—
168	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
169	C₂×C₄×C₅	C₂²⋊Q₈	1	1	yes	characteristic
170	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×SD₁₆	1	1	yes	characteristic
172	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
174	C₅×C₈	—	4	2	no	—
175	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
176	C₅×C₈	C₂²⋊Q₈	1	1	yes	characteristic
177	C₅×C₈	C₂×SD₁₆	1	4	yes	—
178	C₅×C₈	C₂²×Q₈	1	1	yes	characteristic
179	C₅×C₈	C₂×#41	1	1	yes	characteristic
180	C₂×#41	—	2	1	no	—
181	#135	—	2	1	no	—
182	#135	—	2	1	no	—
183	#135	C₅×Q₈	1	2	yes	complemented
184	#135	C₂³×C₅	1	2	yes	—
185	C₂×Q₁₆	C₅×Q₈	1	1	yes	characteristic, complemented
186	C₂×SD₁₆	—	2	1	no	—
187	C₂×D₈	C₅×Q₈	1	1	yes	characteristic, complemented
188	#131	—	2	4	no	—
189	#131	—	4	2	no	—
190	C₂×M₁₆	—	2	1	no	—
191	C₂×M₁₆	—	4	2	no	—
192	C₂×M₁₆	C₅×D₄	1	1	yes	characteristic
193	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
194	C₄⋊Q₈	—	2	2	no	—
195	C₄⋊Q₈	—	4	1	no	—
196	#125	—	2	2	no	—
197	#125	—	4	1	no	—
198	C₂²⋊Q₈	—	4	2	no	—
199	#117	—	2	1	no	—
200	#117	C₅×D₄	1	1	yes	characteristic
201	C₂×C₄⋊C₄	—	2	1	no	—
202	M₃₂	—	2	1	no	—
203	M₃₂	—	2	1	no	—
204	M₃₂	C₂³×C₅	1	2	yes	—
205	M₃₂	C₂³×C₅	1	2	yes	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	#108	—	2	1	no	—
209	#108	C₂³×C₅	1	2	yes	—
210	#107	—	2	1	no	—
211	#107	—	2	2	no	—
212	#106	—	2	1	no	—
213	#106	—	2	2	no	—
214	#106	C₂³×C₅	1	2	yes	—
215	#106	C₂³×C₅	1	1	yes	characteristic
216	#106	C₂³×C₅	1	1	yes	characteristic
217	#104	—	2	4	no	—
218	#97	C₅×D₄	1	2	yes	—
219	C₄×C₈	C₂³×C₅	1	2	yes	—
220	C₂²×C₅	—	4	1	no	—
221	C₂²×C₅	—	4	2	no	—
222	C₂²×C₅	—	16	1	no	—
223	C₂²×C₅	#476	1	1	yes	characteristic
224	C₂²×C₅	#476	1	1	yes	characteristic
225	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	—	4	2	no	—
228	C₄×C₅	—	8	1	no	—
229	C₄×C₅	—	8	2	no	—
230	C₄×C₅	—	8	1	no	—
231	C₄×C₅	#476	1	1	yes	characteristic
232	C₄×C₅	#476	1	1	yes	characteristic
233	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
234	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	#41	—	4	1	no	—
238	#41	—	4	1	no	—
239	C₂×Q₈	—	2	1	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×D₄	—	2	1	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	1	no	—
245	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
246	Q₁₆	—	2	1	no	—
247	Q₁₆	C₂×C₅×Q₈	1	2	yes	—
248	SD₁₆	—	2	2	no	—
249	SD₁₆	—	2	2	no	—
250	D₈	—	2	1	no	—
251	D₈	C₂×C₅×Q₈	1	2	yes	—
252	M₁₆	—	2	2	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	4	4	no	—
255	M₁₆	C₅×SD₁₆	1	2	yes	—
256	M₁₆	C₅×SD₁₆	1	2	yes	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	4	no	—
259	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
260	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
261	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
262	C₂×C₈	C₂×C₅×Q₈	1	1	yes	characteristic
263	C₂×C₈	C₅×#41	1	1	yes	characteristic
264	C₂×C₈	C₅×#41	1	1	yes	characteristic
265	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
266	C₄⋊C₄	—	2	1	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₄²	—	4	1	no	—
277	C₄²	C₂×C₅×D₄	1	2	yes	—
278	C₁₆	—	2	2	no	—
279	C₁₆	—	2	2	no	—
280	C₂×C₅	—	2	1	no	—
281	C₂×C₅	—	2	1	no	—
282	C₂×C₅	—	2	1	no	—
283	C₂×C₅	—	8	2	no	—
284	C₂×C₅	C₂×#476	1	1	yes	characteristic
285	C₂³	—	4	1	no	—
286	C₂³	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	16	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	4	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	1	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	16	1	no	—
302	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
303	C₂×C₄	C₅×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
305	C₂×C₄	C₂×C₅×SD₁₆	1	1	yes	characteristic
306	C₂×C₄	C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
308	C₈	—	4	2	no	—
309	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
310	C₈	C₅×C₂²⋊Q₈	1	1	yes	characteristic
311	C₈	C₂×C₅×SD₁₆	1	4	yes	—
312	C₈	C₂²×C₅×Q₈	1	1	yes	characteristic
313	C₈	C₂×C₅×#41	1	1	yes	characteristic
314	C₅	#33638	1	1	yes	characteristic, direct product factor, Sylow
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	C₅×#476	1	1	yes	characteristic
319	C₂²	C₅×#476	1	1	yes	characteristic
320	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	1	no	—
326	C₄	C₅×#476	1	1	yes	characteristic
327	C₄	C₅×#476	1	1	yes	characteristic
328	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
329	C₄	C₂²×C₅×SD₁₆	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₅×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₅	C₂×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₅
prime radical	C₂×C₅×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅ ◁ C₂²×C₄×C₅ ◁ C₅×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3163	C₂	semidirect
3. C₅×#2231	C₂	semidirect
4. C₅×#2230	C₂	semidirect
5. C₅×#2155	C₂	semidirect
6. C₂×C₅×#365	C₂	semidirect
7. C₅×#2153	C₂	semidirect
16. C₅×#499	C₂²	semidirect
17. C₅×#498	C₂²	semidirect
26. C₅×#365	C₂²	semidirect
27. C₅×#365	C₂²	semidirect
28. C₅×#365	C₂²	semidirect
29. C₅×#364	C₂²	semidirect
33. #33638	C₅	direct (N × H)
37. C₅×#135	Q₈	semidirect
39. C₂×C₅×Q₁₆	Q₈	semidirect
41. C₂×C₅×D₈	Q₈	semidirect
74. #3163	C₂×C₅	semidirect
76. #2231	C₂×C₅	semidirect
77. #2230	C₂×C₅	semidirect
78. #2155	C₂×C₅	semidirect
79. C₂×#365	C₂×C₅	semidirect
80. #2153	C₂×C₅	semidirect
134. #499	C₂²×C₅	semidirect
135. #498	C₂²×C₅	semidirect
144. #365	C₂²×C₅	semidirect
145. #365	C₂²×C₅	semidirect
146. #365	C₂²×C₅	semidirect
147. #364	C₂²×C₅	semidirect
183. #135	C₅×Q₈	semidirect
185. C₂×Q₁₆	C₅×Q₈	semidirect
187. C₂×D₈	C₅×Q₈	semidirect
314. C₅	#33638	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×#3163	2
C₅×#2957	2
C₅×#2231	2
C₅×#2230	2
C₅×#2155	2
C₂×C₅×#365	2
C₅×#2153	2
C₅×#2145	2
C₂×C₅×#360	2
#33638	5

Subgroups — C5×#33638 #11956358

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33638 #11956358