Subgroups — C₅×#33763 #11956483

Order 1280.

592 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#3050	C₂	1	2	yes	maximal, complemented
2	C₅×#2237	C₂	1	2	yes	maximal, complemented
3	C₅×#2234	C₂	1	2	yes	maximal, complemented
4	C₅×#2220	C₂	1	2	yes	maximal, complemented
5	C₅×#2219	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#2218	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅×#2189	C₂	1	2	yes	maximal, complemented
8	C₅×#2186	C₂	1	2	yes	maximal, complemented
9	C₅×#2112	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₅×#135	C₂²	1	2	yes	—
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₅×#525	—	2	2	no	—
13	C₂×C₅×Q₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₅×SD₃₂	C₂²	1	2	yes	complemented
15	C₂×C₅×D₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
18	C₅×#467	C₂²	1	2	yes	—
19	C₅×#467	C₂²	1	2	yes	—
20	C₅×#466	C₂²	1	2	yes	—
21	C₅×#466	C₂²	1	2	yes	—
22	C₅×#465	—	2	2	no	—
23	C₅×#465	—	2	2	no	—
24	C₅×#427	C₂²	1	2	yes	—
25	C₅×#416	—	2	2	no	—
26	C₅×#367	C₂²	1	1	yes	characteristic, complemented
27	C₅×#367	C₂²	1	1	yes	characteristic, complemented
28	C₅×#366	C₂²	1	2	yes	complemented
29	C₅×#358	C₂²	1	2	yes	complemented
30	C₅×#358	C₂²	1	2	yes	complemented
31	C₅×#357	C₂²	1	2	yes	complemented
32	C₅×#357	C₂²	1	2	yes	complemented
33	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
34	C₅×C₂²⋊C₁₆	C₂²	1	2	yes	—
35	#33763	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂×C₅×#41	—	2	2	no	—
37	C₂³×C₄×C₅	D₄	1	1	yes	characteristic
38	C₅×#135	—	4	2	no	—
39	C₅×#135	—	4	2	no	—
40	C₂×C₅×Q₁₆	C₂³	1	2	yes	—
41	C₂×C₅×SD₁₆	—	2	2	no	—
42	C₂×C₅×D₈	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	—	4	1	no	—
44	C₂²×C₅×C₈	—	4	1	no	—
45	C₂²×C₅×C₈	D₄	1	2	yes	—
46	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
47	C₂²×C₅×C₈	C₂³	1	2	yes	—
48	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
49	C₅×#123	—	4	2	no	—
50	C₅×C₂²⋊Q₈	—	2	2	no	—
51	C₅×C₂²⋊Q₈	—	2	2	no	—
52	C₅×#121	—	2	2	no	—
53	C₅×#121	—	2	2	no	—
54	C₅×#120	—	4	2	no	—
55	C₄×C₅×D₄	—	4	2	no	—
56	C₄×C₅×D₄	—	4	2	no	—
57	C₅×#117	—	2	2	no	—
58	C₅×Q₃₂	—	4	1	no	—
59	C₅×SD₃₂	—	4	2	no	—
60	C₅×D₁₆	—	4	1	no	—
61	C₅×M₃₂	—	4	1	no	—
62	C₅×M₃₂	—	4	1	no	—
63	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₅×C₁₆	C₂³	1	2	yes	—
65	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
66	C₅×#107	C₂³	1	2	yes	—
67	C₅×#107	C₂³	1	2	yes	—
68	C₅×#106	—	2	2	no	—
69	C₅×#103	—	2	2	no	—
70	C₅×#103	—	2	2	no	—
71	C₅×#102	—	2	2	no	—
72	C₅×#102	—	2	2	no	—
73	#3050	C₂×C₅	1	2	yes	complemented
74	#2237	C₂×C₅	1	2	yes	complemented
75	#2234	C₂×C₅	1	2	yes	complemented
76	#2220	C₂×C₅	1	2	yes	complemented
77	#2219	C₂×C₅	1	1	yes	characteristic, complemented
78	#2218	C₂×C₅	1	1	yes	characteristic, complemented
79	#2189	C₂×C₅	1	2	yes	complemented
80	#2186	C₂×C₅	1	2	yes	complemented
81	#2112	C₂×C₅	1	1	yes	characteristic, complemented
82	C₂⁴×C₅	D₈	1	1	yes	characteristic
83	C₅×#41	—	4	4	no	—
84	C₅×#41	—	4	4	no	—
85	C₂×C₅×Q₈	—	2	2	no	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂×C₅×D₄	—	4	2	no	—
89	C₂×C₅×D₄	—	4	2	no	—
90	C₂²×C₄×C₅	—	4	1	no	—
91	C₂²×C₄×C₅	—	4	1	no	—
92	C₂²×C₄×C₅	—	4	2	no	—
93	C₂²×C₄×C₅	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₅	D₈	1	2	yes	—
95	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
96	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
97	C₅×Q₁₆	—	2	2	no	—
98	C₅×Q₁₆	—	2	2	no	—
99	C₅×SD₁₆	—	4	4	no	—
100	C₅×D₈	—	2	2	no	—
101	C₅×D₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	2	1	no	—
105	C₂×C₅×C₈	—	2	1	no	—
106	C₂×C₅×C₈	—	4	2	no	—
107	C₂×C₅×C₈	—	4	1	no	—
108	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
109	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
110	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
111	C₅×C₄⋊C₄	—	2	2	no	—
112	C₅×C₄⋊C₄	—	2	2	no	—
113	C₅×C₄⋊C₄	—	4	2	no	—
114	C₅×C₄⋊C₄	—	4	2	no	—
115	C₅×C₂²⋊C₄	—	4	2	no	—
116	C₅×C₂²⋊C₄	—	4	2	no	—
117	C₅×C₂²⋊C₄	—	4	2	no	—
118	C₅×C₂²⋊C₄	—	4	2	no	—
119	C₅×C₂²⋊C₄	—	4	2	no	—
120	C₄²×C₅	—	4	2	no	—
121	C₅×C₁₆	—	2	1	no	—
122	C₅×C₁₆	—	2	2	no	—
123	C₅×C₁₆	—	2	1	no	—
124	C₂×#135	C₂²×C₅	1	2	yes	—
125	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
126	#525	—	2	2	no	—
127	C₂×Q₃₂	C₂²×C₅	1	1	yes	characteristic, complemented
128	C₂×SD₃₂	C₂²×C₅	1	2	yes	complemented
129	C₂×D₁₆	C₂²×C₅	1	1	yes	characteristic, complemented
130	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
131	C₂×M₃₂	C₂²×C₅	1	1	yes	characteristic
132	#467	C₂²×C₅	1	2	yes	—
133	#467	C₂²×C₅	1	2	yes	—
134	#466	C₂²×C₅	1	2	yes	—
135	#466	C₂²×C₅	1	2	yes	—
136	#465	—	2	2	no	—
137	#465	—	2	2	no	—
138	#427	C₂²×C₅	1	2	yes	—
139	#416	—	2	2	no	—
140	#367	C₂²×C₅	1	1	yes	characteristic, complemented
141	#367	C₂²×C₅	1	1	yes	characteristic, complemented
142	#366	C₂²×C₅	1	2	yes	complemented
143	#358	C₂²×C₅	1	2	yes	complemented
144	#358	C₂²×C₅	1	2	yes	complemented
145	#357	C₂²×C₅	1	2	yes	complemented
146	#357	C₂²×C₅	1	2	yes	complemented
147	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
148	C₂²⋊C₁₆	C₂²×C₅	1	2	yes	—
149	C₂³×C₅	—	4	1	no	—
150	C₂³×C₅	—	4	2	no	—
151	C₂³×C₅	—	4	2	no	—
152	C₂³×C₅	C₂×D₈	1	1	yes	characteristic
153	C₂³×C₅	C₂×D₈	1	2	yes	—
154	C₅×Q₈	—	4	2	no	—
155	C₅×Q₈	—	4	2	no	—
156	C₅×D₄	—	4	2	no	—
157	C₅×D₄	—	4	2	no	—
158	C₅×D₄	—	8	4	no	—
159	C₅×D₄	—	8	4	no	—
160	C₅×D₄	—	8	4	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	4	1	no	—
166	C₂×C₄×C₅	—	4	2	no	—
167	C₂×C₄×C₅	—	4	2	no	—
168	C₂×C₄×C₅	—	4	2	no	—
169	C₂×C₄×C₅	—	4	2	no	—
170	C₂×C₄×C₅	—	8	4	no	—
171	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
172	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
174	C₅×C₈	—	2	1	no	—
175	C₅×C₈	—	2	2	no	—
176	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
177	C₅×C₈	H(5,𝔽₂)	1	1	yes	characteristic
178	C₂×#41	—	2	2	no	—
179	C₂³×C₄	C₅×D₄	1	1	yes	characteristic
180	#135	—	4	2	no	—
181	#135	—	4	2	no	—
182	C₂×Q₁₆	C₂³×C₅	1	2	yes	—
183	C₂×SD₁₆	—	2	2	no	—
184	C₂×D₈	C₂³×C₅	1	2	yes	—
185	C₂²×C₈	—	4	1	no	—
186	C₂²×C₈	—	4	1	no	—
187	C₂²×C₈	C₅×D₄	1	2	yes	—
188	C₂²×C₈	C₅×D₄	1	1	yes	characteristic
189	C₂²×C₈	C₂³×C₅	1	2	yes	—
190	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
191	#123	—	4	2	no	—
192	C₂²⋊Q₈	—	2	2	no	—
193	C₂²⋊Q₈	—	2	2	no	—
194	#121	—	2	2	no	—
195	#121	—	2	2	no	—
196	#120	—	4	2	no	—
197	C₄×D₄	—	4	2	no	—
198	C₄×D₄	—	4	2	no	—
199	#117	—	2	2	no	—
200	Q₃₂	—	4	1	no	—
201	SD₃₂	—	4	2	no	—
202	D₁₆	—	4	1	no	—
203	M₃₂	—	4	1	no	—
204	M₃₂	—	4	1	no	—
205	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
206	C₂×C₁₆	C₂³×C₅	1	2	yes	—
207	C₂×C₁₆	C₂³×C₅	1	1	yes	characteristic
208	#107	C₂³×C₅	1	2	yes	—
209	#107	C₂³×C₅	1	2	yes	—
210	#106	—	2	2	no	—
211	#103	—	2	2	no	—
212	#103	—	2	2	no	—
213	#102	—	2	2	no	—
214	#102	—	2	2	no	—
215	C₂²×C₅	—	2	2	no	—
216	C₂²×C₅	—	2	2	no	—
217	C₂²×C₅	—	2	2	no	—
218	C₂²×C₅	—	2	2	no	—
219	C₂²×C₅	—	2	1	no	—
220	C₂²×C₅	—	4	4	no	—
221	C₂²×C₅	—	8	2	no	—
222	C₂²×C₅	—	8	2	no	—
223	C₂²×C₅	—	8	2	no	—
224	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	2	1	no	—
227	C₄×C₅	—	8	2	no	—
228	C₄×C₅	—	8	2	no	—
229	C₄×C₅	—	8	2	no	—
230	C₄×C₅	#534	1	1	yes	characteristic
231	C₄×C₅	#575	1	1	yes	characteristic
232	C₂⁴	C₅×D₈	1	1	yes	characteristic
233	#41	—	4	4	no	—
234	#41	—	4	4	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂²×C₄	—	4	1	no	—
241	C₂²×C₄	—	4	1	no	—
242	C₂²×C₄	—	4	2	no	—
243	C₂²×C₄	C₅×D₈	1	1	yes	characteristic
244	C₂²×C₄	C₅×D₈	1	2	yes	—
245	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
246	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
247	Q₁₆	—	2	2	no	—
248	Q₁₆	—	2	2	no	—
249	SD₁₆	—	4	4	no	—
250	D₈	—	2	2	no	—
251	D₈	—	2	2	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	2	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	C₂×C₅×D₄	1	1	yes	characteristic
259	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
260	C₂×C₈	C₂⁴×C₅	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₄²	—	4	2	no	—
271	C₁₆	—	2	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	—	2	1	no	—
274	C₂×C₅	—	4	1	no	—
275	C₂×C₅	—	4	2	no	—
276	C₂×C₅	—	16	2	no	—
277	C₂×C₅	#3186	1	1	yes	characteristic
278	C₂×C₅	#3414	1	2	yes	—
279	C₂³	—	4	1	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	4	2	no	—
282	C₂³	C₂×C₅×D₈	1	1	yes	characteristic
283	C₂³	C₂×C₅×D₈	1	2	yes	—
284	Q₈	—	4	2	no	—
285	Q₈	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	4	no	—
290	D₄	—	8	4	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	8	4	no	—
301	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
302	C₂×C₄	C₂×C₅×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	1	no	—
306	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
307	C₈	C₅×H(5,𝔽₂)	1	1	yes	characteristic
308	C₅	#33763	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	4	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	2	no	—
318	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	8	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₅×#534	1	1	yes	characteristic
325	C₄	C₅×#575	1	1	yes	characteristic
326	C₂	—	4	1	no	—
327	C₂	—	4	2	no	—
328	C₂	—	16	2	no	—
329	C₂	C₅×#3186	1	1	yes	characteristic
330	C₂	C₅×#3414	1	2	yes	—

	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 ◁ C₂³×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₅×#3050	C₂	semidirect
2. C₅×#2237	C₂	semidirect
3. C₅×#2234	C₂	semidirect
4. C₅×#2220	C₂	semidirect
5. C₅×#2219	C₂	semidirect
6. C₅×#2218	C₂	semidirect
7. C₅×#2189	C₂	semidirect
8. C₅×#2186	C₂	semidirect
9. C₅×#2112	C₂	semidirect
13. C₂×C₅×Q₃₂	C₂²	semidirect
14. C₂×C₅×SD₃₂	C₂²	semidirect
15. C₂×C₅×D₁₆	C₂²	semidirect
26. C₅×#367	C₂²	semidirect
27. C₅×#367	C₂²	semidirect
28. C₅×#366	C₂²	semidirect
29. C₅×#358	C₂²	semidirect
30. C₅×#358	C₂²	semidirect
31. C₅×#357	C₂²	semidirect
32. C₅×#357	C₂²	semidirect
35. #33763	C₅	direct (N × H)
73. #3050	C₂×C₅	semidirect
74. #2237	C₂×C₅	semidirect
75. #2234	C₂×C₅	semidirect
76. #2220	C₂×C₅	semidirect
77. #2219	C₂×C₅	semidirect
78. #2218	C₂×C₅	semidirect
79. #2189	C₂×C₅	semidirect
80. #2186	C₂×C₅	semidirect
81. #2112	C₂×C₅	semidirect
127. C₂×Q₃₂	C₂²×C₅	semidirect
128. C₂×SD₃₂	C₂²×C₅	semidirect
129. C₂×D₁₆	C₂²×C₅	semidirect
140. #367	C₂²×C₅	semidirect
141. #367	C₂²×C₅	semidirect
142. #366	C₂²×C₅	semidirect
143. #358	C₂²×C₅	semidirect
144. #358	C₂²×C₅	semidirect
145. #357	C₂²×C₅	semidirect
146. #357	C₂²×C₅	semidirect
308. C₅	#33763	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×#3050	2
C₅×#2237	2
C₅×#2234	2
C₅×#2220	2
C₅×#2219	2
C₅×#2218	2
C₅×#2189	2
C₅×#2186	2
C₅×#2112	2
#33763	5

Subgroups — C5×#33763 #11956483

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#33763 #11956483