Subgroups — C₅×#32849 #11955569

Order 1280.

604 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×#2859	C₂	1	1	yes	maximal, characteristic
2	C₅×#2859	C₂	1	1	yes	maximal, characteristic
3	C₅×#2100	C₂	1	2	yes	maximal
4	C₅×#2099	C₂	1	2	yes	maximal
5	C₅×#2088	C₂	1	4	yes	maximal
6	C₅×#2084	C₂	1	2	yes	maximal
7	C₅×#2084	C₂	1	2	yes	maximal
8	C₅×#1831	C₂	1	1	yes	maximal, characteristic
9	C₂×C₅×#125	—	2	1	no	—
10	C₂×C₅×#125	—	2	1	no	—
11	C₂×C₅×#106	C₂²	1	4	yes	—
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₅×#389	—	2	1	no	—
14	C₅×#389	—	2	1	no	—
15	C₅×#389	—	2	1	no	—
16	C₅×#389	—	2	1	no	—
17	C₅×#389	C₂²	1	2	yes	—
18	C₅×#389	C₂²	1	2	yes	—
19	C₅×#389	C₂²	1	2	yes	—
20	C₅×#389	C₂²	1	2	yes	—
21	C₅×#384	C₂²	1	1	yes	characteristic
22	C₅×#384	C₂²	1	1	yes	characteristic
23	C₅×#384	C₂²	1	1	yes	characteristic
24	C₅×#384	C₂²	1	1	yes	characteristic
25	C₅×#382	—	2	1	no	—
26	C₅×#382	—	2	1	no	—
27	C₄³×C₅	C₂²	1	1	yes	characteristic
28	C₅×#340	C₂²	1	4	yes	—
29	C₅×#340	C₂²	1	4	yes	—
30	C₅×#340	C₂²	1	4	yes	—
31	C₅×#336	C₂²	1	2	yes	—
32	C₅×#336	C₂²	1	2	yes	—
33	#32849	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅×C₈	C₂³	1	4	yes	—
35	C₅×#125	—	4	4	no	—
36	C₅×#125	—	4	4	no	—
37	C₂×C₅×C₄⋊C₄	—	2	1	no	—
38	C₂×C₅×C₄⋊C₄	—	2	1	no	—
39	C₂×C₅×C₄⋊C₄	—	2	1	no	—
40	C₂×C₅×C₄⋊C₄	—	2	1	no	—
41	C₂×C₅×C₄⋊C₄	—	2	1	no	—
42	C₂×C₅×C₄⋊C₄	—	2	1	no	—
43	C₂×C₅×C₄⋊C₄	—	2	1	no	—
44	C₂×C₅×C₄⋊C₄	—	2	1	no	—
45	C₂×C₅×C₄⋊C₄	—	2	2	no	—
46	C₂×C₅×C₄⋊C₄	—	2	2	no	—
47	C₂×C₅×C₄⋊C₄	—	2	2	no	—
48	C₂×C₅×C₄⋊C₄	—	2	2	no	—
49	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
60	C₅×#106	—	4	4	no	—
61	C₅×C₄⋊C₈	—	4	2	no	—
62	C₅×#95	—	2	2	no	—
63	C₅×#95	—	2	2	no	—
64	C₅×#95	—	2	2	no	—
65	C₅×#95	—	2	2	no	—
66	#2859	C₂×C₅	1	1	yes	characteristic
67	#2859	C₂×C₅	1	1	yes	characteristic
68	#2100	C₂×C₅	1	2	yes	—
69	#2099	C₂×C₅	1	2	yes	—
70	#2088	C₂×C₅	1	4	yes	—
71	#2084	C₂×C₅	1	2	yes	—
72	#2084	C₂×C₅	1	2	yes	—
73	#1831	C₂×C₅	1	1	yes	characteristic
74	C₂²×C₄×C₅	—	2	2	no	—
75	C₂²×C₄×C₅	—	2	2	no	—
76	C₂²×C₄×C₅	—	2	2	no	—
77	C₂²×C₄×C₅	—	2	2	no	—
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
85	C₂×C₅×C₈	—	2	4	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	—	2	4	no	—
88	C₅×C₄⋊C₄	—	2	2	no	—
89	C₅×C₄⋊C₄	—	2	2	no	—
90	C₅×C₄⋊C₄	—	2	4	no	—
91	C₅×C₄⋊C₄	—	2	4	no	—
92	C₅×C₄⋊C₄	—	2	2	no	—
93	C₅×C₄⋊C₄	—	2	2	no	—
94	C₅×C₄⋊C₄	—	4	2	no	—
95	C₅×C₄⋊C₄	—	4	2	no	—
96	C₅×C₄⋊C₄	—	4	2	no	—
97	C₅×C₄⋊C₄	—	4	2	no	—
98	C₅×C₄⋊C₄	—	4	4	no	—
99	C₅×C₄⋊C₄	—	4	2	no	—
100	C₅×C₄⋊C₄	—	4	2	no	—
101	C₅×C₄⋊C₄	—	4	4	no	—
102	C₅×C₄⋊C₄	—	4	4	no	—
103	C₅×C₄⋊C₄	—	4	2	no	—
104	C₅×C₄⋊C₄	—	4	4	no	—
105	C₅×C₄⋊C₄	—	4	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	2	1	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	2	1	no	—
112	C₄²×C₅	—	4	1	no	—
113	C₄²×C₅	SD₁₆	1	4	yes	—
114	C₂×#125	—	2	1	no	—
115	C₂×#125	—	2	1	no	—
116	C₂×#106	C₂²×C₅	1	4	yes	—
117	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
118	#389	—	2	1	no	—
119	#389	—	2	1	no	—
120	#389	—	2	1	no	—
121	#389	—	2	1	no	—
122	#389	C₂²×C₅	1	2	yes	—
123	#389	C₂²×C₅	1	2	yes	—
124	#389	C₂²×C₅	1	2	yes	—
125	#389	C₂²×C₅	1	2	yes	—
126	#384	C₂²×C₅	1	1	yes	characteristic
127	#384	C₂²×C₅	1	1	yes	characteristic
128	#384	C₂²×C₅	1	1	yes	characteristic
129	#384	C₂²×C₅	1	1	yes	characteristic
130	#382	—	2	1	no	—
131	#382	—	2	1	no	—
132	C₄³	C₂²×C₅	1	1	yes	characteristic
133	#340	C₂²×C₅	1	4	yes	—
134	#340	C₂²×C₅	1	4	yes	—
135	#340	C₂²×C₅	1	4	yes	—
136	#336	C₂²×C₅	1	2	yes	—
137	#336	C₂²×C₅	1	2	yes	—
138	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅	—	2	1	no	—
140	C₂×C₄×C₅	—	2	1	no	—
141	C₂×C₄×C₅	—	2	1	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	1	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	4	2	no	—
153	C₂×C₄×C₅	—	4	2	no	—
154	C₂×C₄×C₅	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	2	no	—
157	C₂×C₄×C₅	—	4	2	no	—
158	C₂×C₄×C₅	—	4	2	no	—
159	C₂×C₄×C₅	—	4	2	no	—
160	C₂×C₄×C₅	—	4	2	no	—
161	C₂×C₄×C₅	—	4	2	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
165	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
166	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
167	C₂×C₄×C₅	AGL(1,ℤ/8)	1	2	yes	—
168	C₂×C₄×C₅	#137	1	2	yes	—
169	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
171	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
172	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
173	C₅×C₈	—	4	4	no	—
174	C₂²×C₈	C₂³×C₅	1	4	yes	—
175	#125	—	4	4	no	—
176	#125	—	4	4	no	—
177	C₂×C₄⋊C₄	—	2	1	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	1	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	1	no	—
186	C₂×C₄⋊C₄	—	2	1	no	—
187	C₂×C₄⋊C₄	—	2	1	no	—
188	C₂×C₄⋊C₄	—	2	1	no	—
189	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
190	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
191	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
192	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
193	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
194	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
195	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
196	C₂×C₄²	C₅×D₄	1	1	yes	characteristic
197	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
199	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
200	#106	—	4	4	no	—
201	C₄⋊C₈	—	4	2	no	—
202	#95	—	2	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	C₂²×C₅	#536	1	1	yes	characteristic
207	C₂²×C₅	#536	1	1	yes	characteristic
208	C₂²×C₅	#537	1	1	yes	characteristic
209	C₂²×C₅	C₂²×SD₁₆	1	1	yes	characteristic
210	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
211	C₂²×C₅	C₂×#137	1	1	yes	characteristic
212	C₂²×C₅	#575	1	1	yes	characteristic
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	—	2	2	no	—
215	C₄×C₅	—	2	2	no	—
216	C₄×C₅	—	2	2	no	—
217	C₄×C₅	—	2	2	no	—
218	C₄×C₅	—	4	1	no	—
219	C₄×C₅	—	8	2	no	—
220	C₄×C₅	—	8	2	no	—
221	C₄×C₅	—	8	2	no	—
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	#563	1	2	yes	—
224	C₄×C₅	#564	1	2	yes	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
230	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
231	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
234	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
235	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	4	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	4	4	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	4	1	no	—
264	C₄²	C₅×SD₁₆	1	4	yes	—
265	C₂×C₅	#2822	1	1	yes	characteristic
266	C₂×C₅	#3227	1	1	yes	characteristic
267	C₂×C₅	#3229	1	1	yes	characteristic
268	C₂×C₅	#3234	1	1	yes	characteristic
269	C₂×C₅	#3236	1	1	yes	characteristic
270	C₂×C₅	#3246	1	1	yes	characteristic
271	C₂×C₅	#3254	1	1	yes	characteristic
272	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
299	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
300	C₂×C₄	C₂×C₅×SD₁₆	1	2	yes	—
301	C₂×C₄	C₅×AGL(1,ℤ/8)	1	2	yes	—
302	C₂×C₄	C₅×#137	1	2	yes	—
303	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
304	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
305	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
306	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
307	C₈	—	4	4	no	—
308	C₅	#32849	1	1	yes	characteristic, direct product factor, Sylow
309	C₂²	C₅×#536	1	1	yes	characteristic
310	C₂²	C₅×#536	1	1	yes	characteristic
311	C₂²	C₅×#537	1	1	yes	characteristic
312	C₂²	C₂²×C₅×SD₁₆	1	1	yes	characteristic
313	C₂²	C₂×C₅×AGL(1,ℤ/8)	1	1	yes	characteristic
314	C₂²	C₂×C₅×#137	1	1	yes	characteristic
315	C₂²	C₅×#575	1	1	yes	characteristic
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	2	no	—
326	C₄	C₅×#563	1	2	yes	—
327	C₄	C₅×#564	1	2	yes	—
328	C₂	C₅×#2822	1	1	yes	characteristic
329	C₂	C₅×#3227	1	1	yes	characteristic
330	C₂	C₅×#3229	1	1	yes	characteristic
331	C₂	C₅×#3234	1	1	yes	characteristic
332	C₂	C₅×#3236	1	1	yes	characteristic
333	C₂	C₅×#3246	1	1	yes	characteristic
334	C₂	C₅×#3254	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₅	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄³×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
33. #32849	C₅	direct (N × H)
308. C₅	#32849	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×#2859	2
C₅×#2100	2
C₅×#2099	2
C₅×#2088	2
C₅×#2084	2
C₅×#1831	2
#32849	5

Subgroups — C5×#32849 #11955569

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#32849 #11955569