Subgroups — C₅×#32828 #11955548

Order 1280.

570 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2647	C₂	1	1	yes	maximal, characteristic
2	C₅×#2642	C₂	1	1	yes	maximal, characteristic
3	C₅×#2097	C₂	1	2	yes	maximal
4	C₅×#2072	C₂	1	2	yes	maximal
5	C₅×#2068	C₂	1	2	yes	maximal
6	C₅×#2066	C₂	1	2	yes	maximal
7	C₅×#2059	C₂	1	2	yes	maximal
8	C₅×#2043	C₂	1	2	yes	maximal
9	C₅×#1832	C₂	1	1	yes	maximal, characteristic
10	C₂×C₅×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₂×C₅×#125	C₂²	1	1	yes	characteristic
12	C₂×C₅×#107	C₂²	1	2	yes	—
13	C₂×C₅×#106	C₂²	1	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₅×#103	C₂²	1	2	yes	—
17	C₂×C₅×#103	C₂²	1	2	yes	—
18	C₂×C₅×#103	C₂²	1	2	yes	—
19	C₂×C₅×#103	C₂²	1	2	yes	—
20	C₅×#400	—	2	2	no	—
21	C₅×#400	—	2	2	no	—
22	C₅×#400	—	2	2	no	—
23	C₅×#398	—	2	2	no	—
24	C₅×#391	C₂²	1	2	yes	—
25	C₅×#391	C₂²	1	2	yes	—
26	C₅×#389	C₂²	1	2	yes	—
27	C₅×#389	C₂²	1	2	yes	—
28	C₅×#384	C₂²	1	1	yes	characteristic
29	C₅×#382	C₂²	1	1	yes	characteristic
30	C₅×#376	C₂²	1	1	yes	characteristic
31	C₅×#340	C₂²	1	2	yes	—
32	C₅×#340	C₂²	1	2	yes	—
33	C₅×#336	C₂²	1	2	yes	—
34	C₅×#336	C₂²	1	2	yes	—
35	#32828	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅×Q₈	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	C₂³	1	2	yes	—
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
39	C₅×C₄⋊Q₈	—	4	1	no	—
40	C₅×C₄⋊Q₈	—	4	1	no	—
41	C₅×#125	—	4	1	no	—
42	C₅×#125	—	4	1	no	—
43	C₂×C₅×C₄⋊C₄	—	2	2	no	—
44	C₂×C₅×C₄⋊C₄	—	2	2	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	C₂³	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₅×#107	—	4	2	no	—
55	C₅×#106	—	4	2	no	—
56	C₅×C₄⋊C₈	—	4	1	no	—
57	C₅×C₄⋊C₈	—	4	1	no	—
58	C₅×#103	—	4	2	no	—
59	C₅×#103	—	4	2	no	—
60	C₅×#103	—	4	2	no	—
61	C₅×#103	—	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	C₅×#95	—	2	4	no	—
67	C₅×#95	—	2	2	no	—
68	C₅×#95	D₄	1	2	yes	—
69	C₅×#95	D₄	1	2	yes	—
70	#2647	C₂×C₅	1	1	yes	characteristic
71	#2642	C₂×C₅	1	1	yes	characteristic
72	#2097	C₂×C₅	1	2	yes	—
73	#2072	C₂×C₅	1	2	yes	—
74	#2068	C₂×C₅	1	2	yes	—
75	#2066	C₂×C₅	1	2	yes	—
76	#2059	C₂×C₅	1	2	yes	—
77	#2043	C₂×C₅	1	2	yes	—
78	#1832	C₂×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	2	no	—
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	—	2	2	no	—
82	C₂×C₅×Q₈	—	2	2	no	—
83	C₂×C₅×Q₈	—	2	2	no	—
84	C₂×C₅×Q₈	—	2	2	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	2	no	—
87	C₂²×C₄×C₅	—	2	2	no	—
88	C₂²×C₄×C₅	—	2	2	no	—
89	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
94	C₂×C₅×C₈	—	2	2	no	—
95	C₂×C₅×C₈	—	2	2	no	—
96	C₂×C₅×C₈	—	2	2	no	—
97	C₂×C₅×C₈	—	2	2	no	—
98	C₂×C₅×C₈	—	2	2	no	—
99	C₂×C₅×C₈	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	2	no	—
102	C₅×C₄⋊C₄	—	2	2	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	—	2	2	no	—
105	C₅×C₄⋊C₄	—	2	2	no	—
106	C₅×C₄⋊C₄	—	4	4	no	—
107	C₅×C₄⋊C₄	—	4	4	no	—
108	C₅×C₄⋊C₄	—	8	2	no	—
109	C₅×C₄⋊C₄	—	8	2	no	—
110	C₅×C₄⋊C₄	—	8	2	no	—
111	C₄²×C₅	—	2	1	no	—
112	C₄²×C₅	—	2	1	no	—
113	C₄²×C₅	—	2	1	no	—
114	C₄²×C₅	—	2	1	no	—
115	C₄²×C₅	—	2	1	no	—
116	C₄²×C₅	—	2	1	no	—
117	C₂×C₄⋊Q₈	C₂²×C₅	1	1	yes	characteristic
118	C₂×#125	C₂²×C₅	1	1	yes	characteristic
119	C₂×#107	C₂²×C₅	1	2	yes	—
120	C₂×#106	C₂²×C₅	1	2	yes	—
121	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
123	C₂×#103	C₂²×C₅	1	2	yes	—
124	C₂×#103	C₂²×C₅	1	2	yes	—
125	C₂×#103	C₂²×C₅	1	2	yes	—
126	C₂×#103	C₂²×C₅	1	2	yes	—
127	#400	—	2	2	no	—
128	#400	—	2	2	no	—
129	#400	—	2	2	no	—
130	#398	—	2	2	no	—
131	#391	C₂²×C₅	1	2	yes	—
132	#391	C₂²×C₅	1	2	yes	—
133	#389	C₂²×C₅	1	2	yes	—
134	#389	C₂²×C₅	1	2	yes	—
135	#384	C₂²×C₅	1	1	yes	characteristic
136	#382	C₂²×C₅	1	1	yes	characteristic
137	#376	C₂²×C₅	1	1	yes	characteristic
138	#340	C₂²×C₅	1	2	yes	—
139	#340	C₂²×C₅	1	2	yes	—
140	#336	C₂²×C₅	1	2	yes	—
141	#336	C₂²×C₅	1	2	yes	—
142	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
143	C₅×Q₈	—	4	2	no	—
144	C₅×Q₈	—	4	2	no	—
145	C₅×Q₈	—	4	2	no	—
146	C₅×Q₈	—	4	2	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	1	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	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	—	4	2	no	—
165	C₂×C₄×C₅	—	4	2	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	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
173	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
174	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	—	4	2	no	—
177	C₂²×Q₈	C₂³×C₅	1	2	yes	—
178	C₂²×C₈	C₂³×C₅	1	2	yes	—
179	C₂²×C₈	C₂³×C₅	1	2	yes	—
180	C₄⋊Q₈	—	4	1	no	—
181	C₄⋊Q₈	—	4	1	no	—
182	#125	—	4	1	no	—
183	#125	—	4	1	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	2	2	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×C₄²	C₂³×C₅	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
194	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
195	#107	—	4	2	no	—
196	#106	—	4	2	no	—
197	C₄⋊C₈	—	4	1	no	—
198	C₄⋊C₈	—	4	1	no	—
199	#103	—	4	2	no	—
200	#103	—	4	2	no	—
201	#103	—	4	2	no	—
202	#103	—	4	2	no	—
203	#95	—	2	2	no	—
204	#95	—	2	2	no	—
205	#95	—	2	2	no	—
206	#95	—	2	2	no	—
207	#95	—	2	4	no	—
208	#95	—	2	2	no	—
209	#95	C₅×D₄	1	2	yes	—
210	#95	C₅×D₄	1	2	yes	—
211	C₂²×C₅	#535	1	1	yes	characteristic
212	C₂²×C₅	#536	1	1	yes	characteristic
213	C₂²×C₅	#537	1	1	yes	characteristic
214	C₂²×C₅	#577	1	2	yes	—
215	C₂²×C₅	#578	1	2	yes	—
216	C₄×C₅	—	4	2	no	—
217	C₄×C₅	—	4	2	no	—
218	C₄×C₅	—	4	1	no	—
219	C₄×C₅	—	4	1	no	—
220	C₄×C₅	—	8	2	no	—
221	C₄×C₅	—	8	2	no	—
222	C₄×C₅	—	8	2	no	—
223	C₄×C₅	—	8	2	no	—
224	C₄×C₅	#562	1	1	yes	characteristic
225	C₄×C₅	#562	1	1	yes	characteristic
226	C₄×C₅	#563	1	1	yes	characteristic
227	C₄×C₅	#563	1	1	yes	characteristic
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	2	2	no	—
234	C₂²×C₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
239	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
240	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
242	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	4	no	—
256	C₄⋊C₄	—	4	4	no	—
257	C₄⋊C₄	—	8	2	no	—
258	C₄⋊C₄	—	8	2	no	—
259	C₄⋊C₄	—	8	2	no	—
260	C₄²	—	2	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	2	1	no	—
266	C₂×C₅	#2820	1	1	yes	characteristic
267	C₂×C₅	#3225	1	2	yes	—
268	C₂×C₅	#3242	1	2	yes	—
269	C₂×C₅	#3260	1	1	yes	characteristic
270	C₂×C₅	#3261	1	1	yes	characteristic
271	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
272	Q₈	—	4	2	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	Q₈	—	4	2	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	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	C₅×H(5,𝔽₂)	1	1	yes	characteristic
299	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
300	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
301	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
302	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
303	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
304	C₈	—	4	2	no	—
305	C₈	—	4	2	no	—
306	C₅	#32828	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²	C₅×#535	1	1	yes	characteristic
308	C₂²	C₅×#536	1	1	yes	characteristic
309	C₂²	C₅×#537	1	1	yes	characteristic
310	C₂²	C₅×#577	1	2	yes	—
311	C₂²	C₅×#578	1	2	yes	—
312	C₄	—	4	2	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	2	no	—
316	C₄	—	8	2	no	—
317	C₄	—	8	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₅×#562	1	1	yes	characteristic
321	C₄	C₅×#562	1	1	yes	characteristic
322	C₄	C₅×#563	1	1	yes	characteristic
323	C₄	C₅×#563	1	1	yes	characteristic
324	C₂	C₅×#2820	1	1	yes	characteristic
325	C₂	C₅×#3225	1	2	yes	—
326	C₂	C₅×#3242	1	2	yes	—
327	C₂	C₅×#3260	1	1	yes	characteristic
328	C₂	C₅×#3261	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 ◁ C₅×#376 ◁ G

Complements

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

N	H ≅ G/N	type
35. #32828	C₅	direct (N × H)
306. C₅	#32828	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×#2647	2
C₅×#2642	2
C₅×#2097	2
C₅×#2072	2
C₅×#2068	2
C₅×#2066	2
C₅×#2059	2
C₅×#2043	2
C₅×#1832	2
#32828	5

Subgroups — C5×#32828 #11955548

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#32828 #11955548