Subgroups — C₅×#2798 #10616393

Order 640.

386 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#400	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅×#400	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#399	C₂	1	1	yes	maximal, characteristic
4	C₅×#397	C₂	1	1	yes	maximal, characteristic
5	C₅×#397	C₂	1	1	yes	maximal, characteristic
6	C₅×#393	C₂	1	1	yes	maximal, characteristic
7	C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅×#391	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#389	C₂	1	1	yes	maximal, characteristic, complemented
10	C₅×#388	C₂	1	1	yes	maximal, characteristic
11	C₅×#388	C₂	1	1	yes	maximal, characteristic
12	C₅×#388	C₂	1	1	yes	maximal, characteristic
13	C₅×#387	C₂	1	1	yes	maximal, characteristic, complemented
14	C₅×#387	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅×#387	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×C₅×Q₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
20	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
23	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
24	C₂×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
31	C₂×C₅×C₂²⋊C₄	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
33	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂²	1	1	yes	characteristic
35	C₅×#95	C₂²	1	1	yes	characteristic
36	C₅×#95	C₂²	1	1	yes	characteristic
37	C₅×#95	C₂²	1	1	yes	characteristic
38	C₅×#95	C₂²	1	1	yes	characteristic
39	C₅×#95	C₂²	1	1	yes	characteristic
40	C₅×#95	C₂²	1	1	yes	characteristic
41	C₅×#95	C₂²	1	1	yes	characteristic
42	C₅×#95	C₂²	1	1	yes	characteristic
43	C₅×#95	C₂²	1	1	yes	characteristic
44	C₅×#95	C₂²	1	1	yes	characteristic
45	C₅×#95	C₂²	1	1	yes	characteristic
46	C₅×#95	C₂²	1	1	yes	characteristic
47	C₅×#95	C₂²	1	1	yes	characteristic
48	C₅×#95	C₂²	1	1	yes	characteristic
49	C₅×#95	C₂²	1	1	yes	characteristic
50	C₅×#95	C₂²	1	1	yes	characteristic
51	#2798	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂⁴×C₅	C₂³	1	1	yes	characteristic
53	C₂×C₅×Q₈	—	2	2	no	—
54	C₂×C₅×Q₈	—	2	2	no	—
55	C₂×C₅×Q₈	—	4	1	no	—
56	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₂³	1	1	yes	characteristic
70	C₅×C₄⋊C₄	—	2	2	no	—
71	C₅×C₄⋊C₄	—	2	2	no	—
72	C₅×C₄⋊C₄	—	2	2	no	—
73	C₅×C₄⋊C₄	—	2	2	no	—
74	C₅×C₄⋊C₄	—	4	1	no	—
75	C₅×C₄⋊C₄	—	4	1	no	—
76	C₅×C₄⋊C₄	—	4	1	no	—
77	C₅×C₄⋊C₄	—	4	1	no	—
78	C₅×C₂²⋊C₄	—	2	2	no	—
79	C₅×C₂²⋊C₄	—	2	2	no	—
80	C₅×C₂²⋊C₄	—	2	2	no	—
81	C₅×C₂²⋊C₄	—	4	1	no	—
82	C₅×C₂²⋊C₄	—	4	1	no	—
83	C₅×C₂²⋊C₄	—	4	1	no	—
84	C₅×C₂²⋊C₄	—	4	1	no	—
85	C₄²×C₅	—	4	1	no	—
86	C₄²×C₅	—	4	1	no	—
87	C₄²×C₅	—	4	1	no	—
88	#400	C₂×C₅	1	1	yes	characteristic, complemented
89	#400	C₂×C₅	1	1	yes	characteristic, complemented
90	#399	C₂×C₅	1	1	yes	characteristic
91	#397	C₂×C₅	1	1	yes	characteristic
92	#397	C₂×C₅	1	1	yes	characteristic
93	#393	C₂×C₅	1	1	yes	characteristic
94	#391	C₂×C₅	1	1	yes	characteristic, complemented
95	#391	C₂×C₅	1	1	yes	characteristic, complemented
96	#389	C₂×C₅	1	1	yes	characteristic, complemented
97	#388	C₂×C₅	1	1	yes	characteristic
98	#388	C₂×C₅	1	1	yes	characteristic
99	#388	C₂×C₅	1	1	yes	characteristic
100	#387	C₂×C₅	1	1	yes	characteristic, complemented
101	#387	C₂×C₅	1	1	yes	characteristic, complemented
102	#387	C₂×C₅	1	1	yes	characteristic, complemented
103	C₂³×C₅	—	2	1	no	—
104	C₂³×C₅	—	2	1	no	—
105	C₂³×C₅	—	2	1	no	—
106	C₂³×C₅	—	2	1	no	—
107	C₂³×C₅	—	2	1	no	—
108	C₂³×C₅	—	2	1	no	—
109	C₂³×C₅	—	2	1	no	—
110	C₂³×C₅	C₂⁴	1	1	yes	characteristic
111	C₅×Q₈	—	4	4	no	—
112	C₂×C₄×C₅	—	2	1	no	—
113	C₂×C₄×C₅	—	2	1	no	—
114	C₂×C₄×C₅	—	2	1	no	—
115	C₂×C₄×C₅	—	2	1	no	—
116	C₂×C₄×C₅	—	2	1	no	—
117	C₂×C₄×C₅	—	2	1	no	—
118	C₂×C₄×C₅	—	2	1	no	—
119	C₂×C₄×C₅	—	2	1	no	—
120	C₂×C₄×C₅	—	2	1	no	—
121	C₂×C₄×C₅	—	2	1	no	—
122	C₂×C₄×C₅	—	2	1	no	—
123	C₂×C₄×C₅	—	2	1	no	—
124	C₂×C₄×C₅	—	2	1	no	—
125	C₂×C₄×C₅	—	2	1	no	—
126	C₂×C₄×C₅	—	2	1	no	—
127	C₂×C₄×C₅	—	2	1	no	—
128	C₂×C₄×C₅	—	2	1	no	—
129	C₂×C₄×C₅	—	2	1	no	—
130	C₂×C₄×C₅	—	2	1	no	—
131	C₂×C₄×C₅	—	2	1	no	—
132	C₂×C₄×C₅	—	2	1	no	—
133	C₂×C₄×C₅	—	2	1	no	—
134	C₂×C₄×C₅	—	2	1	no	—
135	C₂×C₄×C₅	—	2	1	no	—
136	C₂×C₄×C₅	—	2	1	no	—
137	C₂×C₄×C₅	—	2	1	no	—
138	C₂×C₄×C₅	—	2	1	no	—
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	#41	1	2	yes	—
152	C₂×C₄×C₅	#41	1	2	yes	—
153	C₂×C₄×C₅	#41	1	2	yes	—
154	C₂²×Q₈	C₂²×C₅	1	1	yes	characteristic
155	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
156	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
157	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
158	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
159	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
160	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
161	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
162	C₂×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
163	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
166	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
167	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
168	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	C₂²×C₅	1	1	yes	characteristic
170	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
171	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
172	C₂×C₄²	C₂²×C₅	1	1	yes	characteristic
173	#95	C₂²×C₅	1	1	yes	characteristic
174	#95	C₂²×C₅	1	1	yes	characteristic
175	#95	C₂²×C₅	1	1	yes	characteristic
176	#95	C₂²×C₅	1	1	yes	characteristic
177	#95	C₂²×C₅	1	1	yes	characteristic
178	#95	C₂²×C₅	1	1	yes	characteristic
179	#95	C₂²×C₅	1	1	yes	characteristic
180	#95	C₂²×C₅	1	1	yes	characteristic
181	#95	C₂²×C₅	1	1	yes	characteristic
182	#95	C₂²×C₅	1	1	yes	characteristic
183	#95	C₂²×C₅	1	1	yes	characteristic
184	#95	C₂²×C₅	1	1	yes	characteristic
185	#95	C₂²×C₅	1	1	yes	characteristic
186	#95	C₂²×C₅	1	1	yes	characteristic
187	#95	C₂²×C₅	1	1	yes	characteristic
188	#95	C₂²×C₅	1	1	yes	characteristic
189	C₂²×C₅	—	4	1	no	—
190	C₂²×C₅	—	4	1	no	—
191	C₂²×C₅	—	4	1	no	—
192	C₂²×C₅	—	4	1	no	—
193	C₂²×C₅	—	4	1	no	—
194	C₂²×C₅	—	4	1	no	—
195	C₂²×C₅	—	4	1	no	—
196	C₂²×C₅	C₂×#41	1	1	yes	characteristic
197	C₂²×C₅	C₂×#41	1	1	yes	characteristic
198	C₂²×C₅	C₂×#41	1	1	yes	characteristic
199	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
200	C₂²×C₅	H(5,𝔽₂)	1	1	yes	characteristic
201	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
202	C₂²×C₅	2₋¹⁺⁴	1	1	yes	characteristic
203	C₄×C₅	—	2	2	no	—
204	C₄×C₅	—	2	2	no	—
205	C₄×C₅	—	2	2	no	—
206	C₄×C₅	—	4	1	no	—
207	C₄×C₅	—	4	1	no	—
208	C₄×C₅	—	4	1	no	—
209	C₄×C₅	—	4	1	no	—
210	C₄×C₅	—	4	1	no	—
211	C₄×C₅	—	4	1	no	—
212	C₄×C₅	—	4	1	no	—
213	C₄×C₅	—	4	1	no	—
214	C₄×C₅	—	4	1	no	—
215	C₄×C₅	—	4	1	no	—
216	C₄×C₅	—	4	1	no	—
217	C₂⁴	C₂³×C₅	1	1	yes	characteristic
218	C₂×Q₈	—	2	2	no	—
219	C₂×Q₈	—	2	2	no	—
220	C₂×Q₈	—	4	1	no	—
221	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
222	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
223	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
224	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
225	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
226	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
227	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
228	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
229	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
230	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
231	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
232	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
233	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
234	C₂²×C₄	C₂³×C₅	1	1	yes	characteristic
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	4	1	no	—
240	C₄⋊C₄	—	4	1	no	—
241	C₄⋊C₄	—	4	1	no	—
242	C₄⋊C₄	—	4	1	no	—
243	C₂²⋊C₄	—	2	2	no	—
244	C₂²⋊C₄	—	2	2	no	—
245	C₂²⋊C₄	—	2	2	no	—
246	C₂²⋊C₄	—	4	1	no	—
247	C₂²⋊C₄	—	4	1	no	—
248	C₂²⋊C₄	—	4	1	no	—
249	C₂²⋊C₄	—	4	1	no	—
250	C₄²	—	4	1	no	—
251	C₄²	—	4	1	no	—
252	C₄²	—	4	1	no	—
253	C₂×C₅	—	8	1	no	—
254	C₂×C₅	#539	1	1	yes	characteristic
255	C₂×C₅	#541	1	1	yes	characteristic
256	C₂×C₅	#542	1	1	yes	characteristic
257	C₂×C₅	#551	1	1	yes	characteristic
258	C₂×C₅	#556	1	1	yes	characteristic
259	C₂×C₅	#559	1	1	yes	characteristic
260	C₂×C₅	#562	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂⁴×C₅	1	1	yes	characteristic
269	Q₈	—	4	4	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
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	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	C₅×#41	1	2	yes	—
310	C₂×C₄	C₅×#41	1	2	yes	—
311	C₂×C₄	C₅×#41	1	2	yes	—
312	C₅	#2798	1	1	yes	characteristic, direct product factor, Sylow
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	C₂×C₅×#41	1	1	yes	characteristic
321	C₂²	C₂×C₅×#41	1	1	yes	characteristic
322	C₂²	C₂×C₅×#41	1	1	yes	characteristic
323	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
324	C₂²	C₅×H(5,𝔽₂)	1	1	yes	characteristic
325	C₂²	C₅×2₋¹⁺⁴	1	1	yes	characteristic
326	C₂²	C₅×2₋¹⁺⁴	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₂	—	8	1	no	—
342	C₂	C₅×#539	1	1	yes	characteristic
343	C₂	C₅×#541	1	1	yes	characteristic
344	C₂	C₅×#542	1	1	yes	characteristic
345	C₂	C₅×#551	1	1	yes	characteristic
346	C₂	C₅×#556	1	1	yes	characteristic
347	C₂	C₅×#559	1	1	yes	characteristic
348	C₂	C₅×#562	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₅×#400	C₂	semidirect
2. C₅×#400	C₂	semidirect
7. C₅×#391	C₂	semidirect
8. C₅×#391	C₂	semidirect
9. C₅×#389	C₂	semidirect
13. C₅×#387	C₂	semidirect
14. C₅×#387	C₂	semidirect
15. C₅×#387	C₂	semidirect
51. #2798	C₅	direct (N × H)
88. #400	C₂×C₅	semidirect
89. #400	C₂×C₅	semidirect
94. #391	C₂×C₅	semidirect
95. #391	C₂×C₅	semidirect
96. #389	C₂×C₅	semidirect
100. #387	C₂×C₅	semidirect
101. #387	C₂×C₅	semidirect
102. #387	C₂×C₅	semidirect
312. C₅	#2798	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×#400	2
C₅×#399	2
C₅×#397	2
C₅×#393	2
C₅×#391	2
C₅×#389	2
C₅×#388	2
C₅×#387	2
#2798	5

Subgroups — C5×#2798 #10616393

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#2798 #10616393