Subgroups — C₅×#22213 #11944933

Order 1280.

822 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#446	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅×C₈×D₄	C₂	1	1	yes	maximal, characteristic
3	C₅×#2398	C₂	1	1	yes	maximal, characteristic
4	C₅×#1985	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#1955	C₂	1	1	yes	maximal, characteristic
6	C₅×#1924	C₂	1	4	yes	maximal, complemented
7	C₅×#1850	C₂	1	4	yes	maximal, complemented
8	C₅×#1843	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅×#1746	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×C₅×C₈	C₂²	1	2	yes	—
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₅×D₄	C₂²	1	1	yes	characteristic
13	C₅×#446	—	2	4	no	—
14	C₅×C₈×D₄	—	2	4	no	—
15	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
19	C₂×C₅×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
21	C₂×C₅×#97	C₂²	1	2	yes	—
22	C₂×C₅×#97	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
24	C₅×#400	C₄	1	2	yes	—
25	C₅×#397	C₄	1	2	yes	—
26	C₅×#394	C₄	1	2	yes	—
27	C₅×#393	C₄	1	2	yes	—
28	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
29	C₅×#376	C₂²	1	1	yes	characteristic
30	C₅×#336	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	4	yes	—
32	C₅×#336	C₂²	1	2	yes	—
33	#22213	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×D₄	C₂×C₄	1	1	yes	characteristic
36	C₂³×C₄×C₅	C₂³	1	2	yes	—
37	C₂²×C₅×C₈	—	2	4	no	—
38	C₂²×C₅×C₈	—	4	4	no	—
39	C₂²×C₅×C₈	C₂³	1	2	yes	—
40	C₂²×C₅×C₈	C₂³	1	2	yes	—
41	C₂²×C₅×C₈	C₂³	1	2	yes	—
42	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
44	C₄×C₅×Q₈	—	2	2	no	—
45	C₄×C₅×Q₈	—	2	2	no	—
46	C₄×C₅×D₄	—	2	2	no	—
47	C₄×C₅×D₄	—	2	2	no	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
52	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₅	C₂³	1	2	yes	—
55	C₂×C₄²×C₅	C₂³	1	2	yes	—
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₅×C₄⋊C₈	—	2	4	no	—
58	C₅×C₄⋊C₈	—	2	2	no	—
59	C₅×C₄⋊C₈	—	2	2	no	—
60	C₅×C₄⋊C₈	—	4	4	no	—
61	C₅×C₂²⋊C₈	—	2	4	no	—
62	C₅×C₂²⋊C₈	—	4	4	no	—
63	C₅×#97	—	2	4	no	—
64	C₅×#97	—	2	2	no	—
65	C₄×C₅×C₈	D₄	1	4	yes	—
66	C₅×#95	C₂×C₄	1	2	yes	—
67	C₅×#95	C₂×C₄	1	2	yes	—
68	C₅×#95	C₂×C₄	1	4	yes	—
69	C₅×#95	C₂×C₄	1	4	yes	—
70	C₂×#446	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂×C₈×D₄	C₂×C₅	1	1	yes	characteristic
72	#2398	C₂×C₅	1	1	yes	characteristic
73	#1985	C₂×C₅	1	1	yes	characteristic, complemented
74	#1955	C₂×C₅	1	1	yes	characteristic
75	#1924	C₂×C₅	1	4	yes	complemented
76	#1850	C₂×C₅	1	4	yes	complemented
77	#1843	C₂×C₅	1	1	yes	characteristic, complemented
78	#1746	C₂×C₅	1	1	yes	characteristic, complemented
79	C₂⁴×C₅	C₂²×C₄	1	2	yes	—
80	C₂×C₅×Q₈	—	2	2	no	—
81	C₂×C₅×Q₈	—	4	2	no	—
82	C₂×C₅×D₄	—	2	4	no	—
83	C₂×C₅×D₄	—	2	2	no	—
84	C₂²×C₄×C₅	—	2	2	no	—
85	C₂²×C₄×C₅	—	2	2	no	—
86	C₂²×C₄×C₅	—	2	4	no	—
87	C₂²×C₄×C₅	—	2	4	no	—
88	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
89	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
91	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
92	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
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	C₂×C₅×C₈	—	2	4	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	4	no	—
103	C₂×C₅×C₈	—	2	2	no	—
104	C₂×C₅×C₈	—	4	8	no	—
105	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
106	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
107	C₅×C₄⋊C₄	—	2	2	no	—
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	—	2	4	no	—
110	C₅×C₂²⋊C₄	—	2	8	no	—
111	C₅×C₂²⋊C₄	—	2	4	no	—
112	C₅×C₂²⋊C₄	—	4	4	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	—	2	2	no	—
116	C₄²×C₅	—	2	2	no	—
117	C₄²×C₅	C₂×D₄	1	2	yes	—
118	C₄²×C₅	#41	1	2	yes	—
119	C₂³×C₈	C₂²×C₅	1	2	yes	—
120	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄×D₄	C₂²×C₅	1	1	yes	characteristic
122	#446	—	2	4	no	—
123	C₈×D₄	—	2	4	no	—
124	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
125	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
126	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
127	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
128	C₂×C₂²⋊C₈	C₂²×C₅	1	2	yes	—
129	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
130	C₂×#97	C₂²×C₅	1	2	yes	—
131	C₂×#97	C₂²×C₅	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
133	#400	C₄×C₅	1	2	yes	—
134	#397	C₄×C₅	1	2	yes	—
135	#394	C₄×C₅	1	2	yes	—
136	#393	C₄×C₅	1	2	yes	—
137	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
138	#376	C₂²×C₅	1	1	yes	characteristic
139	#336	C₂²×C₅	1	2	yes	—
140	#336	C₂²×C₅	1	4	yes	—
141	#336	C₂²×C₅	1	2	yes	—
142	C₂³×C₅	—	2	2	no	—
143	C₂³×C₅	—	2	4	no	—
144	C₂³×C₅	—	2	2	no	—
145	C₂³×C₅	—	2	4	no	—
146	C₂³×C₅	#131	1	4	yes	—
147	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
148	C₅×Q₈	—	4	4	no	—
149	C₅×D₄	—	2	8	no	—
150	C₂×C₄×C₅	—	2	2	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	4	no	—
156	C₂×C₄×C₅	—	2	4	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	—	2	4	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	4	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	4	2	no	—
163	C₂×C₄×C₅	—	4	2	no	—
164	C₂×C₄×C₅	—	4	4	no	—
165	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
170	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
171	C₅×C₈	—	2	2	no	—
172	C₅×C₈	—	2	2	no	—
173	C₅×C₈	—	4	2	no	—
174	C₅×C₈	—	4	2	no	—
175	C₅×C₈	—	4	2	no	—
176	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
177	C₂²×D₄	C₂×C₄×C₅	1	1	yes	characteristic
178	C₂³×C₄	C₂³×C₅	1	2	yes	—
179	C₂²×C₈	—	2	4	no	—
180	C₂²×C₈	—	4	4	no	—
181	C₂²×C₈	C₂³×C₅	1	2	yes	—
182	C₂²×C₈	C₂³×C₅	1	2	yes	—
183	C₂²×C₈	C₂³×C₅	1	2	yes	—
184	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
186	C₄×Q₈	—	2	2	no	—
187	C₄×Q₈	—	2	2	no	—
188	C₄×D₄	—	2	2	no	—
189	C₄×D₄	—	2	2	no	—
190	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
193	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	2	yes	—
194	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
195	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	4	yes	—
196	C₂×C₄²	C₂³×C₅	1	2	yes	—
197	C₂×C₄²	C₂³×C₅	1	2	yes	—
198	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	2	2	no	—
201	C₄⋊C₈	—	2	2	no	—
202	C₄⋊C₈	—	4	4	no	—
203	C₂²⋊C₈	—	2	4	no	—
204	C₂²⋊C₈	—	4	4	no	—
205	#97	—	2	4	no	—
206	#97	—	2	2	no	—
207	C₄×C₈	C₅×D₄	1	4	yes	—
208	#95	C₂×C₄×C₅	1	2	yes	—
209	#95	C₂×C₄×C₅	1	2	yes	—
210	#95	C₂×C₄×C₅	1	4	yes	—
211	#95	C₂×C₄×C₅	1	4	yes	—
212	C₂²×C₅	—	2	4	no	—
213	C₂²×C₅	—	2	8	no	—
214	C₂²×C₅	—	4	2	no	—
215	C₂²×C₅	—	4	2	no	—
216	C₂²×C₅	—	4	4	no	—
217	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
218	C₂²×C₅	#520	1	2	yes	—
219	C₂²×C₅	C₂×#131	1	2	yes	—
220	C₂²×C₅	#568	1	2	yes	—
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	—	2	2	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	—	4	2	no	—
226	C₄×C₅	—	4	4	no	—
227	C₄×C₅	#535	1	1	yes	characteristic
228	C₄×C₅	#536	1	1	yes	characteristic
229	C₄×C₅	#542	1	2	yes	—
230	C₂⁴	C₂²×C₄×C₅	1	2	yes	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	—	4	2	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	2	no	—
237	C₂²×C₄	—	2	4	no	—
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
240	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
241	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
243	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	4	8	no	—
256	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
257	C₂×C₈	C₂×C₅×D₄	1	2	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₂²⋊C₄	—	2	8	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	—	4	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	C₂×C₅×D₄	1	2	yes	—
269	C₄²	C₅×#41	1	2	yes	—
270	C₂×C₅	—	4	4	no	—
271	C₂×C₅	#2326	1	1	yes	characteristic
272	C₂×C₅	#2929	1	1	yes	characteristic
273	C₂×C₅	#2930	1	1	yes	characteristic
274	C₂×C₅	#2992	1	4	yes	—
275	C₂³	—	2	2	no	—
276	C₂³	—	2	4	no	—
277	C₂³	—	2	2	no	—
278	C₂³	—	2	4	no	—
279	C₂³	C₅×#131	1	4	yes	—
280	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
281	Q₈	—	4	4	no	—
282	D₄	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
299	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
300	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
301	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
302	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
303	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	2	no	—
308	C₈	—	4	2	no	—
309	C₅	#22213	1	1	yes	characteristic, direct product factor, Sylow
310	C₂²	—	2	8	no	—
311	C₂²	—	2	4	no	—
312	C₂²	—	4	4	no	—
313	C₂²	—	4	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
316	C₂²	C₅×#520	1	2	yes	—
317	C₂²	C₂×C₅×#131	1	2	yes	—
318	C₂²	C₅×#568	1	2	yes	—
319	C₄	—	2	2	no	—
320	C₄	—	2	2	no	—
321	C₄	—	4	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	2	no	—
324	C₄	—	4	2	no	—
325	C₄	C₅×#535	1	1	yes	characteristic
326	C₄	C₅×#536	1	1	yes	characteristic
327	C₄	C₅×#542	1	2	yes	—
328	C₂	—	4	4	no	—
329	C₂	C₅×#2326	1	1	yes	characteristic
330	C₂	C₅×#2929	1	1	yes	characteristic
331	C₂	C₅×#2930	1	1	yes	characteristic
332	C₂	C₅×#2992	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#446	C₂	semidirect
4. C₅×#1985	C₂	semidirect
6. C₅×#1924	C₂	semidirect
7. C₅×#1850	C₂	semidirect
8. C₅×#1843	C₂	semidirect
9. C₅×#1746	C₂	semidirect
33. #22213	C₅	direct (N × H)
70. C₂×#446	C₂×C₅	semidirect
73. #1985	C₂×C₅	semidirect
75. #1924	C₂×C₅	semidirect
76. #1850	C₂×C₅	semidirect
77. #1843	C₂×C₅	semidirect
78. #1746	C₂×C₅	semidirect
309. C₅	#22213	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₅×#446	2
C₂×C₅×C₈×D₄	2
C₅×#2398	2
C₅×#1985	2
C₅×#1955	2
C₅×#1924	2
C₅×#1850	2
C₅×#1843	2
C₅×#1746	2
#22213	5

Subgroups — C5×#22213 #11944933

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22213 #11944933