Subgroups — C₅×#22807 #11945527

Order 1280.

934 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#2354	C₂	1	2	yes	maximal, complemented
3	C₅×#1970	C₂	1	2	yes	maximal
4	C₅×#1925	C₂	1	4	yes	maximal, complemented
5	C₅×#1845	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#1802	C₂	1	2	yes	maximal, complemented
7	C₄×C₅×#107	C₂	1	2	yes	maximal, complemented
8	C₅×C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
11	C₂×C₅×#117	—	2	2	no	—
12	C₄×C₅×D₈	C₄	1	8	yes	complemented
13	C₂×C₅×#107	C₂²	1	2	yes	—
14	C₂×C₅×#107	C₂²	1	2	yes	—
15	C₂×C₅×#107	C₂²	1	1	yes	characteristic
16	C₂×C₅×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×#102	C₂²	1	4	yes	—
18	C₂×C₅×#102	C₂²	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	C₅×#390	C₂²	1	4	yes	—
22	C₅×#388	—	2	4	no	—
23	C₅×#384	C₂²	1	2	yes	—
24	C₅×#380	—	2	2	no	—
25	C₄×C₅×C₄⋊C₄	C₂²	1	4	yes	—
26	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
27	C₅×#340	C₂²	1	4	yes	—
28	C₅×#336	C₂²	1	2	yes	—
29	#22807	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂²×C₅×D₄	C₂³	1	2	yes	—
31	C₂³×C₄×C₅	—	2	2	no	—
32	C₂×C₅×D₈	—	2	2	no	—
33	C₂×C₅×D₈	—	2	2	no	—
34	C₂×C₅×D₈	C₂×C₄	1	4	yes	—
35	C₂²×C₅×C₈	C₂³	1	2	yes	—
36	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
37	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
38	C₄×C₅×D₄	—	2	4	no	—
39	C₄×C₅×D₄	C₂×C₄	1	8	yes	—
40	C₅×#117	—	4	8	no	—
41	C₂×C₅×C₄⋊C₄	—	2	2	no	—
42	C₂×C₅×C₄⋊C₄	—	2	2	no	—
43	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
44	C₂×C₅×C₄⋊C₄	D₄	1	1	yes	characteristic
45	C₂×C₅×C₄⋊C₄	C₂³	1	4	yes	—
46	C₂×C₅×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
49	C₂×C₅×C₂²⋊C₄	—	2	4	no	—
50	C₂×C₄²×C₅	—	2	4	no	—
51	C₂×C₄²×C₅	C₂³	1	2	yes	—
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
53	C₅×#107	—	2	4	no	—
54	C₅×#107	—	4	2	no	—
55	C₅×#107	C₂×C₄	1	4	yes	—
56	C₅×C₄⋊C₈	—	2	2	no	—
57	C₅×#102	—	4	4	no	—
58	C₅×#102	C₂×C₄	1	8	yes	—
59	C₄×C₅×C₈	—	2	4	no	—
60	C₄×C₅×C₈	C₂×C₄	1	4	yes	—
61	C₅×#95	—	2	4	no	—
62	C₅×#95	D₄	1	2	yes	—
63	C₂×C₄×D₈	C₂×C₅	1	1	yes	characteristic
64	#2354	C₂×C₅	1	2	yes	complemented
65	#1970	C₂×C₅	1	2	yes	—
66	#1925	C₂×C₅	1	4	yes	complemented
67	#1845	C₂×C₅	1	1	yes	characteristic, complemented
68	#1802	C₂×C₅	1	2	yes	complemented
69	C₄×#107	C₂×C₅	1	2	yes	complemented
70	C₈×C₄⋊C₄	C₂×C₅	1	1	yes	characteristic, complemented
71	C₂⁴×C₅	—	2	2	no	—
72	C₂×C₅×D₄	—	2	2	no	—
73	C₂×C₅×D₄	—	2	2	no	—
74	C₂×C₅×D₄	—	2	2	no	—
75	C₂×C₅×D₄	—	2	2	no	—
76	C₂×C₅×D₄	—	2	2	no	—
77	C₂×C₅×D₄	C₂²×C₄	1	4	yes	—
78	C₂²×C₄×C₅	—	2	2	no	—
79	C₂²×C₄×C₅	—	2	8	no	—
80	C₂²×C₄×C₅	—	2	4	no	—
81	C₂²×C₄×C₅	—	4	4	no	—
82	C₂²×C₄×C₅	—	4	4	no	—
83	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
88	C₅×D₈	—	2	8	no	—
89	C₂×C₅×C₈	—	2	1	no	—
90	C₂×C₅×C₈	—	2	1	no	—
91	C₂×C₅×C₈	—	2	2	no	—
92	C₂×C₅×C₈	—	2	2	no	—
93	C₂×C₅×C₈	—	2	2	no	—
94	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₅×C₈	#41	1	2	yes	—
97	C₂×C₅×C₈	#41	1	2	yes	—
98	C₅×C₄⋊C₄	—	2	2	no	—
99	C₅×C₄⋊C₄	—	2	2	no	—
100	C₅×C₄⋊C₄	—	2	4	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	4	4	no	—
103	C₅×C₄⋊C₄	—	8	2	no	—
104	C₅×C₄⋊C₄	D₈	1	4	yes	complemented
105	C₅×C₄⋊C₄	C₂²×C₄	1	4	yes	—
106	C₅×C₂²⋊C₄	—	2	8	no	—
107	C₅×C₂²⋊C₄	—	4	8	no	—
108	C₅×C₂²⋊C₄	—	8	4	no	—
109	C₄²×C₅	—	2	2	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	4	8	no	—
112	C₄²×C₅	C₂²×C₄	1	2	yes	—
113	C₄²×C₅	#41	1	2	yes	—
114	C₂²×D₈	C₂²×C₅	1	1	yes	characteristic
115	C₂×C₄×D₄	C₂²×C₅	1	2	yes	—
116	C₂×#117	—	2	2	no	—
117	C₄×D₈	C₄×C₅	1	8	yes	complemented
118	C₂×#107	C₂²×C₅	1	2	yes	—
119	C₂×#107	C₂²×C₅	1	2	yes	—
120	C₂×#107	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄⋊C₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×#102	C₂²×C₅	1	4	yes	—
123	C₂×#102	C₂²×C₅	1	2	yes	—
124	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
125	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
126	#390	C₂²×C₅	1	4	yes	—
127	#388	—	2	4	no	—
128	#384	C₂²×C₅	1	2	yes	—
129	#380	—	2	2	no	—
130	C₄×C₄⋊C₄	C₂²×C₅	1	4	yes	—
131	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
132	#340	C₂²×C₅	1	4	yes	—
133	#336	C₂²×C₅	1	2	yes	—
134	C₂³×C₅	—	2	4	no	—
135	C₂³×C₅	—	4	2	no	—
136	C₂³×C₅	—	4	2	no	—
137	C₂³×C₅	—	4	2	no	—
138	C₂³×C₅	—	4	2	no	—
139	C₂³×C₅	—	4	2	no	—
140	C₂³×C₅	—	4	2	no	—
141	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
142	C₅×D₄	—	2	4	no	—
143	C₅×D₄	—	2	4	no	—
144	C₅×D₄	—	4	2	no	—
145	C₅×D₄	—	4	2	no	—
146	C₂×C₄×C₅	—	2	2	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	4	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	—	4	2	no	—
155	C₂×C₄×C₅	—	4	2	no	—
156	C₂×C₄×C₅	—	4	16	no	—
157	C₂×C₄×C₅	—	4	4	no	—
158	C₂×C₄×C₅	—	4	4	no	—
159	C₂×C₄×C₅	—	4	4	no	—
160	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
161	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
162	C₂×C₄×C₅	C₂×D₈	1	4	yes	—
163	C₂×C₄×C₅	C₂×D₈	1	2	yes	—
164	C₂×C₄×C₅	#135	1	2	yes	—
165	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
166	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
167	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
170	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
171	C₅×C₈	—	2	2	no	—
172	C₅×C₈	—	4	2	no	—
173	C₅×C₈	#117	1	4	yes	complemented
174	C₂²×D₄	C₂³×C₅	1	2	yes	—
175	C₂³×C₄	—	2	2	no	—
176	C₂×D₈	—	2	2	no	—
177	C₂×D₈	—	2	2	no	—
178	C₂×D₈	C₂×C₄×C₅	1	4	yes	—
179	C₂²×C₈	C₂³×C₅	1	2	yes	—
180	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₅	1	1	yes	characteristic
182	C₄×D₄	—	2	4	no	—
183	C₄×D₄	C₂×C₄×C₅	1	8	yes	—
184	#117	—	4	8	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
188	C₂×C₄⋊C₄	C₅×D₄	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂³×C₅	1	4	yes	—
190	C₂×C₄⋊C₄	C₂³×C₅	1	2	yes	—
191	C₂×C₂²⋊C₄	—	2	2	no	—
192	C₂×C₂²⋊C₄	—	2	4	no	—
193	C₂×C₂²⋊C₄	—	2	4	no	—
194	C₂×C₄²	—	2	4	no	—
195	C₂×C₄²	C₂³×C₅	1	2	yes	—
196	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
197	#107	—	2	4	no	—
198	#107	—	4	2	no	—
199	#107	C₂×C₄×C₅	1	4	yes	—
200	C₄⋊C₈	—	2	2	no	—
201	#102	—	4	4	no	—
202	#102	C₂×C₄×C₅	1	8	yes	—
203	C₄×C₈	—	2	4	no	—
204	C₄×C₈	C₂×C₄×C₅	1	4	yes	—
205	#95	—	2	4	no	—
206	#95	C₅×D₄	1	2	yes	—
207	C₂²×C₅	—	4	4	no	—
208	C₂²×C₅	—	4	4	no	—
209	C₂²×C₅	—	4	4	no	—
210	C₂²×C₅	—	8	2	no	—
211	C₂²×C₅	—	8	2	no	—
212	C₂²×C₅	—	8	2	no	—
213	C₂²×C₅	—	8	2	no	—
214	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
215	C₂²×C₅	#547	1	1	yes	characteristic
216	C₂²×C₅	#550	1	1	yes	characteristic
217	C₂²×C₅	C₂²×D₈	1	1	yes	characteristic
218	C₂²×C₅	C₂×#135	1	1	yes	characteristic
219	C₂²×C₅	#576	1	1	yes	characteristic
220	C₂²×C₅	#578	1	1	yes	characteristic
221	C₄×C₅	—	2	4	no	—
222	C₄×C₅	—	2	2	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	—	4	4	no	—
225	C₄×C₅	—	8	4	no	—
226	C₄×C₅	C₄×D₈	1	4	yes	—
227	C₄×C₅	C₂×#117	1	1	yes	characteristic
228	C₄×C₅	#520	1	1	yes	characteristic
229	C₄×C₅	#555	1	1	yes	characteristic
230	C₄×C₅	#556	1	1	yes	characteristic
231	C₂⁴	—	2	2	no	—
232	C₂×D₄	—	2	2	no	—
233	C₂×D₄	—	2	2	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	C₂²×C₄×C₅	1	4	yes	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	8	no	—
240	C₂²×C₄	—	2	4	no	—
241	C₂²×C₄	—	4	4	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
244	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
245	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
246	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
247	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
248	D₈	—	2	8	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
255	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
256	C₂×C₈	C₅×#41	1	2	yes	—
257	C₂×C₈	C₅×#41	1	2	yes	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	4	4	no	—
263	C₄⋊C₄	—	8	2	no	—
264	C₄⋊C₄	C₅×D₈	1	4	yes	complemented
265	C₄⋊C₄	C₂²×C₄×C₅	1	4	yes	—
266	C₂²⋊C₄	—	2	8	no	—
267	C₂²⋊C₄	—	4	8	no	—
268	C₂²⋊C₄	—	8	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	4	8	no	—
272	C₄²	C₂²×C₄×C₅	1	2	yes	—
273	C₄²	C₅×#41	1	2	yes	—
274	C₂×C₅	—	8	4	no	—
275	C₂×C₅	#2354	1	1	yes	characteristic
276	C₂×C₅	C₂×C₄×D₈	1	1	yes	characteristic
277	C₂×C₅	#2951	1	1	yes	characteristic
278	C₂×C₅	#3334	1	1	yes	characteristic
279	C₂×C₅	#3341	1	1	yes	characteristic
280	C₂×C₅	#3391	1	1	yes	characteristic
281	C₂×C₅	#3393	1	1	yes	characteristic
282	C₂³	—	2	4	no	—
283	C₂³	—	4	2	no	—
284	C₂³	—	4	2	no	—
285	C₂³	—	4	2	no	—
286	C₂³	—	4	2	no	—
287	C₂³	—	4	2	no	—
288	C₂³	—	4	2	no	—
289	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
290	D₄	—	2	4	no	—
291	D₄	—	2	4	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	4	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	16	no	—
305	C₂×C₄	—	4	4	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
309	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
310	C₂×C₄	C₂×C₅×D₈	1	4	yes	—
311	C₂×C₄	C₂×C₅×D₈	1	2	yes	—
312	C₂×C₄	C₅×#135	1	2	yes	—
313	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
314	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
315	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
316	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
317	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
318	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
319	C₈	—	2	2	no	—
320	C₈	—	4	2	no	—
321	C₈	C₅×#117	1	4	yes	complemented
322	C₅	#22807	1	1	yes	characteristic, direct product factor, Sylow
323	C₂²	—	4	4	no	—
324	C₂²	—	4	4	no	—
325	C₂²	—	4	4	no	—
326	C₂²	—	8	2	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
331	C₂²	C₅×#547	1	1	yes	characteristic
332	C₂²	C₅×#550	1	1	yes	characteristic
333	C₂²	C₂²×C₅×D₈	1	1	yes	characteristic
334	C₂²	C₂×C₅×#135	1	1	yes	characteristic
335	C₂²	C₅×#576	1	1	yes	characteristic
336	C₂²	C₅×#578	1	1	yes	characteristic
337	C₄	—	2	4	no	—
338	C₄	—	2	2	no	—
339	C₄	—	4	2	no	—
340	C₄	—	4	4	no	—
341	C₄	—	8	4	no	—
342	C₄	C₄×C₅×D₈	1	4	yes	—
343	C₄	C₂×C₅×#117	1	1	yes	characteristic
344	C₄	C₅×#520	1	1	yes	characteristic
345	C₄	C₅×#555	1	1	yes	characteristic
346	C₄	C₅×#556	1	1	yes	characteristic
347	C₂	—	8	4	no	—
348	C₂	C₅×#2354	1	1	yes	characteristic
349	C₂	C₂×C₄×C₅×D₈	1	1	yes	characteristic
350	C₂	C₅×#2951	1	1	yes	characteristic
351	C₂	C₅×#3334	1	1	yes	characteristic
352	C₂	C₅×#3341	1	1	yes	characteristic
353	C₂	C₅×#3391	1	1	yes	characteristic
354	C₂	C₅×#3393	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
prime radical	C₂²×C₅×D₈	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₅ ◁ C₄×C₅×C₄⋊C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₅×#2354	C₂	semidirect
4. C₅×#1925	C₂	semidirect
5. C₅×#1845	C₂	semidirect
6. C₅×#1802	C₂	semidirect
7. C₄×C₅×#107	C₂	semidirect
8. C₅×C₈×C₄⋊C₄	C₂	semidirect
12. C₄×C₅×D₈	C₄	semidirect
29. #22807	C₅	direct (N × H)
64. #2354	C₂×C₅	semidirect
66. #1925	C₂×C₅	semidirect
67. #1845	C₂×C₅	semidirect
68. #1802	C₂×C₅	semidirect
69. C₄×#107	C₂×C₅	semidirect
70. C₈×C₄⋊C₄	C₂×C₅	semidirect
104. C₅×C₄⋊C₄	D₈	semidirect
117. C₄×D₈	C₄×C₅	semidirect
173. C₅×C₈	#117	semidirect
264. C₄⋊C₄	C₅×D₈	semidirect
321. C₈	C₅×#117	semidirect
322. C₅	#22807	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×C₄×C₅×D₈	2
C₅×#2354	2
C₅×#1970	2
C₅×#1925	2
C₅×#1845	2
C₅×#1802	2
C₄×C₅×#107	2
C₅×C₈×C₄⋊C₄	2
#22807	5

Subgroups — C5×#22807 #11945527

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22807 #11945527