Subgroups — C₅×#22022 #11944742

Order 1280.

706 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄²×C₅×Q₈	C₂	1	1	yes	maximal, characteristic
2	C₅×#1984	C₂	1	2	yes	maximal
3	C₅×#1939	C₂	1	2	yes	maximal
4	C₅×#1917	C₂	1	4	yes	maximal
5	C₅×#1839	C₂	1	2	yes	maximal
6	C₅×#1831	C₂	1	2	yes	maximal
7	C₅×#1767	C₂	1	1	yes	maximal, characteristic
8	C₅×#1744	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×Q₈	C₄	1	1	yes	characteristic
11	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
14	C₂×C₅×#97	C₂²	1	2	yes	—
15	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
16	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
17	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
18	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
19	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
20	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
21	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₄³×C₅	C₂²	1	1	yes	characteristic
23	C₄³×C₅	C₂²	1	2	yes	—
24	C₅×#336	C₂²	1	2	yes	—
25	C₅×#336	C₂²	1	1	yes	characteristic
26	C₅×#336	C₂²	1	4	yes	—
27	C₅×#336	C₂²	1	1	yes	characteristic
28	C₅×#336	C₂²	1	4	yes	—
29	C₅×#336	C₂²	1	4	yes	—
30	C₅×#336	C₂²	1	2	yes	—
31	#22022	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₅×C₈	C₂³	1	2	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	4	yes	—
36	C₄×C₅×Q₈	—	2	4	no	—
37	C₄×C₅×Q₈	—	2	4	no	—
38	C₄×C₅×Q₈	—	2	4	no	—
39	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
42	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
53	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅	C₂³	1	2	yes	—
56	C₂×C₄²×C₅	C₂³	1	2	yes	—
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₅×C₄⋊C₈	—	2	4	no	—
59	C₅×C₄⋊C₈	—	4	4	no	—
60	C₅×#97	—	4	2	no	—
61	C₄×C₅×C₈	—	2	4	no	—
62	C₅×#95	C₂×C₄	1	2	yes	—
63	C₅×#95	C₂×C₄	1	2	yes	—
64	C₅×#95	C₂×C₄	1	2	yes	—
65	C₄²×Q₈	C₂×C₅	1	1	yes	characteristic
66	#1984	C₂×C₅	1	2	yes	—
67	#1939	C₂×C₅	1	2	yes	—
68	#1917	C₂×C₅	1	4	yes	—
69	#1839	C₂×C₅	1	2	yes	—
70	#1831	C₂×C₅	1	2	yes	—
71	#1767	C₂×C₅	1	1	yes	characteristic
72	#1744	C₂×C₅	1	1	yes	characteristic
73	C₂×C₅×Q₈	—	2	2	no	—
74	C₂×C₅×Q₈	—	2	2	no	—
75	C₂×C₅×Q₈	M₁₆	1	4	yes	—
76	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
87	C₂×C₅×C₈	—	2	2	no	—
88	C₂×C₅×C₈	—	2	2	no	—
89	C₂×C₅×C₈	—	2	4	no	—
90	C₂×C₅×C₈	—	2	4	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	—	2	2	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₅×C₄⋊C₄	—	2	4	no	—
97	C₅×C₄⋊C₄	—	2	2	no	—
98	C₅×C₄⋊C₄	—	2	4	no	—
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
102	C₄²×C₅	—	2	4	no	—
103	C₄²×C₅	—	2	4	no	—
104	C₄²×C₅	—	2	2	no	—
105	C₄²×C₅	—	2	2	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	2	4	no	—
109	C₄²×C₅	—	2	4	no	—
110	C₄²×C₅	—	2	4	no	—
111	C₄²×C₅	—	2	1	no	—
112	C₄²×C₅	—	2	2	no	—
113	C₄²×C₅	—	2	2	no	—
114	C₄²×C₅	—	2	2	no	—
115	C₄²×C₅	—	2	1	no	—
116	C₄²×C₅	#41	1	4	yes	—
117	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
118	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
119	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
120	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
121	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
122	C₂×#97	C₂²×C₅	1	2	yes	—
123	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
124	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
125	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
126	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
127	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
128	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
129	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
130	C₄³	C₂²×C₅	1	2	yes	—
131	C₄³	C₂²×C₅	1	1	yes	characteristic
132	#336	C₂²×C₅	1	1	yes	characteristic
133	#336	C₂²×C₅	1	2	yes	—
134	#336	C₂²×C₅	1	4	yes	—
135	#336	C₂²×C₅	1	1	yes	characteristic
136	#336	C₂²×C₅	1	4	yes	—
137	#336	C₂²×C₅	1	4	yes	—
138	#336	C₂²×C₅	1	2	yes	—
139	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
140	C₅×Q₈	—	2	8	no	—
141	C₂×C₄×C₅	—	2	2	no	—
142	C₂×C₄×C₅	—	2	1	no	—
143	C₂×C₄×C₅	—	2	1	no	—
144	C₂×C₄×C₅	—	2	2	no	—
145	C₂×C₄×C₅	—	2	1	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	2	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	2	no	—
151	C₂×C₄×C₅	—	2	2	no	—
152	C₂×C₄×C₅	—	2	1	no	—
153	C₂×C₄×C₅	—	2	1	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	1	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	1	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	1	no	—
161	C₂×C₄×C₅	—	2	1	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	#117	1	2	yes	—
164	C₂×C₄×C₅	#117	1	2	yes	—
165	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
166	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
167	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
168	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
169	C₂×C₄×C₅	#131	1	2	yes	—
170	C₂×C₄×C₅	#131	1	2	yes	—
171	C₂×C₄×C₅	C₂×#41	1	2	yes	—
172	C₂×C₄×C₅	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₄×C₅	2₋¹⁺⁴	1	1	yes	characteristic
174	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	—	4	2	no	—
177	C₅×C₈	—	4	4	no	—
178	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
179	C₂²×C₈	C₂³×C₅	1	2	yes	—
180	C₂²×C₈	C₂³×C₅	1	2	yes	—
181	C₂²×C₈	C₂³×C₅	1	4	yes	—
182	C₄×Q₈	—	2	4	no	—
183	C₄×Q₈	—	2	4	no	—
184	C₄×Q₈	—	2	4	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
187	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
189	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
190	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
192	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
194	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
195	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
196	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
197	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
198	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
199	C₂×C₄²	C₂³×C₅	1	2	yes	—
200	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
201	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
202	C₂×C₄²	C₂³×C₅	1	2	yes	—
203	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
204	C₄⋊C₈	—	2	4	no	—
205	C₄⋊C₈	—	4	4	no	—
206	#97	—	4	2	no	—
207	C₄×C₈	—	2	4	no	—
208	#95	C₂×C₄×C₅	1	2	yes	—
209	#95	C₂×C₄×C₅	1	2	yes	—
210	#95	C₂×C₄×C₅	1	2	yes	—
211	C₂²×C₅	C₂×#117	1	1	yes	characteristic
212	C₂²×C₅	#519	1	1	yes	characteristic
213	C₂²×C₅	#520	1	1	yes	characteristic
214	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
215	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
216	C₂²×C₅	C₂×#131	1	1	yes	characteristic
217	C₂²×C₅	C₂×#131	1	1	yes	characteristic
218	C₄×C₅	—	2	4	no	—
219	C₄×C₅	—	2	2	no	—
220	C₄×C₅	—	2	2	no	—
221	C₄×C₅	—	2	2	no	—
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	—	4	1	no	—
225	C₄×C₅	—	4	1	no	—
226	C₄×C₅	—	4	2	no	—
227	C₄×C₅	#432	1	4	yes	—
228	C₄×C₅	#541	1	2	yes	—
229	C₄×C₅	#542	1	2	yes	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	2	no	—
232	C₂×Q₈	C₅×M₁₆	1	4	yes	—
233	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
234	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
235	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
236	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
237	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
238	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
240	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
242	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
243	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	4	no	—
247	C₂×C₈	—	2	4	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	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	4	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	C₅×M₁₆	1	4	yes	—
259	C₄²	—	2	4	no	—
260	C₄²	—	2	4	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	2	4	no	—
268	C₄²	—	2	1	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	2	no	—
271	C₄²	—	2	2	no	—
272	C₄²	—	2	1	no	—
273	C₄²	C₅×#41	1	4	yes	—
274	C₂×C₅	#2320	1	1	yes	characteristic
275	C₂×C₅	C₂×#432	1	1	yes	characteristic
276	C₂×C₅	#2922	1	1	yes	characteristic
277	C₂×C₅	#2984	1	1	yes	characteristic
278	C₂×C₅	#2984	1	1	yes	characteristic
279	C₂×C₅	#2990	1	1	yes	characteristic
280	C₂×C₅	#2991	1	1	yes	characteristic
281	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
282	Q₈	—	2	8	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	C₅×#117	1	2	yes	—
306	C₂×C₄	C₅×#117	1	2	yes	—
307	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
308	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
309	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
310	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
311	C₂×C₄	C₅×#131	1	2	yes	—
312	C₂×C₄	C₅×#131	1	2	yes	—
313	C₂×C₄	C₂×C₅×#41	1	2	yes	—
314	C₂×C₄	C₅×H(5,𝔽₂)	1	1	yes	characteristic
315	C₂×C₄	C₅×2₋¹⁺⁴	1	1	yes	characteristic
316	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
317	C₈	—	4	2	no	—
318	C₈	—	4	2	no	—
319	C₈	—	4	4	no	—
320	C₅	#22022	1	1	yes	characteristic, direct product factor, Sylow
321	C₂²	C₂×C₅×#117	1	1	yes	characteristic
322	C₂²	C₅×#519	1	1	yes	characteristic
323	C₂²	C₅×#520	1	1	yes	characteristic
324	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
325	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
326	C₂²	C₂×C₅×#131	1	1	yes	characteristic
327	C₂²	C₂×C₅×#131	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	4	no	—
331	C₄	—	2	4	no	—
332	C₄	—	2	2	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	2	no	—
336	C₄	—	4	2	no	—
337	C₄	C₅×#432	1	4	yes	—
338	C₄	C₅×#541	1	2	yes	—
339	C₄	C₅×#542	1	2	yes	—
340	C₂	C₅×#2320	1	1	yes	characteristic
341	C₂	C₂×C₅×#432	1	1	yes	characteristic
342	C₂	C₅×#2922	1	1	yes	characteristic
343	C₂	C₅×#2984	1	1	yes	characteristic
344	C₂	C₅×#2984	1	1	yes	characteristic
345	C₂	C₅×#2990	1	1	yes	characteristic
346	C₂	C₅×#2991	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₅	C₂⁴
commutator	C₂³	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle, prime radical	C₂³×C₅	C₂³×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
31. #22022	C₅	direct (N × H)
320. C₅	#22022	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄²×C₅×Q₈	2
C₅×#1984	2
C₅×#1939	2
C₅×#1917	2
C₅×#1839	2
C₅×#1831	2
C₅×#1767	2
C₅×#1744	2
#22022	5

Subgroups — C5×#22022 #11944742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22022 #11944742