Subgroups — C₅×#21345 #11944065

Order 1280.

746 conjugacy classes of subgroups.

The subgroup lattice has 352 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×#1767	C₂	1	1	yes	maximal, characteristic
6	C₅×#1767	C₂	1	2	yes	maximal
7	C₅×#1745	C₂	1	2	yes	maximal
8	C₄×C₅×#97	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	4	yes	—
13	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₅×#97	C₂²	1	1	yes	characteristic
15	C₂×C₅×#97	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	2	yes	—
17	C₂×C₅×#97	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
20	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
21	C₄×C₅×C₄⋊C₄	C₄	1	2	yes	—
22	C₄×C₅×C₄⋊C₄	C₄	1	1	yes	characteristic
23	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₅×C₄⋊C₄	C₂²	1	1	yes	characteristic
25	C₄³×C₅	C₂²	1	1	yes	characteristic
26	C₄³×C₅	C₂²	1	2	yes	—
27	C₅×#336	C₂²	1	2	yes	—
28	C₅×#336	C₂²	1	4	yes	—
29	C₅×#336	C₂²	1	4	yes	—
30	C₅×#336	C₂²	1	4	yes	—
31	#21345	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	4	yes	—
34	C₂²×C₅×C₈	C₂³	1	2	yes	—
35	C₂²×C₅×C₈	C₂³	1	2	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	2	yes	—
55	C₂×C₄²×C₅	C₂³	1	2	yes	—
56	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
58	C₅×C₄⋊C₈	—	2	8	no	—
59	C₅×C₄⋊C₈	—	2	4	no	—
60	C₅×#97	—	2	4	no	—
61	C₅×#97	—	2	2	no	—
62	C₅×#97	—	2	4	no	—
63	C₅×#97	Q₈	1	4	yes	complemented
64	C₄×C₅×C₈	—	2	4	no	—
65	C₅×#95	C₂×C₄	1	2	yes	—
66	C₅×#95	C₂×C₄	1	2	yes	—
67	C₅×#95	C₂×C₄	1	2	yes	—
68	C₄²×Q₈	C₂×C₅	1	1	yes	characteristic
69	#1984	C₂×C₅	1	2	yes	—
70	#1939	C₂×C₅	1	2	yes	—
71	#1917	C₂×C₅	1	4	yes	—
72	#1767	C₂×C₅	1	1	yes	characteristic
73	#1767	C₂×C₅	1	2	yes	—
74	#1745	C₂×C₅	1	2	yes	—
75	C₄×#97	C₂×C₅	1	1	yes	characteristic
76	C₂×C₅×Q₈	—	2	2	no	—
77	C₂×C₅×Q₈	—	2	2	no	—
78	C₂×C₅×Q₈	M₁₆	1	4	yes	—
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	1	yes	characteristic
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	C₂²×C₄	1	2	yes	—
87	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
90	C₂×C₅×C₈	—	2	2	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	4	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	—	2	4	no	—
97	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
98	C₂×C₅×C₈	#41	1	4	yes	—
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	—	2	2	no	—
101	C₅×C₄⋊C₄	—	2	4	no	—
102	C₅×C₄⋊C₄	—	2	4	no	—
103	C₅×C₄⋊C₄	—	2	2	no	—
104	C₅×C₄⋊C₄	M₁₆	1	4	yes	—
105	C₄²×C₅	—	2	4	no	—
106	C₄²×C₅	—	2	4	no	—
107	C₄²×C₅	—	2	2	no	—
108	C₄²×C₅	—	2	2	no	—
109	C₄²×C₅	—	2	4	no	—
110	C₄²×C₅	—	2	2	no	—
111	C₄²×C₅	—	2	4	no	—
112	C₄²×C₅	—	4	1	no	—
113	C₄²×C₅	—	4	2	no	—
114	C₄²×C₅	—	4	1	no	—
115	C₄²×C₅	—	4	2	no	—
116	C₄²×C₅	C₂×Q₈	1	2	yes	—
117	C₄²×C₅	#41	1	4	yes	—
118	C₄²×C₅	#41	1	2	yes	—
119	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
120	C₂×C₄×Q₈	C₄×C₅	1	1	yes	characteristic
121	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
123	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
124	C₂×#97	C₂²×C₅	1	2	yes	—
125	C₂×#97	C₂²×C₅	1	1	yes	characteristic
126	C₂×#97	C₂²×C₅	1	2	yes	—
127	C₂×#97	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
129	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
130	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
131	C₄×C₄⋊C₄	C₄×C₅	1	2	yes	—
132	C₄×C₄⋊C₄	C₄×C₅	1	1	yes	characteristic
133	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
134	C₄×C₄⋊C₄	C₂²×C₅	1	1	yes	characteristic
135	C₄³	C₂²×C₅	1	1	yes	characteristic
136	C₄³	C₂²×C₅	1	2	yes	—
137	#336	C₂²×C₅	1	2	yes	—
138	#336	C₂²×C₅	1	4	yes	—
139	#336	C₂²×C₅	1	4	yes	—
140	#336	C₂²×C₅	1	4	yes	—
141	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
142	C₅×Q₈	—	2	8	no	—
143	C₂×C₄×C₅	—	2	2	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	1	no	—
151	C₂×C₄×C₅	—	2	1	no	—
152	C₂×C₄×C₅	—	2	2	no	—
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	1	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	1	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	#117	1	2	yes	—
166	C₂×C₄×C₅	#117	1	2	yes	—
167	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
168	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
169	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
170	C₂×C₄×C₅	C₂×M₁₆	1	4	yes	—
171	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
172	C₂×C₄×C₅	C₂×M₁₆	1	2	yes	—
173	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×#41	1	2	yes	—
176	C₂×C₄×C₅	C₂×#41	1	2	yes	—
177	C₅×C₈	—	2	4	no	—
178	C₅×C₈	—	2	4	no	—
179	C₅×C₈	—	4	4	no	—
180	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
181	C₂²×C₈	C₂³×C₅	1	4	yes	—
182	C₂²×C₈	C₂³×C₅	1	2	yes	—
183	C₂²×C₈	C₂³×C₅	1	2	yes	—
184	C₄×Q₈	—	2	4	no	—
185	C₄×Q₈	—	2	4	no	—
186	C₄×Q₈	—	2	4	no	—
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×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
192	C₂×C₄⋊C₄	C₂×C₄×C₅	1	1	yes	characteristic
193	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
194	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
195	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
196	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
197	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
198	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
199	C₂×C₄²	C₂×C₄×C₅	1	1	yes	characteristic
200	C₂×C₄²	C₂×C₄×C₅	1	2	yes	—
201	C₂×C₄²	C₂³×C₅	1	2	yes	—
202	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
203	C₂×C₄²	C₂³×C₅	1	2	yes	—
204	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
205	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
206	C₄⋊C₈	—	2	8	no	—
207	C₄⋊C₈	—	2	4	no	—
208	#97	—	2	4	no	—
209	#97	—	2	2	no	—
210	#97	—	2	4	no	—
211	#97	C₅×Q₈	1	4	yes	complemented
212	C₄×C₈	—	2	4	no	—
213	#95	C₂×C₄×C₅	1	2	yes	—
214	#95	C₂×C₄×C₅	1	2	yes	—
215	#95	C₂×C₄×C₅	1	2	yes	—
216	C₂²×C₅	C₂×#117	1	1	yes	characteristic
217	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
218	C₂²×C₅	C₄×#41	1	1	yes	characteristic
219	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
220	C₂²×C₅	C₂²×M₁₆	1	1	yes	characteristic
221	C₂²×C₅	#568	1	1	yes	characteristic
222	C₂²×C₅	#568	1	1	yes	characteristic
223	C₄×C₅	—	2	4	no	—
224	C₄×C₅	—	2	2	no	—
225	C₄×C₅	—	2	2	no	—
226	C₄×C₅	—	2	2	no	—
227	C₄×C₅	—	2	4	no	—
228	C₄×C₅	—	4	2	no	—
229	C₄×C₅	—	4	1	no	—
230	C₄×C₅	—	4	1	no	—
231	C₄×C₅	—	4	2	no	—
232	C₄×C₅	#432	1	4	yes	—
233	C₄×C₅	#529	1	2	yes	—
234	C₄×C₅	#533	1	2	yes	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	C₅×M₁₆	1	4	yes	—
238	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
239	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
240	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
241	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
242	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
244	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
245	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
246	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
247	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
248	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	4	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
257	C₂×C₈	C₅×#41	1	4	yes	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	C₅×M₁₆	1	4	yes	—
264	C₄²	—	2	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	2	2	no	—
268	C₄²	—	2	2	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	4	no	—
271	C₄²	—	4	1	no	—
272	C₄²	—	4	2	no	—
273	C₄²	—	4	1	no	—
274	C₄²	—	4	2	no	—
275	C₄²	C₂×C₅×Q₈	1	2	yes	—
276	C₄²	C₅×#41	1	4	yes	—
277	C₄²	C₅×#41	1	2	yes	—
278	C₂×C₅	#2295	1	1	yes	characteristic
279	C₂×C₅	C₂×#432	1	1	yes	characteristic
280	C₂×C₅	#2923	1	1	yes	characteristic
281	C₂×C₅	Q₈×M₁₆	1	1	yes	characteristic
282	C₂×C₅	Q₈×M₁₆	1	1	yes	characteristic
283	C₂×C₅	#2970	1	1	yes	characteristic
284	C₂×C₅	#2971	1	1	yes	characteristic
285	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
286	Q₈	—	2	8	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	2	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	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	C₅×#117	1	2	yes	—
310	C₂×C₄	C₅×#117	1	2	yes	—
311	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
312	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
313	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
314	C₂×C₄	C₂×C₅×M₁₆	1	4	yes	—
315	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
316	C₂×C₄	C₂×C₅×M₁₆	1	2	yes	—
317	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
319	C₂×C₄	C₂×C₅×#41	1	2	yes	—
320	C₂×C₄	C₂×C₅×#41	1	2	yes	—
321	C₈	—	2	4	no	—
322	C₈	—	2	4	no	—
323	C₈	—	4	4	no	—
324	C₅	#21345	1	1	yes	characteristic, direct product factor, Sylow
325	C₂²	C₂×C₅×#117	1	1	yes	characteristic
326	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
327	C₂²	C₄×C₅×#41	1	1	yes	characteristic
328	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
329	C₂²	C₂²×C₅×M₁₆	1	1	yes	characteristic
330	C₂²	C₅×#568	1	1	yes	characteristic
331	C₂²	C₅×#568	1	1	yes	characteristic
332	C₄	—	2	4	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	2	no	—
335	C₄	—	2	2	no	—
336	C₄	—	2	4	no	—
337	C₄	—	4	2	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	2	no	—
341	C₄	C₅×#432	1	4	yes	—
342	C₄	C₅×#529	1	2	yes	—
343	C₄	C₅×#533	1	2	yes	—
344	C₂	C₅×#2295	1	1	yes	characteristic
345	C₂	C₂×C₅×#432	1	1	yes	characteristic
346	C₂	C₅×#2923	1	1	yes	characteristic
347	C₂	C₅×Q₈×M₁₆	1	1	yes	characteristic
348	C₂	C₅×Q₈×M₁₆	1	1	yes	characteristic
349	C₂	C₅×#2970	1	1	yes	characteristic
350	C₂	C₅×#2971	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. #21345	C₅	direct (N × H)
63. C₅×#97	Q₈	semidirect
211. #97	C₅×Q₈	semidirect
324. C₅	#21345	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×#1767	2
C₅×#1745	2
C₄×C₅×#97	2
#21345	5

Subgroups — C5×#21345 #11944065

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21345 #11944065