Subgroups — C₅×#22316 #11945036

Order 1280.

630 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×#2399	C₂	1	1	yes	maximal, characteristic
2	C₅×#1985	C₂	1	2	yes	maximal
3	C₅×#1984	C₂	1	2	yes	maximal
4	C₅×#1939	C₂	1	2	yes	maximal
5	C₅×#1917	C₂	1	2	yes	maximal
6	C₅×#1916	C₂	1	2	yes	maximal
7	C₅×#1832	C₂	1	1	yes	maximal, characteristic
8	C₅×#1771	C₂	1	2	yes	maximal
9	C₅×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₄×C₅×Q₈	C₂²	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	2	yes	—
14	C₂×C₅×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₅×#97	C₂²	1	2	yes	—
16	C₂×C₅×#97	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
19	C₅×#398	C₄	1	2	yes	—
20	C₅×#398	C₄	1	2	yes	—
21	C₅×#395	C₄	1	2	yes	—
22	C₅×#395	C₄	1	2	yes	—
23	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
24	C₄×C₅×C₄⋊C₄	C₂²	1	2	yes	—
25	C₅×#376	C₂²	1	1	yes	characteristic
26	C₅×#336	C₂²	1	2	yes	—
27	C₅×#336	C₂²	1	2	yes	—
28	C₅×#336	C₂²	1	1	yes	characteristic
29	C₅×#336	C₂²	1	1	yes	characteristic
30	C₅×#336	C₂²	1	2	yes	—
31	C₅×#336	C₂²	1	2	yes	—
32	C₅×#336	C₂²	1	2	yes	—
33	C₅×#336	C₂²	1	2	yes	—
34	C₅×#336	C₂²	1	2	yes	—
35	#22316	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
36	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₅×C₈	C₂³	1	2	yes	—
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	C₄×C₅×Q₈	—	4	2	no	—
43	C₄×C₅×Q₈	—	4	2	no	—
44	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
45	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
46	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
47	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
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	C₅×C₄⋊C₈	—	2	4	no	—
58	C₅×C₄⋊C₈	—	4	2	no	—
59	C₅×C₄⋊C₈	—	4	2	no	—
60	C₅×#97	—	4	2	no	—
61	C₅×#97	Q₈	1	4	yes	complemented
62	C₄×C₅×C₈	—	2	2	no	—
63	C₄×C₅×C₈	—	4	2	no	—
64	C₅×#95	C₂×C₄	1	2	yes	—
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	C₅×#95	C₂×C₄	1	2	yes	—
69	C₅×#95	C₂×C₄	1	2	yes	—
70	#2399	C₂×C₅	1	1	yes	characteristic
71	#1985	C₂×C₅	1	2	yes	—
72	#1984	C₂×C₅	1	2	yes	—
73	#1939	C₂×C₅	1	2	yes	—
74	#1917	C₂×C₅	1	2	yes	—
75	#1916	C₂×C₅	1	2	yes	—
76	#1832	C₂×C₅	1	1	yes	characteristic
77	#1771	C₂×C₅	1	2	yes	—
78	#1746	C₂×C₅	1	1	yes	characteristic
79	C₂×C₅×Q₈	—	2	4	no	—
80	C₂×C₅×Q₈	—	4	1	no	—
81	C₂×C₅×Q₈	—	4	1	no	—
82	C₂×C₅×Q₈	—	4	2	no	—
83	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
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	2	yes	—
91	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
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	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	2	no	—
101	C₂×C₅×C₈	—	2	2	no	—
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
104	C₅×C₄⋊C₄	—	2	4	no	—
105	C₅×C₄⋊C₄	—	2	4	no	—
106	C₅×C₄⋊C₄	—	4	2	no	—
107	C₅×C₄⋊C₄	—	4	2	no	—
108	C₅×C₄⋊C₄	—	4	2	no	—
109	C₅×C₄⋊C₄	—	4	1	no	—
110	C₅×C₄⋊C₄	—	4	1	no	—
111	C₅×C₄⋊C₄	—	4	2	no	—
112	C₄²×C₅	—	2	2	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	—	2	2	no	—
118	C₄²×C₅	C₂×Q₈	1	2	yes	—
119	C₄²×C₅	#41	1	2	yes	—
120	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
121	C₂×C₄×Q₈	C₂²×C₅	1	1	yes	characteristic
122	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
123	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
124	C₂×C₄⋊C₈	C₂²×C₅	1	2	yes	—
125	C₂×#97	C₂²×C₅	1	2	yes	—
126	C₂×#97	C₂²×C₅	1	1	yes	characteristic
127	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
128	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
129	#398	C₄×C₅	1	2	yes	—
130	#398	C₄×C₅	1	2	yes	—
131	#395	C₄×C₅	1	2	yes	—
132	#395	C₄×C₅	1	2	yes	—
133	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
134	C₄×C₄⋊C₄	C₂²×C₅	1	2	yes	—
135	#376	C₂²×C₅	1	1	yes	characteristic
136	#336	C₂²×C₅	1	1	yes	characteristic
137	#336	C₂²×C₅	1	1	yes	characteristic
138	#336	C₂²×C₅	1	2	yes	—
139	#336	C₂²×C₅	1	2	yes	—
140	#336	C₂²×C₅	1	2	yes	—
141	#336	C₂²×C₅	1	2	yes	—
142	#336	C₂²×C₅	1	2	yes	—
143	#336	C₂²×C₅	1	2	yes	—
144	#336	C₂²×C₅	1	2	yes	—
145	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
146	C₅×Q₈	—	4	4	no	—
147	C₅×Q₈	—	4	4	no	—
148	C₂×C₄×C₅	—	2	2	no	—
149	C₂×C₄×C₅	—	2	2	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	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	2	2	no	—
168	C₂×C₄×C₅	C₄×Q₈	1	4	yes	—
169	C₂×C₄×C₅	#131	1	4	yes	—
170	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
172	C₂×C₄×C₅	H(5,𝔽₂)	1	2	yes	—
173	C₂×C₄×C₅	2₋¹⁺⁴	1	2	yes	—
174	C₅×C₈	—	2	4	no	—
175	C₅×C₈	—	4	2	no	—
176	C₅×C₈	—	4	2	no	—
177	C₅×C₈	—	4	2	no	—
178	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
179	C₂²×Q₈	C₂×C₄×C₅	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₅	1	2	yes	—
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	C₄×Q₈	—	4	2	no	—
185	C₄×Q₈	—	4	2	no	—
186	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
188	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
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	1	yes	characteristic
192	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
193	C₂×C₄⋊C₄	C₂×C₄×C₅	1	2	yes	—
194	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₅	1	2	yes	—
196	C₂×C₄²	C₂³×C₅	1	2	yes	—
197	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
198	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
199	C₄⋊C₈	—	2	4	no	—
200	C₄⋊C₈	—	4	2	no	—
201	C₄⋊C₈	—	4	2	no	—
202	#97	—	4	2	no	—
203	#97	C₅×Q₈	1	4	yes	complemented
204	C₄×C₈	—	2	2	no	—
205	C₄×C₈	—	4	2	no	—
206	#95	C₂×C₄×C₅	1	2	yes	—
207	#95	C₂×C₄×C₅	1	2	yes	—
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	#95	C₂×C₄×C₅	1	2	yes	—
212	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
213	C₂²×C₅	#518	1	1	yes	characteristic
214	C₂²×C₅	#519	1	1	yes	characteristic
215	C₂²×C₅	C₂×#131	1	2	yes	—
216	C₂²×C₅	#568	1	2	yes	—
217	C₄×C₅	—	2	4	no	—
218	C₄×C₅	—	2	4	no	—
219	C₄×C₅	—	4	2	no	—
220	C₄×C₅	—	4	1	no	—
221	C₄×C₅	—	4	2	no	—
222	C₄×C₅	—	4	1	no	—
223	C₄×C₅	—	4	2	no	—
224	C₄×C₅	—	4	2	no	—
225	C₄×C₅	#542	1	2	yes	—
226	C₄×C₅	#544	1	2	yes	—
227	C₂×Q₈	—	2	4	no	—
228	C₂×Q₈	—	4	1	no	—
229	C₂×Q₈	—	4	1	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
232	C₂²×C₄	C₂²×C₄×C₅	1	2	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	2	yes	—
238	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
239	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
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	C₂×C₅×Q₈	1	4	yes	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	4	2	no	—
255	C₄⋊C₄	—	4	1	no	—
256	C₄⋊C₄	—	4	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	2	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	C₂×C₅×Q₈	1	2	yes	—
267	C₄²	C₅×#41	1	2	yes	—
268	C₂×C₅	#2331	1	1	yes	characteristic
269	C₂×C₅	#2962	1	2	yes	—
270	C₂×C₅	#2977	1	1	yes	characteristic
271	C₂×C₅	#2978	1	1	yes	characteristic
272	C₂×C₅	#2985	1	2	yes	—
273	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
274	Q₈	—	4	4	no	—
275	Q₈	—	4	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	2	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	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	C₄×C₅×Q₈	1	4	yes	—
297	C₂×C₄	C₅×#131	1	4	yes	—
298	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
299	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
300	C₂×C₄	C₅×H(5,𝔽₂)	1	2	yes	—
301	C₂×C₄	C₅×2₋¹⁺⁴	1	2	yes	—
302	C₈	—	2	4	no	—
303	C₈	—	4	2	no	—
304	C₈	—	4	2	no	—
305	C₈	—	4	2	no	—
306	C₅	#22316	1	1	yes	characteristic, direct product factor, Sylow
307	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
308	C₂²	C₅×#518	1	1	yes	characteristic
309	C₂²	C₅×#519	1	1	yes	characteristic
310	C₂²	C₂×C₅×#131	1	2	yes	—
311	C₂²	C₅×#568	1	2	yes	—
312	C₄	—	2	4	no	—
313	C₄	—	2	4	no	—
314	C₄	—	4	2	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	2	no	—
320	C₄	C₅×#542	1	2	yes	—
321	C₄	C₅×#544	1	2	yes	—
322	C₂	C₅×#2331	1	1	yes	characteristic
323	C₂	C₅×#2962	1	2	yes	—
324	C₂	C₅×#2977	1	1	yes	characteristic
325	C₂	C₅×#2978	1	1	yes	characteristic
326	C₂	C₅×#2985	1	2	yes	—

	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
35. #22316	C₅	direct (N × H)
61. C₅×#97	Q₈	semidirect
203. #97	C₅×Q₈	semidirect
306. C₅	#22316	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×#2399	2
C₅×#1985	2
C₅×#1984	2
C₅×#1939	2
C₅×#1917	2
C₅×#1916	2
C₅×#1832	2
C₅×#1771	2
C₅×#1746	2
#22316	5

Subgroups — C5×#22316 #11945036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#22316 #11945036