Subgroups — C₅×#21178 #11943898

Order 1280.

866 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#430	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
3	C₅×#2905	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅×#424	C₂	1	1	yes	maximal, characteristic
5	C₅×#2873	C₂	1	1	yes	maximal, characteristic
6	C₅×#1950	C₂	1	8	yes	maximal, complemented
7	C₅×#1738	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅×#341	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
12	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
13	C₂²×C₅×M₁₆	C₂²	1	1	yes	characteristic
14	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₅×#117	C₂²	1	1	yes	characteristic
16	C₅×#430	—	2	4	no	—
17	C₂×C₅×#108	—	2	4	no	—
18	C₂×C₅×#108	C₂²	1	4	yes	—
19	C₂×C₅×#108	C₂²	1	4	yes	—
20	C₅×#424	—	2	2	no	—
21	C₅×#424	—	2	2	no	—
22	C₅×#424	C₄	1	4	yes	—
23	C₅×#424	C₄	1	4	yes	—
24	C₅×#423	C₂²	1	4	yes	—
25	C₂×C₅×C₄⋊C₈	C₂²	1	4	yes	—
26	C₅×#405	—	2	2	no	—
27	C₅×#405	—	2	2	no	—
28	C₄×C₅×M₁₆	C₂²	1	4	yes	—
29	C₅×#341	C₂²	1	4	yes	—
30	C₅×#341	C₂²	1	4	yes	—
31	#21178	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
32	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
33	C₂×C₅×M₁₆	—	2	1	no	—
34	C₂×C₅×M₁₆	—	2	1	no	—
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂×C₅×M₁₆	—	2	2	no	—
37	C₂×C₅×M₁₆	—	2	2	no	—
38	C₂×C₅×M₁₆	—	2	2	no	—
39	C₂×C₅×M₁₆	—	2	1	no	—
40	C₂×C₅×M₁₆	—	2	1	no	—
41	C₂×C₅×M₁₆	—	2	2	no	—
42	C₂×C₅×M₁₆	—	2	2	no	—
43	C₂×C₅×M₁₆	—	2	2	no	—
44	C₂×C₅×M₁₆	—	2	2	no	—
45	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
47	C₂×C₅×M₁₆	D₄	1	4	yes	—
48	C₂×C₅×M₁₆	Q₈	1	4	yes	—
49	C₂×C₅×M₁₆	C₂³	1	2	yes	—
50	C₂×C₅×M₁₆	C₂³	1	2	yes	—
51	C₂×C₅×M₁₆	C₂³	1	2	yes	—
52	C₂×C₅×M₁₆	C₂³	1	2	yes	—
53	C₂²×C₅×C₈	—	2	1	no	—
54	C₂²×C₅×C₈	—	2	1	no	—
55	C₂²×C₅×C₈	—	2	2	no	—
56	C₂²×C₅×C₈	—	2	2	no	—
57	C₂²×C₅×C₈	—	2	1	no	—
58	C₂²×C₅×C₈	—	2	1	no	—
59	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
60	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
61	C₂²×C₅×C₈	C₂³	1	2	yes	—
62	C₂²×C₅×C₈	C₂³	1	2	yes	—
63	C₅×#117	C₂×C₄	1	2	yes	—
64	C₅×#117	C₂×C₄	1	2	yes	—
65	C₅×#117	D₄	1	2	yes	—
66	C₅×#117	D₄	1	2	yes	—
67	C₂×C₅×C₄⋊C₄	Q₈	1	2	yes	—
68	C₂×C₅×C₂²⋊C₄	Q₈	1	2	yes	—
69	C₂×C₄²×C₅	C₂³	1	2	yes	—
70	C₅×#108	—	2	8	no	—
71	C₅×#108	—	2	8	no	—
72	C₅×C₄⋊C₈	—	2	8	no	—
73	C₅×C₄⋊C₈	C₂×C₄	1	16	yes	—
74	C₅×#97	—	2	4	no	—
75	C₄×C₅×C₈	—	2	4	no	—
76	C₂×#430	C₂×C₅	1	1	yes	characteristic
77	C₂²×#108	C₂×C₅	1	1	yes	characteristic
78	#2905	C₂×C₅	1	1	yes	characteristic
79	C₂×#424	C₂×C₅	1	1	yes	characteristic
80	#2873	C₂×C₅	1	1	yes	characteristic
81	#1950	C₂×C₅	1	8	yes	complemented
82	#1738	C₂×C₅	1	1	yes	characteristic
83	C₂×#341	C₂×C₅	1	1	yes	characteristic
84	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₅	C₂×Q₈	1	4	yes	—
91	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₅	#41	1	2	yes	—
94	C₂²×C₄×C₅	#41	1	2	yes	—
95	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
96	C₅×M₁₆	—	2	4	no	—
97	C₅×M₁₆	—	2	4	no	—
98	C₅×M₁₆	—	2	4	no	—
99	C₅×M₁₆	—	2	4	no	—
100	C₅×M₁₆	—	2	2	no	—
101	C₅×M₁₆	—	2	2	no	—
102	C₅×M₁₆	—	2	2	no	—
103	C₅×M₁₆	—	2	2	no	—
104	C₅×M₁₆	—	2	2	no	—
105	C₅×M₁₆	—	2	2	no	—
106	C₅×M₁₆	—	4	2	no	—
107	C₅×M₁₆	—	4	2	no	—
108	C₅×M₁₆	C₂×D₄	1	4	yes	—
109	C₅×M₁₆	C₂×Q₈	1	4	yes	—
110	C₂×C₅×C₈	—	2	4	no	—
111	C₂×C₅×C₈	—	2	4	no	—
112	C₂×C₅×C₈	—	2	4	no	—
113	C₂×C₅×C₈	—	2	4	no	—
114	C₂×C₅×C₈	—	2	2	no	—
115	C₂×C₅×C₈	—	2	2	no	—
116	C₂×C₅×C₈	—	2	2	no	—
117	C₂×C₅×C₈	—	2	2	no	—
118	C₂×C₅×C₈	—	2	2	no	—
119	C₂×C₅×C₈	—	2	2	no	—
120	C₂×C₅×C₈	—	2	2	no	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	4	2	no	—
123	C₂×C₅×C₈	—	4	2	no	—
124	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
125	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
126	C₅×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
127	C₅×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
128	C₅×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
129	C₅×C₂²⋊C₄	C₄⋊C₄	1	4	yes	—
130	C₄²×C₅	C₂²×C₄	1	4	yes	—
131	C₄²×C₅	C₂×D₄	1	4	yes	—
132	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
133	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
134	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
135	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
136	C₂²×M₁₆	C₂²×C₅	1	1	yes	characteristic
137	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
138	C₂×#117	C₂²×C₅	1	1	yes	characteristic
139	#430	—	2	4	no	—
140	C₂×#108	—	2	4	no	—
141	C₂×#108	C₂²×C₅	1	4	yes	—
142	C₂×#108	C₂²×C₅	1	4	yes	—
143	#424	—	2	2	no	—
144	#424	—	2	2	no	—
145	#424	C₄×C₅	1	4	yes	—
146	#424	C₄×C₅	1	4	yes	—
147	#423	C₂²×C₅	1	4	yes	—
148	C₂×C₄⋊C₈	C₂²×C₅	1	4	yes	—
149	#405	—	2	2	no	—
150	#405	—	2	2	no	—
151	C₄×M₁₆	C₂²×C₅	1	4	yes	—
152	#341	C₂²×C₅	1	4	yes	—
153	#341	C₂²×C₅	1	4	yes	—
154	C₂³×C₅	—	4	2	no	—
155	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
156	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
157	C₂³×C₅	#125	1	2	yes	—
158	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
159	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₅	—	4	4	no	—
161	C₂×C₄×C₅	C₂×C₄⋊C₄	1	8	yes	—
162	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
163	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
165	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
166	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
167	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
168	C₂×C₄×C₅	#121	1	2	yes	—
169	C₂×C₄×C₅	#121	1	2	yes	—
170	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
171	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
172	C₂×C₄×C₅	#125	1	2	yes	—
173	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
174	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
180	C₅×C₈	—	2	4	no	—
181	C₅×C₈	—	2	4	no	—
182	C₅×C₈	—	2	4	no	—
183	C₅×C₈	—	2	4	no	—
184	C₅×C₈	—	4	2	no	—
185	C₅×C₈	—	4	2	no	—
186	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
187	C₂×M₁₆	—	2	1	no	—
188	C₂×M₁₆	—	2	1	no	—
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	2	no	—
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	1	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	—	2	2	no	—
198	C₂×M₁₆	—	2	2	no	—
199	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
200	C₂×M₁₆	C₂×C₄×C₅	1	2	yes	—
201	C₂×M₁₆	C₅×D₄	1	4	yes	—
202	C₂×M₁₆	C₅×Q₈	1	4	yes	—
203	C₂×M₁₆	C₂³×C₅	1	2	yes	—
204	C₂×M₁₆	C₂³×C₅	1	2	yes	—
205	C₂×M₁₆	C₂³×C₅	1	2	yes	—
206	C₂×M₁₆	C₂³×C₅	1	2	yes	—
207	C₂²×C₈	—	2	2	no	—
208	C₂²×C₈	—	2	1	no	—
209	C₂²×C₈	—	2	1	no	—
210	C₂²×C₈	—	2	2	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	2	1	no	—
213	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
214	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
215	C₂²×C₈	C₂³×C₅	1	2	yes	—
216	C₂²×C₈	C₂³×C₅	1	2	yes	—
217	#117	C₂×C₄×C₅	1	2	yes	—
218	#117	C₂×C₄×C₅	1	2	yes	—
219	#117	C₅×D₄	1	2	yes	—
220	#117	C₅×D₄	1	2	yes	—
221	C₂×C₄⋊C₄	C₅×Q₈	1	2	yes	—
222	C₂×C₂²⋊C₄	C₅×Q₈	1	2	yes	—
223	C₂×C₄²	C₂³×C₅	1	2	yes	—
224	#108	—	2	8	no	—
225	#108	—	2	8	no	—
226	C₄⋊C₈	—	2	8	no	—
227	C₄⋊C₈	C₂×C₄×C₅	1	16	yes	—
228	#97	—	2	4	no	—
229	C₄×C₈	—	2	4	no	—
230	C₂²×C₅	—	2	2	no	—
231	C₂²×C₅	—	2	2	no	—
232	C₂²×C₅	—	2	2	no	—
233	C₂²×C₅	—	4	2	no	—
234	C₂²×C₅	—	4	2	no	—
235	C₂²×C₅	#389	1	2	yes	—
236	C₂²×C₅	#389	1	2	yes	—
237	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
238	C₂²×C₅	C₂×#125	1	1	yes	characteristic
239	C₂²×C₅	C₂×C₄⋊Q₈	1	1	yes	characteristic
240	C₄×C₅	—	2	8	no	—
241	C₄×C₅	#389	1	2	yes	—
242	C₄×C₅	#389	1	2	yes	—
243	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
244	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
245	C₄×C₅	C₂×#121	1	1	yes	characteristic
246	C₄×C₅	C₂×C₂²⋊Q₈	1	1	yes	characteristic
247	C₂⁴	C₂×C₅×Q₈	1	1	yes	characteristic
248	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
250	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
251	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
252	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
253	C₂²×C₄	C₂×C₅×Q₈	1	4	yes	—
254	C₂²×C₄	C₂×C₅×Q₈	1	1	yes	characteristic
255	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
256	C₂²×C₄	C₅×#41	1	2	yes	—
257	C₂²×C₄	C₅×#41	1	2	yes	—
258	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
259	M₁₆	—	2	4	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	2	4	no	—
262	M₁₆	—	2	4	no	—
263	M₁₆	—	2	2	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	2	2	no	—
266	M₁₆	—	2	2	no	—
267	M₁₆	—	2	2	no	—
268	M₁₆	—	2	2	no	—
269	M₁₆	—	4	2	no	—
270	M₁₆	—	4	2	no	—
271	M₁₆	C₂×C₅×D₄	1	4	yes	—
272	M₁₆	C₂×C₅×Q₈	1	4	yes	—
273	C₂×C₈	—	2	4	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	2	4	no	—
276	C₂×C₈	—	2	4	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	—	2	2	no	—
279	C₂×C₈	—	2	2	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	—	4	2	no	—
287	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
288	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
289	C₄⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
290	C₄⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
291	C₂²⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
292	C₂²⋊C₄	C₅×C₄⋊C₄	1	4	yes	—
293	C₄²	C₂²×C₄×C₅	1	4	yes	—
294	C₄²	C₂×C₅×D₄	1	4	yes	—
295	C₂×C₅	—	2	1	no	—
296	C₂×C₅	—	2	1	no	—
297	C₂×C₅	—	2	2	no	—
298	C₂×C₅	—	2	2	no	—
299	C₂×C₅	C₂×#389	1	1	yes	characteristic
300	C₂×C₅	#2916	1	2	yes	—
301	C₂³	—	4	2	no	—
302	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
303	C₂³	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
304	C₂³	C₅×#125	1	2	yes	—
305	C₂³	C₅×C₄⋊Q₈	1	2	yes	—
306	C₂³	C₂²×C₅×Q₈	1	1	yes	characteristic
307	C₂×C₄	—	4	4	no	—
308	C₂×C₄	C₂×C₅×C₄⋊C₄	1	8	yes	—
309	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
310	C₂×C₄	C₂×C₅×C₄⋊C₄	1	1	yes	characteristic
311	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
312	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
313	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
314	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
315	C₂×C₄	C₅×#121	1	2	yes	—
316	C₂×C₄	C₅×#121	1	2	yes	—
317	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
318	C₂×C₄	C₅×C₂²⋊Q₈	1	2	yes	—
319	C₂×C₄	C₅×#125	1	2	yes	—
320	C₂×C₄	C₅×C₄⋊Q₈	1	2	yes	—
321	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
323	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
325	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
326	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
327	C₈	—	2	4	no	—
328	C₈	—	2	4	no	—
329	C₈	—	2	4	no	—
330	C₈	—	2	4	no	—
331	C₈	—	4	2	no	—
332	C₈	—	4	2	no	—
333	C₅	#21178	1	1	yes	characteristic, direct product factor, Sylow
334	C₂²	—	2	2	no	—
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	2	no	—
338	C₂²	—	4	2	no	—
339	C₂²	C₅×#389	1	2	yes	—
340	C₂²	C₅×#389	1	2	yes	—
341	C₂²	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
342	C₂²	C₂×C₅×#125	1	1	yes	characteristic
343	C₂²	C₂×C₅×C₄⋊Q₈	1	1	yes	characteristic
344	C₄	—	2	8	no	—
345	C₄	C₅×#389	1	2	yes	—
346	C₄	C₅×#389	1	2	yes	—
347	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
348	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
349	C₄	C₂×C₅×#121	1	1	yes	characteristic
350	C₄	C₂×C₅×C₂²⋊Q₈	1	1	yes	characteristic
351	C₂	—	2	2	no	—
352	C₂	—	2	2	no	—
353	C₂	—	2	1	no	—
354	C₂	—	2	1	no	—
355	C₂	C₂×C₅×#389	1	1	yes	characteristic
356	C₂	C₅×#2916	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅ ◁ C₂×C₅×#117 ◁ G

Complements

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

N	H ≅ G/N	type
6. C₅×#1950	C₂	semidirect
31. #21178	C₅	direct (N × H)
81. #1950	C₂×C₅	semidirect
333. C₅	#21178	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×#430	2
C₂²×C₅×#108	2
C₅×#2905	2
C₂×C₅×#424	2
C₅×#2873	2
C₅×#1950	2
C₅×#1738	2
C₂×C₅×#341	2
#21178	5

Subgroups — C5×#21178 #11943898

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅×#21178 #11943898