Subgroups — C₂×C₅×#3147 #11983352

Order 1280.

3366 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#135	C₂	1	4	yes	maximal, complemented
2	C₂×C₅×#532	C₂	1	2	yes	maximal, complemented
3	C₅×#3147	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₅×#495	C₂	1	4	yes	maximal, complemented
5	C₂×C₅×#494	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₅×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅×#492	C₂	1	2	yes	maximal, complemented
8	C₂²×C₄×C₅×C₈	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×#41	C₂²	1	4	yes	—
10	C₂×C₅×#135	—	2	32	no	—
11	C₂×C₅×#135	C₂²	1	32	yes	complemented
12	C₂²×C₅×Q₁₆	C₂²	1	4	yes	—
13	C₂²×C₅×SD₁₆	C₂²	1	8	yes	—
14	C₂²×C₅×D₈	C₂²	1	4	yes	—
15	C₂³×C₅×C₈	C₂²	1	2	yes	—
16	C₅×#532	—	2	8	no	—
17	C₅×#532	C₂²	1	16	yes	complemented
18	C₂×C₅×C₄⋊Q₈	C₂²	1	2	yes	—
19	C₂×C₅×#127	C₂²	1	2	yes	—
20	C₂×C₅×#124	C₂²	1	4	yes	—
21	C₂×C₅×#121	—	2	4	no	—
22	C₂×C₄×C₅×D₄	—	2	4	no	—
23	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
24	C₅×#495	C₂²	1	32	yes	complemented
25	C₅×#494	C₂²	1	8	yes	complemented
26	C₅×#493	C₂²	1	8	yes	complemented
27	C₅×#492	C₂²	1	16	yes	complemented
28	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₅×C₈	C₂²	1	2	yes	—
31	C₂×C₄×C₅×C₈	C₂²	1	8	yes	complemented
32	C₂×#3147	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂×C₅×#41	—	2	8	no	—
34	C₂×C₅×#41	—	2	32	no	—
35	C₂×C₅×#41	C₂³	1	16	yes	—
36	C₂²×C₅×Q₈	C₂³	1	4	yes	—
37	C₂²×C₅×D₄	—	2	4	no	—
38	C₂²×C₅×D₄	—	2	4	no	—
39	C₂²×C₅×D₄	C₂³	1	4	yes	—
40	C₂³×C₄×C₅	—	2	4	no	—
41	C₂³×C₄×C₅	D₄	1	2	yes	—
42	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
43	C₅×#135	—	2	128	no	—
44	C₂×C₅×Q₁₆	—	2	16	no	—
45	C₂×C₅×Q₁₆	C₂³	1	16	yes	—
46	C₂×C₅×SD₁₆	—	2	32	no	—
47	C₂×C₅×SD₁₆	C₂³	1	32	yes	—
48	C₂×C₅×D₈	—	2	16	no	—
49	C₂×C₅×D₈	C₂³	1	16	yes	—
50	C₂²×C₅×C₈	D₄	1	16	yes	complemented
51	C₂²×C₅×C₈	C₂³	1	2	yes	—
52	C₂²×C₅×C₈	C₂³	1	2	yes	—
53	C₂²×C₅×C₈	C₂³	1	8	yes	—
54	C₅×C₄⋊Q₈	—	2	4	no	—
55	C₅×C₄⋊Q₈	C₂³	1	8	yes	—
56	C₅×#127	—	2	4	no	—
57	C₅×#127	C₂³	1	8	yes	—
58	C₅×#124	—	2	8	no	—
59	C₅×#124	C₂³	1	16	yes	—
60	C₅×#121	—	2	32	no	—
61	C₄×C₅×D₄	—	2	32	no	—
62	C₂×C₅×C₄⋊C₄	—	2	4	no	—
63	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
64	C₂×C₄²×C₅	D₄	1	4	yes	—
65	C₂×C₄²×C₅	D₄	1	2	yes	—
66	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
67	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₅	C₂³	1	4	yes	—
69	C₄×C₅×C₈	C₂³	1	4	yes	—
70	C₄×C₅×C₈	C₂³	1	4	yes	—
71	C₄×C₅×C₈	C₂³	1	8	yes	—
72	C₂²×#135	C₂×C₅	1	4	yes	complemented
73	C₂×#532	C₂×C₅	1	2	yes	complemented
74	#3147	C₂×C₅	1	16	yes	direct product factor
75	C₂×#495	C₂×C₅	1	4	yes	complemented
76	C₂×#494	C₂×C₅	1	1	yes	characteristic, complemented
77	C₂×#493	C₂×C₅	1	1	yes	characteristic, complemented
78	C₂×#492	C₂×C₅	1	2	yes	complemented
79	C₂²×C₄×C₈	C₂×C₅	1	1	yes	characteristic, complemented
80	C₂⁴×C₅	—	2	4	no	—
81	C₂⁴×C₅	C₂×D₄	1	1	yes	characteristic
82	C₅×#41	—	2	64	no	—
83	C₅×#41	—	4	32	no	—
84	C₂×C₅×Q₈	—	2	4	no	—
85	C₂×C₅×Q₈	—	2	16	no	—
86	C₂×C₅×Q₈	C₂⁴	1	8	yes	—
87	C₂×C₅×D₄	—	2	16	no	—
88	C₂×C₅×D₄	—	2	16	no	—
89	C₂×C₅×D₄	—	2	16	no	—
90	C₂×C₅×D₄	—	2	4	no	—
91	C₂×C₅×D₄	—	4	16	no	—
92	C₂×C₅×D₄	—	4	16	no	—
93	C₂×C₅×D₄	C₂⁴	1	8	yes	—
94	C₂²×C₄×C₅	—	2	8	no	—
95	C₂²×C₄×C₅	—	2	4	no	—
96	C₂²×C₄×C₅	—	2	16	no	—
97	C₂²×C₄×C₅	—	4	16	no	—
98	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
99	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
100	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₅	C₂×D₄	1	8	yes	—
102	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
105	C₂²×C₄×C₅	C₂⁴	1	2	yes	—
106	C₅×Q₁₆	—	2	32	no	—
107	C₅×SD₁₆	—	2	64	no	—
108	C₅×D₈	—	2	32	no	—
109	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
110	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
111	C₂×C₅×C₈	C₂×D₄	1	32	yes	complemented
112	C₂×C₅×C₈	C₂⁴	1	4	yes	—
113	C₂×C₅×C₈	C₂⁴	1	4	yes	—
114	C₅×C₄⋊C₄	—	2	16	no	—
115	C₅×C₂²⋊C₄	—	2	32	no	—
116	C₄²×C₅	C₂×D₄	1	8	yes	—
117	C₄²×C₅	C₂×D₄	1	2	yes	—
118	C₄²×C₅	C₂×D₄	1	2	yes	—
119	C₄²×C₅	C₂⁴	1	2	yes	—
120	C₄²×C₅	C₂⁴	1	2	yes	—
121	C₂²×#41	C₂²×C₅	1	4	yes	—
122	C₂×#135	—	2	32	no	—
123	C₂×#135	C₂²×C₅	1	32	yes	complemented
124	C₂²×Q₁₆	C₂²×C₅	1	4	yes	—
125	C₂²×SD₁₆	C₂²×C₅	1	8	yes	—
126	C₂²×D₈	C₂²×C₅	1	4	yes	—
127	C₂³×C₈	C₂²×C₅	1	2	yes	—
128	#532	—	2	8	no	—
129	#532	C₂²×C₅	1	16	yes	complemented
130	C₂×C₄⋊Q₈	C₂²×C₅	1	2	yes	—
131	C₂×#127	C₂²×C₅	1	2	yes	—
132	C₂×#124	C₂²×C₅	1	4	yes	—
133	C₂×#121	—	2	4	no	—
134	C₂×C₄×D₄	—	2	4	no	—
135	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
136	#495	C₂²×C₅	1	32	yes	complemented
137	#494	C₂²×C₅	1	8	yes	complemented
138	#493	C₂²×C₅	1	8	yes	complemented
139	#492	C₂²×C₅	1	16	yes	complemented
140	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
141	C₂×C₄×C₈	C₂²×C₅	1	1	yes	characteristic
142	C₂×C₄×C₈	C₂²×C₅	1	2	yes	—
143	C₂×C₄×C₈	C₂²×C₅	1	8	yes	complemented
144	C₂³×C₅	—	2	4	no	—
145	C₂³×C₅	—	2	8	no	—
146	C₂³×C₅	—	4	8	no	—
147	C₂³×C₅	—	4	16	no	—
148	C₂³×C₅	#127	1	4	yes	—
149	C₂³×C₅	C₂²×D₄	1	2	yes	—
150	C₂³×C₅	C₂²×D₄	1	1	yes	characteristic
151	C₅×Q₈	—	2	16	no	—
152	C₅×Q₈	—	4	8	no	—
153	C₅×D₄	—	2	16	no	—
154	C₅×D₄	—	4	32	no	—
155	C₅×D₄	—	4	8	no	—
156	C₅×D₄	—	4	32	no	—
157	C₂×C₄×C₅	—	2	4	no	—
158	C₂×C₄×C₅	—	2	16	no	—
159	C₂×C₄×C₅	—	2	8	no	—
160	C₂×C₄×C₅	—	4	8	no	—
161	C₂×C₄×C₅	—	4	32	no	—
162	C₂×C₄×C₅	#127	1	4	yes	—
163	C₂×C₄×C₅	#127	1	8	yes	—
164	C₂×C₄×C₅	#135	1	8	yes	complemented
165	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
166	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
167	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
168	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
169	C₂×C₄×C₅	C₂²×D₄	1	8	yes	—
170	C₂×C₄×C₅	C₂²×D₄	1	4	yes	—
171	C₂×C₄×C₅	C₂⁵	1	1	yes	characteristic
172	C₅×C₈	C₂²×D₄	1	8	yes	—
173	C₅×C₈	C₂²×D₄	1	8	yes	—
174	C₂×#41	—	2	8	no	—
175	C₂×#41	—	2	32	no	—
176	C₂×#41	C₂³×C₅	1	16	yes	—
177	C₂²×Q₈	C₂³×C₅	1	4	yes	—
178	C₂²×D₄	—	2	4	no	—
179	C₂²×D₄	—	2	4	no	—
180	C₂²×D₄	C₂³×C₅	1	4	yes	—
181	C₂³×C₄	—	2	4	no	—
182	C₂³×C₄	C₅×D₄	1	2	yes	—
183	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
184	#135	—	2	128	no	—
185	C₂×Q₁₆	—	2	16	no	—
186	C₂×Q₁₆	C₂³×C₅	1	16	yes	—
187	C₂×SD₁₆	—	2	32	no	—
188	C₂×SD₁₆	C₂³×C₅	1	32	yes	—
189	C₂×D₈	—	2	16	no	—
190	C₂×D₈	C₂³×C₅	1	16	yes	—
191	C₂²×C₈	C₅×D₄	1	16	yes	complemented
192	C₂²×C₈	C₂³×C₅	1	2	yes	—
193	C₂²×C₈	C₂³×C₅	1	2	yes	—
194	C₂²×C₈	C₂³×C₅	1	8	yes	—
195	C₄⋊Q₈	—	2	4	no	—
196	C₄⋊Q₈	C₂³×C₅	1	8	yes	—
197	#127	—	2	4	no	—
198	#127	C₂³×C₅	1	8	yes	—
199	#124	—	2	8	no	—
200	#124	C₂³×C₅	1	16	yes	—
201	#121	—	2	32	no	—
202	C₄×D₄	—	2	32	no	—
203	C₂×C₄⋊C₄	—	2	4	no	—
204	C₂×C₂²⋊C₄	—	2	8	no	—
205	C₂×C₄²	C₅×D₄	1	2	yes	—
206	C₂×C₄²	C₅×D₄	1	4	yes	—
207	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
208	C₂×C₄²	C₂³×C₅	1	1	yes	characteristic
209	C₂×C₄²	C₂³×C₅	1	4	yes	—
210	C₄×C₈	C₂³×C₅	1	4	yes	—
211	C₄×C₈	C₂³×C₅	1	4	yes	—
212	C₄×C₈	C₂³×C₅	1	8	yes	—
213	C₂²×C₅	—	2	8	no	—
214	C₂²×C₅	—	2	4	no	—
215	C₂²×C₅	—	4	8	no	—
216	C₂²×C₅	—	4	16	no	—
217	C₂²×C₅	—	8	16	no	—
218	C₂²×C₅	C₂×#127	1	4	yes	—
219	C₂²×C₅	C₂×#127	1	2	yes	—
220	C₂²×C₅	C₂×#135	1	4	yes	—
221	C₂²×C₅	C₂³×D₄	1	1	yes	characteristic
222	C₄×C₅	—	2	4	no	—
223	C₄×C₅	—	4	8	no	—
224	C₄×C₅	C₂×#127	1	4	yes	—
225	C₄×C₅	C₂×#127	1	2	yes	—
226	C₄×C₅	C₂×#135	1	8	yes	complemented
227	C₄×C₅	C₂³×D₄	1	2	yes	—
228	C₂⁴	—	2	4	no	—
229	C₂⁴	C₂×C₅×D₄	1	1	yes	characteristic
230	#41	—	2	64	no	—
231	#41	—	4	32	no	—
232	C₂×Q₈	—	2	4	no	—
233	C₂×Q₈	—	2	16	no	—
234	C₂×Q₈	C₂⁴×C₅	1	8	yes	—
235	C₂×D₄	—	2	16	no	—
236	C₂×D₄	—	2	16	no	—
237	C₂×D₄	—	2	16	no	—
238	C₂×D₄	—	2	4	no	—
239	C₂×D₄	—	4	16	no	—
240	C₂×D₄	—	4	16	no	—
241	C₂×D₄	C₂⁴×C₅	1	8	yes	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	16	no	—
245	C₂²×C₄	—	4	16	no	—
246	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
247	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
248	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
249	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
250	C₂²×C₄	C₂×C₅×D₄	1	8	yes	—
251	C₂²×C₄	C₂×C₅×D₄	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
253	C₂²×C₄	C₂⁴×C₅	1	2	yes	—
254	Q₁₆	—	2	32	no	—
255	SD₁₆	—	2	64	no	—
256	D₈	—	2	32	no	—
257	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
258	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
259	C₂×C₈	C₂×C₅×D₄	1	32	yes	complemented
260	C₂×C₈	C₂⁴×C₅	1	4	yes	—
261	C₂×C₈	C₂⁴×C₅	1	4	yes	—
262	C₄⋊C₄	—	2	16	no	—
263	C₂²⋊C₄	—	2	32	no	—
264	C₄²	C₂×C₅×D₄	1	8	yes	—
265	C₄²	C₂×C₅×D₄	1	2	yes	—
266	C₄²	C₂×C₅×D₄	1	2	yes	—
267	C₄²	C₂⁴×C₅	1	2	yes	—
268	C₄²	C₂⁴×C₅	1	2	yes	—
269	C₂×C₅	—	2	4	no	—
270	C₂×C₅	—	8	8	no	—
271	C₂×C₅	#3147	1	4	yes	direct product factor
272	C₂×C₅	C₂²×#127	1	1	yes	characteristic
273	C₂×C₅	C₂²×#135	1	2	yes	—
274	C₂³	—	2	4	no	—
275	C₂³	—	2	8	no	—
276	C₂³	—	4	8	no	—
277	C₂³	—	4	16	no	—
278	C₂³	C₅×#127	1	4	yes	—
279	C₂³	C₂²×C₅×D₄	1	2	yes	—
280	C₂³	C₂²×C₅×D₄	1	1	yes	characteristic
281	Q₈	—	2	16	no	—
282	Q₈	—	4	8	no	—
283	D₄	—	2	16	no	—
284	D₄	—	4	32	no	—
285	D₄	—	4	8	no	—
286	D₄	—	4	32	no	—
287	C₂×C₄	—	2	4	no	—
288	C₂×C₄	—	2	16	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	—	4	32	no	—
292	C₂×C₄	C₅×#127	1	4	yes	—
293	C₂×C₄	C₅×#127	1	8	yes	—
294	C₂×C₄	C₅×#135	1	8	yes	complemented
295	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
296	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
297	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
298	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
299	C₂×C₄	C₂²×C₅×D₄	1	8	yes	—
300	C₂×C₄	C₂²×C₅×D₄	1	4	yes	—
301	C₂×C₄	C₂⁵×C₅	1	1	yes	characteristic
302	C₈	C₂²×C₅×D₄	1	8	yes	—
303	C₈	C₂²×C₅×D₄	1	8	yes	—
304	C₅	C₂×#3147	1	1	yes	characteristic, direct product factor, Sylow
305	C₂²	—	2	8	no	—
306	C₂²	—	2	4	no	—
307	C₂²	—	4	8	no	—
308	C₂²	—	4	16	no	—
309	C₂²	—	8	16	no	—
310	C₂²	C₂×C₅×#127	1	4	yes	—
311	C₂²	C₂×C₅×#127	1	2	yes	—
312	C₂²	C₂×C₅×#135	1	4	yes	—
313	C₂²	C₂³×C₅×D₄	1	1	yes	characteristic
314	C₄	—	2	4	no	—
315	C₄	—	4	8	no	—
316	C₄	C₂×C₅×#127	1	4	yes	—
317	C₄	C₂×C₅×#127	1	2	yes	—
318	C₄	C₂×C₅×#135	1	8	yes	complemented
319	C₄	C₂³×C₅×D₄	1	2	yes	—
320	C₂	—	2	4	no	—
321	C₂	—	8	8	no	—
322	C₂	C₅×#3147	1	4	yes	direct product factor
323	C₂	C₂²×C₅×#127	1	1	yes	characteristic
324	C₂	C₂²×C₅×#135	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₅
socle	C₂³×C₅	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂²×C₄²×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₅×#135	C₂	semidirect
2. C₂×C₅×#532	C₂	semidirect
3. C₅×#3147	C₂	direct (N × H)
4. C₂×C₅×#495	C₂	semidirect
5. C₂×C₅×#494	C₂	semidirect
6. C₂×C₅×#493	C₂	semidirect
7. C₂×C₅×#492	C₂	semidirect
8. C₂²×C₄×C₅×C₈	C₂	semidirect
11. C₂×C₅×#135	C₂²	semidirect
17. C₅×#532	C₂²	semidirect
24. C₅×#495	C₂²	semidirect
25. C₅×#494	C₂²	semidirect
26. C₅×#493	C₂²	semidirect
27. C₅×#492	C₂²	semidirect
31. C₂×C₄×C₅×C₈	C₂²	semidirect
32. C₂×#3147	C₅	direct (N × H)
50. C₂²×C₅×C₈	D₄	semidirect
72. C₂²×#135	C₂×C₅	semidirect
73. C₂×#532	C₂×C₅	semidirect
74. #3147	C₂×C₅	direct (N × H)
75. C₂×#495	C₂×C₅	semidirect
76. C₂×#494	C₂×C₅	semidirect
77. C₂×#493	C₂×C₅	semidirect
78. C₂×#492	C₂×C₅	semidirect
79. C₂²×C₄×C₈	C₂×C₅	semidirect
111. C₂×C₅×C₈	C₂×D₄	semidirect
123. C₂×#135	C₂²×C₅	semidirect
129. #532	C₂²×C₅	semidirect
136. #495	C₂²×C₅	semidirect
137. #494	C₂²×C₅	semidirect
138. #493	C₂²×C₅	semidirect
139. #492	C₂²×C₅	semidirect
143. C₂×C₄×C₈	C₂²×C₅	semidirect
164. C₂×C₄×C₅	#135	semidirect
191. C₂²×C₈	C₅×D₄	semidirect
226. C₄×C₅	C₂×#135	semidirect
259. C₂×C₈	C₂×C₅×D₄	semidirect
271. C₂×C₅	#3147	direct (N × H)
294. C₂×C₄	C₅×#135	semidirect
304. C₅	C₂×#3147	direct (N × H)
318. C₄	C₂×C₅×#135	semidirect
322. C₂	C₅×#3147	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₅×#135	2
C₂×C₅×#532	2
C₅×#3147	2
C₂×C₅×#495	2
C₂×C₅×#494	2
C₂×C₅×#493	2
C₂×C₅×#492	2
C₂²×C₄×C₅×C₈	2
C₂×#3147	5

Subgroups — C2×C5×#3147 #11983352

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#3147 #11983352