Subgroups — C₂×C₅×#1773 #11943163

Order 1280.

1062 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×#108	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅×#427	C₂	1	1	yes	maximal, characteristic
3	C₂×C₅×#405	C₂	1	2	yes	maximal
4	C₂²×C₅×#97	C₂	1	1	yes	maximal, characteristic
5	C₅×#1773	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#339	C₂	1	2	yes	maximal
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
9	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₅×#117	C₂²	1	2	yes	—
11	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
12	C₂×C₅×#108	C₄	1	8	yes	complemented
13	C₂×C₅×#108	C₂²	1	4	yes	—
14	C₅×#427	C₄	1	4	yes	complemented
15	C₅×#427	C₂²	1	4	yes	—
16	C₂×C₅×#107	C₄	1	2	yes	complemented
17	C₂×C₅×#106	C₄	1	2	yes	complemented
18	C₅×#405	—	2	4	no	—
19	C₅×#405	C₂²	1	8	yes	—
20	C₂×C₅×#97	—	2	2	no	—
21	C₂×C₅×#97	C₄	1	8	yes	complemented
22	C₂×C₅×#97	C₂²	1	4	yes	—
23	C₂×C₄×C₅×C₈	—	2	2	no	—
24	C₅×#339	C₂²	1	8	yes	—
25	C₂×#1773	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	C₂³×C₄×C₅	—	2	1	no	—
27	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
28	C₂×C₅×M₁₆	—	2	2	no	—
29	C₂×C₅×M₁₆	—	2	4	no	—
30	C₂×C₅×M₁₆	C₂×C₄	1	8	yes	—
31	C₂×C₅×M₁₆	C₂³	1	4	yes	—
32	C₂²×C₅×C₈	—	2	2	no	—
33	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
34	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₅×C₈	C₂×C₄	1	2	yes	—
37	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
38	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
39	C₂²×C₅×C₈	D₄	1	2	yes	—
40	C₂²×C₅×C₈	D₄	1	2	yes	—
41	C₂²×C₅×C₈	Q₈	1	4	yes	—
42	C₂²×C₅×C₈	C₂³	1	2	yes	—
43	C₂²×C₅×C₈	C₂³	1	2	yes	—
44	C₅×#117	—	2	2	no	—
45	C₅×#117	—	2	2	no	—
46	C₅×#117	C₂×C₄	1	4	yes	—
47	C₅×#117	C₂³	1	4	yes	—
48	C₂×C₅×C₄⋊C₄	C₂×C₄	1	4	yes	—
49	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₄²×C₅	—	2	2	no	—
51	C₂×C₄²×C₅	—	2	1	no	—
52	C₂×C₄²×C₅	—	2	2	no	—
53	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
54	C₂×C₄²×C₅	C₂³	1	2	yes	—
55	C₅×#108	C₂×C₄	1	16	yes	complemented
56	C₅×#107	C₂×C₄	1	8	yes	complemented
57	C₅×#106	C₂×C₄	1	8	yes	complemented
58	C₅×#97	—	2	8	no	—
59	C₅×#97	C₂×C₄	1	16	yes	complemented
60	C₄×C₅×C₈	—	2	8	no	—
61	C₂²×#108	C₂×C₅	1	1	yes	characteristic
62	C₂×#427	C₂×C₅	1	1	yes	characteristic
63	C₂×#405	C₂×C₅	1	2	yes	—
64	C₂²×#97	C₂×C₅	1	1	yes	characteristic
65	#1773	C₂×C₅	1	8	yes	direct product factor
66	C₂×#339	C₂×C₅	1	2	yes	—
67	C₂⁴×C₅	#41	1	1	yes	characteristic
68	C₂²×C₄×C₅	—	2	4	no	—
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	2	no	—
71	C₂²×C₄×C₅	—	4	4	no	—
72	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
73	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₅	#41	1	1	yes	characteristic
80	C₂²×C₄×C₅	#41	1	2	yes	—
81	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
82	C₅×M₁₆	—	2	8	no	—
83	C₅×M₁₆	—	2	4	no	—
84	C₅×M₁₆	C₂²×C₄	1	8	yes	—
85	C₂×C₅×C₈	—	2	8	no	—
86	C₂×C₅×C₈	—	2	4	no	—
87	C₂×C₅×C₈	C₄²	1	4	yes	—
88	C₂×C₅×C₈	C₄²	1	4	yes	—
89	C₂×C₅×C₈	C₄²	1	8	yes	—
90	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
91	C₂×C₅×C₈	C₄⋊C₄	1	8	yes	—
92	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
93	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
94	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
95	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
96	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
97	C₂×C₅×C₈	C₂×D₄	1	4	yes	—
98	C₂×C₅×C₈	C₂×Q₈	1	4	yes	—
99	C₅×C₄⋊C₄	—	2	4	no	—
100	C₅×C₄⋊C₄	C₂²×C₄	1	8	yes	—
101	C₅×C₂²⋊C₄	—	2	4	no	—
102	C₅×C₂²⋊C₄	—	2	4	no	—
103	C₄²×C₅	—	2	4	no	—
104	C₄²×C₅	—	2	4	no	—
105	C₄²×C₅	—	2	4	no	—
106	C₄²×C₅	—	2	2	no	—
107	C₄²×C₅	C₂²×C₄	1	4	yes	—
108	C₂²×M₁₆	C₂²×C₅	1	2	yes	—
109	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
110	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂×#117	C₂²×C₅	1	2	yes	—
112	C₂²×C₄²	C₂²×C₅	1	1	yes	characteristic
113	C₂×#108	C₄×C₅	1	8	yes	complemented
114	C₂×#108	C₂²×C₅	1	4	yes	—
115	#427	C₄×C₅	1	4	yes	complemented
116	#427	C₂²×C₅	1	4	yes	—
117	C₂×#107	C₄×C₅	1	2	yes	complemented
118	C₂×#106	C₄×C₅	1	2	yes	complemented
119	#405	—	2	4	no	—
120	#405	C₂²×C₅	1	8	yes	—
121	C₂×#97	—	2	2	no	—
122	C₂×#97	C₄×C₅	1	8	yes	complemented
123	C₂×#97	C₂²×C₅	1	4	yes	—
124	C₂×C₄×C₈	—	2	2	no	—
125	#339	C₂²×C₅	1	8	yes	—
126	C₂³×C₅	—	2	4	no	—
127	C₂³×C₅	#117	1	2	yes	—
128	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
129	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
130	C₂³×C₅	C₄×Q₈	1	2	yes	—
131	C₂³×C₅	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₅	—	2	4	no	—
133	C₂×C₄×C₅	—	2	8	no	—
134	C₂×C₄×C₅	—	4	4	no	—
135	C₂×C₄×C₅	—	4	8	no	—
136	C₂×C₄×C₅	—	4	8	no	—
137	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
138	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
139	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
140	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
141	C₂×C₄×C₅	#117	1	2	yes	—
142	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
143	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
144	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
147	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
148	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
149	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
150	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
151	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
152	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂²×Q₈	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
155	C₅×C₈	—	2	8	no	—
156	C₅×C₈	C₂×C₄²	1	4	yes	—
157	C₅×C₈	C₂×C₄²	1	4	yes	—
158	C₅×C₈	C₂×C₄⋊C₄	1	8	yes	—
159	C₂³×C₄	—	2	1	no	—
160	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	2	4	no	—
163	C₂×M₁₆	C₂×C₄×C₅	1	8	yes	—
164	C₂×M₁₆	C₂³×C₅	1	4	yes	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
167	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
168	C₂²×C₈	C₂×C₄×C₅	1	1	yes	characteristic
169	C₂²×C₈	C₂×C₄×C₅	1	2	yes	—
170	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
171	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
172	C₂²×C₈	C₅×D₄	1	2	yes	—
173	C₂²×C₈	C₅×D₄	1	2	yes	—
174	C₂²×C₈	C₅×Q₈	1	4	yes	—
175	C₂²×C₈	C₂³×C₅	1	2	yes	—
176	C₂²×C₈	C₂³×C₅	1	2	yes	—
177	#117	—	2	2	no	—
178	#117	—	2	2	no	—
179	#117	C₂×C₄×C₅	1	4	yes	—
180	#117	C₂³×C₅	1	4	yes	—
181	C₂×C₄⋊C₄	C₂×C₄×C₅	1	4	yes	—
182	C₂×C₂²⋊C₄	—	2	2	no	—
183	C₂×C₄²	—	2	2	no	—
184	C₂×C₄²	—	2	1	no	—
185	C₂×C₄²	—	2	2	no	—
186	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
187	C₂×C₄²	C₂³×C₅	1	2	yes	—
188	#108	C₂×C₄×C₅	1	16	yes	complemented
189	#107	C₂×C₄×C₅	1	8	yes	complemented
190	#106	C₂×C₄×C₅	1	8	yes	complemented
191	#97	—	2	8	no	—
192	#97	C₂×C₄×C₅	1	16	yes	complemented
193	C₄×C₈	—	2	8	no	—
194	C₂²×C₅	—	2	2	no	—
195	C₂²×C₅	—	2	2	no	—
196	C₂²×C₅	—	2	4	no	—
197	C₂²×C₅	—	2	4	no	—
198	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
199	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
200	C₂²×C₅	#444	1	4	yes	—
201	C₂²×C₅	C₂×#117	1	1	yes	characteristic
202	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂²×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
204	C₄×C₅	—	4	4	no	—
205	C₄×C₅	—	4	8	no	—
206	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
207	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
208	C₄×C₅	C₂²×C₄²	1	1	yes	characteristic
209	C₄×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
210	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
211	C₄×C₅	C₂×C₄×Q₈	1	1	yes	characteristic
212	C₂⁴	C₅×#41	1	1	yes	characteristic
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	4	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	4	4	no	—
217	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
218	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
219	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
220	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
222	C₂²×C₄	C₂×C₅×D₄	1	1	yes	characteristic
223	C₂²×C₄	C₂×C₅×Q₈	1	2	yes	—
224	C₂²×C₄	C₅×#41	1	1	yes	characteristic
225	C₂²×C₄	C₅×#41	1	2	yes	—
226	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
227	M₁₆	—	2	8	no	—
228	M₁₆	—	2	4	no	—
229	M₁₆	C₂²×C₄×C₅	1	8	yes	—
230	C₂×C₈	—	2	8	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	C₄²×C₅	1	4	yes	—
233	C₂×C₈	C₄²×C₅	1	4	yes	—
234	C₂×C₈	C₄²×C₅	1	8	yes	—
235	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
236	C₂×C₈	C₅×C₄⋊C₄	1	8	yes	—
237	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
238	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
239	C₂×C₈	C₂²×C₄×C₅	1	2	yes	—
240	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
241	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
242	C₂×C₈	C₂×C₅×D₄	1	4	yes	—
243	C₂×C₈	C₂×C₅×Q₈	1	4	yes	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	C₂²×C₄×C₅	1	8	yes	—
246	C₂²⋊C₄	—	2	4	no	—
247	C₂²⋊C₄	—	2	4	no	—
248	C₄²	—	2	4	no	—
249	C₄²	—	2	4	no	—
250	C₄²	—	2	4	no	—
251	C₄²	—	2	2	no	—
252	C₄²	C₂²×C₄×C₅	1	4	yes	—
253	C₂×C₅	—	2	1	no	—
254	C₂×C₅	—	2	1	no	—
255	C₂×C₅	—	2	2	no	—
256	C₂×C₅	#1773	1	4	yes	direct product factor
257	C₂×C₅	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
258	C₂×C₅	C₂×#444	1	2	yes	—
259	C₂³	—	2	4	no	—
260	C₂³	C₅×#117	1	2	yes	—
261	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
262	C₂³	C₄×C₅×D₄	1	1	yes	characteristic
263	C₂³	C₄×C₅×Q₈	1	2	yes	—
264	C₂³	C₂×C₅×#41	1	1	yes	characteristic
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	8	no	—
267	C₂×C₄	—	4	8	no	—
268	C₂×C₄	—	4	4	no	—
269	C₂×C₄	—	4	8	no	—
270	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
271	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
272	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
273	C₂×C₄	C₂×C₅×C₄⋊C₄	1	2	yes	—
274	C₂×C₄	C₅×#117	1	2	yes	—
275	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
276	C₂×C₄	C₄×C₅×D₄	1	1	yes	characteristic
277	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
278	C₂×C₄	C₄×C₅×D₄	1	2	yes	—
279	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
280	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
281	C₂×C₄	C₄×C₅×Q₈	1	2	yes	—
282	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
283	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
284	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
285	C₂×C₄	C₂²×C₅×D₄	1	1	yes	characteristic
286	C₂×C₄	C₂²×C₅×Q₈	1	1	yes	characteristic
287	C₂×C₄	C₂×C₅×#41	1	1	yes	characteristic
288	C₈	—	2	8	no	—
289	C₈	C₂×C₄²×C₅	1	4	yes	—
290	C₈	C₂×C₄²×C₅	1	4	yes	—
291	C₈	C₂×C₅×C₄⋊C₄	1	8	yes	—
292	C₅	C₂×#1773	1	1	yes	characteristic, direct product factor, Sylow
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
298	C₂²	C₄×C₅×C₄⋊C₄	1	2	yes	—
299	C₂²	C₅×#444	1	4	yes	—
300	C₂²	C₂×C₅×#117	1	1	yes	characteristic
301	C₂²	C₂×C₄×C₅×D₄	1	1	yes	characteristic
302	C₂²	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
303	C₄	—	4	4	no	—
304	C₄	—	4	8	no	—
305	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
306	C₄	C₄×C₅×C₄⋊C₄	1	2	yes	—
307	C₄	C₂²×C₄²×C₅	1	1	yes	characteristic
308	C₄	C₂²×C₅×C₄⋊C₄	1	1	yes	characteristic
309	C₄	C₂×C₄×C₅×D₄	1	1	yes	characteristic
310	C₄	C₂×C₄×C₅×Q₈	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	2	no	—
313	C₂	—	2	1	no	—
314	C₂	C₅×#1773	1	4	yes	direct product factor
315	C₂	C₂×C₄×C₅×C₄⋊C₄	1	1	yes	characteristic
316	C₂	C₂×C₅×#444	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₅	C₂×D₄
commutator	C₄	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂³×C₅	C₄×D₄
prime radical	C₂⁴×C₅	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₅ ◁ C₂³×C₅×C₈ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₅×#1773	C₂	direct (N × H)
12. C₂×C₅×#108	C₄	semidirect
14. C₅×#427	C₄	semidirect
16. C₂×C₅×#107	C₄	semidirect
17. C₂×C₅×#106	C₄	semidirect
21. C₂×C₅×#97	C₄	semidirect
25. C₂×#1773	C₅	direct (N × H)
55. C₅×#108	C₂×C₄	semidirect
56. C₅×#107	C₂×C₄	semidirect
57. C₅×#106	C₂×C₄	semidirect
59. C₅×#97	C₂×C₄	semidirect
65. #1773	C₂×C₅	direct (N × H)
113. C₂×#108	C₄×C₅	semidirect
115. #427	C₄×C₅	semidirect
117. C₂×#107	C₄×C₅	semidirect
118. C₂×#106	C₄×C₅	semidirect
122. C₂×#97	C₄×C₅	semidirect
188. #108	C₂×C₄×C₅	semidirect
189. #107	C₂×C₄×C₅	semidirect
190. #106	C₂×C₄×C₅	semidirect
192. #97	C₂×C₄×C₅	semidirect
256. C₂×C₅	#1773	direct (N × H)
292. C₅	C₂×#1773	direct (N × H)
314. C₂	C₅×#1773	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×#108	2
C₂×C₅×#427	2
C₂×C₅×#405	2
C₂²×C₅×#97	2
C₅×#1773	2
C₂×C₅×#339	2
C₂×#1773	5

Subgroups — C2×C5×#1773 #11943163

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1773 #11943163