Subgroups — C₂×C₅×#1752 #11943102

Order 1280.

1918 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₅×C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₅×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₅×#97	C₂	1	2	yes	maximal, complemented
4	C₂×C₄×C₅×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
5	C₅×#1752	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₅×#336	C₂	1	2	yes	maximal, complemented
7	C₂⁴×C₄×C₅	C₂²	1	1	yes	characteristic
8	C₂³×C₅×C₈	—	2	8	no	—
9	C₂³×C₅×C₈	C₄	1	8	yes	complemented
10	C₂³×C₅×C₈	C₂²	1	2	yes	—
11	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
12	C₂³×C₅×C₈	C₂²	1	4	yes	—
13	C₂³×C₅×C₈	C₂²	1	1	yes	characteristic
14	C₂²×C₅×C₂²⋊C₄	C₄	1	2	yes	—
15	C₂²×C₄²×C₅	C₂²	1	2	yes	—
16	C₂×C₅×C₂²⋊C₈	C₄	1	8	yes	complemented
17	C₂×C₅×C₂²⋊C₈	C₂²	1	4	yes	—
18	C₂×C₅×#97	—	2	8	no	—
19	C₂×C₅×#97	C₂²	1	8	yes	complemented
20	C₄×C₅×C₂²⋊C₄	C₄	1	4	yes	—
21	C₄×C₅×C₂²⋊C₄	C₂²	1	4	yes	—
22	C₂×C₅×#95	C₄	1	2	yes	—
23	C₅×#336	C₂²	1	8	yes	complemented
24	C₂×#1752	C₅	1	1	yes	maximal, characteristic, direct product factor, Sylow
25	C₂⁵×C₅	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₅	—	2	4	no	—
27	C₂³×C₄×C₅	—	2	4	no	—
28	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
29	C₂³×C₄×C₅	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
31	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
32	C₂³×C₄×C₅	C₂×C₄	1	4	yes	—
33	C₂³×C₄×C₅	C₂³	1	2	yes	—
34	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂²×C₅×C₈	—	2	4	no	—
36	C₂²×C₅×C₈	—	2	4	no	—
37	C₂²×C₅×C₈	—	2	4	no	—
38	C₂²×C₅×C₈	—	2	32	no	—
39	C₂²×C₅×C₈	—	2	16	no	—
40	C₂²×C₅×C₈	C₂×C₄	1	8	yes	—
41	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
42	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
43	C₂²×C₅×C₈	C₂×C₄	1	16	yes	complemented
44	C₂²×C₅×C₈	D₄	1	8	yes	—
45	C₂²×C₅×C₈	C₂³	1	4	yes	—
46	C₂²×C₅×C₈	C₂³	1	2	yes	—
47	C₂²×C₅×C₈	C₂³	1	2	yes	—
48	C₂×C₅×C₂²⋊C₄	—	2	8	no	—
49	C₂×C₅×C₂²⋊C₄	C₂×C₄	1	8	yes	—
50	C₂×C₄²×C₅	—	2	4	no	—
51	C₂×C₄²×C₅	—	2	4	no	—
52	C₂×C₄²×C₅	C₂×C₄	1	4	yes	—
53	C₂×C₄²×C₅	C₂³	1	4	yes	—
54	C₅×C₂²⋊C₈	C₂×C₄	1	16	yes	complemented
55	C₅×#97	—	2	16	no	—
56	C₅×#95	C₂×C₄	1	8	yes	—
57	C₂⁴×C₈	C₂×C₅	1	1	yes	characteristic
58	C₂²×C₂²⋊C₈	C₂×C₅	1	1	yes	characteristic
59	C₂²×#97	C₂×C₅	1	2	yes	complemented
60	C₂×C₄×C₂²⋊C₄	C₂×C₅	1	1	yes	characteristic
61	#1752	C₂×C₅	1	8	yes	direct product factor
62	C₂×#336	C₂×C₅	1	2	yes	complemented
63	C₂⁴×C₅	—	2	4	no	—
64	C₂⁴×C₅	—	2	4	no	—
65	C₂⁴×C₅	C₄²	1	4	yes	—
66	C₂⁴×C₅	M₁₆	1	8	yes	—
67	C₂⁴×C₅	C₂²×C₄	1	2	yes	—
68	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	—	2	4	no	—
70	C₂²×C₄×C₅	—	2	8	no	—
71	C₂²×C₄×C₅	—	2	8	no	—
72	C₂²×C₄×C₅	—	2	8	no	—
73	C₂²×C₄×C₅	—	2	16	no	—
74	C₂²×C₄×C₅	—	2	4	no	—
75	C₂²×C₄×C₅	—	2	4	no	—
76	C₂²×C₄×C₅	—	2	8	no	—
77	C₂²×C₄×C₅	—	2	8	no	—
78	C₂²×C₄×C₅	—	2	8	no	—
79	C₂²×C₄×C₅	C₄²	1	4	yes	—
80	C₂²×C₄×C₅	C₄²	1	8	yes	—
81	C₂²×C₄×C₅	C₂²×C₄	1	8	yes	—
82	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
84	C₂²×C₄×C₅	C₂²×C₄	1	4	yes	—
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	C₂×D₄	1	4	yes	—
89	C₂²×C₄×C₅	#41	1	4	yes	—
90	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
91	C₂×C₅×C₈	—	2	8	no	—
92	C₂×C₅×C₈	—	2	16	no	—
93	C₂×C₅×C₈	—	2	4	no	—
94	C₂×C₅×C₈	—	2	4	no	—
95	C₂×C₅×C₈	—	2	4	no	—
96	C₂×C₅×C₈	—	2	32	no	—
97	C₂×C₅×C₈	C₂²⋊C₄	1	16	yes	complemented
98	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
99	C₂×C₅×C₈	C₂²×C₄	1	8	yes	—
100	C₂×C₅×C₈	C₂²×C₄	1	4	yes	—
101	C₂×C₅×C₈	C₂×D₄	1	8	yes	—
102	C₅×C₂²⋊C₄	—	2	16	no	—
103	C₄²×C₅	—	2	8	no	—
104	C₄²×C₅	—	2	4	no	—
105	C₄²×C₅	—	2	4	no	—
106	C₂⁴×C₄	C₂²×C₅	1	1	yes	characteristic
107	C₂³×C₈	—	2	8	no	—
108	C₂³×C₈	C₄×C₅	1	8	yes	complemented
109	C₂³×C₈	C₂²×C₅	1	2	yes	—
110	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
111	C₂³×C₈	C₂²×C₅	1	4	yes	—
112	C₂³×C₈	C₂²×C₅	1	1	yes	characteristic
113	C₂²×C₂²⋊C₄	C₄×C₅	1	2	yes	—
114	C₂²×C₄²	C₂²×C₅	1	2	yes	—
115	C₂×C₂²⋊C₈	C₄×C₅	1	8	yes	complemented
116	C₂×C₂²⋊C₈	C₂²×C₅	1	4	yes	—
117	C₂×#97	—	2	8	no	—
118	C₂×#97	C₂²×C₅	1	8	yes	complemented
119	C₄×C₂²⋊C₄	C₄×C₅	1	4	yes	—
120	C₄×C₂²⋊C₄	C₂²×C₅	1	4	yes	—
121	C₂×#95	C₄×C₅	1	2	yes	—
122	#336	C₂²×C₅	1	8	yes	complemented
123	C₂³×C₅	—	2	4	no	—
124	C₂³×C₅	—	2	4	no	—
125	C₂³×C₅	—	2	8	no	—
126	C₂³×C₅	—	2	8	no	—
127	C₂³×C₅	—	2	32	no	—
128	C₂³×C₅	#97	1	16	yes	complemented
129	C₂³×C₅	C₂×C₄²	1	4	yes	—
130	C₂³×C₅	C₂×C₄²	1	2	yes	—
131	C₂³×C₅	C₄×D₄	1	4	yes	—
132	C₂³×C₅	C₂×M₁₆	1	8	yes	—
133	C₂³×C₅	C₂×M₁₆	1	4	yes	—
134	C₂³×C₅	#131	1	4	yes	—
135	C₂³×C₅	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₅	—	2	4	no	—
137	C₂×C₄×C₅	—	2	8	no	—
138	C₂×C₄×C₅	—	2	4	no	—
139	C₂×C₄×C₅	—	2	4	no	—
140	C₂×C₄×C₅	—	2	8	no	—
141	C₂×C₄×C₅	—	2	16	no	—
142	C₂×C₄×C₅	—	2	8	no	—
143	C₂×C₄×C₅	—	2	8	no	—
144	C₂×C₄×C₅	—	2	8	no	—
145	C₂×C₄×C₅	—	4	16	no	—
146	C₂×C₄×C₅	C₂×C₄²	1	4	yes	—
147	C₂×C₄×C₅	C₂×C₄²	1	2	yes	—
148	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	4	yes	—
149	C₂×C₄×C₅	#117	1	4	yes	—
150	C₂×C₄×C₅	C₄×D₄	1	8	yes	—
151	C₂×C₄×C₅	C₄×D₄	1	4	yes	—
152	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
153	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅	C₂²×D₄	1	2	yes	—
155	C₂×C₄×C₅	C₂×#41	1	2	yes	—
156	C₅×C₈	—	2	4	no	—
157	C₅×C₈	—	2	8	no	—
158	C₅×C₈	C₂×C₂²⋊C₄	1	8	yes	complemented
159	C₂⁵	C₂×C₄×C₅	1	1	yes	characteristic
160	C₂³×C₄	—	2	4	no	—
161	C₂³×C₄	—	2	4	no	—
162	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
163	C₂³×C₄	C₂×C₄×C₅	1	1	yes	characteristic
164	C₂³×C₄	C₂×C₄×C₅	1	2	yes	—
165	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
166	C₂³×C₄	C₂×C₄×C₅	1	4	yes	—
167	C₂³×C₄	C₂³×C₅	1	2	yes	—
168	C₂³×C₄	C₂³×C₅	1	1	yes	characteristic
169	C₂²×C₈	—	2	4	no	—
170	C₂²×C₈	—	2	4	no	—
171	C₂²×C₈	—	2	4	no	—
172	C₂²×C₈	—	2	32	no	—
173	C₂²×C₈	—	2	16	no	—
174	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
175	C₂²×C₈	C₂×C₄×C₅	1	8	yes	—
176	C₂²×C₈	C₂×C₄×C₅	1	4	yes	—
177	C₂²×C₈	C₂×C₄×C₅	1	16	yes	complemented
178	C₂²×C₈	C₅×D₄	1	8	yes	—
179	C₂²×C₈	C₂³×C₅	1	2	yes	—
180	C₂²×C₈	C₂³×C₅	1	4	yes	—
181	C₂²×C₈	C₂³×C₅	1	2	yes	—
182	C₂×C₂²⋊C₄	—	2	8	no	—
183	C₂×C₂²⋊C₄	C₂×C₄×C₅	1	8	yes	—
184	C₂×C₄²	—	2	4	no	—
185	C₂×C₄²	—	2	4	no	—
186	C₂×C₄²	C₂×C₄×C₅	1	4	yes	—
187	C₂×C₄²	C₂³×C₅	1	4	yes	—
188	C₂²⋊C₈	C₂×C₄×C₅	1	16	yes	complemented
189	#97	—	2	16	no	—
190	#95	C₂×C₄×C₅	1	8	yes	—
191	C₂²×C₅	—	2	4	no	—
192	C₂²×C₅	—	2	16	no	—
193	C₂²×C₅	—	2	32	no	—
194	C₂²×C₅	—	2	4	no	—
195	C₂²×C₅	C₄×C₂²⋊C₄	1	4	yes	—
196	C₂²×C₅	C₂×#97	1	4	yes	—
197	C₂²×C₅	C₂×#97	1	8	yes	complemented
198	C₂²×C₅	#405	1	4	yes	—
199	C₂²×C₅	#435	1	16	yes	—
200	C₂²×C₅	C₂²×C₄²	1	1	yes	characteristic
201	C₂²×C₅	C₂×C₄×D₄	1	2	yes	—
202	C₂²×C₅	C₂²×M₁₆	1	2	yes	—
203	C₂²×C₅	C₂×#131	1	2	yes	—
204	C₄×C₅	—	2	4	no	—
205	C₄×C₅	—	4	8	no	—
206	C₄×C₅	C₄×C₂²⋊C₄	1	4	yes	—
207	C₄×C₅	C₂²×C₂²⋊C₄	1	1	yes	characteristic
208	C₄×C₅	C₂×#117	1	1	yes	characteristic
209	C₄×C₅	C₂×C₄×D₄	1	2	yes	—
210	C₂⁴	—	2	4	no	—
211	C₂⁴	—	2	4	no	—
212	C₂⁴	C₄²×C₅	1	4	yes	—
213	C₂⁴	C₅×M₁₆	1	8	yes	—
214	C₂⁴	C₂²×C₄×C₅	1	2	yes	—
215	C₂⁴	C₂²×C₄×C₅	1	1	yes	characteristic
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	2	8	no	—
218	C₂²×C₄	—	2	8	no	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	2	8	no	—
221	C₂²×C₄	—	2	4	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	8	no	—
224	C₂²×C₄	—	2	8	no	—
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	C₄²×C₅	1	4	yes	—
227	C₂²×C₄	C₄²×C₅	1	8	yes	—
228	C₂²×C₄	C₂²×C₄×C₅	1	8	yes	—
229	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
231	C₂²×C₄	C₂²×C₄×C₅	1	4	yes	—
232	C₂²×C₄	C₂²×C₄×C₅	1	1	yes	characteristic
233	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
234	C₂²×C₄	C₂²×C₄×C₅	1	2	yes	—
235	C₂²×C₄	C₂×C₅×D₄	1	4	yes	—
236	C₂²×C₄	C₅×#41	1	4	yes	—
237	C₂²×C₄	C₂⁴×C₅	1	1	yes	characteristic
238	C₂×C₈	—	2	8	no	—
239	C₂×C₈	—	2	16	no	—
240	C₂×C₈	—	2	4	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	32	no	—
244	C₂×C₈	C₅×C₂²⋊C₄	1	16	yes	complemented
245	C₂×C₈	C₂²×C₄×C₅	1	8	yes	—
246	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
247	C₂×C₈	C₂²×C₄×C₅	1	4	yes	—
248	C₂×C₈	C₂×C₅×D₄	1	8	yes	—
249	C₂²⋊C₄	—	2	16	no	—
250	C₄²	—	2	8	no	—
251	C₄²	—	2	4	no	—
252	C₄²	—	2	4	no	—
253	C₂×C₅	—	2	8	no	—
254	C₂×C₅	#1752	1	8	yes	direct product factor
255	C₂×C₅	C₂×C₄×C₂²⋊C₄	1	1	yes	characteristic
256	C₂×C₅	C₂²×#97	1	1	yes	characteristic
257	C₂×C₅	C₂×#405	1	1	yes	characteristic
258	C₂×C₅	C₂×#435	1	4	yes	—
259	C₂³	—	2	8	no	—
260	C₂³	—	2	8	no	—
261	C₂³	—	2	32	no	—
262	C₂³	—	2	4	no	—
263	C₂³	—	2	4	no	—
264	C₂³	C₅×#97	1	16	yes	complemented
265	C₂³	C₂×C₄²×C₅	1	4	yes	—
266	C₂³	C₂×C₄²×C₅	1	2	yes	—
267	C₂³	C₄×C₅×D₄	1	4	yes	—
268	C₂³	C₂×C₅×M₁₆	1	8	yes	—
269	C₂³	C₂×C₅×M₁₆	1	4	yes	—
270	C₂³	C₅×#131	1	4	yes	—
271	C₂³	C₂³×C₄×C₅	1	1	yes	characteristic
272	C₂×C₄	—	2	8	no	—
273	C₂×C₄	—	2	4	no	—
274	C₂×C₄	—	2	16	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	8	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	16	no	—
282	C₂×C₄	C₂×C₄²×C₅	1	4	yes	—
283	C₂×C₄	C₂×C₄²×C₅	1	2	yes	—
284	C₂×C₄	C₂×C₅×C₂²⋊C₄	1	4	yes	—
285	C₂×C₄	C₅×#117	1	4	yes	—
286	C₂×C₄	C₄×C₅×D₄	1	4	yes	—
287	C₂×C₄	C₄×C₅×D₄	1	8	yes	—
288	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
289	C₂×C₄	C₂³×C₄×C₅	1	1	yes	characteristic
290	C₂×C₄	C₂²×C₅×D₄	1	2	yes	—
291	C₂×C₄	C₂×C₅×#41	1	2	yes	—
292	C₈	—	2	4	no	—
293	C₈	—	2	8	no	—
294	C₈	C₂×C₅×C₂²⋊C₄	1	8	yes	complemented
295	C₅	C₂×#1752	1	1	yes	characteristic, direct product factor, Sylow
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	16	no	—
299	C₂²	—	2	32	no	—
300	C₂²	C₄×C₅×C₂²⋊C₄	1	4	yes	—
301	C₂²	C₂×C₅×#97	1	4	yes	—
302	C₂²	C₂×C₅×#97	1	8	yes	complemented
303	C₂²	C₅×#405	1	4	yes	—
304	C₂²	C₅×#435	1	16	yes	—
305	C₂²	C₂²×C₄²×C₅	1	1	yes	characteristic
306	C₂²	C₂×C₄×C₅×D₄	1	2	yes	—
307	C₂²	C₂²×C₅×M₁₆	1	2	yes	—
308	C₂²	C₂×C₅×#131	1	2	yes	—
309	C₄	—	2	4	no	—
310	C₄	—	4	8	no	—
311	C₄	C₄×C₅×C₂²⋊C₄	1	4	yes	—
312	C₄	C₂²×C₅×C₂²⋊C₄	1	1	yes	characteristic
313	C₄	C₂×C₅×#117	1	1	yes	characteristic
314	C₄	C₂×C₄×C₅×D₄	1	2	yes	—
315	C₂	—	2	8	no	—
316	C₂	C₅×#1752	1	8	yes	direct product factor
317	C₂	C₂×C₄×C₅×C₂²⋊C₄	1	1	yes	characteristic
318	C₂	C₂²×C₅×#97	1	1	yes	characteristic
319	C₂	C₂×C₅×#405	1	1	yes	characteristic
320	C₂	C₂×C₅×#435	1	4	yes	—

	subgroup	G/H
center	C₂³×C₄×C₅	C₂³
commutator	C₂²	C₂²×C₄²×C₅
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅
socle	C₂⁴×C₅	C₂²×C₄
prime radical	C₂⁵×C₅	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₅ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₂²×C₅×#97	C₂	semidirect
5. C₅×#1752	C₂	direct (N × H)
6. C₂×C₅×#336	C₂	semidirect
9. C₂³×C₅×C₈	C₄	semidirect
16. C₂×C₅×C₂²⋊C₈	C₄	semidirect
19. C₂×C₅×#97	C₂²	semidirect
23. C₅×#336	C₂²	semidirect
24. C₂×#1752	C₅	direct (N × H)
43. C₂²×C₅×C₈	C₂×C₄	semidirect
54. C₅×C₂²⋊C₈	C₂×C₄	semidirect
59. C₂²×#97	C₂×C₅	semidirect
61. #1752	C₂×C₅	direct (N × H)
62. C₂×#336	C₂×C₅	semidirect
97. C₂×C₅×C₈	C₂²⋊C₄	semidirect
108. C₂³×C₈	C₄×C₅	semidirect
115. C₂×C₂²⋊C₈	C₄×C₅	semidirect
118. C₂×#97	C₂²×C₅	semidirect
122. #336	C₂²×C₅	semidirect
128. C₂³×C₅	#97	semidirect
158. C₅×C₈	C₂×C₂²⋊C₄	semidirect
177. C₂²×C₈	C₂×C₄×C₅	semidirect
188. C₂²⋊C₈	C₂×C₄×C₅	semidirect
197. C₂²×C₅	C₂×#97	semidirect
244. C₂×C₈	C₅×C₂²⋊C₄	semidirect
254. C₂×C₅	#1752	direct (N × H)
264. C₂³	C₅×#97	semidirect
294. C₈	C₂×C₅×C₂²⋊C₄	semidirect
295. C₅	C₂×#1752	direct (N × H)
302. C₂²	C₂×C₅×#97	semidirect
316. C₂	C₅×#1752	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₅×C₈	2
C₂²×C₅×C₂²⋊C₈	2
C₂²×C₅×#97	2
C₂×C₄×C₅×C₂²⋊C₄	2
C₅×#1752	2
C₂×C₅×#336	2
C₂×#1752	5

Subgroups — C2×C5×#1752 #11943102

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₅×#1752 #11943102