Subgroups — #13101541

Order 1600.

624 conjugacy classes of subgroups.

The subgroup lattice has 345 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅×Dic₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×#1039	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₅×#685	C₂	1	1	yes	maximal, characteristic
4	C₂×#87778	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅²	C₂	1	1	yes	maximal, characteristic, complemented
6	C₅×#3963	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716620	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₅×D₅	C₂²	1	1	yes	characteristic
9	C₂²×C₄×C₅²	C₂²	1	1	yes	characteristic
10	C₂²×#1039	C₂²	1	1	yes	characteristic
11	C₂²×#1039	C₂²	1	1	yes	characteristic
12	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
13	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
14	C₂²×C₅×Dic₅	C₂²	1	1	yes	characteristic
15	C₂×D₅×Dic₅	—	2	2	no	—
16	C₂×D₅×Dic₅	—	2	2	no	—
17	C₂×D₅×Dic₅	C₄	1	4	yes	complemented
18	C₄×#1039	—	2	2	no	—
19	C₅×#685	C₄	1	4	yes	complemented
20	#87778	C₄	1	4	yes	complemented
21	Dic₅²	—	2	2	no	—
22	C₂²⋊C₄×Dic₅	—	5	1	no	maximal
23	#64955	—	5	1	no	maximal
24	C₂³×C₅²	C₂³	1	1	yes	characteristic
25	C₂²×C₅×D₅	—	2	1	no	—
26	C₂²×C₅×D₅	—	2	1	no	—
27	C₂²×C₅×D₅	—	2	1	no	—
28	C₂²×C₅×D₅	—	2	1	no	—
29	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
30	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
31	C₂²×C₅×D₅	C₂×C₄	1	2	yes	—
32	C₂×C₄×C₅²	—	2	1	no	—
33	C₂×C₄×C₅²	—	2	1	no	—
34	C₂×C₄×C₅²	C₂×C₄	1	2	yes	complemented
35	C₂×#1039	—	2	1	no	—
36	C₂×#1039	—	2	1	no	—
37	C₂×#1039	C₂×C₄	1	2	yes	—
38	C₂×#1039	C₂×C₄	1	2	yes	complemented
39	C₂×#1039	D₄	1	2	yes	—
40	C₂×#1039	D₄	1	2	yes	—
41	C₂×C₅×Dic₅	—	2	1	no	—
42	C₂×C₅×Dic₅	—	2	1	no	—
43	C₂×C₅×Dic₅	—	2	1	no	—
44	C₂×C₅×Dic₅	—	2	1	no	—
45	C₂×C₅×Dic₅	—	2	1	no	—
46	C₂×C₅×Dic₅	—	2	1	no	—
47	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	—
48	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	complemented
49	C₂×C₅×Dic₅	C₂×C₄	1	2	yes	complemented
50	D₅×Dic₅	—	2	8	no	—
51	C₂³×Dic₅	—	5	1	no	—
52	C₂²×C₄×D₅	—	5	1	no	—
53	C₂×C₅×C₂²⋊C₄	—	5	1	no	—
54	C₂×C₂²⋊Dic₅	—	5	1	no	—
55	C₂×#685	—	5	1	no	—
56	C₂×#685	D₅	1	1	yes	characteristic
57	C₂×C₄×Dic₅	—	5	1	no	—
58	C₂×C₄×Dic₅	—	5	1	no	—
59	C₂×C₄×Dic₅	—	5	1	no	—
60	C₂×C₄×Dic₅	—	5	1	no	—
61	C₂×C₄×Dic₅	—	10	2	no	—
62	C₅×#95	—	5	1	no	—
63	#3963	—	5	1	no	—
64	#3963	—	5	1	no	—
65	#3963	D₅	1	1	yes	characteristic, complemented
66	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₅²	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₅²	#41	1	1	yes	characteristic
72	C₂²×C₅²	#41	1	1	yes	characteristic
73	C₂×C₅×D₅	—	2	2	no	—
74	C₂×C₅×D₅	—	2	2	no	—
75	C₂×C₅×D₅	—	2	2	no	—
76	C₂×C₅×D₅	—	2	2	no	—
77	C₂×C₅×D₅	—	2	2	no	—
78	C₂×C₅×D₅	—	2	2	no	—
79	C₂×C₅×D₅	C₄²	1	4	yes	complemented
80	C₄×C₅²	—	2	2	no	—
81	#1039	—	2	2	no	—
82	#1039	C₂²⋊C₄	1	4	yes	complemented
83	C₅×Dic₅	—	2	2	no	—
84	C₅×Dic₅	—	2	2	no	—
85	C₅×Dic₅	—	2	2	no	—
86	C₂⁴×C₅	—	5	1	no	—
87	C₂³×D₅	C₂×D₅	1	1	yes	characteristic
88	C₂²×C₄×C₅	—	2	2	no	—
89	C₂²×C₄×C₅	—	5	1	no	—
90	C₂²×C₄×C₅	—	5	1	no	—
91	C₂²×C₄×C₅	—	5	1	no	—
92	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×D₅	1	1	yes	characteristic
94	C₂²×Dic₅	—	5	1	no	—
95	C₂²×Dic₅	—	5	1	no	—
96	C₂²×Dic₅	—	5	1	no	—
97	C₂²×Dic₅	—	5	1	no	—
98	C₂²×Dic₅	—	5	4	no	—
99	C₂²×Dic₅	—	10	2	no	—
100	C₂²×Dic₅	—	10	2	no	—
101	C₂²×Dic₅	—	10	2	no	—
102	C₂²×Dic₅	—	10	2	no	—
103	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
104	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
105	C₂²×Dic₅	C₂×D₅	1	1	yes	characteristic
106	C₂×C₄×D₅	—	5	4	no	—
107	C₂×C₄×D₅	—	10	2	no	—
108	C₂×C₄×D₅	—	10	2	no	—
109	C₅×C₂²⋊C₄	—	5	4	no	—
110	C₂²⋊Dic₅	—	5	4	no	—
111	#685	—	5	4	no	—
112	#685	Dic₅	1	4	yes	complemented
113	C₄×Dic₅	—	10	2	no	—
114	C₄×Dic₅	—	10	2	no	—
115	C₄×Dic₅	—	10	2	no	—
116	C₄×Dic₅	—	10	2	no	—
117	C₄×Dic₅	—	10	8	no	—
118	C₄×C₂²⋊C₄	—	25	1	no	Sylow
119	C₂×C₅²	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₅²	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₅²	#117	1	1	yes	characteristic
122	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
123	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
124	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
125	C₂×C₅²	C₄×D₄	1	1	yes	characteristic
126	C₅×D₅	—	2	4	no	—
127	C₂³×C₅	—	2	2	no	—
128	C₂³×C₅	—	5	2	no	—
129	C₂³×C₅	—	5	2	no	—
130	C₂³×C₅	—	5	2	no	—
131	C₂³×C₅	—	10	1	no	—
132	C₂³×C₅	—	10	1	no	—
133	C₂³×C₅	—	10	1	no	—
134	C₂³×C₅	—	10	1	no	—
135	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
136	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
137	C₂²×D₅	—	2	1	no	—
138	C₂²×D₅	—	2	1	no	—
139	C₂²×D₅	—	2	1	no	—
140	C₂²×D₅	—	2	1	no	—
141	C₂²×D₅	C₄×D₅	1	2	yes	—
142	C₂²×D₅	C₄×D₅	1	2	yes	—
143	C₂²×D₅	C₂×Dic₅	1	2	yes	—
144	C₂×C₄×C₅	—	2	4	no	—
145	C₂×C₄×C₅	—	2	4	no	—
146	C₂×C₄×C₅	—	2	1	no	—
147	C₂×C₄×C₅	—	2	1	no	—
148	C₂×C₄×C₅	—	2	1	no	—
149	C₂×C₄×C₅	—	2	1	no	—
150	C₂×C₄×C₅	—	2	4	no	—
151	C₂×C₄×C₅	—	5	2	no	—
152	C₂×C₄×C₅	—	5	2	no	—
153	C₂×C₄×C₅	—	5	2	no	—
154	C₂×C₄×C₅	—	10	1	no	—
155	C₂×C₄×C₅	—	10	1	no	—
156	C₂×C₄×C₅	—	10	1	no	—
157	C₂×C₄×C₅	—	10	1	no	—
158	C₂×C₄×C₅	—	10	1	no	—
159	C₂×C₄×C₅	—	10	1	no	—
160	C₂×C₄×C₅	C₄×D₅	1	2	yes	complemented
161	C₂×C₄×C₅	C₂×Dic₅	1	2	yes	complemented
162	C₂×Dic₅	—	2	1	no	—
163	C₂×Dic₅	—	2	1	no	—
164	C₂×Dic₅	—	2	1	no	—
165	C₂×Dic₅	—	2	1	no	—
166	C₂×Dic₅	—	2	1	no	—
167	C₂×Dic₅	—	2	1	no	—
168	C₂×Dic₅	—	5	2	no	—
169	C₂×Dic₅	—	5	2	no	—
170	C₂×Dic₅	—	5	2	no	—
171	C₂×Dic₅	—	5	2	no	—
172	C₂×Dic₅	—	5	2	no	—
173	C₂×Dic₅	—	5	2	no	—
174	C₂×Dic₅	—	5	2	no	—
175	C₂×Dic₅	—	5	2	no	—
176	C₂×Dic₅	—	10	4	no	—
177	C₂×Dic₅	—	10	4	no	—
178	C₂×Dic₅	—	10	4	no	—
179	C₂×Dic₅	—	10	4	no	—
180	C₂×Dic₅	—	10	8	no	—
181	C₂×Dic₅	—	10	4	no	—
182	C₂×Dic₅	—	10	4	no	—
183	C₂×Dic₅	—	10	1	no	—
184	C₂×Dic₅	—	10	1	no	—
185	C₂×Dic₅	—	10	1	no	—
186	C₂×Dic₅	—	10	1	no	—
187	C₂×Dic₅	C₄×D₅	1	2	yes	—
188	C₂×Dic₅	C₄×D₅	1	2	yes	complemented
189	C₂×Dic₅	C₂×Dic₅	1	2	yes	complemented
190	C₄×D₅	—	10	8	no	—
191	C₂³×C₄	—	25	1	no	—
192	C₂×C₂²⋊C₄	—	5	1	no	—
193	C₂×C₂²⋊C₄	—	25	1	no	—
194	C₂×C₄²	—	25	1	no	—
195	C₂×C₄²	—	25	1	no	—
196	#95	—	5	1	no	—
197	#95	—	25	1	no	—
198	C₅²	C₄×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
199	C₂²×C₅	—	2	2	no	—
200	C₂²×C₅	—	2	2	no	—
201	C₂²×C₅	—	2	2	no	—
202	C₂²×C₅	—	2	2	no	—
203	C₂²×C₅	—	2	2	no	—
204	C₂²×C₅	—	2	2	no	—
205	C₂²×C₅	—	2	2	no	—
206	C₂²×C₅	—	5	4	no	—
207	C₂²×C₅	—	10	2	no	—
208	C₂²×C₅	—	10	2	no	—
209	C₂²×C₅	—	10	2	no	—
210	C₂²×C₅	—	10	2	no	—
211	C₂²×C₅	—	10	2	no	—
212	C₂²×C₅	—	10	2	no	—
213	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
214	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
215	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
216	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
217	C₂²×C₅	C₂×C₄×D₅	1	1	yes	characteristic
218	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
219	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
220	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
221	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
222	C₂²×C₅	#711	1	1	yes	characteristic
223	C₂²×C₅	#711	1	1	yes	characteristic
224	C₂²×C₅	#711	1	1	yes	characteristic
225	C₂²×C₅	#713	1	1	yes	characteristic
226	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
227	C₂×D₅	—	2	2	no	—
228	C₂×D₅	—	2	2	no	—
229	C₂×D₅	—	2	2	no	—
230	C₂×D₅	—	2	2	no	—
231	C₂×D₅	—	2	2	no	—
232	C₂×D₅	—	2	2	no	—
233	C₂×D₅	C₄×Dic₅	1	4	yes	complemented
234	C₄×C₅	—	2	8	no	—
235	C₄×C₅	—	2	2	no	—
236	C₄×C₅	—	2	2	no	—
237	C₄×C₅	—	10	2	no	—
238	C₄×C₅	—	10	2	no	—
239	C₄×C₅	—	10	2	no	—
240	Dic₅	—	2	2	no	—
241	Dic₅	—	2	2	no	—
242	Dic₅	—	2	2	no	—
243	Dic₅	—	5	4	no	—
244	Dic₅	—	5	4	no	—
245	Dic₅	—	10	8	no	—
246	Dic₅	—	10	8	no	—
247	Dic₅	—	10	2	no	—
248	Dic₅	—	10	2	no	—
249	C₂⁴	—	5	1	no	—
250	C₂²×C₄	—	5	1	no	—
251	C₂²×C₄	—	5	1	no	—
252	C₂²×C₄	—	5	1	no	—
253	C₂²×C₄	—	25	4	no	—
254	C₂²×C₄	—	25	1	no	—
255	C₂²×C₄	—	25	1	no	—
256	C₂²×C₄	—	50	2	no	—
257	C₂²×C₄	—	50	2	no	—
258	C₂²×C₄	D₅²	1	1	yes	characteristic
259	C₂²⋊C₄	—	5	4	no	—
260	C₂²⋊C₄	—	25	4	no	—
261	C₄²	—	50	2	no	—
262	C₄²	—	50	2	no	—
263	C₂×C₅	—	2	2	no	—
264	C₂×C₅	—	2	2	no	—
265	C₂×C₅	—	2	2	no	—
266	C₂×C₅	—	2	2	no	—
267	C₂×C₅	—	2	2	no	—
268	C₂×C₅	—	2	2	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	—	10	4	no	—
271	C₂×C₅	C₄²×D₅	1	1	yes	characteristic
272	C₂×C₅	#4023	1	1	yes	characteristic
273	C₂×C₅	D₅×C₂²⋊C₄	1	1	yes	characteristic
274	C₂×C₅	D₅×C₂²⋊C₄	1	1	yes	characteristic
275	C₂×C₅	#4027	1	1	yes	characteristic
276	C₂×C₅	#4027	1	1	yes	characteristic
277	C₂×C₅	#4027	1	1	yes	characteristic
278	C₂×C₅	#4027	1	1	yes	characteristic
279	C₂×C₅	#4038	1	1	yes	characteristic
280	C₂×C₅	#4039	1	1	yes	characteristic
281	C₂×C₅	#4039	1	1	yes	characteristic
282	C₂×C₅	C₂×C₄×Dic₅	1	1	yes	characteristic
283	C₂×C₅	D₄×Dic₅	1	1	yes	characteristic
284	C₂×C₅	D₄×Dic₅	1	1	yes	characteristic
285	D₅	—	2	4	no	—
286	C₂³	—	5	2	no	—
287	C₂³	—	5	2	no	—
288	C₂³	—	5	2	no	—
289	C₂³	—	10	1	no	—
290	C₂³	—	10	1	no	—
291	C₂³	—	10	1	no	—
292	C₂³	—	10	1	no	—
293	C₂³	C₂×D₅²	1	1	yes	characteristic
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	5	2	no	—
297	C₂×C₄	—	5	2	no	—
298	C₂×C₄	—	5	2	no	—
299	C₂×C₄	—	10	1	no	—
300	C₂×C₄	—	10	1	no	—
301	C₂×C₄	—	10	1	no	—
302	C₂×C₄	—	10	1	no	—
303	C₂×C₄	—	10	1	no	—
304	C₂×C₄	—	10	1	no	—
305	C₂×C₄	—	25	2	no	—
306	C₂×C₄	—	25	2	no	—
307	C₂×C₄	—	25	2	no	—
308	C₂×C₄	—	25	2	no	—
309	C₂×C₄	—	50	8	no	—
310	C₂×C₄	—	50	1	no	—
311	C₂×C₄	—	50	1	no	—
312	C₂×C₄	D₅×Dic₅	1	2	yes	complemented
313	C₅	—	2	2	no	—
314	C₅	#64955	1	1	yes	characteristic, complemented
315	C₅	C₂²⋊C₄×Dic₅	1	1	yes	characteristic, complemented
316	C₂²	—	5	4	no	—
317	C₂²	—	10	2	no	—
318	C₂²	—	10	2	no	—
319	C₂²	—	10	2	no	—
320	C₂²	—	10	2	no	—
321	C₂²	—	10	2	no	—
322	C₂²	—	10	2	no	—
323	C₂²	#87871	1	1	yes	characteristic
324	C₂²	C₄×D₅²	1	1	yes	characteristic
325	C₂²	C₄×D₅²	1	1	yes	characteristic
326	C₂²	#87877	1	1	yes	characteristic
327	C₂²	C₂×D₅×Dic₅	1	1	yes	characteristic
328	C₂²	#87880	1	1	yes	characteristic
329	C₂²	#87886	1	1	yes	characteristic
330	C₄	—	2	2	no	—
331	C₄	—	10	2	no	—
332	C₄	—	10	2	no	—
333	C₄	—	10	2	no	—
334	C₄	—	25	4	no	—
335	C₄	—	50	2	no	—
336	C₂	—	10	4	no	—
337	C₂	C₄×D₅×Dic₅	1	1	yes	characteristic
338	C₂	#11716996	1	1	yes	characteristic
339	C₂	#11716999	1	1	yes	characteristic
340	C₂	#11717017	1	1	yes	characteristic
341	C₂	#11717020	1	1	yes	characteristic
342	C₂	#11717039	1	1	yes	characteristic
343	C₂	#11717091	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂×D₅²
commutator	C₂×C₅²	C₂×C₄²
Fitting	C₂²×C₄×C₅²	C₂²
socle, cosocle	C₂³×C₅²	C₂³
prime radical	C₂³×C₅×D₅	C₂²
hypercenter	C₂²×C₄	D₅²
hypocenter	C₅²	C₄×C₂²⋊C₄
LCS	G ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×#1039	C₂	semidirect
5. C₂×Dic₅²	C₂	semidirect
6. C₅×#3963	C₂	semidirect
7. #11716620	C₂	semidirect
17. C₂×D₅×Dic₅	C₄	semidirect
19. C₅×#685	C₄	semidirect
20. #87778	C₄	semidirect
34. C₂×C₄×C₅²	C₂×C₄	semidirect
38. C₂×#1039	C₂×C₄	semidirect
48. C₂×C₅×Dic₅	C₂×C₄	semidirect
49. C₂×C₅×Dic₅	C₂×C₄	semidirect
65. #3963	D₅	semidirect
79. C₂×C₅×D₅	C₄²	semidirect
82. #1039	C₂²⋊C₄	semidirect
112. #685	Dic₅	semidirect
160. C₂×C₄×C₅	C₄×D₅	semidirect
161. C₂×C₄×C₅	C₂×Dic₅	semidirect
188. C₂×Dic₅	C₄×D₅	semidirect
189. C₂×Dic₅	C₂×Dic₅	semidirect
198. C₅²	C₄×C₂²⋊C₄	semidirect
233. C₂×D₅	C₄×Dic₅	semidirect
312. C₂×C₄	D₅×Dic₅	semidirect
314. C₅	#64955	semidirect
315. C₅	C₂²⋊C₄×Dic₅	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₅×Dic₅	2
C₂×C₄×#1039	2
C₂×C₅×#685	2
C₂×#87778	2
C₂×Dic₅²	2
C₅×#3963	2
#11716620	2
C₂²⋊C₄×Dic₅	5
#64955	5