Subgroups — #13105094

Order 1600.

744 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#87870	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717195	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×C₅×D₂₀	C₂	1	1	yes	maximal, characteristic, complemented
6	#11717040	C₂	1	2	yes	maximal, complemented
7	D₅×#685	C₂	1	2	yes	maximal, complemented
8	#11717027	C₂	1	1	yes	maximal, characteristic, complemented
9	#11717024	C₂	1	2	yes	maximal, complemented
10	#11717011	C₂	1	2	yes	maximal, complemented
11	#11716975	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₅²	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
14	C₂×#6045	C₂²	1	1	yes	characteristic, complemented
15	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
16	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₅×D₅	C₂²	1	1	yes	characteristic, complemented
19	C₂×C₄×C₅×D₅	C₂²	1	2	yes	complemented
20	C₂×#6038	C₂²	1	2	yes	—
21	C₂×#6038	C₂²	1	2	yes	complemented
22	C₂×#6037	C₂²	1	2	yes	complemented
23	C₂×D₅×Dic₅	C₂²	1	2	yes	complemented
24	D₅×D₂₀	—	2	4	no	—
25	#87870	—	2	4	no	—
26	C₄²×C₅²	C₂²	1	1	yes	characteristic, complemented
27	#87808	C₂²	1	2	yes	complemented
28	#87808	C₂²	1	2	yes	complemented
29	C₅×#685	C₂²	1	1	yes	characteristic
30	C₅×#685	C₂²	1	2	yes	complemented
31	C₅×#685	C₂²	1	1	yes	characteristic, complemented
32	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
33	C₅×C₄⋊Dic₅	C₂²	1	1	yes	characteristic, complemented
34	#87780	C₂²	1	2	yes	complemented
35	#87779	C₂²	1	2	yes	complemented
36	#87778	C₂²	1	2	yes	—
37	#65888	—	5	1	no	maximal
38	#65884	—	5	1	no	maximal
39	C₂²×C₅⋊D₅	C₂³	1	2	yes	—
40	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
41	C₂²×C₅×D₅	C₂³	1	2	yes	—
42	C₂²×C₅×D₅	C₂³	1	1	yes	characteristic
43	C₂×D₅²	—	2	4	no	—
44	C₂×D₅²	—	2	4	no	—
45	C₂×D₅²	—	4	2	no	—
46	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
47	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
49	C₂×#1039	C₂³	1	2	yes	—
50	C₄⋊(C₅⋊D₅)	—	2	2	no	—
51	#6045	—	2	2	no	—
52	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
53	C₂×C₅×Dic₅	C₂³	1	1	yes	characteristic
54	C₂×C₅×Dic₅	C₂³	1	2	yes	—
55	C₅×D₂₀	—	2	2	no	—
56	C₅×D₂₀	D₄	1	4	yes	complemented
57	C₄×C₅×D₅	—	2	2	no	—
58	C₄×C₅×D₅	—	2	2	no	—
59	C₄×C₅×D₅	—	4	2	no	—
60	#6038	—	2	4	no	—
61	#6038	—	4	2	no	—
62	#6037	—	2	4	no	—
63	D₅×Dic₅	—	2	4	no	—
64	C₂×#711	—	5	1	no	—
65	C₂×D₄×D₅	—	5	1	no	—
66	C₂×#709	—	5	1	no	—
67	C₂²×D₂₀	—	5	1	no	—
68	C₄×C₅×D₄	—	5	1	no	—
69	C₄×C₅×D₄	—	5	1	no	—
70	#4083	—	5	2	no	—
71	#4076	—	5	2	no	—
72	#4076	—	5	1	no	—
73	C₂×#685	—	5	2	no	—
74	#4070	—	5	1	no	—
75	#4043	—	5	1	no	—
76	#4041	—	5	1	no	—
77	#4040	—	5	2	no	—
78	#4032	—	5	2	no	—
79	#4030	—	5	2	no	—
80	#4028	—	5	2	no	—
81	D₅×C₂²⋊C₄	—	5	2	no	—
82	#4021	—	5	1	no	—
83	#4021	—	5	1	no	—
84	#4021	—	10	2	no	—
85	C₄×D₂₀	D₅	1	1	yes	characteristic, complemented
86	C₄×D₂₀	D₅	1	1	yes	characteristic, complemented
87	C₂²×C₅²	C₂⁴	1	1	yes	characteristic
88	C₂×C₅⋊D₅	—	2	2	no	—
89	C₂×C₅⋊D₅	—	2	2	no	—
90	C₂×C₅⋊D₅	—	2	2	no	—
91	C₂×C₅×D₅	—	2	1	no	—
92	C₂×C₅×D₅	—	2	1	no	—
93	C₂×C₅×D₅	—	2	2	no	—
94	C₂×C₅×D₅	—	2	1	no	—
95	C₂×C₅×D₅	—	2	2	no	—
96	C₂×C₅×D₅	—	2	1	no	—
97	C₂×C₅×D₅	—	2	1	no	—
98	C₂×C₅×D₅	C₂×D₄	1	4	yes	—
99	C₂×C₅×D₅	#41	1	2	yes	complemented
100	D₅²	—	4	8	no	—
101	C₄×C₅²	—	2	1	no	—
102	C₄×C₅²	—	2	1	no	—
103	C₄×C₅²	C₂×D₄	1	2	yes	—
104	#1039	—	2	2	no	—
105	C₅×Dic₅	—	2	1	no	—
106	C₅×Dic₅	—	2	1	no	—
107	C₅×Dic₅	—	2	2	no	—
108	C₂³×D₅	—	5	2	no	—
109	C₂³×D₅	—	5	2	no	—
110	C₂×C₅×D₄	—	5	1	no	—
111	C₂×C₅×D₄	—	5	1	no	—
112	C₂²×C₄×C₅	—	5	1	no	—
113	C₂²×C₄×C₅	—	5	1	no	—
114	C₂²×C₄×C₅	—	5	2	no	—
115	C₂×C₂²⋊D₅	—	5	2	no	—
116	C₂×C₂²⋊D₅	—	5	2	no	—
117	C₂×C₂²⋊D₅	—	5	2	no	—
118	C₂×C₂²⋊D₅	—	5	2	no	—
119	C₂²×Dic₅	—	5	2	no	—
120	#711	—	10	4	no	—
121	D₄×D₅	—	10	4	no	—
122	#709	—	10	4	no	—
123	C₂×D₂₀	—	5	2	no	—
124	C₂×D₂₀	—	5	1	no	—
125	C₂×D₂₀	—	5	1	no	—
126	C₂×D₂₀	—	5	2	no	—
127	C₂×D₂₀	—	10	2	no	—
128	C₂×D₂₀	—	10	4	no	—
129	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
130	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
131	C₂×C₄×D₅	—	5	2	no	—
132	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic
133	C₂×C₄×D₅	C₂×D₅	1	1	yes	characteristic, complemented
134	C₂×C₄×D₅	C₂×D₅	1	2	yes	complemented
135	C₂×Dic₁₀	—	5	1	no	—
136	C₂×Dic₁₀	—	5	1	no	—
137	C₂×Dic₁₀	—	10	2	no	—
138	C₅×C₄⋊C₄	—	5	1	no	—
139	C₅×C₄⋊C₄	—	5	1	no	—
140	C₅×C₂²⋊C₄	—	5	1	no	—
141	C₅×C₂²⋊C₄	—	5	2	no	—
142	C₅×C₂²⋊C₄	—	5	1	no	—
143	C₄²×C₅	—	2	2	no	—
144	C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
145	C₄²×C₅	C₂×D₅	1	1	yes	characteristic, complemented
146	C₂²⋊Dic₅	—	5	2	no	—
147	#685	—	5	2	no	—
148	#685	—	5	2	no	—
149	#685	—	5	2	no	—
150	#685	—	5	2	no	—
151	#685	—	5	2	no	—
152	#685	—	5	2	no	—
153	#685	—	5	2	no	—
154	#685	—	10	4	no	—
155	#685	—	10	4	no	—
156	#685	C₂×D₅	1	1	yes	characteristic
157	#685	C₂×D₅	1	2	yes	complemented
158	#685	C₂×D₅	1	1	yes	characteristic, complemented
159	C₄⋊Dic₅	—	5	2	no	—
160	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic, complemented
161	C₄⋊Dic₅	C₂×D₅	1	1	yes	characteristic, complemented
162	#683	—	5	2	no	—
163	#546	—	25	1	no	Sylow
164	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₅²	C₂×#41	1	1	yes	characteristic
166	C₂×C₅²	H(5,𝔽₂)	1	1	yes	characteristic
167	C₅⋊D₅	—	4	2	no	—
168	C₅×D₅	—	2	2	no	—
169	C₅×D₅	—	2	4	no	—
170	C₅×D₅	—	4	1	no	—
171	C₂³×C₅	—	5	1	no	—
172	C₂³×C₅	—	5	2	no	—
173	C₂³×C₅	—	5	1	no	—
174	C₂²×D₅	—	5	2	no	—
175	C₂²×D₅	—	5	2	no	—
176	C₂²×D₅	—	10	4	no	—
177	C₂²×D₅	—	10	4	no	—
178	C₂²×D₅	—	10	4	no	—
179	C₂²×D₅	—	10	4	no	—
180	C₂²×D₅	—	10	4	no	—
181	C₂²×D₅	—	20	2	no	—
182	C₂²×D₅	—	20	2	no	—
183	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
184	C₂²×D₅	C₂²×D₅	1	2	yes	—
185	C₂²×D₅	C₂²×D₅	1	1	yes	characteristic
186	C₅×D₄	—	5	4	no	—
187	C₅×D₄	—	10	2	no	—
188	C₂×C₄×C₅	—	2	2	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	—	5	1	no	—
192	C₂×C₄×C₅	—	5	2	no	—
193	C₂×C₄×C₅	—	5	1	no	—
194	C₂×C₄×C₅	—	10	2	no	—
195	C₂×C₄×C₅	—	10	2	no	—
196	C₂×C₄×C₅	—	20	2	no	—
197	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
199	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
200	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
201	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
202	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
203	C₂²⋊D₅	—	10	4	no	—
204	C₂²⋊D₅	—	10	4	no	—
205	C₂²⋊D₅	—	10	4	no	—
206	C₂²⋊D₅	—	20	2	no	—
207	C₂×Dic₅	—	5	2	no	—
208	C₂×Dic₅	—	5	2	no	—
209	C₂×Dic₅	—	10	4	no	—
210	C₂×Dic₅	—	10	4	no	—
211	C₂×Dic₅	C₂²×D₅	1	2	yes	—
212	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
213	C₂×Dic₅	C₂²×D₅	1	1	yes	characteristic
214	D₂₀	—	2	2	no	—
215	D₂₀	—	10	4	no	—
216	D₂₀	—	10	2	no	—
217	D₂₀	—	10	2	no	—
218	D₂₀	—	20	4	no	—
219	D₂₀	—	20	2	no	—
220	D₂₀	D₂₀	1	4	yes	complemented
221	C₄×D₅	—	2	2	no	—
222	C₄×D₅	—	2	2	no	—
223	C₄×D₅	—	4	2	no	—
224	C₄×D₅	—	10	4	no	—
225	Dic₁₀	—	10	2	no	—
226	Dic₁₀	—	10	2	no	—
227	Dic₁₀	—	20	4	no	—
228	C₂×#41	—	25	1	no	—
229	C₂²×D₄	—	25	1	no	—
230	#124	—	25	1	no	—
231	#123	—	25	2	no	—
232	C₂²⋊Q₈	—	25	1	no	—
233	#121	—	25	2	no	—
234	#121	—	25	1	no	—
235	#120	—	25	2	no	—
236	C₄×D₄	—	5	1	no	—
237	C₄×D₄	—	5	1	no	—
238	C₂×C₂²⋊C₄	—	25	2	no	—
239	C₅²	#546	1	1	yes	characteristic, complemented, Sylow
240	C₂²×C₅	—	2	2	no	—
241	C₂²×C₅	—	5	4	no	—
242	C₂²×C₅	—	5	2	no	—
243	C₂²×C₅	—	10	1	no	—
244	C₂²×C₅	—	10	2	no	—
245	C₂²×C₅	—	10	1	no	—
246	C₂²×C₅	—	10	2	no	—
247	C₂²×C₅	—	10	1	no	—
248	C₂²×C₅	—	10	1	no	—
249	C₂²×C₅	—	10	1	no	—
250	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
251	C₂²×C₅	C₂³×D₅	1	1	yes	characteristic
252	C₂×D₅	—	2	1	no	—
253	C₂×D₅	—	2	2	no	—
254	C₂×D₅	—	2	2	no	—
255	C₂×D₅	—	2	1	no	—
256	C₂×D₅	—	2	1	no	—
257	C₂×D₅	—	2	1	no	—
258	C₂×D₅	—	2	1	no	—
259	C₂×D₅	—	10	2	no	—
260	C₂×D₅	—	10	2	no	—
261	C₂×D₅	—	10	2	no	—
262	C₂×D₅	—	10	2	no	—
263	C₂×D₅	—	10	2	no	—
264	C₂×D₅	—	10	2	no	—
265	C₂×D₅	—	20	4	no	—
266	C₂×D₅	—	20	4	no	—
267	C₂×D₅	—	20	4	no	—
268	C₂×D₅	—	20	8	no	—
269	C₂×D₅	—	20	8	no	—
270	C₂×D₅	C₂×D₂₀	1	4	yes	—
271	C₂×D₅	#709	1	2	yes	complemented
272	C₄×C₅	—	2	1	no	—
273	C₄×C₅	—	2	1	no	—
274	C₄×C₅	—	2	1	no	—
275	C₄×C₅	—	2	1	no	—
276	C₄×C₅	—	2	4	no	—
277	C₄×C₅	—	4	2	no	—
278	C₄×C₅	—	4	2	no	—
279	C₄×C₅	—	10	1	no	—
280	C₄×C₅	—	10	1	no	—
281	C₄×C₅	—	10	2	no	—
282	C₄×C₅	C₂×D₂₀	1	2	yes	—
283	C₄×C₅	D₄×D₅	1	2	yes	—
284	Dic₅	—	2	1	no	—
285	Dic₅	—	2	1	no	—
286	Dic₅	—	2	2	no	—
287	Dic₅	—	10	2	no	—
288	Dic₅	—	10	2	no	—
289	Dic₅	—	20	4	no	—
290	C₂⁴	—	25	2	no	—
291	#41	—	50	4	no	—
292	C₂×Q₈	—	25	1	no	—
293	C₂×D₄	—	5	1	no	—
294	C₂×D₄	—	5	1	no	—
295	C₂×D₄	—	25	2	no	—
296	C₂×D₄	—	25	2	no	—
297	C₂×D₄	—	25	2	no	—
298	C₂×D₄	—	25	1	no	—
299	C₂×D₄	—	50	4	no	—
300	C₂²×C₄	—	5	1	no	—
301	C₂²×C₄	—	5	1	no	—
302	C₂²×C₄	—	5	2	no	—
303	C₂²×C₄	—	25	2	no	—
304	C₄⋊C₄	—	5	1	no	—
305	C₄⋊C₄	—	5	1	no	—
306	C₄⋊C₄	—	25	2	no	—
307	C₂²⋊C₄	—	5	1	no	—
308	C₂²⋊C₄	—	5	2	no	—
309	C₂²⋊C₄	—	5	1	no	—
310	C₂²⋊C₄	—	25	2	no	—
311	C₂²⋊C₄	—	25	2	no	—
312	C₂²⋊C₄	—	25	2	no	—
313	C₂²⋊C₄	—	25	2	no	—
314	C₄²	D₅²	1	1	yes	characteristic, complemented
315	C₂×C₅	—	2	2	no	—
316	C₂×C₅	—	2	2	no	—
317	C₂×C₅	—	2	2	no	—
318	C₂×C₅	—	10	4	no	—
319	C₂×C₅	—	10	2	no	—
320	C₂×C₅	—	20	1	no	—
321	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
322	C₂×C₅	C₂×#709	1	1	yes	characteristic
323	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
324	C₂×C₅	#4144	1	1	yes	characteristic
325	C₂×C₅	D₅×#41	1	1	yes	characteristic
326	C₂×C₅	#4149	1	1	yes	characteristic
327	D₅	—	2	2	no	—
328	D₅	—	2	4	no	—
329	D₅	—	4	1	no	—
330	D₅	—	20	2	no	—
331	D₅	—	20	2	no	—
332	D₅	—	40	4	no	—
333	C₂³	—	5	1	no	—
334	C₂³	—	5	2	no	—
335	C₂³	—	5	1	no	—
336	C₂³	—	25	2	no	—
337	C₂³	—	50	4	no	—
338	C₂³	—	50	4	no	—
339	C₂³	—	100	2	no	—
340	Q₈	—	50	2	no	—
341	D₄	—	5	4	no	—
342	D₄	—	10	2	no	—
343	D₄	—	50	4	no	—
344	D₄	—	50	4	no	—
345	D₄	—	50	2	no	—
346	D₄	—	100	2	no	—
347	C₂×C₄	—	5	2	no	—
348	C₂×C₄	—	5	1	no	—
349	C₂×C₄	—	5	1	no	—
350	C₂×C₄	—	10	2	no	—
351	C₂×C₄	—	10	2	no	—
352	C₂×C₄	—	20	2	no	—
353	C₂×C₄	—	25	2	no	—
354	C₂×C₄	—	50	4	no	—
355	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
356	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
357	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
358	C₅	—	2	2	no	—
359	C₅	#65884	1	1	yes	characteristic, complemented
360	C₅	#65888	1	1	yes	characteristic, complemented
361	C₂²	—	5	4	no	—
362	C₂²	—	5	2	no	—
363	C₂²	—	10	1	no	—
364	C₂²	—	10	2	no	—
365	C₂²	—	10	2	no	—
366	C₂²	—	10	1	no	—
367	C₂²	—	10	1	no	—
368	C₂²	—	10	1	no	—
369	C₂²	—	10	1	no	—
370	C₂²	—	50	2	no	—
371	C₂²	—	50	2	no	—
372	C₂²	—	50	2	no	—
373	C₂²	—	100	8	no	—
374	C₂²	C₂²×D₅²	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	10	1	no	—
378	C₄	—	10	1	no	—
379	C₄	—	10	2	no	—
380	C₄	—	50	2	no	—
381	C₄	D₅×D₂₀	1	2	yes	—
382	C₂	—	10	2	no	—
383	C₂	—	10	4	no	—
384	C₂	—	20	1	no	—
385	C₂	—	100	2	no	—
386	C₂	C₂×D₅×D₂₀	1	1	yes	characteristic
387	C₂	D₅×#709	1	1	yes	characteristic
388	C₂	#11717472	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂²×D₅²
commutator, socle, cosocle	C₂²×C₅²	C₂⁴
Fitting	C₄²×C₅²	C₂²
hypercenter	C₄²	D₅²
hypocenter	C₅²	#546
LCS	G ▷ C₂²×C₅² ▷ C₅²
UCS	C₁ ◁ C₂² ◁ C₄²

Complements

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

N	H ≅ G/N	type
1. C₂×D₅×D₂₀	C₂	semidirect
2. C₂×#87870	C₂	semidirect
3. #11717195	C₂	semidirect
4. C₄×C₅×D₂₀	C₂	semidirect
5. C₄×C₅×D₂₀	C₂	semidirect
6. #11717040	C₂	semidirect
7. D₅×#685	C₂	semidirect
8. #11717027	C₂	semidirect
9. #11717024	C₂	semidirect
10. #11717011	C₂	semidirect
11. #11716975	C₂	semidirect
14. C₂×#6045	C₂²	semidirect
18. C₂×C₄×C₅×D₅	C₂²	semidirect
19. C₂×C₄×C₅×D₅	C₂²	semidirect
21. C₂×#6038	C₂²	semidirect
22. C₂×#6037	C₂²	semidirect
23. C₂×D₅×Dic₅	C₂²	semidirect
26. C₄²×C₅²	C₂²	semidirect
27. #87808	C₂²	semidirect
28. #87808	C₂²	semidirect
30. C₅×#685	C₂²	semidirect
31. C₅×#685	C₂²	semidirect
32. C₅×C₄⋊Dic₅	C₂²	semidirect
33. C₅×C₄⋊Dic₅	C₂²	semidirect
34. #87780	C₂²	semidirect
35. #87779	C₂²	semidirect
56. C₅×D₂₀	D₄	semidirect
85. C₄×D₂₀	D₅	semidirect
86. C₄×D₂₀	D₅	semidirect
99. C₂×C₅×D₅	#41	semidirect
133. C₂×C₄×D₅	C₂×D₅	semidirect
134. C₂×C₄×D₅	C₂×D₅	semidirect
144. C₄²×C₅	C₂×D₅	semidirect
145. C₄²×C₅	C₂×D₅	semidirect
157. #685	C₂×D₅	semidirect
158. #685	C₂×D₅	semidirect
160. C₄⋊Dic₅	C₂×D₅	semidirect
161. C₄⋊Dic₅	C₂×D₅	semidirect
220. D₂₀	D₂₀	semidirect
239. C₅²	#546	semidirect
271. C₂×D₅	#709	semidirect
314. C₄²	D₅²	semidirect
359. C₅	#65884	semidirect
360. C₅	#65888	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₅×D₂₀	2
C₂×#87870	2
#11717195	2
C₄×C₅×D₂₀	2
#11717040	2
D₅×#685	2
#11717027	2
#11717024	2
#11717011	2
#11716975	2
#65888	5
#65884	5