Subgroups — #13105770

Order 1600.

584 conjugacy classes of subgroups.

The subgroup lattice has 389 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717473	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717463	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#87800	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4050	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅×#4047	C₂	1	1	yes	maximal, characteristic, complemented
6	#11716949	C₂	1	1	yes	maximal, characteristic, complemented
7	#11716943	C₂	1	1	yes	maximal, characteristic, complemented
8	#11716924	C₂	1	2	yes	maximal, complemented
9	#11716919	C₂	1	2	yes	maximal, complemented
10	#11716913	C₂	1	2	yes	maximal, complemented
11	D₅×#677	C₂	1	2	yes	maximal, complemented
12	C₂×C₄⋊(C₅⋊D₅)	C₂²	1	1	yes	characteristic
13	C₂×#6045	C₂²	1	1	yes	characteristic, complemented
14	C₅×#709	C₂²	1	1	yes	characteristic
15	C₅×#709	C₂²	1	1	yes	characteristic
16	C₅×#709	C₂²	1	1	yes	characteristic, complemented
17	#87886	—	2	2	no	—
18	#87880	—	2	2	no	—
19	D₅×D₂₀	—	2	1	no	—
20	D₅×D₂₀	C₂²	1	2	yes	—
21	#87874	—	2	1	no	—
22	#87874	C₂²	1	2	yes	—
23	#87870	—	2	1	no	—
24	#87870	C₂²	1	2	yes	complemented
25	D₅×Dic₁₀	—	2	1	no	—
26	D₅×Dic₁₀	C₂²	1	2	yes	complemented
27	C₂×C₅²×C₈	C₂²	1	1	yes	characteristic, complemented
28	#87800	C₂²	1	2	yes	complemented
29	#87800	C₂²	1	2	yes	complemented
30	C₅×#681	C₂²	1	1	yes	characteristic, complemented
31	C₅×Dic₂₀	C₂²	1	1	yes	characteristic, complemented
32	C₅×D₄₀	C₂²	1	1	yes	characteristic, complemented
33	C₅×#677	C₂²	1	1	yes	characteristic, complemented
34	C₅×#677	C₂²	1	1	yes	characteristic, complemented
35	C₅×#676	C₂²	1	2	yes	complemented
36	C₅×C₈×D₅	C₂²	1	2	yes	complemented
37	#87776	C₂²	1	2	yes	complemented
38	#87774	C₂²	1	2	yes	complemented
39	#87773	C₂²	1	2	yes	complemented
40	#87771	C₂²	1	2	yes	complemented
41	#66104	—	5	1	no	maximal
42	#66085	—	5	1	no	maximal
43	C₂²×C₅⋊D₅	—	2	1	no	—
44	C₂×D₅²	—	2	2	no	—
45	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
46	C₂×#1039	—	2	1	no	—
47	C₄⋊(C₅⋊D₅)	—	2	1	no	—
48	C₄⋊(C₅⋊D₅)	C₂³	1	2	yes	—
49	#6045	—	2	1	no	—
50	#6045	C₂³	1	2	yes	—
51	C₅×C₂²⋊D₅	—	2	1	no	—
52	C₅×C₂²⋊D₅	—	2	1	no	—
53	C₅×C₂²⋊D₅	D₄	1	2	yes	—
54	C₅×D₂₀	D₄	1	1	yes	characteristic
55	C₅×D₂₀	C₂³	1	1	yes	characteristic
56	C₅×D₂₀	C₂³	1	1	yes	characteristic
57	C₄×C₅×D₅	—	2	1	no	—
58	C₄×C₅×D₅	—	2	1	no	—
59	C₄×C₅×D₅	C₂³	1	2	yes	—
60	C₅×Dic₁₀	D₄	1	1	yes	characteristic
61	C₅×Dic₁₀	C₂³	1	1	yes	characteristic
62	C₅×Dic₁₀	C₂³	1	1	yes	characteristic
63	#6039	—	2	2	no	—
64	#6038	—	2	2	no	—
65	#6038	—	2	2	no	—
66	#6037	—	2	2	no	—
67	#6036	—	2	2	no	—
68	D₅×Dic₅	—	2	2	no	—
69	D₅×Dic₅	—	2	2	no	—
70	C₅²×C₈	C₂³	1	1	yes	characteristic
71	C₅²×C₈	C₂³	1	1	yes	characteristic
72	C₅×#168	C₂³	1	2	yes	—
73	#4150	—	5	1	no	—
74	#4150	—	5	1	no	—
75	#4149	—	5	1	no	—
76	#4149	—	5	1	no	—
77	C₅×#135	—	5	1	no	—
78	C₅×#131	—	5	1	no	—
79	#4097	—	5	1	no	—
80	#4095	—	5	1	no	—
81	#4064	—	5	2	no	—
82	#4062	—	5	2	no	—
83	D₅×SD₁₆	—	5	2	no	—
84	#4057	—	5	2	no	—
85	#4055	—	5	1	no	—
86	#4055	—	5	2	no	—
87	#4054	—	5	1	no	—
88	#4054	—	5	2	no	—
89	#4050	—	5	2	no	—
90	#4050	D₅	1	1	yes	characteristic, complemented
91	C₂×#677	—	5	1	no	—
92	C₂×#677	—	5	1	no	—
93	C₂×#677	—	5	2	no	—
94	C₂×#677	—	10	2	no	—
95	#4047	D₅	1	1	yes	characteristic, complemented
96	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅⋊D₅	—	2	2	no	—
98	C₂×C₅⋊D₅	—	8	1	no	—
99	C₂×C₅×D₅	—	2	1	no	—
100	C₂×C₅×D₅	—	2	1	no	—
101	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
102	D₅²	—	8	2	no	—
103	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
104	C₄×C₅²	C₂⁴	1	1	yes	characteristic
105	#1039	—	2	2	no	—
106	C₅×Dic₅	—	2	1	no	—
107	C₅×Dic₅	—	2	1	no	—
108	C₅×Dic₅	C₂×D₄	1	2	yes	—
109	C₅×#41	—	5	1	no	—
110	C₅×#41	—	5	1	no	—
111	C₅×#41	—	5	1	no	—
112	#713	—	5	2	no	—
113	#713	—	10	1	no	—
114	Q₈×D₅	—	5	2	no	—
115	Q₈×D₅	—	10	1	no	—
116	#711	—	5	2	no	—
117	#711	—	10	1	no	—
118	#711	—	10	2	no	—
119	#711	—	10	2	no	—
120	D₄×D₅	—	5	2	no	—
121	D₄×D₅	—	10	1	no	—
122	D₄×D₅	—	10	2	no	—
123	D₄×D₅	—	10	2	no	—
124	#709	—	5	2	no	—
125	#709	—	5	2	no	—
126	#709	—	10	1	no	—
127	#709	—	10	1	no	—
128	#709	C₂×D₅	1	1	yes	characteristic
129	#709	C₂×D₅	1	1	yes	characteristic
130	#709	C₂×D₅	1	1	yes	characteristic, complemented
131	C₂×D₂₀	—	5	1	no	—
132	C₂×D₂₀	—	5	1	no	—
133	C₂×D₂₀	—	5	2	no	—
134	C₂×D₂₀	—	10	2	no	—
135	C₂×D₂₀	—	10	1	no	—
136	C₂×Dic₁₀	—	5	1	no	—
137	C₂×Dic₁₀	—	5	1	no	—
138	C₂×Dic₁₀	—	5	2	no	—
139	C₂×Dic₁₀	—	10	2	no	—
140	C₂×Dic₁₀	—	10	1	no	—
141	C₅×Q₁₆	—	5	1	no	—
142	C₅×SD₁₆	—	5	1	no	—
143	C₅×SD₁₆	—	5	1	no	—
144	C₅×D₈	—	5	1	no	—
145	C₅×M₁₆	—	5	1	no	—
146	C₅×M₁₆	—	5	2	no	—
147	C₂×C₅×C₈	—	2	2	no	—
148	C₂×C₅×C₈	—	5	2	no	—
149	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
150	C₂×C₅×C₈	C₂×D₅	1	1	yes	characteristic, complemented
151	#689	—	5	2	no	—
152	#688	—	5	2	no	—
153	#687	—	5	2	no	—
154	#686	—	5	2	no	—
155	#681	C₂×D₅	1	1	yes	characteristic, complemented
156	Dic₂₀	—	5	2	no	—
157	Dic₂₀	C₂×D₅	1	1	yes	characteristic, complemented
158	D₄₀	—	5	2	no	—
159	D₄₀	C₂×D₅	1	1	yes	characteristic, complemented
160	#677	—	5	2	no	—
161	#677	—	5	2	no	—
162	#677	—	5	2	no	—
163	#677	—	5	2	no	—
164	#677	—	5	2	no	—
165	#677	—	5	2	no	—
166	#677	—	10	4	no	—
167	#677	—	10	4	no	—
168	#677	C₂×D₅	1	1	yes	characteristic, complemented
169	#677	C₂×D₅	1	1	yes	characteristic, complemented
170	#676	C₂×D₅	1	2	yes	complemented
171	C₈×D₅	C₂×D₅	1	2	yes	complemented
172	#577	—	25	1	no	Sylow
173	C₂×C₅²	—	2	1	no	—
174	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
175	C₅⋊D₅	—	4	2	no	—
176	C₅×D₅	—	2	2	no	—
177	C₅×D₅	—	4	1	no	—
178	C₅×D₅	—	4	1	no	—
179	C₂²×D₅	—	10	2	no	—
180	C₂²×D₅	—	10	1	no	—
181	C₂²×D₅	—	10	1	no	—
182	C₂²×D₅	—	10	2	no	—
183	C₂²×D₅	—	20	2	no	—
184	C₅×Q₈	—	5	1	no	—
185	C₅×Q₈	—	5	1	no	—
186	C₅×Q₈	—	5	1	no	—
187	C₅×D₄	—	5	2	no	—
188	C₅×D₄	—	5	1	no	—
189	C₅×D₄	—	5	1	no	—
190	C₅×D₄	—	5	1	no	—
191	C₅×D₄	—	10	1	no	—
192	C₅×D₄	—	10	1	no	—
193	C₂×C₄×C₅	—	2	2	no	—
194	C₂×C₄×C₅	—	5	2	no	—
195	C₂×C₄×C₅	—	10	1	no	—
196	C₂×C₄×C₅	—	10	1	no	—
197	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
198	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
199	C₂²⋊D₅	—	2	1	no	—
200	C₂²⋊D₅	—	2	1	no	—
201	C₂²⋊D₅	—	10	2	no	—
202	C₂²⋊D₅	—	10	2	no	—
203	C₂²⋊D₅	—	10	2	no	—
204	C₂²⋊D₅	—	10	2	no	—
205	C₂²⋊D₅	D₂₀	1	2	yes	—
206	C₂×Dic₅	—	10	2	no	—
207	C₂×Dic₅	—	10	1	no	—
208	C₂×Dic₅	—	10	1	no	—
209	C₂×Dic₅	—	10	2	no	—
210	C₂×Dic₅	—	20	2	no	—
211	D₂₀	—	5	2	no	—
212	D₂₀	—	5	2	no	—
213	D₂₀	—	10	4	no	—
214	D₂₀	—	10	2	no	—
215	D₂₀	—	10	2	no	—
216	D₂₀	—	10	1	no	—
217	D₂₀	—	10	1	no	—
218	D₂₀	—	20	2	no	—
219	D₂₀	D₂₀	1	1	yes	characteristic
220	D₂₀	C₂²×D₅	1	1	yes	characteristic
221	D₂₀	C₂²×D₅	1	1	yes	characteristic
222	C₄×D₅	—	2	1	no	—
223	C₄×D₅	—	2	1	no	—
224	C₄×D₅	—	10	2	no	—
225	C₄×D₅	—	10	2	no	—
226	C₄×D₅	—	10	2	no	—
227	C₄×D₅	—	10	2	no	—
228	C₄×D₅	C₂²×D₅	1	2	yes	—
229	Dic₁₀	—	5	2	no	—
230	Dic₁₀	—	5	2	no	—
231	Dic₁₀	—	10	4	no	—
232	Dic₁₀	—	10	2	no	—
233	Dic₁₀	—	10	1	no	—
234	Dic₁₀	—	10	1	no	—
235	Dic₁₀	—	10	2	no	—
236	Dic₁₀	—	20	2	no	—
237	Dic₁₀	D₂₀	1	1	yes	characteristic
238	Dic₁₀	C₂²×D₅	1	1	yes	characteristic
239	Dic₁₀	C₂²×D₅	1	1	yes	characteristic
240	C₅×C₈	—	2	2	no	—
241	C₅×C₈	—	2	2	no	—
242	C₅×C₈	—	5	2	no	—
243	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
244	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
245	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
246	C₅×C₈	C₂²×D₅	1	1	yes	characteristic
247	#168	C₂²×D₅	1	2	yes	—
248	2₋¹⁺⁴	—	25	1	no	—
249	H(5,𝔽₂)	—	25	1	no	—
250	#137	—	25	2	no	—
251	#137	—	25	1	no	—
252	AGL(1,ℤ/8)	—	25	2	no	—
253	AGL(1,ℤ/8)	—	25	1	no	—
254	#135	—	5	1	no	—
255	#135	—	25	2	no	—
256	C₂×SD₁₆	—	25	2	no	—
257	C₂×SD₁₆	—	25	1	no	—
258	#131	—	5	1	no	—
259	C₅²	#577	1	1	yes	characteristic, complemented, Sylow
260	C₂²×C₅	—	2	2	no	—
261	C₂²×C₅	—	5	2	no	—
262	C₂²×C₅	—	10	1	no	—
263	C₂²×C₅	—	10	1	no	—
264	C₂²×C₅	C₂×D₂₀	1	1	yes	characteristic
265	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
266	C₂×D₅	—	2	1	no	—
267	C₂×D₅	—	2	1	no	—
268	C₂×D₅	—	10	2	no	—
269	C₂×D₅	—	10	2	no	—
270	C₂×D₅	—	20	4	no	—
271	C₂×D₅	—	40	2	no	—
272	C₂×D₅	—	40	2	no	—
273	C₂×D₅	—	40	1	no	—
274	C₂×D₅	—	40	1	no	—
275	C₂×D₅	—	40	4	no	—
276	C₂×D₅	C₂×D₂₀	1	2	yes	—
277	C₄×C₅	—	2	2	no	—
278	C₄×C₅	—	2	2	no	—
279	C₄×C₅	—	5	2	no	—
280	C₄×C₅	—	10	1	no	—
281	C₄×C₅	—	10	1	no	—
282	C₄×C₅	C₂×D₂₀	1	1	yes	characteristic
283	C₄×C₅	D₄×D₅	1	1	yes	characteristic
284	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
285	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
286	Dic₅	—	2	1	no	—
287	Dic₅	—	2	1	no	—
288	Dic₅	—	10	2	no	—
289	Dic₅	—	10	2	no	—
290	Dic₅	—	20	4	no	—
291	Dic₅	C₂×D₂₀	1	2	yes	—
292	#41	—	5	1	no	—
293	#41	—	5	1	no	—
294	#41	—	5	1	no	—
295	#41	—	25	2	no	—
296	#41	—	25	2	no	—
297	#41	—	50	1	no	—
298	#41	—	50	1	no	—
299	#41	—	50	2	no	—
300	C₂×Q₈	—	25	2	no	—
301	C₂×Q₈	—	25	1	no	—
302	C₂×Q₈	—	50	1	no	—
303	C₂×D₄	—	25	2	no	—
304	C₂×D₄	—	25	1	no	—
305	C₂×D₄	—	50	1	no	—
306	C₂×D₄	—	50	2	no	—
307	Q₁₆	—	5	1	no	—
308	Q₁₆	—	25	2	no	—
309	SD₁₆	—	5	1	no	—
310	SD₁₆	—	5	1	no	—
311	SD₁₆	—	25	2	no	—
312	SD₁₆	—	25	2	no	—
313	SD₁₆	—	25	2	no	—
314	SD₁₆	—	25	2	no	—
315	D₈	—	5	1	no	—
316	D₈	—	25	2	no	—
317	M₁₆	—	5	1	no	—
318	M₁₆	—	5	2	no	—
319	C₂×C₈	—	5	2	no	—
320	C₂×C₈	D₅²	1	1	yes	characteristic, complemented
321	C₂×C₅	—	2	4	no	—
322	C₂×C₅	—	2	1	no	—
323	C₂×C₅	—	2	1	no	—
324	C₂×C₅	—	2	2	no	—
325	C₂×C₅	—	10	2	no	—
326	C₂×C₅	—	20	1	no	—
327	C₂×C₅	—	20	1	no	—
328	C₂×C₅	C₂²×D₂₀	1	1	yes	characteristic
329	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
330	D₅	—	2	2	no	—
331	D₅	—	4	1	no	—
332	D₅	—	4	1	no	—
333	D₅	—	20	2	no	—
334	D₅	—	20	2	no	—
335	D₅	—	40	4	no	—
336	C₂³	—	50	2	no	—
337	C₂³	—	50	1	no	—
338	Q₈	—	5	1	no	—
339	Q₈	—	5	1	no	—
340	Q₈	—	5	1	no	—
341	Q₈	—	25	2	no	—
342	Q₈	—	50	2	no	—
343	Q₈	—	50	1	no	—
344	D₄	—	5	2	no	—
345	D₄	—	5	1	no	—
346	D₄	—	5	1	no	—
347	D₄	—	5	1	no	—
348	D₄	—	10	1	no	—
349	D₄	—	10	1	no	—
350	D₄	—	25	2	no	—
351	D₄	—	50	2	no	—
352	D₄	—	50	2	no	—
353	D₄	—	50	2	no	—
354	D₄	—	50	1	no	—
355	C₂×C₄	—	5	2	no	—
356	C₂×C₄	—	10	1	no	—
357	C₂×C₄	—	10	1	no	—
358	C₂×C₄	—	50	2	no	—
359	C₂×C₄	—	50	2	no	—
360	C₂×C₄	—	50	2	no	—
361	C₂×C₄	—	50	1	no	—
362	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
363	C₈	—	5	2	no	—
364	C₈	C₂×D₅²	1	1	yes	characteristic
365	C₈	C₂×D₅²	1	1	yes	characteristic
366	C₅	—	2	2	no	—
367	C₅	#66085	1	1	yes	characteristic, complemented
368	C₅	#66104	1	1	yes	characteristic, complemented
369	C₂²	—	5	2	no	—
370	C₂²	—	10	1	no	—
371	C₂²	—	10	1	no	—
372	C₂²	—	50	2	no	—
373	C₂²	—	200	1	no	—
374	C₂²	—	200	2	no	—
375	C₂²	D₅×D₂₀	1	1	yes	characteristic
376	C₄	—	5	2	no	—
377	C₄	—	10	1	no	—
378	C₄	—	10	1	no	—
379	C₄	—	50	2	no	—
380	C₄	D₅×D₂₀	1	1	yes	characteristic
381	C₄	C₂²×D₅²	1	1	yes	characteristic
382	C₂	—	2	1	no	—
383	C₂	—	10	2	no	—
384	C₂	—	20	1	no	—
385	C₂	—	20	1	no	—
386	C₂	—	100	2	no	—
387	C₂	C₂×D₅×D₂₀	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₅×D₂₀
commutator, cosocle	C₄×C₅²	C₂⁴
Frattini	C₄	C₂²×D₅²
Fitting	C₂×C₅²×C₈	C₂²
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₂×C₈	D₅²
hypocenter	C₅²	#577
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #11717473	C₂	semidirect
2. #11717463	C₂	semidirect
3. C₂×#87800	C₂	semidirect
4. C₅×#4050	C₂	semidirect
5. C₅×#4047	C₂	semidirect
6. #11716949	C₂	semidirect
7. #11716943	C₂	semidirect
8. #11716924	C₂	semidirect
9. #11716919	C₂	semidirect
10. #11716913	C₂	semidirect
11. D₅×#677	C₂	semidirect
13. C₂×#6045	C₂²	semidirect
16. C₅×#709	C₂²	semidirect
24. #87870	C₂²	semidirect
26. D₅×Dic₁₀	C₂²	semidirect
27. C₂×C₅²×C₈	C₂²	semidirect
28. #87800	C₂²	semidirect
29. #87800	C₂²	semidirect
30. C₅×#681	C₂²	semidirect
31. C₅×Dic₂₀	C₂²	semidirect
32. C₅×D₄₀	C₂²	semidirect
33. C₅×#677	C₂²	semidirect
34. C₅×#677	C₂²	semidirect
35. C₅×#676	C₂²	semidirect
36. C₅×C₈×D₅	C₂²	semidirect
37. #87776	C₂²	semidirect
38. #87774	C₂²	semidirect
39. #87773	C₂²	semidirect
40. #87771	C₂²	semidirect
90. #4050	D₅	semidirect
95. #4047	D₅	semidirect
130. #709	C₂×D₅	semidirect
149. C₂×C₅×C₈	C₂×D₅	semidirect
150. C₂×C₅×C₈	C₂×D₅	semidirect
155. #681	C₂×D₅	semidirect
157. Dic₂₀	C₂×D₅	semidirect
159. D₄₀	C₂×D₅	semidirect
168. #677	C₂×D₅	semidirect
169. #677	C₂×D₅	semidirect
170. #676	C₂×D₅	semidirect
171. C₈×D₅	C₂×D₅	semidirect
259. C₅²	#577	semidirect
320. C₂×C₈	D₅²	semidirect
367. C₅	#66085	semidirect
368. C₅	#66104	semidirect

Maximal subgroups

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

subgroup	index
#11717473	2
#11717463	2
C₂×#87800	2
C₅×#4050	2
C₅×#4047	2
#11716949	2
#11716943	2
#11716924	2
#11716919	2
#11716913	2
D₅×#677	2
#66104	5
#66085	5