Subgroups — #13106457

Order 1600.

576 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	C₅×#4147	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717472	C₂	1	1	yes	maximal, characteristic, complemented
3	#11717262	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅×#688	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717065	C₂	1	2	yes	maximal, complemented
6	#11717064	C₂	1	2	yes	maximal, complemented
7	#11717057	C₂	1	2	yes	maximal, complemented
8	D₅×#688	C₂	1	2	yes	maximal, complemented
9	#11716947	C₂	1	1	yes	maximal, characteristic, complemented
10	#11716941	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716932	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅²×Q₈	C₂²	1	1	yes	characteristic, complemented
13	#87900	C₂²	1	1	yes	characteristic
14	C₅×#713	—	2	1	no	—
15	C₅×#713	C₂²	1	2	yes	complemented
16	C₅×Q₈×D₅	—	2	1	no	—
17	C₅×Q₈×D₅	C₂²	1	2	yes	complemented
18	C₅×#709	—	2	2	no	—
19	C₅×#709	C₂²	1	1	yes	characteristic
20	C₅×#709	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
22	D₅×C₂²⋊D₅	—	2	2	no	—
23	#87877	—	2	1	no	—
24	D₅×D₂₀	C₂²	1	2	yes	—
25	#87871	—	2	1	no	—
26	#87870	C₂²	1	2	yes	—
27	#87812	C₂²	1	2	yes	complemented
28	#87811	C₂²	1	2	yes	complemented
29	#87804	C₂²	1	1	yes	characteristic, complemented
30	C₅×#688	C₂²	1	2	yes	complemented
31	C₅×#688	C₂²	1	2	yes	complemented
32	C₂×C₅×#168	C₂²	1	1	yes	characteristic, complemented
33	#87776	C₂²	1	1	yes	characteristic, complemented
34	#87774	C₂²	1	1	yes	characteristic, complemented
35	#87773	C₂²	1	1	yes	characteristic, complemented
36	#87772	C₂²	1	2	yes	complemented
37	#87771	C₂²	1	1	yes	characteristic, complemented
38	#87770	C₂²	1	2	yes	complemented
39	#87768	C₂²	1	2	yes	complemented
40	D₅×#168	C₂²	1	2	yes	complemented
41	#66171	—	5	1	no	maximal
42	#66096	—	5	1	no	maximal
43	C₂²×C₅×D₅	—	2	1	no	—
44	C₂×D₅²	—	2	2	no	—
45	C₅²×Q₈	—	2	1	no	—
46	C₅²×Q₈	C₂³	1	2	yes	—
47	C₂×C₄×C₅²	—	2	1	no	—
48	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
49	C₂²⋊(C₅⋊D₅)	—	2	1	no	—
50	C₄⋊(C₅⋊D₅)	C₂³	1	1	yes	characteristic
51	C₄×C₅⋊D₅	—	2	1	no	—
52	#6045	C₂³	1	1	yes	characteristic
53	C₅×C₂²⋊D₅	—	2	1	no	—
54	C₅×C₂²⋊D₅	D₄	1	2	yes	—
55	C₅×D₂₀	—	2	2	no	—
56	C₅×D₂₀	—	2	1	no	—
57	C₅×D₂₀	D₄	1	1	yes	characteristic
58	C₅×D₂₀	C₂³	1	2	yes	—
59	C₅×D₂₀	C₂³	1	1	yes	characteristic
60	C₄×C₅×D₅	—	2	2	no	—
61	C₄×C₅×D₅	—	2	1	no	—
62	C₄×C₅×D₅	—	2	2	no	—
63	C₄×C₅×D₅	C₂³	1	2	yes	—
64	C₅×Dic₁₀	—	2	2	no	—
65	C₅×Dic₁₀	D₄	1	1	yes	characteristic
66	C₅×Dic₁₀	C₂³	1	1	yes	characteristic
67	#6038	—	2	2	no	—
68	#6037	—	2	2	no	—
69	D₅×Dic₅	—	2	2	no	—
70	#6029	C₂³	1	2	yes	—
71	C₅×#168	C₂³	1	1	yes	characteristic
72	C₅×#168	C₂³	1	1	yes	characteristic
73	C₅×2₋¹⁺⁴	—	5	1	no	—
74	#4149	—	5	1	no	—
75	#4147	D₅	1	1	yes	characteristic, complemented
76	#4144	—	5	1	no	—
77	C₂×C₅×SD₁₆	—	5	1	no	—
78	#4096	—	5	2	no	—
79	#4096	—	5	1	no	—
80	#4095	—	5	1	no	—
81	#4095	—	5	2	no	—
82	#4094	—	5	1	no	—
83	#4088	—	5	1	no	—
84	#4088	—	5	1	no	—
85	#4088	—	5	2	no	—
86	#4088	—	10	2	no	—
87	C₂×#688	—	5	2	no	—
88	C₂×#688	D₅	1	1	yes	characteristic, complemented
89	#4078	—	5	1	no	—
90	#4062	—	5	2	no	—
91	#4061	—	5	2	no	—
92	#4060	—	5	2	no	—
93	D₅×SD₁₆	—	5	2	no	—
94	#4050	—	5	1	no	—
95	#4047	—	5	1	no	—
96	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
97	C₂×C₅⋊D₅	—	2	1	no	—
98	C₂×C₅×D₅	—	2	2	no	—
99	C₂×C₅×D₅	—	2	1	no	—
100	C₂×C₅×D₅	—	8	1	no	—
101	C₂×C₅×D₅	C₂×D₄	1	2	yes	—
102	D₅²	—	8	2	no	—
103	C₄×C₅²	—	2	2	no	—
104	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
105	C₄×C₅²	C₂⁴	1	1	yes	characteristic
106	#1039	—	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	2	no	—
111	C₅×#41	—	5	1	no	—
112	C₅×#41	—	10	2	no	—
113	C₅×#41	—	10	1	no	—
114	C₂×C₅×Q₈	—	2	2	no	—
115	C₂×C₅×Q₈	—	5	2	no	—
116	C₂×C₅×Q₈	—	10	1	no	—
117	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
118	C₂×C₅×Q₈	C₂×D₅	1	1	yes	characteristic, complemented
119	C₂×C₅×D₄	—	5	1	no	—
120	C₂×C₂²⋊D₅	—	10	2	no	—
121	#713	—	2	1	no	—
122	#713	—	10	1	no	—
123	#713	C₂×D₅	1	2	yes	complemented
124	Q₈×D₅	—	2	1	no	—
125	Q₈×D₅	C₂×D₅	1	2	yes	complemented
126	#711	—	5	2	no	—
127	#711	—	10	1	no	—
128	D₄×D₅	—	5	2	no	—
129	D₄×D₅	—	10	1	no	—
130	D₄×D₅	—	10	2	no	—
131	D₄×D₅	—	10	1	no	—
132	#709	—	2	2	no	—
133	#709	—	5	1	no	—
134	#709	—	5	1	no	—
135	#709	—	5	2	no	—
136	#709	—	10	2	no	—
137	#709	C₂×D₅	1	1	yes	characteristic
138	#709	C₂×D₅	1	1	yes	characteristic, complemented
139	C₂×D₂₀	—	5	2	no	—
140	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
141	C₅×SD₁₆	—	5	2	no	—
142	C₅×SD₁₆	—	5	2	no	—
143	C₂×C₅×C₈	—	5	1	no	—
144	#689	—	5	2	no	—
145	#689	—	5	2	no	—
146	#689	—	5	1	no	—
147	#689	—	10	4	no	—
148	#688	—	5	2	no	—
149	#688	—	5	2	no	—
150	#688	—	5	2	no	—
151	#688	—	5	1	no	—
152	#688	—	10	4	no	—
153	#688	C₂×D₅	1	2	yes	complemented
154	#688	C₂×D₅	1	2	yes	complemented
155	#687	—	5	2	no	—
156	#687	—	5	1	no	—
157	#686	—	5	2	no	—
158	#686	—	5	2	no	—
159	#686	—	5	1	no	—
160	#681	—	5	1	no	—
161	#681	—	5	1	no	—
162	#681	—	5	2	no	—
163	#681	—	10	2	no	—
164	C₂×#168	—	5	2	no	—
165	C₂×#168	C₂×D₅	1	1	yes	characteristic, complemented
166	Dic₂₀	—	5	1	no	—
167	D₄₀	—	5	1	no	—
168	#677	—	5	1	no	—
169	#677	—	5	1	no	—
170	#676	—	5	2	no	—
171	C₈×D₅	—	5	2	no	—
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	1	no	—
176	C₅×D₅	—	2	2	no	—
177	C₅×D₅	—	4	2	no	—
178	C₅×D₅	—	4	1	no	—
179	C₂³×C₅	—	10	1	no	—
180	C₂²×D₅	—	2	1	no	—
181	C₂²×D₅	—	10	2	no	—
182	C₂²×D₅	—	10	2	no	—
183	C₅×Q₈	—	2	1	no	—
184	C₅×Q₈	—	2	1	no	—
185	C₅×Q₈	—	2	4	no	—
186	C₅×Q₈	—	4	2	no	—
187	C₅×Q₈	—	5	1	no	—
188	C₅×Q₈	—	5	1	no	—
189	C₅×Q₈	—	10	2	no	—
190	C₅×Q₈	C₂²×D₅	1	2	yes	—
191	C₅×Q₈	C₂²×D₅	1	2	yes	—
192	C₅×D₄	—	5	2	no	—
193	C₅×D₄	—	5	1	no	—
194	C₅×D₄	—	5	1	no	—
195	C₅×D₄	—	5	2	no	—
196	C₅×D₄	—	10	2	no	—
197	C₅×D₄	—	10	1	no	—
198	C₅×D₄	—	10	1	no	—
199	C₂×C₄×C₅	—	2	1	no	—
200	C₂×C₄×C₅	—	2	1	no	—
201	C₂×C₄×C₅	—	2	2	no	—
202	C₂×C₄×C₅	—	4	2	no	—
203	C₂×C₄×C₅	—	5	2	no	—
204	C₂×C₄×C₅	—	10	2	no	—
205	C₂×C₄×C₅	—	10	2	no	—
206	C₂×C₄×C₅	—	10	1	no	—
207	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
208	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
209	C₂²⋊D₅	—	2	1	no	—
210	C₂²⋊D₅	—	10	2	no	—
211	C₂²⋊D₅	—	10	1	no	—
212	C₂²⋊D₅	—	10	1	no	—
213	C₂²⋊D₅	—	10	2	no	—
214	C₂²⋊D₅	—	10	2	no	—
215	C₂²⋊D₅	—	20	2	no	—
216	C₂²⋊D₅	C₂²⋊D₅	1	2	yes	—
217	C₂×Dic₅	—	10	2	no	—
218	D₂₀	—	2	2	no	—
219	D₂₀	—	2	1	no	—
220	D₂₀	—	5	1	no	—
221	D₂₀	—	5	1	no	—
222	D₂₀	—	10	2	no	—
223	D₂₀	—	10	2	no	—
224	D₂₀	C₂²⋊D₅	1	1	yes	characteristic
225	D₂₀	C₂²×D₅	1	1	yes	characteristic
226	D₂₀	C₂²×D₅	1	2	yes	—
227	C₄×D₅	—	2	2	no	—
228	C₄×D₅	—	2	1	no	—
229	C₄×D₅	—	2	2	no	—
230	C₄×D₅	—	10	2	no	—
231	C₄×D₅	—	10	1	no	—
232	C₄×D₅	—	10	1	no	—
233	C₄×D₅	—	20	2	no	—
234	C₄×D₅	C₂²×D₅	1	2	yes	—
235	Dic₁₀	—	2	2	no	—
236	Dic₁₀	—	5	1	no	—
237	Dic₁₀	—	5	1	no	—
238	Dic₁₀	—	10	2	no	—
239	Dic₁₀	C₂²⋊D₅	1	1	yes	characteristic
240	Dic₁₀	C₂²×D₅	1	1	yes	characteristic
241	C₅×C₈	—	5	1	no	—
242	C₅×C₈	—	5	1	no	—
243	#168	—	5	2	no	—
244	#168	—	5	2	no	—
245	#168	—	10	4	no	—
246	#168	C₂²×D₅	1	1	yes	characteristic
247	#168	C₂²×D₅	1	1	yes	characteristic
248	2₋¹⁺⁴	—	5	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	—	25	1	no	—
255	#135	—	25	2	no	—
256	C₂×SD₁₆	—	5	1	no	—
257	C₂×SD₁₆	—	25	2	no	—
258	#131	—	25	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	2	no	—
264	C₂²×C₅	—	40	1	no	—
265	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
266	C₂²×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
267	C₂×D₅	—	2	2	no	—
268	C₂×D₅	—	2	1	no	—
269	C₂×D₅	—	8	1	no	—
270	C₂×D₅	—	10	1	no	—
271	C₂×D₅	—	10	1	no	—
272	C₂×D₅	—	20	2	no	—
273	C₂×D₅	—	40	2	no	—
274	C₂×D₅	—	40	2	no	—
275	C₂×D₅	C₂×C₂²⋊D₅	1	2	yes	—
276	C₄×C₅	—	2	2	no	—
277	C₄×C₅	—	2	2	no	—
278	C₄×C₅	—	2	2	no	—
279	C₄×C₅	—	2	2	no	—
280	C₄×C₅	—	4	4	no	—
281	C₄×C₅	—	5	2	no	—
282	C₄×C₅	—	10	1	no	—
283	C₄×C₅	D₄×D₅	1	1	yes	characteristic
284	C₄×C₅	C₂×C₂²⋊D₅	1	1	yes	characteristic
285	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
286	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
287	Dic₅	—	2	1	no	—
288	Dic₅	—	10	1	no	—
289	Dic₅	—	10	1	no	—
290	Dic₅	—	20	2	no	—
291	Dic₅	C₂×C₂²⋊D₅	1	2	yes	—
292	#41	—	5	1	no	—
293	#41	—	5	2	no	—
294	#41	—	5	1	no	—
295	#41	—	10	2	no	—
296	#41	—	10	1	no	—
297	#41	—	25	2	no	—
298	#41	—	25	1	no	—
299	#41	—	50	1	no	—
300	C₂×Q₈	—	5	2	no	—
301	C₂×Q₈	—	10	1	no	—
302	C₂×Q₈	D₅²	1	1	yes	characteristic, complemented
303	C₂×D₄	—	5	1	no	—
304	C₂×D₄	—	25	2	no	—
305	C₂×D₄	—	50	1	no	—
306	C₂×D₄	—	50	2	no	—
307	Q₁₆	—	25	1	no	—
308	Q₁₆	—	25	2	no	—
309	SD₁₆	—	5	2	no	—
310	SD₁₆	—	5	2	no	—
311	SD₁₆	—	25	1	no	—
312	SD₁₆	—	25	1	no	—
313	SD₁₆	—	25	2	no	—
314	SD₁₆	—	25	2	no	—
315	D₈	—	25	1	no	—
316	D₈	—	25	2	no	—
317	M₁₆	—	25	2	no	—
318	M₁₆	—	25	1	no	—
319	C₂×C₈	—	5	1	no	—
320	C₂×C₈	—	25	2	no	—
321	C₂×C₅	—	2	1	no	—
322	C₂×C₅	—	2	1	no	—
323	C₂×C₅	—	2	2	no	—
324	C₂×C₅	—	4	2	no	—
325	C₂×C₅	—	10	2	no	—
326	C₂×C₅	—	20	1	no	—
327	C₂×C₅	—	20	2	no	—
328	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
329	C₂×C₅	C₂²×C₂²⋊D₅	1	1	yes	characteristic
330	D₅	—	2	2	no	—
331	D₅	—	4	2	no	—
332	D₅	—	4	1	no	—
333	D₅	—	20	1	no	—
334	D₅	—	20	1	no	—
335	D₅	—	40	2	no	—
336	C₂³	—	10	1	no	—
337	C₂³	—	50	2	no	—
338	Q₈	—	2	1	no	—
339	Q₈	—	5	1	no	—
340	Q₈	—	5	1	no	—
341	Q₈	—	10	2	no	—
342	Q₈	—	25	1	no	—
343	Q₈	C₂×D₅²	1	2	yes	—
344	D₄	—	5	2	no	—
345	D₄	—	5	1	no	—
346	D₄	—	5	1	no	—
347	D₄	—	5	2	no	—
348	D₄	—	10	2	no	—
349	D₄	—	10	1	no	—
350	D₄	—	10	1	no	—
351	D₄	—	25	1	no	—
352	D₄	—	50	2	no	—
353	D₄	—	50	2	no	—
354	D₄	—	50	1	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	5	2	no	—
357	C₂×C₄	—	10	2	no	—
358	C₂×C₄	—	10	2	no	—
359	C₂×C₄	—	10	1	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	1	no	—
364	C₈	—	5	1	no	—
365	C₈	—	25	2	no	—
366	C₅	—	2	2	no	—
367	C₅	#66096	1	1	yes	characteristic, complemented
368	C₅	#66171	1	1	yes	characteristic, complemented
369	C₂²	—	5	2	no	—
370	C₂²	—	10	1	no	—
371	C₂²	—	10	2	no	—
372	C₂²	—	40	1	no	—
373	C₂²	—	50	1	no	—
374	C₂²	—	200	2	no	—
375	C₂²	D₅×C₂²⋊D₅	1	1	yes	characteristic
376	C₄	—	2	2	no	—
377	C₄	—	5	2	no	—
378	C₄	—	10	1	no	—
379	C₄	—	50	1	no	—
380	C₄	D₅×C₂²⋊D₅	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	2	no	—
385	C₂	—	20	1	no	—
386	C₂	—	100	1	no	—
387	C₂	C₂×D₅×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₅×C₂²⋊D₅
commutator, cosocle	C₄×C₅²	C₂⁴
Frattini	C₄	C₂²×D₅²
Fitting	C₂×C₅²×Q₈	C₂²
socle	C₂×C₅²	C₂²×D₄
hypercenter	C₂×Q₈	D₅²
hypocenter	C₅²	#577
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. C₅×#4147	C₂	semidirect
2. #11717472	C₂	semidirect
3. #11717262	C₂	semidirect
4. C₂×C₅×#688	C₂	semidirect
5. #11717065	C₂	semidirect
6. #11717064	C₂	semidirect
7. #11717057	C₂	semidirect
8. D₅×#688	C₂	semidirect
9. #11716947	C₂	semidirect
10. #11716941	C₂	semidirect
11. #11716932	C₂	semidirect
12. C₂×C₅²×Q₈	C₂²	semidirect
15. C₅×#713	C₂²	semidirect
17. C₅×Q₈×D₅	C₂²	semidirect
20. C₅×#709	C₂²	semidirect
27. #87812	C₂²	semidirect
28. #87811	C₂²	semidirect
29. #87804	C₂²	semidirect
30. C₅×#688	C₂²	semidirect
31. C₅×#688	C₂²	semidirect
32. C₂×C₅×#168	C₂²	semidirect
33. #87776	C₂²	semidirect
34. #87774	C₂²	semidirect
35. #87773	C₂²	semidirect
36. #87772	C₂²	semidirect
37. #87771	C₂²	semidirect
38. #87770	C₂²	semidirect
39. #87768	C₂²	semidirect
40. D₅×#168	C₂²	semidirect
75. #4147	D₅	semidirect
88. C₂×#688	D₅	semidirect
117. C₂×C₅×Q₈	C₂×D₅	semidirect
118. C₂×C₅×Q₈	C₂×D₅	semidirect
123. #713	C₂×D₅	semidirect
125. Q₈×D₅	C₂×D₅	semidirect
138. #709	C₂×D₅	semidirect
153. #688	C₂×D₅	semidirect
154. #688	C₂×D₅	semidirect
165. C₂×#168	C₂×D₅	semidirect
259. C₅²	#577	semidirect
302. C₂×Q₈	D₅²	semidirect
367. C₅	#66096	semidirect
368. C₅	#66171	semidirect

Maximal subgroups

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

subgroup	index
C₅×#4147	2
#11717472	2
#11717262	2
C₂×C₅×#688	2
#11717065	2
#11717064	2
#11717057	2
D₅×#688	2
#11716947	2
#11716941	2
#11716932	2
#66171	5
#66096	5