Subgroups — #13106553

Order 1600.

662 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11717523	C₂	1	1	yes	maximal, characteristic, complemented
2	#11717472	C₂	1	1	yes	maximal, characteristic, complemented
3	C₅×#4096	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅×#4095	C₂	1	1	yes	maximal, characteristic, complemented
5	#11717069	C₂	1	2	yes	maximal, complemented
6	#11717058	C₂	1	2	yes	maximal, complemented
7	#11717054	C₂	1	2	yes	maximal, complemented
8	#11717045	C₂	1	2	yes	maximal, complemented
9	#11716949	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#87773	C₂	1	1	yes	maximal, characteristic, complemented
11	#11716935	C₂	1	1	yes	maximal, characteristic, complemented
12	C₅²×#41	C₂²	1	1	yes	characteristic, complemented
13	Q₈×C₅⋊D₅	C₂²	1	2	yes	complemented
14	#87902	—	2	2	no	—
15	#87902	C₂²	1	2	yes	complemented
16	#87900	—	2	1	no	—
17	#87900	C₂²	1	1	yes	characteristic
18	C₂×#6045	—	2	1	no	—
19	C₂×#6045	C₂²	1	1	yes	characteristic, complemented
20	C₅×#709	C₂²	1	1	yes	characteristic
21	C₂×C₅×D₂₀	C₂²	1	1	yes	characteristic
22	D₅×C₂²⋊D₅	—	2	2	no	—
23	#87877	C₂²	1	2	yes	—
24	D₅×D₂₀	—	2	1	no	—
25	#87871	C₂²	1	2	yes	—
26	#87870	—	2	1	no	—
27	C₅×#689	C₂²	1	1	yes	characteristic, complemented
28	C₅×#688	C₂²	1	2	yes	complemented
29	C₅×#688	C₂²	1	1	yes	characteristic, complemented
30	C₅×#687	C₂²	1	1	yes	characteristic, complemented
31	C₅×#686	C₂²	1	2	yes	complemented
32	C₅×#686	C₂²	1	1	yes	characteristic, complemented
33	C₅×#681	C₂²	1	1	yes	characteristic, complemented
34	C₂×C₅×#168	C₂²	1	1	yes	characteristic, complemented
35	#87776	C₂²	1	2	yes	complemented
36	#87773	C₂²	1	2	yes	complemented
37	#87773	C₂²	1	2	yes	complemented
38	#87773	C₂²	1	2	yes	complemented
39	#87769	C₂²	1	2	yes	complemented
40	#87767	C₂²	1	2	yes	complemented
41	#66109	—	5	1	no	maximal
42	#66104	—	5	1	no	maximal
43	C₂²×C₅×D₅	—	2	1	no	—
44	C₂×D₅²	—	2	2	no	—
45	C₅²×Q₈	C₂³	1	1	yes	characteristic
46	C₅²×D₄	—	2	1	no	—
47	C₅²×D₄	C₂³	1	1	yes	characteristic
48	C₂×C₄×C₅²	—	2	1	no	—
49	C₂×C₄×C₅²	C₂³	1	1	yes	characteristic
50	C₂²⋊(C₅⋊D₅)	—	2	2	no	—
51	C₂²⋊(C₅⋊D₅)	D₄	1	2	yes	—
52	C₂×#1039	—	2	1	no	—
53	C₂×#1039	—	2	2	no	—
54	C₄⋊(C₅⋊D₅)	D₄	1	1	yes	characteristic
55	C₄×C₅⋊D₅	—	2	2	no	—
56	C₄×C₅⋊D₅	C₂³	1	2	yes	—
57	#6045	—	2	1	no	—
58	#6045	—	2	2	no	—
59	#6045	D₄	1	1	yes	characteristic
60	#6045	C₂³	1	2	yes	—
61	C₅×C₂²⋊D₅	—	2	1	no	—
62	C₅×D₂₀	—	2	1	no	—
63	C₅×D₂₀	C₂³	1	2	yes	—
64	C₅×D₂₀	C₂³	1	1	yes	characteristic
65	C₄×C₅×D₅	—	2	1	no	—
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	C₅×#168	C₂³	1	1	yes	characteristic
71	C₅×#168	C₂³	1	2	yes	—
72	C₅×#168	C₂³	1	1	yes	characteristic
73	#4150	—	5	1	no	—
74	#4150	—	5	1	no	—
75	#4150	—	10	2	no	—
76	#4149	—	5	1	no	—
77	#4144	—	5	1	no	—
78	C₅×AGL(1,ℤ/8)	—	5	1	no	—
79	C₅×#135	—	5	1	no	—
80	#4097	—	5	1	no	—
81	#4096	D₅	1	1	yes	characteristic, complemented
82	#4095	D₅	1	1	yes	characteristic, complemented
83	C₂×#687	—	5	1	no	—
84	#4064	—	5	2	no	—
85	#4062	—	5	2	no	—
86	#4061	—	5	2	no	—
87	D₅×SD₁₆	—	5	2	no	—
88	D₅×SD₁₆	—	5	2	no	—
89	#4058	—	5	2	no	—
90	#4057	—	5	2	no	—
91	#4057	—	5	2	no	—
92	#4055	—	5	1	no	—
93	#4053	—	5	1	no	—
94	C₂×#677	—	5	1	no	—
95	#4047	—	5	1	no	—
96	C₂²×C₅²	—	2	1	no	—
97	C₂²×C₅²	C₂×D₄	1	1	yes	characteristic
98	C₂×C₅⋊D₅	C₂×D₄	1	2	yes	—
99	C₂×C₅×D₅	—	2	2	no	—
100	C₂×C₅×D₅	—	2	1	no	—
101	C₂×C₅×D₅	—	8	1	no	—
102	D₅²	—	8	2	no	—
103	C₄×C₅²	—	2	1	no	—
104	C₄×C₅²	C₂×D₄	1	1	yes	characteristic
105	C₄×C₅²	C₂⁴	1	1	yes	characteristic
106	#1039	—	2	2	no	—
107	#1039	C₂×D₄	1	2	yes	—
108	C₅×Dic₅	—	2	1	no	—
109	C₅×#41	—	2	2	no	—
110	C₅×#41	—	5	1	no	—
111	C₅×#41	C₂×D₅	1	1	yes	characteristic, complemented
112	C₅×#41	C₂×D₅	1	1	yes	characteristic, complemented
113	C₂×C₅×D₄	—	5	1	no	—
114	C₂×C₂²⋊D₅	—	10	2	no	—
115	#713	—	5	2	no	—
116	Q₈×D₅	—	5	2	no	—
117	Q₈×D₅	—	5	2	no	—
118	Q₈×D₅	—	10	4	no	—
119	#711	—	5	2	no	—
120	#711	—	5	2	no	—
121	#711	—	5	2	no	—
122	#711	—	10	4	no	—
123	#711	—	10	1	no	—
124	#711	—	10	2	no	—
125	#711	—	10	2	no	—
126	#711	—	10	8	no	—
127	D₄×D₅	—	5	2	no	—
128	D₄×D₅	—	5	2	no	—
129	D₄×D₅	—	10	1	no	—
130	D₄×D₅	—	10	2	no	—
131	#709	—	5	1	no	—
132	#709	—	5	1	no	—
133	#709	—	10	2	no	—
134	#709	—	10	1	no	—
135	#709	—	10	1	no	—
136	#709	—	10	1	no	—
137	#709	—	10	4	no	—
138	#709	C₂×D₅	1	1	yes	characteristic
139	C₂×D₂₀	—	10	1	no	—
140	C₂×D₂₀	C₂×D₅	1	1	yes	characteristic
141	C₂×Dic₁₀	—	5	1	no	—
142	C₂×Dic₁₀	—	5	1	no	—
143	C₂×Dic₁₀	—	10	2	no	—
144	C₂×Dic₁₀	—	10	1	no	—
145	C₂×Dic₁₀	—	10	1	no	—
146	C₂×Dic₁₀	—	10	4	no	—
147	C₅×Q₁₆	—	5	1	no	—
148	C₅×SD₁₆	—	5	1	no	—
149	C₅×SD₁₆	—	5	2	no	—
150	C₅×SD₁₆	—	5	1	no	—
151	C₅×D₈	—	5	1	no	—
152	C₅×D₈	—	5	2	no	—
153	C₅×M₁₆	—	5	1	no	—
154	C₂×C₅×C₈	—	5	1	no	—
155	#689	—	5	2	no	—
156	#689	C₂×D₅	1	1	yes	characteristic, complemented
157	#688	C₂×D₅	1	1	yes	characteristic, complemented
158	#688	C₂×D₅	1	2	yes	complemented
159	#687	—	5	2	no	—
160	#687	—	5	2	no	—
161	#687	—	5	2	no	—
162	#687	C₂×D₅	1	1	yes	characteristic, complemented
163	#686	C₂×D₅	1	1	yes	characteristic, complemented
164	#686	C₂×D₅	1	2	yes	complemented
165	#681	C₂×D₅	1	1	yes	characteristic, complemented
166	C₂×#168	C₂×D₅	1	1	yes	characteristic, complemented
167	Dic₂₀	—	5	2	no	—
168	#677	—	5	2	no	—
169	#677	—	5	2	no	—
170	#677	—	5	2	no	—
171	#676	—	5	2	no	—
172	#676	—	5	2	no	—
173	C₈×D₅	—	5	2	no	—
174	C₈×D₅	—	5	2	no	—
175	#577	—	25	1	no	Sylow
176	C₂×C₅²	—	2	1	no	—
177	C₂×C₅²	—	4	1	no	—
178	C₂×C₅²	C₂²×D₄	1	1	yes	characteristic
179	C₅⋊D₅	—	2	2	no	—
180	C₅×D₅	—	4	1	no	—
181	C₅×D₅	—	4	2	no	—
182	C₂³×C₅	—	10	1	no	—
183	C₂²×D₅	—	2	1	no	—
184	C₂²×D₅	—	10	2	no	—
185	C₂²×D₅	—	10	2	no	—
186	C₅×Q₈	—	2	2	no	—
187	C₅×Q₈	—	5	1	no	—
188	C₅×Q₈	C₂²×D₅	1	1	yes	characteristic
189	C₅×Q₈	C₂²×D₅	1	1	yes	characteristic
190	C₅×D₄	—	2	4	no	—
191	C₅×D₄	—	2	1	no	—
192	C₅×D₄	—	2	1	no	—
193	C₅×D₄	—	2	2	no	—
194	C₅×D₄	—	5	1	no	—
195	C₅×D₄	—	5	2	no	—
196	C₅×D₄	—	10	1	no	—
197	C₅×D₄	—	10	1	no	—
198	C₅×D₄	C₂²×D₅	1	1	yes	characteristic
199	C₅×D₄	C₂²×D₅	1	1	yes	characteristic
200	C₂×C₄×C₅	—	2	4	no	—
201	C₂×C₄×C₅	—	2	1	no	—
202	C₂×C₄×C₅	—	2	1	no	—
203	C₂×C₄×C₅	—	2	2	no	—
204	C₂×C₄×C₅	—	10	1	no	—
205	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
206	C₂×C₄×C₅	C₂²×D₅	1	1	yes	characteristic
207	C₂²⋊D₅	—	2	1	no	—
208	C₂²⋊D₅	—	5	2	no	—
209	C₂²⋊D₅	—	5	2	no	—
210	C₂²⋊D₅	—	10	4	no	—
211	C₂²⋊D₅	—	10	2	no	—
212	C₂²⋊D₅	—	10	2	no	—
213	C₂²⋊D₅	—	10	2	no	—
214	C₂²⋊D₅	—	10	2	no	—
215	C₂²⋊D₅	—	10	2	no	—
216	C₂²⋊D₅	—	10	8	no	—
217	C₂×Dic₅	—	10	2	no	—
218	C₂×Dic₅	—	10	2	no	—
219	C₂×Dic₅	—	10	2	no	—
220	C₂×Dic₅	—	10	8	no	—
221	C₂×Dic₅	—	10	4	no	—
222	C₂×Dic₅	—	10	1	no	—
223	C₂×Dic₅	—	10	1	no	—
224	D₂₀	—	2	1	no	—
225	D₂₀	—	5	1	no	—
226	D₂₀	—	5	1	no	—
227	D₂₀	—	10	2	no	—
228	D₂₀	—	10	2	no	—
229	D₂₀	C₂²×D₅	1	2	yes	—
230	D₂₀	C₂²×D₅	1	1	yes	characteristic
231	C₄×D₅	—	2	1	no	—
232	C₄×D₅	—	5	2	no	—
233	C₄×D₅	—	5	2	no	—
234	C₄×D₅	—	10	4	no	—
235	C₄×D₅	—	10	2	no	—
236	C₄×D₅	—	10	2	no	—
237	C₄×D₅	—	10	2	no	—
238	C₄×D₅	—	10	8	no	—
239	Dic₁₀	—	5	1	no	—
240	Dic₁₀	—	5	1	no	—
241	Dic₁₀	—	5	2	no	—
242	Dic₁₀	—	5	2	no	—
243	Dic₁₀	—	10	2	no	—
244	Dic₁₀	—	10	4	no	—
245	Dic₁₀	—	10	2	no	—
246	Dic₁₀	—	10	2	no	—
247	Dic₁₀	—	10	4	no	—
248	Dic₁₀	—	10	8	no	—
249	Dic₁₀	—	10	1	no	—
250	Dic₁₀	—	10	1	no	—
251	Dic₁₀	C₂²×D₅	1	1	yes	characteristic
252	C₅×C₈	—	5	1	no	—
253	C₅×C₈	—	5	2	no	—
254	C₅×C₈	—	5	1	no	—
255	#168	C₂²×D₅	1	1	yes	characteristic
256	#168	C₂²×D₅	1	2	yes	—
257	#168	C₂²×D₅	1	1	yes	characteristic
258	2₋¹⁺⁴	—	25	1	no	—
259	H(5,𝔽₂)	—	25	1	no	—
260	#137	—	25	1	no	—
261	#137	—	25	2	no	—
262	AGL(1,ℤ/8)	—	5	1	no	—
263	AGL(1,ℤ/8)	—	25	2	no	—
264	#135	—	5	1	no	—
265	#135	—	25	2	no	—
266	C₂×SD₁₆	—	25	1	no	—
267	C₂×SD₁₆	—	25	2	no	—
268	#131	—	25	1	no	—
269	C₅²	#577	1	1	yes	characteristic, complemented, Sylow
270	C₂²×C₅	—	2	4	no	—
271	C₂²×C₅	—	2	1	no	—
272	C₂²×C₅	—	2	1	no	—
273	C₂²×C₅	—	2	2	no	—
274	C₂²×C₅	—	10	2	no	—
275	C₂²×C₅	—	10	1	no	—
276	C₂²×C₅	—	40	1	no	—
277	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
278	C₂²×C₅	D₄×D₅	1	1	yes	characteristic
279	C₂×D₅	—	2	1	no	—
280	C₂×D₅	—	2	2	no	—
281	C₂×D₅	—	5	2	no	—
282	C₂×D₅	—	5	2	no	—
283	C₂×D₅	—	8	1	no	—
284	C₂×D₅	—	10	4	no	—
285	C₂×D₅	—	40	2	no	—
286	C₂×D₅	—	40	2	no	—
287	C₄×C₅	—	2	4	no	—
288	C₄×C₅	—	2	1	no	—
289	C₄×C₅	—	2	1	no	—
290	C₄×C₅	—	2	2	no	—
291	C₄×C₅	—	2	2	no	—
292	C₄×C₅	—	10	1	no	—
293	C₄×C₅	D₄×D₅	1	1	yes	characteristic
294	C₄×C₅	D₄×D₅	1	1	yes	characteristic
295	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
296	C₄×C₅	C₂³×D₅	1	1	yes	characteristic
297	Dic₅	—	2	1	no	—
298	Dic₅	—	5	2	no	—
299	Dic₅	—	5	2	no	—
300	Dic₅	—	10	4	no	—
301	Dic₅	—	10	8	no	—
302	Dic₅	—	10	2	no	—
303	Dic₅	—	10	2	no	—
304	#41	—	5	1	no	—
305	#41	—	25	1	no	—
306	#41	—	25	2	no	—
307	#41	—	25	2	no	—
308	#41	—	50	1	no	—
309	#41	—	50	1	no	—
310	#41	—	50	2	no	—
311	#41	D₅²	1	1	yes	characteristic, complemented
312	C₂×Q₈	—	25	2	no	—
313	C₂×Q₈	—	25	1	no	—
314	C₂×Q₈	—	50	1	no	—
315	C₂×D₄	—	5	1	no	—
316	C₂×D₄	—	25	2	no	—
317	C₂×D₄	—	50	1	no	—
318	C₂×D₄	—	50	2	no	—
319	Q₁₆	—	5	1	no	—
320	Q₁₆	—	25	2	no	—
321	SD₁₆	—	5	1	no	—
322	SD₁₆	—	5	2	no	—
323	SD₁₆	—	5	1	no	—
324	SD₁₆	—	25	2	no	—
325	SD₁₆	—	25	2	no	—
326	SD₁₆	—	25	2	no	—
327	D₈	—	5	1	no	—
328	D₈	—	5	2	no	—
329	M₁₆	—	5	1	no	—
330	M₁₆	—	25	2	no	—
331	C₂×C₈	—	5	1	no	—
332	C₂×C₈	—	25	2	no	—
333	C₂×C₅	—	2	1	no	—
334	C₂×C₅	—	2	1	no	—
335	C₂×C₅	—	2	2	no	—
336	C₂×C₅	—	4	2	no	—
337	C₂×C₅	—	4	4	no	—
338	C₂×C₅	—	4	1	no	—
339	C₂×C₅	—	4	1	no	—
340	C₂×C₅	—	20	2	no	—
341	C₂×C₅	—	20	1	no	—
342	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
343	C₂×C₅	C₂×D₄×D₅	1	1	yes	characteristic
344	D₅	—	4	1	no	—
345	D₅	—	4	2	no	—
346	D₅	—	10	2	no	—
347	D₅	—	10	2	no	—
348	D₅	—	20	4	no	—
349	C₂³	—	10	1	no	—
350	C₂³	—	50	2	no	—
351	Q₈	—	5	1	no	—
352	Q₈	—	25	1	no	—
353	Q₈	—	25	2	no	—
354	Q₈	—	50	2	no	—
355	Q₈	—	50	1	no	—
356	Q₈	C₂×D₅²	1	1	yes	characteristic
357	D₄	—	2	1	no	—
358	D₄	—	5	2	no	—
359	D₄	—	5	1	no	—
360	D₄	—	10	1	no	—
361	D₄	—	10	1	no	—
362	D₄	—	25	2	no	—
363	D₄	—	25	1	no	—
364	D₄	—	50	2	no	—
365	D₄	—	50	2	no	—
366	D₄	—	50	2	no	—
367	D₄	C₂×D₅²	1	1	yes	characteristic
368	C₂×C₄	—	2	1	no	—
369	C₂×C₄	—	10	1	no	—
370	C₂×C₄	—	25	2	no	—
371	C₂×C₄	—	50	2	no	—
372	C₂×C₄	—	50	2	no	—
373	C₂×C₄	—	50	2	no	—
374	C₂×C₄	—	50	1	no	—
375	C₂×C₄	C₂×D₅²	1	1	yes	characteristic
376	C₈	—	5	1	no	—
377	C₈	—	5	1	no	—
378	C₈	—	5	2	no	—
379	C₅	—	2	2	no	—
380	C₅	#66104	1	1	yes	characteristic, complemented
381	C₅	#66109	1	1	yes	characteristic, complemented
382	C₂²	—	2	1	no	—
383	C₂²	—	10	1	no	—
384	C₂²	—	10	2	no	—
385	C₂²	—	25	2	no	—
386	C₂²	—	40	1	no	—
387	C₂²	—	200	2	no	—
388	C₂²	#87886	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	10	1	no	—
391	C₄	—	25	2	no	—
392	C₄	—	50	2	no	—
393	C₄	#87886	1	1	yes	characteristic
394	C₄	C₂²×D₅²	1	1	yes	characteristic
395	C₂	—	2	1	no	—
396	C₂	—	4	1	no	—
397	C₂	—	20	2	no	—
398	C₂	—	20	1	no	—
399	C₂	—	50	2	no	—
400	C₂	C₂×#87886	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#87886
commutator, cosocle	C₄×C₅²	C₂⁴
Frattini	C₄	C₂²×D₅²
Fitting	C₅²×#41	C₂²
socle	C₂×C₅²	C₂²×D₄
hypercenter	#41	D₅²
hypocenter	C₅²	#577
LCS	G ▷ C₄×C₅² ▷ C₂×C₅² ▷ C₅²
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #41

Complements

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

N	H ≅ G/N	type
1. #11717523	C₂	semidirect
2. #11717472	C₂	semidirect
3. C₅×#4096	C₂	semidirect
4. C₅×#4095	C₂	semidirect
5. #11717069	C₂	semidirect
6. #11717058	C₂	semidirect
7. #11717054	C₂	semidirect
8. #11717045	C₂	semidirect
9. #11716949	C₂	semidirect
10. C₂×#87773	C₂	semidirect
11. #11716935	C₂	semidirect
12. C₅²×#41	C₂²	semidirect
13. Q₈×C₅⋊D₅	C₂²	semidirect
15. #87902	C₂²	semidirect
19. C₂×#6045	C₂²	semidirect
27. C₅×#689	C₂²	semidirect
28. C₅×#688	C₂²	semidirect
29. C₅×#688	C₂²	semidirect
30. C₅×#687	C₂²	semidirect
31. C₅×#686	C₂²	semidirect
32. C₅×#686	C₂²	semidirect
33. C₅×#681	C₂²	semidirect
34. C₂×C₅×#168	C₂²	semidirect
35. #87776	C₂²	semidirect
36. #87773	C₂²	semidirect
37. #87773	C₂²	semidirect
38. #87773	C₂²	semidirect
39. #87769	C₂²	semidirect
40. #87767	C₂²	semidirect
81. #4096	D₅	semidirect
82. #4095	D₅	semidirect
111. C₅×#41	C₂×D₅	semidirect
112. C₅×#41	C₂×D₅	semidirect
156. #689	C₂×D₅	semidirect
157. #688	C₂×D₅	semidirect
158. #688	C₂×D₅	semidirect
162. #687	C₂×D₅	semidirect
163. #686	C₂×D₅	semidirect
164. #686	C₂×D₅	semidirect
165. #681	C₂×D₅	semidirect
166. C₂×#168	C₂×D₅	semidirect
269. C₅²	#577	semidirect
311. #41	D₅²	semidirect
380. C₅	#66104	semidirect
381. C₅	#66109	semidirect

Maximal subgroups

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

subgroup	index
#11717523	2
#11717472	2
C₅×#4096	2
C₅×#4095	2
#11717069	2
#11717058	2
#11717054	2
#11717045	2
#11716949	2
C₂×#87773	2
#11716935	2
#66109	5
#66104	5