Subgroups — #12253693

Order 1280.

696 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#443	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615625	C₂	1	2	yes	maximal, complemented
3	#10610198	C₂	1	1	yes	maximal, characteristic, complemented
4	#10601720	C₂	1	2	yes	maximal, complemented
5	#10601719	C₂	1	2	yes	maximal, complemented
6	#10601718	C₂	1	4	yes	maximal, complemented
7	#10601480	C₂	1	2	yes	maximal, complemented
8	C₂×#64683	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₅×#135	C₂²	1	1	yes	characteristic
10	C₂×C₅×#131	—	2	1	no	—
11	C₂×#4050	—	2	1	no	—
12	C₂×#4050	C₂²	1	2	yes	—
13	#65811	—	2	1	no	—
14	C₅×#443	—	2	1	no	—
15	C₅×#443	—	2	1	no	—
16	C₅×#443	C₂²	1	2	yes	—
17	C₅×#443	C₂²	1	2	yes	—
18	C₂×C₅×#108	C₂²	1	1	yes	characteristic
19	C₂×C₅×#104	—	2	1	no	—
20	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
21	#65484	C₂²	1	4	yes	—
22	#65483	—	2	4	no	—
23	#65482	C₂²	1	4	yes	—
24	#65468	—	2	2	no	—
25	#65436	—	2	2	no	—
26	C₂×#3944	C₂²	1	2	yes	—
27	#64988	C₂²	1	1	yes	characteristic, complemented
28	#64985	—	2	1	no	—
29	#64985	C₂²	1	2	yes	complemented
30	#64984	C₂²	1	1	yes	characteristic, complemented
31	#64982	—	2	1	no	—
32	#64716	C₂²	1	4	yes	complemented
33	#64715	C₂²	1	4	yes	complemented
34	#64683	C₂²	1	2	yes	complemented
35	#64683	C₂²	1	2	yes	complemented
36	#33594	—	5	1	no	maximal, Sylow
37	C₂×C₅×#41	—	2	1	no	—
38	C₂×#709	—	2	2	no	—
39	C₅×#135	—	2	1	no	—
40	C₅×#135	—	2	1	no	—
41	C₅×#135	D₄	1	2	yes	complemented
42	C₅×#135	C₂³	1	2	yes	—
43	C₂×C₅×Q₁₆	D₄	1	1	yes	characteristic, complemented
44	C₂×C₅×SD₁₆	—	2	1	no	—
45	C₂×C₅×D₈	D₄	1	1	yes	characteristic, complemented
46	C₅×#131	—	2	2	no	—
47	C₅×#131	—	2	2	no	—
48	C₅×#131	—	4	1	no	—
49	C₅×#131	—	4	1	no	—
50	C₂×C₅×M₁₆	—	2	1	no	—
51	C₂×C₅×M₁₆	—	4	1	no	—
52	C₂²×C₅×C₈	—	4	1	no	—
53	C₂²×C₅×C₈	D₄	1	1	yes	characteristic
54	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
56	#4076	—	4	2	no	—
57	C₂×Dic₂₀	—	2	1	no	—
58	C₂×Dic₂₀	C₂³	1	2	yes	—
59	#4050	—	2	2	no	—
60	#4050	—	2	2	no	—
61	#4050	—	2	2	no	—
62	#4050	—	2	2	no	—
63	#4050	—	4	1	no	—
64	#4050	—	4	1	no	—
65	#4050	—	4	1	no	—
66	#4050	—	4	1	no	—
67	C₂×D₄₀	—	2	1	no	—
68	C₂×D₄₀	C₂³	1	2	yes	—
69	C₂×#677	—	2	2	no	—
70	C₂×#677	—	2	2	no	—
71	#4021	—	2	2	no	—
72	#4020	—	2	1	no	—
73	C₄×D₂₀	—	4	2	no	—
74	#4015	—	2	1	no	—
75	C₅×#108	—	2	1	no	—
76	C₅×#108	C₂³	1	2	yes	—
77	C₅×#104	—	2	2	no	—
78	C₅×#104	—	2	2	no	—
79	C₄×C₅×C₈	D₄	1	1	yes	characteristic
80	C₄×C₅×C₈	D₄	1	1	yes	characteristic
81	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
82	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
83	#3961	—	2	4	no	—
84	#3960	—	2	4	no	—
85	#3959	—	2	4	no	—
86	#3958	—	2	4	no	—
87	#3944	—	2	2	no	—
88	#3944	C₂³	1	4	yes	—
89	C₂×#672	—	2	2	no	—
90	#3414	—	5	2	no	—
91	#3147	—	5	1	no	—
92	C₂×#443	D₅	1	1	yes	characteristic, complemented
93	#2214	—	5	2	no	—
94	#2213	—	5	2	no	—
95	#2212	—	5	4	no	—
96	C₂×#364	—	5	1	no	—
97	#2148	—	5	2	no	—
98	C₅×#41	—	2	2	no	—
99	C₅×#41	—	2	2	no	—
100	C₅×#41	—	4	1	no	—
101	C₅×#41	—	4	1	no	—
102	C₂×C₅×Q₈	—	2	1	no	—
103	C₂×C₅×D₄	—	2	1	no	—
104	C₂×C₅×D₄	—	4	1	no	—
105	C₂²×C₄×C₅	—	2	1	no	—
106	C₂²×C₄×C₅	—	4	1	no	—
107	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
108	C₂×C₂²⋊D₅	—	4	2	no	—
109	#709	—	4	2	no	—
110	#709	—	4	2	no	—
111	#709	—	4	2	no	—
112	#709	—	4	2	no	—
113	C₂×D₂₀	—	2	2	no	—
114	C₂×D₂₀	—	4	2	no	—
115	C₂×C₄×D₅	—	4	2	no	—
116	C₂×Dic₁₀	—	2	2	no	—
117	C₅×Q₁₆	—	2	1	no	—
118	C₅×Q₁₆	C₂×D₄	1	2	yes	—
119	C₅×SD₁₆	—	2	2	no	—
120	C₅×SD₁₆	—	2	2	no	—
121	C₅×D₈	—	2	1	no	—
122	C₅×D₈	C₂×D₄	1	2	yes	—
123	C₅×M₁₆	—	2	2	no	—
124	C₅×M₁₆	—	4	1	no	—
125	C₅×M₁₆	—	4	2	no	—
126	C₅×M₁₆	—	4	2	no	—
127	C₂×C₅×C₈	—	4	1	no	—
128	C₂×C₅×C₈	—	4	2	no	—
129	C₂×C₅×C₈	—	4	2	no	—
130	C₂×C₅×C₈	D₈	1	1	yes	characteristic
131	C₂×C₅×C₈	D₈	1	1	yes	characteristic
132	C₂×C₅×C₈	D₈	1	1	yes	characteristic
133	C₂×C₅×C₈	D₈	1	1	yes	characteristic
134	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
136	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
137	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
138	C₂×C₅×C₈	C₂×D₄	1	1	yes	characteristic
139	C₂×C₅×C₈	#41	1	1	yes	characteristic
140	C₂×C₅×C₈	#41	1	1	yes	characteristic
141	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
142	C₄²×C₅	—	2	1	no	—
143	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
144	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
145	#685	—	4	4	no	—
146	C₄⋊Dic₅	—	4	2	no	—
147	Dic₂₀	—	2	2	no	—
148	Dic₂₀	—	2	2	no	—
149	Dic₂₀	—	4	1	no	—
150	Dic₂₀	—	4	1	no	—
151	D₄₀	—	2	2	no	—
152	D₄₀	—	2	2	no	—
153	D₄₀	—	4	1	no	—
154	D₄₀	—	4	1	no	—
155	#677	—	4	2	no	—
156	#677	—	4	2	no	—
157	#677	—	4	2	no	—
158	#677	—	4	2	no	—
159	#672	—	2	4	no	—
160	C₂×#135	—	5	2	no	—
161	C₂×#135	—	10	1	no	—
162	C₂×#135	C₂×D₅	1	1	yes	characteristic
163	C₂×#131	—	2	1	no	—
164	#532	—	10	1	no	—
165	#510	—	5	4	no	—
166	#510	—	10	2	no	—
167	#509	—	5	4	no	—
168	#509	—	10	2	no	—
169	#508	—	10	4	no	—
170	C₂×M₃₂	—	5	2	no	—
171	#495	—	5	2	no	—
172	#495	—	10	1	no	—
173	#494	—	5	1	no	—
174	#493	—	5	1	no	—
175	#492	—	10	1	no	—
176	#443	—	2	1	no	—
177	#443	—	2	1	no	—
178	#443	C₂×D₅	1	2	yes	—
179	#443	C₂×D₅	1	2	yes	—
180	C₂×#108	C₂×D₅	1	1	yes	characteristic
181	C₂×#104	—	2	1	no	—
182	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
183	#364	—	5	2	no	—
184	#364	—	5	2	no	—
185	#362	—	5	4	no	—
186	#361	—	5	4	no	—
187	C₂³×C₅	—	4	1	no	—
188	C₂³×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
189	C₂²×D₅	—	4	2	no	—
190	C₅×Q₈	—	2	2	no	—
191	C₅×Q₈	—	4	1	no	—
192	C₅×D₄	—	2	2	no	—
193	C₅×D₄	—	4	2	no	—
194	C₅×D₄	—	4	2	no	—
195	C₅×D₄	—	4	1	no	—
196	C₂×C₄×C₅	—	2	1	no	—
197	C₂×C₄×C₅	—	2	1	no	—
198	C₂×C₄×C₅	—	4	2	no	—
199	C₂×C₄×C₅	—	4	1	no	—
200	C₂×C₄×C₅	—	4	2	no	—
201	C₂×C₄×C₅	—	4	2	no	—
202	C₂×C₄×C₅	#121	1	1	yes	characteristic
203	C₂×C₄×C₅	#121	1	1	yes	characteristic
204	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
205	C₂×C₄×C₅	C₂×D₈	1	1	yes	characteristic
206	C₂×C₄×C₅	AGL(1,ℤ/8)	1	1	yes	characteristic
207	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
208	C₂²⋊D₅	—	8	2	no	—
209	C₂²⋊D₅	—	8	2	no	—
210	C₂×Dic₅	—	4	2	no	—
211	D₂₀	—	4	2	no	—
212	D₂₀	—	4	2	no	—
213	D₂₀	—	8	4	no	—
214	C₄×D₅	—	8	2	no	—
215	C₄×D₅	—	8	2	no	—
216	Dic₁₀	—	4	2	no	—
217	Dic₁₀	—	4	2	no	—
218	C₅×C₈	—	4	2	no	—
219	C₅×C₈	#121	1	1	yes	characteristic
220	C₅×C₈	#121	1	1	yes	characteristic
221	C₅×C₈	C₂×D₈	1	1	yes	characteristic
222	C₅×C₈	C₂×D₈	1	1	yes	characteristic
223	C₅×C₈	C₂×D₈	1	1	yes	characteristic
224	C₅×C₈	C₂×D₈	1	1	yes	characteristic
225	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
226	C₅×C₈	C₂×#41	1	1	yes	characteristic
227	C₂×#41	—	2	1	no	—
228	C₂×#41	—	10	2	no	—
229	#135	—	2	1	no	—
230	#135	—	2	1	no	—
231	#135	—	10	2	no	—
232	#135	—	10	2	no	—
233	#135	—	10	2	no	—
234	#135	—	10	2	no	—
235	#135	—	20	1	no	—
236	#135	—	20	1	no	—
237	#135	—	20	1	no	—
238	#135	—	20	1	no	—
239	#135	D₂₀	1	2	yes	complemented
240	#135	C₂²×D₅	1	2	yes	—
241	C₂×Q₁₆	—	5	2	no	—
242	C₂×Q₁₆	—	10	1	no	—
243	C₂×Q₁₆	D₂₀	1	1	yes	characteristic, complemented
244	C₂×SD₁₆	—	2	1	no	—
245	C₂×SD₁₆	—	10	2	no	—
246	C₂×SD₁₆	—	10	2	no	—
247	C₂×D₈	—	5	2	no	—
248	C₂×D₈	—	10	1	no	—
249	C₂×D₈	D₂₀	1	1	yes	characteristic, complemented
250	#131	—	2	2	no	—
251	#131	—	2	2	no	—
252	#131	—	4	1	no	—
253	#131	—	4	1	no	—
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	4	1	no	—
256	C₂²×C₈	—	4	1	no	—
257	C₂²×C₈	C₂²⋊D₅	1	1	yes	characteristic
258	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
259	C₄⋊Q₈	—	10	1	no	—
260	#127	—	10	1	no	—
261	#124	—	10	2	no	—
262	#121	—	20	2	no	—
263	C₄×D₄	—	20	2	no	—
264	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
265	Q₃₂	—	10	4	no	—
266	SD₃₂	—	10	4	no	—
267	SD₃₂	—	10	4	no	—
268	D₁₆	—	10	4	no	—
269	M₃₂	—	5	4	no	—
270	M₃₂	—	10	2	no	—
271	C₂×C₁₆	—	10	2	no	—
272	#108	—	2	1	no	—
273	#108	C₂²×D₅	1	2	yes	—
274	#104	—	2	2	no	—
275	#104	—	2	2	no	—
276	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
277	C₄×C₈	C₂²⋊D₅	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
280	C₂²×C₅	—	4	1	no	—
281	C₂²×C₅	—	4	2	no	—
282	C₂²×C₅	—	8	2	no	—
283	C₂²×C₅	#459	1	1	yes	characteristic
284	C₂²×C₅	#459	1	1	yes	characteristic
285	C₂²×C₅	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
286	C₂×D₅	—	8	4	no	—
287	C₂×D₅	—	8	2	no	—
288	C₄×C₅	—	2	2	no	—
289	C₄×C₅	—	4	1	no	—
290	C₄×C₅	—	4	2	no	—
291	C₄×C₅	#459	1	1	yes	characteristic
292	C₄×C₅	#459	1	1	yes	characteristic
293	C₄×C₅	C₂×#121	1	1	yes	characteristic
294	C₄×C₅	C₂²×D₈	1	1	yes	characteristic
295	Dic₅	—	8	2	no	—
296	#41	—	2	2	no	—
297	#41	—	2	2	no	—
298	#41	—	4	1	no	—
299	#41	—	4	1	no	—
300	#41	—	20	2	no	—
301	#41	—	20	2	no	—
302	#41	—	20	2	no	—
303	#41	—	20	2	no	—
304	C₂×Q₈	—	2	1	no	—
305	C₂×Q₈	—	10	2	no	—
306	C₂×D₄	—	2	1	no	—
307	C₂×D₄	—	4	1	no	—
308	C₂×D₄	—	10	2	no	—
309	C₂×D₄	—	20	2	no	—
310	C₂×D₄	—	20	2	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	4	1	no	—
313	C₂²×C₄	—	20	2	no	—
314	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
315	Q₁₆	—	2	1	no	—
316	Q₁₆	—	10	2	no	—
317	Q₁₆	—	10	2	no	—
318	Q₁₆	—	20	1	no	—
319	Q₁₆	—	20	1	no	—
320	Q₁₆	C₂×D₂₀	1	2	yes	—
321	SD₁₆	—	2	2	no	—
322	SD₁₆	—	2	2	no	—
323	SD₁₆	—	20	2	no	—
324	SD₁₆	—	20	2	no	—
325	SD₁₆	—	20	2	no	—
326	SD₁₆	—	20	2	no	—
327	D₈	—	2	1	no	—
328	D₈	—	10	2	no	—
329	D₈	—	10	2	no	—
330	D₈	—	20	1	no	—
331	D₈	—	20	1	no	—
332	D₈	C₂×D₂₀	1	2	yes	—
333	M₁₆	—	2	2	no	—
334	M₁₆	—	4	1	no	—
335	M₁₆	—	4	2	no	—
336	M₁₆	—	4	2	no	—
337	C₂×C₈	—	4	1	no	—
338	C₂×C₈	—	4	2	no	—
339	C₂×C₈	—	4	2	no	—
340	C₂×C₈	#686	1	1	yes	characteristic
341	C₂×C₈	#686	1	1	yes	characteristic
342	C₂×C₈	#686	1	1	yes	characteristic
343	C₂×C₈	#686	1	1	yes	characteristic
344	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
345	C₂×C₈	C₂×D₂₀	1	1	yes	characteristic
346	C₂×C₈	#709	1	1	yes	characteristic
347	C₂×C₈	#709	1	1	yes	characteristic
348	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
349	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
350	C₂×C₈	C₂×C₂²⋊D₅	1	1	yes	characteristic
351	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
352	C₄⋊C₄	—	20	2	no	—
353	C₂²⋊C₄	—	20	4	no	—
354	C₄²	—	2	1	no	—
355	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
356	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
357	C₁₆	—	10	4	no	—
358	C₂×C₅	—	2	1	no	—
359	C₂×C₅	—	2	1	no	—
360	C₂×C₅	—	2	1	no	—
361	C₂×C₅	—	8	2	no	—
362	C₂×C₅	C₂×#459	1	1	yes	characteristic
363	D₅	—	16	2	no	—
364	C₂³	—	4	1	no	—
365	C₂³	—	20	2	no	—
366	C₂³	#4095	1	1	yes	characteristic
367	Q₈	—	2	2	no	—
368	Q₈	—	4	1	no	—
369	Q₈	—	20	2	no	—
370	Q₈	—	20	2	no	—
371	D₄	—	2	2	no	—
372	D₄	—	4	2	no	—
373	D₄	—	4	2	no	—
374	D₄	—	4	1	no	—
375	D₄	—	20	2	no	—
376	D₄	—	20	2	no	—
377	D₄	—	40	4	no	—
378	D₄	—	40	2	no	—
379	D₄	—	40	2	no	—
380	C₂×C₄	—	2	1	no	—
381	C₂×C₄	—	2	1	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	20	2	no	—
387	C₂×C₄	—	40	2	no	—
388	C₂×C₄	—	40	2	no	—
389	C₂×C₄	#4076	1	1	yes	characteristic
390	C₂×C₄	#4076	1	1	yes	characteristic
391	C₂×C₄	C₂×#686	1	1	yes	characteristic
392	C₂×C₄	C₂×#686	1	1	yes	characteristic
393	C₂×C₄	#4095	1	1	yes	characteristic
394	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
395	C₈	—	4	2	no	—
396	C₈	#4076	1	1	yes	characteristic
397	C₈	#4076	1	1	yes	characteristic
398	C₈	C₂×#686	1	1	yes	characteristic
399	C₈	C₂×#686	1	1	yes	characteristic
400	C₈	C₂×#686	1	1	yes	characteristic
401	C₈	C₂×#686	1	1	yes	characteristic
402	C₈	C₂²×D₂₀	1	1	yes	characteristic
403	C₈	C₂×#709	1	1	yes	characteristic
404	C₅	#33594	1	1	yes	characteristic, complemented, Sylow
405	C₂²	—	4	1	no	—
406	C₂²	—	4	2	no	—
407	C₂²	—	8	2	no	—
408	C₂²	—	40	4	no	—
409	C₂²	—	40	2	no	—
410	C₂²	#65304	1	1	yes	characteristic
411	C₂²	#65304	1	1	yes	characteristic
412	C₂²	C₂×#4095	1	1	yes	characteristic
413	C₄	—	2	2	no	—
414	C₄	—	4	1	no	—
415	C₄	—	4	2	no	—
416	C₄	—	40	2	no	—
417	C₄	#65304	1	1	yes	characteristic
418	C₄	#65304	1	1	yes	characteristic
419	C₄	C₂×#4076	1	1	yes	characteristic
420	C₄	C₂²×#686	1	1	yes	characteristic
421	C₂	—	2	1	no	—
422	C₂	—	2	1	no	—
423	C₂	—	2	1	no	—
424	C₂	—	8	2	no	—
425	C₂	—	80	2	no	—
426	C₂	C₂×#65304	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#65304
commutator, cosocle	C₂×C₅×C₈	C₂⁴
Frattini	C₂×C₈	C₂³×D₅
Fitting	C₂×C₅×#443	C₂
socle	C₂×C₅	C₂×#459
hypercenter	C₂×#443	D₅
hypocenter	C₅	#33594
LCS	G ▷ C₂×C₅×C₈ ▷ C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#443

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#443	C₂	semidirect
2. #10615625	C₂	semidirect
3. #10610198	C₂	semidirect
4. #10601720	C₂	semidirect
5. #10601719	C₂	semidirect
6. #10601718	C₂	semidirect
7. #10601480	C₂	semidirect
8. C₂×#64683	C₂	semidirect
27. #64988	C₂²	semidirect
29. #64985	C₂²	semidirect
30. #64984	C₂²	semidirect
32. #64716	C₂²	semidirect
33. #64715	C₂²	semidirect
34. #64683	C₂²	semidirect
35. #64683	C₂²	semidirect
41. C₅×#135	D₄	semidirect
43. C₂×C₅×Q₁₆	D₄	semidirect
45. C₂×C₅×D₈	D₄	semidirect
92. C₂×#443	D₅	semidirect
239. #135	D₂₀	semidirect
243. C₂×Q₁₆	D₂₀	semidirect
249. C₂×D₈	D₂₀	semidirect
404. C₅	#33594	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#443	2
#10615625	2
#10610198	2
#10601720	2
#10601719	2
#10601718	2
#10601480	2
C₂×#64683	2
#33594	5