Subgroups — #12252519

Order 1280.

682 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₅×#504	C₂	1	1	yes	maximal, characteristic, complemented
2	#10615517	C₂	1	1	yes	maximal, characteristic
3	C₂×#65196	C₂	1	1	yes	maximal, characteristic, complemented
4	#10602797	C₂	1	1	yes	maximal, characteristic, complemented
5	#10602226	C₂	1	2	yes	maximal, complemented
6	#10602223	C₂	1	2	yes	maximal, complemented
7	Dic₅×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#10602202	C₂	1	1	yes	maximal, characteristic, complemented
9	#10602200	C₂	1	2	yes	maximal, complemented
10	C₁₆×C₂²⋊D₅	C₂	1	2	yes	maximal, complemented
11	C₂×#64721	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₅×#131	C₂²	1	1	yes	characteristic
13	#66166	C₄	1	1	yes	characteristic
14	C₂×#4047	C₂²	1	1	yes	characteristic
15	C₅×#504	—	2	4	no	—
16	C₂×C₅×M₃₂	C₂²	1	2	yes	—
17	C₂×C₅×M₃₂	C₂²	1	1	yes	characteristic
18	C₂²×C₅×C₁₆	C₂²	1	1	yes	characteristic
19	C₂²×C₅×C₁₆	C₂²	1	2	yes	—
20	#65523	C₄	1	1	yes	characteristic
21	#65418	C₄	1	1	yes	characteristic
22	#65417	C₄	1	2	yes	—
23	#65417	C₄	1	2	yes	—
24	#65407	C₄	1	1	yes	characteristic
25	C₈×C₂²⋊D₅	C₂²	1	2	yes	—
26	C₈×C₂²⋊D₅	C₂²	1	2	yes	—
27	C₂×C₈×Dic₅	C₂²	1	1	yes	characteristic
28	C₂×#3944	C₂²	1	1	yes	characteristic
29	C₂²×#672	C₂²	1	1	yes	characteristic
30	#65196	—	2	4	no	—
31	C₂×#3930	C₂²	1	2	yes	—
32	C₂×D₅×C₁₆	C₂²	1	2	yes	—
33	C₂²⋊#672	C₂²	1	2	yes	—
34	C₂²⋊#672	C₂²	1	2	yes	—
35	#64727	C₂²	1	2	yes	—
36	#64727	C₂²	1	2	yes	—
37	#64722	C₂²	1	2	yes	—
38	#64721	C₂²	1	1	yes	characteristic
39	#64721	C₂²	1	1	yes	characteristic
40	#64721	C₂²	1	1	yes	characteristic
41	#64721	C₂²	1	1	yes	characteristic
42	C₁₆×Dic₅	C₂²	1	2	yes	—
43	#33438	—	5	1	no	maximal, Sylow
44	C₂×C₅×#41	C₂×C₄	1	1	yes	characteristic
45	C₂×#709	C₂×C₄	1	1	yes	characteristic
46	C₅×#131	—	2	2	no	—
47	C₅×#131	—	2	2	no	—
48	C₂×C₅×M₁₆	C₂×C₄	1	1	yes	characteristic
49	C₂×C₅×M₁₆	C₂×C₄	1	2	yes	—
50	C₂²×C₅×C₈	C₂³	1	2	yes	—
51	C₂²×C₅×C₈	C₂³	1	1	yes	characteristic
52	#4093	C₈	1	1	yes	characteristic
53	#4090	C₈	1	1	yes	characteristic
54	#4086	C₈	1	1	yes	characteristic
55	#4085	C₈	1	2	yes	—
56	#4085	C₈	1	2	yes	—
57	#4082	C₈	1	1	yes	characteristic
58	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
59	C₄×C₂²⋊D₅	C₂×C₄	1	2	yes	—
60	C₂×C₄×Dic₅	C₂×C₄	1	1	yes	characteristic
61	C₂×#681	C₂×C₄	1	1	yes	characteristic
62	C₂²×#168	C₂×C₄	1	1	yes	characteristic
63	#4047	—	2	2	no	—
64	#4047	—	2	2	no	—
65	C₂×#676	C₂×C₄	1	2	yes	—
66	C₂×C₈×D₅	C₂³	1	2	yes	—
67	C₅×M₃₂	—	2	4	no	—
68	C₅×M₃₂	D₄	1	4	yes	complemented
69	C₂×C₅×C₁₆	—	2	2	no	—
70	C₂×C₅×C₁₆	—	2	4	no	—
71	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
72	C₂×C₅×C₁₆	C₂³	1	1	yes	characteristic
73	C₂×C₅×C₁₆	C₂³	1	2	yes	—
74	C₂²⋊#168	C₂×C₄	1	2	yes	—
75	C₂²⋊#168	C₂×C₄	1	2	yes	—
76	#3952	C₂×C₄	1	2	yes	—
77	#3952	C₂×C₄	1	2	yes	—
78	#3946	C₂×C₄	1	2	yes	—
79	#3946	C₂×C₄	1	2	yes	—
80	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₅	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₅	C₂³	1	1	yes	characteristic
83	C₈×Dic₅	C₂³	1	1	yes	characteristic
84	#3944	—	2	2	no	—
85	C₂×#672	—	2	2	no	—
86	C₂×#672	C₂³	1	1	yes	characteristic
87	C₂×#672	C₂³	1	1	yes	characteristic
88	C₂×#672	C₂³	1	2	yes	—
89	#3930	—	2	4	no	—
90	D₅×C₁₆	—	2	4	no	—
91	C₂×#504	—	5	1	no	—
92	C₂×#504	D₅	1	1	yes	characteristic, complemented
93	#2931	—	5	1	no	—
94	#2169	—	5	2	no	—
95	#2168	—	5	2	no	—
96	#2168	—	5	2	no	—
97	D₄×C₁₆	—	5	2	no	—
98	C₂×C₄⋊C₁₆	—	5	1	no	—
99	#2113	—	5	1	no	—
100	#2113	—	5	1	no	—
101	C₄×M₃₂	—	5	1	no	—
102	C₅×#41	—	2	2	no	—
103	C₅×#41	—	2	2	no	—
104	C₂×C₅×Q₈	C₂×C₈	1	1	yes	characteristic
105	C₂×C₅×D₄	C₂×C₈	1	1	yes	characteristic
106	C₂×C₅×D₄	C₂×C₈	1	2	yes	—
107	C₂²×C₄×C₅	C₂²×C₄	1	2	yes	—
108	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
110	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
111	C₂×C₂²⋊D₅	C₂×C₈	1	2	yes	—
112	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
113	C₂²×Dic₅	C₂×C₈	1	1	yes	characteristic
114	#709	—	2	2	no	—
115	#709	—	2	2	no	—
116	C₂×D₂₀	C₂×C₈	1	1	yes	characteristic
117	C₂×C₄×D₅	C₂²×C₄	1	2	yes	—
118	C₂×Dic₁₀	C₂×C₈	1	1	yes	characteristic
119	C₅×M₁₆	—	2	2	no	—
120	C₅×M₁₆	—	2	4	no	—
121	C₂×C₅×C₈	—	2	2	no	—
122	C₂×C₅×C₈	—	2	2	no	—
123	C₂×C₅×C₈	C₂²×C₄	1	2	yes	—
124	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
126	C₂×C₅×C₈	#41	1	2	yes	—
127	C₂×C₅×C₈	C₂⁴	1	1	yes	characteristic
128	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
129	C₂²⋊Dic₅	C₂×C₈	1	2	yes	—
130	#685	C₂×C₈	1	2	yes	—
131	#685	C₂×C₈	1	2	yes	—
132	#683	C₂×C₈	1	2	yes	—
133	#683	C₂×C₈	1	2	yes	—
134	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
135	C₄×Dic₅	C₂×C₈	1	1	yes	characteristic
136	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
137	C₄×Dic₅	C₂²×C₄	1	1	yes	characteristic
138	#681	—	2	2	no	—
139	C₂×#168	—	2	2	no	—
140	C₂×#168	C₂²×C₄	1	1	yes	characteristic
141	C₂×#168	C₂²×C₄	1	1	yes	characteristic
142	C₂×#168	C₂²×C₄	1	2	yes	—
143	#676	—	2	4	no	—
144	C₈×D₅	—	2	2	no	—
145	C₈×D₅	—	2	2	no	—
146	C₅×C₁₆	—	2	1	no	—
147	C₅×C₁₆	—	2	1	no	—
148	C₅×C₁₆	C₂×D₄	1	4	yes	—
149	#672	—	2	1	no	—
150	#672	—	2	1	no	—
151	#672	—	2	2	no	—
152	C₂×#131	—	5	1	no	—
153	C₂×#131	C₂×D₅	1	1	yes	characteristic
154	#532	—	5	1	no	—
155	#504	—	2	4	no	—
156	#504	—	10	4	no	—
157	C₂×M₃₂	—	5	1	no	—
158	C₂×M₃₂	—	5	2	no	—
159	C₂×M₃₂	C₂×D₅	1	2	yes	—
160	C₂×M₃₂	C₂×D₅	1	1	yes	characteristic
161	C₂²×C₁₆	—	5	1	no	—
162	C₂²×C₁₆	—	5	2	no	—
163	C₂²×C₁₆	C₂×D₅	1	1	yes	characteristic
164	C₂²×C₁₆	C₂×D₅	1	2	yes	—
165	#436	—	5	2	no	—
166	#436	—	5	2	no	—
167	C₈×D₄	—	5	2	no	—
168	C₈×D₄	—	5	2	no	—
169	#423	—	5	1	no	—
170	#408	—	5	1	no	—
171	#408	—	5	1	no	—
172	C₂×C₄×C₈	—	5	1	no	—
173	C₄⋊C₁₆	—	5	1	no	—
174	C₄⋊C₁₆	—	5	1	no	—
175	C₄⋊C₁₆	—	5	1	no	—
176	C₄⋊C₁₆	—	5	1	no	—
177	C₂²⋊C₁₆	—	5	2	no	—
178	C₂²⋊C₁₆	—	5	2	no	—
179	C₂²⋊C₁₆	—	5	2	no	—
180	C₂²⋊C₁₆	—	5	2	no	—
181	#346	—	5	2	no	—
182	C₄×C₁₆	—	5	2	no	—
183	C₂³×C₅	C₂²×C₈	1	2	yes	—
184	C₂³×C₅	C₂²×C₈	1	1	yes	characteristic
185	C₂²×D₅	C₂²×C₈	1	2	yes	—
186	C₅×Q₈	—	2	2	no	—
187	C₅×D₄	—	2	4	no	—
188	C₅×D₄	—	2	2	no	—
189	C₂×C₄×C₅	—	2	2	no	—
190	C₂×C₄×C₅	—	2	2	no	—
191	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₅	C₂²×C₈	1	2	yes	—
193	C₂×C₄×C₅	C₂²×C₈	1	1	yes	characteristic
194	C₂×C₄×C₅	#131	1	2	yes	—
195	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
196	C₂²⋊D₅	—	2	4	no	—
197	C₂²⋊D₅	—	2	4	no	—
198	C₂²⋊D₅	—	4	2	no	—
199	C₂×Dic₅	—	2	2	no	—
200	C₂×Dic₅	—	4	1	no	—
201	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
202	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
203	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
204	C₂×Dic₅	C₂²×C₈	1	2	yes	—
205	C₂×Dic₅	C₂²×C₈	1	1	yes	characteristic
206	D₂₀	—	2	2	no	—
207	C₄×D₅	—	2	2	no	—
208	C₄×D₅	—	2	2	no	—
209	Dic₁₀	—	2	2	no	—
210	C₅×C₈	—	2	2	no	—
211	C₅×C₈	C₄×D₄	1	2	yes	—
212	C₅×C₈	C₂²×D₄	1	1	yes	characteristic
213	C₅×C₈	C₂×#41	1	1	yes	characteristic
214	#168	—	2	2	no	—
215	#168	—	2	1	no	—
216	#168	—	2	1	no	—
217	C₂×#41	—	5	1	no	—
218	C₂×#41	C₄×D₅	1	1	yes	characteristic
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	#131	—	10	2	no	—
222	#131	—	10	2	no	—
223	C₂×M₁₆	—	5	1	no	—
224	C₂×M₁₆	—	5	2	no	—
225	C₂×M₁₆	C₄×D₅	1	1	yes	characteristic
226	C₂×M₁₆	C₄×D₅	1	2	yes	—
227	C₂²×C₈	—	5	1	no	—
228	C₂²×C₈	—	5	2	no	—
229	C₂²×C₈	C₂²×D₅	1	2	yes	—
230	C₂²×C₈	C₂²×D₅	1	1	yes	characteristic
231	C₄⋊Q₈	—	5	1	no	—
232	#127	—	5	1	no	—
233	#124	—	5	1	no	—
234	#124	—	5	1	no	—
235	#121	—	5	2	no	—
236	#121	—	5	2	no	—
237	C₄×D₄	—	5	2	no	—
238	C₄×D₄	—	5	2	no	—
239	C₂×C₄²	—	5	1	no	—
240	M₃₂	—	2	4	no	—
241	M₃₂	—	10	2	no	—
242	M₃₂	—	10	4	no	—
243	M₃₂	C₂²⋊D₅	1	4	yes	complemented
244	C₂×C₁₆	—	2	2	no	—
245	C₂×C₁₆	—	2	4	no	—
246	C₂×C₁₆	—	5	2	no	—
247	C₂×C₁₆	—	5	1	no	—
248	C₂×C₁₆	—	5	1	no	—
249	C₂×C₁₆	—	10	2	no	—
250	C₂×C₁₆	—	10	4	no	—
251	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
252	C₂×C₁₆	C₂²×D₅	1	1	yes	characteristic
253	C₂×C₁₆	C₂²×D₅	1	2	yes	—
254	C₄⋊C₈	—	5	2	no	—
255	C₄⋊C₈	—	5	2	no	—
256	C₂²⋊C₈	—	5	2	no	—
257	C₂²⋊C₈	—	5	2	no	—
258	C₂²⋊C₈	—	5	2	no	—
259	C₂²⋊C₈	—	5	2	no	—
260	C₄×C₈	—	5	1	no	—
261	C₄×C₈	—	5	1	no	—
262	C₄×C₈	—	5	1	no	—
263	C₄×C₈	—	5	1	no	—
264	C₂²×C₅	—	2	1	no	—
265	C₂²×C₅	—	2	1	no	—
266	C₂²×C₅	—	2	2	no	—
267	C₂²×C₅	—	2	2	no	—
268	C₂²×C₅	—	2	2	no	—
269	C₂²×C₅	C₈×D₄	1	2	yes	—
270	C₂²×C₅	C₂³×C₈	1	1	yes	characteristic
271	C₂×D₅	—	2	2	no	—
272	C₂×D₅	—	2	2	no	—
273	C₂×D₅	—	2	2	no	—
274	C₄×C₅	—	2	2	no	—
275	C₄×C₅	C₈×D₄	1	2	yes	—
276	C₄×C₅	C₂×C₄×D₄	1	1	yes	characteristic
277	C₄×C₅	C₂×#131	1	1	yes	characteristic
278	Dic₅	—	2	1	no	—
279	Dic₅	—	2	1	no	—
280	Dic₅	—	2	1	no	—
281	Dic₅	—	2	2	no	—
282	Dic₅	#504	1	2	yes	complemented
283	#41	—	2	2	no	—
284	#41	—	2	2	no	—
285	#41	—	10	2	no	—
286	#41	—	10	2	no	—
287	C₂×Q₈	—	5	1	no	—
288	C₂×Q₈	C₈×D₅	1	1	yes	characteristic
289	C₂×D₄	—	5	2	no	—
290	C₂×D₄	—	5	2	no	—
291	C₂×D₄	—	5	2	no	—
292	C₂×D₄	—	5	1	no	—
293	C₂×D₄	C₈×D₅	1	2	yes	—
294	C₂×D₄	C₈×D₅	1	1	yes	characteristic
295	C₂²×C₄	—	5	1	no	—
296	C₂²×C₄	—	5	1	no	—
297	C₂²×C₄	—	5	2	no	—
298	C₂²×C₄	C₂×C₄×D₅	1	2	yes	—
299	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
300	M₁₆	—	2	2	no	—
301	M₁₆	—	2	4	no	—
302	M₁₆	—	10	2	no	—
303	M₁₆	—	10	4	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	2	2	no	—
306	C₂×C₈	—	5	1	no	—
307	C₂×C₈	—	5	1	no	—
308	C₂×C₈	—	5	2	no	—
309	C₂×C₈	—	10	2	no	—
310	C₂×C₈	—	10	2	no	—
311	C₂×C₈	—	10	2	no	—
312	C₂×C₈	C₂×C₄×D₅	1	2	yes	—
313	C₂×C₈	C₂×C₄×D₅	1	1	yes	characteristic
314	C₂×C₈	#709	1	2	yes	—
315	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
316	C₂×C₈	C₂³×D₅	1	1	yes	characteristic
317	C₄⋊C₄	—	5	2	no	—
318	C₄⋊C₄	—	5	2	no	—
319	C₂²⋊C₄	—	5	2	no	—
320	C₂²⋊C₄	—	5	2	no	—
321	C₂²⋊C₄	—	5	2	no	—
322	C₂²⋊C₄	—	5	2	no	—
323	C₄²	—	5	1	no	—
324	C₄²	—	5	1	no	—
325	C₄²	—	5	1	no	—
326	C₄²	—	5	1	no	—
327	C₁₆	—	2	1	no	—
328	C₁₆	—	2	1	no	—
329	C₁₆	—	10	2	no	—
330	C₁₆	—	10	1	no	—
331	C₁₆	—	10	1	no	—
332	C₁₆	C₂×C₂²⋊D₅	1	4	yes	—
333	C₂×C₅	—	2	2	no	—
334	C₂×C₅	—	4	2	no	—
335	C₂×C₅	C₂×C₈×D₄	1	1	yes	characteristic
336	C₂×C₅	C₂×#504	1	1	yes	characteristic
337	C₂×C₅	#3407	1	1	yes	characteristic
338	D₅	—	4	2	no	—
339	C₂³	—	5	2	no	—
340	C₂³	C₂×C₈×D₅	1	2	yes	—
341	C₂³	C₂×C₈×D₅	1	1	yes	characteristic
342	Q₈	—	2	2	no	—
343	Q₈	—	10	2	no	—
344	D₄	—	2	4	no	—
345	D₄	—	2	2	no	—
346	D₄	—	10	4	no	—
347	D₄	—	10	4	no	—
348	D₄	—	10	2	no	—
349	D₄	—	20	2	no	—
350	C₂×C₄	—	2	2	no	—
351	C₂×C₄	—	2	2	no	—
352	C₂×C₄	—	5	1	no	—
353	C₂×C₄	—	5	1	no	—
354	C₂×C₄	—	5	1	no	—
355	C₂×C₄	—	5	1	no	—
356	C₂×C₄	—	5	2	no	—
357	C₂×C₄	—	10	2	no	—
358	C₂×C₄	—	10	2	no	—
359	C₂×C₄	—	10	2	no	—
360	C₂×C₄	—	20	1	no	—
361	C₂×C₄	C₂×C₈×D₅	1	2	yes	—
362	C₂×C₄	C₂×C₈×D₅	1	1	yes	characteristic
363	C₂×C₄	#4047	1	2	yes	—
364	C₂×C₄	C₄×C₂²⋊D₅	1	2	yes	—
365	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
366	C₈	—	2	2	no	—
367	C₈	—	10	1	no	—
368	C₈	—	10	1	no	—
369	C₈	—	10	2	no	—
370	C₈	C₄×C₂²⋊D₅	1	2	yes	—
371	C₈	C₂×#709	1	1	yes	characteristic
372	C₈	C₂²×C₂²⋊D₅	1	1	yes	characteristic
373	C₅	#33438	1	1	yes	characteristic, complemented, Sylow
374	C₂²	—	2	1	no	—
375	C₂²	—	2	1	no	—
376	C₂²	—	2	2	no	—
377	C₂²	—	2	2	no	—
378	C₂²	—	2	2	no	—
379	C₂²	—	10	2	no	—
380	C₂²	—	10	2	no	—
381	C₂²	—	10	2	no	—
382	C₂²	C₈×C₂²⋊D₅	1	2	yes	—
383	C₂²	C₂²×C₈×D₅	1	1	yes	characteristic
384	C₄	—	2	2	no	—
385	C₄	—	5	2	no	—
386	C₄	—	10	2	no	—
387	C₄	—	10	1	no	—
388	C₄	—	10	1	no	—
389	C₄	—	10	1	no	—
390	C₄	C₈×C₂²⋊D₅	1	2	yes	—
391	C₄	C₂×#4047	1	1	yes	characteristic
392	C₄	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic
393	C₂	—	2	2	no	—
394	C₂	—	4	2	no	—
395	C₂	—	20	2	no	—
396	C₂	D₅×#504	1	1	yes	characteristic
397	C₂	#10613743	1	1	yes	characteristic
398	C₂	C₂×C₈×C₂²⋊D₅	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅
commutator, socle	C₂²×C₅	C₂³×C₈
Fitting	C₂×C₅×#504	C₂
prime radical	#66166	C₄
cosocle	C₂×C₅×C₈	C₂⁴
hypercenter	C₂×#504	D₅
hypocenter	C₅	#33438
LCS	G ▷ C₂²×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂×C₅×#504	C₂	semidirect
3. C₂×#65196	C₂	semidirect
4. #10602797	C₂	semidirect
5. #10602226	C₂	semidirect
6. #10602223	C₂	semidirect
7. Dic₅×M₃₂	C₂	semidirect
8. #10602202	C₂	semidirect
9. #10602200	C₂	semidirect
10. C₁₆×C₂²⋊D₅	C₂	semidirect
11. C₂×#64721	C₂	semidirect
68. C₅×M₃₂	D₄	semidirect
92. C₂×#504	D₅	semidirect
243. M₃₂	C₂²⋊D₅	semidirect
282. Dic₅	#504	semidirect
373. C₅	#33438	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₅×#504	2
#10615517	2
C₂×#65196	2
#10602797	2
#10602226	2
#10602223	2
Dic₅×M₃₂	2
#10602202	2
#10602200	2
C₁₆×C₂²⋊D₅	2
C₂×#64721	2
#33438	5