Subgroups — #12071043

Order 1280.

718 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×D₈	C₂	1	1	yes	maximal, characteristic
2	#10614034	C₂	1	2	yes	maximal, complemented
3	#10602392	C₂	1	2	yes	maximal, complemented
4	#10601675	C₂	1	2	yes	maximal, complemented
5	#10601672	C₂	1	4	yes	maximal, complemented
6	#10601671	C₂	1	2	yes	maximal, complemented
7	#10600525	C₂	1	1	yes	maximal, characteristic, complemented
8	#10600518	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₅×D₈	C₄	1	1	yes	characteristic
10	C₂×C₄×C₅×D₄	C₂²	1	2	yes	—
11	C₄×C₅×D₈	C₄	1	1	yes	characteristic
12	C₄×C₅×D₈	C₄	1	2	yes	—
13	C₄×C₅×D₈	C₄	1	1	yes	characteristic
14	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
15	C₄×C₅×D₈	C₂²	1	2	yes	—
16	C₄×C₅×D₈	C₂²	1	1	yes	characteristic
17	C₂×C₅×#107	C₄	1	1	yes	characteristic
18	C₂×C₅×#102	C₄	1	2	yes	—
19	C₂×C₄×C₅×C₈	C₂²	1	1	yes	characteristic
20	#65496	—	2	2	no	—
21	C₂×C₂²⋊#168	—	2	2	no	—
22	#65301	C₂²	1	4	yes	complemented
23	#65300	—	2	4	no	—
24	D₄×#168	C₂²	1	4	yes	complemented
25	#65215	C₂²	1	2	yes	—
26	#65214	C₂²	1	1	yes	characteristic
27	C₄×#681	C₂²	1	1	yes	characteristic
28	D₄⋊#168	C₂²	1	4	yes	complemented
29	D₄⋊#168	C₂²	1	4	yes	complemented
30	#64678	C₂²	1	2	yes	complemented
31	#64677	C₂²	1	2	yes	complemented
32	#64675	C₂²	1	2	yes	complemented
33	#64674	C₂²	1	2	yes	complemented
34	#18064	—	5	1	no	maximal, Sylow
35	C₂²×C₅×D₄	C₂×C₄	1	2	yes	—
36	C₂³×C₄×C₅	—	2	2	no	—
37	C₂×C₅×D₈	—	2	4	no	—
38	C₂×C₅×D₈	C₂×C₄	1	2	yes	—
39	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₅×D₈	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
42	C₂²×C₅×C₈	C₂×C₄	1	1	yes	characteristic
43	C₄×C₅×D₄	—	2	2	no	—
44	C₄×C₅×D₄	—	2	2	no	—
45	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
46	C₄×C₅×D₄	C₂×C₄	1	2	yes	—
47	C₄×C₅×D₄	C₂³	1	2	yes	—
48	C₄×C₅×D₄	C₂³	1	2	yes	—
49	C₂×C₅×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₅×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
52	C₂×#681	—	2	1	no	—
53	C₂×#681	—	2	4	no	—
54	C₂²×#168	—	2	4	no	—
55	C₅×#107	C₂×C₄	1	2	yes	—
56	C₅×#107	C₂×C₄	1	1	yes	characteristic
57	C₅×#107	C₂×C₄	1	1	yes	characteristic
58	C₅×#102	C₂×C₄	1	2	yes	—
59	C₅×#102	C₂×C₄	1	2	yes	—
60	C₅×#102	C₂×C₄	1	2	yes	—
61	C₅×#102	C₂×C₄	1	2	yes	—
62	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₅×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
65	C₄×C₅×C₈	C₂³	1	1	yes	characteristic
66	C₂²⋊#168	—	2	2	no	—
67	C₂²⋊#168	—	2	4	no	—
68	C₂²⋊#168	—	2	4	no	—
69	C₂²⋊#168	D₄	1	2	yes	—
70	C₄⋊#168	D₄	1	2	yes	—
71	C₄⋊#168	C₂³	1	4	yes	—
72	#3935	—	2	2	no	—
73	#3935	—	2	1	no	—
74	C₄×#168	C₂³	1	2	yes	—
75	C₄×#168	C₂³	1	2	yes	—
76	#2974	—	5	2	no	—
77	C₂×C₄×D₈	D₅	1	1	yes	characteristic
78	#1594	—	5	2	no	—
79	#1581	—	5	4	no	—
80	#1580	—	5	2	no	—
81	#1568	—	5	1	no	—
82	#1490	—	5	2	no	—
83	#1451	—	5	1	no	—
84	C₂⁴×C₅	—	2	2	no	—
85	C₂×C₅×D₄	—	2	4	no	—
86	C₂×C₅×D₄	—	2	2	no	—
87	C₂×C₅×D₄	—	4	2	no	—
88	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
89	C₂×C₅×D₄	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅	—	2	2	no	—
91	C₂²×C₄×C₅	—	2	2	no	—
92	C₂²×C₄×C₅	—	2	2	no	—
93	C₂²×C₄×C₅	—	2	2	no	—
94	C₂²×C₄×C₅	—	2	2	no	—
95	C₂²×C₄×C₅	—	8	4	no	—
96	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₅	#41	1	1	yes	characteristic
99	C₅×D₈	—	2	4	no	—
100	C₅×D₈	—	2	2	no	—
101	C₅×D₈	M₁₆	1	4	yes	complemented
102	C₂×C₅×C₈	—	2	2	no	—
103	C₂×C₅×C₈	—	4	1	no	—
104	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
107	C₂×C₅×C₈	C₂²×C₄	1	1	yes	characteristic
108	C₅×C₄⋊C₄	—	2	2	no	—
109	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
110	C₅×C₄⋊C₄	C₂²×C₄	1	2	yes	—
111	C₅×C₂²⋊C₄	—	2	2	no	—
112	C₅×C₂²⋊C₄	—	2	2	no	—
113	C₅×C₂²⋊C₄	—	2	2	no	—
114	C₅×C₂²⋊C₄	—	2	2	no	—
115	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	#41	1	1	yes	characteristic
118	C₄²×C₅	C₂⁴	1	1	yes	characteristic
119	#681	—	4	2	no	—
120	#681	—	8	4	no	—
121	C₂×#168	—	2	4	no	—
122	C₂×#168	—	2	2	no	—
123	C₂×#168	—	8	4	no	—
124	C₂×#168	C₂×D₄	1	4	yes	—
125	C₂²×D₈	Dic₅	1	1	yes	characteristic
126	C₂×C₄×D₄	C₂×D₅	1	2	yes	—
127	C₄×D₈	Dic₅	1	2	yes	—
128	C₄×D₈	Dic₅	1	1	yes	characteristic
129	C₄×D₈	Dic₅	1	1	yes	characteristic
130	C₄×D₈	C₂×D₅	1	1	yes	characteristic
131	C₄×D₈	C₂×D₅	1	2	yes	—
132	C₄×D₈	C₂×D₅	1	1	yes	characteristic
133	#436	—	5	4	no	—
134	#435	—	10	4	no	—
135	C₈×D₄	—	5	4	no	—
136	#432	—	5	2	no	—
137	#431	—	5	1	no	—
138	C₂×#107	Dic₅	1	1	yes	characteristic
139	C₂×#102	Dic₅	1	2	yes	—
140	#407	—	10	2	no	—
141	C₂×C₂²⋊C₈	—	10	2	no	—
142	C₄×M₁₆	—	5	1	no	—
143	C₂×C₄×C₈	C₂×D₅	1	1	yes	characteristic
144	#335	—	5	2	no	—
145	#334	—	5	2	no	—
146	D₄⋊C₈	—	5	4	no	—
147	D₄⋊C₈	—	5	4	no	—
148	#322	—	5	2	no	—
149	#322	—	5	2	no	—
150	C₂³×C₅	—	2	2	no	—
151	C₂³×C₅	—	2	2	no	—
152	C₂³×C₅	—	4	4	no	—
153	C₂³×C₅	—	4	4	no	—
154	C₂³×C₅	—	8	2	no	—
155	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
156	C₅×D₄	—	2	2	no	—
157	C₅×D₄	—	2	2	no	—
158	C₅×D₄	—	2	2	no	—
159	C₅×D₄	—	4	4	no	—
160	C₅×D₄	C₂×M₁₆	1	4	yes	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	1	no	—
163	C₂×C₄×C₅	—	2	1	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	2	2	no	—
167	C₂×C₄×C₅	—	4	4	no	—
168	C₂×C₄×C₅	—	4	4	no	—
169	C₂×C₄×C₅	—	8	2	no	—
170	C₂×C₄×C₅	—	8	2	no	—
171	C₂×C₄×C₅	—	8	2	no	—
172	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
178	C₅×C₈	—	2	1	no	—
179	C₅×C₈	—	2	1	no	—
180	C₅×C₈	—	2	1	no	—
181	C₅×C₈	C₂×M₁₆	1	2	yes	—
182	#168	—	2	4	no	—
183	#168	—	4	2	no	—
184	#168	—	4	4	no	—
185	C₂²×D₄	C₂×Dic₅	1	2	yes	—
186	C₂³×C₄	—	2	2	no	—
187	C₂×D₈	—	2	4	no	—
188	C₂×D₈	C₂×Dic₅	1	2	yes	—
189	C₂×D₈	C₂×Dic₅	1	1	yes	characteristic
190	C₂×D₈	C₂×Dic₅	1	1	yes	characteristic
191	C₂×M₁₆	—	10	4	no	—
192	C₂×M₁₆	—	10	1	no	—
193	C₂²×C₈	—	10	4	no	—
194	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
195	C₂²×C₈	C₂×Dic₅	1	1	yes	characteristic
196	C₄×D₄	—	2	2	no	—
197	C₄×D₄	—	2	2	no	—
198	C₄×D₄	C₂×Dic₅	1	2	yes	—
199	C₄×D₄	C₂×Dic₅	1	2	yes	—
200	C₄×D₄	C₂²×D₅	1	2	yes	—
201	C₄×D₄	C₂²×D₅	1	2	yes	—
202	C₂×C₄⋊C₄	C₂×Dic₅	1	2	yes	—
203	C₂×C₂²⋊C₄	—	2	2	no	—
204	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
205	#107	C₂×Dic₅	1	2	yes	—
206	#107	C₂×Dic₅	1	1	yes	characteristic
207	#107	C₂×Dic₅	1	1	yes	characteristic
208	C₄⋊C₈	—	5	2	no	—
209	C₄⋊C₈	—	5	4	no	—
210	#102	C₂×Dic₅	1	2	yes	—
211	#102	C₂×Dic₅	1	2	yes	—
212	#102	C₂×Dic₅	1	2	yes	—
213	#102	C₂×Dic₅	1	2	yes	—
214	C₂²⋊C₈	—	5	2	no	—
215	C₂²⋊C₈	—	10	4	no	—
216	C₂²⋊C₈	—	10	4	no	—
217	C₂²⋊C₈	—	10	2	no	—
218	#97	—	10	2	no	—
219	#97	—	10	1	no	—
220	C₄×C₈	—	5	2	no	—
221	C₄×C₈	—	5	2	no	—
222	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
223	C₄×C₈	C₂×Dic₅	1	1	yes	characteristic
224	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
225	C₄×C₈	C₂²×D₅	1	1	yes	characteristic
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	2	4	no	—
229	C₂²×C₅	—	4	2	no	—
230	C₂²×C₅	—	4	2	no	—
231	C₂²×C₅	—	4	2	no	—
232	C₂²×C₅	—	4	2	no	—
233	C₂²×C₅	—	4	2	no	—
234	C₂²×C₅	—	8	8	no	—
235	C₂²×C₅	#444	1	2	yes	—
236	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
237	C₄×C₅	—	2	1	no	—
238	C₄×C₅	—	2	1	no	—
239	C₄×C₅	—	2	1	no	—
240	C₄×C₅	—	2	1	no	—
241	C₄×C₅	—	4	2	no	—
242	C₄×C₅	—	4	2	no	—
243	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
244	C₄×C₅	#568	1	1	yes	characteristic
245	C₄×C₅	#576	1	2	yes	—
246	C₂⁴	—	2	2	no	—
247	C₂×D₄	—	2	4	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	C₂²×Dic₅	1	2	yes	—
251	C₂×D₄	C₂²×Dic₅	1	2	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	8	4	no	—
258	C₂²×C₄	D₄×D₅	1	1	yes	characteristic
259	C₂²×C₄	#711	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
261	D₈	—	2	4	no	—
262	D₈	—	2	2	no	—
263	D₈	#681	1	4	yes	complemented
264	M₁₆	—	20	2	no	—
265	M₁₆	—	40	4	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	5	4	no	—
269	C₂×C₈	—	10	4	no	—
270	C₂×C₈	—	10	2	no	—
271	C₂×C₈	—	40	4	no	—
272	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
274	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
275	C₂×C₈	C₂²×Dic₅	1	1	yes	characteristic
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
278	C₄⋊C₄	C₂²×Dic₅	1	2	yes	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₂²⋊C₄	—	2	2	no	—
282	C₂²⋊C₄	—	2	2	no	—
283	C₄²	D₄×D₅	1	1	yes	characteristic
284	C₄²	#711	1	1	yes	characteristic
285	C₄²	C₂²×Dic₅	1	1	yes	characteristic
286	C₄²	C₂³×D₅	1	1	yes	characteristic
287	C₂×C₅	—	2	2	no	—
288	C₂×C₅	—	4	4	no	—
289	C₂×C₅	—	8	2	no	—
290	C₂×C₅	D₄×M₁₆	1	1	yes	characteristic
291	C₂×C₅	#2948	1	1	yes	characteristic
292	C₂×C₅	C₂×#444	1	1	yes	characteristic
293	C₂³	—	2	2	no	—
294	C₂³	—	2	2	no	—
295	C₂³	—	4	4	no	—
296	C₂³	—	4	4	no	—
297	C₂³	—	8	2	no	—
298	C₂³	D₄×Dic₅	1	1	yes	characteristic
299	D₄	—	2	2	no	—
300	D₄	—	2	2	no	—
301	D₄	—	2	2	no	—
302	D₄	—	4	4	no	—
303	D₄	C₂×#681	1	4	yes	—
304	C₂×C₄	—	2	2	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	4	4	no	—
311	C₂×C₄	—	4	4	no	—
312	C₂×C₄	—	8	2	no	—
313	C₂×C₄	—	8	2	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
316	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
317	C₂×C₄	D₄×Dic₅	1	1	yes	characteristic
318	C₂×C₄	C₂×D₄×D₅	1	1	yes	characteristic
319	C₂×C₄	C₂×#711	1	1	yes	characteristic
320	C₂×C₄	C₂³×Dic₅	1	1	yes	characteristic
321	C₈	—	2	1	no	—
322	C₈	—	2	1	no	—
323	C₈	—	2	1	no	—
324	C₈	—	10	4	no	—
325	C₈	—	20	2	no	—
326	C₈	—	20	4	no	—
327	C₈	C₂×#681	1	2	yes	—
328	C₅	#18064	1	1	yes	characteristic, complemented, Sylow
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	4	no	—
332	C₂²	—	4	2	no	—
333	C₂²	—	4	2	no	—
334	C₂²	—	4	2	no	—
335	C₂²	—	4	2	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	8	8	no	—
338	C₂²	#65486	1	2	yes	—
339	C₂²	C₂×D₄×Dic₅	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	#66091	1	2	yes	—
347	C₄	C₂²×#681	1	1	yes	characteristic
348	C₄	#66153	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	4	no	—
351	C₂	—	8	2	no	—
352	C₂	D₄×#681	1	1	yes	characteristic
353	C₂	#10615554	1	1	yes	characteristic
354	C₂	C₂×#65486	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₂×C₄×C₅×D₈	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×C₅×D₈	C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	C₂×C₄×D₈	D₅
hypocenter	C₅	#18064
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₈

Complements

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

N	H ≅ G/N	type
2. #10614034	C₂	semidirect
3. #10602392	C₂	semidirect
4. #10601675	C₂	semidirect
5. #10601672	C₂	semidirect
6. #10601671	C₂	semidirect
7. #10600525	C₂	semidirect
8. #10600518	C₂	semidirect
22. #65301	C₂²	semidirect
24. D₄×#168	C₂²	semidirect
28. D₄⋊#168	C₂²	semidirect
29. D₄⋊#168	C₂²	semidirect
30. #64678	C₂²	semidirect
31. #64677	C₂²	semidirect
32. #64675	C₂²	semidirect
33. #64674	C₂²	semidirect
101. C₅×D₈	M₁₆	semidirect
263. D₈	#681	semidirect
328. C₅	#18064	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₅×D₈	2
#10614034	2
#10602392	2
#10601675	2
#10601672	2
#10601671	2
#10600525	2
#10600518	2
#18064	5