Subgroups — #12005162

Order 1280.

794 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₅×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#65213	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×#681	C₂	1	1	yes	maximal, characteristic
4	C₂×#64772	C₂	1	4	yes	maximal
5	C₂³×C₅×C₈	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×#681	C₂²	1	2	yes	—
8	C₂²×#681	C₂²	1	2	yes	—
9	C₂×C₅×C₄⋊C₈	—	2	2	no	—
10	C₂×C₅×C₄⋊C₈	—	2	2	no	—
11	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
12	C₂×C₅×C₄⋊C₈	C₄	1	2	yes	—
13	#65213	—	2	2	no	—
14	#65213	C₄	1	4	yes	—
15	C₂×C₄⋊#168	—	2	1	no	—
16	C₂×C₄⋊#168	—	2	1	no	—
17	C₄×#681	—	2	2	no	—
18	C₄×#681	C₄	1	4	yes	—
19	C₂×#3935	—	2	1	no	—
20	C₂×C₄×#168	—	2	1	no	—
21	#64772	—	2	8	no	—
22	#10853	—	5	1	no	maximal, Sylow
23	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
24	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
25	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
26	C₂²×C₅×C₈	—	2	4	no	—
27	C₂²×C₅×C₈	—	2	2	no	—
28	C₂²×C₅×C₈	—	2	2	no	—
29	C₂²×C₅×C₈	—	2	4	no	—
30	C₂²×C₅×C₈	C₂×C₄	1	4	yes	—
31	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	2	yes	—
38	C₂×C₄²×C₅	D₄	1	2	yes	—
39	C₂×#681	—	2	2	no	—
40	C₂×#681	—	2	2	no	—
41	C₂×#681	—	2	2	no	—
42	C₂×#681	—	2	2	no	—
43	C₂×#681	—	2	4	no	—
44	C₂×#681	—	2	4	no	—
45	C₂×#681	C₂×C₄	1	4	yes	—
46	C₂×#681	C₂×C₄	1	4	yes	—
47	C₂×#681	D₄	1	4	yes	—
48	C₂×#681	Q₈	1	4	yes	—
49	C₂²×#168	—	2	2	no	—
50	C₂²×#168	—	2	2	no	—
51	C₅×C₄⋊C₈	—	2	4	no	—
52	C₅×C₄⋊C₈	—	2	4	no	—
53	C₄⋊#168	—	2	4	no	—
54	C₄⋊#168	—	2	4	no	—
55	#3935	—	2	4	no	—
56	C₄×#168	—	2	4	no	—
57	C₂×#423	—	5	1	no	—
58	C₂²×C₄⋊C₈	D₅	1	1	yes	characteristic
59	C₂×C₄×M₁₆	—	5	1	no	—
60	C₂×#341	—	5	4	no	—
61	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
62	C₂²×C₄×C₅	—	2	4	no	—
63	C₂²×C₄×C₅	—	4	2	no	—
64	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
66	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄×C₅	C₄⋊C₄	1	2	yes	—
68	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₅	#41	1	2	yes	—
81	C₂²×C₄×C₅	#41	1	2	yes	—
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	8	no	—
84	C₂×C₅×C₈	—	4	4	no	—
85	C₂×C₅×C₈	—	4	2	no	—
86	C₂×C₅×C₈	—	4	2	no	—
87	C₄²×C₅	C₄²	1	2	yes	—
88	C₄²×C₅	C₄²	1	2	yes	—
89	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
90	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
91	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
93	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
94	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
95	#681	—	2	4	no	—
96	#681	—	2	4	no	—
97	#681	—	2	4	no	—
98	#681	—	2	8	no	—
99	#681	—	4	2	no	—
100	#681	—	4	2	no	—
101	#681	C₄⋊C₄	1	8	yes	—
102	C₂×#168	—	2	4	no	—
103	C₂×#168	—	2	4	no	—
104	C₂×#168	—	2	4	no	—
105	C₂×#168	—	2	4	no	—
106	C₂×#168	—	4	4	no	—
107	C₂²×M₁₆	—	5	2	no	—
108	C₂²×M₁₆	—	5	2	no	—
109	C₂³×C₈	C₂×D₅	1	2	yes	—
110	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
111	#423	—	5	4	no	—
112	#423	—	10	2	no	—
113	C₂×C₄⋊C₈	—	2	2	no	—
114	C₂×C₄⋊C₈	—	2	2	no	—
115	C₂×C₄⋊C₈	—	10	1	no	—
116	C₂×C₄⋊C₈	—	10	1	no	—
117	C₂×C₄⋊C₈	Dic₅	1	2	yes	—
118	C₂×C₄⋊C₈	Dic₅	1	2	yes	—
119	C₄×M₁₆	—	5	4	no	—
120	C₄×M₁₆	—	10	2	no	—
121	C₂×#97	—	10	1	no	—
122	C₂×C₄×C₈	—	10	1	no	—
123	#341	—	10	8	no	—
124	C₂³×C₅	—	2	2	no	—
125	C₂³×C₅	—	2	2	no	—
126	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
129	C₂³×C₅	#125	1	2	yes	—
130	C₂³×C₅	C₄⋊Q₈	1	2	yes	—
131	C₂×C₄×C₅	—	2	2	no	—
132	C₂×C₄×C₅	—	2	2	no	—
133	C₂×C₄×C₅	—	2	2	no	—
134	C₂×C₄×C₅	—	2	4	no	—
135	C₂×C₄×C₅	—	2	4	no	—
136	C₂×C₄×C₅	—	4	2	no	—
137	C₂×C₄×C₅	—	4	2	no	—
138	C₂×C₄×C₅	#95	1	2	yes	—
139	C₂×C₄×C₅	#95	1	2	yes	—
140	C₂×C₄×C₅	#95	1	2	yes	—
141	C₂×C₄×C₅	#95	1	2	yes	—
142	C₂×C₄×C₅	#108	1	4	yes	—
143	C₂×C₄×C₅	C₂×C₄²	1	1	yes	characteristic
144	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
145	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
147	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
148	C₂×C₄×C₅	#117	1	2	yes	—
149	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
150	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
151	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
152	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
153	C₂×C₄×C₅	#121	1	2	yes	—
154	C₂×C₄×C₅	#121	1	2	yes	—
155	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
156	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
157	C₂×C₄×C₅	#124	1	1	yes	characteristic
158	C₂×C₄×C₅	#124	1	1	yes	characteristic
159	C₂×C₄×C₅	#127	1	1	yes	characteristic
160	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
161	C₅×C₈	—	4	4	no	—
162	#168	—	2	8	no	—
163	#168	—	4	4	no	—
164	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
165	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
166	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
167	C₂×M₁₆	—	5	4	no	—
168	C₂×M₁₆	—	5	4	no	—
169	C₂×M₁₆	—	5	4	no	—
170	C₂×M₁₆	—	5	4	no	—
171	C₂×M₁₆	—	10	2	no	—
172	C₂×M₁₆	—	10	2	no	—
173	C₂×M₁₆	—	10	2	no	—
174	C₂×M₁₆	—	10	2	no	—
175	C₂×M₁₆	—	10	4	no	—
176	C₂×M₁₆	—	10	4	no	—
177	C₂²×C₈	—	2	4	no	—
178	C₂²×C₈	—	2	4	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	—	2	2	no	—
181	C₂²×C₈	—	10	2	no	—
182	C₂²×C₈	—	10	2	no	—
183	C₂²×C₈	C₂×Dic₅	1	4	yes	—
184	C₂×C₄²	C₄×D₅	1	2	yes	—
185	C₂×C₄²	C₄×D₅	1	2	yes	—
186	C₂×C₄²	D₂₀	1	1	yes	characteristic
187	C₂×C₄²	D₂₀	1	1	yes	characteristic
188	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
189	C₂×C₄²	C₂×Dic₅	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
191	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
192	C₄⋊C₈	—	2	4	no	—
193	C₄⋊C₈	—	2	4	no	—
194	C₄⋊C₈	—	10	4	no	—
195	C₄⋊C₈	—	10	4	no	—
196	#97	—	10	4	no	—
197	C₄×C₈	—	10	4	no	—
198	C₂²×C₅	—	2	4	no	—
199	C₂²×C₅	—	2	4	no	—
200	C₂²×C₅	—	2	1	no	—
201	C₂²×C₅	—	2	1	no	—
202	C₂²×C₅	—	2	1	no	—
203	C₂²×C₅	—	2	1	no	—
204	C₂²×C₅	C₂×#95	1	1	yes	characteristic
205	C₂²×C₅	#382	1	1	yes	characteristic
206	C₂²×C₅	#384	1	1	yes	characteristic
207	C₂²×C₅	#389	1	2	yes	—
208	C₂²×C₅	#389	1	2	yes	—
209	C₂²×C₅	C₂×#108	1	2	yes	—
210	C₂²×C₅	#430	1	2	yes	—
211	C₄×C₅	—	2	2	no	—
212	C₄×C₅	—	2	2	no	—
213	C₄×C₅	—	2	2	no	—
214	C₄×C₅	#341	1	4	yes	complemented
215	C₄×C₅	C₄×C₄⋊C₄	1	2	yes	—
216	C₄×C₅	#380	1	2	yes	—
217	C₄×C₅	#390	1	1	yes	characteristic
218	C₄×C₅	#390	1	1	yes	characteristic
219	C₄×C₅	#391	1	1	yes	characteristic
220	C₄×C₅	#391	1	1	yes	characteristic
221	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	4	2	no	—
224	C₂²×C₄	#683	1	2	yes	—
225	C₂²×C₄	#683	1	2	yes	—
226	C₂²×C₄	#683	1	2	yes	—
227	C₂²×C₄	#683	1	2	yes	—
228	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
229	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
230	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
231	C₂²×C₄	C₄⋊Dic₅	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
233	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
235	C₂²×C₄	D₄×D₅	1	2	yes	—
236	C₂²×C₄	#711	1	2	yes	—
237	C₂²×C₄	Q₈×D₅	1	2	yes	—
238	C₂²×C₄	#713	1	2	yes	—
239	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
242	M₁₆	—	5	8	no	—
243	M₁₆	—	10	4	no	—
244	M₁₆	—	10	4	no	—
245	M₁₆	—	10	4	no	—
246	M₁₆	—	10	8	no	—
247	M₁₆	—	20	2	no	—
248	M₁₆	—	20	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	8	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	4	4	no	—
253	C₂×C₈	—	4	2	no	—
254	C₂×C₈	—	10	4	no	—
255	C₂×C₈	—	10	4	no	—
256	C₂×C₈	—	10	4	no	—
257	C₂×C₈	—	10	4	no	—
258	C₂×C₈	—	20	4	no	—
259	C₄²	C₄×Dic₅	1	2	yes	—
260	C₄²	C₄×Dic₅	1	2	yes	—
261	C₄²	#685	1	2	yes	—
262	C₄²	#685	1	2	yes	—
263	C₄²	#685	1	2	yes	—
264	C₄²	#685	1	2	yes	—
265	C₄²	C₂²⋊Dic₅	1	2	yes	—
266	C₄²	C₂²⋊Dic₅	1	2	yes	—
267	C₂×C₅	—	2	1	no	—
268	C₂×C₅	—	2	1	no	—
269	C₂×C₅	—	2	2	no	—
270	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
271	C₂×C₅	C₂×#341	1	1	yes	characteristic
272	C₂×C₅	#1738	1	1	yes	characteristic
273	C₂×C₅	#1950	1	4	yes	—
274	C₂³	—	2	2	no	—
275	C₂³	—	2	2	no	—
276	C₂³	#4034	1	2	yes	—
277	C₂³	#4036	1	2	yes	—
278	C₂³	C₂×#683	1	1	yes	characteristic
279	C₂³	C₂×#683	1	1	yes	characteristic
280	C₂³	C₂×C₄⋊Dic₅	1	1	yes	characteristic
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	#3951	1	4	yes	—
289	C₂×C₄	#3963	1	2	yes	—
290	C₂×C₄	#3963	1	2	yes	—
291	C₂×C₄	#3963	1	2	yes	—
292	C₂×C₄	#3963	1	2	yes	—
293	C₂×C₄	#4033	1	2	yes	—
294	C₂×C₄	D₅×C₄⋊C₄	1	2	yes	—
295	C₂×C₄	#4038	1	2	yes	—
296	C₂×C₄	#4039	1	2	yes	—
297	C₂×C₄	#4041	1	2	yes	—
298	C₂×C₄	#4043	1	2	yes	—
299	C₂×C₄	C₂×C₄×Dic₅	1	1	yes	characteristic
300	C₂×C₄	C₂×#685	1	1	yes	characteristic
301	C₂×C₄	C₂×#685	1	1	yes	characteristic
302	C₂×C₄	D₄×Dic₅	1	2	yes	—
303	C₂×C₄	#4082	1	1	yes	characteristic
304	C₂×C₄	#4084	1	2	yes	—
305	C₂×C₄	#4086	1	1	yes	characteristic
306	C₂×C₄	#4090	1	1	yes	characteristic
307	C₂×C₄	Q₈×Dic₅	1	2	yes	—
308	C₂×C₄	#4092	1	2	yes	—
309	C₂×C₄	#4093	1	1	yes	characteristic
310	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
311	C₈	—	4	4	no	—
312	C₈	—	10	8	no	—
313	C₈	—	20	4	no	—
314	C₅	#10853	1	1	yes	characteristic, complemented, Sylow
315	C₂²	—	2	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	4	no	—
320	C₂²	—	2	4	no	—
321	C₂²	#65262	1	2	yes	—
322	C₂²	#65265	1	1	yes	characteristic
323	C₂²	#65266	1	1	yes	characteristic
324	C₂²	#65274	1	2	yes	—
325	C₂²	C₂×#3951	1	2	yes	—
326	C₂²	#65414	1	2	yes	—
327	C₂²	C₂×#3963	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	#64772	1	4	yes	complemented
332	C₄	#65263	1	1	yes	characteristic
333	C₄	C₄⋊C₄×Dic₅	1	2	yes	—
334	C₄	#65276	1	2	yes	—
335	C₄	#65277	1	1	yes	characteristic
336	C₄	#65510	1	1	yes	characteristic
337	C₄	#65519	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	2	1	no	—
340	C₂	—	2	1	no	—
341	C₂	#10601267	1	4	yes	—
342	C₂	#10602356	1	1	yes	characteristic
343	C₂	C₂×#64772	1	1	yes	characteristic
344	C₂	#10602791	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₀
commutator	C₂×C₄×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂²×C₅×C₄⋊C₈	C₂
socle	C₂³×C₅	C₂×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂²×C₄⋊C₈	D₅
hypocenter	C₅	#10853
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
214. C₄×C₅	#341	semidirect
314. C₅	#10853	semidirect
331. C₄	#64772	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₅×C₄⋊C₈	2
C₂×#65213	2
C₂×C₄×#681	2
C₂×#64772	2
#10853	5