Subgroups — #12005177

Order 1280.

918 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₅×M₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#65224	C₂	1	1	yes	maximal, characteristic
3	C₂×#65213	C₂	1	1	yes	maximal, characteristic
4	C₂×#64774	C₂	1	4	yes	maximal
5	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
6	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
7	C₂²×C₄⋊Dic₅	C₄	1	4	yes	—
8	C₂²×C₄⋊Dic₅	C₂²	1	2	yes	—
9	C₂²×#681	C₂²	1	2	yes	—
10	C₄×C₅×M₁₆	—	2	2	no	—
11	C₄×C₅×M₁₆	C₄	1	4	yes	complemented
12	C₂×C₅×#97	—	2	1	no	—
13	C₂×C₄×C₅×C₈	—	2	1	no	—
14	#65224	—	2	2	no	—
15	#65224	—	2	1	no	—
16	#65224	—	2	1	no	—
17	#65213	—	2	2	no	—
18	#65213	C₄	1	4	yes	complemented
19	C₂×C₄⋊#168	—	2	2	no	—
20	#64774	—	2	8	no	—
21	#10857	—	5	1	no	maximal, Sylow
22	C₂³×C₄×C₅	C₂×C₄	1	2	yes	—
23	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
24	C₂³×Dic₅	C₂×C₄	1	4	yes	—
25	C₂×C₅×M₁₆	—	2	2	no	—
26	C₂×C₅×M₁₆	—	2	2	no	—
27	C₂×C₅×M₁₆	—	2	2	no	—
28	C₂×C₅×M₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₅×M₁₆	D₄	1	4	yes	—
30	C₂×C₅×M₁₆	Q₈	1	4	yes	—
31	C₂²×C₅×C₈	—	2	2	no	—
32	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅	C₂×C₄	1	2	yes	—
34	C₂×C₄²×C₅	D₄	1	2	yes	—
35	C₂×C₄²×C₅	D₄	1	2	yes	—
36	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
37	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
38	C₂×C₄²×C₅	Q₈	1	2	yes	—
39	C₂×C₄⋊Dic₅	—	2	4	no	—
40	C₂×C₄⋊Dic₅	—	2	4	no	—
41	C₂×C₄⋊Dic₅	—	2	4	no	—
42	C₂×C₄⋊Dic₅	—	2	2	no	—
43	C₂×C₄⋊Dic₅	—	2	2	no	—
44	C₂×C₄⋊Dic₅	—	2	4	no	—
45	C₂×C₄⋊Dic₅	—	4	8	no	—
46	C₂×#681	—	2	2	no	—
47	C₂×#681	—	2	4	no	—
48	C₂×#681	—	2	4	no	—
49	C₂×#681	C₂×C₄	1	4	yes	—
50	C₂²×#168	—	2	2	no	—
51	C₅×#97	—	2	4	no	—
52	C₄×C₅×C₈	—	2	4	no	—
53	C₄⋊#168	—	2	8	no	—
54	C₂×#423	—	5	1	no	—
55	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
56	C₂×#384	—	5	1	no	—
57	C₂×#343	—	5	4	no	—
58	C₂⁴×C₅	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅	—	2	4	no	—
60	C₂²×C₄×C₅	—	2	4	no	—
61	C₂²×C₄×C₅	C₄²	1	4	yes	—
62	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅	C₄⋊C₄	1	4	yes	—
65	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅	C₂×D₄	1	2	yes	—
69	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
72	C₂²×C₄×C₅	C₂×Q₈	1	2	yes	—
73	C₂²×C₄×C₅	#41	1	2	yes	—
74	C₂²×C₄×C₅	#41	1	2	yes	—
75	C₂²×Dic₅	—	2	4	no	—
76	C₂²×Dic₅	—	2	8	no	—
77	C₂²×Dic₅	—	4	8	no	—
78	C₅×M₁₆	—	2	4	no	—
79	C₅×M₁₆	—	2	4	no	—
80	C₅×M₁₆	—	2	4	no	—
81	C₅×M₁₆	C₄⋊C₄	1	8	yes	complemented
82	C₂×C₅×C₈	—	2	4	no	—
83	C₂×C₅×C₈	—	2	4	no	—
84	C₂×C₅×C₈	—	2	4	no	—
85	C₄²×C₅	C₂²⋊C₄	1	4	yes	—
86	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
87	C₄²×C₅	C₂²⋊C₄	1	2	yes	—
88	C₄²×C₅	C₄⋊C₄	1	2	yes	—
89	C₄²×C₅	C₄⋊C₄	1	2	yes	—
90	C₄²×C₅	C₄⋊C₄	1	4	yes	—
91	C₄⋊Dic₅	—	4	8	no	—
92	C₄⋊Dic₅	—	4	8	no	—
93	C₄⋊Dic₅	—	4	4	no	—
94	C₄⋊Dic₅	—	4	2	no	—
95	C₄⋊Dic₅	—	4	2	no	—
96	#681	—	2	8	no	—
97	#681	—	4	4	no	—
98	C₂×#168	—	2	4	no	—
99	C₂×#168	—	4	2	no	—
100	C₂×#168	—	4	2	no	—
101	C₂²×M₁₆	—	5	2	no	—
102	C₂²×M₁₆	C₂×D₅	1	2	yes	—
103	C₂²×C₄⋊C₄	—	5	2	no	—
104	C₂²×C₄⋊C₄	—	5	4	no	—
105	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
106	#423	—	5	4	no	—
107	#423	—	10	2	no	—
108	C₂×C₄⋊C₈	—	10	2	no	—
109	C₄×M₁₆	—	2	2	no	—
110	C₄×M₁₆	Dic₅	1	4	yes	complemented
111	C₂×#97	—	2	1	no	—
112	C₂×C₄×C₈	—	2	1	no	—
113	#384	—	10	2	no	—
114	#384	—	10	1	no	—
115	#384	—	10	1	no	—
116	#343	—	10	8	no	—
117	C₂³×C₅	—	2	2	no	—
118	C₂³×C₅	—	2	2	no	—
119	C₂³×C₅	C₂×C₄²	1	1	yes	characteristic
120	C₂³×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂³×C₅	C₄×D₄	1	2	yes	—
123	C₂³×C₅	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅	—	2	2	no	—
125	C₂×C₄×C₅	—	2	2	no	—
126	C₂×C₄×C₅	—	2	8	no	—
127	C₂×C₄×C₅	—	2	8	no	—
128	C₂×C₄×C₅	#95	1	4	yes	—
129	C₂×C₄×C₅	#95	1	4	yes	—
130	C₂×C₄×C₅	#100	1	4	yes	—
131	C₂×C₄×C₅	#101	1	4	yes	—
132	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅	C₂×C₄⋊C₄	1	2	yes	—
136	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₄×C₅	#117	1	2	yes	—
138	C₂×C₄×C₅	C₄×D₄	1	2	yes	—
139	C₂×C₄×C₅	C₄×Q₈	1	2	yes	—
140	C₂×C₄×C₅	#121	1	2	yes	—
141	C₂×C₄×C₅	#121	1	2	yes	—
142	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₅	#124	1	1	yes	characteristic
145	C₂×C₄×C₅	#124	1	1	yes	characteristic
146	C₂×C₄×C₅	#125	1	2	yes	—
147	C₂×C₄×C₅	#127	1	1	yes	characteristic
148	C₂×C₄×C₅	C₄⋊Q₈	1	2	yes	—
149	C₂×C₄×C₅	C₄⋊Q₈	1	1	yes	characteristic
150	C₂×Dic₅	—	4	4	no	—
151	C₂×Dic₅	—	4	8	no	—
152	C₂×Dic₅	—	8	8	no	—
153	C₅×C₈	—	2	8	no	—
154	#168	—	4	4	no	—
155	C₂³×C₄	—	5	4	no	—
156	C₂³×C₄	C₄×D₅	1	2	yes	—
157	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	—	2	2	no	—
160	C₂×M₁₆	—	2	2	no	—
161	C₂×M₁₆	—	5	4	no	—
162	C₂×M₁₆	—	10	2	no	—
163	C₂×M₁₆	—	10	4	no	—
164	C₂×M₁₆	—	10	4	no	—
165	C₂×M₁₆	Dic₁₀	1	4	yes	—
166	C₂×M₁₆	D₂₀	1	4	yes	—
167	C₂×M₁₆	C₂×Dic₅	1	4	yes	—
168	C₂²×C₈	—	2	2	no	—
169	C₂²×C₈	—	10	2	no	—
170	C₂×C₄⋊C₄	—	10	4	no	—
171	C₂×C₄⋊C₄	—	10	4	no	—
172	C₂×C₄⋊C₄	—	10	4	no	—
173	C₂×C₄⋊C₄	—	10	2	no	—
174	C₂×C₄⋊C₄	—	10	4	no	—
175	C₂×C₄⋊C₄	—	10	2	no	—
176	C₂×C₄⋊C₄	—	20	8	no	—
177	C₂×C₄²	Dic₁₀	1	2	yes	—
178	C₂×C₄²	C₄×D₅	1	2	yes	—
179	C₂×C₄²	D₂₀	1	2	yes	—
180	C₂×C₄²	C₂×Dic₅	1	2	yes	—
181	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
182	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
183	C₂×C₄²	C₂²⋊D₅	1	2	yes	—
184	C₄⋊C₈	—	10	8	no	—
185	#97	—	2	4	no	—
186	C₄×C₈	—	2	4	no	—
187	C₂²×C₅	—	2	2	no	—
188	C₂²×C₅	—	2	2	no	—
189	C₂²×C₅	—	2	2	no	—
190	C₂²×C₅	—	2	2	no	—
191	C₂²×C₅	—	2	4	no	—
192	C₂²×C₅	C₂×#95	1	1	yes	characteristic
193	C₂²×C₅	C₄×C₄⋊C₄	1	2	yes	—
194	C₂²×C₅	#389	1	2	yes	—
195	C₂²×C₅	#390	1	1	yes	characteristic
196	C₂²×C₅	#391	1	1	yes	characteristic
197	C₂²×C₅	C₂×#100	1	2	yes	—
198	C₂²×C₅	C₂×#101	1	2	yes	—
199	C₄×C₅	—	2	4	no	—
200	C₄×C₅	#343	1	8	yes	complemented
201	C₄×C₅	#380	1	2	yes	—
202	C₄×C₅	#382	1	1	yes	characteristic
203	C₄×C₅	#384	1	1	yes	characteristic
204	C₄×C₅	#389	1	2	yes	—
205	C₄×C₅	#390	1	1	yes	characteristic
206	C₄×C₅	#391	1	1	yes	characteristic
207	Dic₅	—	8	4	no	—
208	C₂⁴	C₂×C₄×D₅	1	1	yes	characteristic
209	C₂²×C₄	—	2	4	no	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	10	4	no	—
212	C₂²×C₄	—	10	8	no	—
213	C₂²×C₄	—	20	8	no	—
214	C₂²×C₄	C₄×Dic₅	1	4	yes	—
215	C₂²×C₄	#683	1	4	yes	—
216	C₂²×C₄	#685	1	2	yes	—
217	C₂²×C₄	#685	1	2	yes	—
218	C₂²×C₄	C₂×Dic₁₀	1	2	yes	—
219	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₂₀	1	2	yes	—
222	C₂²×C₄	C₂×D₂₀	1	1	yes	characteristic
223	C₂²×C₄	#709	1	2	yes	—
224	C₂²×C₄	#709	1	2	yes	—
225	C₂²×C₄	C₂²×Dic₅	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
228	M₁₆	—	2	4	no	—
229	M₁₆	—	2	4	no	—
230	M₁₆	—	2	4	no	—
231	M₁₆	—	10	8	no	—
232	M₁₆	—	20	4	no	—
233	M₁₆	C₄⋊Dic₅	1	8	yes	complemented
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	4	no	—
236	C₂×C₈	—	2	4	no	—
237	C₂×C₈	—	10	4	no	—
238	C₂×C₈	—	20	2	no	—
239	C₂×C₈	—	20	2	no	—
240	C₄⋊C₄	—	20	8	no	—
241	C₄⋊C₄	—	20	8	no	—
242	C₄⋊C₄	—	20	2	no	—
243	C₄⋊C₄	—	20	2	no	—
244	C₄⋊C₄	—	20	4	no	—
245	C₄²	#683	1	2	yes	—
246	C₄²	#683	1	2	yes	—
247	C₄²	C₄⋊Dic₅	1	4	yes	—
248	C₄²	#685	1	4	yes	—
249	C₄²	C₂²⋊Dic₅	1	2	yes	—
250	C₄²	C₂²⋊Dic₅	1	2	yes	—
251	C₂×C₅	—	2	2	no	—
252	C₂×C₅	—	2	2	no	—
253	C₂×C₅	C₄⋊#95	1	1	yes	characteristic
254	C₂×C₅	C₂×#343	1	2	yes	—
255	C₂×C₅	#1965	1	1	yes	characteristic
256	C₂×C₅	#1966	1	1	yes	characteristic
257	C₂×C₅	#1997	1	2	yes	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₄×Dic₁₀	1	2	yes	—
261	C₂³	C₄×D₂₀	1	2	yes	—
262	C₂³	C₂×C₄×Dic₅	1	1	yes	characteristic
263	C₂³	C₂×#683	1	1	yes	characteristic
264	C₂³	C₂×#685	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	8	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	20	8	no	—
270	C₂×C₄	—	20	4	no	—
271	C₂×C₄	—	40	8	no	—
272	C₂×C₄	#3955	1	4	yes	—
273	C₂×C₄	#3956	1	4	yes	—
274	C₂×C₄	#3963	1	4	yes	—
275	C₂×C₄	#3963	1	4	yes	—
276	C₂×C₄	C₄×Dic₁₀	1	2	yes	—
277	C₂×C₄	#4015	1	2	yes	—
278	C₂×C₄	#4015	1	1	yes	characteristic
279	C₂×C₄	#4016	1	2	yes	—
280	C₂×C₄	C₄×D₂₀	1	2	yes	—
281	C₂×C₄	#4020	1	1	yes	characteristic
282	C₂×C₄	#4021	1	1	yes	characteristic
283	C₂×C₄	#4021	1	1	yes	characteristic
284	C₂×C₄	C₂×#683	1	1	yes	characteristic
285	C₂×C₄	#4070	1	2	yes	—
286	C₂×C₄	#4070	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₅	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊Dic₅	1	2	yes	—
289	C₂×C₄	#4072	1	2	yes	—
290	C₂×C₄	C₂×#685	1	1	yes	characteristic
291	C₂×C₄	#4076	1	2	yes	—
292	C₂×C₄	#4076	1	2	yes	—
293	C₂×C₄	C₂×C₂²⋊Dic₅	1	1	yes	characteristic
294	C₈	—	2	8	no	—
295	C₈	—	20	4	no	—
296	C₅	#10857	1	1	yes	characteristic, complemented, Sylow
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	2	no	—
300	C₂²	—	2	2	no	—
301	C₂²	—	2	4	no	—
302	C₂²	#65217	1	2	yes	—
303	C₂²	#65218	1	1	yes	characteristic
304	C₂²	C₄×C₄⋊Dic₅	1	2	yes	—
305	C₂²	#65228	1	1	yes	characteristic
306	C₂²	C₂×#3955	1	2	yes	—
307	C₂²	C₂×#3956	1	2	yes	—
308	C₂²	C₂×#3963	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	40	4	no	—
311	C₄	#64774	1	8	yes	complemented
312	C₄	#65217	1	2	yes	—
313	C₄	#65218	1	1	yes	characteristic
314	C₄	#65224	1	1	yes	characteristic
315	C₄	#65225	1	1	yes	characteristic
316	C₄	#65228	1	1	yes	characteristic
317	C₄	#65501	1	2	yes	—
318	C₂	—	2	2	no	—
319	C₂	—	2	2	no	—
320	C₂	#10600669	1	1	yes	characteristic
321	C₂	#10600670	1	1	yes	characteristic
322	C₂	#10600672	1	2	yes	—
323	C₂	#10602328	1	1	yes	characteristic
324	C₂	C₂×#64774	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×#685
commutator	C₂³×C₅	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₅
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂³×C₅	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₅	C₂²×C₄
cosocle	C₂³×C₄×C₅	C₂³
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#10857
LCS	G ▷ C₂³×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
11. C₄×C₅×M₁₆	C₄	semidirect
18. #65213	C₄	semidirect
81. C₅×M₁₆	C₄⋊C₄	semidirect
110. C₄×M₁₆	Dic₅	semidirect
200. C₄×C₅	#343	semidirect
233. M₁₆	C₄⋊Dic₅	semidirect
296. C₅	#10857	semidirect
311. C₄	#64774	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×M₁₆	2
C₂×#65224	2
C₂×#65213	2
C₂×#64774	2
#10857	5