Subgroups — #12104237

Order 1280.

780 conjugacy classes of subgroups.

The subgroup lattice has 370 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	#10613780	C₂	1	1	yes	maximal, characteristic
3	C₂×#65214	C₂	1	1	yes	maximal, characteristic
4	#10602794	C₂	1	4	yes	maximal
5	#10600668	C₂	1	4	yes	maximal, complemented
6	C₄×#3954	C₂	1	4	yes	maximal, complemented
7	C₂²×C₅×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₄²×C₅	C₂²	1	1	yes	characteristic
9	C₄×C₅×M₁₆	—	2	1	no	—
10	C₄×C₅×M₁₆	—	2	1	no	—
11	C₄×C₅×M₁₆	C₂²	1	2	yes	—
12	C₄×C₅×M₁₆	C₂²	1	2	yes	—
13	C₂×C₅×#97	—	2	1	no	—
14	C₂×C₄×C₅×C₈	—	2	1	no	—
15	#65496	C₂²	1	2	yes	—
16	C₂×C₂²⋊#168	C₂²	1	2	yes	—
17	C₂×#3954	C₂²	1	8	yes	—
18	#65215	—	2	1	no	—
19	#65215	—	2	1	no	—
20	#65214	—	2	1	no	—
21	#65214	—	2	1	no	—
22	#65214	C₄	1	2	yes	—
23	#65214	C₄	1	2	yes	—
24	#65214	C₄	1	2	yes	—
25	#65214	C₄	1	2	yes	—
26	#65213	C₂²	1	2	yes	—
27	C₂×C₄⋊#168	C₂²	1	2	yes	—
28	C₄×#681	C₂²	1	2	yes	—
29	C₂×C₄×#168	C₂²	1	2	yes	—
30	#64778	C₂²	1	8	yes	—
31	#21370	—	5	1	no	maximal, Sylow
32	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
33	C₂³×C₄×C₅	Q₈	1	1	yes	characteristic
34	C₂³×C₄×C₅	C₂³	1	1	yes	characteristic
35	C₂×C₅×M₁₆	—	2	2	no	—
36	C₂×C₅×M₁₆	—	2	8	no	—
37	C₂×C₅×M₁₆	C₂³	1	4	yes	—
38	C₂²×C₅×C₈	—	2	2	no	—
39	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
48	C₂×C₄²×C₅	Q₈	1	1	yes	characteristic
49	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
51	C₂×#681	C₂³	1	4	yes	—
52	C₂²×#168	C₂³	1	4	yes	—
53	C₅×#97	—	2	2	no	—
54	C₅×#97	—	2	2	no	—
55	C₄×C₅×C₈	—	2	2	no	—
56	C₄×C₅×C₈	—	2	2	no	—
57	C₂²⋊#168	—	2	2	no	—
58	C₂²⋊#168	—	2	2	no	—
59	C₂²⋊#168	C₂×C₄	1	4	yes	—
60	C₂²⋊#168	C₂×C₄	1	4	yes	—
61	#3954	—	2	16	no	—
62	C₄⋊#168	—	2	2	no	—
63	C₄⋊#168	—	2	2	no	—
64	C₄⋊#168	C₂×C₄	1	4	yes	—
65	C₄⋊#168	C₂×C₄	1	4	yes	—
66	#3935	—	2	2	no	—
67	C₄×#168	—	2	2	no	—
68	C₄×#168	C₂×C₄	1	4	yes	—
69	C₄×#168	C₂×C₄	1	4	yes	—
70	#2920	—	5	1	no	—
71	C₂×#431	—	5	1	no	—
72	C₂×C₄×M₁₆	D₅	1	1	yes	characteristic
73	#1950	—	5	4	no	—
74	#1855	—	5	4	no	—
75	C₄×#108	—	5	4	no	—
76	C₂⁴×C₅	C₂×Q₈	1	1	yes	characteristic
77	C₂²×C₄×C₅	—	2	4	no	—
78	C₂²×C₄×C₅	—	2	4	no	—
79	C₂²×C₄×C₅	—	4	2	no	—
80	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
86	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
87	C₂²×C₄×C₅	C₄⋊C₄	1	1	yes	characteristic
88	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
91	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
92	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
93	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
94	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
95	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
96	C₂²×C₄×C₅	C₂×Q₈	1	1	yes	characteristic
97	C₂²×C₄×C₅	C₂⁴	1	1	yes	characteristic
98	C₅×M₁₆	—	2	8	no	—
99	C₅×M₁₆	—	2	4	no	—
100	C₅×M₁₆	#41	1	8	yes	—
101	C₂×C₅×C₈	—	2	8	no	—
102	C₂×C₅×C₈	—	2	4	no	—
103	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
104	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
105	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
106	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
107	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
108	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
109	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
110	C₄²×C₅	C₄⋊C₄	1	1	yes	characteristic
111	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
112	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
113	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
115	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
118	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
119	#681	—	2	8	no	—
120	C₂×#168	—	2	8	no	—
121	C₂×#168	—	2	4	no	—
122	C₂×#168	C₂²×C₄	1	8	yes	—
123	C₂²×M₁₆	C₂×D₅	1	2	yes	—
124	C₂²×C₄²	C₂×D₅	1	1	yes	characteristic
125	#432	—	10	1	no	—
126	#432	—	10	1	no	—
127	#431	—	5	2	no	—
128	#431	—	5	2	no	—
129	#431	—	5	2	no	—
130	#431	—	5	2	no	—
131	#431	—	10	1	no	—
132	#431	—	10	1	no	—
133	C₂×#108	—	5	8	no	—
134	#423	—	5	2	no	—
135	C₂×C₄⋊C₈	—	5	2	no	—
136	#407	—	5	2	no	—
137	C₂×C₂²⋊C₈	—	5	2	no	—
138	C₄×M₁₆	—	2	1	no	—
139	C₄×M₁₆	—	2	1	no	—
140	C₄×M₁₆	—	5	2	no	—
141	C₄×M₁₆	C₂×D₅	1	2	yes	—
142	C₄×M₁₆	C₂×D₅	1	2	yes	—
143	C₂×#97	—	2	1	no	—
144	C₂×C₄×C₈	—	2	1	no	—
145	C₂×C₄×C₈	—	5	2	no	—
146	#341	—	5	8	no	—
147	C₂³×C₅	—	2	2	no	—
148	C₂³×C₅	—	2	2	no	—
149	C₂³×C₅	—	4	1	no	—
150	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂³×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
152	C₂³×C₅	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₅	—	2	2	no	—
154	C₂×C₄×C₅	—	2	2	no	—
155	C₂×C₄×C₅	—	2	2	no	—
156	C₂×C₄×C₅	—	2	2	no	—
157	C₂×C₄×C₅	—	2	2	no	—
158	C₂×C₄×C₅	—	2	2	no	—
159	C₂×C₄×C₅	—	2	2	no	—
160	C₂×C₄×C₅	—	2	2	no	—
161	C₂×C₄×C₅	—	2	2	no	—
162	C₂×C₄×C₅	—	2	2	no	—
163	C₂×C₄×C₅	—	2	2	no	—
164	C₂×C₄×C₅	—	2	2	no	—
165	C₂×C₄×C₅	—	2	2	no	—
166	C₂×C₄×C₅	—	4	1	no	—
167	C₂×C₄×C₅	—	4	1	no	—
168	C₂×C₄×C₅	—	4	1	no	—
169	C₂×C₄×C₅	#108	1	4	yes	complemented
170	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
171	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
173	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
174	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
175	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
176	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
177	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
179	C₂×C₄×C₅	C₂×C₄⋊C₄	1	1	yes	characteristic
180	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
181	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
182	C₂×C₄×C₅	C₂×#41	1	4	yes	—
183	C₅×C₈	—	2	8	no	—
184	#168	—	2	8	no	—
185	#168	—	4	4	no	—
186	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
187	C₂³×C₄	Dic₁₀	1	1	yes	characteristic
188	C₂³×C₄	C₂²×D₅	1	1	yes	characteristic
189	C₂×M₁₆	—	2	2	no	—
190	C₂×M₁₆	—	2	8	no	—
191	C₂×M₁₆	—	5	4	no	—
192	C₂×M₁₆	C₂²×D₅	1	4	yes	—
193	C₂²×C₈	—	2	2	no	—
194	C₂²×C₈	—	5	4	no	—
195	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
196	C₂×C₄²	Dic₁₀	1	1	yes	characteristic
197	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
198	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
199	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
200	C₂×C₄²	C₄×D₅	1	1	yes	characteristic
201	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
202	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
203	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
204	C₂×C₄²	C₂²⋊D₅	1	1	yes	characteristic
205	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
206	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
207	#108	—	10	16	no	—
208	C₄⋊C₈	—	5	4	no	—
209	C₄⋊C₈	—	5	4	no	—
210	C₄⋊C₈	—	10	2	no	—
211	C₄⋊C₈	—	10	2	no	—
212	C₂²⋊C₈	—	5	4	no	—
213	C₂²⋊C₈	—	5	4	no	—
214	C₂²⋊C₈	—	10	2	no	—
215	C₂²⋊C₈	—	10	2	no	—
216	#97	—	2	2	no	—
217	#97	—	2	2	no	—
218	#97	—	10	2	no	—
219	C₄×C₈	—	2	2	no	—
220	C₄×C₈	—	2	2	no	—
221	C₄×C₈	—	5	4	no	—
222	C₄×C₈	—	5	4	no	—
223	C₄×C₈	—	10	2	no	—
224	C₂²×C₅	—	2	1	no	—
225	C₂²×C₅	—	2	1	no	—
226	C₂²×C₅	—	2	1	no	—
227	C₂²×C₅	—	2	1	no	—
228	C₂²×C₅	—	2	1	no	—
229	C₂²×C₅	—	2	1	no	—
230	C₂²×C₅	—	2	1	no	—
231	C₂²×C₅	—	4	4	no	—
232	C₂²×C₅	C₂×#108	1	2	yes	—
233	C₂²×C₅	C₂²×C₄⋊C₄	1	1	yes	characteristic
234	C₂²×C₅	#533	1	2	yes	—
235	C₄×C₅	—	2	1	no	—
236	C₄×C₅	—	2	1	no	—
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	—	2	1	no	—
242	C₄×C₅	C₂×#108	1	2	yes	—
243	C₄×C₅	C₂×#108	1	2	yes	—
244	C₄×C₅	#430	1	2	yes	—
245	C₄×C₅	C₄×#41	1	2	yes	—
246	C₄×C₅	#525	1	2	yes	—
247	C₂⁴	C₂×Dic₁₀	1	1	yes	characteristic
248	C₂²×C₄	—	2	4	no	—
249	C₂²×C₄	—	2	4	no	—
250	C₂²×C₄	—	4	2	no	—
251	C₂²×C₄	#683	1	1	yes	characteristic
252	C₂²×C₄	#683	1	1	yes	characteristic
253	C₂²×C₄	#683	1	1	yes	characteristic
254	C₂²×C₄	#683	1	1	yes	characteristic
255	C₂²×C₄	#683	1	1	yes	characteristic
256	C₂²×C₄	#683	1	1	yes	characteristic
257	C₂²×C₄	#683	1	1	yes	characteristic
258	C₂²×C₄	#683	1	1	yes	characteristic
259	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
260	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
261	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
262	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
263	C₂²×C₄	C₂×Dic₁₀	1	1	yes	characteristic
264	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
265	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
266	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
268	C₂²×C₄	C₂³×D₅	1	1	yes	characteristic
269	M₁₆	—	2	8	no	—
270	M₁₆	—	2	4	no	—
271	M₁₆	—	10	8	no	—
272	M₁₆	#709	1	8	yes	—
273	C₂×C₈	—	2	8	no	—
274	C₂×C₈	—	2	4	no	—
275	C₂×C₈	—	5	8	no	—
276	C₂×C₈	—	10	8	no	—
277	C₂×C₈	—	10	4	no	—
278	C₄²	#683	1	1	yes	characteristic
279	C₄²	#683	1	1	yes	characteristic
280	C₄²	#683	1	1	yes	characteristic
281	C₄²	#683	1	1	yes	characteristic
282	C₄²	#683	1	1	yes	characteristic
283	C₄²	#683	1	1	yes	characteristic
284	C₄²	#683	1	1	yes	characteristic
285	C₄²	#683	1	1	yes	characteristic
286	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
287	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
288	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
289	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
290	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
291	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
292	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
293	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
294	C₂×C₅	—	2	2	no	—
295	C₂×C₅	—	2	2	no	—
296	C₂×C₅	—	4	1	no	—
297	C₂×C₅	#2296	1	1	yes	characteristic
298	C₂×C₅	C₂²×#108	1	1	yes	characteristic
299	C₂×C₅	C₂×#430	1	1	yes	characteristic
300	C₂³	—	2	2	no	—
301	C₂³	—	2	2	no	—
302	C₂³	—	4	1	no	—
303	C₂³	C₂×#683	1	1	yes	characteristic
304	C₂³	C₂×#683	1	1	yes	characteristic
305	C₂³	C₂²×Dic₁₀	1	1	yes	characteristic
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	#3954	1	4	yes	complemented
323	C₂×C₄	C₂×#683	1	1	yes	characteristic
324	C₂×C₄	C₂×#683	1	1	yes	characteristic
325	C₂×C₄	C₂×#683	1	1	yes	characteristic
326	C₂×C₄	C₂×#683	1	1	yes	characteristic
327	C₂×C₄	C₂×#683	1	1	yes	characteristic
328	C₂×C₄	C₂×#683	1	1	yes	characteristic
329	C₂×C₄	C₂×#683	1	1	yes	characteristic
330	C₂×C₄	C₂×#683	1	1	yes	characteristic
331	C₂×C₄	C₂×#683	1	1	yes	characteristic
332	C₂×C₄	C₂×#683	1	1	yes	characteristic
333	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
334	C₂×C₄	C₂×#709	1	4	yes	—
335	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
336	C₈	—	2	8	no	—
337	C₈	—	10	8	no	—
338	C₈	—	20	4	no	—
339	C₅	#21370	1	1	yes	characteristic, complemented, Sylow
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	4	4	no	—
348	C₂²	C₂×#3954	1	2	yes	—
349	C₂²	#65804	1	2	yes	—
350	C₂²	C₂²×#683	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	2	1	no	—
357	C₄	—	2	1	no	—
358	C₄	C₂×#3954	1	2	yes	—
359	C₄	C₂×#3954	1	2	yes	—
360	C₄	#65413	1	2	yes	—
361	C₄	C₄×#709	1	2	yes	—
362	C₄	#66125	1	2	yes	—
363	C₂	—	2	2	no	—
364	C₂	—	2	2	no	—
365	C₂	—	4	1	no	—
366	C₂	#10613800	1	1	yes	characteristic
367	C₂	C₂²×#3954	1	1	yes	characteristic
368	C₂	C₂×#65413	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	C₂×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅
Fitting	C₂×C₄×C₅×M₁₆	C₂
socle	C₂²×C₅	C₂²×C₄⋊C₄
prime radical	C₂⁴×C₅	C₂×Q₈
cosocle	C₂²×C₄×C₅	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅
hypocenter	C₅	#21370
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
5. #10600668	C₂	semidirect
6. C₄×#3954	C₂	semidirect
169. C₂×C₄×C₅	#108	semidirect
322. C₂×C₄	#3954	semidirect
339. C₅	#21370	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
#10613780	2
C₂×#65214	2
#10602794	2
#10600668	2
C₄×#3954	2
#21370	5