Subgroups — M₁₆×#688 #12070740

Order 1280.

716 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅×Q₈×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#688	C₂	1	1	yes	maximal, characteristic
3	M₁₆×D₂₀	C₂	1	1	yes	maximal, characteristic
4	#10602415	C₂	1	1	yes	maximal, characteristic, complemented
5	#10601746	C₂	1	2	yes	maximal, complemented
6	#10601743	C₂	1	2	yes	maximal, complemented
7	#10601726	C₂	1	2	yes	maximal, complemented
8	C₈×#688	C₂	1	2	yes	maximal, complemented
9	#10601289	C₂	1	1	yes	maximal, characteristic
10	#10601283	C₂	1	1	yes	maximal, characteristic, complemented
11	M₁₆×#168	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄×C₅×Q₈	C₂²	1	1	yes	characteristic
13	C₂²×#688	C₄	1	1	yes	characteristic
14	C₂×D₅×M₁₆	—	2	1	no	—
15	C₂×C₄×D₂₀	C₂²	1	1	yes	characteristic
16	C₅×#446	—	2	2	no	—
17	C₅×#446	C₂²	1	2	yes	complemented
18	C₅×C₈×Q₈	C₂²	1	2	yes	complemented
19	C₅×#423	—	2	1	no	—
20	C₅×#423	C₂²	1	1	yes	characteristic
21	C₄×C₅×M₁₆	—	2	1	no	—
22	C₄×C₅×M₁₆	C₂²	1	1	yes	characteristic
23	C₂×#3967	C₄	1	1	yes	characteristic
24	#65415	—	2	1	no	—
25	C₄×#688	C₄	1	1	yes	characteristic
26	C₄×#688	C₄	1	1	yes	characteristic
27	C₄×#688	C₄	1	1	yes	characteristic
28	C₄×#688	C₄	1	1	yes	characteristic
29	C₄×#688	C₂²	1	1	yes	characteristic
30	C₄×#688	C₂²	1	1	yes	characteristic
31	C₄×#688	C₂²	1	1	yes	characteristic
32	C₄×#688	C₂²	1	1	yes	characteristic
33	C₂×#3941	C₄	1	1	yes	characteristic
34	C₂×#3940	C₄	1	1	yes	characteristic
35	#65214	C₂²	1	1	yes	characteristic
36	#65213	C₂²	1	1	yes	characteristic
37	C₂×C₄×#168	C₂²	1	1	yes	characteristic
38	#64995	—	2	2	no	—
39	#64977	C₂²	1	2	yes	—
40	C₈×D₂₀	C₂²	1	2	yes	—
41	Q₈⋊#168	C₂²	1	2	yes	complemented
42	Q₈⋊#168	C₂²	1	2	yes	complemented
43	#64703	C₂²	1	2	yes	—
44	#64703	C₂²	1	2	yes	—
45	#64700	C₂²	1	2	yes	complemented
46	#64700	C₂²	1	2	yes	complemented
47	#64675	C₂²	1	2	yes	complemented
48	C₈×#168	C₂²	1	2	yes	complemented
49	M₁₆×SD₁₆	—	5	1	no	maximal, Sylow
50	C₂²×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
51	C₂²×D₂₀	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×D₅	—	2	1	no	—
53	C₂×C₅×M₁₆	—	2	1	no	—
54	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
55	C₂×C₅×M₁₆	D₄	1	1	yes	characteristic
56	C₄×C₅×Q₈	—	2	1	no	—
57	C₄×C₅×Q₈	—	2	1	no	—
58	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₅×Q₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
61	C₄×C₅×Q₈	C₂³	1	1	yes	characteristic
62	C₂×C₅×C₄⋊C₄	—	2	1	no	—
63	C₂×C₅×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅	—	2	1	no	—
65	C₂×C₄²×C₅	C₂³	1	1	yes	characteristic
66	C₂×#688	—	2	4	no	—
67	C₂×#688	C₂×C₄	1	1	yes	characteristic
68	C₂×#688	C₂×C₄	1	1	yes	characteristic
69	C₂×#688	C₂×C₄	1	1	yes	characteristic
70	C₂×#688	C₂×C₄	1	1	yes	characteristic
71	C₂×#685	—	2	1	no	—
72	C₂×C₄⋊Dic₅	C₂×C₄	1	1	yes	characteristic
73	C₂×#681	—	2	1	no	—
74	C₂²×#168	C₂×C₄	1	1	yes	characteristic
75	C₂²×#168	C₂×C₄	1	1	yes	characteristic
76	D₅×M₁₆	—	4	4	no	—
77	C₂×#676	—	2	2	no	—
78	C₂×C₈×D₅	—	2	2	no	—
79	C₄×D₂₀	—	2	1	no	—
80	C₄×D₂₀	—	2	1	no	—
81	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
82	C₄×D₂₀	C₂×C₄	1	1	yes	characteristic
83	C₄×D₂₀	C₂³	1	1	yes	characteristic
84	C₄×D₂₀	C₂³	1	1	yes	characteristic
85	C₅×C₄⋊C₈	—	2	2	no	—
86	C₅×C₄⋊C₈	—	2	2	no	—
87	C₅×C₄⋊C₈	—	2	1	no	—
88	C₅×C₄⋊C₈	C₂³	1	2	yes	—
89	C₅×#97	—	2	2	no	—
90	C₅×#97	D₄	1	2	yes	—
91	C₄×C₅×C₈	—	2	2	no	—
92	C₄×C₅×C₈	C₂³	1	2	yes	—
93	#3967	C₂×C₄	1	1	yes	characteristic
94	#3967	C₂×C₄	1	1	yes	characteristic
95	#3967	C₂×C₄	1	1	yes	characteristic
96	#3967	C₂×C₄	1	1	yes	characteristic
97	C₂²⋊#168	—	2	1	no	—
98	#3952	—	2	2	no	—
99	#3952	—	2	2	no	—
100	#3941	C₂×C₄	1	1	yes	characteristic
101	#3941	C₂×C₄	1	1	yes	characteristic
102	#3941	C₂×C₄	1	1	yes	characteristic
103	#3941	C₂×C₄	1	1	yes	characteristic
104	#3940	C₂×C₄	1	1	yes	characteristic
105	#3940	C₂×C₄	1	1	yes	characteristic
106	#3940	C₂×C₄	1	1	yes	characteristic
107	#3940	C₂×C₄	1	1	yes	characteristic
108	C₄⋊#168	—	2	1	no	—
109	C₄⋊#168	—	2	1	no	—
110	C₄⋊#168	C₂³	1	2	yes	—
111	C₄×#168	C₂×C₄	1	1	yes	characteristic
112	C₄×#168	C₂×C₄	1	1	yes	characteristic
113	C₄×#168	C₂³	1	2	yes	—
114	C₄×#168	C₂³	1	1	yes	characteristic
115	C₄×#168	C₂³	1	1	yes	characteristic
116	Q₈×M₁₆	D₅	1	1	yes	characteristic, complemented
117	C₂×C₄×SD₁₆	—	5	1	no	—
118	D₄×M₁₆	—	5	1	no	—
119	#1590	—	5	2	no	—
120	#1583	—	5	2	no	—
121	#1582	—	5	2	no	—
122	C₈×SD₁₆	—	5	2	no	—
123	#1566	—	5	1	no	—
124	#1487	—	5	1	no	—
125	#1486	—	5	1	no	—
126	C₈×M₁₆	—	5	1	no	—
127	C₂³×D₅	—	2	1	no	—
128	C₂×C₅×Q₈	—	2	1	no	—
129	C₂×C₅×Q₈	—	2	2	no	—
130	C₂×C₅×Q₈	—	4	1	no	—
131	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₅×Q₈	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₅	—	2	1	no	—
134	C₂²×C₄×C₅	—	2	1	no	—
135	C₂²×C₄×C₅	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₅	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₅	#41	1	1	yes	characteristic
138	C₂²×Dic₅	—	2	1	no	—
139	C₂×D₂₀	—	2	2	no	—
140	C₂×D₂₀	—	2	1	no	—
141	C₂×D₂₀	—	4	1	no	—
142	C₂×D₂₀	C₂²×C₄	1	1	yes	characteristic
143	C₂×D₂₀	C₂²×C₄	1	1	yes	characteristic
144	C₂×C₄×D₅	—	2	1	no	—
145	C₂×C₄×D₅	—	2	1	no	—
146	C₂×C₄×D₅	—	2	1	no	—
147	C₂×C₄×D₅	—	2	1	no	—
148	C₂×C₄×D₅	—	4	2	no	—
149	C₂×C₄×D₅	—	4	2	no	—
150	C₅×M₁₆	—	2	2	no	—
151	C₅×M₁₆	—	4	2	no	—
152	C₅×M₁₆	SD₁₆	1	4	yes	complemented
153	C₂×C₅×C₈	—	2	2	no	—
154	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
155	C₂×C₅×C₈	C₂×D₄	1	2	yes	—
156	C₅×C₄⋊C₄	—	2	1	no	—
157	C₅×C₄⋊C₄	—	2	1	no	—
158	C₅×C₄⋊C₄	—	2	1	no	—
159	C₅×C₄⋊C₄	—	2	1	no	—
160	C₅×C₄⋊C₄	—	2	1	no	—
161	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
162	C₅×C₄⋊C₄	C₂²×C₄	1	1	yes	characteristic
163	C₄²×C₅	—	2	1	no	—
164	C₄²×C₅	—	2	1	no	—
165	C₄²×C₅	—	2	1	no	—
166	C₄²×C₅	—	2	1	no	—
167	C₄²×C₅	C₂²×C₄	1	1	yes	characteristic
168	C₄²×C₅	C₂×D₄	1	1	yes	characteristic
169	C₄²×C₅	#41	1	1	yes	characteristic
170	C₄²×C₅	C₂⁴	1	1	yes	characteristic
171	#688	—	2	2	no	—
172	#688	—	2	2	no	—
173	#688	—	2	2	no	—
174	#688	M₁₆	1	4	yes	direct product factor
175	#685	—	2	1	no	—
176	#685	—	2	1	no	—
177	#685	—	2	1	no	—
178	#685	—	2	1	no	—
179	C₄⋊Dic₅	—	2	1	no	—
180	C₄⋊Dic₅	C₂²×C₄	1	1	yes	characteristic
181	C₄⋊Dic₅	C₂²×C₄	1	1	yes	characteristic
182	#681	—	4	2	no	—
183	C₂×#168	—	2	2	no	—
184	C₂×#168	—	2	2	no	—
185	C₂×#168	—	2	2	no	—
186	C₂×#168	—	4	1	no	—
187	C₂×#168	C₂²×C₄	1	1	yes	characteristic
188	C₂×#168	C₂²×C₄	1	1	yes	characteristic
189	C₂×#168	C₂²×C₄	1	1	yes	characteristic
190	C₂×#168	C₂²×C₄	1	1	yes	characteristic
191	#676	—	4	4	no	—
192	C₈×D₅	—	4	4	no	—
193	C₂²×SD₁₆	—	5	1	no	—
194	C₂²×M₁₆	—	10	1	no	—
195	C₂×C₄×Q₈	C₂×D₅	1	1	yes	characteristic
196	C₂×C₄×D₄	—	5	1	no	—
197	#446	—	2	2	no	—
198	#446	C₂×D₅	1	2	yes	complemented
199	C₈×Q₈	C₂×D₅	1	2	yes	complemented
200	C₄×SD₁₆	—	5	1	no	—
201	C₄×SD₁₆	—	5	1	no	—
202	C₄×SD₁₆	—	5	1	no	—
203	C₄×SD₁₆	—	5	1	no	—
204	C₄×SD₁₆	—	5	1	no	—
205	C₄×SD₁₆	—	5	1	no	—
206	C₄×SD₁₆	—	5	1	no	—
207	C₄×SD₁₆	—	5	1	no	—
208	#436	—	5	2	no	—
209	#435	—	10	2	no	—
210	C₈×D₄	—	5	2	no	—
211	#431	—	5	1	no	—
212	C₂×#106	—	5	1	no	—
213	#423	—	2	1	no	—
214	#423	—	5	1	no	—
215	#423	C₂×D₅	1	1	yes	characteristic
216	C₂×#103	—	5	1	no	—
217	C₂×#102	—	5	1	no	—
218	#407	—	10	1	no	—
219	C₄×M₁₆	—	2	1	no	—
220	C₄×M₁₆	C₂×D₅	1	1	yes	characteristic
221	C₂×C₄×C₈	—	5	1	no	—
222	#334	—	5	2	no	—
223	#334	—	5	2	no	—
224	Q₈⋊C₈	—	5	2	no	—
225	Q₈⋊C₈	—	5	2	no	—
226	D₄⋊C₈	—	5	2	no	—
227	D₄⋊C₈	—	5	2	no	—
228	#322	—	5	2	no	—
229	C₈²	—	5	2	no	—
230	C₂³×C₅	C₄×D₄	1	1	yes	characteristic
231	C₂²×D₅	—	2	1	no	—
232	C₂²×D₅	—	2	1	no	—
233	C₂²×D₅	—	4	2	no	—
234	C₂²×D₅	—	4	2	no	—
235	C₂²×D₅	—	8	1	no	—
236	C₅×Q₈	—	2	1	no	—
237	C₅×Q₈	—	2	1	no	—
238	C₅×Q₈	—	2	1	no	—
239	C₅×Q₈	—	4	2	no	—
240	C₅×Q₈	C₂×M₁₆	1	2	yes	complemented
241	C₂×C₄×C₅	—	2	1	no	—
242	C₂×C₄×C₅	—	2	1	no	—
243	C₂×C₄×C₅	—	2	1	no	—
244	C₂×C₄×C₅	—	2	1	no	—
245	C₂×C₄×C₅	—	2	1	no	—
246	C₂×C₄×C₅	—	2	1	no	—
247	C₂×C₄×C₅	—	4	1	no	—
248	C₂×C₄×C₅	—	4	2	no	—
249	C₂×C₄×C₅	—	8	1	no	—
250	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
251	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
252	C₂×C₄×C₅	C₄×D₄	1	1	yes	characteristic
253	C₂×C₄×C₅	C₂×SD₁₆	1	2	yes	—
254	C₂×C₄×C₅	#135	1	2	yes	—
255	C₂×C₄×C₅	C₂³×C₄	1	1	yes	characteristic
256	C₂×C₄×C₅	C₂²×D₄	1	1	yes	characteristic
257	C₂×C₄×C₅	C₂×#41	1	1	yes	characteristic
258	C₂×Dic₅	—	2	1	no	—
259	C₂×Dic₅	—	2	1	no	—
260	C₂×Dic₅	—	4	2	no	—
261	D₂₀	—	2	1	no	—
262	D₂₀	—	2	1	no	—
263	D₂₀	—	2	1	no	—
264	D₂₀	—	4	2	no	—
265	D₂₀	C₂×M₁₆	1	2	yes	—
266	C₄×D₅	—	4	1	no	—
267	C₄×D₅	—	4	1	no	—
268	C₄×D₅	—	4	1	no	—
269	C₄×D₅	—	4	1	no	—
270	C₄×D₅	—	4	2	no	—
271	C₄×D₅	—	4	2	no	—
272	C₅×C₈	—	2	2	no	—
273	C₅×C₈	—	4	2	no	—
274	C₅×C₈	C₂×SD₁₆	1	4	yes	complemented
275	#168	—	2	1	no	—
276	#168	—	2	1	no	—
277	#168	—	2	1	no	—
278	#168	—	4	2	no	—
279	#168	—	4	2	no	—
280	#168	C₂×M₁₆	1	2	yes	complemented
281	C₂²×Q₈	C₄×D₅	1	1	yes	characteristic
282	C₂²×D₄	—	5	1	no	—
283	C₂³×C₄	—	10	1	no	—
284	C₂×SD₁₆	—	5	1	no	—
285	C₂×SD₁₆	—	5	1	no	—
286	C₂×SD₁₆	—	5	1	no	—
287	C₂×SD₁₆	—	5	1	no	—
288	C₂×SD₁₆	—	10	4	no	—
289	C₂×M₁₆	—	2	1	no	—
290	C₂×M₁₆	—	10	1	no	—
291	C₂×M₁₆	—	10	2	no	—
292	C₂×M₁₆	—	20	4	no	—
293	C₂×M₁₆	C₂²⋊D₅	1	1	yes	characteristic
294	C₂×M₁₆	C₂²⋊D₅	1	1	yes	characteristic
295	C₂²×C₈	—	5	1	no	—
296	C₂²×C₈	—	5	1	no	—
297	C₂²×C₈	—	10	2	no	—
298	C₄×Q₈	—	2	1	no	—
299	C₄×Q₈	—	2	1	no	—
300	C₄×Q₈	C₄×D₅	1	1	yes	characteristic
301	C₄×Q₈	C₄×D₅	1	1	yes	characteristic
302	C₄×Q₈	C₂²×D₅	1	1	yes	characteristic
303	C₄×Q₈	C₂²×D₅	1	1	yes	characteristic
304	C₄×D₄	—	5	1	no	—
305	C₄×D₄	—	5	1	no	—
306	C₄×D₄	—	5	1	no	—
307	C₄×D₄	—	5	1	no	—
308	C₄×D₄	—	10	1	no	—
309	C₄×D₄	—	10	1	no	—
310	C₂×C₄⋊C₄	—	2	1	no	—
311	C₂×C₄⋊C₄	—	5	1	no	—
312	C₂×C₄⋊C₄	C₄×D₅	1	1	yes	characteristic
313	C₂×C₂²⋊C₄	—	10	1	no	—
314	C₂×C₄²	—	2	1	no	—
315	C₂×C₄²	C₂²×D₅	1	1	yes	characteristic
316	#106	—	5	1	no	—
317	#106	—	5	1	no	—
318	#106	—	5	1	no	—
319	#106	—	5	1	no	—
320	C₄⋊C₈	—	2	2	no	—
321	C₄⋊C₈	—	2	2	no	—
322	C₄⋊C₈	—	2	1	no	—
323	C₄⋊C₈	—	5	2	no	—
324	C₄⋊C₈	—	10	1	no	—
325	C₄⋊C₈	—	10	1	no	—
326	C₄⋊C₈	C₂²×D₅	1	2	yes	—
327	#103	—	5	1	no	—
328	#103	—	5	1	no	—
329	#103	—	5	1	no	—
330	#103	—	5	1	no	—
331	#102	—	5	1	no	—
332	#102	—	5	1	no	—
333	#102	—	5	1	no	—
334	#102	—	5	1	no	—
335	C₂²⋊C₈	—	10	1	no	—
336	C₂²⋊C₈	—	10	2	no	—
337	C₂²⋊C₈	—	10	2	no	—
338	#97	—	2	2	no	—
339	#97	C₂²⋊D₅	1	2	yes	—
340	C₄×C₈	—	2	2	no	—
341	C₄×C₈	—	5	1	no	—
342	C₄×C₈	—	5	1	no	—
343	C₄×C₈	—	5	1	no	—
344	C₄×C₈	—	5	1	no	—
345	C₄×C₈	—	5	2	no	—
346	C₄×C₈	C₂²×D₅	1	2	yes	—
347	C₂²×C₅	—	2	1	no	—
348	C₂²×C₅	—	2	1	no	—
349	C₂²×C₅	C₄×SD₁₆	1	2	yes	—
350	C₂²×C₅	C₂×C₄×D₄	1	1	yes	characteristic
351	C₂×D₅	—	2	2	no	—
352	C₂×D₅	—	4	1	no	—
353	C₂×D₅	—	4	1	no	—
354	C₂×D₅	—	4	1	no	—
355	C₂×D₅	—	4	1	no	—
356	C₂×D₅	—	4	1	no	—
357	C₂×D₅	—	8	4	no	—
358	C₄×C₅	—	2	1	no	—
359	C₄×C₅	—	2	2	no	—
360	C₄×C₅	—	2	1	no	—
361	C₄×C₅	—	2	1	no	—
362	C₄×C₅	—	4	1	no	—
363	C₄×C₅	—	4	1	no	—
364	C₄×C₅	—	4	1	no	—
365	C₄×C₅	C₄×SD₁₆	1	2	yes	—
366	C₄×C₅	C₂²×M₁₆	1	1	yes	characteristic
367	C₄×C₅	#568	1	1	yes	characteristic
368	C₄×C₅	C₂²×SD₁₆	1	1	yes	characteristic
369	C₄×C₅	C₂×#135	1	1	yes	characteristic
370	Dic₅	—	4	1	no	—
371	Dic₅	—	4	1	no	—
372	C₂⁴	—	10	1	no	—
373	C₂×Q₈	—	2	1	no	—
374	C₂×Q₈	—	2	2	no	—
375	C₂×Q₈	—	4	1	no	—
376	C₂×Q₈	C₂×C₄×D₅	1	1	yes	characteristic
377	C₂×Q₈	C₂×C₄×D₅	1	1	yes	characteristic
378	C₂×D₄	—	5	1	no	—
379	C₂×D₄	—	5	1	no	—
380	C₂×D₄	—	10	2	no	—
381	C₂×D₄	—	10	1	no	—
382	C₂×D₄	—	20	1	no	—
383	C₂²×C₄	—	2	1	no	—
384	C₂²×C₄	—	2	1	no	—
385	C₂²×C₄	—	10	1	no	—
386	C₂²×C₄	—	10	1	no	—
387	C₂²×C₄	—	10	1	no	—
388	C₂²×C₄	—	10	1	no	—
389	C₂²×C₄	—	10	1	no	—
390	C₂²×C₄	—	20	2	no	—
391	C₂²×C₄	—	20	2	no	—
392	C₂²×C₄	C₂×C₄×D₅	1	1	yes	characteristic
393	C₂²×C₄	#709	1	1	yes	characteristic
394	C₂²×C₄	C₂×C₂²⋊D₅	1	1	yes	characteristic
395	SD₁₆	—	5	4	no	—
396	SD₁₆	—	10	2	no	—
397	SD₁₆	—	10	2	no	—
398	SD₁₆	—	10	2	no	—
399	M₁₆	—	2	2	no	—
400	M₁₆	—	4	2	no	—
401	M₁₆	—	20	2	no	—
402	M₁₆	—	20	4	no	—
403	M₁₆	#688	1	4	yes	direct product factor
404	C₂×C₈	—	2	2	no	—
405	C₂×C₈	—	5	1	no	—
406	C₂×C₈	—	5	1	no	—
407	C₂×C₈	—	5	1	no	—
408	C₂×C₈	—	5	1	no	—
409	C₂×C₈	—	10	2	no	—
410	C₂×C₈	—	10	2	no	—
411	C₂×C₈	—	10	2	no	—
412	C₂×C₈	—	20	1	no	—
413	C₂×C₈	—	20	4	no	—
414	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
415	C₂×C₈	C₂×C₂²⋊D₅	1	2	yes	—
416	C₄⋊C₄	—	2	1	no	—
417	C₄⋊C₄	—	2	1	no	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	1	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	5	1	no	—
422	C₄⋊C₄	—	5	1	no	—
423	C₄⋊C₄	—	10	1	no	—
424	C₄⋊C₄	C₂×C₄×D₅	1	1	yes	characteristic
425	C₄⋊C₄	C₂×C₄×D₅	1	1	yes	characteristic
426	C₂²⋊C₄	—	10	1	no	—
427	C₂²⋊C₄	—	10	1	no	—
428	C₂²⋊C₄	—	10	1	no	—
429	C₂²⋊C₄	—	10	1	no	—
430	C₄²	—	2	1	no	—
431	C₄²	—	2	1	no	—
432	C₄²	—	2	1	no	—
433	C₄²	—	2	1	no	—
434	C₄²	C₂×C₄×D₅	1	1	yes	characteristic
435	C₄²	#709	1	1	yes	characteristic
436	C₄²	C₂×C₂²⋊D₅	1	1	yes	characteristic
437	C₄²	C₂³×D₅	1	1	yes	characteristic
438	C₂×C₅	—	2	2	no	—
439	C₂×C₅	D₄×M₁₆	1	1	yes	characteristic
440	C₂×C₅	C₂×C₄×SD₁₆	1	1	yes	characteristic
441	C₂×C₅	#2957	1	1	yes	characteristic
442	D₅	—	4	2	no	—
443	D₅	—	8	1	no	—
444	C₂³	—	10	1	no	—
445	C₂³	—	10	1	no	—
446	C₂³	—	20	2	no	—
447	C₂³	—	20	2	no	—
448	C₂³	—	40	1	no	—
449	C₂³	C₄×C₂²⋊D₅	1	1	yes	characteristic
450	Q₈	—	2	1	no	—
451	Q₈	—	2	1	no	—
452	Q₈	—	2	1	no	—
453	Q₈	—	4	2	no	—
454	Q₈	D₅×M₁₆	1	2	yes	complemented
455	D₄	—	5	2	no	—
456	D₄	—	10	1	no	—
457	D₄	—	10	1	no	—
458	D₄	—	10	1	no	—
459	D₄	—	20	2	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	4	1	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	8	1	no	—
469	C₂×C₄	—	10	1	no	—
470	C₂×C₄	—	10	1	no	—
471	C₂×C₄	—	20	2	no	—
472	C₂×C₄	—	20	2	no	—
473	C₂×C₄	—	20	2	no	—
474	C₂×C₄	—	20	1	no	—
475	C₂×C₄	—	20	1	no	—
476	C₂×C₄	—	20	1	no	—
477	C₂×C₄	—	20	1	no	—
478	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
479	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
480	C₂×C₄	C₄×C₂²⋊D₅	1	1	yes	characteristic
481	C₂×C₄	C₂×#688	1	2	yes	—
482	C₂×C₄	#4096	1	2	yes	—
483	C₂×C₄	C₂²×C₄×D₅	1	1	yes	characteristic
484	C₂×C₄	C₂×#709	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₂²⋊D₅	1	1	yes	characteristic
486	C₈	—	2	2	no	—
487	C₈	—	4	2	no	—
488	C₈	—	5	2	no	—
489	C₈	—	10	1	no	—
490	C₈	—	10	1	no	—
491	C₈	—	10	1	no	—
492	C₈	—	20	2	no	—
493	C₈	—	20	2	no	—
494	C₈	C₂×#688	1	4	yes	complemented
495	C₅	M₁₆×SD₁₆	1	1	yes	characteristic, complemented, Sylow
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	10	2	no	—
499	C₂²	—	20	1	no	—
500	C₂²	—	20	1	no	—
501	C₂²	—	20	1	no	—
502	C₂²	—	20	1	no	—
503	C₂²	—	20	1	no	—
504	C₂²	—	40	4	no	—
505	C₂²	C₄×#688	1	2	yes	—
506	C₂²	C₂×C₄×C₂²⋊D₅	1	1	yes	characteristic
507	C₄	—	2	1	no	—
508	C₄	—	2	2	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	1	no	—
511	C₄	—	4	1	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	20	1	no	—
515	C₄	—	20	1	no	—
516	C₄	C₄×#688	1	2	yes	—
517	C₄	C₂×D₅×M₁₆	1	1	yes	characteristic
518	C₄	#66079	1	1	yes	characteristic
519	C₄	C₂²×#688	1	1	yes	characteristic
520	C₄	C₂×#4096	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	20	2	no	—
523	C₂	—	40	1	no	—
524	C₂	C₂×C₄×#688	1	1	yes	characteristic
525	C₂	M₁₆×C₂²⋊D₅	1	1	yes	characteristic
526	C₂	#10615543	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₅
commutator	C₂×C₄×C₅	C₂³×C₄
Frattini	C₄²	C₂³×D₅
Fitting	C₅×Q₈×M₁₆	C₂
socle	C₂²×C₅	C₂×C₄×D₄
prime radical	C₂²×D₂₀	C₂×C₄
cosocle	C₄²×C₅	C₂⁴
hypercenter	Q₈×M₁₆	D₅
hypocenter	C₅	M₁₆×SD₁₆
LCS	G ▷ C₂×C₄×C₅ ▷ C₂×C₅ ▷ C₅
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
1. C₅×Q₈×M₁₆	C₂	semidirect
4. #10602415	C₂	semidirect
5. #10601746	C₂	semidirect
6. #10601743	C₂	semidirect
7. #10601726	C₂	semidirect
8. C₈×#688	C₂	semidirect
10. #10601283	C₂	semidirect
11. M₁₆×#168	C₂	semidirect
17. C₅×#446	C₂²	semidirect
18. C₅×C₈×Q₈	C₂²	semidirect
41. Q₈⋊#168	C₂²	semidirect
42. Q₈⋊#168	C₂²	semidirect
45. #64700	C₂²	semidirect
46. #64700	C₂²	semidirect
47. #64675	C₂²	semidirect
48. C₈×#168	C₂²	semidirect
116. Q₈×M₁₆	D₅	semidirect
152. C₅×M₁₆	SD₁₆	semidirect
174. #688	M₁₆	direct (N × H)
198. #446	C₂×D₅	semidirect
199. C₈×Q₈	C₂×D₅	semidirect
240. C₅×Q₈	C₂×M₁₆	semidirect
274. C₅×C₈	C₂×SD₁₆	semidirect
280. #168	C₂×M₁₆	semidirect
403. M₁₆	#688	direct (N × H)
454. Q₈	D₅×M₁₆	semidirect
494. C₈	C₂×#688	semidirect
495. C₅	M₁₆×SD₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₅×Q₈×M₁₆	2
C₂×C₄×#688	2
M₁₆×D₂₀	2
#10602415	2
#10601746	2
#10601743	2
#10601726	2
C₈×#688	2
#10601289	2
#10601283	2
M₁₆×#168	2
M₁₆×SD₁₆	5

Subgroups — M16×#688 #12070740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×#688 #12070740