Subgroups — #35320340

Order 21248.

450 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24915885	C₂	1	1	yes	maximal, characteristic
2	#24909218	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×Dic₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#19127486	C₂	1	1	yes	maximal, characteristic
5	C₂×#19127484	C₂	1	1	yes	maximal, characteristic
6	#24905552	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1379	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581800	C₄	1	1	yes	characteristic
9	C₂×#14581798	C₂²	1	1	yes	characteristic
10	#19128169	C₄	1	1	yes	characteristic
11	C₂×#14581732	C₂²	1	1	yes	characteristic
12	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
13	#19127658	—	2	1	no	—
14	#19127657	C₄	1	1	yes	characteristic
15	#19127655	—	2	1	no	—
16	#19127653	—	2	1	no	—
17	M₁₆×Dic₈₃	C₄	1	1	yes	characteristic
18	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×D₈₃	—	2	1	no	—
20	Dic₈₃⋊C₁₆	—	2	1	no	—
21	Dic₈₃⋊C₁₆	—	2	1	no	—
22	Dic₈₃⋊C₁₆	—	2	1	no	—
23	Dic₈₃⋊C₁₆	—	2	1	no	—
24	#19127486	—	2	2	no	—
25	#19127484	—	2	2	no	—
26	C₂×M₃₂×C₈₃	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
28	#405×C₈₃	C₂²	1	1	yes	characteristic
29	C₂×#13050590	C₂×C₄	1	1	yes	characteristic
30	#14581800	—	2	2	no	—
31	#14581800	—	2	2	no	—
32	#14581798	—	2	1	no	—
33	#14581798	—	2	1	no	—
34	#14581798	D₄	1	2	yes	—
35	#14581798	Q₈	1	2	yes	—
36	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
37	C₂×#13050573	—	2	1	no	—
38	C₂×#13050573	—	2	1	no	—
39	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₈₃	—	2	1	no	—
41	C₂×C₈×D₈₃	—	2	1	no	—
42	#14581757	C₈	1	1	yes	characteristic
43	#14581757	C₈	1	1	yes	characteristic
44	#14581754	C₈	1	1	yes	characteristic
45	#14581754	C₈	1	1	yes	characteristic
46	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
47	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
48	#14581749	—	2	1	no	—
49	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
50	C₄²×D₈₃	—	2	1	no	—
51	#14581732	—	2	1	no	—
52	#14581732	—	2	1	no	—
53	C₂×C₈₃⋊C₁₆	—	2	1	no	—
54	C₂×C₈₃⋊C₁₆	—	2	1	no	—
55	C₂×C₈₃⋊C₁₆	—	2	1	no	—
56	C₂×C₈₃⋊C₁₆	—	2	1	no	—
57	#14581703	—	2	1	no	—
58	#14581702	—	2	1	no	—
59	C₄×C₈₃⋊C₈	—	2	1	no	—
60	C₈×Dic₈₃	—	2	1	no	—
61	C₈×Dic₈₃	D₄	1	1	yes	characteristic
62	C₈×Dic₈₃	D₄	1	1	yes	characteristic
63	C₈×Dic₈₃	D₄	1	1	yes	characteristic
64	C₈×Dic₈₃	D₄	1	1	yes	characteristic
65	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
67	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
68	M₃₂×C₈₃	—	2	1	no	—
69	M₃₂×C₈₃	—	2	1	no	—
70	C₂×C₁₆×C₈₃	—	2	1	no	—
71	C₂×C₁₆×C₈₃	—	2	1	no	—
72	C₂×C₁₆×C₈₃	—	2	1	no	—
73	C₂×C₁₆×C₈₃	—	2	1	no	—
74	#97×C₈₃	—	2	1	no	—
75	C₄×C₈×C₈₃	—	2	1	no	—
76	#13050590	—	2	1	no	—
77	#13050590	—	2	1	no	—
78	#13050590	C₄⋊C₄	1	2	yes	—
79	#13050590	M₁₆	1	2	yes	—
80	C₂×Dic₁₆₆	C₂×C₈	1	1	yes	characteristic
81	C₂×D₃₃₂	C₂×C₈	1	1	yes	characteristic
82	C₂×C₂²⋊D₈₃	—	2	1	no	—
83	C₂²×Dic₈₃	M₁₆	1	1	yes	characteristic
84	C₂²×Dic₈₃	M₁₆	1	1	yes	characteristic
85	C₂×C₄×D₈₃	—	2	1	no	—
86	C₂×C₄×D₈₃	—	2	1	no	—
87	C₂×C₄×D₈₃	—	2	1	no	—
88	#13050574	—	2	1	no	—
89	#13050574	C₄⋊C₄	1	2	yes	—
90	#13050573	—	2	4	no	—
91	#13050573	—	2	2	no	—
92	#13050573	—	2	2	no	—
93	C₂×C₈₃⋊C₈	—	2	1	no	—
94	C₂×C₈₃⋊C₈	—	2	1	no	—
95	C₂×C₈₃⋊C₈	—	2	1	no	—
96	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₈×D₈₃	—	2	4	no	—
99	C₈×D₈₃	—	2	2	no	—
100	C₈×D₈₃	—	2	2	no	—
101	Dic₈₃⋊C₄	—	2	1	no	—
102	Dic₈₃⋊C₄	—	2	1	no	—
103	(C₂×D₈₃)⋊C₄	—	2	1	no	—
104	(C₂×D₈₃)⋊C₄	—	2	1	no	—
105	C₄×Dic₈₃	—	2	1	no	—
106	C₄×Dic₈₃	—	2	1	no	—
107	C₄×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
108	C₄×Dic₈₃	C₂²⋊C₄	1	1	yes	characteristic
109	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
110	C₄×Dic₈₃	C₂×C₈	1	1	yes	characteristic
111	C₈₃⋊C₁₆	—	4	1	no	—
112	C₈₃⋊C₁₆	—	4	1	no	—
113	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
114	M₁₆×C₈₃	—	2	1	no	—
115	M₁₆×C₈₃	M₁₆	1	2	yes	—
116	C₂×C₈×C₈₃	—	2	1	no	—
117	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
121	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
122	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
123	C₄⋊C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
124	C₄⋊C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
125	C₂²⋊C₄×C₈₃	—	2	1	no	—
126	C₄²×C₈₃	—	2	1	no	—
127	C₁₆×C₈₃	—	4	1	no	—
128	C₁₆×C₈₃	—	4	1	no	—
129	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
130	C₂²×D₈₃	—	2	1	no	—
131	C₂×C₄×C₈₃	—	2	1	no	—
132	C₂×C₄×C₈₃	—	2	1	no	—
133	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
137	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
139	C₂²⋊D₈₃	—	2	2	no	—
140	C₂²⋊D₈₃	—	2	2	no	—
141	C₂×Dic₈₃	—	2	1	no	—
142	C₂×Dic₈₃	—	2	2	no	—
143	C₂×Dic₈₃	—	4	1	no	—
144	C₂×Dic₈₃	C₂²⋊C₈	1	1	yes	characteristic
145	C₂×Dic₈₃	C₂²⋊C₈	1	1	yes	characteristic
146	C₂×Dic₈₃	C₂²⋊C₈	1	1	yes	characteristic
147	C₂×Dic₈₃	C₂²⋊C₈	1	1	yes	characteristic
148	D₃₃₂	—	2	1	no	—
149	D₃₃₂	C₄⋊C₈	1	2	yes	—
150	C₄×D₈₃	—	2	2	no	—
151	C₄×D₈₃	—	2	2	no	—
152	C₄×D₈₃	—	4	2	no	—
153	C₄×D₈₃	—	4	2	no	—
154	Dic₁₆₆	—	2	1	no	—
155	Dic₁₆₆	C₄⋊C₈	1	2	yes	—
156	C₈×C₈₃	—	2	2	no	—
157	C₈×C₈₃	#121	1	1	yes	characteristic
158	C₈×C₈₃	#121	1	1	yes	characteristic
159	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
160	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
161	C₈₃⋊C₈	—	2	2	no	—
162	C₈₃⋊C₈	—	2	1	no	—
163	C₈₃⋊C₈	—	2	1	no	—
164	C₂²×C₈₃	—	2	1	no	—
165	C₂²×C₈₃	—	2	1	no	—
166	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
167	C₂²×C₈₃	#423	1	1	yes	characteristic
168	C₂²×C₈₃	#432	1	1	yes	characteristic
169	C₂×D₈₃	—	2	2	no	—
170	C₂×D₈₃	—	4	1	no	—
171	C₂×D₈₃	—	4	1	no	—
172	C₄×C₈₃	—	4	1	no	—
173	C₄×C₈₃	—	4	1	no	—
174	C₄×C₈₃	#380	1	1	yes	characteristic
175	C₄×C₈₃	#407	1	1	yes	characteristic
176	C₄×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
177	C₄×C₈₃	#431	1	1	yes	characteristic
178	Dic₈₃	—	2	2	no	—
179	Dic₈₃	—	2	1	no	—
180	Dic₈₃	—	2	1	no	—
181	Dic₈₃	—	2	1	no	—
182	Dic₈₃	#350	1	2	yes	complemented
183	#12787	—	83	1	no	maximal, Sylow
184	C₂×C₈₃	—	2	1	no	—
185	C₂×C₈₃	—	2	1	no	—
186	C₂×C₈₃	#1797	1	1	yes	characteristic
187	C₂×C₈₃	C₂×#350	1	1	yes	characteristic
188	C₂×C₈₃	#2117	1	1	yes	characteristic
189	D₈₃	—	4	2	no	—
190	C₄×#131	—	83	1	no	—
191	#2150	—	83	1	no	—
192	C₂×C₄⋊C₁₆	—	83	1	no	—
193	C₂×#350	—	83	1	no	—
194	C₂×#350	—	83	1	no	—
195	#1379	—	83	1	no	—
196	#1379	D₈₃	1	1	yes	characteristic, complemented
197	C₈₃	#12787	1	1	yes	characteristic, complemented, Sylow
198	C₂×#131	—	83	1	no	—
199	C₂×#131	—	83	1	no	—
200	C₄×#41	—	83	1	no	—
201	C₂×M₃₂	—	83	1	no	—
202	C₂×M₃₂	C₂×D₈₃	1	1	yes	characteristic
203	C₂²×C₁₆	—	83	1	no	—
204	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
205	#405	—	83	1	no	—
206	#405	—	166	1	no	—
207	#405	—	166	1	no	—
208	#405	C₂×D₈₃	1	1	yes	characteristic
209	C₄×M₁₆	—	83	1	no	—
210	C₄×M₁₆	—	166	1	no	—
211	C₂×C₄×C₈	—	83	1	no	—
212	C₂×C₄×C₈	—	166	1	no	—
213	C₄⋊C₁₆	—	166	1	no	—
214	C₄⋊C₁₆	—	166	1	no	—
215	C₄⋊C₁₆	—	166	1	no	—
216	C₄⋊C₁₆	—	166	1	no	—
217	#350	—	166	2	no	—
218	#350	—	166	2	no	—
219	C₂×#41	—	83	1	no	—
220	#131	—	83	2	no	—
221	#131	—	83	2	no	—
222	#131	—	166	2	no	—
223	#131	—	166	2	no	—
224	#131	—	166	1	no	—
225	#131	—	166	1	no	—
226	C₂×M₁₆	—	83	1	no	—
227	C₂×M₁₆	—	166	1	no	—
228	C₂×M₁₆	—	166	1	no	—
229	C₂×M₁₆	C₄×D₈₃	1	1	yes	characteristic
230	C₂²×C₈	—	83	1	no	—
231	C₂²×C₈	—	166	1	no	—
232	C₂²×C₈	—	166	1	no	—
233	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
234	C₄×Q₈	—	83	1	no	—
235	C₄×Q₈	—	83	1	no	—
236	C₄×D₄	—	83	1	no	—
237	C₄×D₄	—	83	1	no	—
238	C₄×D₄	—	166	1	no	—
239	C₄×D₄	—	166	1	no	—
240	#117	—	166	1	no	—
241	#117	C₄×D₈₃	1	1	yes	characteristic
242	C₂×C₄²	—	83	1	no	—
243	C₂×C₄²	—	166	1	no	—
244	M₃₂	—	2	1	no	—
245	M₃₂	—	2	1	no	—
246	M₃₂	—	166	1	no	—
247	M₃₂	—	166	1	no	—
248	C₂×C₁₆	—	2	1	no	—
249	C₂×C₁₆	—	2	1	no	—
250	C₂×C₁₆	—	2	1	no	—
251	C₂×C₁₆	—	2	1	no	—
252	C₂×C₁₆	—	166	1	no	—
253	C₂×C₁₆	—	166	1	no	—
254	C₂×C₁₆	—	166	1	no	—
255	C₂×C₁₆	—	166	1	no	—
256	#97	—	2	1	no	—
257	#97	—	166	1	no	—
258	#97	—	166	1	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	83	1	no	—
261	C₄×C₈	—	83	1	no	—
262	C₄×C₈	—	83	1	no	—
263	C₄×C₈	—	83	1	no	—
264	C₄×C₈	—	166	1	no	—
265	C₄×C₈	—	166	1	no	—
266	#41	—	83	2	no	—
267	#41	—	83	2	no	—
268	#41	—	166	1	no	—
269	#41	—	166	1	no	—
270	C₂×Q₈	—	83	1	no	—
271	C₂×D₄	—	83	1	no	—
272	C₂×D₄	—	166	1	no	—
273	C₂²×C₄	—	83	1	no	—
274	C₂²×C₄	—	83	1	no	—
275	C₂²×C₄	—	166	1	no	—
276	C₂²×C₄	—	166	1	no	—
277	C₂²×C₄	—	166	1	no	—
278	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
279	M₁₆	—	2	1	no	—
280	M₁₆	—	83	2	no	—
281	M₁₆	—	166	4	no	—
282	M₁₆	—	166	1	no	—
283	M₁₆	—	166	2	no	—
284	M₁₆	—	166	2	no	—
285	M₁₆	#13050573	1	2	yes	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	83	1	no	—
288	C₂×C₈	—	83	1	no	—
289	C₂×C₈	—	166	4	no	—
290	C₂×C₈	—	166	1	no	—
291	C₂×C₈	—	166	1	no	—
292	C₂×C₈	—	166	1	no	—
293	C₂×C₈	—	166	2	no	—
294	C₂×C₈	—	166	2	no	—
295	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
296	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
297	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
298	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
299	C₂×C₈	#13050590	1	1	yes	characteristic
300	C₂×C₈	#13050590	1	1	yes	characteristic
301	C₄⋊C₄	—	166	1	no	—
302	C₄⋊C₄	—	166	1	no	—
303	C₄⋊C₄	C₈×D₈₃	1	1	yes	characteristic
304	C₄⋊C₄	C₈×D₈₃	1	1	yes	characteristic
305	C₂²⋊C₄	—	2	1	no	—
306	C₂²⋊C₄	—	166	1	no	—
307	C₂²⋊C₄	—	166	1	no	—
308	C₄²	—	2	1	no	—
309	C₄²	—	83	1	no	—
310	C₄²	—	83	1	no	—
311	C₄²	—	83	1	no	—
312	C₄²	—	83	1	no	—
313	C₄²	—	166	1	no	—
314	C₄²	—	166	1	no	—
315	C₁₆	—	4	1	no	—
316	C₁₆	—	4	1	no	—
317	C₁₆	—	332	1	no	—
318	C₁₆	—	332	1	no	—
319	C₂³	—	166	1	no	—
320	C₂³	M₁₆×D₈₃	1	1	yes	characteristic
321	Q₈	—	83	2	no	—
322	Q₈	—	166	1	no	—
323	D₄	—	83	2	no	—
324	D₄	—	166	2	no	—
325	D₄	—	166	2	no	—
326	D₄	—	166	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	332	2	no	—
330	C₂×C₄	—	332	2	no	—
331	C₂×C₄	—	332	1	no	—
332	C₂×C₄	—	83	1	no	—
333	C₂×C₄	—	83	1	no	—
334	C₂×C₄	—	83	1	no	—
335	C₂×C₄	—	83	1	no	—
336	C₂×C₄	—	166	2	no	—
337	C₂×C₄	—	166	1	no	—
338	C₂×C₄	—	166	2	no	—
339	C₂×C₄	—	166	2	no	—
340	C₂×C₄	C₂²⋊C₄×D₈₃	1	1	yes	characteristic
341	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
342	C₂×C₄	#14581749	1	1	yes	characteristic
343	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
344	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
345	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
346	C₈	—	2	2	no	—
347	C₈	—	166	2	no	—
348	C₈	—	166	1	no	—
349	C₈	—	166	1	no	—
350	C₈	#14581765	1	1	yes	characteristic
351	C₈	#14581765	1	1	yes	characteristic
352	C₈	#14581770	1	1	yes	characteristic
353	C₈	#14581770	1	1	yes	characteristic
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	332	1	no	—
357	C₂²	—	332	1	no	—
358	C₂²	—	166	2	no	—
359	C₂²	C₂²⋊C₈×D₈₃	1	1	yes	characteristic
360	C₂²	#19127723	1	1	yes	characteristic
361	C₂²	#19127756	1	1	yes	characteristic
362	C₄	—	4	1	no	—
363	C₄	—	4	1	no	—
364	C₄	—	83	2	no	—
365	C₄	—	166	1	no	—
366	C₄	—	166	1	no	—
367	C₄	—	166	1	no	—
368	C₄	—	166	2	no	—
369	C₄	#19127560	1	1	yes	characteristic
370	C₄	#19127663	1	1	yes	characteristic
371	C₄	C₄⋊C₈×D₈₃	1	1	yes	characteristic
372	C₄	#19127752	1	1	yes	characteristic
373	C₂	—	2	1	no	—
374	C₂	—	2	1	no	—
375	C₂	—	332	2	no	—
376	C₂	#24907442	1	1	yes	characteristic
377	C₂	#350×D₈₃	1	1	yes	characteristic
378	C₂	#24909089	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1379	C₂
socle	C₂²×C₈₃	C₂×C₂²⋊C₈
prime radical	C₂×#13050590	C₂×C₄
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1379	D₈₃
hypocenter	C₈₃	#12787
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #1379

Complements

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

N	H ≅ G/N	type
2. #24909218	C₂	semidirect
3. C₂×Dic₈₃⋊C₁₆	C₂	semidirect
6. #24905552	C₂	semidirect
7. C₈₃×#1379	C₂	semidirect
182. Dic₈₃	#350	semidirect
196. #1379	D₈₃	semidirect
197. C₈₃	#12787	semidirect

Maximal subgroups

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

subgroup	index
#24915885	2
#24909218	2
C₂×Dic₈₃⋊C₁₆	2
C₂×#19127486	2
C₂×#19127484	2
#24905552	2
C₈₃×#1379	2
#12787	83