Subgroups — #35376092

Order 21248.

626 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24916549	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic
3	#24906360	C₂	1	1	yes	maximal, characteristic, complemented
4	#24906337	C₂	1	2	yes	maximal, complemented
5	#24906332	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₃₃₂	C₂	1	1	yes	maximal, characteristic, complemented
7	#24906269	C₂	1	1	yes	maximal, characteristic, complemented
8	#24906225	C₂	1	1	yes	maximal, characteristic
9	#24905908	C₂	1	2	yes	maximal
10	C₈×#13050573	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#1591	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₈₃	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₈₃	C₂²	1	2	yes	—
14	#19127832	C₂²	1	2	yes	—
15	#19127830	—	2	2	no	—
16	#19127829	—	2	2	no	—
17	C₈×Dic₁₆₆	C₂²	1	2	yes	—
18	#19127825	C₂²	1	2	yes	—
19	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₈₃	C₄	1	1	yes	characteristic
21	#19127795	C₄	1	1	yes	characteristic
22	C₄×#13050580	C₂²	1	2	yes	—
23	#19127793	C₄	1	2	yes	—
24	#19127757	C₂²	1	2	yes	—
25	#19127756	—	2	2	no	—
26	#19127753	—	2	2	no	—
27	#19127752	C₂²	1	1	yes	characteristic
28	#107×D₈₃	C₄	1	1	yes	characteristic
29	#103×D₈₃	C₄	1	2	yes	—
30	C₄×#13050573	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
32	#19127454	C₂²	1	1	yes	characteristic
33	#19127451	C₂²	1	2	yes	—
34	#19127433	C₂²	1	2	yes	—
35	Q₈⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
36	#19127431	C₂²	1	2	yes	—
37	#19127423	C₂²	1	1	yes	characteristic
38	#19127422	C₂²	1	1	yes	characteristic
39	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	#446×C₈₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
42	#335×C₈₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₃	C₂²	1	2	yes	—
44	C₈²×C₈₃	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₈₃	C₂×C₄	1	2	yes	—
46	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
47	C₂×#13050580	C₂×C₄	1	2	yes	—
48	Q₁₆×D₈₃	—	2	4	no	—
49	C₂×#13050573	—	2	1	no	—
50	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
52	Q₈×Dic₈₃	C₂×C₄	1	2	yes	—
53	C₄×Dic₁₆₆	—	2	2	no	—
54	C₄×Dic₁₆₆	C₂³	1	2	yes	—
55	#14581757	—	2	2	no	—
56	#14581757	C₂×C₄	1	2	yes	—
57	C₄⋊C₄×D₈₃	—	2	2	no	—
58	C₄⋊C₄×D₈₃	C₂×C₄	1	2	yes	—
59	C₄²×D₈₃	—	2	2	no	—
60	C₄²×D₈₃	C₂³	1	1	yes	characteristic
61	#14581726	C₂×C₄	1	1	yes	characteristic
62	#14581725	C₂×C₄	1	2	yes	—
63	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
64	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
65	Dic₈₃⋊C₈	—	2	2	no	—
66	Dic₈₃⋊C₈	—	2	2	no	—
67	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
68	Dic₈₃⋊C₈	D₄	1	1	yes	characteristic
69	#14581717	C₂×C₄	1	2	yes	—
70	#14581716	C₂×C₄	1	2	yes	—
71	#14581715	C₂×C₄	1	2	yes	—
72	(C₂×D₈₃)⋊C₈	—	2	2	no	—
73	(C₂×D₈₃)⋊C₈	D₄	1	1	yes	characteristic
74	(C₂×D₈₃)⋊C₈	D₄	1	1	yes	characteristic
75	#14581703	—	2	2	no	—
76	#14581702	—	2	2	no	—
77	#14581702	—	2	1	no	—
78	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₈×Dic₈₃	—	2	2	no	—
82	C₈×Dic₈₃	—	2	2	no	—
83	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
85	C₂×Q₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₈₃	C₂³	1	2	yes	—
87	#107×C₈₃	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₈₃	—	2	2	no	—
89	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
90	#103×C₈₃	C₂×C₄	1	2	yes	—
91	#97×C₈₃	—	2	2	no	—
92	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₆₆	—	2	2	no	—
96	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
97	Q₈×D₈₃	—	2	4	no	—
98	Q₈×D₈₃	—	4	2	no	—
99	C₂×C₄×D₈₃	—	2	2	no	—
100	C₂×C₄×D₈₃	—	2	2	no	—
101	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₈₃	#41	1	1	yes	characteristic
104	Dic₃₃₂	M₁₆	1	4	yes	complemented
105	#13050580	—	2	4	no	—
106	#13050573	—	2	4	no	—
107	C₂×C₈₃⋊C₈	—	2	1	no	—
108	C₂×C₈₃⋊C₈	—	2	2	no	—
109	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
113	C₈×D₈₃	—	2	2	no	—
114	C₈×D₈₃	—	4	1	no	—
115	C₄⋊Dic₈₃	—	2	2	no	—
116	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
117	Dic₈₃⋊C₄	—	2	2	no	—
118	Dic₈₃⋊C₄	—	2	2	no	—
119	Dic₈₃⋊C₄	—	2	2	no	—
120	C₄×Dic₈₃	—	2	2	no	—
121	C₄×Dic₈₃	—	2	2	no	—
122	C₄×Dic₈₃	—	2	2	no	—
123	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
124	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
125	C₄×Dic₈₃	#41	1	1	yes	characteristic
126	C₂×Q₈×C₈₃	C₂²×C₄	1	2	yes	—
127	Q₁₆×C₈₃	—	2	2	no	—
128	C₂×C₈×C₈₃	—	2	1	no	—
129	C₂×C₈×C₈₃	—	2	2	no	—
130	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
134	C₄⋊C₄×C₈₃	—	2	2	no	—
135	C₄⋊C₄×C₈₃	C₂²×C₄	1	2	yes	—
136	C₄²×C₈₃	—	2	2	no	—
137	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
138	C₂²×D₈₃	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₈₃	—	2	2	no	—
140	Q₈×C₈₃	—	2	2	no	—
141	C₂×C₄×C₈₃	—	2	2	no	—
142	C₂×C₄×C₈₃	—	2	2	no	—
143	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₈₃	—	2	2	no	—
147	C₂×Dic₈₃	—	2	2	no	—
148	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
150	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
151	C₄×D₈₃	—	2	2	no	—
152	C₄×D₈₃	—	2	1	no	—
153	C₄×D₈₃	—	2	1	no	—
154	C₄×D₈₃	—	2	1	no	—
155	C₄×D₈₃	—	2	1	no	—
156	C₄×D₈₃	—	4	4	no	—
157	C₄×D₈₃	—	4	4	no	—
158	Dic₁₆₆	—	2	2	no	—
159	Dic₁₆₆	—	4	4	no	—
160	Dic₁₆₆	C₂×M₁₆	1	4	yes	—
161	C₈×C₈₃	—	2	2	no	—
162	C₈×C₈₃	—	2	1	no	—
163	C₈×C₈₃	—	2	1	no	—
164	C₈×C₈₃	—	2	1	no	—
165	C₈×C₈₃	—	4	2	no	—
166	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
167	C₈₃⋊C₈	—	2	2	no	—
168	C₈₃⋊C₈	—	2	1	no	—
169	C₈₃⋊C₈	—	2	1	no	—
170	C₈₃⋊C₈	—	2	1	no	—
171	C₈₃⋊C₈	—	2	1	no	—
172	C₈₃⋊C₈	—	4	2	no	—
173	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×D₈₃	—	2	1	no	—
175	C₂×D₈₃	—	2	1	no	—
176	C₂×D₈₃	#443	1	2	yes	—
177	C₄×C₈₃	—	2	1	no	—
178	C₄×C₈₃	—	4	2	no	—
179	C₄×C₈₃	—	4	2	no	—
180	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
181	C₄×C₈₃	#568	1	1	yes	characteristic
182	C₄×C₈₃	#578	1	1	yes	characteristic
183	C₄×C₈₃	#578	1	1	yes	characteristic
184	Dic₈₃	—	2	4	no	—
185	Dic₈₃	—	2	1	no	—
186	Dic₈₃	—	2	1	no	—
187	Dic₈₃	—	2	1	no	—
188	Dic₈₃	—	4	2	no	—
189	#18058	—	83	1	no	maximal, Sylow
190	C₂×C₈₃	D₄×M₁₆	1	1	yes	characteristic
191	C₂×C₈₃	#2950	1	1	yes	characteristic
192	C₂×C₈₃	C₂×#443	1	1	yes	characteristic
193	D₈₃	—	2	2	no	—
194	#2984	—	83	2	no	—
195	C₂×C₄×Q₁₆	—	83	1	no	—
196	#1591	—	83	1	no	—
197	#1591	D₈₃	1	1	yes	characteristic, complemented
198	#1587	—	83	2	no	—
199	#1587	—	83	2	no	—
200	C₈×Q₁₆	—	83	1	no	—
201	C₈×Q₁₆	—	83	1	no	—
202	#1568	—	83	1	no	—
203	#1491	—	83	2	no	—
204	C₈×M₁₆	—	83	1	no	—
205	C₈₃	#18058	1	1	yes	characteristic, complemented, Sylow
206	C₂²×Q₁₆	—	83	1	no	—
207	C₂×C₄×Q₈	—	83	2	no	—
208	#446	—	83	2	no	—
209	#446	—	166	2	no	—
210	#446	—	166	2	no	—
211	#446	C₂×D₈₃	1	2	yes	—
212	C₈×Q₈	—	83	2	no	—
213	C₈×Q₈	—	83	2	no	—
214	C₄×Q₁₆	—	83	1	no	—
215	C₄×Q₁₆	—	83	2	no	—
216	C₄×Q₁₆	—	83	1	no	—
217	C₄×Q₁₆	—	83	2	no	—
218	C₄×Q₁₆	—	83	1	no	—
219	C₄×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
220	#432	—	83	2	no	—
221	#432	—	166	2	no	—
222	#431	—	83	1	no	—
223	#431	—	166	2	no	—
224	C₂×#107	—	83	1	no	—
225	C₂×#103	—	83	2	no	—
226	C₄×M₁₆	—	83	1	no	—
227	C₂×C₄×C₈	—	83	1	no	—
228	#335	—	83	1	no	—
229	#335	C₂×D₈₃	1	1	yes	characteristic
230	#334	—	83	2	no	—
231	Q₈⋊C₈	—	83	2	no	—
232	Q₈⋊C₈	—	83	2	no	—
233	Q₈⋊C₈	—	83	2	no	—
234	Q₈⋊C₈	C₂×D₈₃	1	2	yes	—
235	#322	—	83	1	no	—
236	#322	—	83	1	no	—
237	C₈²	—	83	1	no	—
238	C₈²	C₂×D₈₃	1	1	yes	characteristic
239	C₂²×Q₈	—	83	2	no	—
240	C₂×Q₁₆	—	83	2	no	—
241	C₂×Q₁₆	—	83	1	no	—
242	C₂×Q₁₆	—	166	4	no	—
243	C₂×Q₁₆	C₄×D₈₃	1	1	yes	characteristic
244	C₂×M₁₆	—	166	1	no	—
245	C₂²×C₈	—	83	1	no	—
246	C₂²×C₈	—	83	1	no	—
247	C₄×Q₈	—	83	2	no	—
248	C₄×Q₈	—	83	2	no	—
249	C₄×Q₈	—	83	2	no	—
250	C₄×Q₈	—	166	2	no	—
251	C₄×Q₈	—	166	2	no	—
252	C₄×Q₈	C₂²×D₈₃	1	2	yes	—
253	C₂×C₄⋊C₄	—	83	2	no	—
254	C₂×C₄⋊C₄	—	166	2	no	—
255	C₂×C₄²	—	83	1	no	—
256	C₂×C₄²	—	166	2	no	—
257	#107	—	83	1	no	—
258	#107	—	83	2	no	—
259	#107	C₄×D₈₃	1	1	yes	characteristic
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	83	1	no	—
262	C₄⋊C₈	—	83	1	no	—
263	C₄⋊C₈	—	83	2	no	—
264	C₄⋊C₈	—	166	2	no	—
265	C₄⋊C₈	—	166	2	no	—
266	C₄⋊C₈	—	166	2	no	—
267	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
268	#103	—	83	2	no	—
269	#103	—	83	2	no	—
270	#103	—	83	2	no	—
271	#103	C₄×D₈₃	1	2	yes	—
272	C₂²⋊C₈	—	83	1	no	—
273	C₂²⋊C₈	—	83	1	no	—
274	C₂²⋊C₈	—	166	2	no	—
275	#97	—	2	2	no	—
276	#97	—	166	2	no	—
277	#97	—	166	1	no	—
278	#97	—	166	2	no	—
279	C₄×C₈	—	83	1	no	—
280	C₄×C₈	—	83	1	no	—
281	C₄×C₈	—	83	1	no	—
282	C₄×C₈	—	83	1	no	—
283	C₄×C₈	—	83	1	no	—
284	C₄×C₈	—	166	2	no	—
285	C₄×C₈	—	166	2	no	—
286	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
289	C₂×Q₈	—	332	2	no	—
290	C₂×Q₈	—	83	2	no	—
291	C₂×Q₈	—	166	2	no	—
292	C₂×Q₈	—	166	4	no	—
293	C₂×Q₈	C₂×C₄×D₈₃	1	2	yes	—
294	C₂²×C₄	—	83	1	no	—
295	C₂²×C₄	—	83	1	no	—
296	C₂²×C₄	—	83	1	no	—
297	C₂²×C₄	—	166	2	no	—
298	C₂²×C₄	—	166	2	no	—
299	Q₁₆	—	2	2	no	—
300	Q₁₆	—	83	4	no	—
301	Q₁₆	—	166	4	no	—
302	M₁₆	—	166	4	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	332	1	no	—
306	C₂×C₈	—	83	1	no	—
307	C₂×C₈	—	83	1	no	—
308	C₂×C₈	—	83	1	no	—
309	C₂×C₈	—	83	1	no	—
310	C₂×C₈	—	166	2	no	—
311	C₂×C₈	—	166	1	no	—
312	C₂×C₈	—	166	2	no	—
313	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
314	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
315	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
316	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	83	2	no	—
319	C₄⋊C₄	—	166	2	no	—
320	C₄⋊C₄	—	166	2	no	—
321	C₄⋊C₄	—	166	2	no	—
322	C₄⋊C₄	—	166	2	no	—
323	C₄⋊C₄	C₂×C₄×D₈₃	1	2	yes	—
324	C₄²	—	2	2	no	—
325	C₄²	—	83	1	no	—
326	C₄²	—	83	1	no	—
327	C₄²	—	83	1	no	—
328	C₄²	—	166	2	no	—
329	C₄²	—	166	2	no	—
330	C₄²	—	166	2	no	—
331	C₄²	C₂³×D₈₃	1	1	yes	characteristic
332	C₂³	—	83	1	no	—
333	Q₈	—	2	2	no	—
334	Q₈	—	2	2	no	—
335	Q₈	—	332	4	no	—
336	Q₈	—	83	4	no	—
337	Q₈	—	166	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	332	4	no	—
341	C₂×C₄	—	332	4	no	—
342	C₂×C₄	—	83	1	no	—
343	C₂×C₄	—	83	1	no	—
344	C₂×C₄	—	83	1	no	—
345	C₂×C₄	—	166	2	no	—
346	C₂×C₄	—	166	2	no	—
347	C₂×C₄	—	166	2	no	—
348	C₂×C₄	—	166	1	no	—
349	C₂×C₄	—	166	1	no	—
350	C₂×C₄	—	166	1	no	—
351	C₂×C₄	—	166	1	no	—
352	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
354	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	2	no	—
360	C₈	—	332	2	no	—
361	C₈	—	166	2	no	—
362	C₈	—	166	1	no	—
363	C₈	—	166	1	no	—
364	C₈	—	166	1	no	—
365	C₈	—	166	1	no	—
366	C₈	M₁₆×D₈₃	1	2	yes	—
367	C₂²	—	83	2	no	—
368	C₂²	—	166	1	no	—
369	C₂²	—	166	1	no	—
370	C₂²	C₄×D₄×D₈₃	1	1	yes	characteristic
371	C₄	—	2	1	no	—
372	C₄	—	4	2	no	—
373	C₄	—	4	2	no	—
374	C₄	—	332	2	no	—
375	C₄	—	166	1	no	—
376	C₄	—	166	1	no	—
377	C₄	—	166	4	no	—
378	C₄	—	166	1	no	—
379	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
380	C₄	#19128466	1	1	yes	characteristic
381	C₄	#19128514	1	1	yes	characteristic
382	C₄	#19128515	1	1	yes	characteristic
383	C₂	—	166	2	no	—
384	C₂	#24916199	1	1	yes	characteristic
385	C₂	#24916326	1	1	yes	characteristic
386	C₂	#443×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1591	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂²×D₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1591	D₈₃
hypocenter	C₈₃	#18058
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1591

Complements

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

N	H ≅ G/N	type
3. #24906360	C₂	semidirect
4. #24906337	C₂	semidirect
5. #24906332	C₂	semidirect
6. C₈×Dic₃₃₂	C₂	semidirect
7. #24906269	C₂	semidirect
11. C₈₃×#1591	C₂	semidirect
104. Dic₃₃₂	M₁₆	semidirect
197. #1591	D₈₃	semidirect
205. C₈₃	#18058	semidirect

Maximal subgroups

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

subgroup	index
#24916549	2
C₄×Q₁₆×D₈₃	2
#24906360	2
#24906337	2
#24906332	2
C₈×Dic₃₃₂	2
#24906269	2
#24906225	2
#24905908	2
C₈×#13050573	2
C₈₃×#1591	2
#18058	83