Subgroups — #35374960

Order 21248.

598 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918046	C₂	1	1	yes	maximal, characteristic
2	#24916394	C₂	1	2	yes	maximal
3	#24906373	C₂	1	1	yes	maximal, characteristic, complemented
4	#24906370	C₂	1	2	yes	maximal, complemented
5	#24906282	C₂	1	1	yes	maximal, characteristic, complemented
6	#24906281	C₂	1	1	yes	maximal, characteristic, complemented
7	#24906279	C₂	1	2	yes	maximal, complemented
8	#24906217	C₂	1	1	yes	maximal, characteristic
9	#24905916	C₂	1	2	yes	maximal
10	#322×D₈₃	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#1593	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128255	C₂²	1	2	yes	—
13	#19128105	C₄	1	1	yes	characteristic
14	#19128079	C₄	1	1	yes	characteristic
15	#19128050	C₄	1	2	yes	—
16	#19127943	C₄	1	1	yes	characteristic
17	#19127941	C₄	1	2	yes	—
18	#19127939	C₄	1	1	yes	characteristic
19	#19127831	C₂²	1	2	yes	—
20	#19127829	—	2	2	no	—
21	Q₈×C₈₃⋊C₈	C₂²	1	2	yes	—
22	#19127825	—	2	2	no	—
23	#19127825	C₂²	1	2	yes	—
24	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
25	C₄×#13050580	C₂²	1	2	yes	—
26	#19127726	—	2	2	no	—
27	#19127723	C₂²	1	2	yes	—
28	#19127658	—	2	2	no	—
29	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
32	#19127452	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	#19127423	C₂²	1	1	yes	characteristic
39	#19127422	C₂²	1	1	yes	characteristic
40	#446×C₈₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
42	#334×C₈₃	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₈₃	C₂²	1	2	yes	—
44	#322×C₈₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#13050580	C₂×C₄	1	2	yes	—
47	C₂×#13050573	—	2	2	no	—
48	C₂×C₈×D₈₃	—	2	1	no	—
49	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₈₃	D₄	1	2	yes	—
52	#14581791	C₂×C₄	1	2	yes	—
53	#14581790	C₂×C₄	1	2	yes	—
54	#14581782	—	2	2	no	—
55	#14581772	C₂×C₄	1	2	yes	—
56	#14581770	—	2	2	no	—
57	#14581768	C₂×C₄	1	2	yes	—
58	C₄×Dic₁₆₆	—	2	2	no	—
59	C₄×Dic₁₆₆	C₂³	1	2	yes	—
60	#14581749	—	2	2	no	—
61	C₄²×D₈₃	C₂³	1	1	yes	characteristic
62	#14581726	C₂×C₄	1	1	yes	characteristic
63	#14581725	C₂×C₄	1	2	yes	—
64	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
65	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
66	Dic₈₃⋊C₈	—	2	2	no	—
67	Dic₈₃⋊C₈	—	2	2	no	—
68	Dic₈₃⋊C₈	—	2	2	no	—
69	#14581717	C₂×C₄	1	2	yes	—
70	#14581716	C₂×C₄	1	2	yes	—
71	#14581715	C₂×C₄	1	2	yes	—
72	#14581702	—	2	2	no	—
73	C₄×C₈₃⋊C₈	—	2	2	no	—
74	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₈×Dic₈₃	—	2	2	no	—
78	C₈×Dic₈₃	—	2	1	no	—
79	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
80	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₈₃	D₄	1	2	yes	—
82	C₂×Q₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
83	C₄×Q₈×C₈₃	C₂³	1	2	yes	—
84	#107×C₈₃	C₂×C₄	1	1	yes	characteristic
85	C₄⋊C₈×C₈₃	—	2	2	no	—
86	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
87	#103×C₈₃	C₂×C₄	1	2	yes	—
88	#97×C₈₃	—	2	2	no	—
89	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
90	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₆₆	—	2	2	no	—
93	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
94	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×D₈₃	#41	1	1	yes	characteristic
97	Dic₃₃₂	—	2	2	no	—
98	#13050580	—	2	4	no	—
99	#13050573	—	4	4	no	—
100	C₂×C₈₃⋊C₈	—	2	2	no	—
101	C₂×C₈₃⋊C₈	—	2	1	no	—
102	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
105	C₈×D₈₃	—	2	4	no	—
106	C₈×D₈₃	—	2	2	no	—
107	C₈×D₈₃	—	2	2	no	—
108	C₈×D₈₃	—	4	2	no	—
109	C₄⋊Dic₈₃	—	2	2	no	—
110	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
111	Dic₈₃⋊C₄	—	2	2	no	—
112	Dic₈₃⋊C₄	—	2	2	no	—
113	Dic₈₃⋊C₄	—	2	2	no	—
114	Dic₈₃⋊C₄	—	2	2	no	—
115	Dic₈₃⋊C₄	—	2	2	no	—
116	(C₂×D₈₃)⋊C₄	—	2	2	no	—
117	(C₂×D₈₃)⋊C₄	—	2	2	no	—
118	C₄×Dic₈₃	—	2	2	no	—
119	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
120	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
121	C₄×Dic₈₃	#41	1	1	yes	characteristic
122	C₂×Q₈×C₈₃	C₂²×C₄	1	2	yes	—
123	Q₁₆×C₈₃	—	2	2	no	—
124	C₂×C₈×C₈₃	—	2	2	no	—
125	C₂×C₈×C₈₃	—	2	1	no	—
126	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
129	C₄⋊C₄×C₈₃	—	2	2	no	—
130	C₄⋊C₄×C₈₃	C₂²×C₄	1	2	yes	—
131	C₄²×C₈₃	—	2	2	no	—
132	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
133	C₂²×D₈₃	C₄×D₄	1	1	yes	characteristic
134	Q₈×C₈₃	—	2	2	no	—
135	Q₈×C₈₃	—	2	2	no	—
136	C₂×C₄×C₈₃	—	2	2	no	—
137	C₂×C₄×C₈₃	—	2	2	no	—
138	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
141	C₂×Dic₈₃	—	2	2	no	—
142	C₂×Dic₈₃	—	2	2	no	—
143	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
144	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
145	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
146	C₄×D₈₃	—	2	1	no	—
147	C₄×D₈₃	—	2	1	no	—
148	C₄×D₈₃	—	2	2	no	—
149	C₄×D₈₃	#137	1	4	yes	—
150	Dic₁₆₆	—	2	2	no	—
151	Dic₁₆₆	—	2	2	no	—
152	Dic₁₆₆	—	4	4	no	—
153	C₈×C₈₃	—	2	1	no	—
154	C₈×C₈₃	—	2	2	no	—
155	C₈×C₈₃	—	4	1	no	—
156	C₈×C₈₃	—	4	2	no	—
157	C₈×C₈₃	#131	1	2	yes	—
158	C₈₃⋊C₈	—	2	1	no	—
159	C₈₃⋊C₈	—	2	2	no	—
160	C₈₃⋊C₈	—	4	1	no	—
161	C₈₃⋊C₈	—	4	2	no	—
162	C₈₃⋊C₈	#131	1	2	yes	—
163	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₈₃	—	2	1	no	—
165	C₂×D₈₃	—	2	1	no	—
166	C₂×D₈₃	#441	1	2	yes	—
167	C₄×C₈₃	—	2	1	no	—
168	C₄×C₈₃	—	4	2	no	—
169	C₄×C₈₃	—	4	2	no	—
170	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
171	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
172	C₄×C₈₃	C₂×#137	1	2	yes	—
173	Dic₈₃	—	2	1	no	—
174	Dic₈₃	—	4	2	no	—
175	Dic₈₃	—	4	2	no	—
176	Dic₈₃	#441	1	2	yes	—
177	Dic₈₃	#444	1	2	yes	—
178	#17961	—	83	1	no	maximal, Sylow
179	C₂×C₈₃	#2931	1	1	yes	characteristic
180	C₂×C₈₃	C₂×#441	1	1	yes	characteristic
181	C₂×C₈₃	C₂×#444	1	1	yes	characteristic
182	D₈₃	—	2	2	no	—
183	#3170	—	83	1	no	—
184	#2960	—	83	2	no	—
185	#1593	—	83	2	no	—
186	#1593	—	83	1	no	—
187	#1593	D₈₃	1	1	yes	characteristic, complemented
188	#1579	—	83	1	no	—
189	#1579	—	83	1	no	—
190	#1579	—	83	2	no	—
191	#1566	—	83	1	no	—
192	#1493	—	83	2	no	—
193	C₂×#322	—	83	1	no	—
194	C₈₃	#17961	1	1	yes	characteristic, complemented, Sylow
195	#533	—	83	2	no	—
196	#500	—	83	1	no	—
197	#494	—	83	1	no	—
198	#487	—	83	2	no	—
199	#467	—	83	1	no	—
200	#467	—	83	1	no	—
201	#467	—	83	2	no	—
202	#446	—	83	2	no	—
203	#446	—	166	2	no	—
204	#446	C₂×D₈₃	1	2	yes	—
205	C₈×Q₈	—	83	2	no	—
206	C₈×Q₈	—	83	2	no	—
207	C₈×Q₈	—	166	2	no	—
208	C₄×Q₁₆	—	83	2	no	—
209	C₄×Q₁₆	—	83	1	no	—
210	C₄×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
211	#424	—	166	2	no	—
212	#423	—	83	2	no	—
213	#405	—	166	2	no	—
214	C₂×C₄×C₈	—	83	1	no	—
215	C₂×C₄×C₈	—	83	1	no	—
216	C₂×C₄×C₈	—	83	1	no	—
217	#334	—	83	2	no	—
218	#334	—	83	1	no	—
219	#334	C₂×D₈₃	1	1	yes	characteristic
220	Q₈⋊C₈	—	83	2	no	—
221	Q₈⋊C₈	—	83	2	no	—
222	Q₈⋊C₈	—	83	2	no	—
223	Q₈⋊C₈	C₂×D₈₃	1	2	yes	—
224	#322	—	83	1	no	—
225	#322	—	83	1	no	—
226	#322	—	83	1	no	—
227	#322	C₂×D₈₃	1	1	yes	characteristic
228	C₂×Q₁₆	—	83	2	no	—
229	C₂×Q₁₆	—	83	1	no	—
230	C₂×Q₁₆	C₄×D₈₃	1	1	yes	characteristic
231	C₂×M₁₆	—	166	2	no	—
232	C₂²×C₈	—	83	1	no	—
233	C₂²×C₈	—	83	1	no	—
234	C₂²×C₈	—	83	2	no	—
235	C₂²×C₈	—	166	1	no	—
236	C₄⋊Q₈	—	83	2	no	—
237	C₄⋊Q₈	—	83	2	no	—
238	#125	—	166	2	no	—
239	C₂²⋊Q₈	—	83	2	no	—
240	C₂²⋊Q₈	—	83	2	no	—
241	C₂²⋊Q₈	—	166	2	no	—
242	C₄×Q₈	—	83	2	no	—
243	C₄×Q₈	—	166	2	no	—
244	C₄×Q₈	C₂²×D₈₃	1	2	yes	—
245	#117	—	166	2	no	—
246	C₂×C₄²	—	83	1	no	—
247	#107	—	83	1	no	—
248	#107	—	83	2	no	—
249	#107	C₄×D₈₃	1	1	yes	characteristic
250	C₄⋊C₈	—	2	2	no	—
251	C₄⋊C₈	—	83	2	no	—
252	C₄⋊C₈	—	166	2	no	—
253	C₄⋊C₈	—	166	2	no	—
254	C₄⋊C₈	—	166	2	no	—
255	C₄⋊C₈	—	166	2	no	—
256	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
257	#103	—	83	2	no	—
258	#103	—	83	2	no	—
259	#103	—	83	2	no	—
260	#103	C₄×D₈₃	1	2	yes	—
261	#97	—	2	2	no	—
262	#97	—	166	2	no	—
263	C₄×C₈	—	83	1	no	—
264	C₄×C₈	—	83	1	no	—
265	C₄×C₈	—	83	2	no	—
266	C₄×C₈	—	83	1	no	—
267	C₄×C₈	—	83	1	no	—
268	C₄×C₈	—	83	1	no	—
269	C₄×C₈	—	166	2	no	—
270	C₄×C₈	—	166	1	no	—
271	C₄×C₈	—	166	2	no	—
272	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
273	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
274	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
275	C₂×Q₈	—	83	2	no	—
276	C₂×Q₈	—	166	2	no	—
277	C₂×Q₈	C₂×C₄×D₈₃	1	2	yes	—
278	C₂²×C₄	—	83	1	no	—
279	C₂²×C₄	—	83	1	no	—
280	C₂²×C₄	—	83	1	no	—
281	Q₁₆	—	2	2	no	—
282	Q₁₆	—	166	4	no	—
283	Q₁₆	—	166	2	no	—
284	M₁₆	—	332	4	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	332	2	no	—
288	C₂×C₈	—	83	1	no	—
289	C₂×C₈	—	83	1	no	—
290	C₂×C₈	—	83	2	no	—
291	C₂×C₈	—	166	2	no	—
292	C₂×C₈	—	166	4	no	—
293	C₂×C₈	—	166	2	no	—
294	C₂×C₈	—	166	1	no	—
295	C₂×C₈	—	166	2	no	—
296	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
297	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
298	C₂×C₈	D₄×D₈₃	1	2	yes	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	83	2	no	—
301	C₄⋊C₄	—	166	2	no	—
302	C₄⋊C₄	—	166	2	no	—
303	C₄⋊C₄	—	166	2	no	—
304	C₄⋊C₄	—	166	2	no	—
305	C₄⋊C₄	—	166	2	no	—
306	C₄⋊C₄	—	166	2	no	—
307	C₄⋊C₄	C₂×C₄×D₈₃	1	2	yes	—
308	C₂²⋊C₄	—	166	2	no	—
309	C₂²⋊C₄	—	166	2	no	—
310	C₄²	—	2	2	no	—
311	C₄²	—	83	1	no	—
312	C₄²	—	83	1	no	—
313	C₄²	—	83	1	no	—
314	C₄²	—	166	2	no	—
315	C₄²	C₂³×D₈₃	1	1	yes	characteristic
316	C₂³	—	83	1	no	—
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	332	4	no	—
320	Q₈	—	166	2	no	—
321	Q₈	—	166	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	83	1	no	—
325	C₂×C₄	—	83	1	no	—
326	C₂×C₄	—	83	4	no	—
327	C₂×C₄	—	83	1	no	—
328	C₂×C₄	—	166	2	no	—
329	C₂×C₄	—	166	2	no	—
330	C₂×C₄	—	166	2	no	—
331	C₂×C₄	—	166	1	no	—
332	C₂×C₄	—	166	1	no	—
333	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
334	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
335	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
336	C₈	—	2	1	no	—
337	C₈	—	2	2	no	—
338	C₈	—	4	1	no	—
339	C₈	—	4	2	no	—
340	C₈	—	332	1	no	—
341	C₈	—	332	2	no	—
342	C₈	—	83	2	no	—
343	C₈	—	166	1	no	—
344	C₈	—	166	2	no	—
345	C₈	#14581800	1	2	yes	—
346	C₂²	—	83	2	no	—
347	C₂²	—	166	1	no	—
348	C₂²	—	166	1	no	—
349	C₂²	C₄×D₄×D₈₃	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	2	no	—
353	C₄	—	332	2	no	—
354	C₄	—	332	2	no	—
355	C₄	—	83	2	no	—
356	C₄	—	83	2	no	—
357	C₄	—	166	1	no	—
358	C₄	#131×D₈₃	1	1	yes	characteristic
359	C₄	C₂×#14581800	1	1	yes	characteristic
360	C₄	#137×D₈₃	1	2	yes	—
361	C₂	—	166	2	no	—
362	C₂	#24916172	1	1	yes	characteristic
363	C₂	#441×D₈₃	1	1	yes	characteristic
364	C₂	#444×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×#1593	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂²×D₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1593	D₈₃
hypocenter	C₈₃	#17961
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1593

Complements

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

N	H ≅ G/N	type
3. #24906373	C₂	semidirect
4. #24906370	C₂	semidirect
5. #24906282	C₂	semidirect
6. #24906281	C₂	semidirect
7. #24906279	C₂	semidirect
11. C₈₃×#1593	C₂	semidirect
187. #1593	D₈₃	semidirect
194. C₈₃	#17961	semidirect

Maximal subgroups

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

subgroup	index
#24918046	2
#24916394	2
#24906373	2
#24906370	2
#24906282	2
#24906281	2
#24906279	2
#24906217	2
#24905916	2
#322×D₈₃	2
C₈₃×#1593	2
#17961	83