Subgroups — #35375081

Order 21248.

560 conjugacy classes of subgroups.

The subgroup lattice has 360 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	#24916554	C₂	1	2	yes	maximal
3	#24906372	C₂	1	1	yes	maximal, characteristic, complemented
4	#24906371	C₂	1	1	yes	maximal, characteristic, complemented
5	#24906357	C₂	1	2	yes	maximal, complemented
6	#24906283	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#13050580	C₂	1	2	yes	maximal, complemented
8	#24906223	C₂	1	1	yes	maximal, characteristic
9	#24905934	C₂	1	2	yes	maximal
10	#24905748	C₂	1	1	yes	maximal, characteristic
11	C₈₃×#1579	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	#19127832	C₂²	1	2	yes	—
20	#19127831	—	2	2	no	—
21	#19127830	—	2	2	no	—
22	#19127828	C₂²	1	2	yes	—
23	C₈×Dic₁₆₆	C₂²	1	2	yes	—
24	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
25	C₄×#13050580	C₂²	1	2	yes	—
26	#19127761	—	2	2	no	—
27	#19127760	—	2	2	no	—
28	#19127757	C₂²	1	2	yes	—
29	#19127752	C₂²	1	1	yes	characteristic
30	#19127752	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
32	#19127453	C₂²	1	2	yes	—
33	#19127452	C₂²	1	1	yes	characteristic
34	#19127433	C₂²	1	2	yes	—
35	Q₈⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
36	#19127431	C₂²	1	2	yes	—
37	#19127422	C₂²	1	1	yes	characteristic
38	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₈×Q₈×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×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
49	#14581791	C₂×C₄	1	2	yes	—
50	#14581790	C₂×C₄	1	2	yes	—
51	#14581782	—	2	2	no	—
52	#14581772	C₂×C₄	1	2	yes	—
53	#14581770	—	2	2	no	—
54	#14581768	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₆₆	—	2	2	no	—
56	C₄×Dic₁₆₆	C₂³	1	2	yes	—
57	#14581749	—	2	2	no	—
58	C₄²×D₈₃	C₂³	1	1	yes	characteristic
59	#14581726	C₂×C₄	1	1	yes	characteristic
60	#14581725	C₂×C₄	1	2	yes	—
61	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
62	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
63	Dic₈₃⋊C₈	—	2	2	no	—
64	Dic₈₃⋊C₈	—	2	1	no	—
65	Dic₈₃⋊C₈	—	2	2	no	—
66	Dic₈₃⋊C₈	D₄	1	2	yes	—
67	#14581717	C₂×C₄	1	2	yes	—
68	#14581716	C₂×C₄	1	2	yes	—
69	#14581715	C₂×C₄	1	2	yes	—
70	(C₂×D₈₃)⋊C₈	—	2	2	no	—
71	(C₂×D₈₃)⋊C₈	—	2	1	no	—
72	(C₂×D₈₃)⋊C₈	D₄	1	2	yes	—
73	#14581703	—	2	2	no	—
74	#14581702	—	2	2	no	—
75	#14581702	—	2	2	no	—
76	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×Dic₈₃	—	2	2	no	—
80	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
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	C₄×C₈×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	C₂×C₈₃⋊C₈	—	2	2	no	—
100	C₂×C₈₃⋊C₈	—	2	1	no	—
101	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
104	C₈×D₈₃	—	2	2	no	—
105	C₈×D₈₃	—	2	2	no	—
106	C₄⋊Dic₈₃	—	2	2	no	—
107	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
108	Dic₈₃⋊C₄	—	2	2	no	—
109	Dic₈₃⋊C₄	—	2	2	no	—
110	Dic₈₃⋊C₄	—	2	2	no	—
111	Dic₈₃⋊C₄	—	2	2	no	—
112	Dic₈₃⋊C₄	—	2	2	no	—
113	(C₂×D₈₃)⋊C₄	—	2	2	no	—
114	(C₂×D₈₃)⋊C₄	—	2	2	no	—
115	C₄×Dic₈₃	—	2	2	no	—
116	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
117	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
118	C₄×Dic₈₃	#41	1	1	yes	characteristic
119	C₂×Q₈×C₈₃	C₂²×C₄	1	2	yes	—
120	Q₁₆×C₈₃	—	2	2	no	—
121	C₂×C₈×C₈₃	—	2	2	no	—
122	C₂×C₈×C₈₃	—	2	1	no	—
123	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
124	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
125	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
126	C₄⋊C₄×C₈₃	—	2	2	no	—
127	C₄⋊C₄×C₈₃	C₂²×C₄	1	2	yes	—
128	C₄²×C₈₃	—	2	2	no	—
129	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
130	C₂²×D₈₃	C₄×D₄	1	1	yes	characteristic
131	Q₈×C₈₃	—	2	2	no	—
132	Q₈×C₈₃	—	2	2	no	—
133	C₂×C₄×C₈₃	—	2	2	no	—
134	C₂×C₄×C₈₃	—	2	2	no	—
135	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
138	C₂×Dic₈₃	—	2	2	no	—
139	C₂×Dic₈₃	—	2	2	no	—
140	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
141	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
142	C₂×Dic₈₃	C₄×D₄	1	1	yes	characteristic
143	C₄×D₈₃	—	2	1	no	—
144	C₄×D₈₃	—	2	1	no	—
145	C₄×D₈₃	—	2	2	no	—
146	C₄×D₈₃	—	4	1	no	—
147	Dic₁₆₆	—	2	2	no	—
148	Dic₁₆₆	—	2	2	no	—
149	Dic₁₆₆	—	8	2	no	—
150	C₈×C₈₃	—	2	1	no	—
151	C₈×C₈₃	—	2	2	no	—
152	C₈×C₈₃	—	4	1	no	—
153	C₈×C₈₃	—	4	2	no	—
154	C₈×C₈₃	#131	1	2	yes	—
155	C₈₃⋊C₈	—	2	1	no	—
156	C₈₃⋊C₈	—	2	2	no	—
157	C₈₃⋊C₈	—	4	1	no	—
158	C₈₃⋊C₈	—	4	2	no	—
159	C₈₃⋊C₈	#131	1	2	yes	—
160	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
161	C₂×D₈₃	—	2	1	no	—
162	C₂×D₈₃	—	2	1	no	—
163	C₂×D₈₃	—	2	1	no	—
164	C₄×C₈₃	—	2	1	no	—
165	C₄×C₈₃	—	4	2	no	—
166	C₄×C₈₃	—	4	2	no	—
167	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
168	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
169	C₄×C₈₃	#578	1	2	yes	—
170	Dic₈₃	—	2	1	no	—
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	4	2	no	—
174	Dic₈₃	—	4	2	no	—
175	#17972	—	83	1	no	maximal, Sylow
176	C₂×C₈₃	#2931	1	1	yes	characteristic
177	C₂×C₈₃	#2950	1	1	yes	characteristic
178	C₂×C₈₃	#2957	1	1	yes	characteristic
179	D₈₃	—	4	1	no	—
180	#3170	—	83	1	no	—
181	#2985	—	83	2	no	—
182	#1593	—	83	1	no	—
183	#1593	—	83	1	no	—
184	#1591	—	83	2	no	—
185	#1579	—	83	1	no	—
186	#1579	D₈₃	1	1	yes	characteristic, complemented
187	C₈×Q₁₆	—	83	2	no	—
188	#1568	—	83	1	no	—
189	#1497	—	83	2	no	—
190	#1450	—	83	1	no	—
191	C₈₃	#17972	1	1	yes	characteristic, complemented, Sylow
192	#533	—	83	2	no	—
193	#500	—	83	1	no	—
194	#494	—	83	1	no	—
195	#487	—	83	2	no	—
196	#467	—	83	1	no	—
197	#467	—	83	1	no	—
198	#467	—	83	2	no	—
199	#446	—	83	2	no	—
200	#446	—	83	2	no	—
201	#446	—	166	2	no	—
202	#446	—	166	2	no	—
203	C₈×Q₈	—	83	2	no	—
204	C₈×Q₈	C₂×D₈₃	1	2	yes	—
205	C₄×Q₁₆	—	83	2	no	—
206	C₄×Q₁₆	—	83	1	no	—
207	C₄×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
208	#433	—	166	2	no	—
209	#433	—	166	2	no	—
210	#432	—	83	2	no	—
211	#431	—	83	1	no	—
212	#431	—	83	1	no	—
213	C₂×C₄×C₈	—	83	1	no	—
214	#335	—	83	2	no	—
215	#334	—	83	1	no	—
216	#334	C₂×D₈₃	1	1	yes	characteristic
217	Q₈⋊C₈	—	83	2	no	—
218	Q₈⋊C₈	—	83	2	no	—
219	Q₈⋊C₈	—	83	2	no	—
220	Q₈⋊C₈	C₂×D₈₃	1	2	yes	—
221	#322	—	83	1	no	—
222	#322	C₂×D₈₃	1	1	yes	characteristic
223	C₈²	—	83	1	no	—
224	C₈²	—	83	1	no	—
225	C₂×Q₁₆	—	83	2	no	—
226	C₂×Q₁₆	—	83	1	no	—
227	C₂×Q₁₆	C₄×D₈₃	1	1	yes	characteristic
228	C₂²×C₈	—	83	1	no	—
229	C₂²×C₈	—	83	1	no	—
230	C₄⋊Q₈	—	83	2	no	—
231	C₄⋊Q₈	—	83	2	no	—
232	#125	—	166	2	no	—
233	C₂²⋊Q₈	—	83	2	no	—
234	C₂²⋊Q₈	—	83	2	no	—
235	C₂²⋊Q₈	—	166	2	no	—
236	C₄×Q₈	—	83	2	no	—
237	C₄×Q₈	—	166	2	no	—
238	C₄×Q₈	C₂²×D₈₃	1	2	yes	—
239	#117	—	166	2	no	—
240	C₂×C₄²	—	83	1	no	—
241	#107	—	83	1	no	—
242	#107	—	83	2	no	—
243	#107	C₄×D₈₃	1	1	yes	characteristic
244	C₄⋊C₈	—	2	2	no	—
245	C₄⋊C₈	—	83	2	no	—
246	C₄⋊C₈	—	83	2	no	—
247	C₄⋊C₈	—	166	2	no	—
248	C₄⋊C₈	—	166	1	no	—
249	C₄⋊C₈	—	166	2	no	—
250	C₄⋊C₈	—	166	2	no	—
251	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
252	#103	—	83	2	no	—
253	#103	—	83	2	no	—
254	#103	—	83	2	no	—
255	#103	C₄×D₈₃	1	2	yes	—
256	C₂²⋊C₈	—	83	2	no	—
257	C₂²⋊C₈	—	166	1	no	—
258	C₂²⋊C₈	—	166	2	no	—
259	#97	—	166	2	no	—
260	#97	—	166	2	no	—
261	#97	—	166	2	no	—
262	C₄×C₈	—	2	2	no	—
263	C₄×C₈	—	83	1	no	—
264	C₄×C₈	—	83	1	no	—
265	C₄×C₈	—	83	1	no	—
266	C₄×C₈	—	83	1	no	—
267	C₄×C₈	—	83	1	no	—
268	C₄×C₈	—	166	2	no	—
269	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
271	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₂×Q₈	—	83	2	no	—
273	C₂×Q₈	—	166	2	no	—
274	C₂×Q₈	C₂×C₄×D₈₃	1	2	yes	—
275	C₂²×C₄	—	83	1	no	—
276	C₂²×C₄	—	83	1	no	—
277	C₂²×C₄	—	83	1	no	—
278	Q₁₆	—	2	2	no	—
279	Q₁₆	—	166	4	no	—
280	Q₁₆	—	166	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	83	1	no	—
284	C₂×C₈	—	83	1	no	—
285	C₂×C₈	—	83	2	no	—
286	C₂×C₈	—	166	2	no	—
287	C₂×C₈	—	166	2	no	—
288	C₂×C₈	—	166	1	no	—
289	C₂×C₈	—	166	2	no	—
290	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
292	C₂×C₈	D₄×D₈₃	1	2	yes	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	83	2	no	—
295	C₄⋊C₄	—	166	2	no	—
296	C₄⋊C₄	—	166	2	no	—
297	C₄⋊C₄	—	166	2	no	—
298	C₄⋊C₄	—	166	2	no	—
299	C₄⋊C₄	—	166	2	no	—
300	C₄⋊C₄	—	166	2	no	—
301	C₄⋊C₄	C₂×C₄×D₈₃	1	2	yes	—
302	C₂²⋊C₄	—	166	2	no	—
303	C₂²⋊C₄	—	166	2	no	—
304	C₄²	—	2	2	no	—
305	C₄²	—	83	1	no	—
306	C₄²	—	83	1	no	—
307	C₄²	—	83	1	no	—
308	C₄²	—	166	2	no	—
309	C₄²	C₂³×D₈₃	1	1	yes	characteristic
310	C₂³	—	83	1	no	—
311	Q₈	—	2	2	no	—
312	Q₈	—	2	2	no	—
313	Q₈	—	664	2	no	—
314	Q₈	—	166	2	no	—
315	Q₈	—	166	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	332	1	no	—
319	C₂×C₄	—	83	1	no	—
320	C₂×C₄	—	83	1	no	—
321	C₂×C₄	—	83	1	no	—
322	C₂×C₄	—	166	2	no	—
323	C₂×C₄	—	166	2	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	—	166	1	no	—
326	C₂×C₄	—	166	2	no	—
327	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
328	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
329	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	2	2	no	—
332	C₈	—	4	1	no	—
333	C₈	—	4	2	no	—
334	C₈	—	332	1	no	—
335	C₈	—	332	2	no	—
336	C₈	—	83	2	no	—
337	C₈	—	166	1	no	—
338	C₈	—	166	2	no	—
339	C₈	#14581798	1	2	yes	—
340	C₂²	—	166	1	no	—
341	C₂²	—	166	1	no	—
342	C₂²	—	166	1	no	—
343	C₂²	C₄×D₄×D₈₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	2	no	—
347	C₄	—	332	2	no	—
348	C₄	—	332	2	no	—
349	C₄	—	166	1	no	—
350	C₄	—	166	1	no	—
351	C₄	—	166	1	no	—
352	C₄	#131×D₈₃	1	1	yes	characteristic
353	C₄	C₂×#14581798	1	1	yes	characteristic
354	C₄	#19128517	1	2	yes	—
355	C₂	—	332	1	no	—
356	C₂	#24916170	1	1	yes	characteristic
357	C₂	#24916328	1	1	yes	characteristic
358	C₂	#24916377	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×#1579	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	C₂²×D₈₃	C₄×D₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1579	D₈₃
hypocenter	C₈₃	#17972
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #1579

Complements

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

N	H ≅ G/N	type
3. #24906372	C₂	semidirect
4. #24906371	C₂	semidirect
5. #24906357	C₂	semidirect
6. #24906283	C₂	semidirect
7. C₈×#13050580	C₂	semidirect
11. C₈₃×#1579	C₂	semidirect
186. #1579	D₈₃	semidirect
191. C₈₃	#17972	semidirect

Maximal subgroups

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

subgroup	index
#24918046	2
#24916554	2
#24906372	2
#24906371	2
#24906357	2
#24906283	2
C₈×#13050580	2
#24906223	2
#24905934	2
#24905748	2
C₈₃×#1579	2
#17972	83