Subgroups — #35302736

Order 21248.

466 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908389	C₂	1	1	yes	maximal, characteristic
2	#24908268	C₂	1	1	yes	maximal, characteristic
3	#24908224	C₂	1	1	yes	maximal, characteristic, complemented
4	#24907328	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127444	C₂	1	1	yes	maximal, characteristic
6	#24905297	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14581789	C₂²	1	1	yes	characteristic
9	C₂×#14581723	—	2	1	no	—
10	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
11	C₂×#14581714	—	2	1	no	—
12	C₂×#14581714	—	2	1	no	—
13	C₂×#14581712	—	2	1	no	—
14	C₂×#14581712	—	2	1	no	—
15	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₈×Dic₈₃	—	2	1	no	—
17	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
18	#19127463	—	2	1	no	—
19	#19127455	—	2	1	no	—
20	#19127444	C₄	1	4	yes	complemented
21	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
23	#384×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²×D₃₃₂	—	2	1	no	—
25	C₂²×D₃₃₂	—	2	1	no	—
26	C₂²×D₃₃₂	—	2	1	no	—
27	C₂²×C₈₃⋊C₈	—	2	1	no	—
28	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
30	#14581789	—	2	1	no	—
31	#14581789	—	2	1	no	—
32	#14581789	—	2	1	no	—
33	#14581789	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
35	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₈₃	—	2	1	no	—
37	#14581723	—	4	2	no	—
38	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	2	yes	—
40	#14581714	—	4	2	no	—
41	#14581714	—	4	2	no	—
42	#14581712	—	4	2	no	—
43	#14581712	—	4	2	no	—
44	C₄×C₈₃⋊C₈	—	2	1	no	—
45	C₄×C₈₃⋊C₈	—	2	1	no	—
46	C₈×Dic₈₃	—	4	2	no	—
47	C₈₃⋊#95	—	2	1	no	—
48	C₂²×C₈×C₈₃	—	2	1	no	—
49	C₂²×C₈×C₈₃	—	2	1	no	—
50	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
51	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
53	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₈₃	—	2	1	no	—
55	C₄⋊C₈×C₈₃	—	2	1	no	—
56	C₄⋊C₈×C₈₃	—	2	1	no	—
57	C₄⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
58	C₂³×D₈₃	—	4	1	no	—
59	C₂×D₃₃₂	—	2	2	no	—
60	C₂×D₃₃₂	—	2	2	no	—
61	C₂×D₃₃₂	—	2	2	no	—
62	C₂×D₃₃₂	—	4	1	no	—
63	C₂×D₃₃₂	—	4	1	no	—
64	C₂×D₃₃₂	—	4	1	no	—
65	C₂×D₃₃₂	—	4	1	no	—
66	C₂×D₃₃₂	—	4	1	no	—
67	C₂×D₃₃₂	—	4	1	no	—
68	C₂×D₃₃₂	—	4	1	no	—
69	C₂×D₃₃₂	—	4	1	no	—
70	C₂×D₃₃₂	—	4	1	no	—
71	C₂×D₃₃₂	—	4	1	no	—
72	C₂×D₃₃₂	—	4	1	no	—
73	C₂×D₃₃₂	—	8	1	no	—
74	C₂×D₃₃₂	—	8	1	no	—
75	C₂²×Dic₈₃	—	2	1	no	—
76	C₂²×Dic₈₃	—	2	1	no	—
77	C₂×C₈₃⋊C₈	—	2	1	no	—
78	C₂×C₈₃⋊C₈	—	2	1	no	—
79	C₂×C₈₃⋊C₈	—	2	2	no	—
80	C₂×C₈₃⋊C₈	—	2	1	no	—
81	C₂×C₈₃⋊C₈	—	2	1	no	—
82	C₂×C₈₃⋊C₈	—	2	1	no	—
83	C₂×C₈₃⋊C₈	—	4	1	no	—
84	C₂×C₈₃⋊C₈	—	4	1	no	—
85	C₂×C₈₃⋊C₈	SD₁₆	1	2	yes	—
86	C₄⋊Dic₈₃	—	2	1	no	—
87	C₄⋊Dic₈₃	—	2	1	no	—
88	C₄⋊Dic₈₃	D₈	1	2	yes	—
89	C₄⋊Dic₈₃	SD₁₆	1	2	yes	—
90	C₄×Dic₈₃	—	4	1	no	—
91	C₄×Dic₈₃	—	4	1	no	—
92	C₂²×C₄×C₈₃	—	4	1	no	—
93	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₈₃	—	2	2	no	—
97	C₂×C₈×C₈₃	—	4	1	no	—
98	C₂×C₈×C₈₃	—	4	1	no	—
99	C₂×C₈×C₈₃	—	4	1	no	—
100	C₂×C₈×C₈₃	—	4	1	no	—
101	C₂×C₈×C₈₃	—	4	1	no	—
102	C₄⋊C₄×C₈₃	—	4	1	no	—
103	C₄⋊C₄×C₈₃	—	4	1	no	—
104	C₄⋊C₄×C₈₃	—	4	1	no	—
105	C₄⋊C₄×C₈₃	—	4	1	no	—
106	C₄⋊C₄×C₈₃	—	8	1	no	—
107	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
108	C₄²×C₈₃	#41	1	1	yes	characteristic
109	C₄²×C₈₃	#41	1	1	yes	characteristic
110	C₄²×C₈₃	#41	1	1	yes	characteristic
111	C₂³×C₈₃	#121	1	1	yes	characteristic
112	C₂²×D₈₃	—	4	2	no	—
113	C₂²×D₈₃	—	8	1	no	—
114	C₂²×D₈₃	—	8	1	no	—
115	C₂²×D₈₃	—	8	1	no	—
116	C₂²×D₈₃	—	8	1	no	—
117	C₂²×D₈₃	—	8	1	no	—
118	C₂²×D₈₃	—	8	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	8	1	no	—
122	C₂×C₄×C₈₃	—	8	1	no	—
123	C₂×C₄×C₈₃	—	8	1	no	—
124	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
127	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
128	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
129	C₂×C₄×C₈₃	#126	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
133	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
134	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
135	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
136	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×Dic₈₃	—	2	2	no	—
139	C₂×Dic₈₃	—	2	2	no	—
140	C₂×Dic₈₃	—	4	1	no	—
141	C₂×Dic₈₃	—	4	1	no	—
142	C₂×Dic₈₃	—	4	1	no	—
143	C₂×Dic₈₃	—	4	1	no	—
144	D₃₃₂	—	4	2	no	—
145	D₃₃₂	—	4	2	no	—
146	D₃₃₂	—	4	2	no	—
147	D₃₃₂	—	4	2	no	—
148	D₃₃₂	—	8	2	no	—
149	D₃₃₂	—	8	2	no	—
150	D₃₃₂	—	8	1	no	—
151	D₃₃₂	—	8	1	no	—
152	D₃₃₂	—	8	1	no	—
153	D₃₃₂	—	8	1	no	—
154	C₈×C₈₃	—	4	2	no	—
155	C₈×C₈₃	—	8	1	no	—
156	C₈₃⋊C₈	—	2	2	no	—
157	C₈₃⋊C₈	—	4	2	no	—
158	C₈₃⋊C₈	—	4	1	no	—
159	C₂²×C₈₃	#388	1	1	yes	characteristic
160	C₂²×C₈₃	#460	1	1	yes	characteristic
161	C₂²×C₈₃	#463	1	1	yes	characteristic
162	C₂²×C₈₃	#465	1	1	yes	characteristic
163	C₂²×C₈₃	#466	1	1	yes	characteristic
164	C₂²×C₈₃	#471	1	1	yes	characteristic
165	C₂²×C₈₃	#472	1	1	yes	characteristic
166	C₂×D₈₃	—	8	2	no	—
167	C₂×D₈₃	—	8	2	no	—
168	C₂×D₈₃	—	8	2	no	—
169	C₂×D₈₃	—	16	1	no	—
170	C₂×D₈₃	—	16	1	no	—
171	C₂×D₈₃	—	16	1	no	—
172	C₂×D₈₃	—	16	1	no	—
173	C₄×C₈₃	—	2	1	no	—
174	C₄×C₈₃	—	2	1	no	—
175	C₄×C₈₃	—	16	1	no	—
176	C₄×C₈₃	C₄×D₈	1	1	yes	characteristic
177	C₄×C₈₃	C₄×SD₁₆	1	1	yes	characteristic
178	C₄×C₈₃	C₄×SD₁₆	1	1	yes	characteristic
179	C₄×C₈₃	#442	1	1	yes	characteristic
180	C₄×C₈₃	#481	1	1	yes	characteristic
181	C₄×C₈₃	#482	1	1	yes	characteristic
182	C₄×C₈₃	#486	1	1	yes	characteristic
183	C₄×C₈₃	#488	1	1	yes	characteristic
184	Dic₈₃	—	4	2	no	—
185	Dic₈₃	—	4	2	no	—
186	#9564	—	83	1	no	maximal, Sylow
187	C₂×C₈₃	#1666	1	1	yes	characteristic
188	C₂×C₈₃	#1668	1	1	yes	characteristic
189	C₂×C₈₃	#1677	1	1	yes	characteristic
190	C₂×C₈₃	#1681	1	1	yes	characteristic
191	C₂×C₈₃	#1926	1	1	yes	characteristic
192	C₂×C₈₃	#1931	1	1	yes	characteristic
193	C₂×C₈₃	#1934	1	1	yes	characteristic
194	D₈₃	—	16	2	no	—
195	#1982	—	83	1	no	—
196	#1957	—	83	1	no	—
197	#1948	—	83	1	no	—
198	C₈×C₄⋊C₄	—	83	1	no	—
199	C₂×#331	—	83	1	no	—
200	#1296	D₈₃	1	1	yes	characteristic, complemented
201	#1277	—	83	1	no	—
202	C₈₃	#9564	1	1	yes	characteristic, complemented, Sylow
203	C₂×#127	—	83	1	no	—
204	C₂×#106	—	166	1	no	—
205	C₂×C₄⋊C₈	—	83	1	no	—
206	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
207	C₂×C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
208	C₂×#102	—	166	1	no	—
209	C₂×#102	—	166	1	no	—
210	C₂×#102	—	166	1	no	—
211	C₂×#102	—	166	1	no	—
212	C₂×C₄×C₈	—	83	1	no	—
213	C₂×C₄×C₈	—	166	1	no	—
214	#384	C₂×D₈₃	1	1	yes	characteristic
215	C₄×C₄⋊C₄	—	83	1	no	—
216	#340	—	166	1	no	—
217	#336	—	166	1	no	—
218	#331	—	83	4	no	—
219	C₂²×D₄	—	166	1	no	—
220	C₂²×D₄	—	166	1	no	—
221	C₂²×D₄	—	166	1	no	—
222	C₂²×C₈	—	2	1	no	—
223	C₂²×C₈	—	2	1	no	—
224	C₂²×C₈	—	83	1	no	—
225	C₂²×C₈	—	83	1	no	—
226	C₂²×C₈	—	166	1	no	—
227	#127	—	83	2	no	—
228	#127	—	166	1	no	—
229	#127	—	166	1	no	—
230	#127	—	166	1	no	—
231	C₂×C₄⋊C₄	—	2	1	no	—
232	C₂×C₄⋊C₄	—	2	1	no	—
233	C₂×C₄⋊C₄	—	2	1	no	—
234	C₂×C₄⋊C₄	—	83	1	no	—
235	C₂×C₄⋊C₄	—	83	1	no	—
236	C₂×C₄²	—	166	1	no	—
237	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
238	#106	—	332	2	no	—
239	C₄⋊C₈	—	2	1	no	—
240	C₄⋊C₈	—	2	1	no	—
241	C₄⋊C₈	—	2	1	no	—
242	C₄⋊C₈	—	83	2	no	—
243	C₄⋊C₈	—	166	1	no	—
244	C₄⋊C₈	C₄×D₈₃	1	2	yes	—
245	#102	—	332	2	no	—
246	#102	—	332	2	no	—
247	#102	—	332	2	no	—
248	#102	—	332	2	no	—
249	C₄×C₈	—	332	2	no	—
250	C₄×C₈	—	166	1	no	—
251	C₄×C₈	—	166	1	no	—
252	#95	—	166	1	no	—
253	C₂⁴	—	332	1	no	—
254	C₂×D₄	—	332	1	no	—
255	C₂×D₄	—	332	1	no	—
256	C₂×D₄	—	332	1	no	—
257	C₂×D₄	—	332	1	no	—
258	C₂×D₄	—	332	1	no	—
259	C₂×D₄	—	332	1	no	—
260	C₂×D₄	—	332	1	no	—
261	C₂×D₄	—	332	1	no	—
262	C₂×D₄	—	332	1	no	—
263	C₂×D₄	—	332	1	no	—
264	C₂×D₄	—	332	1	no	—
265	C₂×D₄	—	664	1	no	—
266	C₂×D₄	—	664	1	no	—
267	C₂×D₄	—	166	2	no	—
268	C₂×D₄	—	166	2	no	—
269	C₂×D₄	—	166	2	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	—	166	1	no	—
272	C₂²×C₄	—	166	1	no	—
273	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
274	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
275	C₂²×C₄	#13050590	1	1	yes	characteristic
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	4	1	no	—
282	C₂×C₈	—	332	1	no	—
283	C₂×C₈	—	332	1	no	—
284	C₂×C₈	—	83	2	no	—
285	C₂×C₈	—	166	1	no	—
286	C₂×C₈	—	166	1	no	—
287	C₂×C₈	—	166	1	no	—
288	C₂×C₈	—	166	2	no	—
289	C₂×C₈	—	166	1	no	—
290	C₂×C₈	—	166	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	4	1	no	—
295	C₄⋊C₄	—	8	1	no	—
296	C₄⋊C₄	—	83	2	no	—
297	C₄⋊C₄	—	83	2	no	—
298	C₄⋊C₄	—	166	1	no	—
299	C₄⋊C₄	—	166	1	no	—
300	C₄²	—	332	1	no	—
301	C₄²	—	332	1	no	—
302	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
303	C₄²	#13050589	1	1	yes	characteristic
304	C₄²	#13050590	1	1	yes	characteristic
305	C₄²	#13050591	1	1	yes	characteristic
306	C₂³	—	332	2	no	—
307	C₂³	—	664	1	no	—
308	C₂³	—	664	1	no	—
309	C₂³	—	664	1	no	—
310	C₂³	—	664	1	no	—
311	C₂³	—	664	1	no	—
312	C₂³	—	664	1	no	—
313	C₂³	#14581765	1	1	yes	characteristic
314	D₄	—	332	2	no	—
315	D₄	—	332	2	no	—
316	D₄	—	332	2	no	—
317	D₄	—	332	2	no	—
318	D₄	—	664	2	no	—
319	D₄	—	664	2	no	—
320	D₄	—	664	1	no	—
321	D₄	—	664	1	no	—
322	D₄	—	664	1	no	—
323	D₄	—	664	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	8	1	no	—
327	C₂×C₄	—	8	1	no	—
328	C₂×C₄	—	8	1	no	—
329	C₂×C₄	—	332	1	no	—
330	C₂×C₄	—	332	1	no	—
331	C₂×C₄	—	332	1	no	—
332	C₂×C₄	—	332	1	no	—
333	C₂×C₄	—	166	2	no	—
334	C₂×C₄	—	166	2	no	—
335	C₂×C₄	#14581749	1	1	yes	characteristic
336	C₂×C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
337	C₂×C₄	#14581754	1	1	yes	characteristic
338	C₂×C₄	#14581777	1	1	yes	characteristic
339	C₂×C₄	#14581778	1	1	yes	characteristic
340	C₂×C₄	#14581787	1	1	yes	characteristic
341	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
342	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
343	C₂×C₄	SD₁₆×D₈₃	1	1	yes	characteristic
344	C₂×C₄	#14581813	1	1	yes	characteristic
345	C₂×C₄	#14581815	1	1	yes	characteristic
346	C₂×C₄	#14581815	1	1	yes	characteristic
347	C₂×C₄	#14581816	1	1	yes	characteristic
348	C₂×C₄	#14581819	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₈	—	332	1	no	—
352	C₈	—	332	2	no	—
353	C₈	—	166	2	no	—
354	C₂²	—	1328	1	no	—
355	C₂²	—	1328	1	no	—
356	C₂²	—	1328	1	no	—
357	C₂²	—	1328	1	no	—
358	C₂²	—	664	2	no	—
359	C₂²	—	664	2	no	—
360	C₂²	—	664	2	no	—
361	C₂²	#19127590	1	1	yes	characteristic
362	C₂²	#19127900	1	1	yes	characteristic
363	C₂²	#19127919	1	1	yes	characteristic
364	C₂²	#19127933	1	1	yes	characteristic
365	C₂²	#19127936	1	1	yes	characteristic
366	C₂²	#19127967	1	1	yes	characteristic
367	C₂²	#19127970	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	16	1	no	—
371	C₄	—	332	2	no	—
372	C₄	—	332	2	no	—
373	C₄	#19127783	1	1	yes	characteristic
374	C₄	#19127786	1	1	yes	characteristic
375	C₄	#19127788	1	1	yes	characteristic
376	C₄	#19127810	1	1	yes	characteristic
377	C₄	#19128021	1	1	yes	characteristic
378	C₄	#19128028	1	1	yes	characteristic
379	C₄	#19128046	1	1	yes	characteristic
380	C₄	#19128057	1	1	yes	characteristic
381	C₂	—	1328	2	no	—
382	C₂	#24906814	1	1	yes	characteristic
383	C₂	#24906822	1	1	yes	characteristic
384	C₂	#24906875	1	1	yes	characteristic
385	C₂	#24906903	1	1	yes	characteristic
386	C₂	#24908088	1	1	yes	characteristic
387	C₂	#24908123	1	1	yes	characteristic
388	C₂	#24908144	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581765
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1296	C₂
socle	C₂³×C₈₃	#121
prime radical	C₂×#14581789	C₂²
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1296	D₈₃
hypocenter	C₈₃	#9564
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
3. #24908224	C₂	semidirect
4. #24907328	C₂	semidirect
6. #24905297	C₂	semidirect
7. C₈₃×#1296	C₂	semidirect
20. #19127444	C₄	semidirect
200. #1296	D₈₃	semidirect
202. C₈₃	#9564	semidirect

Maximal subgroups

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

subgroup	index
#24908389	2
#24908268	2
#24908224	2
#24907328	2
C₂×#19127444	2
#24905297	2
C₈₃×#1296	2
#9564	83