Subgroups — C₈₃×#19645 #35247469

Order 21248.

772 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#3228	C₂	1	2	yes	maximal, complemented
2	C₈₃×#3182	C₂	1	2	yes	maximal, complemented
3	C₈₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₃×#3147	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1719	C₂	1	2	yes	maximal, complemented
6	C₈₃×#1714	C₂	1	2	yes	maximal, complemented
7	C₈₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₃×#1709	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₃×#1707	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×M₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#135×C₈₃	—	2	2	no	—
13	#533×C₈₃	C₂²	1	2	yes	—
14	#532×C₈₃	C₂²	1	2	yes	—
15	#500×C₈₃	C₂²	1	1	yes	characteristic
16	#500×C₈₃	C₂²	1	2	yes	—
17	#500×C₈₃	C₂²	1	2	yes	—
18	#498×C₈₃	C₂²	1	2	yes	—
19	#495×C₈₃	—	2	2	no	—
20	#494×C₈₃	C₂²	1	1	yes	characteristic
21	#494×C₈₃	C₂²	1	1	yes	characteristic
22	#493×C₈₃	C₂²	1	1	yes	characteristic
23	#492×C₈₃	C₂²	1	2	yes	—
24	#492×C₈₃	C₂²	1	2	yes	—
25	#487×C₈₃	C₂²	1	2	yes	—
26	#487×C₈₃	C₂²	1	2	yes	—
27	#486×C₈₃	C₂²	1	2	yes	—
28	#486×C₈₃	C₂²	1	2	yes	—
29	#486×C₈₃	C₂²	1	2	yes	—
30	#482×C₈₃	—	2	4	no	—
31	#478×C₈₃	—	2	4	no	—
32	#468×C₈₃	—	2	4	no	—
33	#467×C₈₃	—	2	2	no	—
34	#440×C₈₃	—	2	4	no	—
35	C₄×Q₁₆×C₈₃	—	2	2	no	—
36	#431×C₈₃	C₂²	1	1	yes	characteristic
37	C₄×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
38	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
39	#322×C₈₃	C₂²	1	1	yes	characteristic
40	#322×C₈₃	C₂²	1	1	yes	characteristic
41	C₈²×C₈₃	C₂²	1	1	yes	characteristic
42	C₈²×C₈₃	C₂²	1	1	yes	characteristic
43	C₂×#41×C₈₃	—	2	2	no	—
44	#135×C₈₃	—	4	8	no	—
45	C₂×Q₁₆×C₈₃	—	2	2	no	—
46	C₂×Q₁₆×C₈₃	—	2	2	no	—
47	C₂×SD₁₆×C₈₃	—	2	4	no	—
48	C₂×SD₁₆×C₈₃	—	2	4	no	—
49	C₂×D₈×C₈₃	—	2	2	no	—
50	C₂×M₁₆×C₈₃	D₄	1	2	yes	—
51	C₂²×C₈×C₈₃	D₄	1	2	yes	—
52	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
53	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
54	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
55	#127×C₈₃	C₂³	1	2	yes	—
56	#125×C₈₃	—	2	2	no	—
57	#124×C₈₃	—	2	2	no	—
58	C₂²⋊Q₈×C₈₃	—	2	2	no	—
59	C₂²⋊Q₈×C₈₃	—	2	2	no	—
60	#121×C₈₃	—	2	2	no	—
61	#121×C₈₃	—	2	2	no	—
62	C₄×Q₈×C₈₃	—	2	2	no	—
63	C₄×Q₈×C₈₃	—	2	2	no	—
64	C₄×D₄×C₈₃	—	2	2	no	—
65	C₄×D₄×C₈₃	—	2	2	no	—
66	#117×C₈₃	—	2	2	no	—
67	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
68	#107×C₈₃	—	2	2	no	—
69	#107×C₈₃	—	2	4	no	—
70	#107×C₈₃	—	2	4	no	—
71	#106×C₈₃	—	2	4	no	—
72	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
73	#103×C₈₃	—	2	4	no	—
74	#103×C₈₃	—	2	4	no	—
75	#102×C₈₃	—	2	4	no	—
76	#102×C₈₃	—	2	4	no	—
77	#102×C₈₃	—	2	4	no	—
78	C₂²⋊C₈×C₈₃	D₄	1	2	yes	—
79	#97×C₈₃	D₄	1	2	yes	—
80	C₄×C₈×C₈₃	D₄	1	2	yes	—
81	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
82	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
83	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
84	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
85	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
87	#41×C₈₃	—	4	8	no	—
88	C₂×Q₈×C₈₃	—	2	2	no	—
89	C₂×Q₈×C₈₃	—	2	2	no	—
90	C₂×Q₈×C₈₃	—	2	2	no	—
91	C₂×D₄×C₈₃	—	2	2	no	—
92	C₂×D₄×C₈₃	—	2	2	no	—
93	C₂×D₄×C₈₃	—	2	2	no	—
94	C₂×D₄×C₈₃	—	4	2	no	—
95	C₂²×C₄×C₈₃	—	4	2	no	—
96	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
99	Q₁₆×C₈₃	—	4	4	no	—
100	Q₁₆×C₈₃	—	4	4	no	—
101	SD₁₆×C₈₃	—	4	8	no	—
102	SD₁₆×C₈₃	—	4	8	no	—
103	D₈×C₈₃	—	4	4	no	—
104	M₁₆×C₈₃	—	2	4	no	—
105	C₂×C₈×C₈₃	—	2	4	no	—
106	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
110	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
111	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
112	C₄⋊C₄×C₈₃	—	2	2	no	—
113	C₄⋊C₄×C₈₃	—	2	2	no	—
114	C₄⋊C₄×C₈₃	—	2	2	no	—
115	C₄⋊C₄×C₈₃	—	2	2	no	—
116	C₄⋊C₄×C₈₃	—	2	2	no	—
117	C₄⋊C₄×C₈₃	—	2	2	no	—
118	C₄⋊C₄×C₈₃	—	4	2	no	—
119	C₄⋊C₄×C₈₃	—	4	2	no	—
120	C₂²⋊C₄×C₈₃	—	4	2	no	—
121	C₂²⋊C₄×C₈₃	—	4	2	no	—
122	C₂²⋊C₄×C₈₃	—	4	2	no	—
123	C₄²×C₈₃	—	4	2	no	—
124	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
127	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
128	C₂³×C₈₃	—	4	2	no	—
129	C₂³×C₈₃	#127	1	1	yes	characteristic
130	Q₈×C₈₃	—	4	4	no	—
131	Q₈×C₈₃	—	4	4	no	—
132	Q₈×C₈₃	—	4	4	no	—
133	D₄×C₈₃	—	4	4	no	—
134	D₄×C₈₃	—	4	4	no	—
135	D₄×C₈₃	—	4	4	no	—
136	D₄×C₈₃	—	8	4	no	—
137	C₂×C₄×C₈₃	—	2	2	no	—
138	C₂×C₄×C₈₃	—	2	2	no	—
139	C₂×C₄×C₈₃	—	2	2	no	—
140	C₂×C₄×C₈₃	—	4	2	no	—
141	C₂×C₄×C₈₃	—	4	2	no	—
142	C₂×C₄×C₈₃	—	4	2	no	—
143	C₂×C₄×C₈₃	—	8	4	no	—
144	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
145	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
146	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
147	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
150	C₈×C₈₃	—	2	2	no	—
151	C₈×C₈₃	—	2	2	no	—
152	C₈×C₈₃	—	2	2	no	—
153	C₈×C₈₃	—	2	2	no	—
154	C₈×C₈₃	#135	1	4	yes	complemented
155	C₈×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
156	C₂²×C₈₃	—	2	1	no	—
157	C₂²×C₈₃	—	2	1	no	—
158	C₂²×C₈₃	—	2	1	no	—
159	C₂²×C₈₃	—	8	2	no	—
160	C₂²×C₈₃	—	8	2	no	—
161	C₂²×C₈₃	—	8	2	no	—
162	C₂²×C₈₃	C₂×#127	1	1	yes	characteristic
163	C₄×C₈₃	—	2	1	no	—
164	C₄×C₈₃	—	2	1	no	—
165	C₄×C₈₃	—	2	1	no	—
166	C₄×C₈₃	—	8	2	no	—
167	C₄×C₈₃	—	8	2	no	—
168	C₄×C₈₃	—	8	2	no	—
169	C₄×C₈₃	C₂×#135	1	2	yes	—
170	C₄×C₈₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
171	C₄×C₈₃	#578	1	2	yes	—
172	#19645	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
173	C₂×C₈₃	—	4	1	no	—
174	C₂×C₈₃	—	16	2	no	—
175	C₂×C₈₃	#3147	1	1	yes	characteristic
176	C₂×C₈₃	#3151	1	1	yes	characteristic
177	C₂×C₈₃	#3156	1	1	yes	characteristic
178	#3228	C₂×C₈₃	1	2	yes	complemented
179	#3182	C₂×C₈₃	1	2	yes	complemented
180	#3170	C₂×C₈₃	1	1	yes	characteristic, complemented
181	#3147	C₂×C₈₃	1	1	yes	characteristic
182	#1719	C₂×C₈₃	1	2	yes	complemented
183	#1714	C₂×C₈₃	1	2	yes	complemented
184	#1713	C₂×C₈₃	1	1	yes	characteristic, complemented
185	#1709	C₂×C₈₃	1	1	yes	characteristic, complemented
186	#1707	C₂×C₈₃	1	1	yes	characteristic, complemented
187	#1706	C₂×C₈₃	1	1	yes	characteristic, complemented
188	C₈×M₁₆	C₂×C₈₃	1	1	yes	characteristic, complemented
189	C₈₃	#19645	1	1	yes	characteristic, direct product factor, Sylow
190	C₂×#135	—	2	2	no	—
191	#533	C₂²×C₈₃	1	2	yes	—
192	#532	C₂²×C₈₃	1	2	yes	—
193	#500	C₂²×C₈₃	1	1	yes	characteristic
194	#500	C₂²×C₈₃	1	2	yes	—
195	#500	C₂²×C₈₃	1	2	yes	—
196	#498	C₂²×C₈₃	1	2	yes	—
197	#495	—	2	2	no	—
198	#494	C₂²×C₈₃	1	1	yes	characteristic
199	#494	C₂²×C₈₃	1	1	yes	characteristic
200	#493	C₂²×C₈₃	1	1	yes	characteristic
201	#492	C₂²×C₈₃	1	2	yes	—
202	#492	C₂²×C₈₃	1	2	yes	—
203	#487	C₂²×C₈₃	1	2	yes	—
204	#487	C₂²×C₈₃	1	2	yes	—
205	#486	C₂²×C₈₃	1	2	yes	—
206	#486	C₂²×C₈₃	1	2	yes	—
207	#486	C₂²×C₈₃	1	2	yes	—
208	#482	—	2	4	no	—
209	#478	—	2	4	no	—
210	#468	—	2	4	no	—
211	#467	—	2	2	no	—
212	#440	—	2	4	no	—
213	C₄×Q₁₆	—	2	2	no	—
214	#431	C₂²×C₈₃	1	1	yes	characteristic
215	C₄×M₁₆	C₂²×C₈₃	1	1	yes	characteristic
216	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
217	#322	C₂²×C₈₃	1	1	yes	characteristic
218	#322	C₂²×C₈₃	1	1	yes	characteristic
219	C₈²	C₂²×C₈₃	1	1	yes	characteristic
220	C₈²	C₂²×C₈₃	1	1	yes	characteristic
221	C₂×#41	—	2	2	no	—
222	#135	—	4	8	no	—
223	C₂×Q₁₆	—	2	2	no	—
224	C₂×Q₁₆	—	2	2	no	—
225	C₂×SD₁₆	—	2	4	no	—
226	C₂×SD₁₆	—	2	4	no	—
227	C₂×D₈	—	2	2	no	—
228	C₂×M₁₆	D₄×C₈₃	1	2	yes	—
229	C₂²×C₈	D₄×C₈₃	1	2	yes	—
230	C₄⋊Q₈	C₂³×C₈₃	1	2	yes	—
231	C₄⋊Q₈	C₂³×C₈₃	1	2	yes	—
232	C₄⋊Q₈	C₂³×C₈₃	1	2	yes	—
233	#127	C₂³×C₈₃	1	2	yes	—
234	#125	—	2	2	no	—
235	#124	—	2	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₂²⋊Q₈	—	2	2	no	—
238	#121	—	2	2	no	—
239	#121	—	2	2	no	—
240	C₄×Q₈	—	2	2	no	—
241	C₄×Q₈	—	2	2	no	—
242	C₄×D₄	—	2	2	no	—
243	C₄×D₄	—	2	2	no	—
244	#117	—	2	2	no	—
245	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
246	#107	—	2	2	no	—
247	#107	—	2	4	no	—
248	#107	—	2	4	no	—
249	#106	—	2	4	no	—
250	C₄⋊C₈	D₄×C₈₃	1	2	yes	—
251	#103	—	2	4	no	—
252	#103	—	2	4	no	—
253	#102	—	2	4	no	—
254	#102	—	2	4	no	—
255	#102	—	2	4	no	—
256	C₂²⋊C₈	D₄×C₈₃	1	2	yes	—
257	#97	D₄×C₈₃	1	2	yes	—
258	C₄×C₈	D₄×C₈₃	1	2	yes	—
259	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
260	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
261	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
262	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
263	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
264	C₄×C₈	C₂³×C₈₃	1	1	yes	characteristic
265	#41	—	4	8	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×Q₈	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	2	2	no	—
271	C₂×D₄	—	2	2	no	—
272	C₂×D₄	—	4	2	no	—
273	C₂²×C₄	—	4	2	no	—
274	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
275	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
276	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
277	Q₁₆	—	4	4	no	—
278	Q₁₆	—	4	4	no	—
279	SD₁₆	—	4	8	no	—
280	SD₁₆	—	4	8	no	—
281	D₈	—	4	4	no	—
282	M₁₆	—	2	4	no	—
283	C₂×C₈	—	2	4	no	—
284	C₂×C₈	C₂×D₄×C₈₃	1	2	yes	—
285	C₂×C₈	C₂×D₄×C₈₃	1	2	yes	—
286	C₂×C₈	C₂×D₄×C₈₃	1	2	yes	—
287	C₂×C₈	C₂×D₄×C₈₃	1	2	yes	—
288	C₂×C₈	C₂×D₄×C₈₃	1	2	yes	—
289	C₂×C₈	C₂×D₄×C₈₃	1	2	yes	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	2	no	—
292	C₄⋊C₄	—	2	2	no	—
293	C₄⋊C₄	—	2	2	no	—
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	4	2	no	—
297	C₄⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	—	4	2	no	—
299	C₂²⋊C₄	—	4	2	no	—
300	C₂²⋊C₄	—	4	2	no	—
301	C₄²	—	4	2	no	—
302	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
303	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
304	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
305	C₄²	C₂⁴×C₈₃	1	1	yes	characteristic
306	C₂³	—	4	2	no	—
307	C₂³	#127×C₈₃	1	1	yes	characteristic
308	Q₈	—	4	4	no	—
309	Q₈	—	4	4	no	—
310	Q₈	—	4	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	4	4	no	—
314	D₄	—	8	4	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	8	4	no	—
322	C₂×C₄	#127×C₈₃	1	1	yes	characteristic
323	C₂×C₄	#127×C₈₃	1	1	yes	characteristic
324	C₂×C₄	#127×C₈₃	1	1	yes	characteristic
325	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
326	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
327	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
328	C₈	—	2	2	no	—
329	C₈	—	2	2	no	—
330	C₈	—	2	2	no	—
331	C₈	—	2	2	no	—
332	C₈	#135×C₈₃	1	4	yes	complemented
333	C₈	AGL(1,ℤ/8)×C₈₃	1	4	yes	—
334	C₂²	—	2	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	1	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	C₂×#127×C₈₃	1	1	yes	characteristic
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	2	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	2	no	—
347	C₄	C₂×#135×C₈₃	1	2	yes	—
348	C₄	C₂×AGL(1,ℤ/8)×C₈₃	1	2	yes	—
349	C₄	#578×C₈₃	1	2	yes	—
350	C₂	—	4	1	no	—
351	C₂	—	16	2	no	—
352	C₂	C₈₃×#3147	1	1	yes	characteristic
353	C₂	C₈₃×#3151	1	1	yes	characteristic
354	C₂	C₈₃×#3156	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₃	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₃
prime radical	C₈₃×#3147	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₃×#3228	C₂	semidirect
2. C₈₃×#3182	C₂	semidirect
3. C₈₃×#3170	C₂	semidirect
5. C₈₃×#1719	C₂	semidirect
6. C₈₃×#1714	C₂	semidirect
7. C₈₃×#1713	C₂	semidirect
8. C₈₃×#1709	C₂	semidirect
9. C₈₃×#1707	C₂	semidirect
10. C₈₃×#1706	C₂	semidirect
11. C₈×M₁₆×C₈₃	C₂	semidirect
154. C₈×C₈₃	#135	semidirect
172. #19645	C₈₃	direct (N × H)
178. #3228	C₂×C₈₃	semidirect
179. #3182	C₂×C₈₃	semidirect
180. #3170	C₂×C₈₃	semidirect
182. #1719	C₂×C₈₃	semidirect
183. #1714	C₂×C₈₃	semidirect
184. #1713	C₂×C₈₃	semidirect
185. #1709	C₂×C₈₃	semidirect
186. #1707	C₂×C₈₃	semidirect
187. #1706	C₂×C₈₃	semidirect
188. C₈×M₁₆	C₂×C₈₃	semidirect
189. C₈₃	#19645	direct (N × H)
332. C₈	#135×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₈₃×#3228	2
C₈₃×#3182	2
C₈₃×#3170	2
C₈₃×#3147	2
C₈₃×#1719	2
C₈₃×#1714	2
C₈₃×#1713	2
C₈₃×#1709	2
C₈₃×#1707	2
C₈₃×#1706	2
C₈×M₁₆×C₈₃	2
#19645	83

Subgroups — C83×#19645 #35247469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#19645 #35247469