Subgroups — C₈₃×#9822 #35237646

Order 21248.

578 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#486×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₈₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₃×#1957	C₂	1	1	yes	maximal, characteristic
4	C₈₃×#1957	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1956	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₃×#1834	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₈²×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
9	C₂×#127×C₈₃	C₂²	1	1	yes	characteristic
10	#486×C₈₃	—	2	2	no	—
11	#486×C₈₃	C₄	1	4	yes	complemented
12	C₂×#103×C₈₃	—	2	2	no	—
13	C₂×#102×C₈₃	—	2	2	no	—
14	C₂×#102×C₈₃	—	2	2	no	—
15	C₂×#102×C₈₃	—	2	2	no	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	#384×C₈₃	C₂²	1	1	yes	characteristic
20	#384×C₈₃	C₂²	1	1	yes	characteristic
21	#340×C₈₃	—	2	2	no	—
22	#340×C₈₃	—	2	2	no	—
23	C₈²×C₈₃	—	2	2	no	—
24	C₂²×Q₈×C₈₃	—	2	1	no	—
25	C₂²×D₄×C₈₃	—	2	1	no	—
26	C₂²×D₄×C₈₃	—	2	1	no	—
27	C₂²×D₄×C₈₃	—	2	1	no	—
28	C₂²×C₈×C₈₃	D₄	1	2	yes	—
29	C₂²×C₈×C₈₃	D₄	1	2	yes	—
30	C₂²×C₈×C₈₃	D₄	1	2	yes	—
31	C₄⋊Q₈×C₈₃	—	2	1	no	—
32	C₄⋊Q₈×C₈₃	—	2	1	no	—
33	C₄⋊Q₈×C₈₃	—	2	1	no	—
34	C₄⋊Q₈×C₈₃	C₂×C₄	1	2	yes	—
35	#127×C₈₃	—	2	1	no	—
36	#127×C₈₃	—	2	1	no	—
37	#127×C₈₃	—	2	1	no	—
38	#127×C₈₃	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
41	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
42	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
43	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
44	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
45	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
46	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
47	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
48	#103×C₈₃	—	4	4	no	—
49	#102×C₈₃	—	2	8	no	—
50	#102×C₈₃	—	4	4	no	—
51	#102×C₈₃	—	4	4	no	—
52	C₄×C₈×C₈₃	—	2	1	no	—
53	C₄×C₈×C₈₃	—	2	1	no	—
54	C₄×C₈×C₈₃	—	2	1	no	—
55	C₄×C₈×C₈₃	—	2	1	no	—
56	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₈₃	D₄	1	2	yes	—
58	C₂⁴×C₈₃	—	4	1	no	—
59	C₂×Q₈×C₈₃	—	2	2	no	—
60	C₂×Q₈×C₈₃	—	4	1	no	—
61	C₂×Q₈×C₈₃	—	4	2	no	—
62	C₂×Q₈×C₈₃	—	4	2	no	—
63	C₂×D₄×C₈₃	—	2	2	no	—
64	C₂×D₄×C₈₃	—	2	2	no	—
65	C₂×D₄×C₈₃	—	2	2	no	—
66	C₂×D₄×C₈₃	—	4	1	no	—
67	C₂×D₄×C₈₃	—	4	1	no	—
68	C₂×D₄×C₈₃	—	4	1	no	—
69	C₂×D₄×C₈₃	—	4	2	no	—
70	C₂×D₄×C₈₃	—	4	2	no	—
71	C₂×D₄×C₈₃	—	4	2	no	—
72	C₂×D₄×C₈₃	—	4	2	no	—
73	C₂×D₄×C₈₃	—	4	2	no	—
74	C₂×D₄×C₈₃	—	4	2	no	—
75	C₂²×C₄×C₈₃	—	4	1	no	—
76	C₂²×C₄×C₈₃	—	4	1	no	—
77	C₂²×C₄×C₈₃	—	4	1	no	—
78	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₃	—	2	2	no	—
82	C₂×C₈×C₈₃	—	2	2	no	—
83	C₂×C₈×C₈₃	—	2	2	no	—
84	C₂×C₈×C₈₃	—	2	2	no	—
85	C₂×C₈×C₈₃	—	2	2	no	—
86	C₂×C₈×C₈₃	—	2	2	no	—
87	C₂×C₈×C₈₃	C₂²⋊C₄	1	4	yes	—
88	C₂×C₈×C₈₃	D₈	1	4	yes	—
89	C₂×C₈×C₈₃	SD₁₆	1	4	yes	—
90	C₄⋊C₄×C₈₃	—	2	2	no	—
91	C₄⋊C₄×C₈₃	—	2	2	no	—
92	C₄⋊C₄×C₈₃	—	4	1	no	—
93	C₄⋊C₄×C₈₃	—	4	1	no	—
94	C₄⋊C₄×C₈₃	—	4	2	no	—
95	C₄⋊C₄×C₈₃	—	4	2	no	—
96	C₄⋊C₄×C₈₃	—	4	2	no	—
97	C₄⋊C₄×C₈₃	—	4	2	no	—
98	C₄⋊C₄×C₈₃	—	4	2	no	—
99	C₄⋊C₄×C₈₃	—	4	2	no	—
100	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₈₃	#41	1	1	yes	characteristic
103	C₄²×C₈₃	#41	1	1	yes	characteristic
104	C₂³×C₈₃	—	4	2	no	—
105	C₂³×C₈₃	—	8	1	no	—
106	C₂³×C₈₃	—	8	1	no	—
107	C₂³×C₈₃	—	8	1	no	—
108	C₂³×C₈₃	—	8	1	no	—
109	C₂³×C₈₃	—	8	1	no	—
110	C₂³×C₈₃	—	8	1	no	—
111	C₂³×C₈₃	#127	1	1	yes	characteristic
112	Q₈×C₈₃	—	4	4	no	—
113	Q₈×C₈₃	—	8	2	no	—
114	D₄×C₈₃	—	4	4	no	—
115	D₄×C₈₃	—	4	4	no	—
116	D₄×C₈₃	—	4	4	no	—
117	D₄×C₈₃	—	8	2	no	—
118	D₄×C₈₃	—	8	2	no	—
119	D₄×C₈₃	—	8	2	no	—
120	C₂×C₄×C₈₃	—	4	2	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	—	8	1	no	—
125	C₂×C₄×C₈₃	—	8	1	no	—
126	C₂×C₄×C₈₃	—	8	1	no	—
127	C₂×C₄×C₈₃	—	8	1	no	—
128	C₂×C₄×C₈₃	—	8	1	no	—
129	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
133	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
134	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
135	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₈₃	#135	1	2	yes	—
138	C₂×C₄×C₈₃	#135	1	2	yes	—
139	C₂×C₄×C₈₃	#135	1	2	yes	—
140	C₂×C₄×C₈₃	#135	1	2	yes	—
141	C₈×C₈₃	—	2	4	no	—
142	C₈×C₈₃	—	2	4	no	—
143	C₈×C₈₃	—	2	4	no	—
144	C₂²×C₈₃	—	8	2	no	—
145	C₂²×C₈₃	—	8	2	no	—
146	C₂²×C₈₃	—	8	2	no	—
147	C₂²×C₈₃	—	16	1	no	—
148	C₂²×C₈₃	—	16	1	no	—
149	C₂²×C₈₃	—	16	1	no	—
150	C₂²×C₈₃	—	16	1	no	—
151	C₂²×C₈₃	#390	1	1	yes	characteristic
152	C₂²×C₈₃	#492	1	1	yes	characteristic
153	C₂²×C₈₃	#493	1	1	yes	characteristic
154	C₂²×C₈₃	#495	1	1	yes	characteristic
155	C₂²×C₈₃	#495	1	1	yes	characteristic
156	C₂²×C₈₃	#495	1	1	yes	characteristic
157	C₂²×C₈₃	#495	1	1	yes	characteristic
158	C₄×C₈₃	—	8	2	no	—
159	C₄×C₈₃	—	16	1	no	—
160	C₄×C₈₃	—	16	1	no	—
161	C₄×C₈₃	#416	1	2	yes	—
162	C₄×C₈₃	C₄×D₈	1	2	yes	—
163	C₄×C₈₃	C₄×SD₁₆	1	2	yes	—
164	C₄×C₈₃	#465	1	2	yes	—
165	C₄×C₈₃	#466	1	2	yes	—
166	C₄×C₈₃	#488	1	2	yes	—
167	#9822	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
168	C₂×C₈₃	—	16	2	no	—
169	C₂×C₈₃	#1706	1	1	yes	characteristic
170	C₂×C₈₃	#1708	1	1	yes	characteristic
171	C₂×C₈₃	#1709	1	1	yes	characteristic
172	C₂×C₈₃	#1711	1	1	yes	characteristic
173	C₂×C₈₃	#1958	1	1	yes	characteristic
174	C₂×C₈₃	#1968	1	1	yes	characteristic
175	C₂×C₈₃	#1970	1	1	yes	characteristic
176	C₂×#486	C₂×C₈₃	1	1	yes	characteristic
177	#1990	C₂×C₈₃	1	1	yes	characteristic, complemented
178	#1957	C₂×C₈₃	1	1	yes	characteristic
179	#1957	C₂×C₈₃	1	1	yes	characteristic
180	#1956	C₂×C₈₃	1	1	yes	characteristic, complemented
181	#1834	C₂×C₈₃	1	1	yes	characteristic, complemented
182	C₂×C₈²	C₂×C₈₃	1	1	yes	characteristic, complemented
183	C₈₃	#9822	1	1	yes	characteristic, direct product factor, Sylow
184	C₂×C₄⋊Q₈	C₂²×C₈₃	1	1	yes	characteristic
185	C₂×#127	C₂²×C₈₃	1	1	yes	characteristic
186	#486	—	2	2	no	—
187	#486	C₄×C₈₃	1	4	yes	complemented
188	C₂×#103	—	2	2	no	—
189	C₂×#102	—	2	2	no	—
190	C₂×#102	—	2	2	no	—
191	C₂×#102	—	2	2	no	—
192	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
194	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
195	#384	C₂²×C₈₃	1	1	yes	characteristic
196	#384	C₂²×C₈₃	1	1	yes	characteristic
197	#340	—	2	2	no	—
198	#340	—	2	2	no	—
199	C₈²	—	2	2	no	—
200	C₂²×Q₈	—	2	1	no	—
201	C₂²×D₄	—	2	1	no	—
202	C₂²×D₄	—	2	1	no	—
203	C₂²×D₄	—	2	1	no	—
204	C₂²×C₈	D₄×C₈₃	1	2	yes	—
205	C₂²×C₈	D₄×C₈₃	1	2	yes	—
206	C₂²×C₈	D₄×C₈₃	1	2	yes	—
207	C₄⋊Q₈	—	2	1	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	C₂×C₄×C₈₃	1	2	yes	—
211	#127	—	2	1	no	—
212	#127	—	2	1	no	—
213	#127	—	2	1	no	—
214	#127	C₂×C₄×C₈₃	1	2	yes	—
215	C₂×C₄⋊C₄	—	2	1	no	—
216	C₂×C₄⋊C₄	—	2	1	no	—
217	C₂×C₄⋊C₄	—	2	1	no	—
218	C₂×C₄⋊C₄	—	2	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	2	1	no	—
222	C₂×C₄⋊C₄	—	2	1	no	—
223	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
224	#103	—	4	4	no	—
225	#102	—	2	8	no	—
226	#102	—	4	4	no	—
227	#102	—	4	4	no	—
228	C₄×C₈	—	2	1	no	—
229	C₄×C₈	—	2	1	no	—
230	C₄×C₈	—	2	1	no	—
231	C₄×C₈	—	2	1	no	—
232	C₄×C₈	C₂×C₄×C₈₃	1	2	yes	—
233	C₄×C₈	D₄×C₈₃	1	2	yes	—
234	C₂⁴	—	4	1	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×Q₈	—	4	2	no	—
239	C₂×D₄	—	2	2	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂×D₄	—	4	2	no	—
247	C₂×D₄	—	4	2	no	—
248	C₂×D₄	—	4	2	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	—	4	2	no	—
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	—	4	1	no	—
253	C₂²×C₄	—	4	1	no	—
254	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
255	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
256	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	2	2	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	C₂²⋊C₄×C₈₃	1	4	yes	—
264	C₂×C₈	D₈×C₈₃	1	4	yes	—
265	C₂×C₈	SD₁₆×C₈₃	1	4	yes	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	2	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄²	C₂²×C₄×C₈₃	1	1	yes	characteristic
277	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
278	C₄²	#41×C₈₃	1	1	yes	characteristic
279	C₄²	#41×C₈₃	1	1	yes	characteristic
280	C₂³	—	4	2	no	—
281	C₂³	—	8	1	no	—
282	C₂³	—	8	1	no	—
283	C₂³	—	8	1	no	—
284	C₂³	—	8	1	no	—
285	C₂³	—	8	1	no	—
286	C₂³	—	8	1	no	—
287	C₂³	#127×C₈₃	1	1	yes	characteristic
288	Q₈	—	4	4	no	—
289	Q₈	—	8	2	no	—
290	D₄	—	4	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	8	2	no	—
294	D₄	—	8	2	no	—
295	D₄	—	8	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	1	no	—
298	C₂×C₄	—	8	1	no	—
299	C₂×C₄	—	8	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	8	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	8	1	no	—
305	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
306	C₂×C₄	C₄×D₄×C₈₃	1	1	yes	characteristic
307	C₂×C₄	C₄×D₄×C₈₃	1	1	yes	characteristic
308	C₂×C₄	#121×C₈₃	1	1	yes	characteristic
309	C₂×C₄	#121×C₈₃	1	1	yes	characteristic
310	C₂×C₄	#124×C₈₃	1	1	yes	characteristic
311	C₂×C₄	C₂×D₈×C₈₃	1	2	yes	—
312	C₂×C₄	C₂×SD₁₆×C₈₃	1	2	yes	—
313	C₂×C₄	#135×C₈₃	1	2	yes	—
314	C₂×C₄	#135×C₈₃	1	2	yes	—
315	C₂×C₄	#135×C₈₃	1	2	yes	—
316	C₂×C₄	#135×C₈₃	1	2	yes	—
317	C₈	—	2	4	no	—
318	C₈	—	2	4	no	—
319	C₈	—	2	4	no	—
320	C₂²	—	8	2	no	—
321	C₂²	—	8	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	16	1	no	—
324	C₂²	—	16	1	no	—
325	C₂²	—	16	1	no	—
326	C₂²	—	16	1	no	—
327	C₂²	#390×C₈₃	1	1	yes	characteristic
328	C₂²	#492×C₈₃	1	1	yes	characteristic
329	C₂²	#493×C₈₃	1	1	yes	characteristic
330	C₂²	#495×C₈₃	1	1	yes	characteristic
331	C₂²	#495×C₈₃	1	1	yes	characteristic
332	C₂²	#495×C₈₃	1	1	yes	characteristic
333	C₂²	#495×C₈₃	1	1	yes	characteristic
334	C₄	—	8	2	no	—
335	C₄	—	16	1	no	—
336	C₄	—	16	1	no	—
337	C₄	#416×C₈₃	1	2	yes	—
338	C₄	C₄×D₈×C₈₃	1	2	yes	—
339	C₄	C₄×SD₁₆×C₈₃	1	2	yes	—
340	C₄	#465×C₈₃	1	2	yes	—
341	C₄	#466×C₈₃	1	2	yes	—
342	C₄	#488×C₈₃	1	2	yes	—
343	C₂	—	16	2	no	—
344	C₂	C₈₃×#1706	1	1	yes	characteristic
345	C₂	C₈₃×#1708	1	1	yes	characteristic
346	C₂	C₈₃×#1709	1	1	yes	characteristic
347	C₂	C₈₃×#1711	1	1	yes	characteristic
348	C₂	C₈₃×#1958	1	1	yes	characteristic
349	C₂	C₈₃×#1968	1	1	yes	characteristic
350	C₂	C₈₃×#1970	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₃	#127
commutator	C₄²	C₂²×C₄×C₈₃
Frattini, cosocle	C₂×C₄²	C₂³×C₈₃
prime radical	C₂×#127×C₈₃	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₃×#1990	C₂	semidirect
5. C₈₃×#1956	C₂	semidirect
6. C₈₃×#1834	C₂	semidirect
7. C₂×C₈²×C₈₃	C₂	semidirect
11. #486×C₈₃	C₄	semidirect
167. #9822	C₈₃	direct (N × H)
177. #1990	C₂×C₈₃	semidirect
180. #1956	C₂×C₈₃	semidirect
181. #1834	C₂×C₈₃	semidirect
182. C₂×C₈²	C₂×C₈₃	semidirect
183. C₈₃	#9822	direct (N × H)
187. #486	C₄×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#486×C₈₃	2
C₈₃×#1990	2
C₈₃×#1957	2
C₈₃×#1956	2
C₈₃×#1834	2
C₂×C₈²×C₈₃	2
#9822	83

Subgroups — C83×#9822 #35237646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#9822 #35237646