Subgroups — C₈₃×#11470 #35239294

Order 21248.

392 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#2929	C₂	1	1	yes	maximal, characteristic
2	C₈₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₂²⋊C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₃×#1323	C₂	1	2	yes	maximal, complemented
5	C₈₃×#1315	C₂	1	2	yes	maximal, complemented
6	C₂×#131×C₈₃	C₄	1	1	yes	characteristic
7	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
8	#525×C₈₃	C₄	1	1	yes	characteristic
9	C₂×M₃₂×C₈₃	—	2	1	no	—
10	C₂²×C₁₆×C₈₃	—	2	1	no	—
11	#435×C₈₃	—	2	1	no	—
12	#435×C₈₃	—	2	1	no	—
13	C₈×D₄×C₈₃	—	2	1	no	—
14	C₈×D₄×C₈₃	—	2	1	no	—
15	#424×C₈₃	C₂²	1	1	yes	characteristic
16	#408×C₈₃	C₂²	1	1	yes	characteristic
17	#407×C₈₃	C₄	1	1	yes	characteristic
18	#405×C₈₃	C₄	1	1	yes	characteristic
19	C₂²⋊C₁₆×C₈₃	—	2	1	no	—
20	C₂²⋊C₁₆×C₈₃	—	2	1	no	—
21	C₂²⋊C₁₆×C₈₃	C₂²	1	2	yes	—
22	C₂²⋊C₁₆×C₈₃	C₂²	1	2	yes	—
23	C₂×#41×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂³×C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
25	#131×C₈₃	—	4	2	no	—
26	C₂×M₁₆×C₈₃	—	2	1	no	—
27	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₈×C₈₃	—	2	1	no	—
30	C₂²×C₈×C₈₃	—	4	2	no	—
31	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
37	#123×C₈₃	—	2	1	no	—
38	C₂²⋊Q₈×C₈₃	C₈	1	1	yes	characteristic
39	C₂²⋊Q₈×C₈₃	C₈	1	1	yes	characteristic
40	#121×C₈₃	C₈	1	1	yes	characteristic
41	#121×C₈₃	C₈	1	1	yes	characteristic
42	#120×C₈₃	—	2	1	no	—
43	C₄×D₄×C₈₃	—	2	1	no	—
44	C₄×D₄×C₈₃	—	2	1	no	—
45	#117×C₈₃	C₂×C₄	1	1	yes	characteristic
46	M₃₂×C₈₃	—	4	2	no	—
47	C₂×C₁₆×C₈₃	—	2	2	no	—
48	C₂×C₁₆×C₈₃	—	2	2	no	—
49	C₂×C₁₆×C₈₃	—	4	2	no	—
50	C₄⋊C₈×C₈₃	—	2	1	no	—
51	C₄⋊C₈×C₈₃	—	2	1	no	—
52	C₂²⋊C₈×C₈₃	—	2	1	no	—
53	C₂²⋊C₈×C₈₃	—	2	1	no	—
54	C₂²⋊C₈×C₈₃	—	2	1	no	—
55	C₂²⋊C₈×C₈₃	—	2	1	no	—
56	#97×C₈₃	—	2	1	no	—
57	C₄×C₈×C₈₃	—	2	1	no	—
58	C₂⁴×C₈₃	M₁₆	1	1	yes	characteristic
59	#41×C₈₃	—	4	1	no	—
60	#41×C₈₃	—	4	1	no	—
61	C₂×Q₈×C₈₃	C₂×C₈	1	1	yes	characteristic
62	C₂×D₄×C₈₃	—	2	1	no	—
63	C₂×D₄×C₈₃	—	2	1	no	—
64	C₂×D₄×C₈₃	—	2	1	no	—
65	C₂×D₄×C₈₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	—	2	1	no	—
67	C₂²×C₄×C₈₃	—	4	1	no	—
68	C₂²×C₄×C₈₃	—	4	1	no	—
69	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
74	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
75	M₁₆×C₈₃	—	4	2	no	—
76	M₁₆×C₈₃	—	4	1	no	—
77	M₁₆×C₈₃	—	4	1	no	—
78	C₂×C₈×C₈₃	—	2	2	no	—
79	C₂×C₈×C₈₃	—	2	2	no	—
80	C₂×C₈×C₈₃	—	2	2	no	—
81	C₂×C₈×C₈₃	—	2	1	no	—
82	C₂×C₈×C₈₃	—	2	1	no	—
83	C₂×C₈×C₈₃	—	4	1	no	—
84	C₂×C₈×C₈₃	—	4	1	no	—
85	C₂×C₈×C₈₃	—	4	1	no	—
86	C₂×C₈×C₈₃	—	4	2	no	—
87	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
88	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₄⋊C₄×C₈₃	—	2	1	no	—
92	C₄⋊C₄×C₈₃	—	2	1	no	—
93	C₄⋊C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
94	C₄⋊C₄×C₈₃	C₂×C₈	1	1	yes	characteristic
95	C₂²⋊C₄×C₈₃	—	2	1	no	—
96	C₂²⋊C₄×C₈₃	—	2	1	no	—
97	C₂²⋊C₄×C₈₃	—	2	1	no	—
98	C₂²⋊C₄×C₈₃	—	2	1	no	—
99	C₂²⋊C₄×C₈₃	—	2	1	no	—
100	C₄²×C₈₃	—	2	1	no	—
101	C₁₆×C₈₃	—	4	2	no	—
102	C₁₆×C₈₃	—	4	2	no	—
103	C₂³×C₈₃	—	2	2	no	—
104	C₂³×C₈₃	—	2	1	no	—
105	C₂³×C₈₃	—	4	1	no	—
106	C₂³×C₈₃	—	4	1	no	—
107	C₂³×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
108	C₂³×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
109	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
110	Q₈×C₈₃	—	4	1	no	—
111	D₄×C₈₃	—	4	1	no	—
112	D₄×C₈₃	—	4	2	no	—
113	D₄×C₈₃	—	4	2	no	—
114	D₄×C₈₃	—	4	2	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	2	1	no	—
124	C₂×C₄×C₈₃	—	4	1	no	—
125	C₂×C₄×C₈₃	—	4	1	no	—
126	C₂×C₄×C₈₃	—	4	1	no	—
127	C₂×C₄×C₈₃	—	4	1	no	—
128	C₂×C₄×C₈₃	—	4	1	no	—
129	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
130	C₂×C₄×C₈₃	C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
133	C₈×C₈₃	—	2	1	no	—
134	C₈×C₈₃	—	2	1	no	—
135	C₈×C₈₃	—	2	1	no	—
136	C₈×C₈₃	—	4	1	no	—
137	C₈×C₈₃	—	4	1	no	—
138	C₈×C₈₃	#99	1	2	yes	—
139	C₂²×C₈₃	—	2	1	no	—
140	C₂²×C₈₃	—	2	1	no	—
141	C₂²×C₈₃	—	2	1	no	—
142	C₂²×C₈₃	—	2	1	no	—
143	C₂²×C₈₃	—	2	1	no	—
144	C₂²×C₈₃	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	4	4	no	—
148	C₂²×C₈₃	—	4	1	no	—
149	C₂²×C₈₃	—	4	1	no	—
150	C₂²×C₈₃	—	4	1	no	—
151	C₂²×C₈₃	#350	1	2	yes	—
152	C₂²×C₈₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
153	C₄×C₈₃	—	2	1	no	—
154	C₄×C₈₃	—	2	1	no	—
155	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	C₂×#99	1	1	yes	characteristic
160	C₄×C₈₃	#413	1	1	yes	characteristic
161	#11470	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₈₃	—	2	2	no	—
163	C₂×C₈₃	—	4	1	no	—
164	C₂×C₈₃	—	4	1	no	—
165	C₂×C₈₃	—	8	1	no	—
166	C₂×C₈₃	#1459	1	1	yes	characteristic
167	C₂×C₈₃	C₂×#350	1	1	yes	characteristic
168	C₂×C₈₃	#2117	1	1	yes	characteristic
169	#2929	C₂×C₈₃	1	1	yes	characteristic
170	#2112	C₂×C₈₃	1	1	yes	characteristic, complemented
171	C₂×C₂²⋊C₁₆	C₂×C₈₃	1	1	yes	characteristic, complemented
172	#1323	C₂×C₈₃	1	2	yes	complemented
173	#1315	C₂×C₈₃	1	2	yes	complemented
174	C₈₃	#11470	1	1	yes	characteristic, direct product factor, Sylow
175	C₂×#131	C₄×C₈₃	1	1	yes	characteristic
176	C₂³×C₈	C₂²×C₈₃	1	1	yes	characteristic
177	#525	C₄×C₈₃	1	1	yes	characteristic
178	C₂×M₃₂	—	2	1	no	—
179	C₂²×C₁₆	—	2	1	no	—
180	#435	—	2	1	no	—
181	#435	—	2	1	no	—
182	C₈×D₄	—	2	1	no	—
183	C₈×D₄	—	2	1	no	—
184	#424	C₂²×C₈₃	1	1	yes	characteristic
185	#408	C₂²×C₈₃	1	1	yes	characteristic
186	#407	C₄×C₈₃	1	1	yes	characteristic
187	#405	C₄×C₈₃	1	1	yes	characteristic
188	C₂²⋊C₁₆	—	2	1	no	—
189	C₂²⋊C₁₆	—	2	1	no	—
190	C₂²⋊C₁₆	C₂²×C₈₃	1	2	yes	—
191	C₂²⋊C₁₆	C₂²×C₈₃	1	2	yes	—
192	C₂×#41	C₂×C₄×C₈₃	1	1	yes	characteristic
193	C₂³×C₄	C₂×C₄×C₈₃	1	1	yes	characteristic
194	#131	—	4	2	no	—
195	C₂×M₁₆	—	2	1	no	—
196	C₂×M₁₆	C₂×C₄×C₈₃	1	1	yes	characteristic
197	C₂×M₁₆	C₂×C₄×C₈₃	1	1	yes	characteristic
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	4	2	no	—
200	C₂²×C₈	C₂×C₄×C₈₃	1	1	yes	characteristic
201	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
202	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
203	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
204	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
205	C₂²×C₈	C₂³×C₈₃	1	1	yes	characteristic
206	#123	—	2	1	no	—
207	C₂²⋊Q₈	C₈×C₈₃	1	1	yes	characteristic
208	C₂²⋊Q₈	C₈×C₈₃	1	1	yes	characteristic
209	#121	C₈×C₈₃	1	1	yes	characteristic
210	#121	C₈×C₈₃	1	1	yes	characteristic
211	#120	—	2	1	no	—
212	C₄×D₄	—	2	1	no	—
213	C₄×D₄	—	2	1	no	—
214	#117	C₂×C₄×C₈₃	1	1	yes	characteristic
215	M₃₂	—	4	2	no	—
216	C₂×C₁₆	—	2	2	no	—
217	C₂×C₁₆	—	2	2	no	—
218	C₂×C₁₆	—	4	2	no	—
219	C₄⋊C₈	—	2	1	no	—
220	C₄⋊C₈	—	2	1	no	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	2	1	no	—
223	C₂²⋊C₈	—	2	1	no	—
224	C₂²⋊C₈	—	2	1	no	—
225	#97	—	2	1	no	—
226	C₄×C₈	—	2	1	no	—
227	C₂⁴	M₁₆×C₈₃	1	1	yes	characteristic
228	#41	—	4	1	no	—
229	#41	—	4	1	no	—
230	C₂×Q₈	C₂×C₈×C₈₃	1	1	yes	characteristic
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	C₂×C₈×C₈₃	1	1	yes	characteristic
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	—	4	1	no	—
238	C₂²×C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₈×C₈₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₈×C₈₃	1	1	yes	characteristic
242	C₂²×C₄	M₁₆×C₈₃	1	1	yes	characteristic
243	C₂²×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
244	M₁₆	—	4	2	no	—
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	2	no	—
256	C₂×C₈	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
257	C₂×C₈	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₈₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₈₃	1	1	yes	characteristic
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	C₂×C₈×C₈₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂×C₈×C₈₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₄²	—	2	1	no	—
270	C₁₆	—	4	2	no	—
271	C₁₆	—	4	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	—	4	1	no	—
276	C₂³	C₂²⋊C₈×C₈₃	1	1	yes	characteristic
277	C₂³	C₂²⋊C₈×C₈₃	1	1	yes	characteristic
278	C₂³	C₂×M₁₆×C₈₃	1	1	yes	characteristic
279	Q₈	—	4	1	no	—
280	D₄	—	4	1	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	4	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	C₂²⋊C₈×C₈₃	1	1	yes	characteristic
299	C₂×C₄	C₂²⋊C₈×C₈₃	1	1	yes	characteristic
300	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
301	C₂×C₄	C₂²×C₈×C₈₃	1	1	yes	characteristic
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	#99×C₈₃	1	2	yes	—
308	C₂²	—	2	1	no	—
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	4	4	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	#350×C₈₃	1	2	yes	—
321	C₂²	C₂×C₂²⋊C₈×C₈₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	C₂×#99×C₈₃	1	1	yes	characteristic
329	C₄	#413×C₈₃	1	1	yes	characteristic
330	C₂	—	2	2	no	—
331	C₂	—	4	1	no	—
332	C₂	—	4	1	no	—
333	C₂	—	8	1	no	—
334	C₂	C₈₃×#1459	1	1	yes	characteristic
335	C₂	C₂×#350×C₈₃	1	1	yes	characteristic
336	C₂	C₈₃×#2117	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈₃	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₈×C₈₃
Frattini, cosocle	C₂²×C₈	C₂³×C₈₃
socle	C₂²×C₈₃	C₂×C₂²⋊C₈
prime radical	#525×C₈₃	C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₃ ◁ C₂³×C₈×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₃×#2112	C₂	semidirect
3. C₂×C₂²⋊C₁₆×C₈₃	C₂	semidirect
4. C₈₃×#1323	C₂	semidirect
5. C₈₃×#1315	C₂	semidirect
161. #11470	C₈₃	direct (N × H)
170. #2112	C₂×C₈₃	semidirect
171. C₂×C₂²⋊C₁₆	C₂×C₈₃	semidirect
172. #1323	C₂×C₈₃	semidirect
173. #1315	C₂×C₈₃	semidirect
174. C₈₃	#11470	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₃×#2929	2
C₈₃×#2112	2
C₂×C₂²⋊C₁₆×C₈₃	2
C₈₃×#1323	2
C₈₃×#1315	2
#11470	83

Subgroups — C83×#11470 #35239294

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#11470 #35239294