Subgroups — C₈₃×#20368 #35248192

Order 21248.

1042 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#405×C₈₃	C₂	1	2	yes	maximal
2	C₄×#117×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#1746	C₂	1	4	yes	maximal, complemented
4	C₈₃×#1745	C₂	1	2	yes	maximal, complemented
5	C₈₃×#1744	C₂	1	2	yes	maximal, complemented
6	C₈₃×#1729	C₂	1	2	yes	maximal
7	C₈₃×#1728	C₂	1	1	yes	maximal, characteristic
8	C₂×#336×C₈₃	C₂	1	1	yes	maximal, characteristic
9	C₂²×M₁₆×C₈₃	C₂²	1	2	yes	—
10	C₂³×C₈×C₈₃	C₂²	1	2	yes	—
11	C₂×#117×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×#117×C₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
14	#405×C₈₃	C₄	1	16	yes	complemented
15	C₂×#97×C₈₃	C₂²	1	4	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	4	yes	—
17	C₄×C₄⋊C₄×C₈₃	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₈₃	C₄	1	2	yes	—
19	C₄×C₂²⋊C₄×C₈₃	C₄	1	4	yes	—
20	#376×C₈₃	C₂²	1	2	yes	—
21	C₄³×C₈₃	C₂²	1	2	yes	—
22	#336×C₈₃	C₂²	1	1	yes	characteristic
23	#336×C₈₃	C₂²	1	2	yes	—
24	#336×C₈₃	C₂²	1	4	yes	—
25	#336×C₈₃	C₂²	1	1	yes	characteristic
26	#336×C₈₃	C₂²	1	4	yes	—
27	#336×C₈₃	C₂²	1	2	yes	—
28	#336×C₈₃	C₂²	1	2	yes	—
29	C₂³×C₄×C₈₃	C₂×C₄	1	2	yes	—
30	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×M₁₆×C₈₃	—	2	4	no	—
32	C₂×M₁₆×C₈₃	—	2	4	no	—
33	C₂×M₁₆×C₈₃	C₂×C₄	1	8	yes	—
34	C₂²×C₈×C₈₃	—	2	4	no	—
35	C₂²×C₈×C₈₃	—	2	4	no	—
36	C₂²×C₈×C₈₃	C₂×C₄	1	8	yes	—
37	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
39	C₂²×C₈×C₈₃	C₂³	1	4	yes	—
40	#117×C₈₃	—	2	2	no	—
41	#117×C₈₃	—	2	2	no	—
42	#117×C₈₃	C₂×C₄	1	8	yes	—
43	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
44	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
47	C₂×C₂²⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
48	C₂×C₄²×C₈₃	—	2	4	no	—
49	C₂×C₄²×C₈₃	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₈₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₈₃	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₃	C₂³	1	2	yes	—
55	C₂×C₄²×C₈₃	C₂³	1	2	yes	—
56	#97×C₈₃	C₂×C₄	1	16	yes	complemented
57	C₄×C₈×C₈₃	C₂×C₄	1	16	yes	complemented
58	#95×C₈₃	C₂×C₄	1	4	yes	—
59	#95×C₈₃	C₂×C₄	1	4	yes	—
60	C₂⁴×C₈₃	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	—	2	4	no	—
62	C₂²×C₄×C₈₃	—	2	4	no	—
63	C₂²×C₄×C₈₃	—	4	2	no	—
64	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₈₃	C₂²×C₄	1	4	yes	—
66	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
68	C₂²×C₄×C₈₃	C₂²×C₄	1	2	yes	—
69	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₈₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₈₃	#41	1	4	yes	—
72	C₂²×C₄×C₈₃	#41	1	2	yes	—
73	C₂²×C₄×C₈₃	#41	1	2	yes	—
74	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
75	M₁₆×C₈₃	—	2	16	no	—
76	C₂×C₈×C₈₃	—	2	16	no	—
77	C₂×C₈×C₈₃	—	2	2	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	4	no	—
82	C₂×C₈×C₈₃	—	2	4	no	—
83	C₂×C₈×C₈₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₈₃	C₂²×C₄	1	8	yes	—
85	C₂×C₈×C₈₃	C₂²×C₄	1	4	yes	—
86	C₄⋊C₄×C₈₃	—	2	4	no	—
87	C₄⋊C₄×C₈₃	C₄²	1	4	yes	—
88	C₄⋊C₄×C₈₃	C₄²	1	4	yes	—
89	C₂²⋊C₄×C₈₃	—	2	4	no	—
90	C₂²⋊C₄×C₈₃	C₄²	1	8	yes	—
91	C₄²×C₈₃	—	2	4	no	—
92	C₄²×C₈₃	—	2	8	no	—
93	C₄²×C₈₃	—	2	2	no	—
94	C₄²×C₈₃	—	2	2	no	—
95	C₄²×C₈₃	—	2	2	no	—
96	C₄²×C₈₃	—	2	2	no	—
97	C₄²×C₈₃	—	4	4	no	—
98	C₄²×C₈₃	C₂²×C₄	1	8	yes	—
99	C₂³×C₈₃	—	2	2	no	—
100	C₂³×C₈₃	—	2	2	no	—
101	C₂³×C₈₃	C₂×C₄²	1	2	yes	—
102	C₂³×C₈₃	#117	1	2	yes	—
103	C₂³×C₈₃	#117	1	2	yes	—
104	C₂³×C₈₃	C₂³×C₄	1	1	yes	characteristic
105	C₂×C₄×C₈₃	—	2	2	no	—
106	C₂×C₄×C₈₃	—	2	4	no	—
107	C₂×C₄×C₈₃	—	2	2	no	—
108	C₂×C₄×C₈₃	—	2	4	no	—
109	C₂×C₄×C₈₃	—	2	2	no	—
110	C₂×C₄×C₈₃	—	2	4	no	—
111	C₂×C₄×C₈₃	—	2	4	no	—
112	C₂×C₄×C₈₃	—	2	2	no	—
113	C₂×C₄×C₈₃	—	2	4	no	—
114	C₂×C₄×C₈₃	—	2	2	no	—
115	C₂×C₄×C₈₃	—	4	8	no	—
116	C₂×C₄×C₈₃	C₂×C₄²	1	4	yes	—
117	C₂×C₄×C₈₃	C₂×C₄²	1	4	yes	—
118	C₂×C₄×C₈₃	C₂×C₄²	1	2	yes	—
119	C₂×C₄×C₈₃	#117	1	8	yes	—
120	C₂×C₄×C₈₃	#117	1	8	yes	—
121	C₂×C₄×C₈₃	#117	1	2	yes	—
122	C₂×C₄×C₈₃	#117	1	2	yes	—
123	C₂×C₄×C₈₃	#131	1	4	yes	—
124	C₂×C₄×C₈₃	C₂³×C₄	1	2	yes	—
125	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
126	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
128	C₈×C₈₃	—	2	4	no	—
129	C₈×C₈₃	—	2	4	no	—
130	C₈×C₈₃	—	2	8	no	—
131	C₂²×C₈₃	—	2	2	no	—
132	C₂²×C₈₃	—	2	2	no	—
133	C₂²×C₈₃	—	2	4	no	—
134	C₂²×C₈₃	—	2	4	no	—
135	C₂²×C₈₃	#376	1	4	yes	—
136	C₂²×C₈₃	C₂²×C₄²	1	1	yes	characteristic
137	C₂²×C₈₃	C₂×#117	1	1	yes	characteristic
138	C₂²×C₈₃	C₂×#117	1	1	yes	characteristic
139	C₂²×C₈₃	C₂×#131	1	2	yes	—
140	C₂²×C₈₃	#568	1	2	yes	—
141	C₄×C₈₃	—	2	4	no	—
142	C₄×C₈₃	—	2	4	no	—
143	C₄×C₈₃	—	2	4	no	—
144	C₄×C₈₃	—	4	2	no	—
145	C₄×C₈₃	—	4	4	no	—
146	C₄×C₈₃	#376	1	4	yes	—
147	C₄×C₈₃	C₂×#117	1	2	yes	—
148	C₄×C₈₃	C₂×#117	1	2	yes	—
149	#20368	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
150	C₂×C₈₃	—	2	4	no	—
151	C₂×C₈₃	C₂×#376	1	1	yes	characteristic
152	C₂×C₈₃	C₄×#131	1	1	yes	characteristic
153	C₂×C₈₃	#2875	1	1	yes	characteristic
154	C₂×C₈₃	#2924	1	2	yes	—
155	C₂×C₈₃	#2925	1	2	yes	—
156	C₂×#405	C₂×C₈₃	1	2	yes	—
157	C₄×#117	C₂×C₈₃	1	1	yes	characteristic
158	#1746	C₂×C₈₃	1	4	yes	complemented
159	#1745	C₂×C₈₃	1	2	yes	complemented
160	#1744	C₂×C₈₃	1	2	yes	complemented
161	#1729	C₂×C₈₃	1	2	yes	—
162	#1728	C₂×C₈₃	1	1	yes	characteristic
163	C₂×#336	C₂×C₈₃	1	1	yes	characteristic
164	C₈₃	#20368	1	1	yes	characteristic, direct product factor, Sylow
165	C₂²×M₁₆	C₂²×C₈₃	1	2	yes	—
166	C₂³×C₈	C₂²×C₈₃	1	2	yes	—
167	C₂×#117	C₂²×C₈₃	1	1	yes	characteristic
168	C₂×#117	C₂²×C₈₃	1	1	yes	characteristic
169	C₂²×C₄²	C₂²×C₈₃	1	1	yes	characteristic
170	#405	C₄×C₈₃	1	16	yes	complemented
171	C₂×#97	C₂²×C₈₃	1	4	yes	—
172	C₂×C₄×C₈	C₂²×C₈₃	1	4	yes	—
173	C₄×C₄⋊C₄	C₄×C₈₃	1	2	yes	—
174	C₄×C₄⋊C₄	C₄×C₈₃	1	2	yes	—
175	C₄×C₂²⋊C₄	C₄×C₈₃	1	4	yes	—
176	#376	C₂²×C₈₃	1	2	yes	—
177	C₄³	C₂²×C₈₃	1	2	yes	—
178	#336	C₂²×C₈₃	1	1	yes	characteristic
179	#336	C₂²×C₈₃	1	1	yes	characteristic
180	#336	C₂²×C₈₃	1	2	yes	—
181	#336	C₂²×C₈₃	1	2	yes	—
182	#336	C₂²×C₈₃	1	2	yes	—
183	#336	C₂²×C₈₃	1	4	yes	—
184	#336	C₂²×C₈₃	1	4	yes	—
185	C₂³×C₄	C₂×C₄×C₈₃	1	2	yes	—
186	C₂³×C₄	C₂³×C₈₃	1	1	yes	characteristic
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	—	2	4	no	—
189	C₂×M₁₆	C₂×C₄×C₈₃	1	8	yes	—
190	C₂²×C₈	—	2	4	no	—
191	C₂²×C₈	—	2	4	no	—
192	C₂²×C₈	C₂×C₄×C₈₃	1	8	yes	—
193	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
194	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
195	C₂²×C₈	C₂³×C₈₃	1	4	yes	—
196	#117	—	2	2	no	—
197	#117	—	2	2	no	—
198	#117	C₂×C₄×C₈₃	1	8	yes	—
199	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	2	yes	—
200	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	1	yes	characteristic
201	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	1	yes	characteristic
202	C₂×C₂²⋊C₄	C₂×C₄×C₈₃	1	2	yes	—
203	C₂×C₂²⋊C₄	C₂×C₄×C₈₃	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	C₂×C₄×C₈₃	1	4	yes	—
206	C₂×C₄²	C₂×C₄×C₈₃	1	2	yes	—
207	C₂×C₄²	C₂×C₄×C₈₃	1	4	yes	—
208	C₂×C₄²	C₂³×C₈₃	1	2	yes	—
209	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
210	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₈₃	1	2	yes	—
212	#97	C₂×C₄×C₈₃	1	16	yes	complemented
213	C₄×C₈	C₂×C₄×C₈₃	1	16	yes	complemented
214	#95	C₂×C₄×C₈₃	1	4	yes	—
215	#95	C₂×C₄×C₈₃	1	4	yes	—
216	C₂⁴	C₂²×C₄×C₈₃	1	1	yes	characteristic
217	C₂²×C₄	—	2	4	no	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	4	2	no	—
220	C₂²×C₄	C₂²×C₄×C₈₃	1	2	yes	—
221	C₂²×C₄	C₂²×C₄×C₈₃	1	4	yes	—
222	C₂²×C₄	C₂²×C₄×C₈₃	1	2	yes	—
223	C₂²×C₄	C₂²×C₄×C₈₃	1	2	yes	—
224	C₂²×C₄	C₂²×C₄×C₈₃	1	2	yes	—
225	C₂²×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²×C₄×C₈₃	1	4	yes	—
227	C₂²×C₄	#41×C₈₃	1	4	yes	—
228	C₂²×C₄	#41×C₈₃	1	2	yes	—
229	C₂²×C₄	#41×C₈₃	1	2	yes	—
230	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
231	M₁₆	—	2	16	no	—
232	C₂×C₈	—	2	16	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	C₂²×C₄×C₈₃	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₈₃	1	8	yes	—
241	C₂×C₈	C₂²×C₄×C₈₃	1	4	yes	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	C₄²×C₈₃	1	4	yes	—
244	C₄⋊C₄	C₄²×C₈₃	1	4	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	C₄²×C₈₃	1	8	yes	—
247	C₄²	—	2	4	no	—
248	C₄²	—	2	8	no	—
249	C₄²	—	2	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	2	no	—
253	C₄²	—	4	4	no	—
254	C₄²	C₂²×C₄×C₈₃	1	8	yes	—
255	C₂³	—	2	2	no	—
256	C₂³	—	2	2	no	—
257	C₂³	C₂×C₄²×C₈₃	1	2	yes	—
258	C₂³	#117×C₈₃	1	2	yes	—
259	C₂³	#117×C₈₃	1	2	yes	—
260	C₂³	C₂³×C₄×C₈₃	1	1	yes	characteristic
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	8	no	—
272	C₂×C₄	C₂×C₄²×C₈₃	1	4	yes	—
273	C₂×C₄	C₂×C₄²×C₈₃	1	4	yes	—
274	C₂×C₄	C₂×C₄²×C₈₃	1	2	yes	—
275	C₂×C₄	#117×C₈₃	1	8	yes	—
276	C₂×C₄	#117×C₈₃	1	8	yes	—
277	C₂×C₄	#117×C₈₃	1	2	yes	—
278	C₂×C₄	#117×C₈₃	1	2	yes	—
279	C₂×C₄	#131×C₈₃	1	4	yes	—
280	C₂×C₄	C₂³×C₄×C₈₃	1	2	yes	—
281	C₂×C₄	C₂×#41×C₈₃	1	2	yes	—
282	C₂×C₄	C₂×#41×C₈₃	1	1	yes	characteristic
283	C₂×C₄	C₂×#41×C₈₃	1	1	yes	characteristic
284	C₈	—	2	4	no	—
285	C₈	—	2	4	no	—
286	C₈	—	2	8	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	4	no	—
290	C₂²	—	2	4	no	—
291	C₂²	#376×C₈₃	1	4	yes	—
292	C₂²	C₂²×C₄²×C₈₃	1	1	yes	characteristic
293	C₂²	C₂×#117×C₈₃	1	1	yes	characteristic
294	C₂²	C₂×#117×C₈₃	1	1	yes	characteristic
295	C₂²	C₂×#131×C₈₃	1	2	yes	—
296	C₂²	#568×C₈₃	1	2	yes	—
297	C₄	—	2	4	no	—
298	C₄	—	2	4	no	—
299	C₄	—	2	4	no	—
300	C₄	—	4	2	no	—
301	C₄	—	4	4	no	—
302	C₄	#376×C₈₃	1	4	yes	—
303	C₄	C₂×#117×C₈₃	1	2	yes	—
304	C₄	C₂×#117×C₈₃	1	2	yes	—
305	C₂	—	2	4	no	—
306	C₂	C₂×#376×C₈₃	1	1	yes	characteristic
307	C₂	C₄×#131×C₈₃	1	1	yes	characteristic
308	C₂	C₈₃×#2875	1	1	yes	characteristic
309	C₂	C₈₃×#2924	1	2	yes	—
310	C₂	C₈₃×#2925	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₈₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₈₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₃
socle	C₂³×C₈₃	C₂³×C₄
prime radical	C₂⁴×C₈₃	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₃×#1746	C₂	semidirect
4. C₈₃×#1745	C₂	semidirect
5. C₈₃×#1744	C₂	semidirect
14. #405×C₈₃	C₄	semidirect
56. #97×C₈₃	C₂×C₄	semidirect
57. C₄×C₈×C₈₃	C₂×C₄	semidirect
149. #20368	C₈₃	direct (N × H)
158. #1746	C₂×C₈₃	semidirect
159. #1745	C₂×C₈₃	semidirect
160. #1744	C₂×C₈₃	semidirect
164. C₈₃	#20368	direct (N × H)
170. #405	C₄×C₈₃	semidirect
212. #97	C₂×C₄×C₈₃	semidirect
213. C₄×C₈	C₂×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×#405×C₈₃	2
C₄×#117×C₈₃	2
C₈₃×#1746	2
C₈₃×#1745	2
C₈₃×#1744	2
C₈₃×#1729	2
C₈₃×#1728	2
C₂×#336×C₈₃	2
#20368	83

Subgroups — C83×#20368 #35248192

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#20368 #35248192