Subgroups — C₄×#490×C₈₃ #35249474

Order 21248.

970 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#490×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₄×#127×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₄×#125×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₃×#1992	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1961	C₂	1	1	yes	maximal, characteristic
6	C₈₃×#1936	C₂	1	4	yes	maximal, complemented
7	C₈₃×#1836	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#102×C₈₃	C₂	1	4	yes	maximal, complemented
9	C₄×#97×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#127×C₈₃	C₂²	1	1	yes	characteristic
11	C₂×#125×C₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₄×D₄×C₈₃	—	2	2	no	—
13	C₂×C₄×D₄×C₈₃	C₂²	1	2	yes	—
14	#490×C₈₃	C₄	1	8	yes	direct product factor
15	C₂×#102×C₈₃	C₂²	1	4	yes	—
16	C₂×#102×C₈₃	C₂²	1	4	yes	—
17	C₂×#97×C₈₃	C₂²	1	2	yes	—
18	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₈₃	C₂²	1	2	yes	—
21	#390×C₈₃	—	2	2	no	—
22	#390×C₈₃	C₂²	1	2	yes	—
23	#389×C₈₃	C₂²	1	2	yes	—
24	#387×C₈₃	—	2	2	no	—
25	#384×C₈₃	C₂²	1	1	yes	characteristic
26	#382×C₈₃	C₂²	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₈₃	—	2	2	no	—
28	C₄×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
29	C₄³×C₈₃	C₂²	1	1	yes	characteristic
30	#340×C₈₃	C₂²	1	4	yes	—
31	#340×C₈₃	C₂²	1	4	yes	—
32	C₂²×D₄×C₈₃	—	2	2	no	—
33	C₂²×D₄×C₈₃	C₂³	1	2	yes	—
34	C₂³×C₄×C₈₃	—	2	2	no	—
35	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
36	C₂²×C₈×C₈₃	C₂³	1	2	yes	—
37	#127×C₈₃	—	2	2	no	—
38	#127×C₈₃	C₂×C₄	1	4	yes	—
39	#125×C₈₃	—	2	2	no	—
40	#125×C₈₃	C₂×C₄	1	4	yes	complemented
41	C₄×D₄×C₈₃	—	2	8	no	—
42	C₄×D₄×C₈₃	—	4	8	no	—
43	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
46	C₂×C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
47	C₂×C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
48	C₂×C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
49	C₂×C₂²⋊C₄×C₈₃	—	2	4	no	—
50	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
51	C₂×C₄²×C₈₃	—	2	2	no	—
52	C₂×C₄²×C₈₃	D₄	1	2	yes	—
53	C₂×C₄²×C₈₃	D₄	1	2	yes	—
54	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
57	#102×C₈₃	—	2	8	no	—
58	#102×C₈₃	C₂×C₄	1	16	yes	complemented
59	#97×C₈₃	—	2	4	no	—
60	#97×C₈₃	—	2	2	no	—
61	#97×C₈₃	C₂×C₄	1	4	yes	complemented
62	C₄×C₈×C₈₃	—	2	4	no	—
63	#95×C₈₃	—	2	2	no	—
64	#95×C₈₃	—	2	2	no	—
65	#95×C₈₃	—	2	2	no	—
66	C₂⁴×C₈₃	—	2	2	no	—
67	C₂×D₄×C₈₃	—	2	8	no	—
68	C₂×D₄×C₈₃	—	2	4	no	—
69	C₂×D₄×C₈₃	—	2	4	no	—
70	C₂×D₄×C₈₃	—	2	2	no	—
71	C₂×D₄×C₈₃	—	4	4	no	—
72	C₂×D₄×C₈₃	—	4	4	no	—
73	C₂×D₄×C₈₃	C₂²×C₄	1	4	yes	—
74	C₂²×C₄×C₈₃	—	2	2	no	—
75	C₂²×C₄×C₈₃	—	2	2	no	—
76	C₂²×C₄×C₈₃	—	2	2	no	—
77	C₂²×C₄×C₈₃	—	4	4	no	—
78	C₂²×C₄×C₈₃	—	4	8	no	—
79	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
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	—	2	2	no	—
88	C₂×C₈×C₈₃	C₂²×C₄	1	4	yes	—
89	C₂×C₈×C₈₃	#41	1	4	yes	—
90	C₄⋊C₄×C₈₃	—	2	8	no	—
91	C₄⋊C₄×C₈₃	—	2	2	no	—
92	C₄⋊C₄×C₈₃	—	2	4	no	—
93	C₄⋊C₄×C₈₃	—	2	4	no	—
94	C₄⋊C₄×C₈₃	—	4	4	no	—
95	C₄⋊C₄×C₈₃	—	4	4	no	—
96	C₄⋊C₄×C₈₃	C₂²×C₄	1	4	yes	—
97	C₂²⋊C₄×C₈₃	—	4	8	no	—
98	C₂²⋊C₄×C₈₃	—	4	4	no	—
99	C₄²×C₈₃	—	2	4	no	—
100	C₄²×C₈₃	—	2	2	no	—
101	C₄²×C₈₃	—	4	2	no	—
102	C₄²×C₈₃	—	4	4	no	—
103	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
104	C₄²×C₈₃	#41	1	2	yes	—
105	C₄²×C₈₃	#41	1	4	yes	—
106	C₂³×C₈₃	—	2	4	no	—
107	C₂³×C₈₃	—	4	2	no	—
108	C₂³×C₈₃	—	4	2	no	—
109	C₂³×C₈₃	—	4	4	no	—
110	C₂³×C₈₃	—	4	4	no	—
111	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
112	D₄×C₈₃	—	2	8	no	—
113	D₄×C₈₃	—	4	16	no	—
114	D₄×C₈₃	—	4	4	no	—
115	C₂×C₄×C₈₃	—	2	2	no	—
116	C₂×C₄×C₈₃	—	2	2	no	—
117	C₂×C₄×C₈₃	—	2	2	no	—
118	C₂×C₄×C₈₃	—	2	4	no	—
119	C₂×C₄×C₈₃	—	2	2	no	—
120	C₂×C₄×C₈₃	—	2	2	no	—
121	C₂×C₄×C₈₃	—	2	2	no	—
122	C₂×C₄×C₈₃	—	4	2	no	—
123	C₂×C₄×C₈₃	—	4	2	no	—
124	C₂×C₄×C₈₃	—	4	2	no	—
125	C₂×C₄×C₈₃	—	4	2	no	—
126	C₂×C₄×C₈₃	—	4	2	no	—
127	C₂×C₄×C₈₃	—	4	4	no	—
128	C₂×C₄×C₈₃	—	4	2	no	—
129	C₂×C₄×C₈₃	—	8	8	no	—
130	C₂×C₄×C₈₃	C₄×D₄	1	4	yes	—
131	C₂×C₄×C₈₃	#124	1	2	yes	—
132	C₂×C₄×C₈₃	#124	1	2	yes	—
133	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	4	yes	—
134	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
135	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
136	C₂×C₄×C₈₃	C₂×#41	1	2	yes	—
137	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
138	C₈×C₈₃	—	2	4	no	—
139	C₈×C₈₃	—	2	4	no	—
140	C₂²×C₈₃	—	4	8	no	—
141	C₂²×C₈₃	—	4	4	no	—
142	C₂²×C₈₃	—	8	2	no	—
143	C₂²×C₈₃	—	8	2	no	—
144	C₂²×C₈₃	—	8	4	no	—
145	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
146	C₂²×C₈₃	C₂×#124	1	1	yes	characteristic
147	C₂²×C₈₃	#532	1	1	yes	characteristic
148	C₂²×C₈₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
149	C₂²×C₈₃	#575	1	2	yes	—
150	C₄×C₈₃	—	2	2	no	—
151	C₄×C₈₃	—	2	4	no	—
152	C₄×C₈₃	—	4	2	no	—
153	C₄×C₈₃	—	4	4	no	—
154	C₄×C₈₃	—	8	2	no	—
155	C₄×C₈₃	—	8	2	no	—
156	C₄×C₈₃	#490	1	4	yes	direct product factor
157	C₄×C₈₃	C₄×#41	1	2	yes	—
158	C₄×C₈₃	#529	1	2	yes	—
159	C₄×#490	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
160	C₂×C₈₃	—	8	4	no	—
161	C₂×C₈₃	C₄×#124	1	1	yes	characteristic
162	C₂×C₈₃	C₄×AGL(1,ℤ/8)	1	2	yes	—
163	C₂×C₈₃	C₂×#490	1	1	yes	characteristic
164	C₂×C₈₃	#3135	1	1	yes	characteristic
165	C₂×C₈₃	#3171	1	2	yes	—
166	C₂×#490	C₂×C₈₃	1	1	yes	characteristic
167	C₄×#127	C₂×C₈₃	1	1	yes	characteristic
168	C₄×#125	C₂×C₈₃	1	1	yes	characteristic, complemented
169	#1992	C₂×C₈₃	1	1	yes	characteristic, complemented
170	#1961	C₂×C₈₃	1	1	yes	characteristic
171	#1936	C₂×C₈₃	1	4	yes	complemented
172	#1836	C₂×C₈₃	1	1	yes	characteristic, complemented
173	C₄×#102	C₂×C₈₃	1	4	yes	complemented
174	C₄×#97	C₂×C₈₃	1	1	yes	characteristic, complemented
175	C₈₃	C₄×#490	1	1	yes	characteristic, direct product factor, Sylow
176	C₂×#127	C₂²×C₈₃	1	1	yes	characteristic
177	C₂×#125	C₂²×C₈₃	1	1	yes	characteristic
178	C₂×C₄×D₄	—	2	2	no	—
179	C₂×C₄×D₄	C₂²×C₈₃	1	2	yes	—
180	#490	C₄×C₈₃	1	8	yes	direct product factor
181	C₂×#102	C₂²×C₈₃	1	4	yes	—
182	C₂×#102	C₂²×C₈₃	1	4	yes	—
183	C₂×#97	C₂²×C₈₃	1	2	yes	—
184	C₂×#97	C₂²×C₈₃	1	1	yes	characteristic
185	C₂×#97	C₂²×C₈₃	1	1	yes	characteristic
186	C₂×C₄×C₈	C₂²×C₈₃	1	2	yes	—
187	#390	—	2	2	no	—
188	#390	C₂²×C₈₃	1	2	yes	—
189	#389	C₂²×C₈₃	1	2	yes	—
190	#387	—	2	2	no	—
191	#384	C₂²×C₈₃	1	1	yes	characteristic
192	#382	C₂²×C₈₃	1	1	yes	characteristic
193	C₄×C₄⋊C₄	—	2	2	no	—
194	C₄×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
195	C₄³	C₂²×C₈₃	1	1	yes	characteristic
196	#340	C₂²×C₈₃	1	4	yes	—
197	#340	C₂²×C₈₃	1	4	yes	—
198	C₂²×D₄	—	2	2	no	—
199	C₂²×D₄	C₂³×C₈₃	1	2	yes	—
200	C₂³×C₄	—	2	2	no	—
201	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
202	C₂²×C₈	C₂³×C₈₃	1	2	yes	—
203	#127	—	2	2	no	—
204	#127	C₂×C₄×C₈₃	1	4	yes	—
205	#125	—	2	2	no	—
206	#125	C₂×C₄×C₈₃	1	4	yes	complemented
207	C₄×D₄	—	2	8	no	—
208	C₄×D₄	—	4	8	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	C₂³×C₈₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₈₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₈₃	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	4	no	—
216	C₂×C₂²⋊C₄	—	2	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	D₄×C₈₃	1	2	yes	—
219	C₂×C₄²	D₄×C₈₃	1	2	yes	—
220	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
221	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
222	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
223	#102	—	2	8	no	—
224	#102	C₂×C₄×C₈₃	1	16	yes	complemented
225	#97	—	2	4	no	—
226	#97	—	2	2	no	—
227	#97	C₂×C₄×C₈₃	1	4	yes	complemented
228	C₄×C₈	—	2	4	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	#95	—	2	2	no	—
232	C₂⁴	—	2	2	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	4	no	—
238	C₂×D₄	—	4	4	no	—
239	C₂×D₄	C₂²×C₄×C₈₃	1	4	yes	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	4	4	no	—
244	C₂²×C₄	—	4	8	no	—
245	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
246	C₂²×C₄	C₂×D₄×C₈₃	1	2	yes	—
247	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
249	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂²×C₄×C₈₃	1	4	yes	—
255	C₂×C₈	#41×C₈₃	1	4	yes	—
256	C₄⋊C₄	—	2	8	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	—	4	4	no	—
261	C₄⋊C₄	—	4	4	no	—
262	C₄⋊C₄	C₂²×C₄×C₈₃	1	4	yes	—
263	C₂²⋊C₄	—	4	8	no	—
264	C₂²⋊C₄	—	4	4	no	—
265	C₄²	—	2	4	no	—
266	C₄²	—	2	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	4	no	—
269	C₄²	C₂²×C₄×C₈₃	1	2	yes	—
270	C₄²	#41×C₈₃	1	2	yes	—
271	C₄²	#41×C₈₃	1	4	yes	—
272	C₂³	—	2	4	no	—
273	C₂³	—	4	2	no	—
274	C₂³	—	4	2	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	4	no	—
277	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
278	D₄	—	2	8	no	—
279	D₄	—	4	16	no	—
280	D₄	—	4	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	8	8	no	—
296	C₂×C₄	C₄×D₄×C₈₃	1	4	yes	—
297	C₂×C₄	#124×C₈₃	1	2	yes	—
298	C₂×C₄	#124×C₈₃	1	2	yes	—
299	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	4	yes	—
300	C₂×C₄	C₂³×C₄×C₈₃	1	1	yes	characteristic
301	C₂×C₄	C₂×#41×C₈₃	1	2	yes	—
302	C₂×C₄	C₂×#41×C₈₃	1	2	yes	—
303	C₂×C₄	C₂×#41×C₈₃	1	1	yes	characteristic
304	C₈	—	2	4	no	—
305	C₈	—	2	4	no	—
306	C₂²	—	4	4	no	—
307	C₂²	—	4	8	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	4	no	—
311	C₂²	C₂×C₄×D₄×C₈₃	1	1	yes	characteristic
312	C₂²	C₂×#124×C₈₃	1	1	yes	characteristic
313	C₂²	#532×C₈₃	1	1	yes	characteristic
314	C₂²	C₂×AGL(1,ℤ/8)×C₈₃	1	2	yes	—
315	C₂²	#575×C₈₃	1	2	yes	—
316	C₄	—	2	2	no	—
317	C₄	—	2	4	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	4	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	#490×C₈₃	1	4	yes	direct product factor
323	C₄	C₄×#41×C₈₃	1	2	yes	—
324	C₄	#529×C₈₃	1	2	yes	—
325	C₂	—	8	4	no	—
326	C₂	C₄×#124×C₈₃	1	1	yes	characteristic
327	C₂	C₄×AGL(1,ℤ/8)×C₈₃	1	2	yes	—
328	C₂	C₂×#490×C₈₃	1	1	yes	characteristic
329	C₂	C₈₃×#3135	1	1	yes	characteristic
330	C₂	C₈₃×#3171	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₈₃	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₃
socle	C₂³×C₈₃	C₂²×D₄
prime radical	C₂×#127×C₈₃	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₃ ◁ C₄³×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₄×#125×C₈₃	C₂	semidirect
4. C₈₃×#1992	C₂	semidirect
6. C₈₃×#1936	C₂	semidirect
7. C₈₃×#1836	C₂	semidirect
8. C₄×#102×C₈₃	C₂	semidirect
9. C₄×#97×C₈₃	C₂	semidirect
14. #490×C₈₃	C₄	direct (N × H)
40. #125×C₈₃	C₂×C₄	semidirect
58. #102×C₈₃	C₂×C₄	semidirect
61. #97×C₈₃	C₂×C₄	semidirect
156. C₄×C₈₃	#490	direct (N × H)
159. C₄×#490	C₈₃	direct (N × H)
168. C₄×#125	C₂×C₈₃	semidirect
169. #1992	C₂×C₈₃	semidirect
171. #1936	C₂×C₈₃	semidirect
172. #1836	C₂×C₈₃	semidirect
173. C₄×#102	C₂×C₈₃	semidirect
174. C₄×#97	C₂×C₈₃	semidirect
175. C₈₃	C₄×#490	direct (N × H)
180. #490	C₄×C₈₃	direct (N × H)
206. #125	C₂×C₄×C₈₃	semidirect
224. #102	C₂×C₄×C₈₃	semidirect
227. #97	C₂×C₄×C₈₃	semidirect
322. C₄	#490×C₈₃	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#490×C₈₃	2
C₄×#127×C₈₃	2
C₄×#125×C₈₃	2
C₈₃×#1992	2
C₈₃×#1961	2
C₈₃×#1936	2
C₈₃×#1836	2
C₄×#102×C₈₃	2
C₄×#97×C₈₃	2
C₄×#490	83

Subgroups — C4×#490×C83 #35249474

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#490×C₈₃ #35249474