Subgroups — C₈₃×#61818 #35289642

Order 21248.

1930 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#3502	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×D₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
3	Q₈×SD₁₆×C₈₃	C₂	1	8	yes	maximal, complemented
4	C₈₃×#3359	C₂	1	8	yes	maximal, complemented
5	C₈₃×#3229	C₂	1	6	yes	maximal, complemented
6	C₈₃×#3101	C₂	1	6	yes	maximal
7	C₈₃×#2982	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₈₃	—	2	1	no	—
9	Q₈²×C₈₃	C₂²	1	4	yes	complemented
10	#557×C₈₃	—	2	4	no	—
11	#557×C₈₃	C₂²	1	4	yes	complemented
12	D₄×Q₈×C₈₃	—	2	4	no	—
13	D₄×Q₈×C₈₃	C₂²	1	8	yes	—
14	#544×C₈₃	—	2	3	no	—
15	#533×C₈₃	C₂²	1	6	yes	—
16	C₂×C₄⋊Q₈×C₈₃	C₂²	1	3	yes	—
17	C₂×C₂²⋊Q₈×C₈₃	—	2	3	no	—
18	#519×C₈₃	—	2	1	no	—
19	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₄×D₄×C₈₃	C₂²	1	3	yes	—
21	#498×C₈₃	C₂²	1	12	yes	complemented
22	#487×C₈₃	C₂²	1	12	yes	complemented
23	#483×C₈₃	C₂²	1	12	yes	—
24	#476×C₈₃	C₂²	1	12	yes	—
25	#475×C₈₃	C₂²	1	24	yes	complemented
26	#461×C₈₃	C₂²	1	12	yes	—
27	#450×C₈₃	C₂²	1	4	yes	—
28	C₈×Q₈×C₈₃	C₂²	1	8	yes	complemented
29	C₄×SD₁₆×C₈₃	C₂²	1	24	yes	—
30	#431×C₈₃	C₂²	1	6	yes	—
31	C₂²×Q₈×C₈₃	—	2	8	no	—
32	C₂²×Q₈×C₈₃	—	2	3	no	—
33	C₂²×Q₈×C₈₃	—	4	8	no	—
34	C₂²×Q₈×C₈₃	D₄	1	1	yes	characteristic
35	C₂²×D₄×C₈₃	C₂³	1	1	yes	characteristic
36	C₂³×C₄×C₈₃	—	2	3	no	—
37	C₂×SD₁₆×C₈₃	C₂³	1	8	yes	—
38	C₄⋊Q₈×C₈₃	—	2	12	no	—
39	C₄⋊Q₈×C₈₃	—	2	6	no	—
40	C₄⋊Q₈×C₈₃	C₂³	1	6	yes	—
41	C₄⋊Q₈×C₈₃	C₂³	1	12	yes	—
42	C₄⋊Q₈×C₈₃	C₂³	1	6	yes	—
43	#125×C₈₃	—	2	12	no	—
44	#125×C₈₃	—	2	6	no	—
45	C₂²⋊Q₈×C₈₃	—	2	6	no	—
46	C₂²⋊Q₈×C₈₃	—	2	24	no	—
47	C₂²⋊Q₈×C₈₃	C₂³	1	4	yes	—
48	C₄×Q₈×C₈₃	—	2	4	no	—
49	C₄×Q₈×C₈₃	—	2	4	no	—
50	C₄×Q₈×C₈₃	—	2	6	no	—
51	C₄×Q₈×C₈₃	D₄	1	4	yes	—
52	C₄×Q₈×C₈₃	C₂³	1	12	yes	—
53	C₄×Q₈×C₈₃	C₂³	1	4	yes	—
54	C₄×D₄×C₈₃	—	2	6	no	—
55	C₄×D₄×C₈₃	C₂³	1	12	yes	—
56	#117×C₈₃	—	2	6	no	—
57	C₂×C₄⋊C₄×C₈₃	—	2	3	no	—
58	C₂×C₄⋊C₄×C₈₃	D₄	1	3	yes	—
59	C₂×C₄⋊C₄×C₈₃	C₂³	1	3	yes	—
60	C₂×C₂²⋊C₄×C₈₃	—	2	3	no	—
61	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
62	#106×C₈₃	C₂³	1	24	yes	—
63	C₄⋊C₈×C₈₃	C₂³	1	12	yes	—
64	#103×C₈₃	C₂³	1	24	yes	—
65	#102×C₈₃	C₂³	1	8	yes	—
66	C₂²⋊C₈×C₈₃	C₂³	1	4	yes	—
67	C₄×C₈×C₈₃	C₂³	1	12	yes	—
68	C₂⁴×C₈₃	—	2	1	no	—
69	C₂×Q₈×C₈₃	—	2	12	no	—
70	C₂×Q₈×C₈₃	—	2	6	no	—
71	C₂×Q₈×C₈₃	—	4	32	no	—
72	C₂×Q₈×C₈₃	—	4	12	no	—
73	C₂×Q₈×C₈₃	SD₁₆	1	8	yes	complemented
74	C₂×Q₈×C₈₃	C₂×D₄	1	4	yes	—
75	C₂×Q₈×C₈₃	C₂⁴	1	4	yes	—
76	C₂×D₄×C₈₃	—	2	4	no	—
77	C₂×D₄×C₈₃	—	2	1	no	—
78	C₂×D₄×C₈₃	C₂⁴	1	2	yes	—
79	C₂²×C₄×C₈₃	—	2	3	no	—
80	C₂²×C₄×C₈₃	—	2	12	no	—
81	C₂²×C₄×C₈₃	—	4	12	no	—
82	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
83	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
84	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
85	SD₁₆×C₈₃	—	2	16	no	—
86	C₂×C₈×C₈₃	C₂⁴	1	8	yes	—
87	C₄⋊C₄×C₈₃	—	2	12	no	—
88	C₄⋊C₄×C₈₃	—	2	3	no	—
89	C₄⋊C₄×C₈₃	—	2	6	no	—
90	C₄⋊C₄×C₈₃	—	2	6	no	—
91	C₄⋊C₄×C₈₃	—	2	4	no	—
92	C₄⋊C₄×C₈₃	—	2	6	no	—
93	C₄⋊C₄×C₈₃	—	2	12	no	—
94	C₄⋊C₄×C₈₃	C₂×D₄	1	12	yes	—
95	C₄⋊C₄×C₈₃	C₂⁴	1	4	yes	—
96	C₄⋊C₄×C₈₃	C₂⁴	1	6	yes	—
97	C₂²⋊C₄×C₈₃	—	2	4	no	—
98	C₂²⋊C₄×C₈₃	—	2	12	no	—
99	C₄²×C₈₃	—	2	12	no	—
100	C₄²×C₈₃	C₂×D₄	1	6	yes	—
101	C₄²×C₈₃	C₂⁴	1	6	yes	—
102	C₂³×C₈₃	—	2	4	no	—
103	C₂³×C₈₃	—	2	2	no	—
104	C₂³×C₈₃	—	4	2	no	—
105	C₂³×C₈₃	—	4	2	no	—
106	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₈₃	—	2	8	no	—
108	Q₈×C₈₃	—	4	24	no	—
109	Q₈×C₈₃	—	8	6	no	—
110	Q₈×C₈₃	C₂×SD₁₆	1	16	yes	—
111	D₄×C₈₃	—	2	4	no	—
112	D₄×C₈₃	—	2	2	no	—
113	D₄×C₈₃	2₋¹⁺⁴	1	4	yes	—
114	C₂×C₄×C₈₃	—	2	2	no	—
115	C₂×C₄×C₈₃	—	2	4	no	—
116	C₂×C₄×C₈₃	—	2	4	no	—
117	C₂×C₄×C₈₃	—	2	6	no	—
118	C₂×C₄×C₈₃	—	4	3	no	—
119	C₂×C₄×C₈₃	—	4	6	no	—
120	C₂×C₄×C₈₃	—	4	24	no	—
121	C₂×C₄×C₈₃	C₂×SD₁₆	1	12	yes	—
122	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₈₃	C₂²×D₄	1	6	yes	—
124	C₂×C₄×C₈₃	C₂²×D₄	1	6	yes	—
125	C₂×C₄×C₈₃	C₂⁵	1	1	yes	characteristic
126	C₈×C₈₃	—	2	8	no	—
127	C₂²×C₈₃	—	2	1	no	—
128	C₂²×C₈₃	—	2	1	no	—
129	C₂²×C₈₃	—	2	4	no	—
130	C₂²×C₈₃	—	4	8	no	—
131	C₂²×C₈₃	—	4	2	no	—
132	C₂²×C₈₃	—	4	2	no	—
133	C₂²×C₈₃	C₂²×SD₁₆	1	2	yes	—
134	C₂²×C₈₃	C₂³×D₄	1	1	yes	characteristic
135	C₄×C₈₃	—	2	6	no	—
136	C₄×C₈₃	—	2	1	no	—
137	C₄×C₈₃	—	4	4	no	—
138	C₄×C₈₃	—	4	4	no	—
139	C₄×C₈₃	—	4	6	no	—
140	C₄×C₈₃	C₂²×SD₁₆	1	12	yes	—
141	C₄×C₈₃	C₂×2₋¹⁺⁴	1	2	yes	—
142	#61818	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₈₃	—	2	2	no	—
144	C₂×C₈₃	—	4	4	no	—
145	C₂×C₈₃	#3486	1	1	yes	characteristic
146	C₂×C₈₃	C₂³×SD₁₆	1	1	yes	characteristic
147	C₂×C₈₃	#3586	1	1	yes	characteristic
148	#3502	C₂×C₈₃	1	1	yes	characteristic, complemented
149	C₂×D₄×Q₈	C₂×C₈₃	1	1	yes	characteristic
150	Q₈×SD₁₆	C₂×C₈₃	1	8	yes	complemented
151	#3359	C₂×C₈₃	1	8	yes	complemented
152	#3229	C₂×C₈₃	1	6	yes	complemented
153	#3101	C₂×C₈₃	1	6	yes	—
154	#2982	C₂×C₈₃	1	1	yes	characteristic, complemented
155	C₈₃	#61818	1	1	yes	characteristic, direct product factor, Sylow
156	C₂³×Q₈	—	2	1	no	—
157	Q₈²	C₂²×C₈₃	1	4	yes	complemented
158	#557	—	2	4	no	—
159	#557	C₂²×C₈₃	1	4	yes	complemented
160	D₄×Q₈	—	2	4	no	—
161	D₄×Q₈	C₂²×C₈₃	1	8	yes	—
162	#544	—	2	3	no	—
163	#533	C₂²×C₈₃	1	6	yes	—
164	C₂×C₄⋊Q₈	C₂²×C₈₃	1	3	yes	—
165	C₂×C₂²⋊Q₈	—	2	3	no	—
166	#519	—	2	1	no	—
167	C₂×C₄×Q₈	C₂²×C₈₃	1	1	yes	characteristic
168	C₂×C₄×D₄	C₂²×C₈₃	1	3	yes	—
169	#498	C₂²×C₈₃	1	12	yes	complemented
170	#487	C₂²×C₈₃	1	12	yes	complemented
171	#483	C₂²×C₈₃	1	12	yes	—
172	#476	C₂²×C₈₃	1	12	yes	—
173	#475	C₂²×C₈₃	1	24	yes	complemented
174	#461	C₂²×C₈₃	1	12	yes	—
175	#450	C₂²×C₈₃	1	4	yes	—
176	C₈×Q₈	C₂²×C₈₃	1	8	yes	complemented
177	C₄×SD₁₆	C₂²×C₈₃	1	24	yes	—
178	#431	C₂²×C₈₃	1	6	yes	—
179	C₂²×Q₈	—	2	8	no	—
180	C₂²×Q₈	—	2	3	no	—
181	C₂²×Q₈	—	4	8	no	—
182	C₂²×Q₈	D₄×C₈₃	1	1	yes	characteristic
183	C₂²×D₄	C₂³×C₈₃	1	1	yes	characteristic
184	C₂³×C₄	—	2	3	no	—
185	C₂×SD₁₆	C₂³×C₈₃	1	8	yes	—
186	C₄⋊Q₈	—	2	12	no	—
187	C₄⋊Q₈	—	2	6	no	—
188	C₄⋊Q₈	C₂³×C₈₃	1	6	yes	—
189	C₄⋊Q₈	C₂³×C₈₃	1	12	yes	—
190	C₄⋊Q₈	C₂³×C₈₃	1	6	yes	—
191	#125	—	2	12	no	—
192	#125	—	2	6	no	—
193	C₂²⋊Q₈	—	2	6	no	—
194	C₂²⋊Q₈	—	2	24	no	—
195	C₂²⋊Q₈	C₂³×C₈₃	1	4	yes	—
196	C₄×Q₈	—	2	4	no	—
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	6	no	—
199	C₄×Q₈	D₄×C₈₃	1	4	yes	—
200	C₄×Q₈	C₂³×C₈₃	1	12	yes	—
201	C₄×Q₈	C₂³×C₈₃	1	4	yes	—
202	C₄×D₄	—	2	6	no	—
203	C₄×D₄	C₂³×C₈₃	1	12	yes	—
204	#117	—	2	6	no	—
205	C₂×C₄⋊C₄	—	2	3	no	—
206	C₂×C₄⋊C₄	D₄×C₈₃	1	3	yes	—
207	C₂×C₄⋊C₄	C₂³×C₈₃	1	3	yes	—
208	C₂×C₂²⋊C₄	—	2	3	no	—
209	C₂×C₄²	C₂³×C₈₃	1	3	yes	—
210	#106	C₂³×C₈₃	1	24	yes	—
211	C₄⋊C₈	C₂³×C₈₃	1	12	yes	—
212	#103	C₂³×C₈₃	1	24	yes	—
213	#102	C₂³×C₈₃	1	8	yes	—
214	C₂²⋊C₈	C₂³×C₈₃	1	4	yes	—
215	C₄×C₈	C₂³×C₈₃	1	12	yes	—
216	C₂⁴	—	2	1	no	—
217	C₂×Q₈	—	2	12	no	—
218	C₂×Q₈	—	2	6	no	—
219	C₂×Q₈	—	4	32	no	—
220	C₂×Q₈	—	4	12	no	—
221	C₂×Q₈	SD₁₆×C₈₃	1	8	yes	complemented
222	C₂×Q₈	C₂×D₄×C₈₃	1	4	yes	—
223	C₂×Q₈	C₂⁴×C₈₃	1	4	yes	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	C₂⁴×C₈₃	1	2	yes	—
227	C₂²×C₄	—	2	3	no	—
228	C₂²×C₄	—	2	12	no	—
229	C₂²×C₄	—	4	12	no	—
230	C₂²×C₄	C₂×D₄×C₈₃	1	3	yes	—
231	C₂²×C₄	C₂×D₄×C₈₃	1	3	yes	—
232	C₂²×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
233	SD₁₆	—	2	16	no	—
234	C₂×C₈	C₂⁴×C₈₃	1	8	yes	—
235	C₄⋊C₄	—	2	12	no	—
236	C₄⋊C₄	—	2	3	no	—
237	C₄⋊C₄	—	2	6	no	—
238	C₄⋊C₄	—	2	6	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	2	6	no	—
241	C₄⋊C₄	—	2	12	no	—
242	C₄⋊C₄	C₂×D₄×C₈₃	1	12	yes	—
243	C₄⋊C₄	C₂⁴×C₈₃	1	4	yes	—
244	C₄⋊C₄	C₂⁴×C₈₃	1	6	yes	—
245	C₂²⋊C₄	—	2	4	no	—
246	C₂²⋊C₄	—	2	12	no	—
247	C₄²	—	2	12	no	—
248	C₄²	C₂×D₄×C₈₃	1	6	yes	—
249	C₄²	C₂⁴×C₈₃	1	6	yes	—
250	C₂³	—	2	4	no	—
251	C₂³	—	2	2	no	—
252	C₂³	—	4	2	no	—
253	C₂³	—	4	2	no	—
254	C₂³	C₂²×D₄×C₈₃	1	1	yes	characteristic
255	Q₈	—	2	8	no	—
256	Q₈	—	4	24	no	—
257	Q₈	—	8	6	no	—
258	Q₈	C₂×SD₁₆×C₈₃	1	16	yes	—
259	D₄	—	2	4	no	—
260	D₄	—	2	2	no	—
261	D₄	2₋¹⁺⁴×C₈₃	1	4	yes	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	4	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	6	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	4	6	no	—
268	C₂×C₄	—	4	24	no	—
269	C₂×C₄	C₂×SD₁₆×C₈₃	1	12	yes	—
270	C₂×C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
271	C₂×C₄	C₂²×D₄×C₈₃	1	6	yes	—
272	C₂×C₄	C₂²×D₄×C₈₃	1	6	yes	—
273	C₂×C₄	C₂⁵×C₈₃	1	1	yes	characteristic
274	C₈	—	2	8	no	—
275	C₂²	—	2	1	no	—
276	C₂²	—	2	1	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	8	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	C₂²×SD₁₆×C₈₃	1	2	yes	—
282	C₂²	C₂³×D₄×C₈₃	1	1	yes	characteristic
283	C₄	—	2	6	no	—
284	C₄	—	2	1	no	—
285	C₄	—	4	6	no	—
286	C₄	—	4	4	no	—
287	C₄	—	4	4	no	—
288	C₄	C₂²×SD₁₆×C₈₃	1	12	yes	—
289	C₄	C₂×2₋¹⁺⁴×C₈₃	1	2	yes	—
290	C₂	—	2	2	no	—
291	C₂	—	4	4	no	—
292	C₂	C₈₃×#3486	1	1	yes	characteristic
293	C₂	C₂³×SD₁₆×C₈₃	1	1	yes	characteristic
294	C₂	C₈₃×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₃	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₃
prime radical	C₂²×D₄×C₈₃	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₃ ◁ C₂×C₄×Q₈×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₃×#3502	C₂	semidirect
3. Q₈×SD₁₆×C₈₃	C₂	semidirect
4. C₈₃×#3359	C₂	semidirect
5. C₈₃×#3229	C₂	semidirect
7. C₈₃×#2982	C₂	semidirect
9. Q₈²×C₈₃	C₂²	semidirect
11. #557×C₈₃	C₂²	semidirect
21. #498×C₈₃	C₂²	semidirect
22. #487×C₈₃	C₂²	semidirect
25. #475×C₈₃	C₂²	semidirect
28. C₈×Q₈×C₈₃	C₂²	semidirect
73. C₂×Q₈×C₈₃	SD₁₆	semidirect
142. #61818	C₈₃	direct (N × H)
148. #3502	C₂×C₈₃	semidirect
150. Q₈×SD₁₆	C₂×C₈₃	semidirect
151. #3359	C₂×C₈₃	semidirect
152. #3229	C₂×C₈₃	semidirect
154. #2982	C₂×C₈₃	semidirect
155. C₈₃	#61818	direct (N × H)
157. Q₈²	C₂²×C₈₃	semidirect
159. #557	C₂²×C₈₃	semidirect
169. #498	C₂²×C₈₃	semidirect
170. #487	C₂²×C₈₃	semidirect
173. #475	C₂²×C₈₃	semidirect
176. C₈×Q₈	C₂²×C₈₃	semidirect
221. C₂×Q₈	SD₁₆×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₈₃×#3502	2
C₂×D₄×Q₈×C₈₃	2
Q₈×SD₁₆×C₈₃	2
C₈₃×#3359	2
C₈₃×#3229	2
C₈₃×#3101	2
C₈₃×#2982	2
#61818	83

Subgroups — C83×#61818 #35289642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#61818 #35289642