Subgroups — C₈₃×#8878 #35236702

Order 21248.

432 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₃×#2932	C₂	1	1	yes	maximal, characteristic
2	C₈₃×#1940	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₃×#1571	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₃×#1502	C₂	1	1	yes	maximal, characteristic
5	C₈₃×#1480	C₂	1	1	yes	maximal, characteristic
6	C₈₃×#1303	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₃×#1281	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#131×C₈₃	—	2	1	no	—
9	#532×C₈₃	C₂²	1	1	yes	characteristic
10	#435×C₈₃	—	2	2	no	—
11	#435×C₈₃	—	2	2	no	—
12	#431×C₈₃	C₂²	1	1	yes	characteristic
13	#431×C₈₃	C₂²	1	1	yes	characteristic
14	#423×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₈₃	—	2	1	no	—
16	C₂×C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#408×C₈₃	—	2	1	no	—
18	C₂×#97×C₈₃	C₂²	1	1	yes	characteristic
19	C₄×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
20	#336×C₈₃	—	2	1	no	—
21	#335×C₈₃	—	2	2	no	—
22	#327×C₈₃	C₄	1	2	yes	—
23	#327×C₈₃	C₄	1	2	yes	—
24	D₄⋊C₈×C₈₃	—	2	2	no	—
25	C₂×#41×C₈₃	—	2	1	no	—
26	#131×C₈₃	—	4	2	no	—
27	#131×C₈₃	—	4	2	no	—
28	C₂×M₁₆×C₈₃	—	2	2	no	—
29	C₂×M₁₆×C₈₃	—	2	1	no	—
30	C₂²×C₈×C₈₃	—	2	2	no	—
31	C₂²×C₈×C₈₃	—	2	1	no	—
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
34	C₄⋊Q₈×C₈₃	Q₈	1	1	yes	characteristic
35	#127×C₈₃	Q₈	1	1	yes	characteristic
36	#124×C₈₃	—	2	1	no	—
37	#121×C₈₃	—	2	1	no	—
38	#121×C₈₃	C₂×C₄	1	2	yes	—
39	C₄×D₄×C₈₃	—	2	1	no	—
40	C₄×D₄×C₈₃	D₄	1	2	yes	—
41	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
42	C₂×C₄²×C₈₃	—	2	1	no	—
43	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₈₃	—	2	1	no	—
45	C₄⋊C₈×C₈₃	—	2	1	no	—
46	C₄⋊C₈×C₈₃	—	2	1	no	—
47	C₄⋊C₈×C₈₃	—	2	1	no	—
48	C₄⋊C₈×C₈₃	—	2	1	no	—
49	C₄⋊C₈×C₈₃	—	4	2	no	—
50	C₄⋊C₈×C₈₃	D₄	1	2	yes	—
51	C₂²⋊C₈×C₈₃	—	2	4	no	—
52	C₂²⋊C₈×C₈₃	—	2	1	no	—
53	C₂²⋊C₈×C₈₃	C₂×C₄	1	2	yes	—
54	#97×C₈₃	—	2	1	no	—
55	#97×C₈₃	—	2	1	no	—
56	C₄×C₈×C₈₃	—	2	1	no	—
57	C₄×C₈×C₈₃	—	2	1	no	—
58	#95×C₈₃	C₂×C₄	1	1	yes	characteristic
59	#95×C₈₃	C₂×C₄	1	1	yes	characteristic
60	#41×C₈₃	—	4	1	no	—
61	#41×C₈₃	—	4	1	no	—
62	#41×C₈₃	—	4	1	no	—
63	#41×C₈₃	—	4	1	no	—
64	C₂×Q₈×C₈₃	—	2	1	no	—
65	C₂×D₄×C₈₃	—	2	2	no	—
66	C₂×D₄×C₈₃	—	2	1	no	—
67	C₂×D₄×C₈₃	—	2	1	no	—
68	C₂×D₄×C₈₃	C₄⋊C₄	1	2	yes	—
69	C₂²×C₄×C₈₃	—	2	1	no	—
70	C₂²×C₄×C₈₃	—	2	2	no	—
71	C₂²×C₄×C₈₃	—	2	1	no	—
72	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
75	M₁₆×C₈₃	—	4	4	no	—
76	M₁₆×C₈₃	—	4	1	no	—
77	M₁₆×C₈₃	—	4	1	no	—
78	C₂×C₈×C₈₃	—	2	1	no	—
79	C₂×C₈×C₈₃	—	2	1	no	—
80	C₂×C₈×C₈₃	—	2	2	no	—
81	C₂×C₈×C₈₃	—	2	1	no	—
82	C₂×C₈×C₈₃	—	2	2	no	—
83	C₂×C₈×C₈₃	—	2	1	no	—
84	C₂×C₈×C₈₃	—	2	1	no	—
85	C₂×C₈×C₈₃	—	4	4	no	—
86	C₂×C₈×C₈₃	—	4	1	no	—
87	C₂×C₈×C₈₃	—	4	1	no	—
88	C₂×C₈×C₈₃	—	4	1	no	—
89	C₂×C₈×C₈₃	—	4	1	no	—
90	C₂×C₈×C₈₃	D₈	1	2	yes	—
91	C₄⋊C₄×C₈₃	—	2	1	no	—
92	C₄⋊C₄×C₈₃	—	4	2	no	—
93	C₄⋊C₄×C₈₃	C₄⋊C₄	1	2	yes	—
94	C₂²⋊C₄×C₈₃	—	2	2	no	—
95	C₂²⋊C₄×C₈₃	—	2	2	no	—
96	C₄²×C₈₃	—	4	1	no	—
97	C₄²×C₈₃	—	4	1	no	—
98	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
99	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
100	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
101	C₄²×C₈₃	#41	1	1	yes	characteristic
102	C₂³×C₈₃	—	2	2	no	—
103	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
104	Q₈×C₈₃	—	4	1	no	—
105	Q₈×C₈₃	—	4	1	no	—
106	D₄×C₈₃	—	4	2	no	—
107	D₄×C₈₃	—	4	2	no	—
108	D₄×C₈₃	—	4	2	no	—
109	D₄×C₈₃	—	4	1	no	—
110	D₄×C₈₃	—	4	1	no	—
111	D₄×C₈₃	—	8	1	no	—
112	C₂×C₄×C₈₃	—	2	2	no	—
113	C₂×C₄×C₈₃	—	2	2	no	—
114	C₂×C₄×C₈₃	—	4	1	no	—
115	C₂×C₄×C₈₃	—	4	1	no	—
116	C₂×C₄×C₈₃	—	4	1	no	—
117	C₂×C₄×C₈₃	—	4	1	no	—
118	C₂×C₄×C₈₃	—	4	2	no	—
119	C₂×C₄×C₈₃	—	4	2	no	—
120	C₂×C₄×C₈₃	—	4	1	no	—
121	C₂×C₄×C₈₃	—	4	1	no	—
122	C₂×C₄×C₈₃	—	4	1	no	—
123	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
129	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
131	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
133	C₈×C₈₃	—	2	2	no	—
134	C₈×C₈₃	—	4	2	no	—
135	C₈×C₈₃	—	4	2	no	—
136	C₈×C₈₃	—	4	1	no	—
137	C₈×C₈₃	—	8	1	no	—
138	C₈×C₈₃	—	8	1	no	—
139	C₂²×C₈₃	—	2	1	no	—
140	C₂²×C₈₃	—	2	1	no	—
141	C₂²×C₈₃	—	4	2	no	—
142	C₂²×C₈₃	—	4	2	no	—
143	C₂²×C₈₃	—	4	2	no	—
144	C₂²×C₈₃	#385	1	1	yes	characteristic
145	C₂²×C₈₃	C₄×D₈	1	1	yes	characteristic
146	C₂²×C₈₃	#440	1	1	yes	characteristic
147	C₄×C₈₃	—	2	1	no	—
148	C₄×C₈₃	—	2	1	no	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	4	2	no	—
152	C₄×C₈₃	—	4	2	no	—
153	C₄×C₈₃	—	8	1	no	—
154	C₄×C₈₃	#447	1	1	yes	characteristic
155	C₄×C₈₃	#452	1	1	yes	characteristic
156	C₄×C₈₃	#474	1	1	yes	characteristic
157	C₄×C₈₃	#478	1	1	yes	characteristic
158	#8878	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂×C₈₃	—	2	1	no	—
160	C₂×C₈₃	—	2	1	no	—
161	C₂×C₈₃	—	8	2	no	—
162	C₂×C₈₃	#1861	1	1	yes	characteristic
163	C₂×C₈₃	#1862	1	1	yes	characteristic
164	C₂×C₈₃	#1868	1	1	yes	characteristic
165	#2932	C₂×C₈₃	1	1	yes	characteristic
166	#1940	C₂×C₈₃	1	1	yes	characteristic, complemented
167	#1571	C₂×C₈₃	1	1	yes	characteristic, complemented
168	#1502	C₂×C₈₃	1	1	yes	characteristic
169	#1480	C₂×C₈₃	1	1	yes	characteristic
170	#1303	C₂×C₈₃	1	1	yes	characteristic, complemented
171	#1281	C₂×C₈₃	1	1	yes	characteristic, complemented
172	C₈₃	#8878	1	1	yes	characteristic, direct product factor, Sylow
173	C₂×#131	—	2	1	no	—
174	#532	C₂²×C₈₃	1	1	yes	characteristic
175	#435	—	2	2	no	—
176	#435	—	2	2	no	—
177	#431	C₂²×C₈₃	1	1	yes	characteristic
178	#431	C₂²×C₈₃	1	1	yes	characteristic
179	#423	C₂²×C₈₃	1	1	yes	characteristic
180	C₂×C₄⋊C₈	—	2	1	no	—
181	C₂×C₄⋊C₈	C₂²×C₈₃	1	1	yes	characteristic
182	#408	—	2	1	no	—
183	C₂×#97	C₂²×C₈₃	1	1	yes	characteristic
184	C₄×C₄⋊C₄	C₂²×C₈₃	1	1	yes	characteristic
185	#336	—	2	1	no	—
186	#335	—	2	2	no	—
187	#327	C₄×C₈₃	1	2	yes	—
188	#327	C₄×C₈₃	1	2	yes	—
189	D₄⋊C₈	—	2	2	no	—
190	C₂×#41	—	2	1	no	—
191	#131	—	4	2	no	—
192	#131	—	4	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	1	no	—
195	C₂²×C₈	—	2	2	no	—
196	C₂²×C₈	—	2	1	no	—
197	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
198	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
199	C₄⋊Q₈	Q₈×C₈₃	1	1	yes	characteristic
200	#127	Q₈×C₈₃	1	1	yes	characteristic
201	#124	—	2	1	no	—
202	#121	—	2	1	no	—
203	#121	C₂×C₄×C₈₃	1	2	yes	—
204	C₄×D₄	—	2	1	no	—
205	C₄×D₄	D₄×C₈₃	1	2	yes	—
206	C₂×C₄⋊C₄	—	2	1	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	C₂³×C₈₃	1	1	yes	characteristic
209	C₄⋊C₈	—	2	1	no	—
210	C₄⋊C₈	—	2	1	no	—
211	C₄⋊C₈	—	2	1	no	—
212	C₄⋊C₈	—	2	1	no	—
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	4	2	no	—
215	C₄⋊C₈	D₄×C₈₃	1	2	yes	—
216	C₂²⋊C₈	—	2	4	no	—
217	C₂²⋊C₈	—	2	1	no	—
218	C₂²⋊C₈	C₂×C₄×C₈₃	1	2	yes	—
219	#97	—	2	1	no	—
220	#97	—	2	1	no	—
221	C₄×C₈	—	2	1	no	—
222	C₄×C₈	—	2	1	no	—
223	#95	C₂×C₄×C₈₃	1	1	yes	characteristic
224	#95	C₂×C₄×C₈₃	1	1	yes	characteristic
225	#41	—	4	1	no	—
226	#41	—	4	1	no	—
227	#41	—	4	1	no	—
228	#41	—	4	1	no	—
229	C₂×Q₈	—	2	1	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	2	1	no	—
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₄⋊C₄×C₈₃	1	2	yes	—
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	2	2	no	—
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
239	C₂²×C₄	#41×C₈₃	1	1	yes	characteristic
240	M₁₆	—	4	4	no	—
241	M₁₆	—	4	1	no	—
242	M₁₆	—	4	1	no	—
243	C₂×C₈	—	2	1	no	—
244	C₂×C₈	—	2	1	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	4	4	no	—
251	C₂×C₈	—	4	1	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	D₈×C₈₃	1	2	yes	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	C₄⋊C₄×C₈₃	1	2	yes	—
259	C₂²⋊C₄	—	2	2	no	—
260	C₂²⋊C₄	—	2	2	no	—
261	C₄²	—	4	1	no	—
262	C₄²	—	4	1	no	—
263	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₈₃	1	1	yes	characteristic
265	C₄²	C₂×Q₈×C₈₃	1	1	yes	characteristic
266	C₄²	#41×C₈₃	1	1	yes	characteristic
267	C₂³	—	2	2	no	—
268	C₂³	C₄×D₄×C₈₃	1	1	yes	characteristic
269	Q₈	—	4	1	no	—
270	Q₈	—	4	1	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	4	2	no	—
274	D₄	—	4	1	no	—
275	D₄	—	4	1	no	—
276	D₄	—	8	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	4	1	no	—
280	C₂×C₄	—	4	1	no	—
281	C₂×C₄	—	4	1	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
289	C₂×C₄	C₄×D₄×C₈₃	1	1	yes	characteristic
290	C₂×C₄	#120×C₈₃	1	1	yes	characteristic
291	C₂×C₄	C₂²⋊Q₈×C₈₃	1	1	yes	characteristic
292	C₂×C₄	C₂²⋊Q₈×C₈₃	1	1	yes	characteristic
293	C₂×C₄	#123×C₈₃	1	1	yes	characteristic
294	C₂×C₄	C₂×D₈×C₈₃	1	1	yes	characteristic
295	C₂×C₄	#135×C₈₃	1	1	yes	characteristic
296	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	1	yes	characteristic
297	C₂×C₄	#137×C₈₃	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	4	2	no	—
300	C₈	—	4	2	no	—
301	C₈	—	4	1	no	—
302	C₈	—	8	1	no	—
303	C₈	—	8	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	4	2	no	—
307	C₂²	—	4	2	no	—
308	C₂²	—	4	2	no	—
309	C₂²	#385×C₈₃	1	1	yes	characteristic
310	C₂²	C₄×D₈×C₈₃	1	1	yes	characteristic
311	C₂²	#440×C₈₃	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	1	no	—
319	C₄	#447×C₈₃	1	1	yes	characteristic
320	C₄	#452×C₈₃	1	1	yes	characteristic
321	C₄	#474×C₈₃	1	1	yes	characteristic
322	C₄	#478×C₈₃	1	1	yes	characteristic
323	C₂	—	2	1	no	—
324	C₂	—	2	1	no	—
325	C₂	—	8	2	no	—
326	C₂	C₈₃×#1861	1	1	yes	characteristic
327	C₂	C₈₃×#1862	1	1	yes	characteristic
328	C₂	C₈₃×#1868	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈₃	#120
commutator	C₂²×C₄	C₂²×C₄×C₈₃
Frattini, cosocle	C₂×C₄²	C₂³×C₈₃
socle	C₂²×C₈₃	#385
prime radical	#532×C₈₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₃ ◁ C₂×C₄²×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₃×#1940	C₂	semidirect
3. C₈₃×#1571	C₂	semidirect
6. C₈₃×#1303	C₂	semidirect
7. C₈₃×#1281	C₂	semidirect
158. #8878	C₈₃	direct (N × H)
166. #1940	C₂×C₈₃	semidirect
167. #1571	C₂×C₈₃	semidirect
170. #1303	C₂×C₈₃	semidirect
171. #1281	C₂×C₈₃	semidirect
172. C₈₃	#8878	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×#2932	2
C₈₃×#1940	2
C₈₃×#1571	2
C₈₃×#1502	2
C₈₃×#1480	2
C₈₃×#1303	2
C₈₃×#1281	2
#8878	83

Subgroups — C83×#8878 #35236702

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#8878 #35236702