Subgroups — C₈₃×#11376 #35239200

Order 21248.

654 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	C₂×#427×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#341×C₈₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#339×C₈₃	C₂	1	2	yes	maximal
4	C₈₃×#1729	C₂	1	2	yes	maximal
5	C₂×#336×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×M₁₆×C₈₃	C₂²	1	2	yes	—
7	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
9	C₂×#117×C₈₃	C₂²	1	2	yes	—
10	C₂²×C₄²×C₈₃	C₂²	1	1	yes	characteristic
11	#427×C₈₃	—	2	2	no	—
12	#427×C₈₃	—	2	2	no	—
13	C₂×#107×C₈₃	C₄	1	1	yes	characteristic, complemented
14	C₂×#107×C₈₃	C₄	1	1	yes	characteristic, complemented
15	C₂×#106×C₈₃	C₄	1	2	yes	complemented
16	#341×C₈₃	—	2	2	no	—
17	#339×C₈₃	—	2	4	no	—
18	#336×C₈₃	—	2	2	no	—
19	#336×C₈₃	—	2	4	no	—
20	C₂³×C₄×C₈₃	—	2	1	no	—
21	C₂³×C₄×C₈₃	C₂³	1	1	yes	characteristic
22	C₂×M₁₆×C₈₃	—	2	2	no	—
23	C₂×M₁₆×C₈₃	—	2	2	no	—
24	C₂×M₁₆×C₈₃	—	2	2	no	—
25	C₂×M₁₆×C₈₃	—	2	2	no	—
26	C₂×M₁₆×C₈₃	—	2	2	no	—
27	C₂×M₁₆×C₈₃	Q₈	1	4	yes	—
28	C₂²×C₈×C₈₃	—	2	2	no	—
29	C₂²×C₈×C₈₃	—	2	2	no	—
30	C₂²×C₈×C₈₃	—	2	1	no	—
31	C₂²×C₈×C₈₃	—	2	2	no	—
32	C₂²×C₈×C₈₃	—	2	1	no	—
33	C₂²×C₈×C₈₃	—	2	2	no	—
34	C₂²×C₈×C₈₃	—	2	2	no	—
35	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	D₄	1	2	yes	—
38	C₂²×C₈×C₈₃	D₄	1	2	yes	—
39	C₂²×C₈×C₈₃	D₄	1	2	yes	—
40	#117×C₈₃	—	2	2	no	—
41	#117×C₈₃	—	2	2	no	—
42	#117×C₈₃	—	2	2	no	—
43	#117×C₈₃	—	2	2	no	—
44	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
45	C₂×C₄⋊C₄×C₈₃	C₂×C₄	1	2	yes	—
46	C₂×C₂²⋊C₄×C₈₃	—	2	2	no	—
47	C₂×C₄²×C₈₃	—	2	2	no	—
48	C₂×C₄²×C₈₃	—	2	2	no	—
49	C₂×C₄²×C₈₃	—	2	1	no	—
50	C₂×C₄²×C₈₃	—	2	2	no	—
51	C₂×C₄²×C₈₃	D₄	1	2	yes	—
52	#107×C₈₃	—	2	2	no	—
53	#107×C₈₃	—	2	2	no	—
54	#106×C₈₃	—	2	4	no	—
55	C₂⁴×C₈₃	#41	1	1	yes	characteristic
56	C₂²×C₄×C₈₃	—	2	4	no	—
57	C₂²×C₄×C₈₃	—	2	2	no	—
58	C₂²×C₄×C₈₃	—	2	2	no	—
59	C₂²×C₄×C₈₃	—	2	2	no	—
60	C₂²×C₄×C₈₃	—	4	2	no	—
61	C₂²×C₄×C₈₃	—	4	2	no	—
62	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₈₃	#41	1	2	yes	—
69	C₂²×C₄×C₈₃	#41	1	2	yes	—
70	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₈₃	#41	1	2	yes	—
72	M₁₆×C₈₃	—	2	4	no	—
73	M₁₆×C₈₃	—	2	4	no	—
74	M₁₆×C₈₃	—	4	2	no	—
75	M₁₆×C₈₃	—	4	2	no	—
76	C₂×C₈×C₈₃	—	2	2	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	4	no	—
81	C₂×C₈×C₈₃	—	2	2	no	—
82	C₂×C₈×C₈₃	—	2	4	no	—
83	C₂×C₈×C₈₃	—	2	4	no	—
84	C₂×C₈×C₈₃	—	4	1	no	—
85	C₂×C₈×C₈₃	—	4	2	no	—
86	C₂×C₈×C₈₃	—	4	1	no	—
87	C₂×C₈×C₈₃	—	4	2	no	—
88	C₂×C₈×C₈₃	—	4	2	no	—
89	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
90	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
91	C₄⋊C₄×C₈₃	—	2	2	no	—
92	C₄⋊C₄×C₈₃	—	2	2	no	—
93	C₄⋊C₄×C₈₃	—	2	2	no	—
94	C₄⋊C₄×C₈₃	—	2	2	no	—
95	C₂²⋊C₄×C₈₃	—	4	2	no	—
96	C₂²⋊C₄×C₈₃	—	4	2	no	—
97	C₄²×C₈₃	—	2	4	no	—
98	C₄²×C₈₃	—	4	2	no	—
99	C₄²×C₈₃	—	4	1	no	—
100	C₄²×C₈₃	—	4	1	no	—
101	C₄²×C₈₃	—	4	2	no	—
102	C₂³×C₈₃	—	2	2	no	—
103	C₂³×C₈₃	—	2	2	no	—
104	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
105	C₂³×C₈₃	#121	1	1	yes	characteristic
106	C₂³×C₈₃	#124	1	1	yes	characteristic
107	C₂³×C₈₃	#125	1	2	yes	—
108	C₂³×C₈₃	#126	1	2	yes	—
109	C₂×C₄×C₈₃	—	2	4	no	—
110	C₂×C₄×C₈₃	—	4	8	no	—
111	C₂×C₄×C₈₃	—	4	2	no	—
112	C₂×C₄×C₈₃	—	4	2	no	—
113	C₂×C₄×C₈₃	—	4	2	no	—
114	C₂×C₄×C₈₃	—	4	2	no	—
115	C₂×C₄×C₈₃	—	4	4	no	—
116	C₂×C₄×C₈₃	—	4	4	no	—
117	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	#117	1	2	yes	—
119	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₄×Q₈	1	2	yes	—
121	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
122	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
124	C₂×C₄×C₈₃	C₂²⋊Q₈	1	2	yes	—
125	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
126	C₂×C₄×C₈₃	#123	1	1	yes	characteristic
127	C₂×C₄×C₈₃	#123	1	2	yes	—
128	C₂×C₄×C₈₃	#123	1	2	yes	—
129	C₂×C₄×C₈₃	#123	1	2	yes	—
130	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
131	C₂×C₄×C₈₃	#125	1	2	yes	—
132	C₂×C₄×C₈₃	#126	1	2	yes	—
133	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
135	C₈×C₈₃	—	2	2	no	—
136	C₈×C₈₃	—	2	2	no	—
137	C₈×C₈₃	—	4	2	no	—
138	C₈×C₈₃	—	4	4	no	—
139	C₂²×C₈₃	—	2	4	no	—
140	C₂²×C₈₃	—	2	4	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	#387	1	2	yes	—
146	C₂²×C₈₃	#390	1	1	yes	characteristic
147	C₂²×C₈₃	#397	1	1	yes	characteristic
148	C₂²×C₈₃	#398	1	1	yes	characteristic
149	C₂²×C₈₃	#400	1	2	yes	—
150	C₂²×C₈₃	#443	1	2	yes	—
151	C₂²×C₈₃	#444	1	2	yes	—
152	C₄×C₈₃	—	4	4	no	—
153	C₄×C₈₃	—	8	2	no	—
154	C₄×C₈₃	—	8	2	no	—
155	C₄×C₈₃	#381	1	1	yes	characteristic
156	C₄×C₈₃	#387	1	2	yes	—
157	C₄×C₈₃	#391	1	1	yes	characteristic
158	C₄×C₈₃	#397	1	1	yes	characteristic
159	C₄×C₈₃	#398	1	1	yes	characteristic
160	C₄×C₈₃	#399	1	2	yes	—
161	#11376	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂×C₈₃	—	2	1	no	—
163	C₂×C₈₃	—	2	1	no	—
164	C₂×C₈₃	—	2	2	no	—
165	C₂×C₈₃	#1445	1	1	yes	characteristic
166	C₂×C₈₃	#1922	1	4	yes	—
167	C₂×C₈₃	#1974	1	1	yes	characteristic
168	C₂×C₈₃	#1975	1	1	yes	characteristic
169	C₂×#427	C₂×C₈₃	1	1	yes	characteristic
170	C₂×#341	C₂×C₈₃	1	1	yes	characteristic
171	C₂×#339	C₂×C₈₃	1	2	yes	—
172	#1729	C₂×C₈₃	1	2	yes	—
173	C₂×#336	C₂×C₈₃	1	1	yes	characteristic
174	C₈₃	#11376	1	1	yes	characteristic, direct product factor, Sylow
175	C₂²×M₁₆	C₂²×C₈₃	1	2	yes	—
176	C₂³×C₈	C₂²×C₈₃	1	1	yes	characteristic
177	C₂³×C₈	C₂²×C₈₃	1	1	yes	characteristic
178	C₂×#117	C₂²×C₈₃	1	2	yes	—
179	C₂²×C₄²	C₂²×C₈₃	1	1	yes	characteristic
180	#427	—	2	2	no	—
181	#427	—	2	2	no	—
182	C₂×#107	C₄×C₈₃	1	1	yes	characteristic, complemented
183	C₂×#107	C₄×C₈₃	1	1	yes	characteristic, complemented
184	C₂×#106	C₄×C₈₃	1	2	yes	complemented
185	#341	—	2	2	no	—
186	#339	—	2	4	no	—
187	#336	—	2	2	no	—
188	#336	—	2	4	no	—
189	C₂³×C₄	—	2	1	no	—
190	C₂³×C₄	C₂³×C₈₃	1	1	yes	characteristic
191	C₂×M₁₆	—	2	2	no	—
192	C₂×M₁₆	—	2	2	no	—
193	C₂×M₁₆	—	2	2	no	—
194	C₂×M₁₆	—	2	2	no	—
195	C₂×M₁₆	—	2	2	no	—
196	C₂×M₁₆	Q₈×C₈₃	1	4	yes	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	2	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	—	2	2	no	—
204	C₂²×C₈	C₂×C₄×C₈₃	1	1	yes	characteristic
205	C₂²×C₈	C₂×C₄×C₈₃	1	1	yes	characteristic
206	C₂²×C₈	D₄×C₈₃	1	2	yes	—
207	C₂²×C₈	D₄×C₈₃	1	2	yes	—
208	C₂²×C₈	D₄×C₈₃	1	2	yes	—
209	#117	—	2	2	no	—
210	#117	—	2	2	no	—
211	#117	—	2	2	no	—
212	#117	—	2	2	no	—
213	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂×C₄×C₈₃	1	2	yes	—
215	C₂×C₂²⋊C₄	—	2	2	no	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	—	2	2	no	—
218	C₂×C₄²	—	2	2	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	D₄×C₈₃	1	2	yes	—
221	#107	—	2	2	no	—
222	#107	—	2	2	no	—
223	#106	—	2	4	no	—
224	C₂⁴	#41×C₈₃	1	1	yes	characteristic
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	4	2	no	—
230	C₂²×C₄	—	4	2	no	—
231	C₂²×C₄	C₂²×C₄×C₈₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×Q₈×C₈₃	1	2	yes	—
237	C₂²×C₄	#41×C₈₃	1	2	yes	—
238	C₂²×C₄	#41×C₈₃	1	2	yes	—
239	C₂²×C₄	#41×C₈₃	1	1	yes	characteristic
240	C₂²×C₄	#41×C₈₃	1	2	yes	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	4	2	no	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	2	2	no	—
247	C₂×C₈	—	2	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	2	4	no	—
250	C₂×C₈	—	2	2	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	2	4	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	2	no	—
257	C₂×C₈	—	4	2	no	—
258	C₂×C₈	C₂²⋊C₄×C₈₃	1	2	yes	—
259	C₂×C₈	C₂²⋊C₄×C₈₃	1	2	yes	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₄²	—	2	4	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	2	no	—
271	C₂³	—	2	2	no	—
272	C₂³	—	2	2	no	—
273	C₂³	C₄×D₄×C₈₃	1	1	yes	characteristic
274	C₂³	#121×C₈₃	1	1	yes	characteristic
275	C₂³	#124×C₈₃	1	1	yes	characteristic
276	C₂³	#125×C₈₃	1	2	yes	—
277	C₂³	#126×C₈₃	1	2	yes	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	4	8	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
287	C₂×C₄	#117×C₈₃	1	2	yes	—
288	C₂×C₄	C₄×D₄×C₈₃	1	1	yes	characteristic
289	C₂×C₄	C₄×Q₈×C₈₃	1	2	yes	—
290	C₂×C₄	#121×C₈₃	1	1	yes	characteristic
291	C₂×C₄	C₂²⋊Q₈×C₈₃	1	2	yes	—
292	C₂×C₄	C₂²⋊Q₈×C₈₃	1	2	yes	—
293	C₂×C₄	C₂²⋊Q₈×C₈₃	1	2	yes	—
294	C₂×C₄	#123×C₈₃	1	1	yes	characteristic
295	C₂×C₄	#123×C₈₃	1	1	yes	characteristic
296	C₂×C₄	#123×C₈₃	1	2	yes	—
297	C₂×C₄	#123×C₈₃	1	2	yes	—
298	C₂×C₄	#123×C₈₃	1	2	yes	—
299	C₂×C₄	#124×C₈₃	1	1	yes	characteristic
300	C₂×C₄	#125×C₈₃	1	2	yes	—
301	C₂×C₄	#126×C₈₃	1	2	yes	—
302	C₂×C₄	#127×C₈₃	1	1	yes	characteristic
303	C₂×C₄	C₄⋊Q₈×C₈₃	1	1	yes	characteristic
304	C₈	—	2	2	no	—
305	C₈	—	2	2	no	—
306	C₈	—	4	2	no	—
307	C₈	—	4	4	no	—
308	C₂²	—	2	4	no	—
309	C₂²	—	2	4	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	#387×C₈₃	1	2	yes	—
315	C₂²	#390×C₈₃	1	1	yes	characteristic
316	C₂²	#397×C₈₃	1	1	yes	characteristic
317	C₂²	#398×C₈₃	1	1	yes	characteristic
318	C₂²	#400×C₈₃	1	2	yes	—
319	C₂²	#443×C₈₃	1	2	yes	—
320	C₂²	#444×C₈₃	1	2	yes	—
321	C₄	—	4	4	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	#381×C₈₃	1	1	yes	characteristic
325	C₄	#387×C₈₃	1	2	yes	—
326	C₄	#391×C₈₃	1	1	yes	characteristic
327	C₄	#397×C₈₃	1	1	yes	characteristic
328	C₄	#398×C₈₃	1	1	yes	characteristic
329	C₄	#399×C₈₃	1	2	yes	—
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	2	no	—
333	C₂	C₈₃×#1445	1	1	yes	characteristic
334	C₂	C₈₃×#1922	1	4	yes	—
335	C₂	C₈₃×#1974	1	1	yes	characteristic
336	C₂	C₈₃×#1975	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₈₃	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₈₃
Frattini, cosocle	C₂³×C₄	C₂³×C₈₃
socle	C₂³×C₈₃	C₄×D₄
prime radical	C₂⁴×C₈₃	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₃ ◁ C₂³×C₈×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#107×C₈₃	C₄	semidirect
14. C₂×#107×C₈₃	C₄	semidirect
15. C₂×#106×C₈₃	C₄	semidirect
161. #11376	C₈₃	direct (N × H)
174. C₈₃	#11376	direct (N × H)
182. C₂×#107	C₄×C₈₃	semidirect
183. C₂×#107	C₄×C₈₃	semidirect
184. C₂×#106	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×#427×C₈₃	2
C₂×#341×C₈₃	2
C₂×#339×C₈₃	2
C₈₃×#1729	2
C₂×#336×C₈₃	2
#11376	83

Subgroups — C83×#11376 #35239200

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#11376 #35239200