Subgroups — #35557538

Order 21248.

524 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24915880	C₂	1	1	yes	maximal, characteristic
2	#24909278	C₂	1	2	yes	maximal, complemented
3	#24909277	C₂	1	2	yes	maximal, complemented
4	#24909275	C₂	1	1	yes	maximal, characteristic, complemented
5	#24909272	C₂	1	2	yes	maximal, complemented
6	#24909227	C₂	1	2	yes	maximal, complemented
7	#349×D₈₃	C₂	1	2	yes	maximal
8	#24909066	C₂	1	2	yes	maximal, complemented
9	C₈₃×#2165	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×M₁₆×D₈₃	C₄	1	1	yes	characteristic
11	C₂×M₁₆×D₈₃	C₂²	1	1	yes	characteristic
12	C₂×#14581749	C₄	1	1	yes	characteristic
13	M₃₂×D₈₃	C₂²	1	4	yes	—
14	#19127658	C₄	1	1	yes	characteristic
15	#19127658	C₄	1	1	yes	characteristic
16	#19127658	C₂²	1	1	yes	characteristic
17	#19127658	C₂²	1	1	yes	characteristic
18	#19127655	C₄	1	1	yes	characteristic
19	#19127655	C₄	1	1	yes	characteristic
20	#19127655	C₂²	1	1	yes	characteristic
21	#19127655	C₂²	1	1	yes	characteristic
22	C₄×#13050574	C₄	1	1	yes	characteristic
23	M₁₆×Dic₈₃	C₄	1	1	yes	characteristic
24	M₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
25	#19127522	C₂²	1	2	yes	complemented
26	#19127521	C₂²	1	4	yes	complemented
27	#19127483	C₂²	1	2	yes	—
28	#19127482	C₂²	1	4	yes	—
29	#19127479	C₂²	1	2	yes	complemented
30	#19127478	C₂²	1	4	yes	complemented
31	C₄×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
32	#364×C₈₃	C₂²	1	2	yes	complemented
33	#349×C₈₃	C₂²	1	2	yes	—
34	#347×C₈₃	C₂²	1	2	yes	complemented
35	C₂²×C₄×D₈₃	C₂×C₄	1	1	yes	characteristic
36	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
37	C₂×#13050574	C₂×C₄	1	1	yes	characteristic
38	C₂×#13050573	C₂×C₄	1	1	yes	characteristic
39	C₂×#13050573	C₂×C₄	1	1	yes	characteristic
40	C₂×#13050573	C₂×C₄	1	1	yes	characteristic
41	C₂×#13050573	C₂³	1	1	yes	characteristic
42	M₁₆×D₈₃	—	2	4	no	—
43	M₁₆×D₈₃	—	2	4	no	—
44	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₈×D₈₃	C₂³	1	1	yes	characteristic
48	#14581749	—	2	1	no	—
49	#14581749	—	2	1	no	—
50	#14581749	C₂×C₄	1	1	yes	characteristic
51	#14581749	C₂×C₄	1	1	yes	characteristic
52	#14581749	C₂×C₄	1	1	yes	characteristic
53	#14581749	C₂×C₄	1	1	yes	characteristic
54	C₂×Dic₈₃⋊C₄	—	2	1	no	—
55	C₂×(C₂×D₈₃)⋊C₄	—	2	1	no	—
56	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
57	#14581732	C₂³	1	4	yes	—
58	#14581731	—	2	4	no	—
59	C₁₆×D₈₃	—	2	4	no	—
60	#14581703	C₂×C₄	1	1	yes	characteristic
61	#14581703	C₂×C₄	1	1	yes	characteristic
62	#14581702	C₂×C₄	1	1	yes	characteristic
63	#14581702	C₂×C₄	1	1	yes	characteristic
64	#14581702	C₂×C₄	1	1	yes	characteristic
65	#14581702	C₂³	1	1	yes	characteristic
66	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
67	C₄×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
68	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
69	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
70	C₈×Dic₈₃	C₂×C₄	1	1	yes	characteristic
71	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
72	C₂×M₁₆×C₈₃	C₂×C₄	1	1	yes	characteristic
73	C₂×M₁₆×C₈₃	C₂³	1	1	yes	characteristic
74	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
75	M₃₂×C₈₃	C₂³	1	4	yes	—
76	#97×C₈₃	C₂×C₄	1	1	yes	characteristic
77	#97×C₈₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
79	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
80	C₂³×D₈₃	M₁₆	1	1	yes	characteristic
81	C₂²×Dic₈₃	—	2	1	no	—
82	C₂²×Dic₈₃	M₁₆	1	1	yes	characteristic
83	C₂×C₄×D₈₃	—	2	2	no	—
84	C₂×C₄×D₈₃	—	2	2	no	—
85	C₂×C₄×D₈₃	M₁₆	1	1	yes	characteristic
86	C₂×C₄×D₈₃	M₁₆	1	1	yes	characteristic
87	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×D₈₃	C₂²×C₄	1	1	yes	characteristic
89	#13050574	—	2	2	no	—
90	#13050574	—	2	2	no	—
91	#13050573	—	2	2	no	—
92	#13050573	—	2	2	no	—
93	#13050573	—	2	2	no	—
94	#13050573	M₁₆	1	4	yes	—
95	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
99	C₈×D₈₃	—	2	2	no	—
100	C₈×D₈₃	—	2	2	no	—
101	C₈×D₈₃	—	2	2	no	—
102	C₈×D₈₃	#41	1	4	yes	—
103	Dic₈₃⋊C₄	—	2	1	no	—
104	Dic₈₃⋊C₄	—	2	1	no	—
105	Dic₈₃⋊C₄	—	2	1	no	—
106	Dic₈₃⋊C₄	—	2	1	no	—
107	(C₂×D₈₃)⋊C₄	—	2	1	no	—
108	(C₂×D₈₃)⋊C₄	—	2	1	no	—
109	(C₂×D₈₃)⋊C₄	—	2	1	no	—
110	(C₂×D₈₃)⋊C₄	—	2	1	no	—
111	C₄×Dic₈₃	—	2	1	no	—
112	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
113	C₄×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
114	C₈₃⋊C₁₆	—	2	4	no	—
115	C₂²×C₄×C₈₃	—	2	1	no	—
116	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
117	M₁₆×C₈₃	—	2	2	no	—
118	M₁₆×C₈₃	—	2	2	no	—
119	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
123	C₄²×C₈₃	—	2	1	no	—
124	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
126	C₁₆×C₈₃	—	2	4	no	—
127	C₂³×C₈₃	C₂×M₁₆	1	1	yes	characteristic
128	C₂²×D₈₃	—	2	2	no	—
129	C₂²×D₈₃	—	8	1	no	—
130	C₂²×D₈₃	C₂×M₁₆	1	1	yes	characteristic
131	C₂²×D₈₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₈₃	—	2	1	no	—
133	C₂×C₄×C₈₃	—	2	1	no	—
134	C₂×C₄×C₈₃	—	2	1	no	—
135	C₂×C₄×C₈₃	—	2	1	no	—
136	C₂×C₄×C₈₃	—	8	1	no	—
137	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
138	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×Dic₈₃	—	2	2	no	—
140	C₂×Dic₈₃	—	2	1	no	—
141	C₂×Dic₈₃	—	2	1	no	—
142	C₂×Dic₈₃	—	8	1	no	—
143	C₂×Dic₈₃	C₂×M₁₆	1	1	yes	characteristic
144	C₂×Dic₈₃	C₂×M₁₆	1	1	yes	characteristic
145	C₄×D₈₃	—	2	2	no	—
146	C₄×D₈₃	—	2	1	no	—
147	C₄×D₈₃	—	2	1	no	—
148	C₄×D₈₃	—	2	2	no	—
149	C₄×D₈₃	#117	1	2	yes	—
150	C₄×D₈₃	C₂×M₁₆	1	2	yes	—
151	C₈×C₈₃	—	2	1	no	—
152	C₈×C₈₃	—	2	1	no	—
153	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
154	C₈×C₈₃	C₂×#41	1	2	yes	—
155	C₈₃⋊C₈	—	2	1	no	—
156	C₈₃⋊C₈	—	2	1	no	—
157	C₈₃⋊C₈	#117	1	2	yes	—
158	C₈₃⋊C₈	C₂×M₁₆	1	2	yes	—
159	C₂²×C₈₃	—	2	1	no	—
160	C₂²×C₈₃	—	4	1	no	—
161	C₂²×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
162	C₂×D₈₃	—	2	1	no	—
163	C₂×D₈₃	—	4	1	no	—
164	C₂×D₈₃	—	4	1	no	—
165	C₂×D₈₃	—	8	2	no	—
166	C₂×D₈₃	#432	1	2	yes	—
167	C₄×C₈₃	—	2	1	no	—
168	C₄×C₈₃	—	4	1	no	—
169	C₄×C₈₃	—	4	1	no	—
170	C₄×C₈₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
172	Dic₈₃	—	2	1	no	—
173	Dic₈₃	—	4	1	no	—
174	Dic₈₃	—	4	1	no	—
175	Dic₈₃	#432	1	2	yes	—
176	#33427	—	83	1	no	maximal, Sylow
177	C₂×C₈₃	—	2	1	no	—
178	C₂×C₈₃	—	4	1	no	—
179	C₂×C₈₃	C₂×#432	1	1	yes	characteristic
180	D₈₃	—	2	2	no	—
181	D₈₃	—	4	1	no	—
182	#2873	—	83	1	no	—
183	#2166	—	83	2	no	—
184	#2166	—	83	2	no	—
185	#2165	—	83	1	no	—
186	#2165	—	83	2	no	—
187	#2165	D₈₃	1	1	yes	characteristic, complemented
188	#2153	—	83	2	no	—
189	C₂×#349	—	83	2	no	—
190	#2110	—	83	2	no	—
191	C₈₃	#33427	1	1	yes	characteristic, complemented, Sylow
192	C₂²×M₁₆	—	83	1	no	—
193	C₂²×M₁₆	—	83	1	no	—
194	C₂×#117	—	83	1	no	—
195	C₂×M₃₂	—	83	4	no	—
196	#405	—	83	1	no	—
197	#405	—	83	1	no	—
198	#405	—	83	1	no	—
199	#405	—	83	1	no	—
200	#405	—	83	1	no	—
201	#405	—	83	1	no	—
202	#405	—	83	1	no	—
203	#405	—	83	1	no	—
204	C₄×M₁₆	—	83	1	no	—
205	C₄×M₁₆	—	83	1	no	—
206	C₄×M₁₆	—	83	1	no	—
207	C₄×M₁₆	C₂×D₈₃	1	1	yes	characteristic
208	#364	—	83	4	no	—
209	#364	—	83	2	no	—
210	#364	C₂×D₈₃	1	2	yes	complemented
211	#349	—	83	2	no	—
212	#349	—	83	4	no	—
213	#349	C₂×D₈₃	1	2	yes	—
214	#347	—	83	4	no	—
215	#347	—	83	2	no	—
216	#347	C₂×D₈₃	1	2	yes	complemented
217	C₂³×C₄	—	83	1	no	—
218	C₂×M₁₆	—	83	1	no	—
219	C₂×M₁₆	—	83	1	no	—
220	C₂×M₁₆	—	83	1	no	—
221	C₂×M₁₆	—	83	1	no	—
222	C₂×M₁₆	—	83	1	no	—
223	C₂×M₁₆	—	83	1	no	—
224	C₂×M₁₆	—	166	4	no	—
225	C₂×M₁₆	—	166	4	no	—
226	C₂×M₁₆	C₄×D₈₃	1	1	yes	characteristic
227	C₂×M₁₆	C₂²×D₈₃	1	1	yes	characteristic
228	C₂²×C₈	—	83	1	no	—
229	C₂²×C₈	—	83	1	no	—
230	C₂²×C₈	—	83	1	no	—
231	C₂²×C₈	—	83	1	no	—
232	#117	—	83	1	no	—
233	#117	—	83	1	no	—
234	#117	—	83	1	no	—
235	#117	—	83	1	no	—
236	#117	—	166	1	no	—
237	#117	—	166	1	no	—
238	C₂×C₄⋊C₄	—	166	1	no	—
239	C₂×C₂²⋊C₄	—	166	1	no	—
240	C₂×C₄²	—	83	1	no	—
241	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
242	M₃₂	—	83	4	no	—
243	M₃₂	—	166	4	no	—
244	M₃₂	C₂²×D₈₃	1	4	yes	—
245	C₂×C₁₆	—	166	4	no	—
246	#97	—	83	1	no	—
247	#97	—	83	1	no	—
248	#97	—	83	1	no	—
249	#97	—	83	1	no	—
250	#97	—	83	1	no	—
251	#97	—	83	1	no	—
252	#97	C₄×D₈₃	1	1	yes	characteristic
253	#97	C₂²×D₈₃	1	1	yes	characteristic
254	C₄×C₈	—	83	1	no	—
255	C₄×C₈	—	83	1	no	—
256	C₄×C₈	—	83	1	no	—
257	C₄×C₈	—	83	1	no	—
258	C₄×C₈	—	83	1	no	—
259	C₄×C₈	—	83	1	no	—
260	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
261	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₂⁴	—	83	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	—	83	1	no	—
265	C₂²×C₄	—	83	1	no	—
266	C₂²×C₄	—	83	1	no	—
267	C₂²×C₄	—	83	1	no	—
268	C₂²×C₄	—	83	1	no	—
269	C₂²×C₄	—	166	2	no	—
270	C₂²×C₄	—	166	2	no	—
271	C₂²×C₄	—	166	1	no	—
272	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	83	4	no	—
276	M₁₆	—	166	2	no	—
277	M₁₆	—	166	2	no	—
278	M₁₆	—	166	2	no	—
279	M₁₆	—	166	2	no	—
280	M₁₆	—	166	2	no	—
281	C₂×C₈	—	83	1	no	—
282	C₂×C₈	—	83	1	no	—
283	C₂×C₈	—	83	4	no	—
284	C₂×C₈	—	83	1	no	—
285	C₂×C₈	—	83	1	no	—
286	C₂×C₈	—	166	2	no	—
287	C₂×C₈	—	166	2	no	—
288	C₂×C₈	—	166	2	no	—
289	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
290	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
292	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
293	C₄⋊C₄	—	166	1	no	—
294	C₄⋊C₄	—	166	1	no	—
295	C₄⋊C₄	—	166	1	no	—
296	C₄⋊C₄	—	166	1	no	—
297	C₂²⋊C₄	—	166	1	no	—
298	C₂²⋊C₄	—	166	1	no	—
299	C₂²⋊C₄	—	166	1	no	—
300	C₂²⋊C₄	—	166	1	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	83	1	no	—
303	C₄²	—	83	1	no	—
304	C₄²	—	166	1	no	—
305	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
306	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
307	C₁₆	—	2	4	no	—
308	C₁₆	—	166	4	no	—
309	C₂³	—	83	1	no	—
310	C₂³	—	83	1	no	—
311	C₂³	—	664	1	no	—
312	C₂³	—	166	2	no	—
313	C₂³	M₁₆×D₈₃	1	1	yes	characteristic
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	8	1	no	—
319	C₂×C₄	—	83	1	no	—
320	C₂×C₄	—	83	2	no	—
321	C₂×C₄	—	83	2	no	—
322	C₂×C₄	—	83	1	no	—
323	C₂×C₄	—	664	1	no	—
324	C₂×C₄	—	166	2	no	—
325	C₂×C₄	—	166	2	no	—
326	C₂×C₄	—	166	1	no	—
327	C₂×C₄	—	166	1	no	—
328	C₂×C₄	—	166	2	no	—
329	C₂×C₄	—	166	1	no	—
330	C₂×C₄	—	166	1	no	—
331	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	83	2	no	—
336	C₈	—	83	2	no	—
337	C₈	—	166	1	no	—
338	C₈	—	166	1	no	—
339	C₈	M₁₆×D₈₃	1	2	yes	—
340	C₈	#41×D₈₃	1	2	yes	—
341	C₂²	—	2	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	332	1	no	—
344	C₂²	—	332	1	no	—
345	C₂²	—	83	2	no	—
346	C₂²	—	664	2	no	—
347	C₂²	—	166	1	no	—
348	C₂²	C₂×M₁₆×D₈₃	1	1	yes	characteristic
349	C₄	—	2	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	1	no	—
352	C₄	—	332	1	no	—
353	C₄	—	332	1	no	—
354	C₄	—	83	2	no	—
355	C₄	—	166	1	no	—
356	C₄	#117×D₈₃	1	1	yes	characteristic
357	C₄	C₂×M₁₆×D₈₃	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	4	1	no	—
360	C₂	—	332	1	no	—
361	C₂	—	166	2	no	—
362	C₂	#432×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#117×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2165	C₂
socle	C₂×C₈₃	C₂×#432
prime radical	C₂³×D₈₃	M₁₆
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#2165	D₈₃
hypocenter	C₈₃	#33427
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ #97 ◁ #2165

Complements

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

N	H ≅ G/N	type
2. #24909278	C₂	semidirect
3. #24909277	C₂	semidirect
4. #24909275	C₂	semidirect
5. #24909272	C₂	semidirect
6. #24909227	C₂	semidirect
8. #24909066	C₂	semidirect
9. C₈₃×#2165	C₂	semidirect
25. #19127522	C₂²	semidirect
26. #19127521	C₂²	semidirect
29. #19127479	C₂²	semidirect
30. #19127478	C₂²	semidirect
32. #364×C₈₃	C₂²	semidirect
34. #347×C₈₃	C₂²	semidirect
187. #2165	D₈₃	semidirect
191. C₈₃	#33427	semidirect
210. #364	C₂×D₈₃	semidirect
216. #347	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24915880	2
#24909278	2
#24909277	2
#24909275	2
#24909272	2
#24909227	2
#349×D₈₃	2
#24909066	2
C₈₃×#2165	2
#33427	83