Subgroups — #35395932

Order 21248.

606 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24920331	C₂	1	1	yes	maximal, characteristic, complemented
2	#24919811	C₂	1	1	yes	maximal, characteristic
3	#24919781	C₂	1	2	yes	maximal, complemented
4	#24907073	C₂	1	1	yes	maximal, characteristic, complemented
5	#24907039	C₂	1	2	yes	maximal, complemented
6	#24906862	C₂	1	1	yes	maximal, characteristic, complemented
7	#24906857	C₂	1	1	yes	maximal, characteristic, complemented
8	#24906679	C₂	1	2	yes	maximal, complemented
9	#24906616	C₂	1	2	yes	maximal, complemented
10	#24906349	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128411	C₂²	1	2	yes	—
13	#19128355	C₂²	1	2	yes	—
14	#19128106	C₂²	1	1	yes	characteristic
15	#19128106	C₂²	1	1	yes	characteristic
16	#19128103	—	2	1	no	—
17	#19128052	C₂²	1	2	yes	—
18	#19128052	C₂²	1	2	yes	—
19	#19128033	—	2	2	no	—
20	#19128015	—	2	2	no	—
21	#19128002	—	2	2	no	—
22	#19127992	C₂²	1	2	yes	—
23	#19127991	C₂²	1	2	yes	—
24	#19127982	C₂²	1	2	yes	—
25	#19127958	—	2	2	no	—
26	#19127937	—	2	1	no	—
27	#19127911	C₂²	1	2	yes	—
28	#19127860	—	2	2	no	—
29	C₄×D₆₆₄	—	2	1	no	—
30	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
31	#19127780	C₂²	1	2	yes	—
32	C₂×#14581726	—	2	1	no	—
33	#19127699	—	2	2	no	—
34	#19127454	C₂²	1	1	yes	characteristic
35	#19127449	C₂²	1	2	yes	—
36	#19127446	C₂²	1	2	yes	—
37	#19127430	C₂²	1	2	yes	—
38	C₈×Q₈×C₈₃	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₈₃	C₂²	1	1	yes	characteristic
41	#334×C₈₃	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
43	D₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
44	C₈²×C₈₃	C₂²	1	1	yes	characteristic
45	C₂×#13050589	—	2	2	no	—
46	C₂×#13050580	—	2	2	no	—
47	C₂×D₆₆₄	D₄	1	1	yes	characteristic
48	C₂×#13050573	—	2	2	no	—
49	#14581792	C₂³	1	2	yes	—
50	#14581792	C₂³	1	2	yes	—
51	#14581784	—	2	2	no	—
52	#14581776	—	4	2	no	—
53	#14581772	—	2	2	no	—
54	#14581770	—	4	2	no	—
55	#14581768	—	2	2	no	—
56	#14581766	—	2	2	no	—
57	C₄×D₃₃₂	—	2	2	no	—
58	C₄×D₃₃₂	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₈₃	—	2	2	no	—
60	C₄⋊C₄×D₈₃	—	2	2	no	—
61	#14581726	—	2	2	no	—
62	#14581726	—	2	1	no	—
63	#14581726	—	2	1	no	—
64	#14581726	—	2	1	no	—
65	#14581726	—	2	1	no	—
66	#14581726	D₄	1	1	yes	characteristic
67	#14581724	—	2	2	no	—
68	#14581723	—	2	2	no	—
69	C₄⋊(C₈₃⋊C₈)	C₂³	1	2	yes	—
70	#14581717	—	2	2	no	—
71	#14581717	—	2	2	no	—
72	#14581717	—	4	2	no	—
73	#14581715	—	2	2	no	—
74	#14581714	—	2	2	no	—
75	#14581714	—	2	2	no	—
76	#14581714	D₄	1	2	yes	—
77	#14581713	—	2	2	no	—
78	(C₂×D₈₃)⋊C₈	—	2	2	no	—
79	C₂×SD₁₆×C₈₃	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₈₃	—	2	1	no	—
81	C₄×Q₈×C₈₃	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₈₃	C₂³	1	1	yes	characteristic
83	#106×C₈₃	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₈₃	—	2	1	no	—
85	C₄⋊C₈×C₈₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₈₃	C₂³	1	1	yes	characteristic
87	#103×C₈₃	D₄	1	1	yes	characteristic
88	#102×C₈₃	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₈₃	—	2	1	no	—
90	C₄×C₈×C₈₃	—	2	1	no	—
91	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
94	#13050589	—	4	4	no	—
95	#13050589	—	8	2	no	—
96	C₂×Dic₁₆₆	—	2	2	no	—
97	C₂×Dic₁₆₆	—	2	2	no	—
98	C₂×D₃₃₂	C₂×D₄	1	2	yes	—
99	C₂×C₂²⋊D₈₃	—	4	2	no	—
100	C₂²×Dic₈₃	—	4	2	no	—
101	C₂×C₄×D₈₃	—	2	2	no	—
102	C₂×C₄×D₈₃	—	4	2	no	—
103	#13050580	—	8	2	no	—
104	D₆₆₄	—	2	2	no	—
105	#13050573	—	8	2	no	—
106	C₂×C₈₃⋊C₈	—	2	2	no	—
107	C₄⋊Dic₈₃	—	2	2	no	—
108	C₄⋊Dic₈₃	—	2	2	no	—
109	C₄⋊Dic₈₃	—	2	2	no	—
110	C₄⋊Dic₈₃	—	2	2	no	—
111	C₄⋊Dic₈₃	—	4	2	no	—
112	C₄⋊Dic₈₃	C₂×D₄	1	2	yes	—
113	Dic₈₃⋊C₄	—	4	2	no	—
114	Dic₈₃⋊C₄	—	4	2	no	—
115	(C₂×D₈₃)⋊C₄	—	2	2	no	—
116	(C₂×D₈₃)⋊C₄	—	4	2	no	—
117	(C₂×D₈₃)⋊C₄	—	4	2	no	—
118	C₂×Q₈×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×D₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₈₃	—	2	1	no	—
121	SD₁₆×C₈₃	—	2	2	no	—
122	C₂×C₈×C₈₃	—	2	1	no	—
123	C₂×C₈×C₈₃	—	2	1	no	—
124	C₂×C₈×C₈₃	—	2	1	no	—
125	C₂×C₈×C₈₃	—	4	2	no	—
126	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₈₃	—	2	1	no	—
131	C₄⋊C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
133	C₂²⋊C₄×C₈₃	—	2	1	no	—
134	C₄²×C₈₃	—	2	1	no	—
135	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
136	C₂³×C₈₃	—	2	1	no	—
137	C₂²×D₈₃	—	2	2	no	—
138	Q₈×C₈₃	—	2	1	no	—
139	Q₈×C₈₃	—	2	1	no	—
140	D₄×C₈₃	—	2	1	no	—
141	D₄×C₈₃	#135	1	2	yes	—
142	C₂×C₄×C₈₃	—	2	1	no	—
143	C₂×C₄×C₈₃	—	2	1	no	—
144	C₂×C₄×C₈₃	—	2	1	no	—
145	C₂×C₄×C₈₃	—	4	1	no	—
146	C₂×C₄×C₈₃	—	4	1	no	—
147	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²⋊D₈₃	—	8	4	no	—
151	C₂×Dic₈₃	—	2	2	no	—
152	C₂×Dic₈₃	—	4	2	no	—
153	C₂×Dic₈₃	—	4	2	no	—
154	C₂×Dic₈₃	—	8	4	no	—
155	D₃₃₂	—	2	2	no	—
156	D₃₃₂	—	2	2	no	—
157	C₄×D₈₃	—	4	2	no	—
158	C₄×D₈₃	—	4	2	no	—
159	C₄×D₈₃	—	16	2	no	—
160	Dic₁₆₆	—	4	2	no	—
161	Dic₁₆₆	—	4	2	no	—
162	Dic₁₆₆	—	8	2	no	—
163	C₈×C₈₃	—	2	1	no	—
164	C₈×C₈₃	—	2	1	no	—
165	C₈×C₈₃	—	2	1	no	—
166	C₈×C₈₃	—	4	1	no	—
167	C₈×C₈₃	—	4	1	no	—
168	C₈×C₈₃	—	4	1	no	—
169	C₈×C₈₃	#137	1	2	yes	—
170	C₈₃⋊C₈	—	4	2	no	—
171	C₂²×C₈₃	—	2	2	no	—
172	C₂²×C₈₃	—	4	1	no	—
173	C₂²×C₈₃	—	4	1	no	—
174	C₂²×C₈₃	D₄²	1	1	yes	characteristic
175	C₂×D₈₃	—	4	2	no	—
176	C₂×D₈₃	—	4	2	no	—
177	C₂×D₈₃	—	4	2	no	—
178	C₄×C₈₃	—	2	1	no	—
179	C₄×C₈₃	—	4	1	no	—
180	C₄×C₈₃	—	4	1	no	—
181	C₄×C₈₃	—	4	1	no	—
182	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
183	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
184	C₄×C₈₃	#576	1	1	yes	characteristic
185	C₄×C₈₃	#578	1	1	yes	characteristic
186	Dic₈₃	—	4	2	no	—
187	Dic₈₃	—	8	2	no	—
188	Dic₈₃	—	8	2	no	—
189	#19903	—	83	1	no	maximal, Sylow
190	C₂×C₈₃	—	4	2	no	—
191	C₂×C₈₃	#3276	1	1	yes	characteristic
192	C₂×C₈₃	#3280	1	1	yes	characteristic
193	C₂×C₈₃	#3290	1	1	yes	characteristic
194	D₈₃	—	8	2	no	—
195	#3380	—	83	1	no	—
196	#3334	—	83	1	no	—
197	#3331	—	83	2	no	—
198	#1710	—	83	1	no	—
199	#1703	—	83	2	no	—
200	#1675	—	83	1	no	—
201	#1674	—	83	1	no	—
202	#1644	—	83	2	no	—
203	#1635	—	83	2	no	—
204	#1589	—	83	1	no	—
205	C₈×SD₁₆	D₈₃	1	1	yes	characteristic, complemented
206	C₈₃	#19903	1	1	yes	characteristic, complemented, Sylow
207	#554	—	83	2	no	—
208	#547	—	83	2	no	—
209	#500	—	83	1	no	—
210	#500	—	83	1	no	—
211	#499	—	166	1	no	—
212	#487	—	83	2	no	—
213	#487	—	83	2	no	—
214	#483	—	166	2	no	—
215	#480	—	166	2	no	—
216	#478	—	166	2	no	—
217	#476	—	83	2	no	—
218	#476	—	83	2	no	—
219	#474	—	83	2	no	—
220	#470	—	166	2	no	—
221	#466	—	166	1	no	—
222	#462	—	83	2	no	—
223	#452	—	166	2	no	—
224	C₈×Q₈	C₂×D₈₃	1	1	yes	characteristic
225	C₄×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
226	C₄×D₈	—	83	1	no	—
227	C₄×D₈	—	166	1	no	—
228	#436	—	83	2	no	—
229	C₈×D₄	C₂×D₈₃	1	1	yes	characteristic
230	C₂×#107	—	166	1	no	—
231	#417	—	166	2	no	—
232	#335	—	83	1	no	—
233	#334	C₂×D₈₃	1	1	yes	characteristic
234	#333	—	83	2	no	—
235	#332	—	83	2	no	—
236	Q₈⋊C₈	C₂×D₈₃	1	1	yes	characteristic
237	D₄⋊C₈	—	83	2	no	—
238	D₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
239	C₈²	C₂×D₈₃	1	1	yes	characteristic
240	C₂×#41	—	166	2	no	—
241	C₂×Q₁₆	—	166	2	no	—
242	C₂×SD₁₆	D₃₃₂	1	1	yes	characteristic
243	C₂×D₈	—	83	1	no	—
244	C₂×M₁₆	—	166	2	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	83	2	no	—
247	C₄⋊Q₈	—	83	2	no	—
248	#125	—	166	2	no	—
249	#123	—	332	2	no	—
250	C₂²⋊Q₈	—	332	2	no	—
251	C₂²⋊Q₈	—	166	2	no	—
252	C₂²⋊Q₈	—	166	2	no	—
253	#121	—	166	2	no	—
254	C₄×Q₈	C₂²×D₈₃	1	1	yes	characteristic
255	C₄×D₄	—	83	2	no	—
256	C₄×D₄	—	166	2	no	—
257	C₄×D₄	C₂²×D₈₃	1	1	yes	characteristic
258	C₂×C₄⋊C₄	—	166	2	no	—
259	C₂×C₄⋊C₄	—	166	2	no	—
260	#107	—	83	1	no	—
261	#107	—	166	2	no	—
262	#107	—	166	1	no	—
263	#107	—	166	1	no	—
264	#107	—	166	1	no	—
265	#107	—	166	1	no	—
266	#106	—	166	2	no	—
267	#106	—	166	2	no	—
268	#106	D₃₃₂	1	1	yes	characteristic
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	83	2	no	—
271	C₄⋊C₈	C₂²×D₈₃	1	1	yes	characteristic
272	C₄⋊C₈	C₂²×D₈₃	1	1	yes	characteristic
273	#103	—	332	2	no	—
274	#103	—	166	2	no	—
275	#103	—	166	2	no	—
276	#103	—	166	2	no	—
277	#103	D₃₃₂	1	1	yes	characteristic
278	#102	—	83	2	no	—
279	#102	—	166	2	no	—
280	#102	—	166	2	no	—
281	#102	—	166	2	no	—
282	#102	D₃₃₂	1	1	yes	characteristic
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	166	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
289	#41	—	332	4	no	—
290	#41	—	664	2	no	—
291	C₂×Q₈	—	166	2	no	—
292	C₂×Q₈	—	166	2	no	—
293	C₂×Q₈	C₂×D₃₃₂	1	1	yes	characteristic
294	C₂×D₄	—	332	2	no	—
295	C₂×D₄	—	83	2	no	—
296	C₂×D₄	C₂×D₃₃₂	1	1	yes	characteristic
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	332	2	no	—
299	C₂²×C₄	—	332	2	no	—
300	C₂²×C₄	—	166	2	no	—
301	Q₁₆	—	664	2	no	—
302	SD₁₆	—	2	2	no	—
303	D₈	—	166	2	no	—
304	M₁₆	—	664	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	166	2	no	—
310	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
311	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
312	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
313	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	332	2	no	—
316	C₄⋊C₄	—	332	2	no	—
317	C₄⋊C₄	—	332	2	no	—
318	C₄⋊C₄	—	83	2	no	—
319	C₄⋊C₄	—	166	2	no	—
320	C₄⋊C₄	—	166	2	no	—
321	C₄⋊C₄	—	166	2	no	—
322	C₄⋊C₄	—	166	2	no	—
323	C₄⋊C₄	C₂×D₃₃₂	1	1	yes	characteristic
324	C₄⋊C₄	C₂×D₃₃₂	1	1	yes	characteristic
325	C₂²⋊C₄	—	2	1	no	—
326	C₂²⋊C₄	—	332	2	no	—
327	C₂²⋊C₄	—	332	2	no	—
328	C₂²⋊C₄	—	166	2	no	—
329	C₄²	—	2	1	no	—
330	C₄²	C₂³×D₈₃	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	166	2	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	2	1	no	—
335	Q₈	—	332	2	no	—
336	Q₈	—	332	2	no	—
337	Q₈	—	664	2	no	—
338	D₄	—	2	1	no	—
339	D₄	—	664	4	no	—
340	D₄	—	166	2	no	—
341	D₄	—	166	2	no	—
342	D₄	#14581814	1	2	yes	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	1328	2	no	—
349	C₂×C₄	—	332	2	no	—
350	C₂×C₄	—	332	2	no	—
351	C₂×C₄	—	332	2	no	—
352	C₂×C₄	—	332	2	no	—
353	C₂×C₄	—	664	4	no	—
354	C₂×C₄	—	166	2	no	—
355	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
357	C₂×C₄	#14581838	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	332	2	no	—
365	C₈	#14581821	1	2	yes	—
366	C₂²	—	2	2	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	332	2	no	—
370	C₂²	—	332	2	no	—
371	C₂²	—	332	2	no	—
372	C₂²	D₄×D₃₃₂	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	4	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	332	2	no	—
378	C₄	—	664	2	no	—
379	C₄	—	664	2	no	—
380	C₄	C₂×#14581814	1	1	yes	characteristic
381	C₄	C₂×#14581821	1	1	yes	characteristic
382	C₄	#19128506	1	1	yes	characteristic
383	C₄	#19128515	1	1	yes	characteristic
384	C₂	—	4	2	no	—
385	C₂	—	664	2	no	—
386	C₂	#24919110	1	1	yes	characteristic
387	C₂	#24919150	1	1	yes	characteristic
388	C₂	#24919250	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₃₂
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈×SD₁₆×C₈₃	C₂
socle	C₂²×C₈₃	D₄²
prime radical	#24919811	C₂
hypercenter	C₈×SD₁₆	D₈₃
hypocenter	C₈₃	#19903
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #24920331	C₂	semidirect
3. #24919781	C₂	semidirect
4. #24907073	C₂	semidirect
5. #24907039	C₂	semidirect
6. #24906862	C₂	semidirect
7. #24906857	C₂	semidirect
8. #24906679	C₂	semidirect
9. #24906616	C₂	semidirect
10. #24906349	C₂	semidirect
11. C₈×SD₁₆×C₈₃	C₂	semidirect
205. C₈×SD₁₆	D₈₃	semidirect
206. C₈₃	#19903	semidirect

Maximal subgroups

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

subgroup	index
#24920331	2
#24919811	2
#24919781	2
#24907073	2
#24907039	2
#24906862	2
#24906857	2
#24906679	2
#24906616	2
#24906349	2
C₈×SD₁₆×C₈₃	2
#19903	83