Subgroups — #35395978

Order 21248.

602 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24920452	C₂	1	1	yes	maximal, characteristic
2	#24920315	C₂	1	1	yes	maximal, characteristic
3	#24919749	C₂	1	2	yes	maximal, complemented
4	#24907061	C₂	1	1	yes	maximal, characteristic, complemented
5	#24907009	C₂	1	2	yes	maximal, complemented
6	#24906834	C₂	1	1	yes	maximal, characteristic, complemented
7	#24906731	C₂	1	2	yes	maximal, complemented
8	#24906727	C₂	1	2	yes	maximal, complemented
9	#24906347	C₂	1	1	yes	maximal, characteristic
10	C₈×#13050580	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₃×#1674	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128424	C₂²	1	2	yes	—
13	#19128366	C₂²	1	2	yes	—
14	#19128105	—	2	1	no	—
15	#19128097	C₂²	1	2	yes	—
16	#19128085	—	2	1	no	—
17	#19128081	C₂²	1	1	yes	characteristic
18	#19128051	C₂²	1	2	yes	—
19	#19128040	—	2	2	no	—
20	#19127987	C₂²	1	2	yes	—
21	#19127980	—	2	2	no	—
22	#19127978	C₂²	1	2	yes	—
23	#19127956	—	2	2	no	—
24	#19127924	—	2	2	no	—
25	#19127907	C₂²	1	2	yes	—
26	#19127855	—	2	2	no	—
27	C₈×Dic₁₆₆	C₂²	1	1	yes	characteristic
28	C₄×Dic₃₃₂	—	2	1	no	—
29	C₄×#13050580	C₂²	1	1	yes	characteristic
30	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
31	#19127783	—	2	1	no	—
32	#19127778	C₂²	1	2	yes	—
33	#19127708	—	2	2	no	—
34	#19127453	C₂²	1	1	yes	characteristic
35	#19127446	C₂²	1	2	yes	—
36	Q₈⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
37	#19127431	C₂²	1	1	yes	characteristic
38	#19127428	C₂²	1	2	yes	—
39	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	#500×C₈₃	C₂²	1	1	yes	characteristic
41	#462×C₈₃	C₂²	1	2	yes	—
42	C₈×Q₈×C₈₃	C₂²	1	1	yes	characteristic
43	#335×C₈₃	C₂²	1	1	yes	characteristic
44	#332×C₈₃	C₂²	1	2	yes	—
45	C₂×Q₈×D₈₃	—	2	2	no	—
46	C₂×Dic₃₃₂	—	2	1	no	—
47	C₂×Dic₃₃₂	—	2	1	no	—
48	C₂×#13050580	D₄	1	1	yes	characteristic
49	C₂×#13050579	—	2	2	no	—
50	C₂×D₆₆₄	D₄	1	1	yes	characteristic
51	C₂×#13050573	—	2	2	no	—
52	#14581792	C₂³	1	2	yes	—
53	#14581791	—	2	2	no	—
54	#14581787	—	4	2	no	—
55	#14581780	—	2	2	no	—
56	#14581770	—	4	2	no	—
57	#14581768	—	2	2	no	—
58	#14581768	—	2	2	no	—
59	C₄×Dic₁₆₆	—	2	2	no	—
60	C₄×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₄×D₃₃₂	C₂³	1	2	yes	—
62	#14581754	—	2	2	no	—
63	#14581750	—	2	2	no	—
64	#14581726	D₄	1	1	yes	characteristic
65	#14581725	—	2	2	no	—
66	#14581725	D₄	1	1	yes	characteristic
67	#14581724	—	2	2	no	—
68	#14581723	—	4	2	no	—
69	C₄⋊(C₈₃⋊C₈)	C₂³	1	1	yes	characteristic
70	Dic₈₃⋊C₈	—	2	1	no	—
71	#14581717	—	2	2	no	—
72	#14581717	—	2	2	no	—
73	#14581716	—	2	2	no	—
74	#14581716	—	2	2	no	—
75	#14581716	D₄	1	1	yes	characteristic
76	#14581715	D₄	1	1	yes	characteristic
77	#14581714	D₄	1	2	yes	—
78	#14581712	—	2	2	no	—
79	(C₂×D₈₃)⋊C₈	—	2	2	no	—
80	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₈₃	—	2	1	no	—
83	C₂×Q₁₆×C₈₃	—	2	2	no	—
84	C₄⋊Q₈×C₈₃	C₂³	1	2	yes	—
85	C₄×Q₈×C₈₃	C₂³	1	1	yes	characteristic
86	#107×C₈₃	—	2	1	no	—
87	#107×C₈₃	—	2	1	no	—
88	C₄⋊C₈×C₈₃	—	2	1	no	—
89	C₄⋊C₈×C₈₃	C₂³	1	1	yes	characteristic
90	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
91	#103×C₈₃	—	2	2	no	—
92	C₄×C₈×C₈₃	—	2	1	no	—
93	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
94	C₂×Dic₁₆₆	—	2	2	no	—
95	C₂×Dic₁₆₆	—	4	2	no	—
96	C₂×Dic₁₆₆	C₂×D₄	1	1	yes	characteristic
97	Q₈×D₈₃	—	4	4	no	—
98	Q₈×D₈₃	—	8	2	no	—
99	C₂×D₃₃₂	C₂×D₄	1	2	yes	—
100	C₂×C₄×D₈₃	—	2	2	no	—
101	C₂×C₄×D₈₃	—	4	2	no	—
102	Dic₃₃₂	—	4	2	no	—
103	Dic₃₃₂	—	8	1	no	—
104	#13050580	—	2	2	no	—
105	#13050579	—	8	2	no	—
106	D₆₆₄	—	2	2	no	—
107	#13050573	—	8	2	no	—
108	C₂×C₈₃⋊C₈	—	2	1	no	—
109	C₂×C₈₃⋊C₈	—	2	1	no	—
110	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
112	C₄⋊Dic₈₃	—	2	2	no	—
113	C₄⋊Dic₈₃	—	2	2	no	—
114	C₄⋊Dic₈₃	C₂×D₄	1	1	yes	characteristic
115	C₄⋊Dic₈₃	C₂×D₄	1	2	yes	—
116	Dic₈₃⋊C₄	—	2	1	no	—
117	Dic₈₃⋊C₄	—	4	2	no	—
118	Dic₈₃⋊C₄	—	4	2	no	—
119	(C₂×D₈₃)⋊C₄	—	2	2	no	—
120	(C₂×D₈₃)⋊C₄	—	4	2	no	—
121	(C₂×D₈₃)⋊C₄	—	4	2	no	—
122	(C₂×D₈₃)⋊C₄	—	4	2	no	—
123	C₄×Dic₈₃	—	2	1	no	—
124	C₄×Dic₈₃	—	4	2	no	—
125	C₂×Q₈×C₈₃	—	2	2	no	—
126	C₂×Q₈×C₈₃	C₂×D₄	1	1	yes	characteristic
127	Q₁₆×C₈₃	—	8	2	no	—
128	C₂×C₈×C₈₃	—	2	2	no	—
129	C₂×C₈×C₈₃	—	2	1	no	—
130	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
132	C₄⋊C₄×C₈₃	—	2	1	no	—
133	C₄⋊C₄×C₈₃	—	2	2	no	—
134	C₄⋊C₄×C₈₃	—	2	2	no	—
135	C₄⋊C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
136	C₄²×C₈₃	—	2	1	no	—
137	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
138	C₂²×D₈₃	—	2	2	no	—
139	Q₈×C₈₃	—	2	1	no	—
140	Q₈×C₈₃	—	2	1	no	—
141	Q₈×C₈₃	—	4	2	no	—
142	Q₈×C₈₃	—	4	2	no	—
143	C₂×C₄×C₈₃	—	2	1	no	—
144	C₂×C₄×C₈₃	—	2	1	no	—
145	C₂×C₄×C₈₃	—	4	2	no	—
146	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×Dic₈₃	—	2	1	no	—
150	C₂×Dic₈₃	—	2	1	no	—
151	C₂×Dic₈₃	—	2	2	no	—
152	C₂×Dic₈₃	—	4	2	no	—
153	D₃₃₂	—	2	2	no	—
154	D₃₃₂	—	2	2	no	—
155	C₄×D₈₃	—	4	2	no	—
156	C₄×D₈₃	—	4	2	no	—
157	C₄×D₈₃	—	8	4	no	—
158	Dic₁₆₆	—	2	1	no	—
159	Dic₁₆₆	—	4	2	no	—
160	Dic₁₆₆	—	4	2	no	—
161	Dic₁₆₆	—	8	4	no	—
162	Dic₁₆₆	#135	1	2	yes	—
163	C₈×C₈₃	—	2	1	no	—
164	C₈×C₈₃	—	4	2	no	—
165	C₈×C₈₃	—	4	1	no	—
166	C₈×C₈₃	#137	1	2	yes	—
167	C₈₃⋊C₈	—	2	2	no	—
168	C₈₃⋊C₈	—	2	1	no	—
169	C₈₃⋊C₈	—	2	1	no	—
170	C₈₃⋊C₈	—	4	1	no	—
171	C₂²×C₈₃	D₄²	1	1	yes	characteristic
172	C₂×D₈₃	—	4	2	no	—
173	C₂×D₈₃	—	4	2	no	—
174	C₂×D₈₃	—	4	2	no	—
175	C₄×C₈₃	—	2	1	no	—
176	C₄×C₈₃	—	4	1	no	—
177	C₄×C₈₃	—	4	1	no	—
178	C₄×C₈₃	—	8	2	no	—
179	C₄×C₈₃	C₂×#135	1	1	yes	characteristic
180	C₄×C₈₃	C₂×#137	1	1	yes	characteristic
181	C₄×C₈₃	#576	1	1	yes	characteristic
182	C₄×C₈₃	#577	1	1	yes	characteristic
183	Dic₈₃	—	2	2	no	—
184	Dic₈₃	—	4	2	no	—
185	Dic₈₃	—	4	1	no	—
186	Dic₈₃	—	8	2	no	—
187	#19907	—	83	1	no	maximal, Sylow
188	C₂×C₈₃	#3278	1	1	yes	characteristic
189	C₂×C₈₃	#3280	1	1	yes	characteristic
190	C₂×C₈₃	#3293	1	1	yes	characteristic
191	D₈₃	—	8	2	no	—
192	#3393	—	83	1	no	—
193	Q₈×D₈	—	83	1	no	—
194	#3328	—	83	2	no	—
195	#1707	—	83	1	no	—
196	#1699	—	83	2	no	—
197	#1674	D₈₃	1	1	yes	characteristic, complemented
198	#1670	—	83	1	no	—
199	#1652	—	83	2	no	—
200	#1651	—	83	2	no	—
201	#1589	—	83	1	no	—
202	C₈×Q₁₆	—	83	1	no	—
203	C₈₃	#19907	1	1	yes	characteristic, complemented, Sylow
204	#556	—	83	2	no	—
205	D₄×Q₈	—	83	2	no	—
206	#500	—	166	1	no	—
207	#500	C₂×D₈₃	1	1	yes	characteristic
208	#498	—	83	2	no	—
209	#495	—	166	1	no	—
210	#494	—	83	1	no	—
211	#487	—	83	2	no	—
212	#485	—	166	2	no	—
213	#475	—	83	2	no	—
214	#474	—	83	2	no	—
215	#474	—	166	2	no	—
216	#470	—	166	2	no	—
217	#464	—	166	2	no	—
218	#462	C₂×D₈₃	1	2	yes	—
219	#461	—	83	2	no	—
220	#451	—	166	2	no	—
221	C₈×Q₈	—	83	1	no	—
222	C₈×Q₈	C₂×D₈₃	1	1	yes	characteristic
223	C₄×Q₁₆	—	83	1	no	—
224	C₄×Q₁₆	—	166	1	no	—
225	C₄×D₈	—	83	1	no	—
226	C₄×D₈	—	166	1	no	—
227	#436	—	83	2	no	—
228	#419	—	166	2	no	—
229	#335	—	83	1	no	—
230	#335	C₂×D₈₃	1	1	yes	characteristic
231	#332	—	83	2	no	—
232	#332	C₂×D₈₃	1	2	yes	—
233	Q₈⋊C₈	—	83	1	no	—
234	Q₈⋊C₈	—	83	1	no	—
235	D₄⋊C₈	—	83	2	no	—
236	C₈²	—	83	1	no	—
237	C₂²×Q₈	—	166	2	no	—
238	C₂×Q₁₆	—	2	2	no	—
239	C₂×Q₁₆	—	83	1	no	—
240	C₂×Q₁₆	—	166	1	no	—
241	C₂×Q₁₆	—	166	1	no	—
242	C₂×SD₁₆	—	166	2	no	—
243	C₂×D₈	—	83	1	no	—
244	C₂×M₁₆	—	166	2	no	—
245	C₄⋊Q₈	—	83	2	no	—
246	C₄⋊Q₈	—	166	2	no	—
247	C₄⋊Q₈	C₂²×D₈₃	1	2	yes	—
248	#126	—	332	2	no	—
249	#124	—	166	2	no	—
250	C₂²⋊Q₈	—	332	2	no	—
251	C₂²⋊Q₈	—	166	2	no	—
252	C₂²⋊Q₈	—	166	2	no	—
253	C₄×Q₈	—	83	1	no	—
254	C₄×Q₈	—	166	2	no	—
255	C₄×Q₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₄×D₄	—	83	2	no	—
257	C₄×D₄	—	166	2	no	—
258	#117	—	166	2	no	—
259	#107	—	2	1	no	—
260	#107	—	2	1	no	—
261	#107	—	83	1	no	—
262	#107	—	83	1	no	—
263	#107	—	166	2	no	—
264	#106	—	332	2	no	—
265	#106	—	166	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	83	1	no	—
268	C₄⋊C₈	—	166	1	no	—
269	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
270	C₄⋊C₈	C₂²×D₈₃	1	1	yes	characteristic
271	#103	—	2	2	no	—
272	#103	—	83	1	no	—
273	#103	—	83	1	no	—
274	#103	—	166	2	no	—
275	#103	—	166	2	no	—
276	#103	—	166	2	no	—
277	#103	—	166	2	no	—
278	#102	—	83	2	no	—
279	#102	—	166	2	no	—
280	C₂²⋊C₈	—	166	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	83	1	no	—
283	C₄×C₈	—	83	1	no	—
284	C₄×C₈	—	166	1	no	—
285	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
286	C₂×Q₈	—	2	2	no	—
287	C₂×Q₈	—	332	2	no	—
288	C₂×Q₈	—	332	4	no	—
289	C₂×Q₈	—	83	1	no	—
290	C₂×Q₈	—	664	2	no	—
291	C₂×Q₈	—	166	2	no	—
292	C₂×Q₈	D₄×D₈₃	1	1	yes	characteristic
293	C₂×D₄	—	83	2	no	—
294	C₂²×C₄	—	332	2	no	—
295	C₂²×C₄	—	166	2	no	—
296	Q₁₆	—	8	2	no	—
297	Q₁₆	—	332	2	no	—
298	Q₁₆	—	664	1	no	—
299	Q₁₆	—	166	2	no	—
300	SD₁₆	—	664	2	no	—
301	D₈	—	166	2	no	—
302	M₁₆	—	664	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	2	no	—
305	C₂×C₈	—	83	1	no	—
306	C₂×C₈	—	83	1	no	—
307	C₂×C₈	—	166	1	no	—
308	C₂×C₈	—	166	1	no	—
309	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
310	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	332	2	no	—
315	C₄⋊C₄	—	332	2	no	—
316	C₄⋊C₄	—	83	2	no	—
317	C₄⋊C₄	—	83	1	no	—
318	C₄⋊C₄	—	166	2	no	—
319	C₄⋊C₄	—	166	2	no	—
320	C₄⋊C₄	—	166	1	no	—
321	C₄⋊C₄	D₄×D₈₃	1	1	yes	characteristic
322	C₂²⋊C₄	—	332	2	no	—
323	C₂²⋊C₄	—	332	2	no	—
324	C₂²⋊C₄	—	332	2	no	—
325	C₂²⋊C₄	—	166	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	332	2	no	—
328	C₄²	—	166	1	no	—
329	C₄²	C₂³×D₈₃	1	1	yes	characteristic
330	C₂³	—	166	2	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	2	1	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	332	2	no	—
336	Q₈	—	332	2	no	—
337	Q₈	—	83	2	no	—
338	Q₈	—	664	4	no	—
339	Q₈	—	166	1	no	—
340	D₄	—	166	2	no	—
341	D₄	—	166	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	4	2	no	—
345	C₂×C₄	—	332	2	no	—
346	C₂×C₄	—	332	2	no	—
347	C₂×C₄	—	332	2	no	—
348	C₂×C₄	—	664	4	no	—
349	C₂×C₄	—	166	2	no	—
350	C₂×C₄	—	166	1	no	—
351	C₂×C₄	—	166	1	no	—
352	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
353	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
354	C₂×C₄	#14581838	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	4	2	no	—
357	C₈	—	4	1	no	—
358	C₈	—	332	1	no	—
359	C₈	—	166	2	no	—
360	C₈	—	166	1	no	—
361	C₈	—	166	1	no	—
362	C₈	#14581824	1	2	yes	—
363	C₂²	—	332	2	no	—
364	C₂²	—	332	2	no	—
365	C₂²	—	332	2	no	—
366	C₂²	#19128335	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	8	2	no	—
371	C₄	—	332	2	no	—
372	C₄	—	332	1	no	—
373	C₄	—	664	2	no	—
374	C₄	—	166	2	no	—
375	C₄	#135×D₈₃	1	1	yes	characteristic
376	C₄	C₂×#14581824	1	1	yes	characteristic
377	C₄	#19128502	1	1	yes	characteristic
378	C₄	#19128510	1	1	yes	characteristic
379	C₂	—	664	2	no	—
380	C₂	#24919124	1	1	yes	characteristic
381	C₂	#24919151	1	1	yes	characteristic
382	C₂	#24919266	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#19128335
commutator, cosocle	C₄²×C₈₃	C₂⁴
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1674	C₂
socle	C₂²×C₈₃	D₄²
prime radical	C₂×D₆₆₄	D₄
hypercenter	#1674	D₈₃
hypocenter	C₈₃	#19907
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #1674

Complements

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

N	H ≅ G/N	type
3. #24919749	C₂	semidirect
4. #24907061	C₂	semidirect
5. #24907009	C₂	semidirect
6. #24906834	C₂	semidirect
7. #24906731	C₂	semidirect
8. #24906727	C₂	semidirect
10. C₈×#13050580	C₂	semidirect
11. C₈₃×#1674	C₂	semidirect
197. #1674	D₈₃	semidirect
203. C₈₃	#19907	semidirect

Maximal subgroups

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

subgroup	index
#24920452	2
#24920315	2
#24919749	2
#24907061	2
#24907009	2
#24906834	2
#24906731	2
#24906727	2
#24906347	2
C₈×#13050580	2
C₈₃×#1674	2
#19907	83