Subgroups — #35375102

Order 21248.

592 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24918033	C₂	1	1	yes	maximal, characteristic
2	#24916608	C₂	1	2	yes	maximal, complemented
3	#24906376	C₂	1	1	yes	maximal, characteristic, complemented
4	#24906371	C₂	1	1	yes	maximal, characteristic, complemented
5	#24906352	C₂	1	2	yes	maximal, complemented
6	#24906286	C₂	1	1	yes	maximal, characteristic, complemented
7	#24906281	C₂	1	1	yes	maximal, characteristic, complemented
8	#24906262	C₂	1	2	yes	maximal, complemented
9	#24905926	C₂	1	2	yes	maximal, complemented
10	#24905746	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₃×#1566	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128235	C₂²	1	2	yes	—
13	#19128103	C₄	1	1	yes	characteristic
14	#19128085	C₄	1	1	yes	characteristic
15	#19128056	C₄	1	2	yes	—
16	#19127942	C₄	1	1	yes	characteristic
17	#19127937	C₄	1	1	yes	characteristic
18	#19127930	C₄	1	2	yes	—
19	#19127828	C₂²	1	2	yes	—
20	#19127825	C₂²	1	2	yes	—
21	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
22	C₄×#13050579	C₂²	1	2	yes	—
23	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
24	#19127778	C₂²	1	2	yes	—
25	#19127777	—	2	2	no	—
26	#19127770	—	2	2	no	—
27	#19127766	C₂²	1	2	yes	—
28	#19127761	—	2	2	no	—
29	#19127754	C₂²	1	1	yes	characteristic
30	#19127754	C₂²	1	1	yes	characteristic
31	#19127726	—	2	2	no	—
32	#19127431	C₂²	1	2	yes	—
33	#19127431	C₂²	1	2	yes	—
34	#19127428	C₂²	1	2	yes	—
35	#19127428	C₂²	1	2	yes	—
36	#19127422	C₂²	1	1	yes	characteristic
37	#19127422	C₂²	1	1	yes	characteristic
38	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
40	#423×C₈₃	C₂²	1	2	yes	—
41	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
42	#334×C₈₃	C₂²	1	2	yes	—
43	#334×C₈₃	C₂²	1	1	yes	characteristic
44	#334×C₈₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#13050579	C₂×C₄	1	2	yes	—
47	C₂×D₆₆₄	C₂×C₄	1	1	yes	characteristic
48	C₂×#13050573	—	2	2	no	—
49	C₂×C₈×D₈₃	—	2	2	no	—
50	#14581786	—	2	2	no	—
51	#14581784	C₂×C₄	1	2	yes	—
52	#14581780	C₂×C₄	1	2	yes	—
53	#14581775	—	2	2	no	—
54	#14581771	C₂×C₄	1	2	yes	—
55	#14581766	C₂×C₄	1	2	yes	—
56	C₄×Dic₁₆₆	C₂³	1	2	yes	—
57	C₄×D₃₃₂	C₂³	1	2	yes	—
58	C₄×C₂²⋊D₈₃	—	2	2	no	—
59	#14581749	—	2	2	no	—
60	#14581726	C₂×C₄	1	1	yes	characteristic
61	#14581726	C₂×C₄	1	1	yes	characteristic
62	#14581724	C₂×C₄	1	2	yes	—
63	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
64	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
65	Dic₈₃⋊C₈	—	2	2	no	—
66	Dic₈₃⋊C₈	—	2	2	no	—
67	#14581717	C₂×C₄	1	2	yes	—
68	#14581717	C₂×C₄	1	2	yes	—
69	#14581714	C₂×C₄	1	2	yes	—
70	#14581714	C₂×C₄	1	2	yes	—
71	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
72	C₂²⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
73	(C₂×D₈₃)⋊C₈	—	2	2	no	—
74	(C₂×D₈₃)⋊C₈	—	2	2	no	—
75	#14581702	—	2	2	no	—
76	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₄×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₈×Dic₈₃	—	2	2	no	—
81	C₂×M₁₆×C₈₃	—	2	2	no	—
82	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
83	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
85	C₄⋊C₈×C₈₃	—	2	2	no	—
86	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
87	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
88	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
89	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
90	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
91	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
93	C₂×D₃₃₂	C₂²×C₄	1	2	yes	—
94	C₂×C₂²⋊D₈₃	—	2	2	no	—
95	C₂×C₄×D₈₃	—	2	2	no	—
96	Dic₃₃₂	—	2	2	no	—
97	#13050579	—	2	4	no	—
98	D₆₆₄	—	2	2	no	—
99	#13050573	—	8	2	no	—
100	C₂×C₈₃⋊C₈	—	2	1	no	—
101	C₂×C₈₃⋊C₈	—	2	1	no	—
102	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
103	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
104	C₈×D₈₃	—	8	2	no	—
105	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
106	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
107	Dic₈₃⋊C₄	—	2	2	no	—
108	Dic₈₃⋊C₄	—	2	2	no	—
109	Dic₈₃⋊C₄	—	2	2	no	—
110	Dic₈₃⋊C₄	—	2	2	no	—
111	C₂²⋊Dic₈₃	—	2	2	no	—
112	(C₂×D₈₃)⋊C₄	—	2	2	no	—
113	(C₂×D₈₃)⋊C₄	—	2	2	no	—
114	(C₂×D₈₃)⋊C₄	—	2	2	no	—
115	(C₂×D₈₃)⋊C₄	—	2	2	no	—
116	C₄×Dic₈₃	—	2	2	no	—
117	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
120	M₁₆×C₈₃	—	8	2	no	—
121	C₂×C₈×C₈₃	—	2	2	no	—
122	C₂×C₈×C₈₃	—	2	2	no	—
123	C₂×C₈×C₈₃	—	2	2	no	—
124	C₂×C₈×C₈₃	—	2	2	no	—
125	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
126	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
129	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
130	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
131	C₄²×C₈₃	#41	1	1	yes	characteristic
132	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
133	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
134	C₂²×D₈₃	—	2	2	no	—
135	C₂×C₄×C₈₃	—	2	1	no	—
136	C₂×C₄×C₈₃	—	2	1	no	—
137	C₂×C₄×C₈₃	—	2	2	no	—
138	C₂×C₄×C₈₃	—	4	1	no	—
139	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
141	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
145	C₂²⋊D₈₃	—	8	2	no	—
146	C₂×Dic₈₃	—	2	2	no	—
147	C₂×Dic₈₃	—	2	2	no	—
148	C₂×Dic₈₃	—	2	2	no	—
149	D₃₃₂	—	2	2	no	—
150	D₃₃₂	—	2	2	no	—
151	C₄×D₈₃	—	4	2	no	—
152	C₄×D₈₃	—	4	2	no	—
153	Dic₁₆₆	—	2	2	no	—
154	Dic₁₆₆	—	2	2	no	—
155	C₈×C₈₃	—	2	1	no	—
156	C₈×C₈₃	—	2	1	no	—
157	C₈×C₈₃	—	4	2	no	—
158	C₈×C₈₃	—	4	2	no	—
159	C₈×C₈₃	#131	1	2	yes	—
160	C₈×C₈₃	#131	1	2	yes	—
161	C₈₃⋊C₈	—	2	2	no	—
162	C₈₃⋊C₈	—	2	2	no	—
163	C₈₃⋊C₈	—	4	1	no	—
164	C₈₃⋊C₈	—	4	1	no	—
165	C₂²×C₈₃	—	2	1	no	—
166	C₂²×C₈₃	—	2	1	no	—
167	C₂²×C₈₃	—	2	1	no	—
168	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₂×D₈₃	—	4	2	no	—
170	C₂×D₈₃	—	4	2	no	—
171	C₂×D₈₃	—	4	2	no	—
172	C₄×C₈₃	—	2	1	no	—
173	C₄×C₈₃	—	2	1	no	—
174	C₄×C₈₃	—	2	1	no	—
175	C₄×C₈₃	—	2	1	no	—
176	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
177	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
178	C₄×C₈₃	#577	1	2	yes	—
179	Dic₈₃	—	4	2	no	—
180	Dic₈₃	—	4	2	no	—
181	Dic₈₃	—	4	2	no	—
182	#17974	—	83	1	no	maximal, Sylow
183	C₂×C₈₃	—	4	1	no	—
184	C₂×C₈₃	#2931	1	1	yes	characteristic
185	C₂×C₈₃	#2952	1	1	yes	characteristic
186	C₂×C₈₃	#2957	1	1	yes	characteristic
187	D₈₃	—	8	2	no	—
188	#3168	—	83	1	no	—
189	#2993	—	83	2	no	—
190	#1594	—	83	1	no	—
191	#1593	—	83	1	no	—
192	#1590	—	83	2	no	—
193	#1580	—	83	1	no	—
194	#1579	—	83	1	no	—
195	C₈×SD₁₆	—	83	2	no	—
196	#1566	D₈₃	1	1	yes	characteristic, complemented
197	#1495	—	83	2	no	—
198	#1450	—	83	1	no	—
199	C₈₃	#17974	1	1	yes	characteristic, complemented, Sylow
200	#529	—	83	2	no	—
201	#499	—	83	1	no	—
202	#495	—	83	1	no	—
203	#488	—	83	2	no	—
204	#467	—	83	1	no	—
205	#466	—	83	1	no	—
206	#465	—	83	2	no	—
207	#446	—	83	2	no	—
208	C₈×Q₈	—	83	2	no	—
209	C₄×Q₁₆	—	83	1	no	—
210	C₄×SD₁₆	—	83	2	no	—
211	C₄×D₈	—	83	1	no	—
212	#436	—	83	2	no	—
213	#436	—	166	2	no	—
214	#435	—	166	2	no	—
215	C₈×D₄	—	83	2	no	—
216	#433	—	166	2	no	—
217	#431	—	83	1	no	—
218	#431	—	83	1	no	—
219	#424	—	166	2	no	—
220	#423	C₂×D₈₃	1	2	yes	—
221	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
222	#334	C₂×D₈₃	1	2	yes	—
223	#334	C₂×D₈₃	1	1	yes	characteristic
224	#334	C₂×D₈₃	1	1	yes	characteristic
225	Q₈⋊C₈	—	83	2	no	—
226	Q₈⋊C₈	—	83	2	no	—
227	D₄⋊C₈	—	83	2	no	—
228	D₄⋊C₈	—	83	2	no	—
229	#322	—	83	1	no	—
230	#322	—	83	1	no	—
231	C₈²	—	83	1	no	—
232	C₈²	—	83	1	no	—
233	C₂×Q₁₆	—	83	1	no	—
234	C₂×SD₁₆	—	83	2	no	—
235	C₂×D₈	—	83	1	no	—
236	C₂×M₁₆	—	2	2	no	—
237	C₂×M₁₆	—	166	2	no	—
238	C₂²×C₈	—	166	2	no	—
239	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
240	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
241	#126	—	166	2	no	—
242	#125	—	83	2	no	—
243	#124	—	83	2	no	—
244	#123	—	166	2	no	—
245	C₂²⋊Q₈	—	83	2	no	—
246	#121	—	83	2	no	—
247	C₄×Q₈	—	83	2	no	—
248	C₄×D₄	—	83	2	no	—
249	C₄×D₄	—	166	2	no	—
250	#117	—	166	2	no	—
251	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
252	#107	—	83	1	no	—
253	#107	—	83	1	no	—
254	#106	—	83	2	no	—
255	C₄⋊C₈	—	2	2	no	—
256	C₄⋊C₈	—	83	2	no	—
257	C₄⋊C₈	—	166	2	no	—
258	C₄⋊C₈	—	166	2	no	—
259	C₄⋊C₈	—	166	1	no	—
260	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
261	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
262	#103	—	83	2	no	—
263	#103	—	83	2	no	—
264	#102	—	83	2	no	—
265	#102	—	83	2	no	—
266	C₂²⋊C₈	—	83	2	no	—
267	C₂²⋊C₈	—	166	2	no	—
268	C₂²⋊C₈	—	166	2	no	—
269	C₂²⋊C₈	—	166	1	no	—
270	#97	—	166	2	no	—
271	C₄×C₈	—	83	1	no	—
272	C₄×C₈	—	83	1	no	—
273	C₄×C₈	—	83	1	no	—
274	C₄×C₈	—	83	1	no	—
275	C₄×C₈	—	166	2	no	—
276	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
277	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
278	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
279	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
280	C₂×Q₈	—	83	2	no	—
281	C₂×D₄	—	83	2	no	—
282	C₂×D₄	—	166	2	no	—
283	C₂²×C₄	—	166	2	no	—
284	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
285	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
286	C₂²×C₄	#13050591	1	1	yes	characteristic
287	Q₁₆	—	166	2	no	—
288	SD₁₆	—	166	4	no	—
289	D₈	—	166	2	no	—
290	M₁₆	—	8	2	no	—
291	M₁₆	—	664	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	83	2	no	—
297	C₂×C₈	—	83	2	no	—
298	C₂×C₈	—	664	2	no	—
299	C₂×C₈	—	166	1	no	—
300	C₂×C₈	—	166	1	no	—
301	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
302	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
303	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
304	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
305	C₄⋊C₄	—	83	2	no	—
306	C₄⋊C₄	—	83	2	no	—
307	C₄⋊C₄	—	166	2	no	—
308	C₄⋊C₄	—	166	2	no	—
309	C₄⋊C₄	—	166	2	no	—
310	C₄⋊C₄	—	166	2	no	—
311	C₂²⋊C₄	—	166	2	no	—
312	C₂²⋊C₄	—	166	2	no	—
313	C₂²⋊C₄	—	166	2	no	—
314	C₂²⋊C₄	—	166	2	no	—
315	C₂²⋊C₄	—	166	2	no	—
316	C₄²	—	166	2	no	—
317	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
318	C₄²	D₄×D₈₃	1	1	yes	characteristic
319	C₄²	#13050591	1	1	yes	characteristic
320	C₄²	C₂³×D₈₃	1	1	yes	characteristic
321	C₂³	—	166	2	no	—
322	C₂³	#14581754	1	1	yes	characteristic
323	Q₈	—	166	2	no	—
324	Q₈	—	166	2	no	—
325	D₄	—	664	2	no	—
326	D₄	—	166	2	no	—
327	D₄	—	166	2	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	332	2	no	—
333	C₂×C₄	—	332	2	no	—
334	C₂×C₄	—	166	2	no	—
335	C₂×C₄	—	166	2	no	—
336	C₂×C₄	—	166	2	no	—
337	C₂×C₄	#14581754	1	1	yes	characteristic
338	C₂×C₄	#14581754	1	1	yes	characteristic
339	C₂×C₄	#14581754	1	1	yes	characteristic
340	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
341	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
342	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
343	C₈	—	2	1	no	—
344	C₈	—	2	1	no	—
345	C₈	—	4	2	no	—
346	C₈	—	4	2	no	—
347	C₈	—	332	1	no	—
348	C₈	—	332	1	no	—
349	C₈	—	166	2	no	—
350	C₈	—	166	2	no	—
351	C₈	#14581800	1	2	yes	—
352	C₈	#14581800	1	2	yes	—
353	C₂²	—	2	1	no	—
354	C₂²	—	2	1	no	—
355	C₂²	—	2	1	no	—
356	C₂²	—	332	2	no	—
357	C₂²	—	332	2	no	—
358	C₂²	—	332	2	no	—
359	C₂²	C₂×#14581754	1	1	yes	characteristic
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	2	1	no	—
363	C₄	—	2	1	no	—
364	C₄	—	332	2	no	—
365	C₄	—	332	2	no	—
366	C₄	—	332	2	no	—
367	C₄	C₂×#14581800	1	1	yes	characteristic
368	C₄	C₂×#14581800	1	1	yes	characteristic
369	C₄	#19128507	1	2	yes	—
370	C₂	—	4	1	no	—
371	C₂	—	664	2	no	—
372	C₂	#24916174	1	1	yes	characteristic
373	C₂	#24916346	1	1	yes	characteristic
374	C₂	#24916379	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₄²	C₂³×D₈₃
Fitting	C₈₃×#1566	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	#19127937	C₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1566	D₈₃
hypocenter	C₈₃	#17974
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

Complements

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

N	H ≅ G/N	type
2. #24916608	C₂	semidirect
3. #24906376	C₂	semidirect
4. #24906371	C₂	semidirect
5. #24906352	C₂	semidirect
6. #24906286	C₂	semidirect
7. #24906281	C₂	semidirect
8. #24906262	C₂	semidirect
9. #24905926	C₂	semidirect
10. #24905746	C₂	semidirect
11. C₈₃×#1566	C₂	semidirect
196. #1566	D₈₃	semidirect
199. C₈₃	#17974	semidirect

Maximal subgroups

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

subgroup	index
#24918033	2
#24916608	2
#24906376	2
#24906371	2
#24906352	2
#24906286	2
#24906281	2
#24906262	2
#24905926	2
#24905746	2
C₈₃×#1566	2
#17974	83