Subgroups — #35376236

Order 21248.

592 conjugacy classes of subgroups.

The subgroup lattice has 368 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	#24916461	C₂	1	2	yes	maximal, complemented
3	#24906364	C₂	1	2	yes	maximal, complemented
4	#24906333	C₂	1	2	yes	maximal, complemented
5	#24906326	C₂	1	2	yes	maximal, complemented
6	#24906274	C₂	1	2	yes	maximal, complemented
7	#24905869	C₂	1	2	yes	maximal, complemented
8	#24905752	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic, complemented
10	#19128235	C₂²	1	2	yes	—
11	#19128103	C₄	1	1	yes	characteristic
12	#19128085	C₄	1	1	yes	characteristic
13	#19128056	C₄	1	2	yes	—
14	#19127942	C₄	1	1	yes	characteristic
15	#19127937	C₄	1	1	yes	characteristic
16	#19127930	C₄	1	2	yes	—
17	#19127828	C₂²	1	2	yes	—
18	#19127825	C₂²	1	2	yes	—
19	C₄×Dic₃₃₂	C₂²	1	1	yes	characteristic
20	C₄×#13050579	C₂²	1	2	yes	—
21	C₄×D₆₆₄	C₂²	1	1	yes	characteristic
22	#19127778	C₂²	1	2	yes	—
23	#19127776	—	2	2	no	—
24	#19127775	—	2	2	no	—
25	#19127766	C₂²	1	2	yes	—
26	#19127760	—	2	2	no	—
27	#19127754	C₂²	1	1	yes	characteristic
28	#19127655	—	2	2	no	—
29	C₄×#13050574	C₂²	1	1	yes	characteristic
30	#19127431	C₂²	1	2	yes	—
31	#19127431	C₂²	1	2	yes	—
32	#19127428	C₂²	1	2	yes	—
33	#19127428	C₂²	1	2	yes	—
34	#19127423	C₂²	1	2	yes	—
35	#19127422	C₂²	1	2	yes	—
36	#431×C₈₃	C₂²	1	2	yes	—
37	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
38	#335×C₈₃	C₂²	1	1	yes	characteristic
39	#335×C₈₃	C₂²	1	1	yes	characteristic
40	#334×C₈₃	C₂²	1	2	yes	—
41	C₂×Dic₃₃₂	C₂×C₄	1	1	yes	characteristic
42	C₂×#13050579	C₂×C₄	1	2	yes	—
43	C₂×D₆₆₄	C₂×C₄	1	1	yes	characteristic
44	C₂×#13050574	D₄	1	1	yes	characteristic
45	C₂×#13050574	D₄	1	1	yes	characteristic
46	C₂×#13050573	—	2	2	no	—
47	C₂×C₈×D₈₃	—	2	2	no	—
48	#14581786	—	2	2	no	—
49	#14581784	C₂×C₄	1	2	yes	—
50	#14581780	C₂×C₄	1	2	yes	—
51	#14581775	—	2	2	no	—
52	#14581771	C₂×C₄	1	2	yes	—
53	#14581766	C₂×C₄	1	2	yes	—
54	C₄×Dic₁₆₆	C₂³	1	2	yes	—
55	C₄×D₃₃₂	C₂³	1	2	yes	—
56	C₄×C₂²⋊D₈₃	—	2	2	no	—
57	#14581749	—	2	2	no	—
58	#14581726	C₂×C₄	1	1	yes	characteristic
59	#14581726	C₂×C₄	1	1	yes	characteristic
60	#14581724	C₂×C₄	1	2	yes	—
61	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
62	Dic₈₃⋊C₈	—	2	2	no	—
63	Dic₈₃⋊C₈	—	2	2	no	—
64	#14581717	C₂×C₄	1	2	yes	—
65	#14581717	C₂×C₄	1	2	yes	—
66	#14581714	C₂×C₄	1	2	yes	—
67	#14581714	C₂×C₄	1	2	yes	—
68	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
69	(C₂×D₈₃)⋊C₈	—	2	2	no	—
70	(C₂×D₈₃)⋊C₈	—	2	2	no	—
71	#14581703	D₄	1	2	yes	—
72	#14581702	—	2	2	no	—
73	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
74	C₄×C₈₃⋊C₈	C₂³	1	2	yes	—
75	C₈×Dic₈₃	—	2	2	no	—
76	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
77	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
80	C₄⋊C₈×C₈₃	C₂³	1	2	yes	—
81	C₂²⋊C₈×C₈₃	—	2	2	no	—
82	C₄×C₈×C₈₃	—	2	2	no	—
83	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
84	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
85	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
86	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
87	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
88	C₂×D₃₃₂	C₂²×C₄	1	2	yes	—
89	C₂×C₂²⋊D₈₃	—	2	2	no	—
90	C₂×C₄×D₈₃	—	2	2	no	—
91	Dic₃₃₂	—	4	1	no	—
92	#13050579	—	2	4	no	—
93	D₆₆₄	—	4	1	no	—
94	#13050574	—	4	1	no	—
95	#13050574	—	4	1	no	—
96	#13050573	—	4	4	no	—
97	C₂×C₈₃⋊C₈	—	2	2	no	—
98	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈₃⋊C₈	C₂×D₄	1	2	yes	—
100	C₈×D₈₃	—	4	4	no	—
101	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₈₃	C₂²×C₄	1	2	yes	—
103	Dic₈₃⋊C₄	—	2	2	no	—
104	Dic₈₃⋊C₄	—	2	2	no	—
105	Dic₈₃⋊C₄	—	2	2	no	—
106	Dic₈₃⋊C₄	—	2	2	no	—
107	C₂²⋊Dic₈₃	—	2	2	no	—
108	(C₂×D₈₃)⋊C₄	—	2	2	no	—
109	(C₂×D₈₃)⋊C₄	—	2	2	no	—
110	(C₂×D₈₃)⋊C₄	—	2	2	no	—
111	(C₂×D₈₃)⋊C₄	—	2	2	no	—
112	C₄×Dic₈₃	—	2	2	no	—
113	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
116	C₂×C₈×C₈₃	—	2	2	no	—
117	C₂×C₈×C₈₃	—	2	2	no	—
118	C₂×C₈×C₈₃	—	2	2	no	—
119	C₂×C₈×C₈₃	—	4	1	no	—
120	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
122	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
123	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
124	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
125	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
126	C₄²×C₈₃	#41	1	1	yes	characteristic
127	C₄²×C₈₃	C₂⁴	1	1	yes	characteristic
128	C₂³×C₈₃	C₄×D₄	1	1	yes	characteristic
129	C₂²×D₈₃	—	2	2	no	—
130	C₂×C₄×C₈₃	—	2	1	no	—
131	C₂×C₄×C₈₃	—	2	1	no	—
132	C₂×C₄×C₈₃	—	2	1	no	—
133	C₂×C₄×C₈₃	—	2	1	no	—
134	C₂×C₄×C₈₃	—	4	1	no	—
135	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
141	C₂²⋊D₈₃	—	8	2	no	—
142	C₂×Dic₈₃	—	2	2	no	—
143	C₂×Dic₈₃	—	2	2	no	—
144	C₂×Dic₈₃	—	2	2	no	—
145	D₃₃₂	—	2	2	no	—
146	D₃₃₂	—	2	2	no	—
147	C₄×D₈₃	—	4	2	no	—
148	C₄×D₈₃	—	4	2	no	—
149	Dic₁₆₆	—	2	2	no	—
150	Dic₁₆₆	—	2	2	no	—
151	C₈×C₈₃	—	2	1	no	—
152	C₈×C₈₃	—	2	1	no	—
153	C₈×C₈₃	—	2	1	no	—
154	C₈×C₈₃	—	4	2	no	—
155	C₈×C₈₃	—	4	2	no	—
156	C₈×C₈₃	#131	1	2	yes	—
157	C₈₃⋊C₈	—	2	2	no	—
158	C₈₃⋊C₈	—	2	2	no	—
159	C₈₃⋊C₈	—	4	2	no	—
160	C₂²×C₈₃	—	2	1	no	—
161	C₂²×C₈₃	—	2	1	no	—
162	C₂²×C₈₃	—	2	1	no	—
163	C₂²×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
164	C₂×D₈₃	—	4	2	no	—
165	C₂×D₈₃	—	4	2	no	—
166	C₂×D₈₃	—	4	2	no	—
167	C₄×C₈₃	—	2	1	no	—
168	C₄×C₈₃	—	2	1	no	—
169	C₄×C₈₃	—	2	1	no	—
170	C₄×C₈₃	—	2	1	no	—
171	C₄×C₈₃	C₂×#131	1	1	yes	characteristic
172	C₄×C₈₃	#568	1	1	yes	characteristic
173	C₄×C₈₃	#575	1	1	yes	characteristic
174	C₄×C₈₃	#575	1	1	yes	characteristic
175	Dic₈₃	—	4	2	no	—
176	Dic₈₃	—	4	2	no	—
177	Dic₈₃	—	4	2	no	—
178	#18074	—	83	1	no	maximal, Sylow
179	C₂×C₈₃	—	4	1	no	—
180	C₂×C₈₃	#2935	1	1	yes	characteristic
181	C₂×C₈₃	#2943	1	1	yes	characteristic
182	C₂×C₈₃	#2957	1	1	yes	characteristic
183	D₈₃	—	8	2	no	—
184	#3168	—	83	1	no	—
185	#2969	—	83	2	no	—
186	#1592	—	83	2	no	—
187	#1587	—	83	2	no	—
188	#1586	—	83	2	no	—
189	SD₁₆⋊C₈	—	83	2	no	—
190	#1564	D₈₃	1	1	yes	characteristic, complemented
191	#1482	—	83	2	no	—
192	#1451	—	83	1	no	—
193	C₈₃	#18074	1	1	yes	characteristic, complemented, Sylow
194	#529	—	83	2	no	—
195	#499	—	83	1	no	—
196	#495	—	83	1	no	—
197	#488	—	83	2	no	—
198	#467	—	83	1	no	—
199	#466	—	83	1	no	—
200	#465	—	83	2	no	—
201	#446	—	83	2	no	—
202	C₈×Q₈	—	83	2	no	—
203	C₄×Q₁₆	—	83	1	no	—
204	C₄×SD₁₆	—	83	2	no	—
205	C₄×D₈	—	83	1	no	—
206	#436	—	83	2	no	—
207	#436	—	166	2	no	—
208	#435	—	166	2	no	—
209	C₈×D₄	—	83	2	no	—
210	#433	—	166	2	no	—
211	#431	—	83	1	no	—
212	#431	C₂×D₈₃	1	2	yes	—
213	#405	—	166	2	no	—
214	C₄×M₁₆	—	83	1	no	—
215	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
216	#335	C₂×D₈₃	1	1	yes	characteristic
217	#335	C₂×D₈₃	1	1	yes	characteristic
218	#334	C₂×D₈₃	1	2	yes	—
219	Q₈⋊C₈	—	83	2	no	—
220	Q₈⋊C₈	—	83	2	no	—
221	D₄⋊C₈	—	83	2	no	—
222	D₄⋊C₈	—	83	2	no	—
223	#322	—	83	2	no	—
224	#322	—	83	2	no	—
225	C₂×Q₁₆	—	83	1	no	—
226	C₂×SD₁₆	—	83	2	no	—
227	C₂×D₈	—	83	1	no	—
228	C₂×M₁₆	—	83	1	no	—
229	C₂×M₁₆	—	83	1	no	—
230	C₂×M₁₆	—	166	2	no	—
231	C₂²×C₈	—	166	2	no	—
232	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
233	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
234	#126	—	166	2	no	—
235	#125	—	83	2	no	—
236	#124	—	83	2	no	—
237	#123	—	166	2	no	—
238	C₂²⋊Q₈	—	83	2	no	—
239	#121	—	83	2	no	—
240	C₄×Q₈	—	83	2	no	—
241	C₄×D₄	—	83	2	no	—
242	C₄×D₄	—	166	2	no	—
243	#117	—	166	2	no	—
244	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
245	#107	—	83	1	no	—
246	#107	—	83	1	no	—
247	#106	—	83	2	no	—
248	C₄⋊C₈	—	166	2	no	—
249	C₄⋊C₈	—	166	2	no	—
250	C₄⋊C₈	—	166	1	no	—
251	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
252	C₄⋊C₈	C₂²×D₈₃	1	2	yes	—
253	#103	—	83	2	no	—
254	#103	—	83	2	no	—
255	#102	—	83	2	no	—
256	#102	—	83	2	no	—
257	C₂²⋊C₈	—	2	2	no	—
258	C₂²⋊C₈	—	166	2	no	—
259	C₂²⋊C₈	—	166	2	no	—
260	C₂²⋊C₈	—	166	1	no	—
261	#97	—	83	2	no	—
262	#97	—	166	2	no	—
263	C₄×C₈	—	2	2	no	—
264	C₄×C₈	—	83	2	no	—
265	C₄×C₈	—	83	2	no	—
266	C₄×C₈	—	166	2	no	—
267	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
268	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
269	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
270	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
271	C₂×Q₈	—	83	2	no	—
272	C₂×D₄	—	83	2	no	—
273	C₂×D₄	—	166	2	no	—
274	C₂²×C₄	—	166	2	no	—
275	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
276	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
277	C₂²×C₄	#13050591	1	1	yes	characteristic
278	Q₁₆	—	332	1	no	—
279	SD₁₆	—	166	4	no	—
280	D₈	—	332	1	no	—
281	M₁₆	—	332	4	no	—
282	M₁₆	—	332	1	no	—
283	M₁₆	—	332	1	no	—
284	C₂×C₈	—	2	2	no	—
285	C₂×C₈	—	2	2	no	—
286	C₂×C₈	—	2	2	no	—
287	C₂×C₈	—	4	1	no	—
288	C₂×C₈	—	332	4	no	—
289	C₂×C₈	—	83	2	no	—
290	C₂×C₈	—	83	2	no	—
291	C₂×C₈	—	166	2	no	—
292	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
293	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
294	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
295	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
296	C₄⋊C₄	—	83	2	no	—
297	C₄⋊C₄	—	83	2	no	—
298	C₄⋊C₄	—	166	2	no	—
299	C₄⋊C₄	—	166	2	no	—
300	C₄⋊C₄	—	166	2	no	—
301	C₄⋊C₄	—	166	2	no	—
302	C₂²⋊C₄	—	166	2	no	—
303	C₂²⋊C₄	—	166	2	no	—
304	C₂²⋊C₄	—	166	2	no	—
305	C₂²⋊C₄	—	166	2	no	—
306	C₂²⋊C₄	—	166	2	no	—
307	C₄²	—	166	2	no	—
308	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
309	C₄²	D₄×D₈₃	1	1	yes	characteristic
310	C₄²	#13050591	1	1	yes	characteristic
311	C₄²	C₂³×D₈₃	1	1	yes	characteristic
312	C₂³	—	166	2	no	—
313	C₂³	#14581754	1	1	yes	characteristic
314	Q₈	—	166	2	no	—
315	Q₈	—	166	2	no	—
316	D₄	—	664	2	no	—
317	D₄	—	166	2	no	—
318	D₄	—	166	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	332	2	no	—
325	C₂×C₄	—	332	2	no	—
326	C₂×C₄	—	166	2	no	—
327	C₂×C₄	—	166	2	no	—
328	C₂×C₄	—	166	2	no	—
329	C₂×C₄	#14581754	1	1	yes	characteristic
330	C₂×C₄	#14581754	1	1	yes	characteristic
331	C₂×C₄	#14581754	1	1	yes	characteristic
332	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
333	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
334	C₂×C₄	C₂×#13050591	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	2	1	no	—
338	C₈	—	4	2	no	—
339	C₈	—	4	2	no	—
340	C₈	—	332	2	no	—
341	C₈	—	166	2	no	—
342	C₈	—	166	2	no	—
343	C₈	#14581798	1	2	yes	—
344	C₂²	—	2	1	no	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	332	2	no	—
348	C₂²	—	332	2	no	—
349	C₂²	—	332	2	no	—
350	C₂²	C₂×#14581754	1	1	yes	characteristic
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	332	2	no	—
356	C₄	—	332	2	no	—
357	C₄	—	332	2	no	—
358	C₄	C₂×#14581798	1	1	yes	characteristic
359	C₄	#19128467	1	1	yes	characteristic
360	C₄	#19128498	1	1	yes	characteristic
361	C₄	#19128498	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	664	2	no	—
364	C₂	#24916206	1	1	yes	characteristic
365	C₂	#24916263	1	1	yes	characteristic
366	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×#1564	C₂
socle	C₂²×C₈₃	C₂×C₄×D₄
prime radical	#19127937	C₄
cosocle	C₄²×C₈₃	C₂⁴
hypercenter	#1564	D₈₃
hypocenter	C₈₃	#18074
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
2. #24916461	C₂	semidirect
3. #24906364	C₂	semidirect
4. #24906333	C₂	semidirect
5. #24906326	C₂	semidirect
6. #24906274	C₂	semidirect
7. #24905869	C₂	semidirect
8. #24905752	C₂	semidirect
9. C₈₃×#1564	C₂	semidirect
190. #1564	D₈₃	semidirect
193. C₈₃	#18074	semidirect

Maximal subgroups

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

subgroup	index
#24918033	2
#24916461	2
#24906364	2
#24906333	2
#24906326	2
#24906274	2
#24905869	2
#24905752	2
C₈₃×#1564	2
#18074	83