Subgroups — #24916341

Order 10624.

484 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	C₂×#14581813	C₂	1	1	yes	maximal, characteristic
2	#19128183	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128180	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127811	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127810	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127805	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#13050579	C₂	1	1	yes	maximal, characteristic, complemented
9	SD₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127788	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127701	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127699	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127655	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×#13050589	C₂²	1	1	yes	characteristic
18	#14581813	C₄	1	8	yes	complemented
19	C₂×#13050580	C₂²	1	1	yes	characteristic
20	C₂×#13050579	C₂²	1	1	yes	characteristic
21	C₂×#13050575	C₂²	1	1	yes	characteristic
22	C₂×#13050573	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
24	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
26	#14581757	C₂²	1	1	yes	characteristic
27	C₄×D₃₃₂	—	2	1	no	—
28	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
29	D₄×Dic₈₃	C₂²	1	1	yes	characteristic
30	#14581754	—	2	1	no	—
31	#14581754	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₈₃	—	2	1	no	—
33	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
34	#14581749	—	2	1	no	—
35	#14581749	C₂²	1	1	yes	characteristic
36	#14581748	—	2	1	no	—
37	C₂×Dic₈₃⋊C₄	—	2	1	no	—
38	C₄⋊C₄×D₈₃	—	2	1	no	—
39	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	#14581723	C₂²	1	1	yes	characteristic
44	#14581717	C₂²	1	1	yes	characteristic
45	#14581716	C₂²	1	1	yes	characteristic
46	#14581715	C₂²	1	1	yes	characteristic
47	#14581714	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581712	C₂²	1	1	yes	characteristic
50	#14581703	C₂²	1	1	yes	characteristic
51	#14581702	C₂²	1	1	yes	characteristic
52	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
56	#106×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#102×C₈₃	C₂²	1	1	yes	characteristic
59	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
60	#13050591	—	2	2	no	—
61	#13050591	C₂×C₄	1	4	yes	complemented
62	#13050589	—	2	2	no	—
63	#13050589	C₂×C₄	1	4	yes	complemented
64	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
65	C₂×D₃₃₂	—	2	1	no	—
66	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₈₃	—	2	1	no	—
68	C₂²×Dic₈₃	—	2	1	no	—
69	C₂²×Dic₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	—	2	1	no	—
73	C₂×C₄×D₈₃	—	2	1	no	—
74	C₂×C₄×D₈₃	—	2	1	no	—
75	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
76	#13050580	C₂×C₄	1	4	yes	complemented
77	#13050579	C₂×C₄	1	4	yes	complemented
78	#13050575	C₂×C₄	1	4	yes	complemented
79	#13050573	—	2	2	no	—
80	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
81	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×D₈₃	C₂×C₄	1	4	yes	complemented
83	C₄⋊Dic₈₃	—	2	1	no	—
84	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
86	Dic₈₃⋊C₄	—	2	1	no	—
87	Dic₈₃⋊C₄	—	2	1	no	—
88	Dic₈₃⋊C₄	—	2	1	no	—
89	Dic₈₃⋊C₄	—	2	1	no	—
90	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
91	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
92	C₂²⋊Dic₈₃	—	2	1	no	—
93	(C₂×D₈₃)⋊C₄	—	2	1	no	—
94	(C₂×D₈₃)⋊C₄	—	2	1	no	—
95	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
96	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₈₃	—	2	1	no	—
98	C₄×Dic₈₃	—	2	1	no	—
99	C₄×Dic₈₃	—	2	1	no	—
100	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
101	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₂²×C₄×C₈₃	—	2	1	no	—
104	SD₁₆×C₈₃	C₂×C₄	1	4	yes	complemented
105	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
106	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
107	C₄⋊C₄×C₈₃	—	2	1	no	—
108	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
109	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
110	C₂²⋊C₄×C₈₃	—	2	1	no	—
111	C₄²×C₈₃	—	2	1	no	—
112	C₄²×C₈₃	C₂³	1	1	yes	characteristic
113	C₂³×C₈₃	—	2	1	no	—
114	C₂²×D₈₃	—	2	1	no	—
115	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₈₃	—	2	1	no	—
117	Q₈×C₈₃	C₂²×C₄	1	2	yes	—
118	D₄×C₈₃	—	2	1	no	—
119	D₄×C₈₃	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₈₃	—	2	1	no	—
121	C₂×C₄×C₈₃	—	2	1	no	—
122	C₂×C₄×C₈₃	—	2	1	no	—
123	C₂×C₄×C₈₃	—	8	1	no	—
124	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₈₃	—	2	4	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	4	no	—
130	C₂×Dic₈₃	—	2	1	no	—
131	C₂×Dic₈₃	—	2	1	no	—
132	C₂×Dic₈₃	—	8	1	no	—
133	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
136	D₃₃₂	—	2	4	no	—
137	D₃₃₂	—	2	1	no	—
138	D₃₃₂	C₂²×C₄	1	2	yes	—
139	C₄×D₈₃	—	2	4	no	—
140	C₄×D₈₃	—	8	1	no	—
141	C₄×D₈₃	—	8	1	no	—
142	C₄×D₈₃	—	8	1	no	—
143	C₄×D₈₃	—	8	1	no	—
144	C₄×D₈₃	C₂²×C₄	1	2	yes	—
145	C₄×D₈₃	#41	1	2	yes	—
146	Dic₁₆₆	—	2	1	no	—
147	Dic₁₆₆	C₂²×C₄	1	2	yes	—
148	C₈×C₈₃	—	2	1	no	—
149	C₈×C₈₃	C₂²×C₄	1	2	yes	—
150	C₈₃⋊C₈	—	2	1	no	—
151	C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
152	C₂²×C₈₃	—	2	2	no	—
153	C₂²×C₈₃	—	4	1	no	—
154	C₂²×C₈₃	—	4	1	no	—
155	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₈₃	—	2	1	no	—
157	C₂×D₈₃	—	2	1	no	—
158	C₂×D₈₃	—	2	2	no	—
159	C₂×D₈₃	—	4	1	no	—
160	C₂×D₈₃	—	4	1	no	—
161	C₂×D₈₃	C₄×D₄	1	2	yes	—
162	C₄×C₈₃	—	2	2	no	—
163	C₄×C₈₃	—	2	1	no	—
164	C₄×C₈₃	—	2	1	no	—
165	C₄×C₈₃	—	4	1	no	—
166	C₄×C₈₃	—	4	1	no	—
167	C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
168	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
169	Dic₈₃	—	2	2	no	—
170	Dic₈₃	—	2	1	no	—
171	Dic₈₃	—	2	1	no	—
172	Dic₈₃	—	4	1	no	—
173	Dic₈₃	—	4	1	no	—
174	Dic₈₃	C₄×D₄	1	2	yes	—
175	C₂×C₈₃	—	4	2	no	—
176	C₂×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₈₃	#577	1	1	yes	characteristic
178	C₂×C₈₃	#577	1	1	yes	characteristic
179	D₈₃	—	2	2	no	—
180	D₈₃	—	4	2	no	—
181	#2952	—	83	1	no	maximal, Sylow
182	C₈₃	#2952	1	1	yes	characteristic, complemented, Sylow
183	C₂×#135	—	83	1	no	—
184	#520	—	83	1	no	—
185	#520	—	83	1	no	—
186	#442	—	83	1	no	—
187	#442	—	83	1	no	—
188	#441	—	83	1	no	—
189	#441	—	83	1	no	—
190	C₄×SD₁₆	—	83	1	no	—
191	C₄×SD₁₆	—	83	1	no	—
192	C₄×SD₁₆	—	83	1	no	—
193	C₄×SD₁₆	D₈₃	1	1	yes	characteristic, complemented
194	C₂×#106	—	83	1	no	—
195	#417	—	83	1	no	—
196	#417	—	83	1	no	—
197	#405	—	83	1	no	—
198	C₂×#41	—	83	1	no	—
199	C₂×#41	—	83	1	no	—
200	#135	—	83	8	no	—
201	C₂×Q₁₆	—	83	1	no	—
202	C₂×SD₁₆	—	83	1	no	—
203	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
204	C₂×D₈	—	83	1	no	—
205	C₂×M₁₆	—	83	1	no	—
206	C₂²×C₈	—	83	1	no	—
207	C₄×Q₈	—	83	1	no	—
208	C₄×Q₈	—	83	1	no	—
209	C₄×Q₈	—	83	1	no	—
210	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
211	C₄×D₄	—	83	1	no	—
212	C₄×D₄	—	83	1	no	—
213	C₄×D₄	—	83	1	no	—
214	C₄×D₄	—	166	1	no	—
215	C₄×D₄	—	166	1	no	—
216	C₄×D₄	—	166	1	no	—
217	C₄×D₄	—	166	1	no	—
218	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
219	#117	—	83	1	no	—
220	#117	—	166	1	no	—
221	#117	—	166	1	no	—
222	C₂×C₄⋊C₄	—	83	1	no	—
223	C₂×C₄⋊C₄	—	83	1	no	—
224	C₂×C₄⋊C₄	—	166	1	no	—
225	C₂×C₄⋊C₄	—	166	1	no	—
226	#106	—	83	1	no	—
227	#106	—	83	1	no	—
228	#106	—	83	1	no	—
229	#106	C₂×D₈₃	1	1	yes	characteristic
230	#103	—	83	1	no	—
231	#103	—	83	1	no	—
232	#103	—	83	1	no	—
233	#103	C₂×D₈₃	1	1	yes	characteristic
234	#102	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	C₂×D₈₃	1	1	yes	characteristic
238	#97	—	83	1	no	—
239	#97	—	83	1	no	—
240	C₄×C₈	—	83	1	no	—
241	C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
242	#41	—	83	4	no	—
243	#41	—	83	4	no	—
244	#41	—	166	2	no	—
245	#41	—	166	2	no	—
246	C₂×Q₈	—	83	1	no	—
247	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₂×D₄	—	83	1	no	—
249	C₂×D₄	—	166	1	no	—
250	C₂×D₄	—	166	1	no	—
251	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	83	1	no	—
254	C₂²×C₄	—	166	1	no	—
255	C₂²×C₄	—	166	1	no	—
256	C₂²×C₄	—	166	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	C₂²×C₄	—	166	1	no	—
259	C₂²×C₄	—	166	1	no	—
260	C₂²×C₄	—	166	1	no	—
261	Q₁₆	—	83	4	no	—
262	SD₁₆	—	83	4	no	—
263	SD₁₆	C₄×D₈₃	1	4	yes	complemented
264	D₈	—	83	4	no	—
265	M₁₆	—	166	2	no	—
266	C₂×C₈	—	83	1	no	—
267	C₂×C₈	—	83	4	no	—
268	C₂×C₈	—	83	1	no	—
269	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	83	1	no	—
273	C₄⋊C₄	—	83	1	no	—
274	C₄⋊C₄	—	83	1	no	—
275	C₄⋊C₄	—	83	1	no	—
276	C₄⋊C₄	—	166	1	no	—
277	C₄⋊C₄	—	166	1	no	—
278	C₄⋊C₄	—	166	1	no	—
279	C₄⋊C₄	—	166	1	no	—
280	C₄⋊C₄	—	166	1	no	—
281	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	83	1	no	—
285	C₂²⋊C₄	—	83	1	no	—
286	C₂²⋊C₄	—	166	1	no	—
287	C₂²⋊C₄	—	166	1	no	—
288	C₂²⋊C₄	—	166	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	83	1	no	—
291	C₄²	—	166	1	no	—
292	C₄²	—	166	1	no	—
293	C₄²	—	166	1	no	—
294	C₄²	C₂²×D₈₃	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	83	1	no	—
297	C₂³	—	166	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	83	2	no	—
300	Q₈	—	166	1	no	—
301	Q₈	C₂×C₄×D₈₃	1	2	yes	—
302	D₄	—	2	1	no	—
303	D₄	—	83	2	no	—
304	D₄	—	166	4	no	—
305	D₄	—	166	4	no	—
306	D₄	—	166	1	no	—
307	D₄	C₂×C₄×D₈₃	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	83	1	no	—
313	C₂×C₄	—	83	2	no	—
314	C₂×C₄	—	83	2	no	—
315	C₂×C₄	—	83	1	no	—
316	C₂×C₄	—	83	1	no	—
317	C₂×C₄	—	664	1	no	—
318	C₂×C₄	—	664	1	no	—
319	C₂×C₄	—	664	1	no	—
320	C₂×C₄	—	664	1	no	—
321	C₂×C₄	—	664	1	no	—
322	C₂×C₄	—	166	4	no	—
323	C₂×C₄	—	166	1	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	—	166	4	no	—
326	C₂×C₄	—	166	1	no	—
327	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
328	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	83	2	no	—
332	C₈	—	166	1	no	—
333	C₈	C₂×C₄×D₈₃	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	332	1	no	—
338	C₂²	—	332	1	no	—
339	C₂²	—	83	2	no	—
340	C₂²	—	166	2	no	—
341	C₂²	—	166	1	no	—
342	C₂²	—	166	1	no	—
343	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	332	1	no	—
350	C₄	—	332	1	no	—
351	C₄	—	83	2	no	—
352	C₄	—	166	2	no	—
353	C₄	—	166	1	no	—
354	C₄	—	166	1	no	—
355	C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
356	C₄	#41×D₈₃	1	1	yes	characteristic
357	C₂	—	4	2	no	—
358	C₂	—	332	2	no	—
359	C₂	—	166	2	no	—
360	C₂	C₄×D₄×D₈₃	1	1	yes	characteristic
361	C₂	#19128507	1	1	yes	characteristic
362	C₂	#19128508	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator	C₄×C₈₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₄×SD₁₆×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#14581813	C₂
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₄×SD₁₆	D₈₃
hypocenter	C₈₃	#2952
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. #19128183	C₂	semidirect
3. #19128180	C₂	semidirect
4. #19127811	C₂	semidirect
5. #19127810	C₂	semidirect
6. #19127806	C₂	semidirect
7. #19127805	C₂	semidirect
8. C₄×#13050579	C₂	semidirect
9. SD₁₆×Dic₈₃	C₂	semidirect
10. #19127788	C₂	semidirect
11. #106×D₈₃	C₂	semidirect
12. #19127701	C₂	semidirect
13. #19127699	C₂	semidirect
14. #19127655	C₂	semidirect
15. C₄×SD₁₆×C₈₃	C₂	semidirect
18. #14581813	C₄	semidirect
61. #13050591	C₂×C₄	semidirect
63. #13050589	C₂×C₄	semidirect
76. #13050580	C₂×C₄	semidirect
77. #13050579	C₂×C₄	semidirect
78. #13050575	C₂×C₄	semidirect
82. C₈×D₈₃	C₂×C₄	semidirect
104. SD₁₆×C₈₃	C₂×C₄	semidirect
182. C₈₃	#2952	semidirect
193. C₄×SD₁₆	D₈₃	semidirect
263. SD₁₆	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14581813	2
#19128183	2
#19128180	2
#19127811	2
#19127810	2
#19127806	2
#19127805	2
C₄×#13050579	2
SD₁₆×Dic₈₃	2
#19127788	2
#106×D₈₃	2
#19127701	2
#19127699	2
#19127655	2
C₄×SD₁₆×C₈₃	2
#2952	83