Subgroups — #24917010

Order 10624.

472 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	C₂×#14581824	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14581815	C₂	1	1	yes	maximal, characteristic
3	#19128276	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128183	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127925	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127923	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127919	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127917	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127909	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127905	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127903	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127726	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127701	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#462×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂×#13050591	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
19	#14581824	—	2	4	no	—
20	#14581815	—	2	4	no	—
21	C₂×#13050580	C₂²	1	1	yes	characteristic
22	C₂×#13050579	C₂²	1	1	yes	characteristic
23	C₂×#13050577	C₂²	1	1	yes	characteristic
24	C₂×#13050577	C₂²	1	1	yes	characteristic
25	C₂×#13050577	C₂²	1	1	yes	characteristic
26	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
27	C₂×#13050573	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
29	#14581790	C₂²	1	1	yes	characteristic
30	#14581781	C₂²	1	1	yes	characteristic
31	#14581780	C₂²	1	1	yes	characteristic
32	#14581777	—	2	1	no	—
33	#14581777	—	2	1	no	—
34	#14581770	—	2	1	no	—
35	#14581770	—	2	1	no	—
36	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
37	C₄×D₃₃₂	—	2	1	no	—
38	C₄×D₃₃₂	C₂²	1	1	yes	characteristic
39	#14581754	—	2	1	no	—
40	#14581754	C₂²	1	1	yes	characteristic
41	#14581749	—	2	1	no	—
42	#14581749	C₂²	1	1	yes	characteristic
43	C₄⋊C₄×D₈₃	—	2	1	no	—
44	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
45	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
46	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
48	#14581717	C₂²	1	1	yes	characteristic
49	#14581716	C₂²	1	1	yes	characteristic
50	#14581716	C₂²	1	1	yes	characteristic
51	#14581715	C₂²	1	1	yes	characteristic
52	#14581714	C₂²	1	1	yes	characteristic
53	#14581712	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
55	C₂×Q₁₆×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#103×C₈₃	C₂²	1	1	yes	characteristic
60	#103×C₈₃	C₂²	1	1	yes	characteristic
61	#13050591	—	2	2	no	—
62	#13050591	—	2	2	no	—
63	#13050591	—	2	2	no	—
64	#13050591	D₄	1	4	yes	—
65	C₂×Dic₁₆₆	—	2	1	no	—
66	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
67	Q₈×D₈₃	—	2	2	no	—
68	Q₈×D₈₃	—	2	2	no	—
69	C₂×D₃₃₂	—	2	1	no	—
70	C₂×D₃₃₂	—	2	1	no	—
71	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
72	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₈₃	—	2	1	no	—
74	C₂×C₄×D₈₃	—	2	1	no	—
75	C₂×C₄×D₈₃	—	2	1	no	—
76	C₂×C₄×D₈₃	—	2	1	no	—
77	C₂×C₄×D₈₃	—	2	1	no	—
78	C₂×C₄×D₈₃	—	2	1	no	—
79	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
80	#13050580	—	2	2	no	—
81	#13050579	—	2	2	no	—
82	#13050577	—	2	2	no	—
83	#13050577	—	2	2	no	—
84	#13050577	—	2	2	no	—
85	D₆₆₄	—	2	2	no	—
86	#13050573	—	2	2	no	—
87	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
88	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
89	C₈×D₈₃	—	2	2	no	—
90	C₄⋊Dic₈₃	—	2	1	no	—
91	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
92	Dic₈₃⋊C₄	—	2	1	no	—
93	Dic₈₃⋊C₄	—	2	1	no	—
94	Dic₈₃⋊C₄	—	2	1	no	—
95	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
96	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
97	(C₂×D₈₃)⋊C₄	—	2	1	no	—
98	(C₂×D₈₃)⋊C₄	—	2	1	no	—
99	(C₂×D₈₃)⋊C₄	—	2	1	no	—
100	(C₂×D₈₃)⋊C₄	—	2	1	no	—
101	(C₂×D₈₃)⋊C₄	—	2	1	no	—
102	(C₂×D₈₃)⋊C₄	—	2	1	no	—
103	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
104	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
105	C₄×Dic₈₃	—	2	1	no	—
106	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
107	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
108	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
109	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
110	Q₁₆×C₈₃	—	2	2	no	—
111	Q₁₆×C₈₃	—	2	2	no	—
112	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
113	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
114	C₄⋊C₄×C₈₃	—	2	1	no	—
115	C₄⋊C₄×C₈₃	—	2	1	no	—
116	C₄⋊C₄×C₈₃	—	2	1	no	—
117	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
118	C₄²×C₈₃	—	2	1	no	—
119	C₄²×C₈₃	C₂³	1	1	yes	characteristic
120	C₂²×D₈₃	—	2	1	no	—
121	C₂²×D₈₃	—	2	1	no	—
122	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
123	Q₈×C₈₃	—	2	1	no	—
124	Q₈×C₈₃	—	2	1	no	—
125	Q₈×C₈₃	—	2	1	no	—
126	Q₈×C₈₃	—	2	1	no	—
127	Q₈×C₈₃	—	2	1	no	—
128	Q₈×C₈₃	C₂×D₄	1	2	yes	—
129	C₂×C₄×C₈₃	—	2	1	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	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
136	C₂×Dic₈₃	—	2	1	no	—
137	C₂×Dic₈₃	—	2	1	no	—
138	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
139	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
141	D₃₃₂	—	2	4	no	—
142	D₃₃₂	—	2	1	no	—
143	D₃₃₂	—	2	1	no	—
144	D₃₃₂	—	2	1	no	—
145	D₃₃₂	—	4	2	no	—
146	D₃₃₂	C₂×D₄	1	2	yes	—
147	C₄×D₈₃	—	2	4	no	—
148	C₄×D₈₃	—	4	2	no	—
149	C₄×D₈₃	—	4	2	no	—
150	C₄×D₈₃	—	8	1	no	—
151	C₄×D₈₃	—	8	1	no	—
152	C₄×D₈₃	—	8	1	no	—
153	C₄×D₈₃	C₂×D₄	1	2	yes	—
154	C₄×D₈₃	#41	1	2	yes	—
155	Dic₁₆₆	—	2	1	no	—
156	Dic₁₆₆	—	2	1	no	—
157	Dic₁₆₆	—	4	2	no	—
158	C₈×C₈₃	—	2	1	no	—
159	C₈×C₈₃	—	2	1	no	—
160	C₈₃⋊C₈	—	2	1	no	—
161	C₈₃⋊C₈	—	2	1	no	—
162	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₈₃	—	2	1	no	—
164	C₂×D₈₃	—	2	1	no	—
165	C₂×D₈₃	—	2	2	no	—
166	C₂×D₈₃	—	4	1	no	—
167	C₂×D₈₃	—	4	1	no	—
168	C₂×D₈₃	—	4	1	no	—
169	C₂×D₈₃	—	4	1	no	—
170	C₂×D₈₃	—	4	1	no	—
171	C₂×D₈₃	#121	1	2	yes	—
172	C₄×C₈₃	—	2	2	no	—
173	C₄×C₈₃	—	2	1	no	—
174	C₄×C₈₃	—	2	1	no	—
175	C₄×C₈₃	—	4	1	no	—
176	C₄×C₈₃	—	4	1	no	—
177	C₄×C₈₃	—	4	1	no	—
178	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
179	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
180	Dic₈₃	—	2	1	no	—
181	Dic₈₃	—	2	1	no	—
182	Dic₈₃	—	4	1	no	—
183	Dic₈₃	—	4	1	no	—
184	Dic₈₃	#121	1	2	yes	—
185	C₂×C₈₃	C₂×#121	1	1	yes	characteristic
186	C₂×C₈₃	#577	1	1	yes	characteristic
187	C₂×C₈₃	#578	1	1	yes	characteristic
188	D₈₃	—	2	2	no	—
189	D₈₃	—	4	2	no	—
190	D₈₃	—	8	1	no	—
191	#3046	—	83	1	no	maximal, Sylow
192	C₈₃	#3046	1	1	yes	characteristic, complemented, Sylow
193	C₂×#137	—	83	1	no	—
194	C₂×#135	—	83	1	no	—
195	#536	—	83	1	no	—
196	#520	—	83	1	no	—
197	#464	—	83	1	no	—
198	#464	—	83	1	no	—
199	#463	—	83	1	no	—
200	#463	—	83	1	no	—
201	#462	—	83	1	no	—
202	#462	D₈₃	1	1	yes	characteristic, complemented
203	#461	—	83	1	no	—
204	#461	—	83	1	no	—
205	#424	—	83	1	no	—
206	#417	—	83	1	no	—
207	C₂×#103	—	83	1	no	—
208	C₂×#41	—	83	1	no	—
209	C₂×#41	—	83	1	no	—
210	C₂²×Q₈	—	83	1	no	—
211	#137	—	166	4	no	—
212	#135	—	166	4	no	—
213	C₂×Q₁₆	—	83	1	no	—
214	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
215	C₂×Q₁₆	C₂×D₈₃	1	1	yes	characteristic
216	C₂×SD₁₆	—	83	1	no	—
217	C₂×SD₁₆	—	83	1	no	—
218	C₂×SD₁₆	—	83	1	no	—
219	C₂×SD₁₆	—	83	1	no	—
220	C₂×D₈	—	83	1	no	—
221	C₂×M₁₆	—	83	1	no	—
222	C₂²×C₈	—	83	1	no	—
223	C₄⋊Q₈	—	83	1	no	—
224	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
225	#124	—	83	1	no	—
226	#124	—	83	1	no	—
227	#123	—	166	1	no	—
228	#123	—	166	1	no	—
229	C₂²⋊Q₈	—	166	1	no	—
230	C₂²⋊Q₈	—	166	1	no	—
231	C₄×Q₈	—	83	1	no	—
232	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
233	C₄×D₄	—	83	1	no	—
234	C₄×D₄	—	83	1	no	—
235	C₄×D₄	—	166	1	no	—
236	C₄×D₄	—	166	1	no	—
237	#117	—	83	1	no	—
238	#117	—	166	1	no	—
239	C₂×C₄⋊C₄	—	83	1	no	—
240	C₂×C₄⋊C₄	—	166	1	no	—
241	C₄⋊C₈	—	83	1	no	—
242	C₄⋊C₈	—	83	1	no	—
243	C₄⋊C₈	—	83	1	no	—
244	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
245	#103	—	83	1	no	—
246	#103	—	83	1	no	—
247	#103	—	83	1	no	—
248	#103	—	83	1	no	—
249	#103	C₂×D₈₃	1	1	yes	characteristic
250	#103	C₂×D₈₃	1	1	yes	characteristic
251	#102	—	83	1	no	—
252	#102	—	83	1	no	—
253	#41	—	83	4	no	—
254	#41	—	166	2	no	—
255	#41	—	166	2	no	—
256	#41	—	166	2	no	—
257	C₂×Q₈	—	83	1	no	—
258	C₂×Q₈	—	166	1	no	—
259	C₂×Q₈	—	166	2	no	—
260	C₂×Q₈	—	166	2	no	—
261	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
263	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
264	C₂×D₄	—	83	1	no	—
265	C₂×D₄	—	83	1	no	—
266	C₂×D₄	—	166	1	no	—
267	C₂×D₄	—	166	1	no	—
268	C₂²×C₄	—	83	1	no	—
269	C₂²×C₄	—	166	1	no	—
270	C₂²×C₄	—	166	1	no	—
271	C₂²×C₄	—	166	1	no	—
272	C₂²×C₄	—	166	1	no	—
273	C₂²×C₄	—	166	1	no	—
274	C₂²×C₄	—	166	1	no	—
275	Q₁₆	—	2	2	no	—
276	Q₁₆	—	2	2	no	—
277	Q₁₆	—	166	2	no	—
278	SD₁₆	—	166	2	no	—
279	SD₁₆	—	166	2	no	—
280	SD₁₆	—	166	2	no	—
281	SD₁₆	—	166	2	no	—
282	D₈	—	166	2	no	—
283	M₁₆	—	166	2	no	—
284	C₂×C₈	—	83	1	no	—
285	C₂×C₈	—	83	1	no	—
286	C₂×C₈	—	166	2	no	—
287	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
288	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
289	C₄⋊C₄	—	2	1	no	—
290	C₄⋊C₄	—	2	1	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	83	1	no	—
293	C₄⋊C₄	—	83	1	no	—
294	C₄⋊C₄	—	83	1	no	—
295	C₄⋊C₄	—	166	1	no	—
296	C₄⋊C₄	—	166	1	no	—
297	C₄⋊C₄	—	166	1	no	—
298	C₄⋊C₄	—	166	1	no	—
299	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
300	C₂²⋊C₄	—	83	1	no	—
301	C₂²⋊C₄	—	83	1	no	—
302	C₂²⋊C₄	—	166	1	no	—
303	C₂²⋊C₄	—	166	1	no	—
304	C₂²⋊C₄	—	166	1	no	—
305	C₂²⋊C₄	—	166	1	no	—
306	C₂²⋊C₄	—	166	1	no	—
307	C₂²⋊C₄	—	166	1	no	—
308	C₄²	—	2	1	no	—
309	C₄²	—	83	1	no	—
310	C₄²	—	166	1	no	—
311	C₄²	C₂²×D₈₃	1	1	yes	characteristic
312	C₂³	—	83	1	no	—
313	C₂³	—	166	1	no	—
314	C₂³	—	166	1	no	—
315	Q₈	—	2	1	no	—
316	Q₈	—	2	1	no	—
317	Q₈	—	2	1	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	332	2	no	—
321	Q₈	—	166	1	no	—
322	Q₈	—	166	1	no	—
323	Q₈	D₄×D₈₃	1	2	yes	—
324	D₄	—	332	2	no	—
325	D₄	—	83	2	no	—
326	D₄	—	166	4	no	—
327	D₄	—	166	1	no	—
328	D₄	—	166	1	no	—
329	D₄	—	166	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	332	2	no	—
335	C₂×C₄	—	332	2	no	—
336	C₂×C₄	—	83	1	no	—
337	C₂×C₄	—	83	2	no	—
338	C₂×C₄	—	83	2	no	—
339	C₂×C₄	—	83	1	no	—
340	C₂×C₄	—	83	1	no	—
341	C₂×C₄	—	664	1	no	—
342	C₂×C₄	—	664	1	no	—
343	C₂×C₄	—	664	1	no	—
344	C₂×C₄	—	166	4	no	—
345	C₂×C₄	—	166	1	no	—
346	C₂×C₄	—	166	1	no	—
347	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
348	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
349	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
350	C₈	—	2	1	no	—
351	C₈	—	2	1	no	—
352	C₈	—	166	1	no	—
353	C₈	—	166	1	no	—
354	C₂²	—	332	1	no	—
355	C₂²	—	332	1	no	—
356	C₂²	—	332	1	no	—
357	C₂²	—	332	1	no	—
358	C₂²	—	332	1	no	—
359	C₂²	—	83	2	no	—
360	C₂²	—	166	2	no	—
361	C₂²	—	166	1	no	—
362	C₂²	—	166	1	no	—
363	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
364	C₄	—	2	2	no	—
365	C₄	—	2	1	no	—
366	C₄	—	2	1	no	—
367	C₄	—	4	1	no	—
368	C₄	—	4	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	332	1	no	—
371	C₄	—	332	1	no	—
372	C₄	—	83	2	no	—
373	C₄	—	166	1	no	—
374	C₄	—	166	1	no	—
375	C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
376	C₄	#41×D₈₃	1	1	yes	characteristic
377	C₂	—	332	2	no	—
378	C₂	—	664	1	no	—
379	C₂	—	166	2	no	—
380	C₂	#121×D₈₃	1	1	yes	characteristic
381	C₂	#19128510	1	1	yes	characteristic
382	C₂	#19128514	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#462×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×#14581815	C₂
hypercenter	#462	D₈₃
hypocenter	C₈₃	#3046
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
1. C₂×#14581824	C₂	semidirect
3. #19128276	C₂	semidirect
4. #19128183	C₂	semidirect
5. #19127925	C₂	semidirect
6. #19127923	C₂	semidirect
7. #19127919	C₂	semidirect
8. #19127917	C₂	semidirect
9. #19127909	C₂	semidirect
10. #19127905	C₂	semidirect
11. #19127903	C₂	semidirect
12. #19127726	C₂	semidirect
13. #19127701	C₂	semidirect
14. #103×D₈₃	C₂	semidirect
15. #462×C₈₃	C₂	semidirect
192. C₈₃	#3046	semidirect
202. #462	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14581824	2
C₂×#14581815	2
#19128276	2
#19128183	2
#19127925	2
#19127923	2
#19127919	2
#19127917	2
#19127909	2
#19127905	2
#19127903	2
#19127726	2
#19127701	2
#103×D₈₃	2
#462×C₈₃	2
#3046	83