Subgroups — #24917395

Order 10624.

414 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128214	C₂	1	1	yes	maximal, characteristic
2	#117×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128041	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128040	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128039	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128028	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128027	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128026	C₂	1	1	yes	maximal, characteristic, complemented
9	#19128025	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127742	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127740	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127701	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127701	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127663	C₂	1	1	yes	maximal, characteristic, complemented
15	#485×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050591	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₈₃	C₂²	1	1	yes	characteristic
18	C₂×#13050573	C₂²	1	1	yes	characteristic
19	C₂×#13050573	C₂²	1	1	yes	characteristic
20	#14581777	—	2	1	no	—
21	#14581768	C₂²	1	1	yes	characteristic
22	#14581767	C₂²	1	1	yes	characteristic
23	#14581766	C₂²	1	1	yes	characteristic
24	#14581762	—	2	1	no	—
25	#14581754	—	2	1	no	—
26	(C₂²⋊D₈₃)⋊C₄	—	2	1	no	—
27	#14581751	C₂²	1	1	yes	characteristic
28	#14581750	C₂²	1	1	yes	characteristic
29	#14581750	C₂²	1	1	yes	characteristic
30	#14581750	C₂²	1	1	yes	characteristic
31	#14581749	—	2	1	no	—
32	#14581748	—	2	1	no	—
33	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
35	C₂²⋊C₄×D₈₃	—	2	1	no	—
36	C₄²×D₈₃	—	2	1	no	—
37	#14581726	C₂²	1	1	yes	characteristic, complemented
38	#14581725	C₂²	1	1	yes	characteristic, complemented
39	#14581725	C₂²	1	1	yes	characteristic, complemented
40	#14581724	C₂²	1	1	yes	characteristic, complemented
41	#14581723	C₂²	1	1	yes	characteristic, complemented
42	#14581723	C₂²	1	1	yes	characteristic, complemented
43	#14581716	C₂²	1	1	yes	characteristic, complemented
44	#14581716	C₂²	1	1	yes	characteristic, complemented
45	#14581714	C₂²	1	1	yes	characteristic, complemented
46	#14581714	C₂²	1	1	yes	characteristic, complemented
47	#14581712	C₂²	1	1	yes	characteristic, complemented
48	#14581712	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	#117×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic, complemented
55	#106×C₈₃	C₂²	1	1	yes	characteristic, complemented
56	#103×C₈₃	C₂²	1	1	yes	characteristic, complemented
57	#103×C₈₃	C₂²	1	1	yes	characteristic, complemented
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	C₂³×D₈₃	D₄	1	1	yes	characteristic
60	#13050591	—	2	2	no	—
61	#13050591	—	2	2	no	—
62	C₂×D₃₃₂	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂²×Dic₈₃	D₄	1	1	yes	characteristic
67	C₂×C₄×D₈₃	—	2	2	no	—
68	C₂×C₄×D₈₃	—	2	2	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	—	2	1	no	—
72	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
73	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
74	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
76	#13050573	—	2	2	no	—
77	#13050573	—	2	2	no	—
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	Dic₈₃⋊C₄	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	(C₂×D₈₃)⋊C₄	—	2	1	no	—
88	(C₂×D₈₃)⋊C₄	—	2	1	no	—
89	(C₂×D₈₃)⋊C₄	—	2	1	no	—
90	(C₂×D₈₃)⋊C₄	—	2	1	no	—
91	(C₂×D₈₃)⋊C₄	—	2	1	no	—
92	(C₂×D₈₃)⋊C₄	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	—	2	1	no	—
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₂²⋊C₄×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	—	2	1	no	—
108	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₃	—	2	2	no	—
110	C₂²×D₈₃	—	2	1	no	—
111	C₂²×D₈₃	—	4	1	no	—
112	C₂²×D₈₃	—	4	1	no	—
113	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₈₃	—	2	1	no	—
116	Q₈×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₈₃	—	4	2	no	—
126	C₂×Dic₈₃	—	2	2	no	—
127	C₂×Dic₈₃	—	2	1	no	—
128	C₂×Dic₈₃	—	2	1	no	—
129	C₂×Dic₈₃	—	2	1	no	—
130	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	—	2	1	no	—
134	D₃₃₂	—	4	2	no	—
135	D₃₃₂	—	4	2	no	—
136	C₄×D₈₃	—	2	2	no	—
137	C₄×D₈₃	—	2	1	no	—
138	C₄×D₈₃	—	2	1	no	—
139	C₄×D₈₃	—	2	2	no	—
140	C₄×D₈₃	—	4	2	no	—
141	C₄×D₈₃	—	4	2	no	—
142	C₄×D₈₃	—	4	2	no	—
143	C₄×D₈₃	#41	1	2	yes	—
144	C₄×D₈₃	#41	1	2	yes	—
145	C₈×C₈₃	—	2	1	no	—
146	C₈×C₈₃	—	2	1	no	—
147	C₈₃⋊C₈	—	2	1	no	—
148	C₈₃⋊C₈	—	2	1	no	—
149	C₂²×C₈₃	—	2	1	no	—
150	C₂²×C₈₃	—	2	1	no	—
151	C₂²×C₈₃	—	2	1	no	—
152	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₈₃	—	2	1	no	—
154	C₂×D₈₃	—	2	1	no	—
155	C₂×D₈₃	—	2	1	no	—
156	C₂×D₈₃	—	2	1	no	—
157	C₂×D₈₃	—	2	1	no	—
158	C₂×D₈₃	—	4	4	no	—
159	C₂×D₈₃	—	4	1	no	—
160	C₂×D₈₃	—	4	1	no	—
161	C₂×D₈₃	—	4	1	no	—
162	C₂×D₈₃	#123	1	2	yes	—
163	C₄×C₈₃	—	2	1	no	—
164	C₄×C₈₃	—	4	1	no	—
165	C₄×C₈₃	—	4	1	no	—
166	C₄×C₈₃	—	4	1	no	—
167	C₄×C₈₃	—	4	1	no	—
168	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
169	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
170	Dic₈₃	—	2	1	no	—
171	Dic₈₃	—	4	1	no	—
172	Dic₈₃	—	4	1	no	—
173	Dic₈₃	—	4	1	no	—
174	Dic₈₃	#123	1	2	yes	—
175	C₂×C₈₃	—	4	1	no	—
176	C₂×C₈₃	C₂×#123	1	1	yes	characteristic
177	C₂×C₈₃	#575	1	1	yes	characteristic
178	C₂×C₈₃	#575	1	1	yes	characteristic
179	D₈₃	—	2	2	no	—
180	D₈₃	—	4	1	no	—
181	D₈₃	—	8	1	no	—
182	#3093	—	83	1	no	maximal, Sylow
183	C₈₃	#3093	1	1	yes	characteristic, complemented, Sylow
184	#525	—	83	1	no	—
185	C₂×#117	—	83	1	no	—
186	#485	—	83	1	no	—
187	#485	—	83	1	no	—
188	#485	—	83	1	no	—
189	#485	D₈₃	1	1	yes	characteristic, complemented
190	#482	—	83	1	no	—
191	#482	—	83	1	no	—
192	#482	—	83	1	no	—
193	#482	—	83	1	no	—
194	#428	—	83	1	no	—
195	#428	—	83	1	no	—
196	#417	—	83	1	no	—
197	#417	—	83	1	no	—
198	#407	—	83	1	no	—
199	C₂×#41	—	83	1	no	—
200	C₂³×C₄	—	83	1	no	—
201	C₂×M₁₆	—	83	1	no	—
202	C₂×M₁₆	—	83	1	no	—
203	#123	—	166	1	no	—
204	C₂²⋊Q₈	—	83	1	no	—
205	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
206	#121	—	83	1	no	—
207	#121	—	83	1	no	—
208	#120	—	166	1	no	—
209	C₄×D₄	—	166	1	no	—
210	C₄×D₄	—	166	1	no	—
211	#117	—	83	1	no	—
212	#117	—	83	1	no	—
213	#117	—	83	1	no	—
214	#117	—	83	1	no	—
215	#117	—	166	1	no	—
216	#117	—	166	1	no	—
217	#117	C₂×D₈₃	1	1	yes	characteristic
218	C₂×C₄⋊C₄	—	83	1	no	—
219	C₂×C₄⋊C₄	—	83	1	no	—
220	C₂×C₂²⋊C₄	—	166	1	no	—
221	C₂×C₄²	—	166	1	no	—
222	#107	—	83	1	no	—
223	#107	—	83	1	no	—
224	#107	—	83	1	no	—
225	#107	C₂×D₈₃	1	1	yes	characteristic, complemented
226	#106	—	83	1	no	—
227	#106	—	83	1	no	—
228	#106	—	83	1	no	—
229	#106	C₂×D₈₃	1	1	yes	characteristic, complemented
230	#103	—	83	1	no	—
231	#103	—	83	1	no	—
232	#103	C₂×D₈₃	1	1	yes	characteristic, complemented
233	#103	C₂×D₈₃	1	1	yes	characteristic, complemented
234	#102	—	83	1	no	—
235	#102	—	83	1	no	—
236	#102	—	83	1	no	—
237	#102	—	83	1	no	—
238	C₂²⋊C₈	—	83	1	no	—
239	C₂²⋊C₈	—	83	1	no	—
240	C₂²⋊C₈	—	83	1	no	—
241	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
242	C₂⁴	—	83	1	no	—
243	#41	—	166	2	no	—
244	#41	—	166	2	no	—
245	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
246	C₂×D₄	—	83	1	no	—
247	C₂×D₄	—	166	1	no	—
248	C₂×D₄	—	166	1	no	—
249	C₂×D₄	—	166	1	no	—
250	C₂²×C₄	—	83	1	no	—
251	C₂²×C₄	—	83	1	no	—
252	C₂²×C₄	—	83	1	no	—
253	C₂²×C₄	—	83	1	no	—
254	C₂²×C₄	—	83	1	no	—
255	C₂²×C₄	—	166	1	no	—
256	C₂²×C₄	—	166	1	no	—
257	C₂²×C₄	—	166	1	no	—
258	C₂²×C₄	—	166	2	no	—
259	C₂²×C₄	—	166	2	no	—
260	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
261	M₁₆	—	166	2	no	—
262	M₁₆	—	166	2	no	—
263	C₂×C₈	—	83	1	no	—
264	C₂×C₈	—	83	1	no	—
265	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	83	1	no	—
269	C₄⋊C₄	—	83	1	no	—
270	C₄⋊C₄	—	83	1	no	—
271	C₄⋊C₄	—	166	1	no	—
272	C₄⋊C₄	—	166	1	no	—
273	C₄⋊C₄	—	166	1	no	—
274	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
275	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	2	1	no	—
279	C₂²⋊C₄	—	166	1	no	—
280	C₂²⋊C₄	—	166	1	no	—
281	C₂²⋊C₄	—	166	1	no	—
282	C₂²⋊C₄	—	166	1	no	—
283	C₂²⋊C₄	—	166	1	no	—
284	C₂²⋊C₄	—	166	1	no	—
285	C₂²⋊C₄	—	166	1	no	—
286	C₄²	—	2	1	no	—
287	C₄²	—	166	1	no	—
288	C₄²	—	166	1	no	—
289	C₄²	—	166	1	no	—
290	C₄²	—	166	1	no	—
291	C₂³	—	332	1	no	—
292	C₂³	—	332	1	no	—
293	C₂³	—	83	1	no	—
294	C₂³	—	83	1	no	—
295	C₂³	—	166	2	no	—
296	C₂³	—	166	1	no	—
297	C₂³	D₄×D₈₃	1	1	yes	characteristic
298	Q₈	—	2	1	no	—
299	Q₈	—	2	1	no	—
300	D₄	—	332	2	no	—
301	D₄	—	332	2	no	—
302	D₄	—	332	2	no	—
303	D₄	—	166	1	no	—
304	D₄	—	166	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	332	2	no	—
312	C₂×C₄	—	332	2	no	—
313	C₂×C₄	—	332	2	no	—
314	C₂×C₄	—	83	1	no	—
315	C₂×C₄	—	83	2	no	—
316	C₂×C₄	—	83	2	no	—
317	C₂×C₄	—	83	1	no	—
318	C₂×C₄	—	166	2	no	—
319	C₂×C₄	—	166	2	no	—
320	C₂×C₄	—	166	1	no	—
321	C₂×C₄	—	166	1	no	—
322	C₂×C₄	—	166	1	no	—
323	C₂×C₄	—	166	2	no	—
324	C₂×C₄	—	166	1	no	—
325	C₂×C₄	—	166	1	no	—
326	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
327	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
328	C₈	—	2	1	no	—
329	C₈	—	2	1	no	—
330	C₈	—	166	1	no	—
331	C₈	—	166	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	—	2	1	no	—
335	C₂²	—	332	4	no	—
336	C₂²	—	332	1	no	—
337	C₂²	—	332	1	no	—
338	C₂²	—	332	1	no	—
339	C₂²	—	83	2	no	—
340	C₂²	—	166	1	no	—
341	C₂²	—	166	1	no	—
342	C₂²	—	166	1	no	—
343	C₂²	—	166	1	no	—
344	C₂²	—	166	1	no	—
345	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	332	1	no	—
352	C₄	—	332	1	no	—
353	C₄	—	332	1	no	—
354	C₄	—	83	2	no	—
355	C₄	—	166	1	no	—
356	C₄	#41×D₈₃	1	1	yes	characteristic
357	C₄	#41×D₈₃	1	1	yes	characteristic
358	C₂	—	4	1	no	—
359	C₂	—	332	1	no	—
360	C₂	—	664	1	no	—
361	C₂	—	166	2	no	—
362	C₂	#123×D₈₃	1	1	yes	characteristic
363	C₂	#19128498	1	1	yes	characteristic
364	C₂	#19128500	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#485×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19128214	C₂
hypercenter	#485	D₈₃
hypocenter	C₈₃	#3093
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
2. #117×D₈₃	C₂	semidirect
3. #19128041	C₂	semidirect
4. #19128040	C₂	semidirect
5. #19128039	C₂	semidirect
6. #19128028	C₂	semidirect
7. #19128027	C₂	semidirect
8. #19128026	C₂	semidirect
9. #19128025	C₂	semidirect
10. #19127742	C₂	semidirect
11. #19127740	C₂	semidirect
12. #19127701	C₂	semidirect
13. #19127701	C₂	semidirect
14. #19127663	C₂	semidirect
15. #485×C₈₃	C₂	semidirect
37. #14581726	C₂²	semidirect
38. #14581725	C₂²	semidirect
39. #14581725	C₂²	semidirect
40. #14581724	C₂²	semidirect
41. #14581723	C₂²	semidirect
42. #14581723	C₂²	semidirect
43. #14581716	C₂²	semidirect
44. #14581716	C₂²	semidirect
45. #14581714	C₂²	semidirect
46. #14581714	C₂²	semidirect
47. #14581712	C₂²	semidirect
48. #14581712	C₂²	semidirect
54. #107×C₈₃	C₂²	semidirect
55. #106×C₈₃	C₂²	semidirect
56. #103×C₈₃	C₂²	semidirect
57. #103×C₈₃	C₂²	semidirect
183. C₈₃	#3093	semidirect
189. #485	D₈₃	semidirect
225. #107	C₂×D₈₃	semidirect
229. #106	C₂×D₈₃	semidirect
232. #103	C₂×D₈₃	semidirect
233. #103	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128214	2
#117×D₈₃	2
#19128041	2
#19128040	2
#19128039	2
#19128028	2
#19128027	2
#19128026	2
#19128025	2
#19127742	2
#19127740	2
#19127701	2
#19127663	2
#485×C₈₃	2
#3093	83