Subgroups — #24918151

Order 10624.

374 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128325	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128307	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128090	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128058	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128049	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127955	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127948	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127938	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127931	C₂	1	1	yes	maximal, characteristic, complemented
10	#19127812	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127800	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127788	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127783	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127658	C₂	1	1	yes	maximal, characteristic, complemented
15	#501×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050579	C₂²	1	1	yes	characteristic
17	C₂×#13050579	C₂²	1	1	yes	characteristic
18	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
19	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
20	C₂×#13050573	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
22	#14581789	C₂²	1	1	yes	characteristic
23	#14581787	—	2	1	no	—
24	#14581781	—	2	1	no	—
25	#14581780	C₂²	1	1	yes	characteristic
26	#14581777	—	2	1	no	—
27	#14581777	—	2	1	no	—
28	#14581777	—	2	1	no	—
29	#14581772	C₂²	1	1	yes	characteristic
30	#14581768	—	2	1	no	—
31	#14581767	—	2	1	no	—
32	#14581767	—	2	1	no	—
33	#14581767	C₂²	1	1	yes	characteristic
34	#14581767	C₂²	1	1	yes	characteristic
35	#14581767	C₂²	1	1	yes	characteristic
36	#14581757	C₂²	1	1	yes	characteristic
37	#14581754	C₂²	1	1	yes	characteristic
38	#14581754	C₂²	1	1	yes	characteristic
39	#14581754	C₂²	1	1	yes	characteristic
40	#14581749	C₂²	1	1	yes	characteristic
41	#14581715	C₂²	1	1	yes	characteristic
42	#14581715	C₂²	1	1	yes	characteristic
43	#14581712	C₂²	1	1	yes	characteristic
44	#14581712	C₂²	1	1	yes	characteristic
45	#14581712	C₂²	1	1	yes	characteristic
46	#14581712	C₂²	1	1	yes	characteristic
47	#14581712	C₂²	1	1	yes	characteristic
48	#14581712	C₂²	1	1	yes	characteristic
49	#14581702	C₂²	1	1	yes	characteristic
50	C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	#125×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#106×C₈₃	C₂²	1	1	yes	characteristic
57	#106×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
60	C₂×D₃₃₂	—	2	1	no	—
61	C₂×D₃₃₂	—	2	1	no	—
62	C₂×D₃₃₂	—	2	1	no	—
63	C₂×D₃₃₂	—	2	1	no	—
64	C₂×D₃₃₂	—	2	1	no	—
65	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
66	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
67	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
72	#13050579	—	2	2	no	—
73	#13050579	—	2	2	no	—
74	D₆₆₄	—	2	2	no	—
75	D₆₆₄	—	2	2	no	—
76	#13050573	—	2	2	no	—
77	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
79	C₈×D₈₃	—	2	2	no	—
80	C₄⋊Dic₈₃	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
83	Dic₈₃⋊C₄	D₄	1	1	yes	characteristic
84	(C₂×D₈₃)⋊C₄	—	2	1	no	—
85	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	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	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
94	(C₂×D₈₃)⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	—	2	1	no	—
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	—	2	1	no	—
98	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
99	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
100	C₄⋊C₄×C₈₃	—	2	1	no	—
101	C₄⋊C₄×C₈₃	—	2	1	no	—
102	C₄⋊C₄×C₈₃	—	2	1	no	—
103	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
106	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂²×D₈₃	—	2	1	no	—
109	C₂²×D₈₃	—	2	1	no	—
110	C₂²×D₈₃	—	2	1	no	—
111	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₈₃	—	8	1	no	—
113	C₂×C₄×C₈₃	—	2	1	no	—
114	C₂×C₄×C₈₃	—	2	1	no	—
115	C₂×C₄×C₈₃	—	2	1	no	—
116	C₂×C₄×C₈₃	—	2	1	no	—
117	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₈₃	—	2	1	no	—
121	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
124	D₃₃₂	—	2	1	no	—
125	D₃₃₂	—	2	1	no	—
126	D₃₃₂	—	2	1	no	—
127	D₃₃₂	—	2	1	no	—
128	D₃₃₂	—	2	1	no	—
129	D₃₃₂	—	2	1	no	—
130	D₃₃₂	—	8	1	no	—
131	D₃₃₂	—	8	1	no	—
132	D₃₃₂	—	8	1	no	—
133	D₃₃₂	—	8	1	no	—
134	D₃₃₂	—	8	1	no	—
135	C₄×D₈₃	—	2	1	no	—
136	C₄×D₈₃	—	2	1	no	—
137	C₄×D₈₃	—	8	1	no	—
138	C₄×D₈₃	—	8	1	no	—
139	C₄×D₈₃	—	8	1	no	—
140	Dic₁₆₆	—	2	1	no	—
141	Dic₁₆₆	—	2	1	no	—
142	C₈×C₈₃	#41	1	2	yes	—
143	C₈×C₈₃	#41	1	2	yes	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₈₃⋊C₈	—	2	1	no	—
146	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₈₃	—	2	1	no	—
148	C₂×D₈₃	—	2	1	no	—
149	C₂×D₈₃	—	2	1	no	—
150	C₂×D₈₃	—	4	1	no	—
151	C₂×D₈₃	—	4	1	no	—
152	C₂×D₈₃	—	4	1	no	—
153	C₂×D₈₃	—	4	1	no	—
154	C₂×D₈₃	—	4	1	no	—
155	C₂×D₈₃	—	4	1	no	—
156	C₂×D₈₃	—	4	1	no	—
157	C₂×D₈₃	—	4	1	no	—
158	C₂×D₈₃	—	4	1	no	—
159	C₄×C₈₃	—	2	1	no	—
160	C₄×C₈₃	—	2	1	no	—
161	C₄×C₈₃	—	4	1	no	—
162	C₄×C₈₃	—	4	1	no	—
163	C₄×C₈₃	—	4	1	no	—
164	C₄×C₈₃	—	4	1	no	—
165	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
166	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
167	Dic₈₃	—	2	1	no	—
168	Dic₈₃	—	2	1	no	—
169	Dic₈₃	—	2	1	no	—
170	Dic₈₃	—	4	1	no	—
171	C₂×C₈₃	#532	1	1	yes	characteristic
172	C₂×C₈₃	#576	1	1	yes	characteristic
173	C₂×C₈₃	#577	1	1	yes	characteristic
174	D₈₃	—	4	1	no	—
175	D₈₃	—	8	1	no	—
176	D₈₃	—	8	1	no	—
177	D₈₃	—	8	1	no	—
178	#3179	—	83	1	no	maximal, Sylow
179	C₈₃	#3179	1	1	yes	characteristic, complemented, Sylow
180	#542	—	83	1	no	—
181	#540	—	83	1	no	—
182	#501	D₈₃	1	1	yes	characteristic, complemented
183	#496	—	83	1	no	—
184	#488	—	83	1	no	—
185	#486	—	83	1	no	—
186	#469	—	83	1	no	—
187	#468	—	83	1	no	—
188	#466	—	83	1	no	—
189	#465	—	83	1	no	—
190	#442	—	83	1	no	—
191	#440	—	83	1	no	—
192	C₄×SD₁₆	—	83	1	no	—
193	C₄×D₈	—	83	1	no	—
194	#405	—	83	1	no	—
195	C₂×SD₁₆	—	83	1	no	—
196	C₂×SD₁₆	—	83	1	no	—
197	C₂×D₈	—	83	1	no	—
198	C₂×D₈	—	83	1	no	—
199	C₂×M₁₆	—	83	1	no	—
200	C₂²×C₈	—	83	1	no	—
201	C₄⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
202	#127	—	83	1	no	—
203	#126	—	166	1	no	—
204	#125	C₂×D₈₃	1	1	yes	characteristic
205	#124	—	83	1	no	—
206	#124	—	166	1	no	—
207	#123	—	166	1	no	—
208	#123	—	166	1	no	—
209	#123	—	166	1	no	—
210	C₂²⋊Q₈	—	83	1	no	—
211	C₂²⋊Q₈	—	166	1	no	—
212	#121	—	83	1	no	—
213	#121	—	83	1	no	—
214	#121	—	83	1	no	—
215	#121	—	166	1	no	—
216	#121	—	166	1	no	—
217	C₄×Q₈	—	83	1	no	—
218	C₄×D₄	—	83	1	no	—
219	C₄×D₄	—	83	1	no	—
220	C₄×D₄	—	83	1	no	—
221	#117	—	83	1	no	—
222	#107	C₂×D₈₃	1	1	yes	characteristic
223	#107	C₂×D₈₃	1	1	yes	characteristic
224	#106	C₂×D₈₃	1	1	yes	characteristic
225	#106	C₂×D₈₃	1	1	yes	characteristic
226	#103	—	83	1	no	—
227	#103	—	83	1	no	—
228	#102	—	83	1	no	—
229	#102	—	83	1	no	—
230	#102	—	83	1	no	—
231	#102	—	83	1	no	—
232	#102	—	83	1	no	—
233	#102	—	83	1	no	—
234	#97	—	83	1	no	—
235	#97	C₂×D₈₃	1	1	yes	characteristic
236	C₄×C₈	—	83	1	no	—
237	C₄×C₈	—	83	1	no	—
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	83	1	no	—
240	C₂×D₄	—	83	1	no	—
241	C₂×D₄	—	83	1	no	—
242	C₂×D₄	—	83	1	no	—
243	C₂×D₄	—	166	1	no	—
244	C₂×D₄	—	166	1	no	—
245	C₂×D₄	—	166	1	no	—
246	C₂×D₄	—	166	1	no	—
247	C₂×D₄	—	166	1	no	—
248	C₂²×C₄	—	83	1	no	—
249	C₂²×C₄	—	166	1	no	—
250	C₂²×C₄	—	166	1	no	—
251	C₂²×C₄	—	166	1	no	—
252	SD₁₆	—	166	2	no	—
253	SD₁₆	—	166	2	no	—
254	D₈	—	166	2	no	—
255	D₈	—	166	2	no	—
256	M₁₆	—	166	2	no	—
257	C₂×C₈	—	83	1	no	—
258	C₂×C₈	—	83	1	no	—
259	C₂×C₈	—	166	2	no	—
260	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	83	1	no	—
266	C₄⋊C₄	—	83	1	no	—
267	C₄⋊C₄	—	166	1	no	—
268	C₄⋊C₄	—	166	1	no	—
269	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
273	C₂²⋊C₄	—	83	1	no	—
274	C₂²⋊C₄	—	83	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₂²⋊C₄	—	166	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	C₄²	—	83	1	no	—
285	C₄²	—	166	1	no	—
286	C₄²	C₂²×D₈₃	1	1	yes	characteristic
287	C₂³	—	83	1	no	—
288	C₂³	—	166	1	no	—
289	C₂³	—	166	1	no	—
290	C₂³	—	166	1	no	—
291	Q₈	—	8	1	no	—
292	Q₈	—	166	1	no	—
293	Q₈	—	166	1	no	—
294	D₄	—	664	1	no	—
295	D₄	—	664	1	no	—
296	D₄	—	664	1	no	—
297	D₄	—	664	1	no	—
298	D₄	—	664	1	no	—
299	D₄	—	166	1	no	—
300	D₄	—	166	1	no	—
301	D₄	—	166	1	no	—
302	D₄	—	166	1	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	—	83	1	no	—
310	C₂×C₄	—	83	1	no	—
311	C₂×C₄	—	83	1	no	—
312	C₂×C₄	—	664	1	no	—
313	C₂×C₄	—	664	1	no	—
314	C₂×C₄	—	664	1	no	—
315	C₂×C₄	—	166	1	no	—
316	C₂×C₄	—	166	1	no	—
317	C₂×C₄	—	166	1	no	—
318	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
319	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
320	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
321	C₈	—	166	1	no	—
322	C₈	—	166	1	no	—
323	C₈	#13050591	1	2	yes	—
324	C₈	#13050591	1	2	yes	—
325	C₂²	—	332	1	no	—
326	C₂²	—	332	1	no	—
327	C₂²	—	332	1	no	—
328	C₂²	—	332	1	no	—
329	C₂²	—	332	1	no	—
330	C₂²	—	332	1	no	—
331	C₂²	—	332	1	no	—
332	C₂²	—	332	1	no	—
333	C₂²	—	332	1	no	—
334	C₂²	—	166	1	no	—
335	C₂²	—	166	1	no	—
336	C₂²	—	166	1	no	—
337	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	4	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	332	1	no	—
345	C₄	—	166	1	no	—
346	C₄	—	166	1	no	—
347	C₄	—	166	1	no	—
348	C₄	C₂×#13050591	1	1	yes	characteristic
349	C₄	C₂×#13050591	1	1	yes	characteristic
350	C₂	—	332	1	no	—
351	C₂	—	664	1	no	—
352	C₂	—	664	1	no	—
353	C₂	—	664	1	no	—
354	C₂	#19128252	1	1	yes	characteristic
355	C₂	#19128505	1	1	yes	characteristic
356	C₂	#19128510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#501×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
hypercenter	#501	D₈₃
hypocenter	C₈₃	#3179
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #501

Complements

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

N	H ≅ G/N	type
1. #19128325	C₂	semidirect
2. #19128307	C₂	semidirect
3. #19128090	C₂	semidirect
4. #19128058	C₂	semidirect
5. #19128049	C₂	semidirect
6. #19127955	C₂	semidirect
7. #19127948	C₂	semidirect
8. #19127938	C₂	semidirect
9. #19127931	C₂	semidirect
10. #19127812	C₂	semidirect
11. #19127800	C₂	semidirect
12. #19127788	C₂	semidirect
13. #19127783	C₂	semidirect
14. #19127658	C₂	semidirect
15. #501×C₈₃	C₂	semidirect
179. C₈₃	#3179	semidirect
182. #501	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128325	2
#19128307	2
#19128090	2
#19128058	2
#19128049	2
#19127955	2
#19127948	2
#19127938	2
#19127931	2
#19127812	2
#19127800	2
#19127788	2
#19127783	2
#19127658	2
#501×C₈₃	2
#3179	83