Subgroups — #24917735

Order 10624.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128324	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128182	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128019	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127990	C₂	1	1	yes	maximal, characteristic, complemented
5	#19127985	C₂	1	1	yes	maximal, characteristic, complemented
6	#19127926	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127919	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127861	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127848	C₂	1	1	yes	maximal, characteristic
10	#19127812	C₂	1	1	yes	maximal, characteristic, complemented
11	#19127805	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127788	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127786	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127759	C₂	1	1	yes	maximal, characteristic, complemented
15	#483×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#13050590	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050579	C₂²	1	1	yes	characteristic
19	C₂×#13050577	C₂²	1	1	yes	characteristic
20	C₂×D₆₆₄	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581787	—	2	1	no	—
23	#14581781	C₂²	1	1	yes	characteristic
24	#14581778	—	2	1	no	—
25	#14581774	—	2	1	no	—
26	#14581772	—	2	1	no	—
27	#14581766	C₂²	1	1	yes	characteristic
28	Q₈×Dic₈₃	C₂²	1	1	yes	characteristic
29	#14581757	C₂²	1	1	yes	characteristic
30	#14581757	C₂²	1	1	yes	characteristic
31	#14581754	C₂²	1	1	yes	characteristic
32	#14581754	C₂²	1	1	yes	characteristic
33	#14581754	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₈₃	—	2	1	no	—
35	C₄×C₂²⋊D₈₃	—	2	1	no	—
36	#14581751	C₂²	1	1	yes	characteristic
37	#14581750	—	2	1	no	—
38	C₄⋊C₄×D₈₃	—	2	1	no	—
39	#14581723	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₈₃⋊C₈	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581715	C₂²	1	1	yes	characteristic
45	#14581714	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	C₂²⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	C₈×Dic₈₃	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
54	#106×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
59	#13050590	—	2	2	no	—
60	#13050590	—	2	2	no	—
61	C₂×Dic₁₆₆	—	2	1	no	—
62	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
64	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₈₃	—	2	1	no	—
66	C₂×C₂²⋊D₈₃	—	2	1	no	—
67	C₂×C₄×D₈₃	—	2	1	no	—
68	C₂×C₄×D₈₃	—	2	1	no	—
69	C₂×C₄×D₈₃	—	2	1	no	—
70	C₂×C₄×D₈₃	—	2	1	no	—
71	#13050580	—	2	2	no	—
72	#13050579	—	2	2	no	—
73	#13050577	—	2	2	no	—
74	D₆₆₄	—	2	2	no	—
75	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	—	2	1	no	—
78	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
79	C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
80	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₈₃	D₄	1	1	yes	characteristic
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	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	—	2	1	no	—
94	C₄×Dic₈₃	—	2	1	no	—
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	—	2	1	no	—
99	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₈₃	—	2	1	no	—
103	C₄⋊C₄×C₈₃	—	2	1	no	—
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	C₂²⋊C₄×C₈₃	—	2	1	no	—
108	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₈₃	—	2	1	no	—
110	C₂²×D₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	Q₈×C₈₃	—	2	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	—	2	1	no	—
118	C₂×C₄×C₈₃	—	2	1	no	—
119	C₂×C₄×C₈₃	—	8	1	no	—
120	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₈₃	—	2	4	no	—
123	C₂²⋊D₈₃	—	8	1	no	—
124	C₂×Dic₈₃	—	2	1	no	—
125	C₂×Dic₈₃	—	2	1	no	—
126	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
130	D₃₃₂	—	2	1	no	—
131	D₃₃₂	—	2	1	no	—
132	D₃₃₂	—	2	1	no	—
133	D₃₃₂	#41	1	2	yes	—
134	C₄×D₈₃	—	2	4	no	—
135	C₄×D₈₃	—	8	1	no	—
136	C₄×D₈₃	—	8	1	no	—
137	C₄×D₈₃	—	8	1	no	—
138	Dic₁₆₆	—	2	1	no	—
139	Dic₁₆₆	—	8	1	no	—
140	Dic₁₆₆	#41	1	2	yes	—
141	C₈×C₈₃	—	2	1	no	—
142	C₈×C₈₃	—	2	1	no	—
143	C₈₃⋊C₈	—	2	1	no	—
144	C₈₃⋊C₈	—	2	1	no	—
145	C₂²×C₈₃	—	2	1	no	—
146	C₂²×C₈₃	—	2	1	no	—
147	C₂²×C₈₃	—	2	1	no	—
148	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₈₃	—	2	2	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	C₄×C₈₃	—	2	1	no	—
156	C₄×C₈₃	—	4	1	no	—
157	C₄×C₈₃	—	4	1	no	—
158	C₄×C₈₃	—	4	1	no	—
159	C₄×C₈₃	—	4	1	no	—
160	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
161	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
162	Dic₈₃	—	2	2	no	—
163	Dic₈₃	—	2	1	no	—
164	Dic₈₃	—	2	1	no	—
165	Dic₈₃	—	2	1	no	—
166	Dic₈₃	—	2	1	no	—
167	Dic₈₃	—	4	1	no	—
168	C₂×C₈₃	—	4	1	no	—
169	C₂×C₈₃	#525	1	1	yes	characteristic
170	C₂×C₈₃	#577	1	1	yes	characteristic
171	C₂×C₈₃	#577	1	1	yes	characteristic
172	D₈₃	—	4	2	no	—
173	D₈₃	—	8	1	no	—
174	#3125	—	83	1	no	maximal, Sylow
175	C₈₃	#3125	1	1	yes	characteristic, complemented, Sylow
176	#542	—	83	1	no	—
177	#520	—	83	1	no	—
178	#483	D₈₃	1	1	yes	characteristic, complemented
179	#481	—	83	1	no	—
180	#476	—	83	1	no	—
181	#475	—	83	1	no	—
182	#464	—	83	1	no	—
183	#463	—	83	1	no	—
184	#452	—	83	1	no	—
185	#449	—	83	1	no	—
186	#442	—	83	1	no	—
187	#441	—	83	1	no	—
188	C₄×SD₁₆	—	83	1	no	—
189	C₄×SD₁₆	—	83	1	no	—
190	#433	—	83	1	no	—
191	C₂×#41	—	83	1	no	—
192	C₂×Q₁₆	—	83	1	no	—
193	C₂×SD₁₆	—	83	1	no	—
194	C₂×SD₁₆	—	83	1	no	—
195	C₂×D₈	—	83	1	no	—
196	C₄⋊Q₈	—	83	1	no	—
197	#126	—	166	1	no	—
198	#124	—	83	1	no	—
199	#124	—	166	1	no	—
200	#123	—	166	1	no	—
201	C₂²⋊Q₈	—	166	1	no	—
202	C₂²⋊Q₈	C₂×D₈₃	1	1	yes	characteristic
203	#121	—	83	1	no	—
204	C₄×Q₈	—	83	1	no	—
205	C₄×Q₈	—	83	1	no	—
206	C₄×Q₈	—	83	1	no	—
207	C₄×D₄	—	83	1	no	—
208	C₄×D₄	—	83	1	no	—
209	C₄×D₄	—	83	1	no	—
210	C₄×D₄	—	166	1	no	—
211	C₄×D₄	—	166	1	no	—
212	#117	—	83	1	no	—
213	#117	—	166	1	no	—
214	C₂×C₄⋊C₄	—	166	1	no	—
215	C₂×C₄⋊C₄	C₂×D₈₃	1	1	yes	characteristic
216	#106	—	83	1	no	—
217	#106	—	83	1	no	—
218	#106	C₂×D₈₃	1	1	yes	characteristic
219	#106	C₂×D₈₃	1	1	yes	characteristic
220	C₄⋊C₈	—	83	1	no	—
221	C₄⋊C₈	—	83	1	no	—
222	#103	—	83	1	no	—
223	#103	—	83	1	no	—
224	#103	C₂×D₈₃	1	1	yes	characteristic
225	#103	C₂×D₈₃	1	1	yes	characteristic
226	#102	—	83	1	no	—
227	#102	—	83	1	no	—
228	#102	—	83	1	no	—
229	#102	—	83	1	no	—
230	C₂²⋊C₈	—	83	1	no	—
231	C₂²⋊C₈	C₂×D₈₃	1	1	yes	characteristic
232	#97	—	83	1	no	—
233	C₄×C₈	—	83	1	no	—
234	#41	—	166	2	no	—
235	#41	—	166	2	no	—
236	C₂×Q₈	—	83	1	no	—
237	C₂×Q₈	—	166	1	no	—
238	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
239	C₂×D₄	—	83	1	no	—
240	C₂×D₄	—	83	1	no	—
241	C₂×D₄	—	166	1	no	—
242	C₂×D₄	—	166	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	166	1	no	—
245	C₂²×C₄	—	166	1	no	—
246	C₂²×C₄	—	166	1	no	—
247	C₂²×C₄	—	166	1	no	—
248	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
249	Q₁₆	—	166	2	no	—
250	SD₁₆	—	166	2	no	—
251	SD₁₆	—	166	2	no	—
252	D₈	—	166	2	no	—
253	C₂×C₈	—	83	1	no	—
254	C₂×C₈	—	83	1	no	—
255	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	1	no	—
259	C₄⋊C₄	—	83	1	no	—
260	C₄⋊C₄	—	83	1	no	—
261	C₄⋊C₄	—	83	1	no	—
262	C₄⋊C₄	—	166	1	no	—
263	C₄⋊C₄	—	166	1	no	—
264	C₄⋊C₄	—	166	1	no	—
265	C₄⋊C₄	—	166	1	no	—
266	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	83	1	no	—
271	C₂²⋊C₄	—	83	1	no	—
272	C₂²⋊C₄	—	166	1	no	—
273	C₂²⋊C₄	—	166	1	no	—
274	C₂²⋊C₄	—	166	1	no	—
275	C₂²⋊C₄	—	166	1	no	—
276	C₂²⋊C₄	—	166	1	no	—
277	C₂²⋊C₄	—	166	1	no	—
278	C₄²	—	83	1	no	—
279	C₄²	—	83	1	no	—
280	C₄²	—	166	1	no	—
281	C₄²	—	166	1	no	—
282	C₄²	—	166	1	no	—
283	C₂³	—	166	1	no	—
284	C₂³	—	166	1	no	—
285	C₂³	D₄×D₈₃	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	83	2	no	—
289	Q₈	—	664	1	no	—
290	Q₈	—	166	1	no	—
291	D₄	—	83	2	no	—
292	D₄	—	664	1	no	—
293	D₄	—	166	4	no	—
294	D₄	—	166	1	no	—
295	D₄	—	166	1	no	—
296	D₄	—	166	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	8	1	no	—
304	C₂×C₄	—	83	1	no	—
305	C₂×C₄	—	83	1	no	—
306	C₂×C₄	—	83	1	no	—
307	C₂×C₄	—	83	1	no	—
308	C₂×C₄	—	664	1	no	—
309	C₂×C₄	—	664	1	no	—
310	C₂×C₄	—	664	1	no	—
311	C₂×C₄	—	166	4	no	—
312	C₂×C₄	—	166	1	no	—
313	C₂×C₄	—	166	1	no	—
314	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	166	1	no	—
319	C₈	—	166	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	332	1	no	—
324	C₂²	—	332	1	no	—
325	C₂²	—	332	1	no	—
326	C₂²	—	332	1	no	—
327	C₂²	—	332	1	no	—
328	C₂²	—	166	2	no	—
329	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	332	1	no	—
336	C₄	—	166	2	no	—
337	C₄	—	166	1	no	—
338	C₄	—	166	1	no	—
339	C₄	—	166	1	no	—
340	C₄	—	166	1	no	—
341	C₄	#41×D₈₃	1	1	yes	characteristic
342	C₄	#41×D₈₃	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	332	2	no	—
345	C₂	—	664	1	no	—
346	C₂	#19128209	1	1	yes	characteristic
347	C₂	#19128507	1	1	yes	characteristic
348	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	#483×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#19127848	C₂
hypercenter	#483	D₈₃
hypocenter	C₈₃	#3125
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
1. #19128324	C₂	semidirect
2. #19128182	C₂	semidirect
3. #19128019	C₂	semidirect
4. #19127990	C₂	semidirect
5. #19127985	C₂	semidirect
6. #19127926	C₂	semidirect
7. #19127919	C₂	semidirect
8. #19127861	C₂	semidirect
10. #19127812	C₂	semidirect
11. #19127805	C₂	semidirect
12. #19127788	C₂	semidirect
13. #19127786	C₂	semidirect
14. #19127759	C₂	semidirect
15. #483×C₈₃	C₂	semidirect
175. C₈₃	#3125	semidirect
178. #483	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128324	2
#19128182	2
#19128019	2
#19127990	2
#19127985	2
#19127926	2
#19127919	2
#19127861	2
#19127848	2
#19127812	2
#19127805	2
#19127788	2
#19127786	2
#19127759	2
#483×C₈₃	2
#3125	83