Subgroups — #24918168

Order 10624.

366 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128255	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128233	C₂	1	1	yes	maximal, characteristic
3	#19128109	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128092	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128068	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128059	C₂	1	1	yes	maximal, characteristic, complemented
7	#19127943	C₂	1	1	yes	maximal, characteristic, complemented
8	#19127941	C₂	1	1	yes	maximal, characteristic, complemented
9	#19127929	C₂	1	1	yes	maximal, characteristic
10	#19127927	C₂	1	1	yes	maximal, characteristic
11	#19127809	C₂	1	1	yes	maximal, characteristic, complemented
12	#19127806	C₂	1	1	yes	maximal, characteristic, complemented
13	#19127801	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₃₃₂	C₂²	1	1	yes	characteristic
17	C₂×#13050580	C₂²	1	1	yes	characteristic
18	C₂×#13050578	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₈₃	C₂²	1	1	yes	characteristic
21	#14581791	C₂²	1	1	yes	characteristic
22	#14581790	C₂²	1	1	yes	characteristic
23	#14581785	—	2	1	no	—
24	#14581783	C₂²	1	1	yes	characteristic
25	#14581782	—	2	1	no	—
26	#14581779	C₂²	1	1	yes	characteristic
27	#14581774	—	2	1	no	—
28	#14581772	C₂²	1	1	yes	characteristic
29	#14581772	C₂²	1	1	yes	characteristic
30	#14581770	—	2	1	no	—
31	#14581768	C₂²	1	1	yes	characteristic
32	#14581763	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₆₆	—	2	1	no	—
34	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₈₃	—	2	1	no	—
37	#14581751	—	2	1	no	—
38	#14581749	—	2	1	no	—
39	C₄²×D₈₃	C₂²	1	1	yes	characteristic
40	#14581725	C₂²	1	1	yes	characteristic
41	#14581724	C₂²	1	1	yes	characteristic
42	#14581723	C₂²	1	1	yes	characteristic
43	#14581717	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	#14581715	C₂²	1	1	yes	characteristic
48	#14581713	C₂²	1	1	yes	characteristic
49	#14581703	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	#14581702	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
55	#107×C₈₃	C₂²	1	1	yes	characteristic
56	#103×C₈₃	C₂²	1	1	yes	characteristic
57	#102×C₈₃	C₂²	1	1	yes	characteristic
58	#97×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₈₃	—	2	1	no	—
63	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
64	C₂×C₄×D₈₃	D₄	1	1	yes	characteristic
65	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
66	Dic₃₃₂	—	2	2	no	—
67	#13050580	—	2	2	no	—
68	#13050578	—	2	2	no	—
69	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
71	C₈×D₈₃	—	2	2	no	—
72	C₈×D₈₃	—	2	2	no	—
73	C₄⋊Dic₈₃	—	2	1	no	—
74	C₄⋊Dic₈₃	—	2	1	no	—
75	C₄⋊Dic₈₃	—	2	1	no	—
76	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
78	Dic₈₃⋊C₄	—	2	1	no	—
79	Dic₈₃⋊C₄	—	2	1	no	—
80	Dic₈₃⋊C₄	—	2	1	no	—
81	Dic₈₃⋊C₄	—	2	1	no	—
82	Dic₈₃⋊C₄	—	2	1	no	—
83	Dic₈₃⋊C₄	—	2	1	no	—
84	Dic₈₃⋊C₄	—	2	1	no	—
85	C₂²⋊Dic₈₃	—	2	1	no	—
86	C₂²⋊Dic₈₃	—	2	1	no	—
87	C₂²⋊Dic₈₃	—	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	C₄×Dic₈₃	—	2	1	no	—
92	C₄×Dic₈₃	—	2	1	no	—
93	C₄×Dic₈₃	D₄	1	1	yes	characteristic
94	C₄×Dic₈₃	D₄	1	1	yes	characteristic
95	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₈₃	—	2	1	no	—
99	SD₁₆×C₈₃	—	2	2	no	—
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	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₈₃	—	2	1	no	—
106	C₄²×C₈₃	—	2	1	no	—
107	C₄²×C₈₃	C₂³	1	1	yes	characteristic
108	C₂³×C₈₃	—	2	1	no	—
109	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₈₃	—	2	1	no	—
111	Q₈×C₈₃	—	2	1	no	—
112	D₄×C₈₃	—	2	1	no	—
113	D₄×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	—	4	2	no	—
118	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₈₃	—	4	2	no	—
122	C₂×Dic₈₃	—	2	1	no	—
123	C₂×Dic₈₃	—	2	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	C₄×D₈₃	—	2	2	no	—
130	C₄×D₈₃	—	2	1	no	—
131	C₄×D₈₃	—	2	1	no	—
132	C₄×D₈₃	—	4	1	no	—
133	Dic₁₆₆	—	2	1	no	—
134	Dic₁₆₆	—	2	1	no	—
135	Dic₁₆₆	—	2	1	no	—
136	Dic₁₆₆	—	2	1	no	—
137	Dic₁₆₆	—	4	2	no	—
138	C₈×C₈₃	—	2	1	no	—
139	C₈×C₈₃	#41	1	2	yes	—
140	C₈₃⋊C₈	—	2	1	no	—
141	C₈₃⋊C₈	#41	1	2	yes	—
142	C₂²×C₈₃	—	4	1	no	—
143	C₂²×C₈₃	—	4	1	no	—
144	C₂²×C₈₃	—	4	1	no	—
145	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₈₃	—	2	1	no	—
147	C₂×D₈₃	—	2	1	no	—
148	C₂×D₈₃	#137	1	2	yes	—
149	C₄×C₈₃	—	2	1	no	—
150	C₄×C₈₃	—	2	1	no	—
151	C₄×C₈₃	—	4	1	no	—
152	C₄×C₈₃	—	4	1	no	—
153	C₄×C₈₃	—	4	1	no	—
154	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
155	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
156	Dic₈₃	—	2	1	no	—
157	Dic₈₃	—	2	1	no	—
158	Dic₈₃	—	2	1	no	—
159	Dic₈₃	—	4	1	no	—
160	Dic₈₃	—	4	1	no	—
161	Dic₈₃	—	4	1	no	—
162	Dic₈₃	—	4	1	no	—
163	C₂×C₈₃	—	8	1	no	—
164	C₂×C₈₃	#532	1	1	yes	characteristic
165	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
166	C₂×C₈₃	#575	1	1	yes	characteristic
167	D₈₃	—	2	2	no	—
168	#3181	—	83	1	no	maximal, Sylow
169	C₈₃	#3181	1	1	yes	characteristic, complemented, Sylow
170	#533	—	83	1	no	—
171	#529	—	83	1	no	—
172	#501	—	83	1	no	—
173	#497	—	83	1	no	—
174	#491	—	83	1	no	—
175	#489	—	83	1	no	—
176	#467	—	83	1	no	—
177	#467	—	83	1	no	—
178	#465	—	83	1	no	—
179	#465	—	83	1	no	—
180	#441	—	83	1	no	—
181	#441	—	83	1	no	—
182	#440	—	83	1	no	—
183	#440	D₈₃	1	1	yes	characteristic, complemented
184	C₂×#97	—	83	1	no	—
185	C₂×Q₁₆	—	83	1	no	—
186	C₂×Q₁₆	—	83	1	no	—
187	C₂×SD₁₆	—	83	1	no	—
188	C₂×SD₁₆	C₂×D₈₃	1	1	yes	characteristic
189	C₂²×C₈	—	83	1	no	—
190	C₂²×C₈	—	83	1	no	—
191	C₄⋊Q₈	—	83	1	no	—
192	C₄⋊Q₈	—	83	1	no	—
193	#126	—	166	1	no	—
194	#125	—	83	1	no	—
195	#125	—	166	1	no	—
196	#124	—	83	1	no	—
197	#123	—	166	1	no	—
198	C₂²⋊Q₈	—	83	1	no	—
199	C₂²⋊Q₈	—	83	1	no	—
200	C₂²⋊Q₈	—	83	1	no	—
201	C₂²⋊Q₈	—	166	1	no	—
202	#121	—	83	1	no	—
203	C₄×Q₈	—	83	1	no	—
204	C₄×Q₈	—	83	1	no	—
205	C₄×Q₈	—	166	1	no	—
206	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
207	C₄×D₄	—	166	1	no	—
208	C₄×D₄	C₂×D₈₃	1	1	yes	characteristic
209	#117	—	166	1	no	—
210	#117	—	166	1	no	—
211	C₂×C₄²	—	83	1	no	—
212	#107	—	83	1	no	—
213	#107	C₂×D₈₃	1	1	yes	characteristic
214	#106	—	83	1	no	—
215	#106	—	83	1	no	—
216	#103	—	83	1	no	—
217	#103	—	83	1	no	—
218	#103	—	83	1	no	—
219	#103	—	83	1	no	—
220	#103	—	83	1	no	—
221	#103	C₂×D₈₃	1	1	yes	characteristic
222	#102	—	83	1	no	—
223	#102	C₂×D₈₃	1	1	yes	characteristic
224	#97	—	83	1	no	—
225	#97	—	83	1	no	—
226	#97	—	83	1	no	—
227	#97	C₂×D₈₃	1	1	yes	characteristic
228	C₂×Q₈	—	83	1	no	—
229	C₂×Q₈	—	83	1	no	—
230	C₂×Q₈	—	166	1	no	—
231	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
232	C₂×D₄	—	166	1	no	—
233	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
234	C₂²×C₄	—	2	1	no	—
235	C₂²×C₄	—	83	1	no	—
236	C₂²×C₄	—	83	1	no	—
237	C₂²×C₄	—	83	1	no	—
238	Q₁₆	—	166	2	no	—
239	Q₁₆	—	166	2	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	166	2	no	—
242	C₂×C₈	—	83	1	no	—
243	C₂×C₈	—	83	1	no	—
244	C₂×C₈	—	166	2	no	—
245	C₂×C₈	—	166	2	no	—
246	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	83	1	no	—
250	C₄⋊C₄	—	83	1	no	—
251	C₄⋊C₄	—	166	1	no	—
252	C₄⋊C₄	—	166	1	no	—
253	C₄⋊C₄	—	166	1	no	—
254	C₄⋊C₄	—	166	1	no	—
255	C₄⋊C₄	—	166	1	no	—
256	C₄⋊C₄	—	166	1	no	—
257	C₄⋊C₄	—	166	1	no	—
258	C₄⋊C₄	—	166	1	no	—
259	C₄⋊C₄	—	166	1	no	—
260	C₄⋊C₄	—	166	1	no	—
261	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	166	1	no	—
265	C₂²⋊C₄	—	166	1	no	—
266	C₂²⋊C₄	—	166	1	no	—
267	C₂²⋊C₄	—	166	1	no	—
268	C₂²⋊C₄	—	166	1	no	—
269	C₂²⋊C₄	—	166	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	83	1	no	—
272	C₄²	—	83	1	no	—
273	C₄²	—	83	1	no	—
274	C₄²	—	166	1	no	—
275	C₄²	—	166	1	no	—
276	C₄²	C₂²×D₈₃	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	83	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	332	2	no	—
282	Q₈	—	166	1	no	—
283	Q₈	—	166	1	no	—
284	Q₈	—	166	1	no	—
285	Q₈	—	166	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	332	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	332	1	no	—
294	C₂×C₄	—	83	1	no	—
295	C₂×C₄	—	83	1	no	—
296	C₂×C₄	—	83	1	no	—
297	C₂×C₄	—	166	2	no	—
298	C₂×C₄	—	166	1	no	—
299	C₂×C₄	—	166	1	no	—
300	C₂×C₄	—	166	1	no	—
301	C₂×C₄	—	166	1	no	—
302	C₂×C₄	—	166	1	no	—
303	C₂×C₄	—	166	1	no	—
304	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
305	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	83	2	no	—
309	C₈	—	166	1	no	—
310	C₈	#13050590	1	2	yes	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	83	2	no	—
315	C₂²	—	166	1	no	—
316	C₂²	—	166	1	no	—
317	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	332	1	no	—
324	C₄	—	332	1	no	—
325	C₄	—	332	1	no	—
326	C₄	—	332	1	no	—
327	C₄	—	166	1	no	—
328	C₄	—	166	1	no	—
329	C₄	—	166	1	no	—
330	C₄	#41×D₈₃	1	1	yes	characteristic
331	C₄	C₂×#13050590	1	1	yes	characteristic
332	C₂	—	8	1	no	—
333	C₂	—	166	2	no	—
334	C₂	#19128248	1	1	yes	characteristic
335	C₂	#137×D₈₃	1	1	yes	characteristic
336	C₂	#19128497	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#440×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	#14581763	C₂²
hypercenter	#440	D₈₃
hypocenter	C₈₃	#3181
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #19128255	C₂	semidirect
3. #19128109	C₂	semidirect
4. #19128092	C₂	semidirect
5. #19128068	C₂	semidirect
6. #19128059	C₂	semidirect
7. #19127943	C₂	semidirect
8. #19127941	C₂	semidirect
11. #19127809	C₂	semidirect
12. #19127806	C₂	semidirect
13. #19127801	C₂	semidirect
14. #97×D₈₃	C₂	semidirect
15. #440×C₈₃	C₂	semidirect
169. C₈₃	#3181	semidirect
183. #440	D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128255	2
#19128233	2
#19128109	2
#19128092	2
#19128068	2
#19128059	2
#19127943	2
#19127941	2
#19127929	2
#19127927	2
#19127809	2
#19127806	2
#19127801	2
#97×D₈₃	2
#440×C₈₃	2
#3181	83