Subgroups — #24918667

Order 10624.

324 conjugacy classes of subgroups.

The subgroup lattice has 312 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
2	#125×D₈₃	C₂	1	1	yes	maximal, characteristic
3	#19128109	C₂	1	1	yes	maximal, characteristic, complemented
4	#19128108	C₂	1	1	yes	maximal, characteristic, complemented
5	#19128068	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128068	C₂	1	1	yes	maximal, characteristic, complemented
7	#19128038	C₂	1	1	yes	maximal, characteristic
8	#19128035	C₂	1	1	yes	maximal, characteristic
9	#19128033	C₂	1	1	yes	maximal, characteristic
10	#19128030	C₂	1	1	yes	maximal, characteristic
11	#19128009	C₂	1	1	yes	maximal, characteristic, complemented
12	#19128008	C₂	1	1	yes	maximal, characteristic, complemented
13	#19128005	C₂	1	1	yes	maximal, characteristic, complemented
14	#19127757	C₂	1	1	yes	maximal, characteristic
15	#479×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
16	#14581791	C₂²	1	1	yes	characteristic
17	#14581790	C₂²	1	1	yes	characteristic
18	#14581784	C₂²	1	1	yes	characteristic
19	#14581783	C₂²	1	1	yes	characteristic
20	#14581783	C₂²	1	1	yes	characteristic
21	#14581782	—	2	1	no	—
22	#14581782	—	2	1	no	—
23	#14581782	—	2	1	no	—
24	#14581772	C₂²	1	1	yes	characteristic
25	#14581770	—	2	1	no	—
26	#14581768	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₆₆	—	2	1	no	—
28	C₄×Dic₁₆₆	C₂²	1	1	yes	characteristic
29	#14581749	—	2	1	no	—
30	C₄⋊C₄×D₈₃	—	2	1	no	—
31	C₄⋊C₄×D₈₃	—	2	1	no	—
32	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
34	C₄²×D₈₃	C₂²	1	1	yes	characteristic
35	#14581726	C₂²	1	1	yes	characteristic
36	#14581725	C₂²	1	1	yes	characteristic
37	#14581725	C₂²	1	1	yes	characteristic
38	#14581724	C₂²	1	1	yes	characteristic
39	#14581723	C₂²	1	1	yes	characteristic
40	#14581723	C₂²	1	1	yes	characteristic
41	C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#14581717	C₂²	1	1	yes	characteristic
43	#14581717	C₂²	1	1	yes	characteristic
44	#14581716	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	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₈₃)⋊C₈	C₂²	1	1	yes	characteristic
50	#14581702	C₂²	1	1	yes	characteristic
51	#14581702	C₂²	1	1	yes	characteristic
52	#125×C₈₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
54	#107×C₈₃	C₂²	1	1	yes	characteristic
55	#106×C₈₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
57	#103×C₈₃	C₂²	1	1	yes	characteristic
58	#103×C₈₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
61	C₂×C₄×D₈₃	—	2	1	no	—
62	C₂×C₄×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	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₈₃	—	2	1	no	—
69	C₄⋊Dic₈₃	—	2	1	no	—
70	C₄⋊Dic₈₃	—	2	1	no	—
71	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
74	Dic₈₃⋊C₄	—	2	1	no	—
75	Dic₈₃⋊C₄	—	2	1	no	—
76	Dic₈₃⋊C₄	—	2	1	no	—
77	Dic₈₃⋊C₄	—	2	1	no	—
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	(C₂×D₈₃)⋊C₄	—	2	1	no	—
86	(C₂×D₈₃)⋊C₄	—	2	1	no	—
87	C₄×Dic₈₃	—	2	1	no	—
88	C₄×Dic₈₃	D₄	1	1	yes	characteristic
89	C₄×Dic₈₃	D₄	1	1	yes	characteristic
90	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
92	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
93	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₈₃	—	2	1	no	—
95	C₄⋊C₄×C₈₃	—	2	1	no	—
96	C₄⋊C₄×C₈₃	—	2	1	no	—
97	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
100	C₄²×C₈₃	—	2	1	no	—
101	C₄²×C₈₃	C₂³	1	1	yes	characteristic
102	C₂²×D₈₃	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₈₃	—	2	1	no	—
104	Q₈×C₈₃	—	2	1	no	—
105	C₂×C₄×C₈₃	—	2	1	no	—
106	C₂×C₄×C₈₃	—	2	1	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
112	C₂×Dic₈₃	—	2	1	no	—
113	C₂×Dic₈₃	—	2	1	no	—
114	C₂×Dic₈₃	—	2	1	no	—
115	C₂×Dic₈₃	—	2	1	no	—
116	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
117	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
118	C₂×Dic₈₃	C₂×D₄	1	1	yes	characteristic
119	C₄×D₈₃	—	2	2	no	—
120	C₄×D₈₃	—	2	1	no	—
121	C₄×D₈₃	—	2	1	no	—
122	C₄×D₈₃	—	4	1	no	—
123	C₄×D₈₃	—	4	2	no	—
124	C₄×D₈₃	—	4	2	no	—
125	Dic₁₆₆	—	2	1	no	—
126	Dic₁₆₆	—	2	1	no	—
127	Dic₁₆₆	—	4	2	no	—
128	C₈×C₈₃	—	2	1	no	—
129	C₈×C₈₃	—	2	1	no	—
130	C₈₃⋊C₈	—	2	1	no	—
131	C₈₃⋊C₈	—	2	1	no	—
132	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
133	C₂×D₈₃	—	2	1	no	—
134	C₂×D₈₃	—	2	1	no	—
135	C₂×D₈₃	#137	1	2	yes	—
136	C₄×C₈₃	—	2	1	no	—
137	C₄×C₈₃	—	2	1	no	—
138	C₄×C₈₃	—	4	1	no	—
139	C₄×C₈₃	—	4	1	no	—
140	C₄×C₈₃	—	4	1	no	—
141	C₄×C₈₃	—	4	1	no	—
142	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
144	Dic₈₃	—	2	1	no	—
145	Dic₈₃	—	2	1	no	—
146	Dic₈₃	—	2	1	no	—
147	Dic₈₃	—	4	1	no	—
148	Dic₈₃	—	4	1	no	—
149	Dic₈₃	—	4	1	no	—
150	Dic₈₃	—	4	1	no	—
151	C₂×C₈₃	#536	1	1	yes	characteristic
152	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
153	C₂×C₈₃	#575	1	1	yes	characteristic
154	D₈₃	—	2	2	no	—
155	#3236	—	83	1	no	maximal, Sylow
156	C₈₃	#3236	1	1	yes	characteristic, complemented, Sylow
157	#533	—	83	1	no	—
158	C₂×#125	—	83	1	no	—
159	#501	—	83	1	no	—
160	#501	—	83	1	no	—
161	#491	—	83	1	no	—
162	#491	—	83	1	no	—
163	#484	—	83	1	no	—
164	#484	—	83	1	no	—
165	#483	—	83	1	no	—
166	#483	—	83	1	no	—
167	#479	—	83	1	no	—
168	#479	—	83	1	no	—
169	#479	—	83	1	no	—
170	#479	D₈₃	1	1	yes	characteristic, complemented
171	#432	—	83	1	no	—
172	C₄⋊Q₈	—	83	1	no	—
173	C₄⋊Q₈	—	83	1	no	—
174	#125	—	83	1	no	—
175	#125	—	83	1	no	—
176	#125	—	83	1	no	—
177	#125	—	166	1	no	—
178	#125	—	166	1	no	—
179	#125	—	166	1	no	—
180	#125	C₂×D₈₃	1	1	yes	characteristic
181	C₂²⋊Q₈	—	83	1	no	—
182	C₂²⋊Q₈	—	83	1	no	—
183	C₂²⋊Q₈	—	166	1	no	—
184	C₄×Q₈	—	83	1	no	—
185	C₄×Q₈	—	166	1	no	—
186	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
187	#117	—	166	1	no	—
188	C₂×C₄⋊C₄	—	83	1	no	—
189	C₂×C₄⋊C₄	—	83	1	no	—
190	C₂×C₄⋊C₄	—	166	1	no	—
191	C₂×C₄⋊C₄	—	166	1	no	—
192	C₂×C₄²	—	83	1	no	—
193	#107	—	83	1	no	—
194	#107	—	83	1	no	—
195	#107	—	83	1	no	—
196	#107	C₂×D₈₃	1	1	yes	characteristic
197	#106	—	83	1	no	—
198	#106	—	83	1	no	—
199	#106	—	83	1	no	—
200	#106	C₂×D₈₃	1	1	yes	characteristic
201	C₄⋊C₈	—	83	1	no	—
202	C₄⋊C₈	C₂×D₈₃	1	1	yes	characteristic
203	#103	—	83	1	no	—
204	#103	—	83	1	no	—
205	#103	—	83	1	no	—
206	#103	—	83	1	no	—
207	#103	—	83	1	no	—
208	#103	—	83	1	no	—
209	#103	C₂×D₈₃	1	1	yes	characteristic
210	#103	C₂×D₈₃	1	1	yes	characteristic
211	C₂²⋊C₈	—	83	1	no	—
212	C₂²⋊C₈	—	83	1	no	—
213	#97	—	83	1	no	—
214	#97	—	83	1	no	—
215	C₂×Q₈	—	83	1	no	—
216	C₂×Q₈	—	166	1	no	—
217	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
218	C₂²×C₄	—	83	1	no	—
219	C₂²×C₄	—	83	1	no	—
220	C₂²×C₄	—	83	1	no	—
221	C₂²×C₄	—	166	1	no	—
222	C₂²×C₄	—	166	1	no	—
223	C₂×C₈	—	83	1	no	—
224	C₂×C₈	—	83	1	no	—
225	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₈₃	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	83	1	no	—
231	C₄⋊C₄	—	83	1	no	—
232	C₄⋊C₄	—	83	1	no	—
233	C₄⋊C₄	—	166	1	no	—
234	C₄⋊C₄	—	166	1	no	—
235	C₄⋊C₄	—	166	1	no	—
236	C₄⋊C₄	—	166	1	no	—
237	C₄⋊C₄	—	166	1	no	—
238	C₄⋊C₄	—	166	1	no	—
239	C₄⋊C₄	—	166	1	no	—
240	C₄⋊C₄	—	166	1	no	—
241	C₄⋊C₄	—	166	1	no	—
242	C₄⋊C₄	—	166	1	no	—
243	C₄⋊C₄	—	166	1	no	—
244	C₄⋊C₄	—	166	1	no	—
245	C₄⋊C₄	—	166	1	no	—
246	C₄⋊C₄	—	166	1	no	—
247	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
250	C₂²⋊C₄	—	166	1	no	—
251	C₂²⋊C₄	—	166	1	no	—
252	C₄²	—	2	1	no	—
253	C₄²	—	83	1	no	—
254	C₄²	—	83	1	no	—
255	C₄²	—	83	1	no	—
256	C₄²	—	166	1	no	—
257	C₄²	C₂²×D₈₃	1	1	yes	characteristic
258	C₂³	—	83	1	no	—
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	332	2	no	—
262	Q₈	—	166	1	no	—
263	Q₈	—	166	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	332	1	no	—
269	C₂×C₄	—	332	2	no	—
270	C₂×C₄	—	332	2	no	—
271	C₂×C₄	—	83	1	no	—
272	C₂×C₄	—	83	1	no	—
273	C₂×C₄	—	83	1	no	—
274	C₂×C₄	—	166	2	no	—
275	C₂×C₄	—	166	1	no	—
276	C₂×C₄	—	166	1	no	—
277	C₂×C₄	—	166	1	no	—
278	C₂×C₄	—	166	1	no	—
279	C₂×C₄	—	166	1	no	—
280	C₂×C₄	—	166	1	no	—
281	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
282	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	166	1	no	—
287	C₈	—	166	1	no	—
288	C₂²	—	83	2	no	—
289	C₂²	—	166	1	no	—
290	C₂²	—	166	1	no	—
291	C₂²	C₂×D₄×D₈₃	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	332	1	no	—
299	C₄	—	332	1	no	—
300	C₄	—	332	1	no	—
301	C₄	—	332	1	no	—
302	C₄	—	166	1	no	—
303	C₄	—	166	1	no	—
304	C₄	—	166	1	no	—
305	C₄	#14581839	1	1	yes	characteristic
306	C₄	#14581840	1	1	yes	characteristic
307	C₂	—	166	2	no	—
308	C₂	#19128274	1	1	yes	characteristic
309	C₂	#137×D₈₃	1	1	yes	characteristic
310	C₂	#19128498	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	#479×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂²×D₈₃	C₂×D₄
hypercenter	#479	D₈₃
hypocenter	C₈₃	#3236
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #19128109	C₂	semidirect
4. #19128108	C₂	semidirect
5. #19128068	C₂	semidirect
6. #19128068	C₂	semidirect
11. #19128009	C₂	semidirect
12. #19128008	C₂	semidirect
13. #19128005	C₂	semidirect
15. #479×C₈₃	C₂	semidirect
156. C₈₃	#3236	semidirect
170. #479	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
#125×D₈₃	2
#19128109	2
#19128108	2
#19128068	2
#19128038	2
#19128035	2
#19128033	2
#19128030	2
#19128009	2
#19128008	2
#19128005	2
#19127757	2
#479×C₈₃	2
#3236	83