Subgroups — #24921360

Order 10624.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19128436	C₂	1	4	yes	maximal, complemented
2	#19128423	C₂	1	4	yes	maximal, complemented
3	#19128375	C₂	1	8	yes	maximal, complemented
4	#19128323	C₂	1	4	yes	maximal, complemented
5	#19128275	C₂	1	1	yes	maximal, characteristic, complemented
6	#19128270	C₂	1	1	yes	maximal, characteristic
7	#19128254	C₂	1	1	yes	maximal, characteristic, complemented
8	#19128238	C₂	1	4	yes	maximal, complemented
9	C₂×#14581781	C₂	1	1	yes	maximal, characteristic, complemented
10	#19128175	C₂	1	1	yes	maximal, characteristic, complemented
11	#19128169	C₂	1	1	yes	maximal, characteristic, complemented
12	#519×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#13050590	C₂²	1	1	yes	characteristic
14	C₂²×D₃₃₂	C₂²	1	1	yes	characteristic
15	#14581791	C₂²	1	2	yes	complemented
16	#14581790	C₂²	1	2	yes	complemented
17	#14581789	C₂²	1	2	yes	—
18	#14581787	C₂²	1	8	yes	complemented
19	#14581786	C₂²	1	8	yes	complemented
20	#14581783	C₂²	1	2	yes	complemented
21	#14581781	C₂²	1	2	yes	—
22	#14581781	C₂²	1	4	yes	—
23	#14581781	C₂²	1	2	yes	—
24	#14581781	C₂²	1	2	yes	complemented
25	#14581780	C₂²	1	2	yes	—
26	#14581780	C₂²	1	8	yes	complemented
27	#14581777	C₂²	1	8	yes	complemented
28	#14581777	C₂²	1	8	yes	complemented
29	#14581775	C₂²	1	4	yes	—
30	#14581770	C₂²	1	8	yes	complemented
31	#14581769	C₂²	1	4	yes	—
32	#14581768	C₂²	1	4	yes	complemented
33	#14581767	C₂²	1	4	yes	—
34	#14581762	C₂²	1	4	yes	—
35	C₄×Dic₁₆₆	C₂²	1	4	yes	complemented
36	#14581757	C₂²	1	2	yes	complemented
37	C₄×D₃₃₂	C₂²	1	4	yes	—
38	C₄×D₃₃₂	C₂²	1	8	yes	complemented
39	#14581754	C₂²	1	4	yes	—
40	#14581754	C₂²	1	2	yes	complemented
41	(C₂²⋊D₈₃)⋊C₄	C₂²	1	4	yes	—
42	#14581751	C₂²	1	1	yes	characteristic
43	#14581749	C₂²	1	2	yes	complemented
44	#14581749	C₂²	1	8	yes	complemented
45	C₂×(C₂×D₈₃)⋊C₄	C₂²	1	4	yes	—
46	C₂²⋊C₄×D₈₃	C₂²	1	4	yes	—
47	C₂×C₄×Dic₈₃	C₂²	1	1	yes	characteristic
48	C₄²×D₈₃	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
50	C₄×Q₈×C₈₃	C₂²	1	4	yes	complemented
51	C₄×Q₈×C₈₃	C₂²	1	4	yes	complemented
52	#117×C₈₃	C₂²	1	4	yes	—
53	#117×C₈₃	C₂²	1	1	yes	characteristic
54	#117×C₈₃	C₂²	1	1	yes	characteristic
55	C₂³×D₈₃	C₂³	1	2	yes	—
56	#13050590	—	2	4	no	—
57	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
58	C₂×Dic₁₆₆	C₂³	1	4	yes	—
59	C₂×D₃₃₂	—	2	4	no	—
60	C₂×D₃₃₂	C₂³	1	2	yes	—
61	C₂×D₃₃₂	C₂³	1	2	yes	—
62	C₂×D₃₃₂	C₂³	1	1	yes	characteristic
63	C₂×D₃₃₂	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₈₃	C₂³	1	2	yes	—
65	C₂²×Dic₈₃	C₂³	1	2	yes	—
66	C₂×C₄×D₈₃	C₂³	1	2	yes	—
67	C₂×C₄×D₈₃	C₂³	1	8	yes	—
68	C₂×C₄×D₈₃	C₂³	1	4	yes	—
69	C₄⋊Dic₈₃	C₂³	1	2	yes	—
70	C₄⋊Dic₈₃	C₂³	1	4	yes	—
71	Dic₈₃⋊C₄	C₂³	1	8	yes	—
72	Dic₈₃⋊C₄	C₂³	1	4	yes	—
73	Dic₈₃⋊C₄	C₂³	1	4	yes	—
74	Dic₈₃⋊C₄	C₂³	1	4	yes	—
75	C₂²⋊Dic₈₃	C₂³	1	2	yes	—
76	(C₂×D₈₃)⋊C₄	C₂³	1	8	yes	—
77	(C₂×D₈₃)⋊C₄	C₂³	1	8	yes	—
78	(C₂×D₈₃)⋊C₄	C₂³	1	8	yes	—
79	(C₂×D₈₃)⋊C₄	C₂³	1	4	yes	—
80	(C₂×D₈₃)⋊C₄	C₂³	1	8	yes	—
81	(C₂×D₈₃)⋊C₄	C₂³	1	8	yes	—
82	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
83	C₄×Dic₈₃	C₂³	1	1	yes	characteristic
84	C₄×Dic₈₃	C₂³	1	2	yes	—
85	C₄×Dic₈₃	C₂³	1	2	yes	—
86	C₄×Dic₈₃	C₂³	1	4	yes	—
87	C₂×Q₈×C₈₃	—	2	4	no	—
88	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
89	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
90	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
91	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
93	C₄⋊C₄×C₈₃	C₂³	1	8	yes	—
94	C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
95	C₂²⋊C₄×C₈₃	C₂³	1	2	yes	—
96	C₂²⋊C₄×C₈₃	C₂³	1	2	yes	—
97	C₄²×C₈₃	C₂³	1	2	yes	—
98	C₄²×C₈₃	C₂³	1	8	yes	—
99	C₄²×C₈₃	C₂³	1	2	yes	—
100	C₂³×C₈₃	C₂⁴	1	1	yes	characteristic
101	C₂²×D₈₃	—	2	4	no	—
102	C₂²×D₈₃	—	4	4	no	—
103	C₂²×D₈₃	C₂⁴	1	4	yes	—
104	C₂²×D₈₃	C₂⁴	1	2	yes	—
105	Q₈×C₈₃	—	2	4	no	—
106	Q₈×C₈₃	—	2	4	no	—
107	C₂×C₄×C₈₃	—	2	4	no	—
108	C₂×C₄×C₈₃	#41	1	4	yes	—
109	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
110	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
111	C₂×C₄×C₈₃	C₂⁴	1	4	yes	—
112	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₈₃	—	2	4	no	—
115	C₂×Dic₈₃	—	4	2	no	—
116	C₂×Dic₈₃	C₂⁴	1	2	yes	—
117	C₂×Dic₈₃	C₂⁴	1	2	yes	—
118	C₂×Dic₈₃	C₂⁴	1	2	yes	—
119	C₂×Dic₈₃	C₂⁴	1	4	yes	—
120	D₃₃₂	—	2	4	no	—
121	D₃₃₂	—	2	4	no	—
122	D₃₃₂	—	2	2	no	—
123	D₃₃₂	—	4	8	no	—
124	C₄×D₈₃	—	2	4	no	—
125	C₄×D₈₃	—	4	8	no	—
126	C₄×D₈₃	—	4	4	no	—
127	Dic₁₆₆	—	2	2	no	—
128	Dic₁₆₆	—	4	4	no	—
129	C₂²×C₈₃	—	2	2	no	—
130	C₂²×C₈₃	C₂×#41	1	2	yes	—
131	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
132	C₂×D₈₃	—	2	2	no	—
133	C₂×D₈₃	—	2	4	no	—
134	C₂×D₈₃	—	2	4	no	—
135	C₂×D₈₃	—	2	8	no	—
136	C₂×D₈₃	—	8	4	no	—
137	C₄×C₈₃	—	2	4	no	—
138	C₄×C₈₃	—	2	4	no	—
139	C₄×C₈₃	—	2	4	no	—
140	C₄×C₈₃	C₂×#41	1	2	yes	—
141	C₄×C₈₃	C₂×#41	1	2	yes	—
142	Dic₈₃	—	2	2	no	—
143	Dic₈₃	—	2	2	no	—
144	Dic₈₃	—	2	4	no	—
145	Dic₈₃	—	2	2	no	—
146	C₂×C₈₃	—	2	2	no	—
147	C₂×C₈₃	C₂²×#41	1	1	yes	characteristic
148	C₂×C₈₃	#585	1	2	yes	—
149	D₈₃	—	4	2	no	—
150	D₈₃	—	4	4	no	—
151	#3510	—	83	1	no	maximal, Sylow
152	C₈₃	#3510	1	1	yes	characteristic, complemented, Sylow
153	#559	—	83	4	no	—
154	#556	—	83	4	no	—
155	#551	—	83	8	no	—
156	#542	—	83	4	no	—
157	#536	—	83	1	no	—
158	#535	—	83	1	no	—
159	#533	—	83	1	no	—
160	#529	—	83	4	no	—
161	C₂×#124	—	83	1	no	—
162	#519	D₈₃	1	1	yes	characteristic, complemented
163	#518	—	83	1	no	—
164	C₄×#41	—	83	1	no	—
165	C₂×#41	—	83	1	no	—
166	C₂²×Q₈	C₂×D₈₃	1	1	yes	characteristic
167	C₂²×D₄	—	83	1	no	—
168	C₄⋊Q₈	—	83	2	no	—
169	C₄⋊Q₈	—	83	2	no	—
170	#127	—	83	2	no	—
171	#126	—	83	8	no	—
172	#126	—	83	8	no	—
173	#125	—	83	2	no	—
174	#124	—	83	2	no	—
175	#124	—	83	2	no	—
176	#124	—	83	4	no	—
177	#124	—	83	2	no	—
178	#124	—	83	2	no	—
179	#124	—	83	8	no	—
180	#123	—	83	4	no	—
181	#123	—	83	8	no	—
182	#123	—	83	8	no	—
183	C₂²⋊Q₈	—	83	4	no	—
184	C₂²⋊Q₈	—	83	8	no	—
185	C₂²⋊Q₈	—	83	4	no	—
186	#121	—	83	4	no	—
187	#120	—	83	4	no	—
188	C₄×Q₈	—	83	2	no	—
189	C₄×Q₈	—	83	4	no	—
190	C₄×Q₈	C₂×D₈₃	1	4	yes	complemented
191	C₄×Q₈	C₂×D₈₃	1	4	yes	complemented
192	C₄×D₄	—	83	2	no	—
193	C₄×D₄	—	83	4	no	—
194	C₄×D₄	—	83	4	no	—
195	C₄×D₄	—	83	4	no	—
196	C₄×D₄	—	83	8	no	—
197	#117	—	83	1	no	—
198	#117	—	83	2	no	—
199	#117	—	83	8	no	—
200	#117	C₂×D₈₃	1	1	yes	characteristic
201	#117	C₂×D₈₃	1	1	yes	characteristic
202	#117	C₂×D₈₃	1	4	yes	—
203	C₂×C₂²⋊C₄	—	83	4	no	—
204	C₂×C₂²⋊C₄	—	83	4	no	—
205	C₂×C₄²	—	83	1	no	—
206	C₂×C₄²	—	83	2	no	—
207	C₂⁴	—	83	2	no	—
208	#41	—	166	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	83	4	no	—
211	C₂×Q₈	—	83	1	no	—
212	C₂×Q₈	C₂²×D₈₃	1	2	yes	—
213	C₂×Q₈	C₂²×D₈₃	1	2	yes	—
214	C₂×D₄	—	83	2	no	—
215	C₂×D₄	—	83	8	no	—
216	C₂×D₄	—	83	2	no	—
217	C₂×D₄	—	83	2	no	—
218	C₂×D₄	—	83	1	no	—
219	C₂×D₄	—	166	4	no	—
220	C₂²×C₄	—	83	2	no	—
221	C₂²×C₄	—	83	4	no	—
222	C₂²×C₄	—	83	8	no	—
223	C₂²×C₄	—	83	2	no	—
224	C₂²×C₄	C₂²×D₈₃	1	2	yes	—
225	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
226	C₄⋊C₄	—	83	4	no	—
227	C₄⋊C₄	—	83	2	no	—
228	C₄⋊C₄	—	83	4	no	—
229	C₄⋊C₄	—	83	8	no	—
230	C₄⋊C₄	—	83	4	no	—
231	C₄⋊C₄	—	83	4	no	—
232	C₄⋊C₄	C₂²×D₈₃	1	2	yes	—
233	C₄⋊C₄	C₂²×D₈₃	1	2	yes	—
234	C₄⋊C₄	C₂²×D₈₃	1	8	yes	—
235	C₂²⋊C₄	—	83	2	no	—
236	C₂²⋊C₄	—	83	8	no	—
237	C₂²⋊C₄	—	83	8	no	—
238	C₂²⋊C₄	—	83	8	no	—
239	C₂²⋊C₄	—	83	4	no	—
240	C₂²⋊C₄	—	83	8	no	—
241	C₂²⋊C₄	—	83	8	no	—
242	C₂²⋊C₄	C₂²×D₈₃	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₈₃	1	2	yes	—
244	C₄²	—	83	1	no	—
245	C₄²	—	83	4	no	—
246	C₄²	—	83	1	no	—
247	C₄²	—	83	2	no	—
248	C₄²	—	83	2	no	—
249	C₄²	C₂²×D₈₃	1	2	yes	—
250	C₄²	C₂²×D₈₃	1	8	yes	—
251	C₄²	C₂²×D₈₃	1	2	yes	—
252	C₂³	—	332	4	no	—
253	C₂³	—	83	4	no	—
254	C₂³	—	83	2	no	—
255	C₂³	—	166	4	no	—
256	C₂³	C₂³×D₈₃	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	332	4	no	—
260	Q₈	—	166	2	no	—
261	D₄	—	332	8	no	—
262	D₄	—	166	4	no	—
263	D₄	—	166	4	no	—
264	D₄	—	166	4	no	—
265	D₄	—	166	2	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	332	4	no	—
268	C₂×C₄	—	332	8	no	—
269	C₂×C₄	—	332	2	no	—
270	C₂×C₄	—	83	2	no	—
271	C₂×C₄	—	83	2	no	—
272	C₂×C₄	—	83	2	no	—
273	C₂×C₄	—	83	4	no	—
274	C₂×C₄	—	166	4	no	—
275	C₂×C₄	#13050590	1	4	yes	—
276	C₂×C₄	C₂³×D₈₃	1	4	yes	—
277	C₂×C₄	C₂³×D₈₃	1	4	yes	—
278	C₂×C₄	C₂³×D₈₃	1	4	yes	—
279	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	664	4	no	—
283	C₂²	—	166	2	no	—
284	C₂²	—	166	4	no	—
285	C₂²	—	166	4	no	—
286	C₂²	—	166	8	no	—
287	C₂²	C₂×#13050590	1	2	yes	—
288	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	166	2	no	—
293	C₄	—	166	2	no	—
294	C₄	—	166	4	no	—
295	C₄	—	166	2	no	—
296	C₄	C₂×#13050590	1	2	yes	—
297	C₄	C₂×#13050590	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	332	4	no	—
300	C₂	—	332	2	no	—
301	C₂	C₂²×#13050590	1	1	yes	characteristic
302	C₂	#19128538	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₃
commutator, socle, cosocle	C₂²×C₈₃	C₂⁵
Fitting	#519×C₈₃	C₂
prime radical	#19128270	C₂
hypercenter	#519	D₈₃
hypocenter	C₈₃	#3510
LCS	G ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #519

Complements

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

N	H ≅ G/N	type
1. #19128436	C₂	semidirect
2. #19128423	C₂	semidirect
3. #19128375	C₂	semidirect
4. #19128323	C₂	semidirect
5. #19128275	C₂	semidirect
7. #19128254	C₂	semidirect
8. #19128238	C₂	semidirect
9. C₂×#14581781	C₂	semidirect
10. #19128175	C₂	semidirect
11. #19128169	C₂	semidirect
12. #519×C₈₃	C₂	semidirect
15. #14581791	C₂²	semidirect
16. #14581790	C₂²	semidirect
18. #14581787	C₂²	semidirect
19. #14581786	C₂²	semidirect
20. #14581783	C₂²	semidirect
24. #14581781	C₂²	semidirect
26. #14581780	C₂²	semidirect
27. #14581777	C₂²	semidirect
28. #14581777	C₂²	semidirect
30. #14581770	C₂²	semidirect
32. #14581768	C₂²	semidirect
35. C₄×Dic₁₆₆	C₂²	semidirect
36. #14581757	C₂²	semidirect
38. C₄×D₃₃₂	C₂²	semidirect
40. #14581754	C₂²	semidirect
43. #14581749	C₂²	semidirect
44. #14581749	C₂²	semidirect
48. C₄²×D₈₃	C₂²	semidirect
50. C₄×Q₈×C₈₃	C₂²	semidirect
51. C₄×Q₈×C₈₃	C₂²	semidirect
152. C₈₃	#3510	semidirect
162. #519	D₈₃	semidirect
190. C₄×Q₈	C₂×D₈₃	semidirect
191. C₄×Q₈	C₂×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
#19128436	2
#19128423	2
#19128375	2
#19128323	2
#19128275	2
#19128270	2
#19128254	2
#19128238	2
C₂×#14581781	2
#19128175	2
#19128169	2
#519×C₈₃	2
#3510	83