Subgroups — #520×D₈₃ #24920655

Order 10624.

1640 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₈₃	C₂	1	1	yes	maximal, characteristic
2	#19128184	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128183	C₂	1	2	yes	maximal, complemented
4	#19128182	C₂	1	3	yes	maximal, complemented
5	#19128181	C₂	1	3	yes	maximal, complemented
6	#19128180	C₂	1	6	yes	maximal, complemented
7	C₄×Q₈×D₈₃	C₂	1	2	yes	maximal, complemented
8	C₄×D₄×D₈₃	C₂	1	6	yes	maximal, complemented
9	#117×D₈₃	C₂	1	3	yes	maximal, complemented
10	C₂×C₄⋊C₄×D₈₃	C₂	1	3	yes	maximal, complemented
11	#520×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#13050591	C₂²	1	1	yes	characteristic
13	C₂×#13050590	C₂²	1	3	yes	—
14	C₂×#13050589	C₂²	1	3	yes	—
15	#41×D₈₃	C₄	1	16	yes	complemented
16	C₂×Q₈×D₈₃	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₈₃	C₂²	1	3	yes	—
18	C₂²×C₄×D₈₃	C₂²	1	6	yes	—
19	C₂²×C₄×D₈₃	C₂²	1	3	yes	—
20	Q₈×Dic₈₃	C₂²	1	2	yes	complemented
21	C₄×Dic₁₆₆	C₂²	1	6	yes	complemented
22	#14581757	C₂²	1	6	yes	complemented
23	C₄×D₃₃₂	C₂²	1	6	yes	complemented
24	D₄×Dic₈₃	C₂²	1	6	yes	complemented
25	#14581754	C₂²	1	6	yes	complemented
26	C₄×C₂²⋊D₈₃	C₂²	1	12	yes	complemented
27	(C₂²⋊D₈₃)⋊C₄	C₂²	1	12	yes	complemented
28	#14581751	C₂²	1	3	yes	complemented
29	#14581750	C₂²	1	3	yes	complemented
30	#14581750	C₂²	1	3	yes	complemented
31	#14581749	C₂²	1	6	yes	complemented
32	#14581748	C₂²	1	6	yes	complemented
33	C₂×C₄⋊Dic₈₃	C₂²	1	3	yes	complemented
34	C₂×Dic₈₃⋊C₄	C₂²	1	6	yes	complemented
35	C₄⋊C₄×D₈₃	C₂²	1	3	yes	—
36	C₄⋊C₄×D₈₃	C₂²	1	6	yes	—
37	C₄⋊C₄×D₈₃	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₈₃	C₂²	1	6	yes	complemented
39	C₄⋊C₄×D₈₃	C₂²	1	6	yes	complemented
40	C₂²⋊C₄×D₈₃	C₂²	1	6	yes	—
41	C₄²×D₈₃	C₂²	1	6	yes	—
42	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
43	C₄×Q₈×C₈₃	C₂²	1	2	yes	complemented
44	C₄×D₄×C₈₃	C₂²	1	6	yes	complemented
45	#117×C₈₃	C₂²	1	3	yes	complemented
46	C₂×C₄⋊C₄×C₈₃	C₂²	1	3	yes	complemented
47	C₂³×D₈₃	C₂³	1	3	yes	—
48	#13050591	C₂×C₄	1	8	yes	complemented
49	#13050590	C₂×C₄	1	24	yes	complemented
50	#13050589	C₂×C₄	1	24	yes	complemented
51	C₂×Dic₁₆₆	C₂³	1	3	yes	—
52	Q₈×D₈₃	C₂×C₄	1	8	yes	complemented
53	C₂×D₃₃₂	C₂³	1	3	yes	—
54	C₂×C₂²⋊D₈₃	C₂³	1	6	yes	—
55	D₄×D₈₃	C₂×C₄	1	24	yes	complemented
56	C₂²×Dic₈₃	C₂³	1	6	yes	—
57	C₂²×Dic₈₃	C₂³	1	3	yes	—
58	C₂×C₄×D₈₃	—	4	12	no	—
59	C₂×C₄×D₈₃	C₂×C₄	1	24	yes	complemented
60	C₂×C₄×D₈₃	C₂³	1	6	yes	—
61	C₂×C₄×D₈₃	C₂³	1	2	yes	—
62	C₂×C₄×D₈₃	C₂³	1	12	yes	—
63	C₂×C₄×D₈₃	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₈₃	C₂³	1	3	yes	—
65	C₂×C₄×D₈₃	C₂³	1	3	yes	—
66	C₂×C₄×D₈₃	C₂³	1	3	yes	—
67	C₄⋊Dic₈₃	C₂³	1	3	yes	—
68	C₄⋊Dic₈₃	C₂³	1	6	yes	—
69	C₄⋊Dic₈₃	C₂³	1	1	yes	characteristic
70	Dic₈₃⋊C₄	C₂³	1	6	yes	—
71	Dic₈₃⋊C₄	C₂³	1	12	yes	—
72	Dic₈₃⋊C₄	C₂³	1	2	yes	—
73	C₂²⋊Dic₈₃	C₂³	1	6	yes	—
74	(C₂×D₈₃)⋊C₄	C₂³	1	6	yes	—
75	(C₂×D₈₃)⋊C₄	C₂³	1	6	yes	—
76	C₄×Dic₈₃	C₂³	1	6	yes	—
77	C₄×Dic₈₃	C₂³	1	6	yes	—
78	C₄×Dic₈₃	C₂³	1	6	yes	—
79	#41×C₈₃	C₂×C₄	1	8	yes	complemented
80	C₂×Q₈×C₈₃	C₂³	1	1	yes	characteristic
81	C₂×D₄×C₈₃	C₂³	1	3	yes	—
82	C₂²×C₄×C₈₃	C₂³	1	6	yes	—
83	C₂²×C₄×C₈₃	C₂³	1	3	yes	—
84	C₄⋊C₄×C₈₃	C₂³	1	3	yes	—
85	C₄⋊C₄×C₈₃	C₂³	1	6	yes	—
86	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
87	C₂²⋊C₄×C₈₃	C₂³	1	6	yes	—
88	C₄²×C₈₃	C₂³	1	6	yes	—
89	C₂³×C₈₃	C₂⁴	1	3	yes	—
90	C₂²×D₈₃	—	2	6	no	—
91	C₂²×D₈₃	—	2	6	no	—
92	C₂²×D₈₃	C₂²×C₄	1	12	yes	—
93	C₂²×D₈₃	C₂⁴	1	3	yes	—
94	C₂²×D₈₃	C₂⁴	1	1	yes	characteristic
95	Q₈×C₈₃	C₂²×C₄	1	4	yes	complemented
96	D₄×C₈₃	C₂²×C₄	1	12	yes	complemented
97	C₂×C₄×C₈₃	—	4	6	no	—
98	C₂×C₄×C₈₃	C₂²×C₄	1	12	yes	complemented
99	C₂×C₄×C₈₃	C₂⁴	1	2	yes	—
100	C₂×C₄×C₈₃	C₂⁴	1	6	yes	—
101	C₂×C₄×C₈₃	C₂⁴	1	3	yes	—
102	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
103	C₂²⋊D₈₃	C₂²×C₄	1	24	yes	complemented
104	C₂×Dic₈₃	—	4	6	no	—
105	C₂×Dic₈₃	C₂²×C₄	1	12	yes	complemented
106	C₂×Dic₈₃	C₂⁴	1	2	yes	—
107	C₂×Dic₈₃	C₂⁴	1	6	yes	—
108	C₂×Dic₈₃	C₂⁴	1	3	yes	—
109	C₂×Dic₈₃	C₂⁴	1	1	yes	characteristic
110	D₃₃₂	C₂²×C₄	1	12	yes	complemented
111	C₄×D₈₃	—	2	12	no	—
112	C₄×D₈₃	—	2	4	no	—
113	C₄×D₈₃	—	4	12	no	—
114	C₄×D₈₃	C₂²×C₄	1	4	yes	—
115	C₄×D₈₃	C₂²×C₄	1	12	yes	—
116	C₄×D₈₃	C₂²×C₄	1	12	yes	complemented
117	C₄×D₈₃	C₂²×C₄	1	12	yes	complemented
118	Dic₁₆₆	C₂²×C₄	1	12	yes	complemented
119	C₂²×C₈₃	—	2	3	no	—
120	C₂²×C₈₃	—	2	3	no	—
121	C₂²×C₈₃	C₂³×C₄	1	6	yes	—
122	C₂²×C₈₃	C₂⁵	1	1	yes	characteristic
123	C₂×D₈₃	—	2	3	no	—
124	C₂×D₈₃	—	2	3	no	—
125	C₂×D₈₃	—	2	24	no	—
126	C₂×D₈₃	C₂³×C₄	1	6	yes	—
127	C₂×D₈₃	C₂³×C₄	1	2	yes	—
128	C₂×D₈₃	H(5,𝔽₂)	1	2	yes	—
129	C₂×D₈₃	2₋¹⁺⁴	1	2	yes	—
130	C₄×C₈₃	—	2	6	no	—
131	C₄×C₈₃	—	2	2	no	—
132	C₄×C₈₃	C₂³×C₄	1	6	yes	—
133	C₄×C₈₃	C₂³×C₄	1	2	yes	—
134	Dic₈₃	—	2	6	no	—
135	Dic₈₃	—	2	2	no	—
136	Dic₈₃	C₂³×C₄	1	6	yes	—
137	Dic₈₃	C₂³×C₄	1	2	yes	—
138	C₂×C₈₃	—	2	6	no	—
139	C₂×C₈₃	C₂⁴×C₄	1	1	yes	characteristic
140	C₂×C₈₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₈₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
142	D₈₃	—	2	6	no	—
143	D₈₃	#520	1	4	yes	direct product factor
144	C₂×#520	—	83	1	no	maximal, Sylow
145	C₈₃	C₂×#520	1	1	yes	characteristic, complemented, Sylow
146	C₂²×#41	—	83	1	no	—
147	#520	—	83	1	no	—
148	#520	—	83	2	no	—
149	#520	—	83	6	no	—
150	#520	—	83	3	no	—
151	#520	—	83	3	no	—
152	#520	D₈₃	1	1	yes	characteristic, direct product factor
153	C₂×C₄×Q₈	—	83	2	no	—
154	C₂×C₄×D₄	—	83	6	no	—
155	C₂×#117	—	83	3	no	—
156	C₂²×C₄⋊C₄	—	83	3	no	—
157	C₂×#41	—	83	16	no	—
158	C₂×#41	—	83	1	no	—
159	C₂×#41	—	83	3	no	—
160	C₂×#41	—	83	3	no	—
161	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
162	C₂²×Q₈	—	83	1	no	—
163	C₂²×D₄	—	83	3	no	—
164	C₂³×C₄	—	83	6	no	—
165	C₂³×C₄	—	83	3	no	—
166	C₄×Q₈	—	83	2	no	—
167	C₄×Q₈	—	83	6	no	—
168	C₄×Q₈	—	83	6	no	—
169	C₄×Q₈	C₂×D₈₃	1	2	yes	complemented
170	C₄×D₄	—	83	6	no	—
171	C₄×D₄	—	83	12	no	—
172	C₄×D₄	—	83	6	no	—
173	C₄×D₄	—	83	12	no	—
174	C₄×D₄	—	83	6	no	—
175	C₄×D₄	C₂×D₈₃	1	6	yes	complemented
176	#117	—	83	3	no	—
177	#117	—	83	3	no	—
178	#117	—	83	3	no	—
179	#117	—	83	6	no	—
180	#117	—	83	6	no	—
181	#117	C₂×D₈₃	1	3	yes	complemented
182	C₂×C₄⋊C₄	—	83	6	no	—
183	C₂×C₄⋊C₄	—	83	3	no	—
184	C₂×C₄⋊C₄	—	83	3	no	—
185	C₂×C₄⋊C₄	—	83	6	no	—
186	C₂×C₄⋊C₄	—	83	6	no	—
187	C₂×C₄⋊C₄	—	83	6	no	—
188	C₂×C₄⋊C₄	—	83	1	no	—
189	C₂×C₄⋊C₄	C₂×D₈₃	1	3	yes	complemented
190	C₂×C₂²⋊C₄	—	83	6	no	—
191	C₂×C₄²	—	83	6	no	—
192	C₂⁴	—	83	3	no	—
193	#41	—	83	8	no	—
194	#41	—	83	24	no	—
195	#41	—	83	24	no	—
196	#41	C₄×D₈₃	1	8	yes	complemented
197	C₂×Q₈	—	83	8	no	—
198	C₂×Q₈	—	83	3	no	—
199	C₂×Q₈	C₂²×D₈₃	1	1	yes	characteristic
200	C₂×D₄	—	83	6	no	—
201	C₂×D₄	—	83	24	no	—
202	C₂×D₄	—	83	3	no	—
203	C₂×D₄	C₂²×D₈₃	1	3	yes	—
204	C₂²×C₄	—	332	12	no	—
205	C₂²×C₄	—	83	6	no	—
206	C₂²×C₄	—	83	12	no	—
207	C₂²×C₄	—	83	3	no	—
208	C₂²×C₄	—	83	24	no	—
209	C₂²×C₄	—	83	3	no	—
210	C₂²×C₄	—	83	1	no	—
211	C₂²×C₄	—	83	3	no	—
212	C₂²×C₄	—	83	3	no	—
213	C₂²×C₄	—	83	2	no	—
214	C₂²×C₄	—	83	6	no	—
215	C₂²×C₄	C₂²×D₈₃	1	6	yes	—
216	C₂²×C₄	C₂²×D₈₃	1	3	yes	—
217	C₄⋊C₄	—	83	6	no	—
218	C₄⋊C₄	—	83	3	no	—
219	C₄⋊C₄	—	83	12	no	—
220	C₄⋊C₄	—	83	6	no	—
221	C₄⋊C₄	—	83	2	no	—
222	C₄⋊C₄	—	83	1	no	—
223	C₄⋊C₄	C₂²×D₈₃	1	3	yes	—
224	C₄⋊C₄	C₂²×D₈₃	1	6	yes	—
225	C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
226	C₂²⋊C₄	—	83	6	no	—
227	C₂²⋊C₄	—	83	6	no	—
228	C₂²⋊C₄	—	83	6	no	—
229	C₂²⋊C₄	C₂²×D₈₃	1	6	yes	—
230	C₄²	—	83	6	no	—
231	C₄²	—	83	6	no	—
232	C₄²	—	83	6	no	—
233	C₄²	C₂²×D₈₃	1	6	yes	—
234	C₂³	—	83	1	no	—
235	C₂³	—	83	12	no	—
236	C₂³	—	83	3	no	—
237	C₂³	—	166	6	no	—
238	C₂³	—	166	6	no	—
239	C₂³	C₂³×D₈₃	1	3	yes	—
240	Q₈	—	83	12	no	—
241	Q₈	C₂×C₄×D₈₃	1	4	yes	complemented
242	D₄	—	83	24	no	—
243	D₄	—	83	12	no	—
244	D₄	C₂×C₄×D₈₃	1	12	yes	complemented
245	C₂×C₄	—	4	6	no	—
246	C₂×C₄	—	332	12	no	—
247	C₂×C₄	—	332	6	no	—
248	C₂×C₄	—	83	12	no	—
249	C₂×C₄	—	83	3	no	—
250	C₂×C₄	—	83	1	no	—
251	C₂×C₄	—	83	4	no	—
252	C₂×C₄	—	83	12	no	—
253	C₂×C₄	—	83	12	no	—
254	C₂×C₄	—	83	12	no	—
255	C₂×C₄	—	83	2	no	—
256	C₂×C₄	—	83	6	no	—
257	C₂×C₄	—	166	12	no	—
258	C₂×C₄	—	166	4	no	—
259	C₂×C₄	C₂×C₄×D₈₃	1	12	yes	complemented
260	C₂×C₄	C₂³×D₈₃	1	2	yes	—
261	C₂×C₄	C₂³×D₈₃	1	6	yes	—
262	C₂×C₄	C₂³×D₈₃	1	3	yes	—
263	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
264	C₂²	—	2	3	no	—
265	C₂²	—	2	3	no	—
266	C₂²	—	83	2	no	—
267	C₂²	—	83	2	no	—
268	C₂²	—	83	2	no	—
269	C₂²	—	83	6	no	—
270	C₂²	—	166	24	no	—
271	C₂²	—	166	3	no	—
272	C₂²	—	166	3	no	—
273	C₂²	C₂²×C₄×D₈₃	1	6	yes	—
274	C₂²	C₂⁴×D₈₃	1	1	yes	characteristic
275	C₄	—	2	6	no	—
276	C₄	—	2	2	no	—
277	C₄	—	83	2	no	—
278	C₄	—	83	6	no	—
279	C₄	—	166	6	no	—
280	C₄	—	166	2	no	—
281	C₄	C₂²×C₄×D₈₃	1	2	yes	—
282	C₄	C₂²×C₄×D₈₃	1	6	yes	—
283	C₂	—	2	6	no	—
284	C₂	—	83	4	no	—
285	C₂	—	166	6	no	—
286	C₂	C₂³×C₄×D₈₃	1	1	yes	characteristic
287	C₂	H(5,𝔽₂)×D₈₃	1	1	yes	characteristic
288	C₂	2₋¹⁺⁴×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₈₃
commutator	C₂×C₈₃	C₂⁴×C₄
Fitting	#520×C₈₃	C₂
socle, cosocle	C₂²×C₈₃	C₂⁵
prime radical	C₂×#41×D₈₃	C₂
hypercenter	#520	D₈₃
hypocenter	C₈₃	C₂×#520
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ #520

Complements

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

N	H ≅ G/N	type
2. #19128184	C₂	semidirect
3. #19128183	C₂	semidirect
4. #19128182	C₂	semidirect
5. #19128181	C₂	semidirect
6. #19128180	C₂	semidirect
7. C₄×Q₈×D₈₃	C₂	semidirect
8. C₄×D₄×D₈₃	C₂	semidirect
9. #117×D₈₃	C₂	semidirect
10. C₂×C₄⋊C₄×D₈₃	C₂	semidirect
11. #520×C₈₃	C₂	semidirect
15. #41×D₈₃	C₄	semidirect
20. Q₈×Dic₈₃	C₂²	semidirect
21. C₄×Dic₁₆₆	C₂²	semidirect
22. #14581757	C₂²	semidirect
23. C₄×D₃₃₂	C₂²	semidirect
24. D₄×Dic₈₃	C₂²	semidirect
25. #14581754	C₂²	semidirect
26. C₄×C₂²⋊D₈₃	C₂²	semidirect
27. (C₂²⋊D₈₃)⋊C₄	C₂²	semidirect
28. #14581751	C₂²	semidirect
29. #14581750	C₂²	semidirect
30. #14581750	C₂²	semidirect
31. #14581749	C₂²	semidirect
32. #14581748	C₂²	semidirect
33. C₂×C₄⋊Dic₈₃	C₂²	semidirect
34. C₂×Dic₈₃⋊C₄	C₂²	semidirect
38. C₄⋊C₄×D₈₃	C₂²	semidirect
39. C₄⋊C₄×D₈₃	C₂²	semidirect
43. C₄×Q₈×C₈₃	C₂²	semidirect
44. C₄×D₄×C₈₃	C₂²	semidirect
45. #117×C₈₃	C₂²	semidirect
46. C₂×C₄⋊C₄×C₈₃	C₂²	semidirect
48. #13050591	C₂×C₄	semidirect
49. #13050590	C₂×C₄	semidirect
50. #13050589	C₂×C₄	semidirect
52. Q₈×D₈₃	C₂×C₄	semidirect
55. D₄×D₈₃	C₂×C₄	semidirect
59. C₂×C₄×D₈₃	C₂×C₄	semidirect
79. #41×C₈₃	C₂×C₄	semidirect
95. Q₈×C₈₃	C₂²×C₄	semidirect
96. D₄×C₈₃	C₂²×C₄	semidirect
98. C₂×C₄×C₈₃	C₂²×C₄	semidirect
103. C₂²⋊D₈₃	C₂²×C₄	semidirect
105. C₂×Dic₈₃	C₂²×C₄	semidirect
110. D₃₃₂	C₂²×C₄	semidirect
116. C₄×D₈₃	C₂²×C₄	semidirect
117. C₄×D₈₃	C₂²×C₄	semidirect
118. Dic₁₆₆	C₂²×C₄	semidirect
143. D₈₃	#520	direct (N × H)
145. C₈₃	C₂×#520	semidirect
152. #520	D₈₃	direct (N × H)
169. C₄×Q₈	C₂×D₈₃	semidirect
175. C₄×D₄	C₂×D₈₃	semidirect
181. #117	C₂×D₈₃	semidirect
189. C₂×C₄⋊C₄	C₂×D₈₃	semidirect
196. #41	C₄×D₈₃	semidirect
241. Q₈	C₂×C₄×D₈₃	semidirect
244. D₄	C₂×C₄×D₈₃	semidirect
259. C₂×C₄	C₂×C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#41×D₈₃	2
#19128184	2
#19128183	2
#19128182	2
#19128181	2
#19128180	2
C₄×Q₈×D₈₃	2
C₄×D₄×D₈₃	2
#117×D₈₃	2
C₂×C₄⋊C₄×D₈₃	2
#520×C₈₃	2
C₂×#520	83

Subgroups — #520×D83 #24920655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #520×D₈₃ #24920655