Subgroups — #576×D₈₃ #24922211

Order 10624.

1512 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	H(5,𝔽₂)×D₈₃	C₂	1	2	yes	maximal, complemented
2	#19128506	C₂	1	1	yes	maximal, characteristic, complemented
3	#19128505	C₂	1	6	yes	maximal, complemented
4	#19128504	C₂	1	2	yes	maximal, complemented
5	#19128503	C₂	1	3	yes	maximal, complemented
6	#19128502	C₂	1	3	yes	maximal, complemented
7	AGL(1,ℤ/8)×D₈₃	C₂	1	6	yes	maximal, complemented
8	#135×D₈₃	C₂	1	3	yes	maximal, complemented
9	C₂×D₈×D₈₃	C₂	1	3	yes	maximal, complemented
10	#131×D₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#576×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14581838	—	2	2	no	—
13	#14581838	C₂²	1	2	yes	complemented
14	#14581838	C₂²	1	6	yes	complemented
15	#14581837	—	2	6	no	—
16	#14581837	C₂²	1	6	yes	complemented
17	#41×D₈₃	—	2	2	no	—
18	#41×D₈₃	C₂²	1	1	yes	characteristic
19	#41×D₈₃	C₂²	1	6	yes	complemented
20	C₂×D₄×D₈₃	—	2	6	no	—
21	C₂×D₄×D₈₃	C₂²	1	6	yes	complemented
22	#14581820	C₂²	1	6	yes	complemented
23	#14581819	C₂²	1	12	yes	complemented
24	#14581818	C₂²	1	6	yes	complemented
25	#14581817	C₂²	1	6	yes	complemented
26	#14581816	C₂²	1	12	yes	complemented
27	#14581815	C₂²	1	3	yes	complemented
28	#14581814	C₂²	1	3	yes	complemented
29	#14581813	C₂²	1	6	yes	complemented
30	#14581812	C₂²	1	6	yes	complemented
31	#14581811	C₂²	1	3	yes	complemented
32	Q₁₆×D₈₃	C₂²	1	1	yes	characteristic, complemented
33	SD₁₆×D₈₃	C₂²	1	6	yes	complemented
34	C₂×D₆₆₄	C₂²	1	3	yes	complemented
35	C₂×#13050575	C₂²	1	6	yes	complemented
36	D₈×D₈₃	C₂²	1	3	yes	complemented
37	D₈×D₈₃	C₂²	1	6	yes	complemented
38	D₈×D₈₃	C₂²	1	6	yes	complemented
39	D₈×D₈₃	C₂²	1	6	yes	complemented
40	#14581800	C₂²	1	3	yes	complemented
41	#14581799	C₂²	1	1	yes	characteristic, complemented
42	#14581798	C₂²	1	3	yes	complemented
43	M₁₆×D₈₃	C₂²	1	3	yes	complemented
44	C₂×C₈×D₈₃	C₂²	1	3	yes	complemented
45	H(5,𝔽₂)×C₈₃	C₂²	1	2	yes	complemented
46	AGL(1,ℤ/8)×C₈₃	C₂²	1	6	yes	complemented
47	#135×C₈₃	C₂²	1	3	yes	complemented
48	C₂×D₈×C₈₃	C₂²	1	3	yes	complemented
49	#131×C₈₃	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₈₃	—	2	6	no	—
51	#13050591	—	2	2	no	—
52	#13050591	D₄	1	1	yes	characteristic
53	#13050591	C₂³	1	6	yes	complemented
54	#13050590	—	2	6	no	—
55	#13050590	—	2	6	no	—
56	#13050590	C₂³	1	3	yes	—
57	#13050590	C₂³	1	6	yes	complemented
58	#13050589	—	2	6	no	—
59	#13050589	—	2	6	no	—
60	#13050589	D₄	1	3	yes	—
61	#13050589	C₂³	1	6	yes	complemented
62	Q₈×D₈₃	D₄	1	1	yes	characteristic
63	Q₈×D₈₃	C₂³	1	2	yes	—
64	C₂×D₃₃₂	—	2	6	no	—
65	C₂×D₃₃₂	C₂³	1	6	yes	complemented
66	C₂×C₂²⋊D₈₃	—	2	12	no	—
67	C₂×C₂²⋊D₈₃	—	2	6	no	—
68	D₄×D₈₃	—	2	6	no	—
69	D₄×D₈₃	—	2	6	no	—
70	D₄×D₈₃	—	2	12	no	—
71	D₄×D₈₃	—	2	6	no	—
72	D₄×D₈₃	—	2	6	no	—
73	D₄×D₈₃	—	2	6	no	—
74	D₄×D₈₃	D₄	1	3	yes	—
75	D₄×D₈₃	C₂³	1	6	yes	—
76	D₄×D₈₃	C₂³	1	12	yes	complemented
77	C₂×C₄×D₈₃	—	2	6	no	—
78	C₂×C₄×D₈₃	C₂³	1	3	yes	—
79	Dic₃₃₂	C₂³	1	1	yes	characteristic, complemented
80	#13050580	C₂³	1	2	yes	complemented
81	#13050579	C₂³	1	6	yes	complemented
82	#13050578	C₂³	1	6	yes	complemented
83	#13050577	C₂³	1	6	yes	complemented
84	D₆₆₄	C₂³	1	3	yes	complemented
85	D₆₆₄	C₂³	1	6	yes	complemented
86	#13050575	C₂³	1	6	yes	complemented
87	#13050575	C₂³	1	12	yes	complemented
88	#13050574	C₂³	1	3	yes	complemented
89	#13050573	C₂³	1	6	yes	complemented
90	C₂×C₈₃⋊C₈	C₂³	1	3	yes	complemented
91	C₈×D₈₃	C₂³	1	1	yes	characteristic
92	C₈×D₈₃	C₂³	1	3	yes	—
93	C₈×D₈₃	C₂³	1	3	yes	complemented
94	C₈×D₈₃	C₂³	1	3	yes	complemented
95	#41×C₈₃	—	2	2	no	—
96	#41×C₈₃	C₂³	1	1	yes	characteristic
97	#41×C₈₃	C₂³	1	6	yes	complemented
98	C₂×D₄×C₈₃	—	2	6	no	—
99	C₂×D₄×C₈₃	C₂³	1	6	yes	complemented
100	Q₁₆×C₈₃	C₂³	1	1	yes	characteristic, complemented
101	SD₁₆×C₈₃	C₂³	1	6	yes	complemented
102	D₈×C₈₃	C₂³	1	3	yes	complemented
103	D₈×C₈₃	C₂³	1	6	yes	complemented
104	M₁₆×C₈₃	C₂³	1	3	yes	complemented
105	C₂×C₈×C₈₃	C₂³	1	3	yes	complemented
106	C₂³×C₈₃	—	2	6	no	—
107	C₂²×D₈₃	—	2	6	no	—
108	C₂²×D₈₃	—	2	6	no	—
109	C₂²×D₈₃	—	2	12	no	—
110	C₂²×D₈₃	—	8	12	no	—
111	C₂²×D₈₃	C₂×D₄	1	3	yes	—
112	Q₈×C₈₃	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₈₃	C₂⁴	1	2	yes	—
114	D₄×C₈₃	—	2	6	no	—
115	D₄×C₈₃	—	2	6	no	—
116	D₄×C₈₃	C₂×D₄	1	3	yes	—
117	D₄×C₈₃	C₂⁴	1	6	yes	—
118	C₂×C₄×C₈₃	—	2	6	no	—
119	C₂×C₄×C₈₃	C₂⁴	1	3	yes	—
120	C₂²⋊D₈₃	—	2	6	no	—
121	C₂²⋊D₈₃	—	2	6	no	—
122	C₂²⋊D₈₃	—	2	6	no	—
123	C₂²⋊D₈₃	—	2	12	no	—
124	C₂²⋊D₈₃	C₂×D₄	1	6	yes	—
125	C₂×Dic₈₃	—	2	6	no	—
126	C₂×Dic₈₃	C₂×D₄	1	3	yes	—
127	D₃₃₂	—	2	6	no	—
128	D₃₃₂	—	2	6	no	—
129	D₃₃₂	C₂×D₄	1	3	yes	—
130	D₃₃₂	C₂⁴	1	6	yes	—
131	C₄×D₈₃	—	2	2	no	—
132	C₄×D₈₃	—	2	6	no	—
133	C₄×D₈₃	—	2	6	no	—
134	C₄×D₈₃	C₂×D₄	1	3	yes	—
135	C₄×D₈₃	C₂×D₄	1	3	yes	—
136	C₄×D₈₃	C₂⁴	1	1	yes	characteristic
137	C₄×D₈₃	C₂⁴	1	3	yes	—
138	Dic₁₆₆	—	2	2	no	—
139	Dic₁₆₆	C₂×D₄	1	3	yes	—
140	Dic₁₆₆	C₂⁴	1	2	yes	—
141	C₈×C₈₃	C₂⁴	1	1	yes	characteristic
142	C₈×C₈₃	C₂⁴	1	3	yes	—
143	C₈₃⋊C₈	C₂⁴	1	1	yes	characteristic
144	C₈₃⋊C₈	C₂⁴	1	3	yes	—
145	C₂²×C₈₃	—	2	6	no	—
146	C₂²×C₈₃	—	8	6	no	—
147	C₂²×C₈₃	C₂²×D₄	1	3	yes	—
148	C₂×D₈₃	—	2	6	no	—
149	C₂×D₈₃	—	2	6	no	—
150	C₂×D₈₃	—	4	12	no	—
151	C₂×D₈₃	—	8	12	no	—
152	C₂×D₈₃	—	8	6	no	—
153	C₂×D₈₃	C₂²×D₄	1	3	yes	—
154	C₂×D₈₃	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₈₃	—	2	2	no	—
156	C₄×C₈₃	C₂²×D₄	1	3	yes	—
157	C₄×C₈₃	C₂⁵	1	1	yes	characteristic
158	Dic₈₃	—	2	2	no	—
159	Dic₈₃	C₂²×D₄	1	1	yes	characteristic
160	Dic₈₃	C₂²×D₄	1	3	yes	—
161	C₂×C₈₃	—	2	3	no	—
162	C₂×C₈₃	—	4	6	no	—
163	C₂×C₈₃	C₂³×D₄	1	1	yes	characteristic
164	D₈₃	—	2	3	no	—
165	D₈₃	—	4	6	no	—
166	D₈₃	#576	1	2	yes	direct product factor
167	C₂×#576	—	83	1	no	maximal, Sylow
168	C₈₃	C₂×#576	1	1	yes	characteristic, complemented, Sylow
169	C₂×H(5,𝔽₂)	—	83	2	no	—
170	#576	—	83	2	no	—
171	#576	—	83	6	no	—
172	#576	—	83	3	no	—
173	#576	—	83	3	no	—
174	#576	—	83	1	no	—
175	#576	D₈₃	1	1	yes	characteristic, direct product factor
176	C₂×AGL(1,ℤ/8)	—	83	6	no	—
177	C₂×#135	—	83	3	no	—
178	C₂²×D₈	—	83	3	no	—
179	C₂×#131	—	83	1	no	—
180	H(5,𝔽₂)	—	83	2	no	—
181	H(5,𝔽₂)	—	83	6	no	—
182	H(5,𝔽₂)	—	83	6	no	—
183	H(5,𝔽₂)	—	166	6	no	—
184	H(5,𝔽₂)	—	166	2	no	—
185	H(5,𝔽₂)	C₂×D₈₃	1	2	yes	complemented
186	C₂×#41	—	83	1	no	—
187	C₂×#41	—	83	6	no	—
188	C₂×#41	—	166	2	no	—
189	C₂²×D₄	—	83	6	no	—
190	C₂²×D₄	—	166	6	no	—
191	AGL(1,ℤ/8)	—	83	6	no	—
192	AGL(1,ℤ/8)	—	83	6	no	—
193	AGL(1,ℤ/8)	—	83	12	no	—
194	AGL(1,ℤ/8)	—	83	12	no	—
195	AGL(1,ℤ/8)	—	83	6	no	—
196	AGL(1,ℤ/8)	C₂×D₈₃	1	6	yes	complemented
197	#135	—	83	6	no	—
198	#135	—	83	3	no	—
199	#135	—	83	6	no	—
200	#135	—	83	3	no	—
201	#135	—	83	3	no	—
202	#135	C₂×D₈₃	1	3	yes	complemented
203	C₂×Q₁₆	—	83	1	no	—
204	C₂×SD₁₆	—	83	6	no	—
205	C₂×D₈	—	83	6	no	—
206	C₂×D₈	—	83	3	no	—
207	C₂×D₈	—	83	6	no	—
208	C₂×D₈	—	83	6	no	—
209	C₂×D₈	—	83	6	no	—
210	C₂×D₈	—	83	3	no	—
211	C₂×D₈	C₂×D₈₃	1	3	yes	complemented
212	#131	—	83	1	no	—
213	#131	—	83	3	no	—
214	#131	—	83	3	no	—
215	#131	C₂×D₈₃	1	1	yes	characteristic, complemented
216	C₂×M₁₆	—	83	3	no	—
217	C₂²×C₈	—	83	3	no	—
218	C₂⁴	—	166	6	no	—
219	#41	—	2	2	no	—
220	#41	—	83	1	no	—
221	#41	—	83	3	no	—
222	#41	—	83	3	no	—
223	#41	—	83	6	no	—
224	#41	—	83	6	no	—
225	#41	—	83	6	no	—
226	#41	—	166	6	no	—
227	#41	—	166	6	no	—
228	#41	—	166	6	no	—
229	#41	—	166	6	no	—
230	#41	—	166	2	no	—
231	#41	C₂²×D₈₃	1	1	yes	characteristic
232	#41	C₂²×D₈₃	1	6	yes	complemented
233	C₂×Q₈	—	83	1	no	—
234	C₂×Q₈	—	83	2	no	—
235	C₂×D₄	—	2	6	no	—
236	C₂×D₄	—	83	3	no	—
237	C₂×D₄	—	83	6	no	—
238	C₂×D₄	—	83	12	no	—
239	C₂×D₄	—	83	6	no	—
240	C₂×D₄	—	166	6	no	—
241	C₂×D₄	—	166	6	no	—
242	C₂×D₄	—	166	6	no	—
243	C₂×D₄	—	166	6	no	—
244	C₂×D₄	—	166	12	no	—
245	C₂×D₄	—	166	6	no	—
246	C₂×D₄	—	166	6	no	—
247	C₂×D₄	—	166	12	no	—
248	C₂×D₄	—	166	6	no	—
249	C₂×D₄	C₂²×D₈₃	1	6	yes	complemented
250	C₂²×C₄	—	83	3	no	—
251	C₂²×C₄	—	166	6	no	—
252	Q₁₆	—	83	2	no	—
253	Q₁₆	—	83	1	no	—
254	Q₁₆	C₂²×D₈₃	1	1	yes	characteristic, complemented
255	SD₁₆	—	83	6	no	—
256	SD₁₆	—	83	6	no	—
257	SD₁₆	—	83	6	no	—
258	SD₁₆	C₂²×D₈₃	1	6	yes	complemented
259	D₈	—	83	6	no	—
260	D₈	—	83	12	no	—
261	D₈	—	83	3	no	—
262	D₈	—	83	6	no	—
263	D₈	C₂²×D₈₃	1	3	yes	complemented
264	D₈	C₂²×D₈₃	1	6	yes	complemented
265	M₁₆	—	83	3	no	—
266	M₁₆	—	83	6	no	—
267	M₁₆	C₂²×D₈₃	1	3	yes	complemented
268	C₂×C₈	—	83	3	no	—
269	C₂×C₈	—	83	1	no	—
270	C₂×C₈	—	83	3	no	—
271	C₂×C₈	—	83	3	no	—
272	C₂×C₈	—	83	3	no	—
273	C₂×C₈	C₂²×D₈₃	1	3	yes	complemented
274	C₂³	—	2	6	no	—
275	C₂³	—	83	3	no	—
276	C₂³	—	664	12	no	—
277	C₂³	—	166	6	no	—
278	C₂³	—	166	12	no	—
279	C₂³	—	166	6	no	—
280	Q₈	—	83	3	no	—
281	Q₈	—	83	2	no	—
282	Q₈	—	166	2	no	—
283	Q₈	D₄×D₈₃	1	1	yes	characteristic
284	Q₈	C₂³×D₈₃	1	2	yes	—
285	D₄	—	2	6	no	—
286	D₄	—	2	6	no	—
287	D₄	—	83	6	no	—
288	D₄	—	83	3	no	—
289	D₄	—	83	6	no	—
290	D₄	—	166	6	no	—
291	D₄	—	166	6	no	—
292	D₄	—	166	12	no	—
293	D₄	—	166	6	no	—
294	D₄	—	166	6	no	—
295	D₄	—	166	6	no	—
296	D₄	D₄×D₈₃	1	3	yes	—
297	D₄	C₂³×D₈₃	1	6	yes	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	83	3	no	—
300	C₂×C₄	—	83	3	no	—
301	C₂×C₄	—	83	3	no	—
302	C₂×C₄	—	83	1	no	—
303	C₂×C₄	—	83	3	no	—
304	C₂×C₄	—	166	6	no	—
305	C₂×C₄	—	166	2	no	—
306	C₂×C₄	—	166	6	no	—
307	C₂×C₄	—	166	6	no	—
308	C₂×C₄	C₂³×D₈₃	1	3	yes	—
309	C₈	—	83	1	no	—
310	C₈	—	83	3	no	—
311	C₈	C₂³×D₈₃	1	1	yes	characteristic
312	C₈	C₂³×D₈₃	1	3	yes	—
313	C₂²	—	2	6	no	—
314	C₂²	—	8	6	no	—
315	C₂²	—	332	12	no	—
316	C₂²	—	83	1	no	—
317	C₂²	—	83	3	no	—
318	C₂²	—	664	6	no	—
319	C₂²	—	664	12	no	—
320	C₂²	—	166	6	no	—
321	C₂²	—	166	6	no	—
322	C₂²	C₂×D₄×D₈₃	1	3	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	83	1	no	—
325	C₄	—	83	3	no	—
326	C₄	—	166	2	no	—
327	C₄	C₂×D₄×D₈₃	1	3	yes	—
328	C₄	C₂⁴×D₈₃	1	1	yes	characteristic
329	C₂	—	2	3	no	—
330	C₂	—	4	6	no	—
331	C₂	—	332	6	no	—
332	C₂	—	83	2	no	—
333	C₂	—	166	3	no	—
334	C₂	C₂²×D₄×D₈₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₈₃
commutator, cosocle	C₄×C₈₃	C₂⁵
Frattini	C₄	C₂⁴×D₈₃
Fitting	#576×C₈₃	C₂
socle	C₂×C₈₃	C₂³×D₄
hypercenter	#576	D₈₃
hypocenter	C₈₃	C₂×#576
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂ ◁ #41 ◁ #576

Complements

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

N	H ≅ G/N	type
1. H(5,𝔽₂)×D₈₃	C₂	semidirect
2. #19128506	C₂	semidirect
3. #19128505	C₂	semidirect
4. #19128504	C₂	semidirect
5. #19128503	C₂	semidirect
6. #19128502	C₂	semidirect
7. AGL(1,ℤ/8)×D₈₃	C₂	semidirect
8. #135×D₈₃	C₂	semidirect
9. C₂×D₈×D₈₃	C₂	semidirect
10. #131×D₈₃	C₂	semidirect
11. #576×C₈₃	C₂	semidirect
13. #14581838	C₂²	semidirect
14. #14581838	C₂²	semidirect
16. #14581837	C₂²	semidirect
19. #41×D₈₃	C₂²	semidirect
21. C₂×D₄×D₈₃	C₂²	semidirect
22. #14581820	C₂²	semidirect
23. #14581819	C₂²	semidirect
24. #14581818	C₂²	semidirect
25. #14581817	C₂²	semidirect
26. #14581816	C₂²	semidirect
27. #14581815	C₂²	semidirect
28. #14581814	C₂²	semidirect
29. #14581813	C₂²	semidirect
30. #14581812	C₂²	semidirect
31. #14581811	C₂²	semidirect
32. Q₁₆×D₈₃	C₂²	semidirect
33. SD₁₆×D₈₃	C₂²	semidirect
34. C₂×D₆₆₄	C₂²	semidirect
35. C₂×#13050575	C₂²	semidirect
36. D₈×D₈₃	C₂²	semidirect
37. D₈×D₈₃	C₂²	semidirect
38. D₈×D₈₃	C₂²	semidirect
39. D₈×D₈₃	C₂²	semidirect
40. #14581800	C₂²	semidirect
41. #14581799	C₂²	semidirect
42. #14581798	C₂²	semidirect
43. M₁₆×D₈₃	C₂²	semidirect
44. C₂×C₈×D₈₃	C₂²	semidirect
45. H(5,𝔽₂)×C₈₃	C₂²	semidirect
46. AGL(1,ℤ/8)×C₈₃	C₂²	semidirect
47. #135×C₈₃	C₂²	semidirect
48. C₂×D₈×C₈₃	C₂²	semidirect
49. #131×C₈₃	C₂²	semidirect
53. #13050591	C₂³	semidirect
57. #13050590	C₂³	semidirect
61. #13050589	C₂³	semidirect
65. C₂×D₃₃₂	C₂³	semidirect
76. D₄×D₈₃	C₂³	semidirect
79. Dic₃₃₂	C₂³	semidirect
80. #13050580	C₂³	semidirect
81. #13050579	C₂³	semidirect
82. #13050578	C₂³	semidirect
83. #13050577	C₂³	semidirect
84. D₆₆₄	C₂³	semidirect
85. D₆₆₄	C₂³	semidirect
86. #13050575	C₂³	semidirect
87. #13050575	C₂³	semidirect
88. #13050574	C₂³	semidirect
89. #13050573	C₂³	semidirect
90. C₂×C₈₃⋊C₈	C₂³	semidirect
93. C₈×D₈₃	C₂³	semidirect
94. C₈×D₈₃	C₂³	semidirect
97. #41×C₈₃	C₂³	semidirect
99. C₂×D₄×C₈₃	C₂³	semidirect
100. Q₁₆×C₈₃	C₂³	semidirect
101. SD₁₆×C₈₃	C₂³	semidirect
102. D₈×C₈₃	C₂³	semidirect
103. D₈×C₈₃	C₂³	semidirect
104. M₁₆×C₈₃	C₂³	semidirect
105. C₂×C₈×C₈₃	C₂³	semidirect
166. D₈₃	#576	direct (N × H)
168. C₈₃	C₂×#576	semidirect
175. #576	D₈₃	direct (N × H)
185. H(5,𝔽₂)	C₂×D₈₃	semidirect
196. AGL(1,ℤ/8)	C₂×D₈₃	semidirect
202. #135	C₂×D₈₃	semidirect
211. C₂×D₈	C₂×D₈₃	semidirect
215. #131	C₂×D₈₃	semidirect
232. #41	C₂²×D₈₃	semidirect
249. C₂×D₄	C₂²×D₈₃	semidirect
254. Q₁₆	C₂²×D₈₃	semidirect
258. SD₁₆	C₂²×D₈₃	semidirect
263. D₈	C₂²×D₈₃	semidirect
264. D₈	C₂²×D₈₃	semidirect
267. M₁₆	C₂²×D₈₃	semidirect
273. C₂×C₈	C₂²×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
H(5,𝔽₂)×D₈₃	2
#19128506	2
#19128505	2
#19128504	2
#19128503	2
#19128502	2
AGL(1,ℤ/8)×D₈₃	2
#135×D₈₃	2
C₂×D₈×D₈₃	2
#131×D₈₃	2
#576×C₈₃	2
C₂×#576	83

Subgroups — #576×D83 #24922211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #576×D₈₃ #24922211