Subgroups — #576×D₅₉ #21438844

Order 7552.

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)×D₅₉	C₂	1	2	yes	maximal, complemented
2	#16838207	C₂	1	1	yes	maximal, characteristic, complemented
3	#16838206	C₂	1	6	yes	maximal, complemented
4	#16838205	C₂	1	2	yes	maximal, complemented
5	#16838204	C₂	1	3	yes	maximal, complemented
6	#16838203	C₂	1	3	yes	maximal, complemented
7	AGL(1,ℤ/8)×D₅₉	C₂	1	6	yes	maximal, complemented
8	#135×D₅₉	C₂	1	3	yes	maximal, complemented
9	C₂×D₈×D₅₉	C₂	1	3	yes	maximal, complemented
10	#131×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#576	C₂	1	1	yes	maximal, characteristic, complemented
12	#14273773	—	2	2	no	—
13	#14273773	C₂²	1	6	yes	complemented
14	#14273773	C₂²	1	2	yes	complemented
15	#14273772	—	2	6	no	—
16	#14273772	C₂²	1	6	yes	complemented
17	#41×D₅₉	—	2	2	no	—
18	#41×D₅₉	C₂²	1	1	yes	characteristic
19	#41×D₅₉	C₂²	1	6	yes	complemented
20	C₂×D₄×D₅₉	—	2	6	no	—
21	C₂×D₄×D₅₉	C₂²	1	6	yes	complemented
22	#14273755	C₂²	1	6	yes	complemented
23	#14273754	C₂²	1	12	yes	complemented
24	#14273753	C₂²	1	6	yes	complemented
25	#14273752	C₂²	1	6	yes	complemented
26	#14273751	C₂²	1	12	yes	complemented
27	#14273750	C₂²	1	3	yes	complemented
28	#14273749	C₂²	1	3	yes	complemented
29	#14273748	C₂²	1	6	yes	complemented
30	#14273747	C₂²	1	6	yes	complemented
31	#14273746	C₂²	1	3	yes	complemented
32	Q₁₆×D₅₉	C₂²	1	1	yes	characteristic, complemented
33	SD₁₆×D₅₉	C₂²	1	6	yes	complemented
34	C₂×D₄₇₂	C₂²	1	3	yes	complemented
35	C₂×#11745669	C₂²	1	6	yes	complemented
36	D₈×D₅₉	C₂²	1	3	yes	complemented
37	D₈×D₅₉	C₂²	1	6	yes	complemented
38	D₈×D₅₉	C₂²	1	6	yes	complemented
39	D₈×D₅₉	C₂²	1	6	yes	complemented
40	#14273735	C₂²	1	3	yes	complemented
41	#14273734	C₂²	1	1	yes	characteristic, complemented
42	#14273733	C₂²	1	3	yes	complemented
43	M₁₆×D₅₉	C₂²	1	3	yes	complemented
44	C₂×C₈×D₅₉	C₂²	1	3	yes	complemented
45	H(5,𝔽₂)×C₅₉	C₂²	1	2	yes	complemented
46	AGL(1,ℤ/8)×C₅₉	C₂²	1	6	yes	complemented
47	#135×C₅₉	C₂²	1	3	yes	complemented
48	C₂×D₈×C₅₉	C₂²	1	3	yes	complemented
49	#131×C₅₉	C₂²	1	1	yes	characteristic, complemented
50	C₂³×D₅₉	—	2	6	no	—
51	#41×C₅₉	—	2	2	no	—
52	#41×C₅₉	C₂³	1	1	yes	characteristic
53	#41×C₅₉	C₂³	1	6	yes	complemented
54	C₂×D₄×C₅₉	—	2	6	no	—
55	C₂×D₄×C₅₉	C₂³	1	6	yes	complemented
56	C₂×C₂²⋊D₅₉	—	2	12	no	—
57	C₂×C₂²⋊D₅₉	—	2	6	no	—
58	#11745689	—	2	2	no	—
59	#11745689	D₄	1	1	yes	characteristic
60	#11745689	C₂³	1	6	yes	complemented
61	Q₈×D₅₉	D₄	1	1	yes	characteristic
62	Q₈×D₅₉	C₂³	1	2	yes	—
63	#11745687	—	2	6	no	—
64	#11745687	—	2	6	no	—
65	#11745687	D₄	1	3	yes	—
66	#11745687	C₂³	1	6	yes	complemented
67	D₄×D₅₉	—	2	6	no	—
68	D₄×D₅₉	—	2	6	no	—
69	D₄×D₅₉	—	2	12	no	—
70	D₄×D₅₉	—	2	6	no	—
71	D₄×D₅₉	—	2	6	no	—
72	D₄×D₅₉	—	2	6	no	—
73	D₄×D₅₉	D₄	1	3	yes	—
74	D₄×D₅₉	C₂³	1	6	yes	—
75	D₄×D₅₉	C₂³	1	12	yes	complemented
76	#11745685	—	2	6	no	—
77	#11745685	—	2	6	no	—
78	#11745685	C₂³	1	3	yes	—
79	#11745685	C₂³	1	6	yes	complemented
80	C₂×D₂₃₆	—	2	6	no	—
81	C₂×D₂₃₆	C₂³	1	6	yes	complemented
82	C₂×C₄×D₅₉	—	2	6	no	—
83	C₂×C₄×D₅₉	C₂³	1	3	yes	—
84	Q₁₆×C₅₉	C₂³	1	1	yes	characteristic, complemented
85	SD₁₆×C₅₉	C₂³	1	6	yes	complemented
86	D₈×C₅₉	C₂³	1	3	yes	complemented
87	D₈×C₅₉	C₂³	1	6	yes	complemented
88	M₁₆×C₅₉	C₂³	1	3	yes	complemented
89	C₂×C₈×C₅₉	C₂³	1	3	yes	complemented
90	#11745672	C₂³	1	2	yes	complemented
91	#11745671	C₂³	1	6	yes	complemented
92	#11745670	C₂³	1	6	yes	complemented
93	#11745669	C₂³	1	6	yes	complemented
94	#11745669	C₂³	1	12	yes	complemented
95	#11745664	C₂³	1	3	yes	complemented
96	C₂×C₅₉⋊C₈	C₂³	1	3	yes	complemented
97	Dic₂₃₆	C₂³	1	1	yes	characteristic, complemented
98	D₄₇₂	C₂³	1	3	yes	complemented
99	D₄₇₂	C₂³	1	6	yes	complemented
100	#11745660	C₂³	1	6	yes	complemented
101	#11745659	C₂³	1	6	yes	complemented
102	C₈×D₅₉	C₂³	1	1	yes	characteristic
103	C₈×D₅₉	C₂³	1	3	yes	—
104	C₈×D₅₉	C₂³	1	3	yes	complemented
105	C₈×D₅₉	C₂³	1	3	yes	complemented
106	C₂³×C₅₉	—	2	6	no	—
107	C₂²×D₅₉	—	2	6	no	—
108	C₂²×D₅₉	—	2	6	no	—
109	C₂²×D₅₉	—	2	12	no	—
110	C₂²×D₅₉	—	8	12	no	—
111	C₂²×D₅₉	C₂×D₄	1	3	yes	—
112	Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₅₉	C₂⁴	1	2	yes	—
114	D₄×C₅₉	—	2	6	no	—
115	D₄×C₅₉	—	2	6	no	—
116	D₄×C₅₉	C₂×D₄	1	3	yes	—
117	D₄×C₅₉	C₂⁴	1	6	yes	—
118	C₂×C₄×C₅₉	—	2	6	no	—
119	C₂×C₄×C₅₉	C₂⁴	1	3	yes	—
120	C₂²⋊D₅₉	—	2	6	no	—
121	C₂²⋊D₅₉	—	2	6	no	—
122	C₂²⋊D₅₉	—	2	6	no	—
123	C₂²⋊D₅₉	—	2	12	no	—
124	C₂²⋊D₅₉	C₂×D₄	1	6	yes	—
125	C₂×Dic₅₉	—	2	6	no	—
126	C₂×Dic₅₉	C₂×D₄	1	3	yes	—
127	D₂₃₆	—	2	6	no	—
128	D₂₃₆	—	2	6	no	—
129	D₂₃₆	C₂×D₄	1	3	yes	—
130	D₂₃₆	C₂⁴	1	6	yes	—
131	C₄×D₅₉	—	2	2	no	—
132	C₄×D₅₉	—	2	6	no	—
133	C₄×D₅₉	—	2	6	no	—
134	C₄×D₅₉	C₂×D₄	1	3	yes	—
135	C₄×D₅₉	C₂×D₄	1	3	yes	—
136	C₄×D₅₉	C₂⁴	1	1	yes	characteristic
137	C₄×D₅₉	C₂⁴	1	3	yes	—
138	Dic₁₁₈	—	2	2	no	—
139	Dic₁₁₈	C₂×D₄	1	3	yes	—
140	Dic₁₁₈	C₂⁴	1	2	yes	—
141	C₈×C₅₉	C₂⁴	1	1	yes	characteristic
142	C₈×C₅₉	C₂⁴	1	3	yes	—
143	C₅₉⋊C₈	C₂⁴	1	1	yes	characteristic
144	C₅₉⋊C₈	C₂⁴	1	3	yes	—
145	C₂²×C₅₉	—	2	6	no	—
146	C₂²×C₅₉	—	8	6	no	—
147	C₂²×C₅₉	C₂²×D₄	1	3	yes	—
148	C₂×D₅₉	—	2	6	no	—
149	C₂×D₅₉	—	2	6	no	—
150	C₂×D₅₉	—	4	12	no	—
151	C₂×D₅₉	—	8	12	no	—
152	C₂×D₅₉	—	8	6	no	—
153	C₂×D₅₉	C₂²×D₄	1	3	yes	—
154	C₂×D₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₄×C₅₉	—	2	2	no	—
156	C₄×C₅₉	C₂²×D₄	1	3	yes	—
157	C₄×C₅₉	C₂⁵	1	1	yes	characteristic
158	Dic₅₉	—	2	2	no	—
159	Dic₅₉	C₂²×D₄	1	1	yes	characteristic
160	Dic₅₉	C₂²×D₄	1	3	yes	—
161	C₂×#576	—	59	1	no	maximal, Sylow
162	C₂×C₅₉	—	2	3	no	—
163	C₂×C₅₉	—	4	6	no	—
164	C₂×C₅₉	C₂³×D₄	1	1	yes	characteristic
165	D₅₉	—	2	3	no	—
166	D₅₉	—	4	6	no	—
167	D₅₉	#576	1	2	yes	direct product factor
168	C₂×H(5,𝔽₂)	—	59	2	no	—
169	#576	—	59	2	no	—
170	#576	—	59	6	no	—
171	#576	—	59	3	no	—
172	#576	—	59	3	no	—
173	#576	—	59	1	no	—
174	#576	D₅₉	1	1	yes	characteristic, direct product factor
175	C₂×AGL(1,ℤ/8)	—	59	6	no	—
176	C₂×#135	—	59	3	no	—
177	C₂²×D₈	—	59	3	no	—
178	C₂×#131	—	59	1	no	—
179	C₅₉	C₂×#576	1	1	yes	characteristic, complemented, Sylow
180	H(5,𝔽₂)	—	59	2	no	—
181	H(5,𝔽₂)	—	59	6	no	—
182	H(5,𝔽₂)	—	59	6	no	—
183	H(5,𝔽₂)	—	118	6	no	—
184	H(5,𝔽₂)	—	118	2	no	—
185	H(5,𝔽₂)	C₂×D₅₉	1	2	yes	complemented
186	C₂×#41	—	59	1	no	—
187	C₂×#41	—	59	6	no	—
188	C₂×#41	—	118	2	no	—
189	C₂²×D₄	—	59	6	no	—
190	C₂²×D₄	—	118	6	no	—
191	AGL(1,ℤ/8)	—	59	6	no	—
192	AGL(1,ℤ/8)	—	59	6	no	—
193	AGL(1,ℤ/8)	—	59	12	no	—
194	AGL(1,ℤ/8)	—	59	12	no	—
195	AGL(1,ℤ/8)	—	59	6	no	—
196	AGL(1,ℤ/8)	C₂×D₅₉	1	6	yes	complemented
197	#135	—	59	6	no	—
198	#135	—	59	3	no	—
199	#135	—	59	6	no	—
200	#135	—	59	3	no	—
201	#135	—	59	3	no	—
202	#135	C₂×D₅₉	1	3	yes	complemented
203	C₂×Q₁₆	—	59	1	no	—
204	C₂×SD₁₆	—	59	6	no	—
205	C₂×D₈	—	59	6	no	—
206	C₂×D₈	—	59	3	no	—
207	C₂×D₈	—	59	6	no	—
208	C₂×D₈	—	59	6	no	—
209	C₂×D₈	—	59	6	no	—
210	C₂×D₈	—	59	3	no	—
211	C₂×D₈	C₂×D₅₉	1	3	yes	complemented
212	#131	—	59	1	no	—
213	#131	—	59	3	no	—
214	#131	—	59	3	no	—
215	#131	C₂×D₅₉	1	1	yes	characteristic, complemented
216	C₂×M₁₆	—	59	3	no	—
217	C₂²×C₈	—	59	3	no	—
218	C₂⁴	—	118	6	no	—
219	#41	—	2	2	no	—
220	#41	—	59	1	no	—
221	#41	—	59	3	no	—
222	#41	—	59	3	no	—
223	#41	—	59	6	no	—
224	#41	—	59	6	no	—
225	#41	—	59	6	no	—
226	#41	—	118	6	no	—
227	#41	—	118	6	no	—
228	#41	—	118	6	no	—
229	#41	—	118	6	no	—
230	#41	—	118	2	no	—
231	#41	C₂²×D₅₉	1	1	yes	characteristic
232	#41	C₂²×D₅₉	1	6	yes	complemented
233	C₂×Q₈	—	59	1	no	—
234	C₂×Q₈	—	59	2	no	—
235	C₂×D₄	—	2	6	no	—
236	C₂×D₄	—	59	3	no	—
237	C₂×D₄	—	59	6	no	—
238	C₂×D₄	—	59	12	no	—
239	C₂×D₄	—	59	6	no	—
240	C₂×D₄	—	118	6	no	—
241	C₂×D₄	—	118	6	no	—
242	C₂×D₄	—	118	6	no	—
243	C₂×D₄	—	118	6	no	—
244	C₂×D₄	—	118	12	no	—
245	C₂×D₄	—	118	6	no	—
246	C₂×D₄	—	118	6	no	—
247	C₂×D₄	—	118	12	no	—
248	C₂×D₄	—	118	6	no	—
249	C₂×D₄	C₂²×D₅₉	1	6	yes	complemented
250	C₂²×C₄	—	59	3	no	—
251	C₂²×C₄	—	118	6	no	—
252	Q₁₆	—	59	2	no	—
253	Q₁₆	—	59	1	no	—
254	Q₁₆	C₂²×D₅₉	1	1	yes	characteristic, complemented
255	SD₁₆	—	59	6	no	—
256	SD₁₆	—	59	6	no	—
257	SD₁₆	—	59	6	no	—
258	SD₁₆	C₂²×D₅₉	1	6	yes	complemented
259	D₈	—	59	6	no	—
260	D₈	—	59	12	no	—
261	D₈	—	59	3	no	—
262	D₈	—	59	6	no	—
263	D₈	C₂²×D₅₉	1	3	yes	complemented
264	D₈	C₂²×D₅₉	1	6	yes	complemented
265	M₁₆	—	59	3	no	—
266	M₁₆	—	59	6	no	—
267	M₁₆	C₂²×D₅₉	1	3	yes	complemented
268	C₂×C₈	—	59	3	no	—
269	C₂×C₈	—	59	1	no	—
270	C₂×C₈	—	59	3	no	—
271	C₂×C₈	—	59	3	no	—
272	C₂×C₈	—	59	3	no	—
273	C₂×C₈	C₂²×D₅₉	1	3	yes	complemented
274	C₂³	—	2	6	no	—
275	C₂³	—	59	3	no	—
276	C₂³	—	118	12	no	—
277	C₂³	—	118	6	no	—
278	C₂³	—	118	6	no	—
279	C₂³	—	472	12	no	—
280	Q₈	—	59	3	no	—
281	Q₈	—	59	2	no	—
282	Q₈	—	118	2	no	—
283	Q₈	D₄×D₅₉	1	1	yes	characteristic
284	Q₈	C₂³×D₅₉	1	2	yes	—
285	D₄	—	2	6	no	—
286	D₄	—	2	6	no	—
287	D₄	—	59	6	no	—
288	D₄	—	59	3	no	—
289	D₄	—	59	6	no	—
290	D₄	—	118	6	no	—
291	D₄	—	118	6	no	—
292	D₄	—	118	12	no	—
293	D₄	—	118	6	no	—
294	D₄	—	118	6	no	—
295	D₄	—	118	6	no	—
296	D₄	D₄×D₅₉	1	3	yes	—
297	D₄	C₂³×D₅₉	1	6	yes	—
298	C₂×C₄	—	2	6	no	—
299	C₂×C₄	—	59	3	no	—
300	C₂×C₄	—	59	3	no	—
301	C₂×C₄	—	59	3	no	—
302	C₂×C₄	—	59	1	no	—
303	C₂×C₄	—	59	3	no	—
304	C₂×C₄	—	118	6	no	—
305	C₂×C₄	—	118	2	no	—
306	C₂×C₄	—	118	6	no	—
307	C₂×C₄	—	118	6	no	—
308	C₂×C₄	C₂³×D₅₉	1	3	yes	—
309	C₈	—	59	1	no	—
310	C₈	—	59	3	no	—
311	C₈	C₂³×D₅₉	1	1	yes	characteristic
312	C₈	C₂³×D₅₉	1	3	yes	—
313	C₂²	—	2	6	no	—
314	C₂²	—	8	6	no	—
315	C₂²	—	59	1	no	—
316	C₂²	—	59	3	no	—
317	C₂²	—	118	6	no	—
318	C₂²	—	118	6	no	—
319	C₂²	—	472	6	no	—
320	C₂²	—	472	12	no	—
321	C₂²	—	236	12	no	—
322	C₂²	C₂×D₄×D₅₉	1	3	yes	—
323	C₄	—	2	2	no	—
324	C₄	—	59	1	no	—
325	C₄	—	59	3	no	—
326	C₄	—	118	2	no	—
327	C₄	C₂×D₄×D₅₉	1	3	yes	—
328	C₄	C₂⁴×D₅₉	1	1	yes	characteristic
329	C₂	—	2	3	no	—
330	C₂	—	4	6	no	—
331	C₂	—	59	2	no	—
332	C₂	—	118	3	no	—
333	C₂	—	236	6	no	—
334	C₂	C₂²×D₄×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×D₄×D₅₉
commutator	C₄×C₅₉	C₂⁵
Frattini	C₄	C₂⁴×D₅₉
Fitting	C₅₉×#576	C₂
socle	C₂×C₅₉	C₂³×D₄
hypercenter	#576	D₅₉
hypocenter	C₅₉	C₂×#576
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
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)×D₅₉	C₂	semidirect
2. #16838207	C₂	semidirect
3. #16838206	C₂	semidirect
4. #16838205	C₂	semidirect
5. #16838204	C₂	semidirect
6. #16838203	C₂	semidirect
7. AGL(1,ℤ/8)×D₅₉	C₂	semidirect
8. #135×D₅₉	C₂	semidirect
9. C₂×D₈×D₅₉	C₂	semidirect
10. #131×D₅₉	C₂	semidirect
11. C₅₉×#576	C₂	semidirect
13. #14273773	C₂²	semidirect
14. #14273773	C₂²	semidirect
16. #14273772	C₂²	semidirect
19. #41×D₅₉	C₂²	semidirect
21. C₂×D₄×D₅₉	C₂²	semidirect
22. #14273755	C₂²	semidirect
23. #14273754	C₂²	semidirect
24. #14273753	C₂²	semidirect
25. #14273752	C₂²	semidirect
26. #14273751	C₂²	semidirect
27. #14273750	C₂²	semidirect
28. #14273749	C₂²	semidirect
29. #14273748	C₂²	semidirect
30. #14273747	C₂²	semidirect
31. #14273746	C₂²	semidirect
32. Q₁₆×D₅₉	C₂²	semidirect
33. SD₁₆×D₅₉	C₂²	semidirect
34. C₂×D₄₇₂	C₂²	semidirect
35. C₂×#11745669	C₂²	semidirect
36. D₈×D₅₉	C₂²	semidirect
37. D₈×D₅₉	C₂²	semidirect
38. D₈×D₅₉	C₂²	semidirect
39. D₈×D₅₉	C₂²	semidirect
40. #14273735	C₂²	semidirect
41. #14273734	C₂²	semidirect
42. #14273733	C₂²	semidirect
43. M₁₆×D₅₉	C₂²	semidirect
44. C₂×C₈×D₅₉	C₂²	semidirect
45. H(5,𝔽₂)×C₅₉	C₂²	semidirect
46. AGL(1,ℤ/8)×C₅₉	C₂²	semidirect
47. #135×C₅₉	C₂²	semidirect
48. C₂×D₈×C₅₉	C₂²	semidirect
49. #131×C₅₉	C₂²	semidirect
53. #41×C₅₉	C₂³	semidirect
55. C₂×D₄×C₅₉	C₂³	semidirect
60. #11745689	C₂³	semidirect
66. #11745687	C₂³	semidirect
75. D₄×D₅₉	C₂³	semidirect
79. #11745685	C₂³	semidirect
81. C₂×D₂₃₆	C₂³	semidirect
84. Q₁₆×C₅₉	C₂³	semidirect
85. SD₁₆×C₅₉	C₂³	semidirect
86. D₈×C₅₉	C₂³	semidirect
87. D₈×C₅₉	C₂³	semidirect
88. M₁₆×C₅₉	C₂³	semidirect
89. C₂×C₈×C₅₉	C₂³	semidirect
90. #11745672	C₂³	semidirect
91. #11745671	C₂³	semidirect
92. #11745670	C₂³	semidirect
93. #11745669	C₂³	semidirect
94. #11745669	C₂³	semidirect
95. #11745664	C₂³	semidirect
96. C₂×C₅₉⋊C₈	C₂³	semidirect
97. Dic₂₃₆	C₂³	semidirect
98. D₄₇₂	C₂³	semidirect
99. D₄₇₂	C₂³	semidirect
100. #11745660	C₂³	semidirect
101. #11745659	C₂³	semidirect
104. C₈×D₅₉	C₂³	semidirect
105. C₈×D₅₉	C₂³	semidirect
167. D₅₉	#576	direct (N × H)
174. #576	D₅₉	direct (N × H)
179. C₅₉	C₂×#576	semidirect
185. H(5,𝔽₂)	C₂×D₅₉	semidirect
196. AGL(1,ℤ/8)	C₂×D₅₉	semidirect
202. #135	C₂×D₅₉	semidirect
211. C₂×D₈	C₂×D₅₉	semidirect
215. #131	C₂×D₅₉	semidirect
232. #41	C₂²×D₅₉	semidirect
249. C₂×D₄	C₂²×D₅₉	semidirect
254. Q₁₆	C₂²×D₅₉	semidirect
258. SD₁₆	C₂²×D₅₉	semidirect
263. D₈	C₂²×D₅₉	semidirect
264. D₈	C₂²×D₅₉	semidirect
267. M₁₆	C₂²×D₅₉	semidirect
273. C₂×C₈	C₂²×D₅₉	semidirect

Subgroups — #576×D59 #21438844

Conjugacy classes of subgroups

Complements

Subgroups — #576×D₅₉ #21438844