Subgroups — #21436621

Order 7552.

336 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838123	C₂	1	1	yes	maximal, characteristic
2	#16838094	C₂	1	1	yes	maximal, characteristic
3	#16837809	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837742	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837741	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837740	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837735	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837707	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837640	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837509	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837471	C₂	1	1	yes	maximal, characteristic
13	#16837444	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837437	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#470	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
17	#14273726	C₂²	1	1	yes	characteristic
18	#14273722	—	2	1	no	—
19	#14273722	—	2	1	no	—
20	#14273720	—	2	1	no	—
21	#14273718	—	2	1	no	—
22	#14273718	C₂²	1	1	yes	characteristic
23	#14273713	—	2	1	no	—
24	#14273711	—	2	1	no	—
25	#14273703	C₂²	1	1	yes	characteristic
26	#14273703	C₂²	1	1	yes	characteristic
27	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
28	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
29	#14273692	—	2	1	no	—
30	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
31	#14273686	C₂²	1	1	yes	characteristic
32	#14273685	C₂²	1	1	yes	characteristic
33	#14273685	C₂²	1	1	yes	characteristic
34	#14273683	—	2	1	no	—
35	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
36	#14273661	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	#14273659	C₂²	1	1	yes	characteristic
42	#14273659	C₂²	1	1	yes	characteristic
43	#14273658	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273651	C₂²	1	1	yes	characteristic
48	#14273651	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	#14273637	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#103×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
63	C₂²×Dic₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
65	C₂×Dic₁₁₈	—	2	1	no	—
66	Q₁₆×C₅₉	—	2	2	no	—
67	M₁₆×C₅₉	—	2	2	no	—
68	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₅₉	—	2	1	no	—
71	C₄⋊C₄×C₅₉	—	2	1	no	—
72	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₅₉	—	2	1	no	—
75	C₂²⋊C₄×C₅₉	—	2	1	no	—
76	C₂²⋊Dic₅₉	—	2	1	no	—
77	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
78	(C₂×D₅₉)⋊C₄	—	2	1	no	—
79	(C₂×D₅₉)⋊C₄	—	2	1	no	—
80	(C₂×D₅₉)⋊C₄	—	2	1	no	—
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
91	Dic₅₉⋊C₄	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₅₉	—	2	1	no	—
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	C₈×D₅₉	—	2	2	no	—
105	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₅₉	—	2	1	no	—
108	Q₈×C₅₉	—	2	1	no	—
109	Q₈×C₅₉	—	2	1	no	—
110	Q₈×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₅₉	—	4	1	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	4	2	no	—
125	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
128	C₄×D₅₉	—	2	1	no	—
129	C₄×D₅₉	—	2	1	no	—
130	Dic₁₁₈	—	4	2	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	#41	1	2	yes	—
133	C₅₉⋊C₈	—	2	1	no	—
134	C₅₉⋊C₈	—	2	1	no	—
135	C₂²×C₅₉	—	2	1	no	—
136	C₂²×C₅₉	—	2	1	no	—
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₅₉	—	2	1	no	—
140	C₂×D₅₉	—	2	1	no	—
141	C₂×D₅₉	—	2	1	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	—	4	1	no	—
146	C₄×C₅₉	—	4	1	no	—
147	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
148	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	2	1	no	—
151	Dic₅₉	—	2	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	—	4	1	no	—
155	Dic₅₉	—	4	1	no	—
156	#3349	—	59	1	no	maximal, Sylow
157	C₂×C₅₉	—	4	1	no	—
158	C₂×C₅₉	#547	1	1	yes	characteristic
159	C₂×C₅₉	#575	1	1	yes	characteristic
160	C₂×C₅₉	#578	1	1	yes	characteristic
161	D₅₉	—	4	1	no	—
162	#556	—	59	1	no	—
163	#552	—	59	1	no	—
164	#501	—	59	1	no	—
165	#485	—	59	1	no	—
166	#485	—	59	1	no	—
167	#485	—	59	1	no	—
168	#484	—	59	1	no	—
169	#479	—	59	1	no	—
170	#477	—	59	1	no	—
171	#470	D₅₉	1	1	yes	characteristic, complemented
172	#467	—	59	1	no	—
173	#441	—	59	1	no	—
174	#435	—	59	1	no	—
175	#428	—	59	1	no	—
176	#427	—	59	1	no	—
177	C₅₉	#3349	1	1	yes	characteristic, complemented, Sylow
178	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
179	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
180	C₂²×C₈	—	59	1	no	—
181	C₄⋊Q₈	—	59	1	no	—
182	#126	—	118	1	no	—
183	#126	—	118	1	no	—
184	#126	—	118	1	no	—
185	#125	—	59	1	no	—
186	#125	—	118	1	no	—
187	#124	—	118	1	no	—
188	#123	—	118	1	no	—
189	C₂²⋊Q₈	—	59	1	no	—
190	C₂²⋊Q₈	—	59	1	no	—
191	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
192	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
193	C₄×Q₈	—	59	1	no	—
194	C₄×Q₈	—	59	1	no	—
195	C₄×Q₈	—	118	1	no	—
196	C₄×D₄	—	59	1	no	—
197	#117	—	59	1	no	—
198	#117	—	59	1	no	—
199	#117	—	59	1	no	—
200	#117	—	118	1	no	—
201	C₂×C₄⋊C₄	—	59	1	no	—
202	#107	—	59	1	no	—
203	#107	—	59	1	no	—
204	#107	—	59	1	no	—
205	#107	—	59	1	no	—
206	#107	—	59	1	no	—
207	#106	—	59	1	no	—
208	#106	—	59	1	no	—
209	#106	—	59	1	no	—
210	#106	C₂×D₅₉	1	1	yes	characteristic
211	C₄⋊C₈	—	59	1	no	—
212	#103	—	59	1	no	—
213	#103	—	59	1	no	—
214	#103	—	59	1	no	—
215	#103	—	59	1	no	—
216	#103	C₂×D₅₉	1	1	yes	characteristic
217	#103	C₂×D₅₉	1	1	yes	characteristic
218	C₂²⋊C₈	—	59	1	no	—
219	C₂²⋊C₈	—	59	1	no	—
220	#97	—	59	1	no	—
221	C₂×Q₈	—	118	1	no	—
222	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
223	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
224	C₂×D₄	—	59	1	no	—
225	C₂²×C₄	—	59	1	no	—
226	C₂²×C₄	—	118	1	no	—
227	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
228	Q₁₆	—	2	2	no	—
229	M₁₆	—	2	2	no	—
230	C₂×C₈	—	59	1	no	—
231	C₂×C₈	—	59	1	no	—
232	C₂×C₈	—	118	2	no	—
233	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	59	1	no	—
238	C₄⋊C₄	—	59	1	no	—
239	C₄⋊C₄	—	59	1	no	—
240	C₄⋊C₄	—	59	1	no	—
241	C₄⋊C₄	—	59	1	no	—
242	C₄⋊C₄	—	118	1	no	—
243	C₄⋊C₄	—	118	1	no	—
244	C₄⋊C₄	—	118	1	no	—
245	C₄⋊C₄	—	118	1	no	—
246	C₄⋊C₄	—	118	1	no	—
247	C₄⋊C₄	—	118	1	no	—
248	C₄⋊C₄	—	118	1	no	—
249	C₄⋊C₄	—	118	1	no	—
250	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	59	1	no	—
255	C₂²⋊C₄	—	59	1	no	—
256	C₂²⋊C₄	—	118	1	no	—
257	C₂²⋊C₄	—	118	1	no	—
258	C₂²⋊C₄	—	118	1	no	—
259	C₂²⋊C₄	—	118	1	no	—
260	C₂²⋊C₄	—	118	1	no	—
261	C₄²	—	59	1	no	—
262	C₄²	—	118	1	no	—
263	C₄²	—	118	1	no	—
264	C₄²	—	118	1	no	—
265	C₄²	—	118	1	no	—
266	C₄²	—	118	1	no	—
267	C₂³	—	59	1	no	—
268	C₂³	D₄×D₅₉	1	1	yes	characteristic
269	Q₈	—	2	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	236	2	no	—
274	D₄	—	236	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	59	1	no	—
282	C₂×C₄	—	59	1	no	—
283	C₂×C₄	—	59	1	no	—
284	C₂×C₄	—	118	1	no	—
285	C₂×C₄	—	118	1	no	—
286	C₂×C₄	—	118	1	no	—
287	C₂×C₄	—	118	1	no	—
288	C₂×C₄	—	118	1	no	—
289	C₂×C₄	—	118	1	no	—
290	C₂×C₄	—	236	2	no	—
291	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	118	1	no	—
295	C₈	—	118	1	no	—
296	C₈	#11745687	1	2	yes	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	118	1	no	—
301	C₂²	—	118	1	no	—
302	C₂²	—	118	1	no	—
303	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	118	1	no	—
310	C₄	—	118	1	no	—
311	C₄	—	118	1	no	—
312	C₄	—	236	1	no	—
313	C₄	—	236	1	no	—
314	C₄	—	236	1	no	—
315	C₄	—	236	1	no	—
316	C₄	C₂×#11745687	1	1	yes	characteristic
317	C₄	#14273774	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	236	1	no	—
320	C₂	#16838052	1	1	yes	characteristic
321	C₂	#16838201	1	1	yes	characteristic
322	C₂	#16838218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#470	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#470	D₅₉
hypocenter	C₅₉	#3349
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

Complements

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

N	H ≅ G/N	type
3. #16837809	C₂	semidirect
4. #16837742	C₂	semidirect
5. #16837741	C₂	semidirect
6. #16837740	C₂	semidirect
7. #16837735	C₂	semidirect
8. #16837707	C₂	semidirect
9. #16837695	C₂	semidirect
10. #16837640	C₂	semidirect
11. #16837509	C₂	semidirect
13. #16837444	C₂	semidirect
14. #16837437	C₂	semidirect
15. C₅₉×#470	C₂	semidirect
171. #470	D₅₉	semidirect
177. C₅₉	#3349	semidirect