Subgroups — #21434115

Order 7552.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837938	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837740	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837727	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837707	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837700	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837626	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837619	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837547	C₂	1	1	yes	maximal, characteristic
10	Q₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837487	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837482	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837452	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#452	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745671	C₂²	1	1	yes	characteristic
17	C₂×#11745669	C₂²	1	1	yes	characteristic
18	#14273722	—	2	1	no	—
19	#14273718	C₂²	1	1	yes	characteristic
20	#14273716	C₂²	1	1	yes	characteristic
21	#14273711	—	2	1	no	—
22	#14273701	C₂²	1	1	yes	characteristic
23	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
26	D₄×Dic₅₉	—	2	1	no	—
27	D₄×Dic₅₉	—	2	1	no	—
28	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
30	#14273689	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
32	#14273686	—	2	1	no	—
33	#14273686	C₂²	1	1	yes	characteristic
34	#14273685	—	2	1	no	—
35	C₂×C₄×Dic₅₉	—	2	1	no	—
36	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273659	C₂²	1	1	yes	characteristic
40	#14273658	C₂²	1	1	yes	characteristic
41	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	#14273651	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273649	C₂²	1	1	yes	characteristic
47	#14273649	C₂²	1	1	yes	characteristic
48	#14273648	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×#41×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
59	#41×C₅₉	—	2	2	no	—
60	#41×C₅₉	—	2	2	no	—
61	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₅₉	—	2	1	no	—
64	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	—	2	1	no	—
66	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊D₅₉	—	2	1	no	—
68	C₂²×Dic₅₉	—	2	1	no	—
69	C₂²×Dic₅₉	—	2	1	no	—
70	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
71	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
72	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₅₉	—	2	1	no	—
74	Q₁₆×C₅₉	—	2	2	no	—
75	SD₁₆×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₅₉	—	2	1	no	—
79	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₂²⋊Dic₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	C₂²⋊Dic₅₉	—	2	1	no	—
84	#11745671	—	2	2	no	—
85	#11745669	—	2	2	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₅₉	—	2	1	no	—
91	C₄⋊Dic₅₉	—	2	1	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	—	2	1	no	—
94	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	Dic₅₉⋊C₄	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	D₄	1	1	yes	characteristic
103	C₄×Dic₅₉	D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
105	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₅₉	—	2	1	no	—
109	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₅₉	—	2	1	no	—
111	Q₈×C₅₉	—	2	1	no	—
112	Q₈×C₅₉	—	2	1	no	—
113	Q₈×C₅₉	—	2	1	no	—
114	Q₈×C₅₉	#41	1	2	yes	—
115	D₄×C₅₉	—	2	4	no	—
116	D₄×C₅₉	—	2	1	no	—
117	D₄×C₅₉	#41	1	2	yes	—
118	C₂×C₄×C₅₉	—	2	4	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
126	C₂²⋊D₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	2	2	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	4	1	no	—
132	C₂×Dic₅₉	—	4	2	no	—
133	C₂×Dic₅₉	—	4	2	no	—
134	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
138	D₂₃₆	—	2	1	no	—
139	D₂₃₆	—	2	1	no	—
140	C₄×D₅₉	—	4	2	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₅₉⋊C₈	—	2	1	no	—
145	C₂²×C₅₉	—	2	1	no	—
146	C₂²×C₅₉	—	2	2	no	—
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	4	1	no	—
150	C₂²×C₅₉	—	4	1	no	—
151	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₅₉	—	4	1	no	—
153	C₂×D₅₉	—	4	1	no	—
154	C₂×D₅₉	—	4	1	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	2	2	no	—
157	C₄×C₅₉	—	4	1	no	—
158	C₄×C₅₉	—	4	1	no	—
159	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	2	1	no	—
163	Dic₅₉	—	2	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	#135	1	2	yes	—
168	#3102	—	59	1	no	maximal, Sylow
169	C₂×C₅₉	—	4	1	no	—
170	C₂×C₅₉	—	4	2	no	—
171	C₂×C₅₉	#525	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
173	C₂×C₅₉	#575	1	1	yes	characteristic
174	D₅₉	—	8	1	no	—
175	#529	—	59	1	no	—
176	C₄×#41	—	59	1	no	—
177	#485	—	59	1	no	—
178	#482	—	59	1	no	—
179	#479	—	59	1	no	—
180	#478	—	59	1	no	—
181	#464	—	59	1	no	—
182	#463	—	59	1	no	—
183	#452	D₅₉	1	1	yes	characteristic, complemented
184	#449	—	59	1	no	—
185	C₄×Q₁₆	—	59	1	no	—
186	C₄×SD₁₆	—	59	1	no	—
187	C₄×SD₁₆	—	59	1	no	—
188	C₄×D₈	—	59	1	no	—
189	#431	—	59	1	no	—
190	C₅₉	#3102	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	C₂×D₅₉	1	1	yes	characteristic
192	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
193	C₂×SD₁₆	—	59	1	no	—
194	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
195	C₂×D₈	—	59	1	no	—
196	#126	—	118	1	no	—
197	#125	—	59	1	no	—
198	#124	—	59	1	no	—
199	#123	—	118	1	no	—
200	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
201	#121	—	59	1	no	—
202	C₄×Q₈	—	59	1	no	—
203	C₄×Q₈	—	59	1	no	—
204	C₄×Q₈	—	59	1	no	—
205	C₄×D₄	—	59	1	no	—
206	C₄×D₄	—	59	1	no	—
207	C₄×D₄	—	59	1	no	—
208	C₄×D₄	—	118	1	no	—
209	C₄×D₄	—	118	1	no	—
210	C₄×D₄	—	118	1	no	—
211	#117	—	59	1	no	—
212	#117	—	118	1	no	—
213	#117	—	118	1	no	—
214	C₂×C₄²	—	59	1	no	—
215	C₂×C₄²	—	118	1	no	—
216	#107	—	59	1	no	—
217	#107	—	59	1	no	—
218	#106	—	59	1	no	—
219	#106	—	59	1	no	—
220	C₄⋊C₈	—	59	1	no	—
221	C₄⋊C₈	—	59	1	no	—
222	#103	—	59	1	no	—
223	#103	—	59	1	no	—
224	#103	—	59	1	no	—
225	#103	C₂×D₅₉	1	1	yes	characteristic
226	#102	—	59	1	no	—
227	#102	—	59	1	no	—
228	#102	—	59	1	no	—
229	#102	C₂×D₅₉	1	1	yes	characteristic
230	C₂²⋊C₈	—	59	1	no	—
231	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
232	C₄×C₈	—	59	1	no	—
233	C₄×C₈	—	59	1	no	—
234	#41	—	2	2	no	—
235	#41	—	2	2	no	—
236	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	59	1	no	—
240	C₂×D₄	—	118	1	no	—
241	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	59	1	no	—
244	C₂²×C₄	—	59	1	no	—
245	C₂²×C₄	—	118	1	no	—
246	C₂²×C₄	—	118	1	no	—
247	C₂²×C₄	—	118	1	no	—
248	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
249	Q₁₆	—	2	2	no	—
250	SD₁₆	—	2	2	no	—
251	SD₁₆	—	118	2	no	—
252	D₈	—	118	2	no	—
253	C₂×C₈	—	59	1	no	—
254	C₂×C₈	—	59	1	no	—
255	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	59	1	no	—
259	C₄⋊C₄	—	59	1	no	—
260	C₄⋊C₄	—	59	1	no	—
261	C₄⋊C₄	—	118	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	—	118	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	—	118	1	no	—
266	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	118	1	no	—
269	C₂²⋊C₄	—	118	1	no	—
270	C₂²⋊C₄	—	118	1	no	—
271	C₂²⋊C₄	—	118	1	no	—
272	C₂²⋊C₄	—	118	1	no	—
273	C₂²⋊C₄	—	118	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₄²	—	59	1	no	—
276	C₄²	—	59	1	no	—
277	C₄²	—	59	1	no	—
278	C₄²	—	59	1	no	—
279	C₄²	—	118	1	no	—
280	C₄²	—	118	1	no	—
281	C₄²	—	118	1	no	—
282	C₄²	—	118	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	118	1	no	—
285	C₂³	D₄×D₅₉	1	1	yes	characteristic
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	#11745687	1	2	yes	—
290	D₄	—	2	4	no	—
291	D₄	—	2	1	no	—
292	D₄	—	118	1	no	—
293	D₄	—	118	1	no	—
294	D₄	—	236	2	no	—
295	D₄	#11745687	1	2	yes	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	59	1	no	—
303	C₂×C₄	—	59	1	no	—
304	C₂×C₄	—	59	1	no	—
305	C₂×C₄	—	59	1	no	—
306	C₂×C₄	—	118	2	no	—
307	C₂×C₄	—	118	1	no	—
308	C₂×C₄	—	118	1	no	—
309	C₂×C₄	—	118	1	no	—
310	C₂×C₄	—	236	2	no	—
311	C₂×C₄	—	236	2	no	—
312	C₂×C₄	—	236	1	no	—
313	C₂×C₄	—	236	2	no	—
314	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	118	1	no	—
319	C₈	—	118	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	236	1	no	—
327	C₂²	—	236	1	no	—
328	C₂²	—	236	1	no	—
329	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	59	2	no	—
335	C₄	—	118	1	no	—
336	C₄	—	118	1	no	—
337	C₄	—	118	1	no	—
338	C₄	—	236	1	no	—
339	C₄	—	236	1	no	—
340	C₄	—	236	1	no	—
341	C₄	C₂×#11745687	1	1	yes	characteristic
342	C₄	C₂×#11745687	1	1	yes	characteristic
343	C₂	—	4	1	no	—
344	C₂	—	4	2	no	—
345	C₂	—	472	1	no	—
346	C₂	#16837908	1	1	yes	characteristic
347	C₂	#135×D₅₉	1	1	yes	characteristic
348	C₂	#16838201	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#452	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#452	D₅₉
hypocenter	C₅₉	#3102
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #452

Complements

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

N	H ≅ G/N	type
1. #16837938	C₂	semidirect
2. #41×Dic₅₉	C₂	semidirect
3. #16837740	C₂	semidirect
4. #16837727	C₂	semidirect
5. #16837707	C₂	semidirect
6. #16837700	C₂	semidirect
7. #16837626	C₂	semidirect
8. #16837619	C₂	semidirect
10. Q₁₆×Dic₅₉	C₂	semidirect
11. SD₁₆×Dic₅₉	C₂	semidirect
12. #16837487	C₂	semidirect
13. #16837482	C₂	semidirect
14. #16837452	C₂	semidirect
15. C₅₉×#452	C₂	semidirect
183. #452	D₅₉	semidirect
190. C₅₉	#3102	semidirect