Subgroups — #21434803

Order 7552.

366 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837956	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837939	C₂	1	1	yes	maximal, characteristic
3	#16837809	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837795	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837769	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837763	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837642	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837640	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837634	C₂	1	1	yes	maximal, characteristic
10	#16837632	C₂	1	1	yes	maximal, characteristic
11	#16837507	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837504	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837500	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745672	C₂²	1	1	yes	characteristic
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×#11745671	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273725	C₂²	1	1	yes	characteristic
23	#14273722	—	2	1	no	—
24	#14273718	C₂²	1	1	yes	characteristic
25	#14273717	—	2	1	no	—
26	#14273716	C₂²	1	1	yes	characteristic
27	#14273712	—	2	1	no	—
28	#14273707	C₂²	1	1	yes	characteristic
29	#14273705	—	2	1	no	—
30	#14273703	C₂²	1	1	yes	characteristic
31	#14273703	C₂²	1	1	yes	characteristic
32	#14273702	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₁₈	—	2	1	no	—
34	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
35	C₄×D₂₃₆	—	2	1	no	—
36	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
37	#14273684	—	2	1	no	—
38	#14273684	—	2	1	no	—
39	C₄²×D₅₉	C₂²	1	1	yes	characteristic
40	#14273661	C₂²	1	1	yes	characteristic
41	#14273660	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273650	C₂²	1	1	yes	characteristic
47	#14273649	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	#14273638	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	#14273637	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#97×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	C₂×D₂₃₆	—	2	1	no	—
62	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₁₈	—	2	1	no	—
68	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
69	Q₁₆×C₅₉	—	2	2	no	—
70	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₅₉	—	2	1	no	—
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
76	C₄²×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	—	2	1	no	—
78	C₄²×C₅₉	C₂³	1	1	yes	characteristic
79	#11745672	—	2	2	no	—
80	#11745671	—	2	2	no	—
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	(C₂×D₅₉)⋊C₄	—	2	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	C₄⋊Dic₅₉	—	2	1	no	—
89	C₄⋊Dic₅₉	—	2	1	no	—
90	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	—	2	1	no	—
96	Dic₅₉⋊C₄	—	2	1	no	—
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	D₄	1	1	yes	characteristic
101	C₄×Dic₅₉	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	#11745660	—	2	2	no	—
106	C₈×D₅₉	—	2	2	no	—
107	C₈×D₅₉	—	2	2	no	—
108	C₂²×D₅₉	—	2	1	no	—
109	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₅₉	—	2	1	no	—
111	Q₈×C₅₉	—	2	1	no	—
112	Q₈×C₅₉	—	2	1	no	—
113	Q₈×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	—	2	1	no	—
118	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
127	D₂₃₆	—	2	1	no	—
128	D₂₃₆	—	2	1	no	—
129	D₂₃₆	—	4	2	no	—
130	C₄×D₅₉	—	2	2	no	—
131	C₄×D₅₉	—	2	1	no	—
132	C₄×D₅₉	—	2	1	no	—
133	C₄×D₅₉	—	4	1	no	—
134	C₄×D₅₉	—	4	2	no	—
135	Dic₁₁₈	—	2	1	no	—
136	Dic₁₁₈	—	2	1	no	—
137	Dic₁₁₈	—	4	2	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₈×C₅₉	#41	1	2	yes	—
140	C₅₉⋊C₈	—	2	1	no	—
141	C₅₉⋊C₈	#41	1	2	yes	—
142	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₅₉	—	2	1	no	—
144	C₂×D₅₉	—	2	1	no	—
145	C₂×D₅₉	—	4	1	no	—
146	C₂×D₅₉	—	4	1	no	—
147	C₂×D₅₉	—	4	1	no	—
148	C₂×D₅₉	#137	1	2	yes	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	4	1	no	—
152	C₄×C₅₉	—	4	1	no	—
153	C₄×C₅₉	—	4	1	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
156	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
157	Dic₅₉	—	2	1	no	—
158	Dic₅₉	—	2	1	no	—
159	Dic₅₉	—	2	1	no	—
160	Dic₅₉	—	4	1	no	—
161	Dic₅₉	—	4	1	no	—
162	Dic₅₉	—	4	1	no	—
163	#3181	—	59	1	no	maximal, Sylow
164	C₂×C₅₉	#532	1	1	yes	characteristic
165	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
166	C₂×C₅₉	#575	1	1	yes	characteristic
167	D₅₉	—	2	2	no	—
168	D₅₉	—	8	1	no	—
169	#533	—	59	1	no	—
170	#529	—	59	1	no	—
171	#501	—	59	1	no	—
172	#497	—	59	1	no	—
173	#491	—	59	1	no	—
174	#489	—	59	1	no	—
175	#467	—	59	1	no	—
176	#467	—	59	1	no	—
177	#465	—	59	1	no	—
178	#465	—	59	1	no	—
179	#441	—	59	1	no	—
180	#441	D₅₉	1	1	yes	characteristic, complemented
181	#440	—	59	1	no	—
182	#440	—	59	1	no	—
183	C₂×#97	—	59	1	no	—
184	C₅₉	#3181	1	1	yes	characteristic, complemented, Sylow
185	C₂×Q₁₆	—	59	1	no	—
186	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
187	C₂×SD₁₆	—	59	1	no	—
188	C₂×SD₁₆	—	59	1	no	—
189	C₂²×C₈	—	59	1	no	—
190	C₂²×C₈	—	59	1	no	—
191	C₄⋊Q₈	—	59	1	no	—
192	C₄⋊Q₈	—	59	1	no	—
193	#126	—	118	1	no	—
194	#125	—	59	1	no	—
195	#125	—	118	1	no	—
196	#124	—	59	1	no	—
197	#123	—	118	1	no	—
198	C₂²⋊Q₈	—	59	1	no	—
199	C₂²⋊Q₈	—	59	1	no	—
200	C₂²⋊Q₈	—	59	1	no	—
201	C₂²⋊Q₈	—	118	1	no	—
202	#121	—	59	1	no	—
203	C₄×Q₈	—	59	1	no	—
204	C₄×Q₈	—	118	1	no	—
205	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
206	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
207	C₄×D₄	—	59	1	no	—
208	C₄×D₄	—	118	1	no	—
209	#117	—	118	1	no	—
210	#117	—	118	1	no	—
211	C₂×C₄²	—	59	1	no	—
212	#107	—	59	1	no	—
213	#107	—	59	1	no	—
214	#106	—	59	1	no	—
215	#106	C₂×D₅₉	1	1	yes	characteristic
216	#103	—	59	1	no	—
217	#103	—	59	1	no	—
218	#103	—	59	1	no	—
219	#103	—	59	1	no	—
220	#103	C₂×D₅₉	1	1	yes	characteristic
221	#103	C₂×D₅₉	1	1	yes	characteristic
222	#102	—	59	1	no	—
223	#102	—	59	1	no	—
224	#97	—	59	1	no	—
225	#97	—	59	1	no	—
226	#97	—	59	1	no	—
227	#97	C₂×D₅₉	1	1	yes	characteristic
228	C₂×Q₈	—	59	1	no	—
229	C₂×Q₈	—	118	1	no	—
230	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
232	C₂×D₄	—	59	1	no	—
233	C₂×D₄	—	118	1	no	—
234	C₂²×C₄	—	59	1	no	—
235	C₂²×C₄	—	59	1	no	—
236	C₂²×C₄	—	59	1	no	—
237	C₂²×C₄	—	118	1	no	—
238	Q₁₆	—	2	2	no	—
239	Q₁₆	—	118	2	no	—
240	SD₁₆	—	118	2	no	—
241	SD₁₆	—	118	2	no	—
242	C₂×C₈	—	59	1	no	—
243	C₂×C₈	—	59	1	no	—
244	C₂×C₈	—	118	2	no	—
245	C₂×C₈	—	118	2	no	—
246	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	59	1	no	—
251	C₄⋊C₄	—	59	1	no	—
252	C₄⋊C₄	—	118	1	no	—
253	C₄⋊C₄	—	118	1	no	—
254	C₄⋊C₄	—	118	1	no	—
255	C₄⋊C₄	—	118	1	no	—
256	C₄⋊C₄	—	118	1	no	—
257	C₄⋊C₄	—	118	1	no	—
258	C₄⋊C₄	—	118	1	no	—
259	C₄⋊C₄	—	118	1	no	—
260	C₄⋊C₄	—	118	1	no	—
261	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
263	C₂²⋊C₄	—	118	1	no	—
264	C₂²⋊C₄	—	118	1	no	—
265	C₂²⋊C₄	—	118	1	no	—
266	C₂²⋊C₄	—	118	1	no	—
267	C₂²⋊C₄	—	118	1	no	—
268	C₂²⋊C₄	—	118	1	no	—
269	C₂²⋊C₄	—	118	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	59	1	no	—
273	C₄²	—	59	1	no	—
274	C₄²	—	59	1	no	—
275	C₄²	—	118	1	no	—
276	C₄²	C₂²×D₅₉	1	1	yes	characteristic
277	C₂³	—	59	1	no	—
278	C₂³	—	118	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	118	1	no	—
284	Q₈	—	118	1	no	—
285	Q₈	—	236	2	no	—
286	D₄	—	118	1	no	—
287	D₄	—	118	1	no	—
288	D₄	—	236	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	59	1	no	—
294	C₂×C₄	—	59	1	no	—
295	C₂×C₄	—	59	1	no	—
296	C₂×C₄	—	118	2	no	—
297	C₂×C₄	—	118	1	no	—
298	C₂×C₄	—	118	1	no	—
299	C₂×C₄	—	118	1	no	—
300	C₂×C₄	—	118	1	no	—
301	C₂×C₄	—	118	1	no	—
302	C₂×C₄	—	236	1	no	—
303	C₂×C₄	—	236	2	no	—
304	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
305	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	59	2	no	—
309	C₈	—	118	1	no	—
310	C₈	#11745685	1	2	yes	—
311	C₂²	—	59	2	no	—
312	C₂²	—	118	1	no	—
313	C₂²	—	118	1	no	—
314	C₂²	—	236	1	no	—
315	C₂²	—	236	1	no	—
316	C₂²	—	236	1	no	—
317	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	118	1	no	—
325	C₄	—	118	1	no	—
326	C₄	—	118	1	no	—
327	C₄	—	236	1	no	—
328	C₄	—	236	1	no	—
329	C₄	—	236	1	no	—
330	C₄	#41×D₅₉	1	1	yes	characteristic
331	C₄	C₂×#11745685	1	1	yes	characteristic
332	C₂	—	118	2	no	—
333	C₂	—	472	1	no	—
334	C₂	#16837949	1	1	yes	characteristic
335	C₂	#137×D₅₉	1	1	yes	characteristic
336	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×#441	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#441	D₅₉
hypocenter	C₅₉	#3181
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. #16837956	C₂	semidirect
3. #16837809	C₂	semidirect
4. #16837795	C₂	semidirect
5. #16837769	C₂	semidirect
6. #16837763	C₂	semidirect
7. #16837642	C₂	semidirect
8. #16837640	C₂	semidirect
11. #16837507	C₂	semidirect
12. #16837504	C₂	semidirect
13. #16837500	C₂	semidirect
14. #97×D₅₉	C₂	semidirect
15. C₅₉×#441	C₂	semidirect
180. #441	D₅₉	semidirect
184. C₅₉	#3181	semidirect