Subgroups — #21434254

Order 7552.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 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	C₄×#11745687	C₂	1	1	yes	maximal, characteristic
3	#16837745	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837741	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837696	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837690	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837611	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837605	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837565	C₂	1	1	yes	maximal, characteristic
10	#16837561	C₂	1	1	yes	maximal, characteristic
11	#16837510	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837507	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837502	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837457	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745687	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
18	C₂×#11745672	C₂²	1	1	yes	characteristic
19	C₂×#11745670	C₂²	1	1	yes	characteristic
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273725	C₂²	1	1	yes	characteristic
22	#14273717	—	2	1	no	—
23	#14273707	C₂²	1	1	yes	characteristic
24	#14273705	—	2	1	no	—
25	#14273703	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₁₈	—	2	1	no	—
27	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
29	#14273692	C₂²	1	1	yes	characteristic
30	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₅₉	—	2	1	no	—
32	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
33	#14273685	C₂²	1	1	yes	characteristic
34	#14273684	—	2	1	no	—
35	#14273683	—	2	1	no	—
36	C₂×C₄×Dic₅₉	—	2	1	no	—
37	C₄²×D₅₉	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	#14273652	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273650	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273648	C₂²	1	1	yes	characteristic
49	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	#14273638	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#102×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×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂²×Dic₅₉	—	2	1	no	—
64	C₂²×Dic₅₉	—	2	1	no	—
65	#11745687	—	2	2	no	—
66	#11745687	—	2	2	no	—
67	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
68	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
70	C₂×Dic₁₁₈	—	2	1	no	—
71	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
73	SD₁₆×C₅₉	—	2	2	no	—
74	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₅₉	—	2	1	no	—
77	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₅₉	—	2	1	no	—
80	C₄²×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	#11745672	—	2	2	no	—
84	#11745670	—	2	2	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	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	—	2	1	no	—
96	Dic₅₉⋊C₄	—	2	1	no	—
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	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
107	Dic₂₃₆	—	2	2	no	—
108	C₂³×C₅₉	—	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	D₄×C₅₉	—	2	1	no	—
113	D₄×C₅₉	#41	1	2	yes	—
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	—	4	2	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	C₂²⋊D₅₉	—	2	4	no	—
122	C₂×Dic₅₉	—	2	4	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
131	C₄×D₅₉	—	2	1	no	—
132	C₄×D₅₉	—	2	1	no	—
133	C₄×D₅₉	—	2	2	no	—
134	C₄×D₅₉	—	4	1	no	—
135	Dic₁₁₈	—	2	1	no	—
136	Dic₁₁₈	—	2	1	no	—
137	Dic₁₁₈	—	2	1	no	—
138	Dic₁₁₈	—	4	2	no	—
139	Dic₁₁₈	#41	1	2	yes	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₂²×C₅₉	—	2	2	no	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	—	4	1	no	—
147	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₅₉	—	2	1	no	—
149	C₂×D₅₉	—	2	1	no	—
150	C₂×D₅₉	—	2	1	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	4	1	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	—	4	1	no	—
156	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
157	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
158	Dic₅₉	—	2	2	no	—
159	Dic₅₉	—	2	1	no	—
160	Dic₅₉	—	2	1	no	—
161	Dic₅₉	—	4	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	#137	1	2	yes	—
165	#3116	—	59	1	no	maximal, Sylow
166	C₂×C₅₉	—	4	2	no	—
167	C₂×C₅₉	#525	1	1	yes	characteristic
168	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
169	C₂×C₅₉	#575	1	1	yes	characteristic
170	D₅₉	—	4	1	no	—
171	#533	—	59	1	no	—
172	C₄×#41	—	59	1	no	—
173	#485	—	59	1	no	—
174	#485	—	59	1	no	—
175	#477	—	59	1	no	—
176	#476	—	59	1	no	—
177	#462	—	59	1	no	—
178	#461	—	59	1	no	—
179	#452	—	59	1	no	—
180	#452	—	59	1	no	—
181	#441	—	59	1	no	—
182	#441	—	59	1	no	—
183	#440	—	59	1	no	—
184	#440	D₅₉	1	1	yes	characteristic, complemented
185	#432	—	59	1	no	—
186	C₅₉	#3116	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	59	1	no	—
188	C₂×Q₁₆	—	59	1	no	—
189	C₂×Q₁₆	—	59	1	no	—
190	C₂×SD₁₆	—	59	1	no	—
191	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
192	C₄⋊Q₈	—	59	1	no	—
193	C₄⋊Q₈	—	59	1	no	—
194	#125	—	118	1	no	—
195	C₂²⋊Q₈	—	59	1	no	—
196	C₂²⋊Q₈	—	59	1	no	—
197	C₂²⋊Q₈	—	118	1	no	—
198	C₄×Q₈	—	59	1	no	—
199	C₄×Q₈	—	59	1	no	—
200	C₄×Q₈	—	59	1	no	—
201	C₄×Q₈	—	118	1	no	—
202	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
203	C₄×D₄	—	59	1	no	—
204	C₄×D₄	—	118	1	no	—
205	C₄×D₄	—	118	1	no	—
206	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
207	#117	—	59	1	no	—
208	#117	—	118	1	no	—
209	#117	—	118	1	no	—
210	C₂×C₄²	—	59	1	no	—
211	C₂×C₄²	—	118	1	no	—
212	#107	—	59	1	no	—
213	#107	C₂×D₅₉	1	1	yes	characteristic
214	#106	—	59	1	no	—
215	#106	—	59	1	no	—
216	C₄⋊C₈	—	59	1	no	—
217	C₄⋊C₈	—	59	1	no	—
218	#103	—	59	1	no	—
219	#103	—	59	1	no	—
220	#103	—	59	1	no	—
221	#103	—	59	1	no	—
222	#103	—	59	1	no	—
223	#103	C₂×D₅₉	1	1	yes	characteristic
224	#102	—	59	1	no	—
225	#102	C₂×D₅₉	1	1	yes	characteristic
226	C₂²⋊C₈	—	59	1	no	—
227	C₂²⋊C₈	—	59	1	no	—
228	#97	—	59	1	no	—
229	#97	C₂×D₅₉	1	1	yes	characteristic
230	#41	—	118	2	no	—
231	#41	—	118	2	no	—
232	C₂×Q₈	—	59	1	no	—
233	C₂×Q₈	—	59	1	no	—
234	C₂×Q₈	—	118	1	no	—
235	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₂×D₄	—	118	1	no	—
237	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
238	C₂²×C₄	—	2	1	no	—
239	C₂²×C₄	—	59	1	no	—
240	C₂²×C₄	—	59	1	no	—
241	C₂²×C₄	—	59	1	no	—
242	C₂²×C₄	—	118	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	Q₁₆	—	118	2	no	—
245	Q₁₆	—	118	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	118	2	no	—
248	C₂×C₈	—	59	1	no	—
249	C₂×C₈	—	59	1	no	—
250	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	59	1	no	—
254	C₄⋊C₄	—	59	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	—	118	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	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	—	59	1	no	—
272	C₄²	—	59	1	no	—
273	C₄²	—	59	1	no	—
274	C₄²	—	118	1	no	—
275	C₄²	—	118	1	no	—
276	C₄²	—	118	1	no	—
277	C₄²	—	118	1	no	—
278	C₄²	C₂²×D₅₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	59	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	59	2	no	—
284	Q₈	—	118	1	no	—
285	Q₈	—	118	1	no	—
286	Q₈	—	118	1	no	—
287	Q₈	—	236	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	118	4	no	—
290	D₄	#11745685	1	2	yes	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	59	1	no	—
296	C₂×C₄	—	59	1	no	—
297	C₂×C₄	—	59	1	no	—
298	C₂×C₄	—	118	2	no	—
299	C₂×C₄	—	118	4	no	—
300	C₂×C₄	—	118	1	no	—
301	C₂×C₄	—	118	1	no	—
302	C₂×C₄	—	118	1	no	—
303	C₂×C₄	—	118	1	no	—
304	C₂×C₄	—	118	1	no	—
305	C₂×C₄	—	118	1	no	—
306	C₂×C₄	—	236	1	no	—
307	C₂×C₄	—	236	2	no	—
308	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	118	1	no	—
314	C₈	—	118	1	no	—
315	C₂²	—	2	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	118	1	no	—
319	C₂²	—	118	1	no	—
320	C₂²	—	118	1	no	—
321	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	59	2	no	—
328	C₄	—	118	1	no	—
329	C₄	—	118	2	no	—
330	C₄	—	118	1	no	—
331	C₄	—	236	1	no	—
332	C₄	—	236	1	no	—
333	C₄	—	236	1	no	—
334	C₄	#41×D₅₉	1	1	yes	characteristic
335	C₄	C₂×#11745685	1	1	yes	characteristic
336	C₂	—	4	2	no	—
337	C₂	—	236	1	no	—
338	C₂	#16837911	1	1	yes	characteristic
339	C₂	#137×D₅₉	1	1	yes	characteristic
340	C₂	#16838198	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#440	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#440	D₅₉
hypocenter	C₅₉	#3116
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
1. #16837956	C₂	semidirect
3. #16837745	C₂	semidirect
4. #16837741	C₂	semidirect
5. #16837696	C₂	semidirect
6. #16837690	C₂	semidirect
7. #16837611	C₂	semidirect
8. #16837605	C₂	semidirect
11. #16837510	C₂	semidirect
12. #16837507	C₂	semidirect
13. #16837502	C₂	semidirect
14. #16837457	C₂	semidirect
15. C₅₉×#440	C₂	semidirect
184. #440	D₅₉	semidirect
186. C₅₉	#3116	semidirect