Subgroups — #21437079

Order 7552.

360 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838124	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838121	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837784	C₂	1	1	yes	maximal, characteristic
4	#16837783	C₂	1	1	yes	maximal, characteristic
5	#16837780	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837626	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837625	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837623	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837606	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837605	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837526	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11745672	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837494	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837482	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#464	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
17	C₂×#11745672	C₂²	1	1	yes	characteristic
18	C₂×#11745672	C₂²	1	1	yes	characteristic
19	C₂×#11745660	C₂²	1	1	yes	characteristic
20	C₂×#11745670	C₂²	1	1	yes	characteristic
21	C₂×#11745671	C₂²	1	1	yes	characteristic
22	C₂×#11745669	C₂²	1	1	yes	characteristic
23	#14273725	C₂²	1	1	yes	characteristic
24	#14273725	C₂²	1	1	yes	characteristic
25	#14273722	—	2	1	no	—
26	#14273721	—	2	1	no	—
27	#14273720	—	2	1	no	—
28	#14273720	—	2	1	no	—
29	#14273716	C₂²	1	1	yes	characteristic
30	#14273715	C₂²	1	1	yes	characteristic
31	#14273714	C₂²	1	1	yes	characteristic
32	#14273705	—	2	1	no	—
33	#14273704	—	2	1	no	—
34	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
37	#14273692	C₂²	1	1	yes	characteristic
38	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
39	#14273689	C₂²	1	1	yes	characteristic
40	#14273684	—	2	1	no	—
41	#14273683	—	2	1	no	—
42	#14273660	C₂²	1	1	yes	characteristic
43	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	#14273652	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273649	C₂²	1	1	yes	characteristic
49	C₄×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×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	#124×C₅₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#102×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×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₅₉	—	2	1	no	—
63	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
66	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
68	Q₁₆×C₅₉	—	4	1	no	—
69	SD₁₆×C₅₉	—	4	1	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	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₅₉	—	2	1	no	—
75	C₂²⋊C₄×C₅₉	—	2	1	no	—
76	C₄²×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₅₉	—	2	1	no	—
79	C₂²⋊Dic₅₉	—	2	1	no	—
80	C₂²⋊Dic₅₉	—	2	1	no	—
81	#11745672	—	2	2	no	—
82	#11745672	—	2	2	no	—
83	#11745671	—	4	1	no	—
84	#11745670	—	4	1	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	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₅₉	—	2	1	no	—
92	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
93	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
101	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
106	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	Dic₂₃₆	—	4	1	no	—
110	#11745660	—	4	1	no	—
111	C₂³×C₅₉	—	2	1	no	—
112	C₂²×D₅₉	—	2	1	no	—
113	Q₈×C₅₉	—	2	1	no	—
114	Q₈×C₅₉	—	2	1	no	—
115	Q₈×C₅₉	—	2	1	no	—
116	Q₈×C₅₉	—	2	1	no	—
117	D₄×C₅₉	—	2	1	no	—
118	D₄×C₅₉	—	2	1	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	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	4	2	no	—
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
133	D₂₃₆	—	2	1	no	—
134	D₂₃₆	—	2	1	no	—
135	C₄×D₅₉	—	4	2	no	—
136	Dic₁₁₈	—	2	2	no	—
137	Dic₁₁₈	—	2	1	no	—
138	Dic₁₁₈	—	2	1	no	—
139	Dic₁₁₈	—	2	1	no	—
140	Dic₁₁₈	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₅₉⋊C₈	#41	1	2	yes	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	—	4	1	no	—
147	C₂²×C₅₉	—	4	1	no	—
148	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₅₉	—	4	1	no	—
150	C₂×D₅₉	—	4	1	no	—
151	C₂×D₅₉	—	4	1	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	—	4	1	no	—
156	C₄×C₅₉	—	4	1	no	—
157	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
158	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
159	Dic₅₉	—	2	1	no	—
160	Dic₅₉	—	2	1	no	—
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	#135	1	2	yes	—
166	#3393	—	59	1	no	maximal, Sylow
167	C₂×C₅₉	—	8	1	no	—
168	C₂×C₅₉	#550	1	1	yes	characteristic
169	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
170	C₂×C₅₉	#578	1	1	yes	characteristic
171	D₅₉	—	8	1	no	—
172	#556	—	59	1	no	—
173	#556	—	59	1	no	—
174	#495	—	59	1	no	—
175	#495	—	59	1	no	—
176	#494	—	59	1	no	—
177	#464	—	59	1	no	—
178	#464	—	59	1	no	—
179	#464	—	59	1	no	—
180	#464	D₅₉	1	1	yes	characteristic, complemented
181	#461	—	59	1	no	—
182	#461	—	59	1	no	—
183	C₈×Q₈	—	59	1	no	—
184	C₄×Q₁₆	—	59	1	no	—
185	C₄×Q₁₆	—	59	1	no	—
186	C₄×D₈	—	59	1	no	—
187	C₅₉	#3393	1	1	yes	characteristic, complemented, Sylow
188	C₂×Q₁₆	—	59	1	no	—
189	C₂×Q₁₆	—	59	1	no	—
190	C₂×Q₁₆	—	59	1	no	—
191	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
192	C₂×SD₁₆	—	59	1	no	—
193	C₂×SD₁₆	—	59	1	no	—
194	C₂×SD₁₆	—	59	1	no	—
195	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
196	C₂×D₈	—	59	1	no	—
197	C₄⋊Q₈	—	59	1	no	—
198	C₄⋊Q₈	—	59	1	no	—
199	#126	—	118	1	no	—
200	#126	—	118	1	no	—
201	#126	—	118	1	no	—
202	#126	—	118	1	no	—
203	#124	—	59	1	no	—
204	#124	—	59	1	no	—
205	#124	—	59	1	no	—
206	#124	C₂×D₅₉	1	1	yes	characteristic
207	C₂²⋊Q₈	—	118	1	no	—
208	C₂²⋊Q₈	—	118	1	no	—
209	C₄×Q₈	—	59	1	no	—
210	C₄×Q₈	—	59	1	no	—
211	C₄×Q₈	—	59	1	no	—
212	C₄×Q₈	—	59	1	no	—
213	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	59	1	no	—
216	#117	—	118	1	no	—
217	#117	—	118	1	no	—
218	#107	—	59	1	no	—
219	C₄⋊C₈	—	59	1	no	—
220	C₄⋊C₈	—	59	1	no	—
221	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
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	C₂×D₅₉	1	1	yes	characteristic
228	C₄×C₈	—	59	1	no	—
229	C₄×C₈	—	59	1	no	—
230	C₄×C₈	—	59	1	no	—
231	C₂×Q₈	—	59	1	no	—
232	C₂×Q₈	—	59	1	no	—
233	C₂×Q₈	—	59	1	no	—
234	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₂×D₄	—	59	1	no	—
237	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
238	C₂²×C₄	—	118	1	no	—
239	C₂²×C₄	—	118	1	no	—
240	Q₁₆	—	4	1	no	—
241	Q₁₆	—	118	2	no	—
242	Q₁₆	—	118	2	no	—
243	Q₁₆	—	236	1	no	—
244	SD₁₆	—	4	1	no	—
245	SD₁₆	—	236	1	no	—
246	SD₁₆	—	236	1	no	—
247	SD₁₆	—	236	1	no	—
248	D₈	—	118	2	no	—
249	C₂×C₈	—	59	1	no	—
250	C₂×C₈	—	59	1	no	—
251	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	59	1	no	—
255	C₄⋊C₄	—	59	1	no	—
256	C₄⋊C₄	—	59	1	no	—
257	C₄⋊C₄	—	59	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	—	118	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
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	C₂²⋊C₄	—	118	1	no	—
276	C₄²	—	2	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	C₂²×D₅₉	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	118	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	2	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	118	2	no	—
290	Q₈	—	118	1	no	—
291	Q₈	—	118	1	no	—
292	Q₈	—	118	1	no	—
293	Q₈	—	118	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	118	1	no	—
297	D₄	—	118	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	59	1	no	—
302	C₂×C₄	—	59	1	no	—
303	C₂×C₄	—	59	1	no	—
304	C₂×C₄	—	59	1	no	—
305	C₂×C₄	—	118	1	no	—
306	C₂×C₄	—	118	1	no	—
307	C₂×C₄	—	118	1	no	—
308	C₂×C₄	—	236	2	no	—
309	C₂×C₄	—	236	2	no	—
310	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
311	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
312	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	59	2	no	—
316	C₈	—	118	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	236	1	no	—
321	C₂²	—	236	1	no	—
322	C₂²	—	236	1	no	—
323	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	59	2	no	—
330	C₄	—	118	1	no	—
331	C₄	—	118	1	no	—
332	C₄	—	118	1	no	—
333	C₄	—	236	1	no	—
334	C₄	—	236	1	no	—
335	C₄	—	236	1	no	—
336	C₄	#41×D₅₉	1	1	yes	characteristic
337	C₄	#14273772	1	1	yes	characteristic
338	C₂	—	8	1	no	—
339	C₂	—	472	1	no	—
340	C₂	#16838070	1	1	yes	characteristic
341	C₂	#135×D₅₉	1	1	yes	characteristic
342	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×#464	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#464	D₅₉
hypocenter	C₅₉	#3393
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
1. #16838124	C₂	semidirect
2. #16838121	C₂	semidirect
5. #16837780	C₂	semidirect
6. #16837626	C₂	semidirect
7. #16837625	C₂	semidirect
8. #16837623	C₂	semidirect
9. #16837606	C₂	semidirect
10. #16837605	C₂	semidirect
11. #16837526	C₂	semidirect
12. C₄×#11745672	C₂	semidirect
13. #16837494	C₂	semidirect
15. C₅₉×#464	C₂	semidirect
180. #464	D₅₉	semidirect
187. C₅₉	#3393	semidirect