Subgroups — #21436375

Order 7552.

374 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	#16838120	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₂²⋊D₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837751	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837743	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837741	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837685	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837660	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837659	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837611	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837559	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837558	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837494	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837478	C₂	1	1	yes	maximal, characteristic
14	#16837409	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#419	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
18	C₂×#11745672	C₂²	1	1	yes	characteristic
19	C₂×#11745672	C₂²	1	1	yes	characteristic
20	C₂×#11745659	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273725	—	2	1	no	—
23	#14273725	C₂²	1	1	yes	characteristic
24	#14273721	—	2	1	no	—
25	#14273720	—	2	1	no	—
26	#14273713	—	2	1	no	—
27	#14273707	C₂²	1	1	yes	characteristic
28	#14273706	—	2	1	no	—
29	#14273706	C₂²	1	1	yes	characteristic
30	#14273706	C₂²	1	1	yes	characteristic
31	#14273703	—	2	1	no	—
32	#14273703	C₂²	1	1	yes	characteristic
33	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₁₈	—	2	1	no	—
35	#14273692	C₂²	1	1	yes	characteristic
36	C₄×C₂²⋊D₅₉	—	2	1	no	—
37	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
38	#14273685	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273659	C₂²	1	1	yes	characteristic
41	#14273658	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	#14273651	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273650	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	#117×C₅₉	C₂²	1	1	yes	characteristic
55	#103×C₅₉	C₂²	1	1	yes	characteristic
56	#103×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	—	2	1	no	—
60	C₂×Q₈×C₅₉	—	2	2	no	—
61	C₂×Q₈×C₅₉	—	4	1	no	—
62	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
63	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	—	2	1	no	—
65	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
66	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
67	Q₈×D₅₉	—	2	2	no	—
68	Q₈×D₅₉	—	4	1	no	—
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
71	C₂×Dic₁₁₈	—	2	1	no	—
72	C₂×Dic₁₁₈	—	2	1	no	—
73	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
75	M₁₆×C₅₉	—	4	1	no	—
76	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	C₂²⋊Dic₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
85	#11745672	—	2	2	no	—
86	#11745672	—	4	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	D₄	1	1	yes	characteristic
91	C₄⋊Dic₅₉	—	2	1	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	—	2	1	no	—
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	#11745659	—	4	1	no	—
111	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
113	Q₈×C₅₉	—	2	1	no	—
114	Q₈×C₅₉	—	2	1	no	—
115	Q₈×C₅₉	—	2	1	no	—
116	Q₈×C₅₉	—	4	2	no	—
117	Q₈×C₅₉	#41	1	2	yes	—
118	C₂×C₄×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	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	4	2	no	—
125	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₅₉	—	2	2	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
135	C₄×D₅₉	—	2	1	no	—
136	C₄×D₅₉	—	2	1	no	—
137	C₄×D₅₉	—	4	2	no	—
138	Dic₁₁₈	—	2	1	no	—
139	Dic₁₁₈	—	2	1	no	—
140	Dic₁₁₈	—	2	1	no	—
141	Dic₁₁₈	—	2	1	no	—
142	Dic₁₁₈	—	4	2	no	—
143	Dic₁₁₈	—	4	2	no	—
144	C₈×C₅₉	—	2	1	no	—
145	C₈×C₅₉	—	2	1	no	—
146	C₅₉⋊C₈	—	2	1	no	—
147	C₅₉⋊C₈	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₅₉	—	2	1	no	—
153	C₂×D₅₉	—	2	1	no	—
154	C₂×D₅₉	—	2	1	no	—
155	C₄×C₅₉	—	2	2	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	4	1	no	—
158	C₄×C₅₉	—	4	1	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
161	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
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	—	4	1	no	—
168	Dic₅₉	#137	1	2	yes	—
169	#3328	—	59	1	no	maximal, Sylow
170	C₂×C₅₉	—	4	1	no	—
171	C₂×C₅₉	#546	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
173	C₂×C₅₉	#578	1	1	yes	characteristic
174	D₅₉	—	4	1	no	—
175	#556	—	59	1	no	—
176	D₄×Q₈	—	59	1	no	—
177	#487	—	59	1	no	—
178	#485	—	59	1	no	—
179	#485	—	59	1	no	—
180	#475	—	59	1	no	—
181	#470	—	59	1	no	—
182	#470	—	59	1	no	—
183	#462	—	59	1	no	—
184	#451	—	59	1	no	—
185	#451	—	59	1	no	—
186	C₄×Q₁₆	—	59	1	no	—
187	#436	—	59	1	no	—
188	#419	—	59	1	no	—
189	#419	D₅₉	1	1	yes	characteristic, complemented
190	C₅₉	#3328	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₈	—	59	1	no	—
192	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
193	C₂×Q₁₆	—	59	1	no	—
194	C₂×Q₁₆	—	59	1	no	—
195	C₂×Q₁₆	—	59	1	no	—
196	C₂×M₁₆	—	59	1	no	—
197	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
198	C₄⋊Q₈	—	59	1	no	—
199	C₄⋊Q₈	—	59	1	no	—
200	C₄⋊Q₈	—	118	1	no	—
201	#126	—	118	1	no	—
202	#126	—	118	1	no	—
203	#124	—	118	1	no	—
204	C₂²⋊Q₈	—	59	1	no	—
205	C₂²⋊Q₈	—	59	1	no	—
206	C₂²⋊Q₈	—	59	1	no	—
207	C₂²⋊Q₈	—	59	1	no	—
208	C₂²⋊Q₈	—	118	1	no	—
209	C₂²⋊Q₈	—	118	1	no	—
210	C₄×Q₈	—	59	1	no	—
211	C₄×Q₈	—	59	1	no	—
212	C₄×Q₈	—	118	1	no	—
213	C₄×D₄	—	59	1	no	—
214	C₄×D₄	—	118	1	no	—
215	#117	—	59	1	no	—
216	#117	C₂×D₅₉	1	1	yes	characteristic
217	#107	—	59	1	no	—
218	#106	—	59	1	no	—
219	#106	—	59	1	no	—
220	C₄⋊C₈	—	59	1	no	—
221	#103	—	59	1	no	—
222	#103	—	59	1	no	—
223	#103	—	59	1	no	—
224	#103	—	59	1	no	—
225	#103	—	59	1	no	—
226	#103	—	59	1	no	—
227	#103	C₂×D₅₉	1	1	yes	characteristic
228	#103	C₂×D₅₉	1	1	yes	characteristic
229	#103	C₂×D₅₉	1	1	yes	characteristic
230	#103	C₂×D₅₉	1	1	yes	characteristic
231	C₂²⋊C₈	—	59	1	no	—
232	C₂²⋊C₈	—	59	1	no	—
233	C₄×C₈	—	59	1	no	—
234	C₂×Q₈	—	2	1	no	—
235	C₂×Q₈	—	2	2	no	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	59	1	no	—
238	C₂×Q₈	—	59	1	no	—
239	C₂×Q₈	—	118	1	no	—
240	C₂×Q₈	—	118	1	no	—
241	C₂×Q₈	—	118	2	no	—
242	C₂×Q₈	—	236	1	no	—
243	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
244	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
245	C₂×D₄	—	59	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	59	1	no	—
248	C₂²×C₄	—	118	1	no	—
249	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
250	Q₁₆	—	118	2	no	—
251	Q₁₆	—	236	1	no	—
252	Q₁₆	—	236	1	no	—
253	M₁₆	—	4	1	no	—
254	M₁₆	—	236	1	no	—
255	C₂×C₈	—	59	1	no	—
256	C₂×C₈	—	59	1	no	—
257	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
259	C₄⋊C₄	—	59	1	no	—
260	C₄⋊C₄	—	59	1	no	—
261	C₄⋊C₄	—	59	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	—	118	1	no	—
267	C₄⋊C₄	—	118	1	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	—	118	1	no	—
270	C₄⋊C₄	—	118	1	no	—
271	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
272	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	59	1	no	—
275	C₂²⋊C₄	—	59	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₂²⋊C₄	—	118	1	no	—
279	C₂²⋊C₄	—	118	1	no	—
280	C₂²⋊C₄	—	118	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	59	1	no	—
283	C₄²	—	118	1	no	—
284	C₄²	—	118	1	no	—
285	C₄²	—	118	1	no	—
286	C₂³	—	59	1	no	—
287	C₂³	D₄×D₅₉	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	4	2	no	—
292	Q₈	—	118	1	no	—
293	Q₈	—	118	1	no	—
294	Q₈	—	118	1	no	—
295	Q₈	—	118	1	no	—
296	Q₈	—	236	2	no	—
297	Q₈	—	236	2	no	—
298	Q₈	#11745685	1	2	yes	—
299	D₄	—	118	2	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	4	2	no	—
307	C₂×C₄	—	59	1	no	—
308	C₂×C₄	—	59	1	no	—
309	C₂×C₄	—	59	1	no	—
310	C₂×C₄	—	118	1	no	—
311	C₂×C₄	—	118	1	no	—
312	C₂×C₄	—	118	1	no	—
313	C₂×C₄	—	118	1	no	—
314	C₂×C₄	—	118	1	no	—
315	C₂×C₄	—	118	1	no	—
316	C₂×C₄	—	236	2	no	—
317	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	118	1	no	—
322	C₈	—	118	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	118	1	no	—
327	C₂²	—	118	1	no	—
328	C₂²	—	118	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	—	4	1	no	—
335	C₄	—	59	2	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	—	236	1	no	—
342	C₄	C₂×#11745685	1	1	yes	characteristic
343	C₄	#14273772	1	1	yes	characteristic
344	C₂	—	4	1	no	—
345	C₂	—	236	1	no	—
346	C₂	#16838042	1	1	yes	characteristic
347	C₂	#137×D₅₉	1	1	yes	characteristic
348	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×#419	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#419	D₅₉
hypocenter	C₅₉	#3328
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #419

Complements

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

N	H ≅ G/N	type
3. #16837751	C₂	semidirect
4. #16837743	C₂	semidirect
5. #16837741	C₂	semidirect
6. #16837685	C₂	semidirect
7. #16837660	C₂	semidirect
8. #16837659	C₂	semidirect
9. #16837611	C₂	semidirect
10. #16837559	C₂	semidirect
11. #16837558	C₂	semidirect
12. #16837494	C₂	semidirect
14. #16837409	C₂	semidirect
15. C₅₉×#419	C₂	semidirect
189. #419	D₅₉	semidirect
190. C₅₉	#3328	semidirect