Subgroups — #21436675

Order 7552.

374 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	#16838123	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838052	C₂	1	1	yes	maximal, characteristic
3	#16837806	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837736	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837735	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837725	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837723	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837685	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837680	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837645	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837478	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837444	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837443	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#468	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745687	—	2	1	no	—
17	C₂×#11745659	C₂²	1	1	yes	characteristic
18	#14273726	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273722	—	2	1	no	—
21	#14273720	—	2	1	no	—
22	#14273713	—	2	1	no	—
23	#14273709	—	2	1	no	—
24	#14273708	—	2	1	no	—
25	#14273704	—	2	1	no	—
26	#14273703	C₂²	1	1	yes	characteristic
27	#14273698	C₂²	1	1	yes	characteristic
28	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
29	#14273692	—	2	1	no	—
30	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
32	#14273686	C₂²	1	1	yes	characteristic
33	#14273685	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
36	#14273661	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	#14273659	C₂²	1	1	yes	characteristic
42	#14273659	C₂²	1	1	yes	characteristic
43	#14273658	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273648	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	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	#121×C₅₉	C₂²	1	1	yes	characteristic
56	#107×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×D₄×C₅₉	—	2	1	no	—
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	2	1	no	—
67	#11745687	—	4	4	no	—
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
70	C₂×Dic₁₁₈	—	2	1	no	—
71	C₂×Dic₁₁₈	—	2	1	no	—
72	SD₁₆×C₅₉	—	2	2	no	—
73	M₁₆×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	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	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
85	(C₂×D₅₉)⋊C₄	—	2	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
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	C₄⋊Dic₅₉	C₂³	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	D₄	1	1	yes	characteristic
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
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	#11745659	—	2	2	no	—
106	C₂³×C₅₉	—	2	1	no	—
107	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₅₉	—	2	1	no	—
110	Q₈×C₅₉	—	2	1	no	—
111	D₄×C₅₉	—	2	1	no	—
112	D₄×C₅₉	—	2	1	no	—
113	D₄×C₅₉	—	4	2	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	—	2	1	no	—
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₅₉	—	2	2	no	—
122	C₂²⋊D₅₉	—	4	2	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	—	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	C₄×D₅₉	—	2	1	no	—
133	C₄×D₅₉	—	2	1	no	—
134	C₄×D₅₉	—	4	2	no	—
135	Dic₁₁₈	—	4	2	no	—
136	Dic₁₁₈	—	4	2	no	—
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	#41	1	2	yes	complemented
139	C₅₉⋊C₈	—	2	1	no	—
140	C₅₉⋊C₈	—	2	1	no	—
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	—	4	1	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	—	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	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₅₉	—	2	1	no	—
158	Dic₅₉	—	2	1	no	—
159	Dic₅₉	—	4	1	no	—
160	Dic₅₉	—	4	1	no	—
161	Dic₅₉	—	4	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
164	#3353	—	59	1	no	maximal, Sylow
165	C₂×C₅₉	—	4	1	no	—
166	C₂×C₅₉	—	8	1	no	—
167	C₂×C₅₉	#547	1	1	yes	characteristic
168	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
169	C₂×C₅₉	#578	1	1	yes	characteristic
170	D₅₉	—	4	1	no	—
171	#556	—	59	1	no	—
172	#547	—	59	1	no	—
173	#500	—	59	1	no	—
174	#484	—	59	1	no	—
175	#484	—	59	1	no	—
176	#482	—	59	1	no	—
177	#482	—	59	1	no	—
178	#475	—	59	1	no	—
179	#474	—	59	1	no	—
180	#468	—	59	1	no	—
181	#468	D₅₉	1	1	yes	characteristic, complemented
182	C₄×SD₁₆	—	59	1	no	—
183	#436	—	59	1	no	—
184	#428	—	59	1	no	—
185	#428	—	59	1	no	—
186	C₅₉	#3353	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	118	1	no	—
188	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
189	C₂×M₁₆	—	59	1	no	—
190	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
191	C₄⋊Q₈	—	59	1	no	—
192	C₄⋊Q₈	—	59	1	no	—
193	#126	—	118	1	no	—
194	#126	—	118	1	no	—
195	#124	—	118	1	no	—
196	#123	—	118	1	no	—
197	#123	—	118	1	no	—
198	C₂²⋊Q₈	—	59	1	no	—
199	C₂²⋊Q₈	—	118	1	no	—
200	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
201	#121	—	59	1	no	—
202	#121	C₂×D₅₉	1	1	yes	characteristic
203	C₄×Q₈	—	59	1	no	—
204	C₄×Q₈	—	118	1	no	—
205	C₄×D₄	—	59	1	no	—
206	C₄×D₄	—	59	1	no	—
207	#117	—	59	1	no	—
208	#117	—	59	1	no	—
209	C₂×C₄⋊C₄	—	59	1	no	—
210	C₂×C₄⋊C₄	—	59	1	no	—
211	#107	—	59	1	no	—
212	#107	—	59	1	no	—
213	#107	—	59	1	no	—
214	#107	—	59	1	no	—
215	#107	—	59	1	no	—
216	#107	C₂×D₅₉	1	1	yes	characteristic
217	#106	—	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	C₂×D₅₉	1	1	yes	characteristic
224	#102	—	59	1	no	—
225	#102	—	59	1	no	—
226	#102	C₂×D₅₉	1	1	yes	characteristic
227	C₂²⋊C₈	—	59	1	no	—
228	C₂²⋊C₈	—	59	1	no	—
229	C₄×C₈	—	59	1	no	—
230	#41	—	236	4	no	—
231	C₂×Q₈	—	118	1	no	—
232	C₂×Q₈	—	118	1	no	—
233	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	59	1	no	—
236	C₂×D₄	—	118	1	no	—
237	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
238	C₂²×C₄	—	59	1	no	—
239	C₂²×C₄	—	118	1	no	—
240	C₂²×C₄	—	118	1	no	—
241	C₂²×C₄	—	118	1	no	—
242	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
243	SD₁₆	—	2	2	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	118	2	no	—
246	C₂×C₈	—	59	1	no	—
247	C₂×C₈	—	59	1	no	—
248	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	59	1	no	—
252	C₄⋊C₄	—	59	1	no	—
253	C₄⋊C₄	—	59	1	no	—
254	C₄⋊C₄	—	59	1	no	—
255	C₄⋊C₄	—	59	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	C₂²×D₅₉	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	59	1	no	—
267	C₂²⋊C₄	—	59	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	C₄²	—	59	1	no	—
274	C₄²	—	118	1	no	—
275	C₄²	—	118	1	no	—
276	C₄²	—	118	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	59	1	no	—
279	C₂³	D₄×D₅₉	1	1	yes	characteristic
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	236	2	no	—
283	Q₈	—	236	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	118	2	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	—	2	1	no	—
294	C₂×C₄	—	59	1	no	—
295	C₂×C₄	—	59	1	no	—
296	C₂×C₄	—	59	1	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	—	118	1	no	—
303	C₂×C₄	—	236	2	no	—
304	C₂×C₄	—	236	2	no	—
305	C₂×C₄	—	236	2	no	—
306	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	118	1	no	—
310	C₈	—	118	1	no	—
311	C₈	#11745687	1	2	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	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	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	59	2	no	—
327	C₄	—	118	1	no	—
328	C₄	—	118	1	no	—
329	C₄	—	236	1	no	—
330	C₄	—	236	1	no	—
331	C₄	—	236	1	no	—
332	C₄	—	236	1	no	—
333	C₄	C₂×#11745687	1	1	yes	characteristic
334	C₄	#14273774	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	8	1	no	—
337	C₂	—	236	1	no	—
338	C₂	#16838052	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
340	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×#468	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#468	D₅₉
hypocenter	C₅₉	#3353
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #16838123	C₂	semidirect
3. #16837806	C₂	semidirect
4. #16837736	C₂	semidirect
5. #16837735	C₂	semidirect
6. #16837725	C₂	semidirect
7. #16837723	C₂	semidirect
8. #16837685	C₂	semidirect
9. #16837680	C₂	semidirect
10. #16837645	C₂	semidirect
11. SD₁₆×Dic₅₉	C₂	semidirect
12. #16837478	C₂	semidirect
13. #16837444	C₂	semidirect
14. #16837443	C₂	semidirect
15. C₅₉×#468	C₂	semidirect
138. C₈×C₅₉	#41	semidirect
181. #468	D₅₉	semidirect
186. C₅₉	#3353	semidirect
311. C₈	#11745687	semidirect