Subgroups — #21435548

Order 7552.

322 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838013	C₂	1	1	yes	maximal, characteristic
2	#16838010	C₂	1	1	yes	maximal, characteristic
3	#16837711	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837709	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837707	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837659	C₂	1	1	yes	maximal, characteristic
8	#16837657	C₂	1	1	yes	maximal, characteristic
9	#16837642	C₂	1	1	yes	maximal, characteristic
10	#16837640	C₂	1	1	yes	maximal, characteristic
11	#16837610	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837610	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837609	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837427	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#462	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745672	C₂²	1	1	yes	characteristic
17	C₂×#11745672	C₂²	1	1	yes	characteristic
18	C₂×#11745659	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
20	#14273725	C₂²	1	1	yes	characteristic
21	#14273722	—	2	1	no	—
22	#14273722	—	2	1	no	—
23	#14273722	—	2	1	no	—
24	#14273721	—	2	1	no	—
25	#14273719	—	2	1	no	—
26	#14273719	C₂²	1	1	yes	characteristic
27	#14273718	C₂²	1	1	yes	characteristic
28	#14273717	—	2	1	no	—
29	#14273717	—	2	1	no	—
30	#14273717	—	2	1	no	—
31	#14273707	C₂²	1	1	yes	characteristic
32	#14273703	C₂²	1	1	yes	characteristic
33	#14273703	C₂²	1	1	yes	characteristic
34	#14273703	C₂²	1	1	yes	characteristic
35	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
36	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
38	#14273684	C₂²	1	1	yes	characteristic
39	#14273661	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	#14273659	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
46	#14273652	C₂²	1	1	yes	characteristic
47	#14273652	C₂²	1	1	yes	characteristic
48	#14273651	C₂²	1	1	yes	characteristic
49	#14273651	C₂²	1	1	yes	characteristic
50	#14273651	C₂²	1	1	yes	characteristic
51	#14273651	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×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	#103×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×Q₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
64	Q₁₆×C₅₉	—	4	1	no	—
65	Q₁₆×C₅₉	—	4	1	no	—
66	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₄⋊C₄×C₅₉	—	2	1	no	—
69	C₄⋊C₄×C₅₉	—	2	1	no	—
70	C₄⋊C₄×C₅₉	—	2	1	no	—
71	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
72	C₄²×C₅₉	—	2	1	no	—
73	C₄²×C₅₉	C₂³	1	1	yes	characteristic
74	#11745672	—	4	1	no	—
75	#11745672	—	4	1	no	—
76	(C₂×D₅₉)⋊C₄	—	2	1	no	—
77	(C₂×D₅₉)⋊C₄	—	2	1	no	—
78	(C₂×D₅₉)⋊C₄	—	2	1	no	—
79	(C₂×D₅₉)⋊C₄	—	2	1	no	—
80	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
81	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
82	C₄⋊Dic₅₉	—	2	1	no	—
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
87	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
89	Dic₅₉⋊C₄	—	2	1	no	—
90	Dic₅₉⋊C₄	—	2	1	no	—
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	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
99	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	#11745659	—	4	1	no	—
107	C₈×D₅₉	—	4	1	no	—
108	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₅₉	—	2	1	no	—
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	Q₈×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	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
129	C₄×D₅₉	—	2	1	no	—
130	C₄×D₅₉	—	2	1	no	—
131	Dic₁₁₈	—	2	1	no	—
132	Dic₁₁₈	—	2	1	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	—	2	1	no	—
135	C₅₉⋊C₈	—	2	1	no	—
136	C₅₉⋊C₈	—	2	1	no	—
137	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₅₉	—	2	1	no	—
139	C₂×D₅₉	—	2	1	no	—
140	C₂×D₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	—	4	1	no	—
146	C₄×C₅₉	—	4	1	no	—
147	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
148	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	2	1	no	—
151	Dic₅₉	—	2	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	—	4	1	no	—
155	Dic₅₉	—	4	1	no	—
156	#3261	—	59	1	no	maximal, Sylow
157	C₂×C₅₉	#537	1	1	yes	characteristic
158	C₂×C₅₉	#578	1	1	yes	characteristic
159	C₂×C₅₉	#578	1	1	yes	characteristic
160	D₅₉	—	4	1	no	—
161	#541	—	59	1	no	—
162	#541	—	59	1	no	—
163	#479	—	59	1	no	—
164	#479	—	59	1	no	—
165	#479	—	59	1	no	—
166	#479	—	59	1	no	—
167	#470	—	59	1	no	—
168	#470	—	59	1	no	—
169	#467	—	59	1	no	—
170	#467	—	59	1	no	—
171	#462	—	59	1	no	—
172	#462	—	59	1	no	—
173	#462	—	59	1	no	—
174	#462	D₅₉	1	1	yes	characteristic, complemented
175	#424	—	59	1	no	—
176	C₅₉	#3261	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	—	59	1	no	—
178	C₂×Q₁₆	—	59	1	no	—
179	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
180	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
181	C₂×M₁₆	—	59	1	no	—
182	C₂²×C₈	—	59	1	no	—
183	C₄⋊Q₈	—	59	1	no	—
184	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
185	#126	—	118	1	no	—
186	#126	—	118	1	no	—
187	#126	—	118	1	no	—
188	#126	—	118	1	no	—
189	#125	—	59	1	no	—
190	#125	—	59	1	no	—
191	#125	—	118	1	no	—
192	#125	—	118	1	no	—
193	#125	—	118	1	no	—
194	#125	—	118	1	no	—
195	C₂²⋊Q₈	—	59	1	no	—
196	C₂²⋊Q₈	—	59	1	no	—
197	C₂²⋊Q₈	—	59	1	no	—
198	C₂²⋊Q₈	—	59	1	no	—
199	C₄×Q₈	—	59	1	no	—
200	C₄×Q₈	—	59	1	no	—
201	C₄×Q₈	—	59	1	no	—
202	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
203	#117	—	59	1	no	—
204	#107	—	59	1	no	—
205	#107	—	59	1	no	—
206	#106	—	59	1	no	—
207	#106	—	59	1	no	—
208	C₄⋊C₈	—	59	1	no	—
209	C₄⋊C₈	—	59	1	no	—
210	C₄⋊C₈	—	59	1	no	—
211	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
212	#103	—	59	1	no	—
213	#103	—	59	1	no	—
214	#103	—	59	1	no	—
215	#103	—	59	1	no	—
216	#103	—	59	1	no	—
217	#103	—	59	1	no	—
218	#103	C₂×D₅₉	1	1	yes	characteristic
219	#103	C₂×D₅₉	1	1	yes	characteristic
220	C₂×Q₈	—	59	1	no	—
221	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
222	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
223	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
224	C₂²×C₄	—	59	1	no	—
225	Q₁₆	—	4	1	no	—
226	Q₁₆	—	4	1	no	—
227	Q₁₆	—	236	1	no	—
228	Q₁₆	—	236	1	no	—
229	M₁₆	—	236	1	no	—
230	C₂×C₈	—	59	1	no	—
231	C₂×C₈	—	59	1	no	—
232	C₂×C₈	—	236	1	no	—
233	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	59	1	no	—
239	C₄⋊C₄	—	59	1	no	—
240	C₄⋊C₄	—	59	1	no	—
241	C₄⋊C₄	—	59	1	no	—
242	C₄⋊C₄	—	59	1	no	—
243	C₄⋊C₄	—	118	1	no	—
244	C₄⋊C₄	—	118	1	no	—
245	C₄⋊C₄	—	118	1	no	—
246	C₄⋊C₄	—	118	1	no	—
247	C₄⋊C₄	—	118	1	no	—
248	C₄⋊C₄	—	118	1	no	—
249	C₄⋊C₄	—	118	1	no	—
250	C₄⋊C₄	—	118	1	no	—
251	C₄⋊C₄	—	118	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	C₂²×D₅₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	59	1	no	—
258	C₂²⋊C₄	—	59	1	no	—
259	C₂²⋊C₄	—	118	1	no	—
260	C₂²⋊C₄	—	118	1	no	—
261	C₂²⋊C₄	—	118	1	no	—
262	C₂²⋊C₄	—	118	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	59	1	no	—
265	C₄²	—	118	1	no	—
266	C₄²	—	118	1	no	—
267	C₄²	—	118	1	no	—
268	C₄²	C₂²×D₅₉	1	1	yes	characteristic
269	C₂³	—	59	1	no	—
270	Q₈	—	2	1	no	—
271	Q₈	—	2	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	118	1	no	—
277	Q₈	—	118	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	59	1	no	—
283	C₂×C₄	—	59	1	no	—
284	C₂×C₄	—	59	1	no	—
285	C₂×C₄	—	118	1	no	—
286	C₂×C₄	—	118	1	no	—
287	C₂×C₄	—	118	1	no	—
288	C₂×C₄	—	118	1	no	—
289	C₂×C₄	—	118	1	no	—
290	C₂×C₄	—	118	1	no	—
291	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
292	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	118	1	no	—
297	C₈	—	118	1	no	—
298	C₂²	—	118	1	no	—
299	C₂²	—	118	1	no	—
300	C₂²	—	118	1	no	—
301	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	118	1	no	—
309	C₄	—	118	1	no	—
310	C₄	—	118	1	no	—
311	C₄	—	236	1	no	—
312	C₄	—	236	1	no	—
313	C₄	—	236	1	no	—
314	C₄	—	236	1	no	—
315	C₄	#14273772	1	1	yes	characteristic
316	C₄	#14273774	1	1	yes	characteristic
317	C₂	—	236	1	no	—
318	C₂	#16837981	1	1	yes	characteristic
319	C₂	#16838215	1	1	yes	characteristic
320	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×#462	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#462	D₅₉
hypocenter	C₅₉	#3261
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
3. #16837711	C₂	semidirect
4. #16837710	C₂	semidirect
5. #16837709	C₂	semidirect
6. #16837707	C₂	semidirect
11. #16837610	C₂	semidirect
12. #16837610	C₂	semidirect
13. #16837609	C₂	semidirect
15. C₅₉×#462	C₂	semidirect
174. #462	D₅₉	semidirect
176. C₅₉	#3261	semidirect