Subgroups — #21435396

Order 7552.

320 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838030	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838023	C₂	1	1	yes	maximal, characteristic
3	#16837811	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837800	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837762	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837757	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837744	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837733	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837727	C₂	1	1	yes	maximal, characteristic
10	#16837721	C₂	1	1	yes	maximal, characteristic
11	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837703	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837693	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837688	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	#14273727	C₂²	1	1	yes	characteristic
17	#14273727	C₂²	1	1	yes	characteristic
18	#14273727	C₂²	1	1	yes	characteristic
19	#14273726	—	2	1	no	—
20	#14273722	—	2	1	no	—
21	#14273719	C₂²	1	1	yes	characteristic
22	#14273719	C₂²	1	1	yes	characteristic
23	#14273718	—	2	1	no	—
24	#14273715	C₂²	1	1	yes	characteristic
25	#14273713	—	2	1	no	—
26	#14273709	—	2	1	no	—
27	#14273706	C₂²	1	1	yes	characteristic
28	#14273705	—	2	1	no	—
29	#14273704	—	2	1	no	—
30	#14273704	—	2	1	no	—
31	#14273701	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
33	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
34	#14273686	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
36	#14273661	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273659	C₂²	1	1	yes	characteristic
39	#14273659	C₂²	1	1	yes	characteristic
40	#14273659	C₂²	1	1	yes	characteristic
41	#14273659	C₂²	1	1	yes	characteristic
42	#14273659	C₂²	1	1	yes	characteristic
43	#14273659	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273652	C₂²	1	1	yes	characteristic
46	#14273652	C₂²	1	1	yes	characteristic
47	#14273652	C₂²	1	1	yes	characteristic
48	#14273649	C₂²	1	1	yes	characteristic
49	#14273649	C₂²	1	1	yes	characteristic
50	#14273649	C₂²	1	1	yes	characteristic
51	#14273649	C₂²	1	1	yes	characteristic
52	#117×C₅₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#97×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₉	—	2	1	no	—
61	C₂²×Dic₅₉	—	2	1	no	—
62	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×Dic₁₁₈	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	—	2	1	no	—
67	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
74	C₂²⋊C₄×C₅₉	D₄	1	1	yes	characteristic
75	C₂²⋊C₄×C₅₉	D₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂³	1	1	yes	characteristic
77	C₄²×C₅₉	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₅₉	—	2	1	no	—
79	C₂²⋊Dic₅₉	—	2	1	no	—
80	(C₂×D₅₉)⋊C₄	—	2	1	no	—
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	—	2	1	no	—
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×D₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
112	C₂²⋊D₅₉	—	8	1	no	—
113	C₂×Dic₅₉	—	2	1	no	—
114	C₂×Dic₅₉	—	2	1	no	—
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	—	2	1	no	—
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	—	8	1	no	—
121	D₂₃₆	—	2	1	no	—
122	D₂₃₆	—	2	1	no	—
123	C₄×D₅₉	—	8	1	no	—
124	Dic₁₁₈	—	2	1	no	—
125	Dic₁₁₈	—	2	1	no	—
126	Dic₁₁₈	—	8	1	no	—
127	Dic₁₁₈	—	8	1	no	—
128	Dic₁₁₈	—	8	1	no	—
129	C₈×C₅₉	—	2	1	no	—
130	C₈×C₅₉	—	2	1	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	—	2	1	no	—
133	C₂²×C₅₉	—	2	1	no	—
134	C₂²×C₅₉	—	2	1	no	—
135	C₂²×C₅₉	—	2	1	no	—
136	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₅₉	—	4	1	no	—
138	C₂×D₅₉	—	4	1	no	—
139	C₂×D₅₉	—	4	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
146	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₅₉	—	4	1	no	—
148	Dic₅₉	—	4	1	no	—
149	Dic₅₉	—	4	1	no	—
150	Dic₅₉	—	4	1	no	—
151	Dic₅₉	—	4	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	—	4	1	no	—
154	#3244	—	59	1	no	maximal, Sylow
155	C₂×C₅₉	—	4	1	no	—
156	C₂×C₅₉	#536	1	1	yes	characteristic
157	C₂×C₅₉	#577	1	1	yes	characteristic
158	C₂×C₅₉	#577	1	1	yes	characteristic
159	D₅₉	—	8	1	no	—
160	#544	—	59	1	no	—
161	#542	—	59	1	no	—
162	#501	—	59	1	no	—
163	#498	—	59	1	no	—
164	#489	—	59	1	no	—
165	#488	—	59	1	no	—
166	#485	—	59	1	no	—
167	#483	—	59	1	no	—
168	#482	—	59	1	no	—
169	#481	—	59	1	no	—
170	#479	—	59	1	no	—
171	#478	—	59	1	no	—
172	#476	—	59	1	no	—
173	#475	—	59	1	no	—
174	#433	D₅₉	1	1	yes	characteristic, complemented
175	C₅₉	#3244	1	1	yes	characteristic, complemented, Sylow
176	C₄⋊Q₈	—	59	1	no	—
177	C₄⋊Q₈	—	59	1	no	—
178	C₄⋊Q₈	—	59	1	no	—
179	C₄⋊Q₈	—	118	1	no	—
180	#126	—	118	1	no	—
181	#125	—	59	1	no	—
182	#125	—	59	1	no	—
183	#125	—	118	1	no	—
184	#124	—	59	1	no	—
185	#124	—	118	1	no	—
186	#123	—	118	1	no	—
187	C₂²⋊Q₈	—	59	1	no	—
188	C₂²⋊Q₈	—	118	1	no	—
189	C₂²⋊Q₈	—	118	1	no	—
190	C₂²⋊Q₈	—	118	1	no	—
191	#121	—	59	1	no	—
192	C₄×Q₈	—	59	1	no	—
193	C₄×D₄	—	59	1	no	—
194	#117	—	59	1	no	—
195	#117	C₂×D₅₉	1	1	yes	characteristic
196	C₂×C₄⋊C₄	—	59	1	no	—
197	#107	—	59	1	no	—
198	#107	—	59	1	no	—
199	#106	—	59	1	no	—
200	#106	—	59	1	no	—
201	#106	—	59	1	no	—
202	#106	—	59	1	no	—
203	#106	—	59	1	no	—
204	#106	—	59	1	no	—
205	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
206	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
207	#103	—	59	1	no	—
208	#103	—	59	1	no	—
209	#103	—	59	1	no	—
210	#103	—	59	1	no	—
211	#102	—	59	1	no	—
212	#102	—	59	1	no	—
213	#102	—	59	1	no	—
214	#102	—	59	1	no	—
215	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
216	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
217	#97	C₂×D₅₉	1	1	yes	characteristic
218	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
219	C₂×Q₈	—	59	1	no	—
220	C₂×Q₈	—	118	1	no	—
221	C₂×Q₈	—	118	1	no	—
222	C₂×Q₈	—	118	1	no	—
223	C₂×D₄	—	59	1	no	—
224	C₂×D₄	—	118	1	no	—
225	C₂²×C₄	—	118	1	no	—
226	C₂²×C₄	—	118	1	no	—
227	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
228	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
229	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
230	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
232	C₄⋊C₄	—	59	1	no	—
233	C₄⋊C₄	—	59	1	no	—
234	C₄⋊C₄	—	59	1	no	—
235	C₄⋊C₄	—	59	1	no	—
236	C₄⋊C₄	—	59	1	no	—
237	C₄⋊C₄	—	59	1	no	—
238	C₄⋊C₄	—	118	1	no	—
239	C₄⋊C₄	—	118	1	no	—
240	C₄⋊C₄	—	118	1	no	—
241	C₄⋊C₄	—	118	1	no	—
242	C₄⋊C₄	—	118	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	D₂₃₆	1	1	yes	characteristic
249	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
250	C₂²⋊C₄	—	118	1	no	—
251	C₂²⋊C₄	—	118	1	no	—
252	C₂²⋊C₄	—	118	1	no	—
253	C₂²⋊C₄	—	118	1	no	—
254	C₂²⋊C₄	—	118	1	no	—
255	C₂²⋊C₄	—	118	1	no	—
256	C₂²⋊C₄	D₂₃₆	1	1	yes	characteristic
257	C₂²⋊C₄	D₂₃₆	1	1	yes	characteristic
258	C₄²	—	118	1	no	—
259	C₄²	—	118	1	no	—
260	C₄²	C₂²×D₅₉	1	1	yes	characteristic
261	C₄²	C₂²×D₅₉	1	1	yes	characteristic
262	C₂³	—	118	1	no	—
263	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
264	Q₈	—	118	1	no	—
265	Q₈	—	118	1	no	—
266	Q₈	—	472	1	no	—
267	Q₈	—	472	1	no	—
268	Q₈	—	472	1	no	—
269	D₄	—	118	1	no	—
270	D₄	—	118	1	no	—
271	D₄	—	472	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	118	1	no	—
275	C₂×C₄	—	118	1	no	—
276	C₂×C₄	—	118	1	no	—
277	C₂×C₄	—	118	1	no	—
278	C₂×C₄	—	118	1	no	—
279	C₂×C₄	—	118	1	no	—
280	C₂×C₄	—	118	1	no	—
281	C₂×C₄	—	472	1	no	—
282	C₂×C₄	—	472	1	no	—
283	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
284	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
285	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
286	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
287	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
289	C₈	—	2	1	no	—
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	236	1	no	—
297	C₂²	—	236	1	no	—
298	C₂²	—	236	1	no	—
299	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	236	1	no	—
306	C₄	—	236	1	no	—
307	C₄	—	236	1	no	—
308	C₄	—	236	1	no	—
309	C₄	—	236	1	no	—
310	C₄	—	236	1	no	—
311	C₄	—	236	1	no	—
312	C₄	#14273774	1	1	yes	characteristic
313	C₄	#14273774	1	1	yes	characteristic
314	C₂	—	4	1	no	—
315	C₂	—	472	1	no	—
316	C₂	#16837978	1	1	yes	characteristic
317	C₂	#16838214	1	1	yes	characteristic
318	C₂	#16838214	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#433	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#433	D₅₉
hypocenter	C₅₉	#3244
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
1. #16838030	C₂	semidirect
3. #16837811	C₂	semidirect
4. #16837800	C₂	semidirect
5. #16837762	C₂	semidirect
6. #16837757	C₂	semidirect
7. #16837744	C₂	semidirect
8. #16837733	C₂	semidirect
11. #16837710	C₂	semidirect
12. #16837703	C₂	semidirect
13. #16837693	C₂	semidirect
14. #16837688	C₂	semidirect
15. C₅₉×#433	C₂	semidirect
174. #433	D₅₉	semidirect
175. C₅₉	#3244	semidirect