Subgroups — #21435307

Order 7552.

322 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837955	C₂	1	1	yes	maximal, characteristic
2	#16837954	C₂	1	1	yes	maximal, characteristic
3	#16837810	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837809	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837769	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837769	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837744	C₂	1	1	yes	maximal, characteristic
8	#16837743	C₂	1	1	yes	maximal, characteristic
9	#16837740	C₂	1	1	yes	maximal, characteristic
10	#16837696	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837686	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837685	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837456	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#485	C₂	1	1	yes	maximal, characteristic
16	#14273726	C₂²	1	1	yes	characteristic
17	#14273726	C₂²	1	1	yes	characteristic
18	#14273726	C₂²	1	1	yes	characteristic
19	#14273725	C₂²	1	1	yes	characteristic
20	#14273718	C₂²	1	1	yes	characteristic
21	#14273718	C₂²	1	1	yes	characteristic
22	#14273717	—	2	1	no	—
23	#14273706	C₂²	1	1	yes	characteristic
24	#14273704	—	2	1	no	—
25	#14273704	—	2	1	no	—
26	#14273704	—	2	1	no	—
27	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₁₈	—	2	1	no	—
29	C₄×Dic₁₁₈	—	2	1	no	—
30	#14273692	—	2	1	no	—
31	#14273692	C₂²	1	1	yes	characteristic
32	#14273686	C₂²	1	1	yes	characteristic
33	#14273683	—	2	1	no	—
34	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
35	#14273661	C₂²	1	1	yes	characteristic
36	#14273660	C₂²	1	1	yes	characteristic
37	#14273660	C₂²	1	1	yes	characteristic
38	#14273659	C₂²	1	1	yes	characteristic
39	#14273658	C₂²	1	1	yes	characteristic
40	#14273658	C₂²	1	1	yes	characteristic
41	Dic₅₉⋊C₈	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	#14273637	C₂²	1	1	yes	characteristic
51	#14273637	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	#117×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
62	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
63	C₂×Dic₁₁₈	—	2	1	no	—
64	C₂×Dic₁₁₈	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₅₉	—	2	1	no	—
70	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
73	C₂²⋊C₄×C₅₉	—	2	1	no	—
74	C₂²⋊C₄×C₅₉	—	2	1	no	—
75	C₄²×C₅₉	—	2	1	no	—
76	C₂²⋊Dic₅₉	—	2	1	no	—
77	C₂²⋊Dic₅₉	—	2	1	no	—
78	C₄⋊Dic₅₉	—	2	1	no	—
79	C₄⋊Dic₅₉	—	2	1	no	—
80	C₄⋊Dic₅₉	—	2	1	no	—
81	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
84	Dic₅₉⋊C₄	—	2	1	no	—
85	Dic₅₉⋊C₄	—	2	1	no	—
86	Dic₅₉⋊C₄	—	2	1	no	—
87	Dic₅₉⋊C₄	—	2	1	no	—
88	Dic₅₉⋊C₄	—	2	1	no	—
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	C₄×Dic₅₉	—	2	1	no	—
94	C₄×Dic₅₉	—	2	1	no	—
95	C₄×Dic₅₉	—	2	1	no	—
96	C₄×Dic₅₉	D₄	1	1	yes	characteristic
97	C₄×Dic₅₉	D₄	1	1	yes	characteristic
98	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
99	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
100	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
103	Q₈×C₅₉	—	2	1	no	—
104	Q₈×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₅₉	—	2	2	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	—	4	1	no	—
119	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
123	Dic₁₁₈	—	2	1	no	—
124	Dic₁₁₈	—	2	1	no	—
125	Dic₁₁₈	—	4	2	no	—
126	Dic₁₁₈	—	4	2	no	—
127	Dic₁₁₈	—	4	2	no	—
128	C₈×C₅₉	—	2	1	no	—
129	C₈×C₅₉	—	2	1	no	—
130	C₅₉⋊C₈	—	2	1	no	—
131	C₅₉⋊C₈	—	2	1	no	—
132	C₂²×C₅₉	—	2	1	no	—
133	C₂²×C₅₉	—	2	1	no	—
134	C₂²×C₅₉	—	2	1	no	—
135	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₅₉	—	2	1	no	—
137	C₄×C₅₉	—	4	1	no	—
138	C₄×C₅₉	—	4	1	no	—
139	C₄×C₅₉	—	4	1	no	—
140	C₄×C₅₉	—	4	1	no	—
141	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
142	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
143	Dic₅₉	—	2	1	no	—
144	Dic₅₉	—	2	1	no	—
145	Dic₅₉	—	2	1	no	—
146	Dic₅₉	—	4	1	no	—
147	Dic₅₉	—	4	1	no	—
148	Dic₅₉	—	4	1	no	—
149	Dic₅₉	—	4	1	no	—
150	Dic₅₉	#137	1	2	yes	—
151	#3237	—	59	1	no	maximal, Sylow
152	C₂×C₅₉	—	4	1	no	—
153	C₂×C₅₉	#536	1	1	yes	characteristic
154	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
155	C₂×C₅₉	#575	1	1	yes	characteristic
156	#533	—	59	1	no	—
157	#533	—	59	1	no	—
158	#501	—	59	1	no	—
159	#501	—	59	1	no	—
160	#491	—	59	1	no	—
161	#491	—	59	1	no	—
162	#485	—	59	1	no	—
163	#485	—	59	1	no	—
164	#485	—	59	1	no	—
165	#485	D₅₉	1	1	yes	characteristic
166	#477	—	59	1	no	—
167	#477	—	59	1	no	—
168	#475	—	59	1	no	—
169	#475	—	59	1	no	—
170	#432	—	59	1	no	—
171	C₅₉	#3237	1	1	yes	characteristic, complemented, Sylow
172	C₄⋊Q₈	—	59	1	no	—
173	C₄⋊Q₈	—	59	1	no	—
174	C₄⋊Q₈	—	59	1	no	—
175	C₄⋊Q₈	—	59	1	no	—
176	#125	—	59	1	no	—
177	#125	—	59	1	no	—
178	#125	—	118	1	no	—
179	C₂²⋊Q₈	—	59	1	no	—
180	C₂²⋊Q₈	—	118	1	no	—
181	C₂²⋊Q₈	—	118	1	no	—
182	C₂²⋊Q₈	—	118	1	no	—
183	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
184	C₄×Q₈	—	59	1	no	—
185	C₄×Q₈	—	59	1	no	—
186	C₄×Q₈	—	118	1	no	—
187	C₄×Q₈	—	118	1	no	—
188	C₄×Q₈	—	118	1	no	—
189	#117	—	59	1	no	—
190	#117	—	118	1	no	—
191	#117	C₂×D₅₉	1	1	yes	characteristic
192	C₂×C₄²	—	59	1	no	—
193	#107	—	59	1	no	—
194	#107	—	59	1	no	—
195	#107	—	59	1	no	—
196	#107	C₂×D₅₉	1	1	yes	characteristic
197	#106	—	59	1	no	—
198	#106	—	59	1	no	—
199	#106	—	59	1	no	—
200	#106	C₂×D₅₉	1	1	yes	characteristic
201	C₄⋊C₈	—	59	1	no	—
202	C₄⋊C₈	—	59	1	no	—
203	#103	—	59	1	no	—
204	#103	—	59	1	no	—
205	#103	—	59	1	no	—
206	#103	—	59	1	no	—
207	#103	—	59	1	no	—
208	#103	—	59	1	no	—
209	#103	C₂×D₅₉	1	1	yes	characteristic
210	#103	C₂×D₅₉	1	1	yes	characteristic
211	C₂²⋊C₈	—	59	1	no	—
212	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
213	#97	—	59	1	no	—
214	#97	—	59	1	no	—
215	C₂×Q₈	—	59	1	no	—
216	C₂×Q₈	—	118	1	no	—
217	C₂×Q₈	—	118	1	no	—
218	C₂×Q₈	—	118	1	no	—
219	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
220	C₂²×C₄	—	59	1	no	—
221	C₂²×C₄	—	59	1	no	—
222	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
223	C₂×C₈	—	59	1	no	—
224	C₂×C₈	—	59	1	no	—
225	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
226	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
227	C₄⋊C₄	—	2	1	no	—
228	C₄⋊C₄	—	59	1	no	—
229	C₄⋊C₄	—	59	1	no	—
230	C₄⋊C₄	—	59	1	no	—
231	C₄⋊C₄	—	118	1	no	—
232	C₄⋊C₄	—	118	1	no	—
233	C₄⋊C₄	—	118	1	no	—
234	C₄⋊C₄	—	118	1	no	—
235	C₄⋊C₄	—	118	1	no	—
236	C₄⋊C₄	—	118	1	no	—
237	C₄⋊C₄	—	118	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	C₂²×D₅₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
246	C₂²⋊C₄	—	2	1	no	—
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	118	1	no	—
249	C₂²⋊C₄	—	118	1	no	—
250	C₄²	—	2	1	no	—
251	C₄²	—	59	1	no	—
252	C₄²	—	59	1	no	—
253	C₄²	—	59	1	no	—
254	C₄²	—	59	1	no	—
255	C₄²	—	118	1	no	—
256	C₄²	—	118	1	no	—
257	C₄²	—	118	1	no	—
258	C₂³	D₄×D₅₉	1	1	yes	characteristic
259	Q₈	—	2	1	no	—
260	Q₈	—	2	1	no	—
261	Q₈	—	118	1	no	—
262	Q₈	—	118	1	no	—
263	Q₈	—	236	2	no	—
264	Q₈	—	236	2	no	—
265	Q₈	—	236	2	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	59	1	no	—
273	C₂×C₄	—	59	1	no	—
274	C₂×C₄	—	59	1	no	—
275	C₂×C₄	—	59	1	no	—
276	C₂×C₄	—	118	2	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	—	236	1	no	—
282	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	118	1	no	—
287	C₈	—	118	1	no	—
288	C₂²	—	2	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	59	2	no	—
298	C₄	—	118	1	no	—
299	C₄	—	118	1	no	—
300	C₄	—	118	1	no	—
301	C₄	—	236	1	no	—
302	C₄	—	236	1	no	—
303	C₄	—	236	1	no	—
304	C₄	—	236	1	no	—
305	C₄	#14273774	1	1	yes	characteristic
306	C₄	#14273774	1	1	yes	characteristic
307	C₂	—	4	1	no	—
308	C₂	#16837974	1	1	yes	characteristic
309	C₂	#137×D₅₉	1	1	yes	characteristic
310	C₂	#16838199	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#485	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#485	D₅₉
hypocenter	C₅₉	#3237
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #485

Complements

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

N	H ≅ G/N	type
3. #16837810	C₂	semidirect
4. #16837809	C₂	semidirect
5. #16837769	C₂	semidirect
6. #16837769	C₂	semidirect
10. #16837696	C₂	semidirect
11. #16837695	C₂	semidirect
12. #16837686	C₂	semidirect
13. #16837685	C₂	semidirect
171. C₅₉	#3237	semidirect