Subgroups — #21436782

Order 7552.

336 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838128	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838124	C₂	1	1	yes	maximal, characteristic
3	#16837758	C₂	1	1	yes	maximal, characteristic
4	#16837757	C₂	1	1	yes	maximal, characteristic
5	#16837751	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837709	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837708	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837681	C₂	1	1	yes	maximal, characteristic
9	#16837626	C₂	1	1	yes	maximal, characteristic
10	#16837624	C₂	1	1	yes	maximal, characteristic
11	#16837610	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837494	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837487	C₂	1	1	yes	maximal, characteristic
15	C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745672	C₂²	1	1	yes	characteristic
17	C₂×#11745671	C₂²	1	1	yes	characteristic
18	#14273726	—	2	1	no	—
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273725	C₂²	1	1	yes	characteristic
21	#14273722	—	2	1	no	—
22	#14273721	—	2	1	no	—
23	#14273719	—	2	1	no	—
24	#14273719	C₂²	1	1	yes	characteristic
25	#14273718	—	2	1	no	—
26	#14273718	C₂²	1	1	yes	characteristic
27	#14273716	C₂²	1	1	yes	characteristic
28	#14273715	C₂²	1	1	yes	characteristic
29	#14273705	—	2	1	no	—
30	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
31	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₁₈	—	2	1	no	—
33	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
34	#14273692	C₂²	1	1	yes	characteristic
35	#14273689	C₂²	1	1	yes	characteristic
36	#14273684	—	2	1	no	—
37	#14273660	C₂²	1	1	yes	characteristic
38	#14273658	C₂²	1	1	yes	characteristic
39	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
42	#14273652	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273650	C₂²	1	1	yes	characteristic
47	#14273649	C₂²	1	1	yes	characteristic
48	#14273649	C₂²	1	1	yes	characteristic
49	#14273647	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×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×D₂₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₅₉	—	2	1	no	—
63	C₂×Dic₁₁₈	—	2	1	no	—
64	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
65	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
66	Q₁₆×C₅₉	—	2	2	no	—
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	—	2	1	no	—
71	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
73	C₄²×C₅₉	—	2	1	no	—
74	C₄²×C₅₉	—	2	1	no	—
75	C₄²×C₅₉	C₂³	1	1	yes	characteristic
76	#11745672	—	2	2	no	—
77	#11745671	—	2	2	no	—
78	(C₂×D₅₉)⋊C₄	—	2	1	no	—
79	(C₂×D₅₉)⋊C₄	—	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	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	D₄	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	D₄	1	1	yes	characteristic
97	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
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	C₂²×D₅₉	—	2	1	no	—
106	Q₈×C₅₉	—	2	1	no	—
107	Q₈×C₅₉	—	2	1	no	—
108	Q₈×C₅₉	—	2	1	no	—
109	Q₈×C₅₉	#41	1	2	yes	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
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	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
124	D₂₃₆	—	2	1	no	—
125	D₂₃₆	—	2	1	no	—
126	C₄×D₅₉	—	4	2	no	—
127	Dic₁₁₈	—	2	2	no	—
128	Dic₁₁₈	—	2	1	no	—
129	Dic₁₁₈	—	2	1	no	—
130	Dic₁₁₈	—	4	2	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	—	2	1	no	—
133	C₅₉⋊C₈	—	2	1	no	—
134	C₅₉⋊C₈	—	2	1	no	—
135	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
136	C₂×D₅₉	—	4	1	no	—
137	C₂×D₅₉	—	4	1	no	—
138	C₂×D₅₉	—	4	1	no	—
139	C₄×C₅₉	—	2	2	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	4	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
146	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₅₉	—	2	1	no	—
148	Dic₅₉	—	2	1	no	—
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	4	1	no	—
151	Dic₅₉	—	4	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	#135	1	2	yes	—
154	#3364	—	59	1	no	maximal, Sylow
155	C₂×C₅₉	#548	1	1	yes	characteristic
156	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
157	C₂×C₅₉	#578	1	1	yes	characteristic
158	D₅₉	—	8	1	no	—
159	#557	—	59	1	no	—
160	#556	—	59	1	no	—
161	#488	—	59	1	no	—
162	#488	—	59	1	no	—
163	#487	—	59	1	no	—
164	#479	—	59	1	no	—
165	#479	—	59	1	no	—
166	#474	—	59	1	no	—
167	#464	—	59	1	no	—
168	#464	—	59	1	no	—
169	#462	—	59	1	no	—
170	C₈×Q₈	—	59	1	no	—
171	C₄×Q₁₆	—	59	1	no	—
172	C₄×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
173	C₄×SD₁₆	—	59	1	no	—
174	C₅₉	#3364	1	1	yes	characteristic, complemented, Sylow
175	C₂×Q₁₆	—	59	1	no	—
176	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
177	C₂×SD₁₆	—	59	1	no	—
178	C₄⋊Q₈	—	59	1	no	—
179	C₄⋊Q₈	—	59	1	no	—
180	C₄⋊Q₈	—	118	1	no	—
181	#126	—	118	1	no	—
182	#126	—	118	1	no	—
183	#125	—	59	1	no	—
184	#125	—	59	1	no	—
185	#125	—	118	1	no	—
186	#125	—	118	1	no	—
187	#124	—	59	1	no	—
188	#124	—	59	1	no	—
189	C₂²⋊Q₈	—	118	1	no	—
190	C₄×Q₈	—	59	1	no	—
191	C₄×Q₈	—	59	1	no	—
192	C₄×Q₈	—	59	1	no	—
193	C₄×Q₈	—	59	1	no	—
194	C₄×Q₈	—	118	1	no	—
195	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
196	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
197	C₄×D₄	—	59	1	no	—
198	#117	—	118	1	no	—
199	#107	—	59	1	no	—
200	#107	C₂×D₅₉	1	1	yes	characteristic
201	#106	—	59	1	no	—
202	C₄⋊C₈	—	59	1	no	—
203	C₄⋊C₈	—	59	1	no	—
204	C₄⋊C₈	—	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	—	59	1	no	—
210	#103	C₂×D₅₉	1	1	yes	characteristic
211	#103	C₂×D₅₉	1	1	yes	characteristic
212	#102	—	59	1	no	—
213	#102	—	59	1	no	—
214	#102	—	59	1	no	—
215	C₄×C₈	—	59	1	no	—
216	C₄×C₈	—	59	1	no	—
217	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
218	C₂×Q₈	—	59	1	no	—
219	C₂×Q₈	—	59	1	no	—
220	C₂×Q₈	—	118	1	no	—
221	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
222	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
223	C₂×D₄	—	59	1	no	—
224	C₂²×C₄	—	118	1	no	—
225	Q₁₆	—	2	2	no	—
226	Q₁₆	—	118	2	no	—
227	SD₁₆	—	118	2	no	—
228	C₂×C₈	—	59	1	no	—
229	C₂×C₈	—	59	1	no	—
230	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	2	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	—	59	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	—	118	1	no	—
249	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
250	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
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	C₄²	—	2	1	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	59	1	no	—
259	C₄²	—	59	1	no	—
260	C₄²	—	118	1	no	—
261	C₄²	—	118	1	no	—
262	C₄²	—	118	1	no	—
263	C₄²	C₂²×D₅₉	1	1	yes	characteristic
264	C₂³	—	118	1	no	—
265	Q₈	—	2	1	no	—
266	Q₈	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	118	2	no	—
269	Q₈	—	118	1	no	—
270	Q₈	—	118	1	no	—
271	Q₈	—	236	2	no	—
272	Q₈	#11745685	1	2	yes	—
273	D₄	—	118	1	no	—
274	D₄	—	118	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	59	1	no	—
280	C₂×C₄	—	59	1	no	—
281	C₂×C₄	—	59	1	no	—
282	C₂×C₄	—	59	1	no	—
283	C₂×C₄	—	118	1	no	—
284	C₂×C₄	—	118	1	no	—
285	C₂×C₄	—	118	1	no	—
286	C₂×C₄	—	236	2	no	—
287	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
288	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	2	1	no	—
292	C₈	—	118	1	no	—
293	C₈	—	118	1	no	—
294	C₂²	—	236	1	no	—
295	C₂²	—	236	1	no	—
296	C₂²	—	236	1	no	—
297	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	2	no	—
301	C₄	—	4	1	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	59	2	no	—
305	C₄	—	118	1	no	—
306	C₄	—	118	1	no	—
307	C₄	—	118	1	no	—
308	C₄	—	236	1	no	—
309	C₄	—	236	1	no	—
310	C₄	—	236	1	no	—
311	C₄	C₂×#11745685	1	1	yes	characteristic
312	C₄	#14273774	1	1	yes	characteristic
313	C₂	—	472	1	no	—
314	C₂	#16838061	1	1	yes	characteristic
315	C₂	#135×D₅₉	1	1	yes	characteristic
316	C₂	#16838215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₄×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₄×Q₁₆	D₅₉
hypocenter	C₅₉	#3364
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
1. #16838128	C₂	semidirect
5. #16837751	C₂	semidirect
6. #16837709	C₂	semidirect
7. #16837708	C₂	semidirect
11. #16837610	C₂	semidirect
12. C₈×Dic₁₁₈	C₂	semidirect
13. #16837494	C₂	semidirect
15. C₄×Q₁₆×C₅₉	C₂	semidirect
172. C₄×Q₁₆	D₅₉	semidirect
174. C₅₉	#3364	semidirect