Subgroups — #21436446

Order 7552.

388 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838110	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838053	C₂	1	1	yes	maximal, characteristic
3	#16837799	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837731	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837722	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837721	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837691	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837685	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837634	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837633	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837487	C₂	1	1	yes	maximal, characteristic, complemented
12	(C₂²⋊D₅₉)⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14273658	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#483	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745685	—	2	1	no	—
17	C₂×#11745671	C₂²	1	1	yes	characteristic
18	C₂²×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273717	—	2	1	no	—
23	#14273712	—	2	1	no	—
24	#14273710	—	2	1	no	—
25	#14273707	—	2	1	no	—
26	#14273705	—	2	1	no	—
27	#14273704	—	2	1	no	—
28	#14273703	C₂²	1	1	yes	characteristic
29	#14273702	C₂²	1	1	yes	characteristic
30	#14273701	C₂²	1	1	yes	characteristic
31	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
32	#14273689	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
34	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
35	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
38	#14273659	C₂²	1	1	yes	characteristic
39	#14273659	C₂²	1	1	yes	characteristic
40	#14273658	C₂²	1	1	yes	characteristic
41	#14273658	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	#14273658	C₂²	1	1	yes	characteristic
44	#14273658	C₂²	1	1	yes	characteristic
45	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
46	#14273651	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	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
54	#106×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	—	2	1	no	—
61	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
64	C₂²×Dic₅₉	—	2	1	no	—
65	#11745685	—	4	4	no	—
66	C₂×D₂₃₆	—	2	1	no	—
67	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
72	C₂×Dic₁₁₈	—	2	1	no	—
73	C₂×Dic₁₁₈	—	2	1	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	—	2	1	no	—
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
83	#11745671	—	2	2	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	(C₂×D₅₉)⋊C₄	—	2	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
89	C₄⋊Dic₅₉	—	2	1	no	—
90	C₄⋊Dic₅₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	—	2	2	no	—
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₈×D₅₉	—	2	2	no	—
106	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₅₉	—	2	1	no	—
108	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₅₉	—	2	1	no	—
110	Q₈×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	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	4	2	no	—
118	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₅₉	—	4	2	no	—
121	C₂²⋊D₅₉	SD₁₆	1	4	yes	complemented
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	8	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	D₂₃₆	—	2	1	no	—
130	D₂₃₆	—	2	1	no	—
131	D₂₃₆	—	4	2	no	—
132	C₄×D₅₉	—	2	1	no	—
133	C₄×D₅₉	—	2	1	no	—
134	C₄×D₅₉	—	4	2	no	—
135	C₄×D₅₉	—	4	2	no	—
136	C₄×D₅₉	—	8	1	no	—
137	Dic₁₁₈	—	4	2	no	—
138	Dic₁₁₈	—	8	1	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₅₉⋊C₈	—	2	1	no	—
142	C₅₉⋊C₈	#41	1	2	yes	complemented
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	—	2	1	no	—
145	C₂²×C₅₉	C₂×SD₁₆	1	2	yes	—
146	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₅₉	—	2	1	no	—
148	C₂×D₅₉	—	2	1	no	—
149	C₂×D₅₉	—	4	1	no	—
150	C₂×D₅₉	—	4	1	no	—
151	C₂×D₅₉	—	4	1	no	—
152	C₂×D₅₉	C₂×SD₁₆	1	2	yes	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	—	4	1	no	—
156	C₄×C₅₉	—	4	1	no	—
157	C₄×C₅₉	—	4	1	no	—
158	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
159	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
160	Dic₅₉	—	2	1	no	—
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	C₂×SD₁₆	1	2	yes	—
166	#3335	—	59	1	no	maximal, Sylow
167	C₂×C₅₉	—	2	2	no	—
168	C₂×C₅₉	#547	1	1	yes	characteristic
169	C₂×C₅₉	C₂²×SD₁₆	1	1	yes	characteristic
170	C₂×C₅₉	#577	1	1	yes	characteristic
171	D₅₉	—	2	2	no	—
172	D₅₉	—	8	1	no	—
173	#554	—	59	1	no	—
174	#547	—	59	1	no	—
175	#498	—	59	1	no	—
176	#483	—	59	1	no	—
177	#483	D₅₉	1	1	yes	characteristic, complemented
178	#481	—	59	1	no	—
179	#481	—	59	1	no	—
180	#476	—	59	1	no	—
181	#475	—	59	1	no	—
182	#465	—	59	1	no	—
183	#465	—	59	1	no	—
184	C₄×SD₁₆	—	59	1	no	—
185	C₈×D₄	—	59	1	no	—
186	C₂×#106	—	59	1	no	—
187	C₂×#106	—	59	1	no	—
188	C₅₉	#3335	1	1	yes	characteristic, complemented, Sylow
189	C₂×#41	—	118	1	no	—
190	C₂×SD₁₆	—	59	1	no	—
191	C₂²×C₈	—	59	1	no	—
192	C₂²×C₈	—	59	1	no	—
193	C₄⋊Q₈	—	59	1	no	—
194	C₄⋊Q₈	—	59	1	no	—
195	#125	—	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	—	118	1	no	—
201	C₂²⋊Q₈	—	118	1	no	—
202	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
203	#121	—	59	1	no	—
204	#121	—	59	1	no	—
205	C₄×Q₈	—	59	1	no	—
206	C₄×D₄	—	59	1	no	—
207	C₄×D₄	—	59	1	no	—
208	C₄×D₄	—	118	1	no	—
209	C₂×C₄⋊C₄	—	59	1	no	—
210	C₂×C₄⋊C₄	—	59	1	no	—
211	C₂×C₄⋊C₄	—	59	1	no	—
212	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
213	#106	—	59	1	no	—
214	#106	—	59	1	no	—
215	#106	—	59	1	no	—
216	#106	—	59	1	no	—
217	#106	—	59	1	no	—
218	#106	—	59	1	no	—
219	#106	—	59	1	no	—
220	#106	C₂×D₅₉	1	1	yes	characteristic
221	#106	C₂×D₅₉	1	1	yes	characteristic
222	C₄⋊C₈	—	59	1	no	—
223	#103	—	59	1	no	—
224	#103	C₂×D₅₉	1	1	yes	characteristic
225	#103	C₂×D₅₉	1	1	yes	characteristic
226	#102	—	59	1	no	—
227	#102	—	59	1	no	—
228	#102	—	59	1	no	—
229	C₂²⋊C₈	—	59	1	no	—
230	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
231	C₄×C₈	—	59	1	no	—
232	#41	—	236	4	no	—
233	C₂×Q₈	—	118	1	no	—
234	C₂×Q₈	—	118	1	no	—
235	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₂×D₄	—	59	1	no	—
237	C₂×D₄	—	59	1	no	—
238	C₂×D₄	—	118	1	no	—
239	C₂×D₄	—	118	1	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	59	1	no	—
242	C₂²×C₄	—	118	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	C₂²×C₄	—	118	1	no	—
245	C₂²×C₄	—	118	1	no	—
246	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
247	SD₁₆	—	118	2	no	—
248	C₂×C₈	—	59	1	no	—
249	C₂×C₈	—	59	1	no	—
250	C₂×C₈	—	118	2	no	—
251	C₂×C₈	—	118	2	no	—
252	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
253	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	59	1	no	—
257	C₄⋊C₄	—	59	1	no	—
258	C₄⋊C₄	—	59	1	no	—
259	C₄⋊C₄	—	59	1	no	—
260	C₄⋊C₄	—	118	1	no	—
261	C₄⋊C₄	—	118	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	—	118	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	—	118	1	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
269	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	59	1	no	—
272	C₂²⋊C₄	—	59	1	no	—
273	C₂²⋊C₄	—	118	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₂²⋊C₄	—	118	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₄²	—	59	1	no	—
278	C₄²	—	118	1	no	—
279	C₂³	—	59	1	no	—
280	C₂³	—	118	1	no	—
281	C₂³	D₄×D₅₉	1	1	yes	characteristic
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	472	1	no	—
285	Q₈	—	236	2	no	—
286	D₄	—	59	4	no	—
287	D₄	—	118	1	no	—
288	D₄	—	118	1	no	—
289	D₄	—	236	2	no	—
290	D₄	—	236	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	59	1	no	—
299	C₂×C₄	—	59	1	no	—
300	C₂×C₄	—	59	1	no	—
301	C₂×C₄	—	118	1	no	—
302	C₂×C₄	—	118	1	no	—
303	C₂×C₄	—	118	1	no	—
304	C₂×C₄	—	118	1	no	—
305	C₂×C₄	—	118	1	no	—
306	C₂×C₄	—	472	1	no	—
307	C₂×C₄	—	472	1	no	—
308	C₂×C₄	—	236	2	no	—
309	C₂×C₄	—	236	2	no	—
310	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	59	2	no	—
315	C₈	—	118	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	59	2	no	—
319	C₂²	—	118	1	no	—
320	C₂²	—	118	1	no	—
321	C₂²	—	236	1	no	—
322	C₂²	—	236	1	no	—
323	C₂²	—	236	1	no	—
324	C₂²	SD₁₆×D₅₉	1	2	yes	—
325	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	59	2	no	—
332	C₄	—	118	1	no	—
333	C₄	—	118	1	no	—
334	C₄	—	236	1	no	—
335	C₄	—	236	1	no	—
336	C₄	—	236	1	no	—
337	C₄	#41×D₅₉	1	1	yes	characteristic
338	C₄	#14273774	1	1	yes	characteristic
339	C₂	—	2	2	no	—
340	C₂	—	118	2	no	—
341	C₂	—	472	1	no	—
342	C₂	#16838050	1	1	yes	characteristic
343	C₂	C₂×SD₁₆×D₅₉	1	1	yes	characteristic
344	C₂	#16838211	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#483	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#483	D₅₉
hypocenter	C₅₉	#3335
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #483

Complements

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

N	H ≅ G/N	type
1. #16838110	C₂	semidirect
3. #16837799	C₂	semidirect
4. #16837731	C₂	semidirect
5. #16837722	C₂	semidirect
6. #16837721	C₂	semidirect
7. #16837691	C₂	semidirect
8. #16837685	C₂	semidirect
9. #16837634	C₂	semidirect
10. #16837633	C₂	semidirect
11. #16837487	C₂	semidirect
12. (C₂²⋊D₅₉)⋊C₈	C₂	semidirect
13. C₂×#14273658	C₂	semidirect
14. #106×D₅₉	C₂	semidirect
15. C₅₉×#483	C₂	semidirect
121. C₂²⋊D₅₉	SD₁₆	semidirect
142. C₅₉⋊C₈	#41	semidirect
177. #483	D₅₉	semidirect
188. C₅₉	#3335	semidirect