Subgroups — #21434239

Order 7552.

450 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837937	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₄×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837734	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837717	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837699	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837618	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837617	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837554	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837537	C₂	1	1	yes	maximal, characteristic
10	#16837513	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837511	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837501	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837499	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837458	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×D₄×D₅₉	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₅₉	—	2	1	no	—
18	C₂×#11745660	C₂²	1	1	yes	characteristic
19	C₂×#11745670	C₂²	1	1	yes	characteristic
20	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
21	C₂×#11745669	C₂²	1	1	yes	characteristic
22	#14273722	—	2	1	no	—
23	#14273715	C₂²	1	1	yes	characteristic, complemented
24	#14273712	—	2	1	no	—
25	#14273707	C₂²	1	1	yes	characteristic
26	#14273702	C₂²	1	1	yes	characteristic
27	#14273692	C₂²	1	1	yes	characteristic, complemented
28	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
29	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
30	#14273689	—	2	1	no	—
31	#14273689	C₂²	1	1	yes	characteristic
32	#14273689	C₂²	1	1	yes	characteristic, complemented
33	C₄×C₂²⋊D₅₉	—	2	1	no	—
34	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
35	#14273685	—	2	1	no	—
36	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₄×D₅₉	—	2	1	no	—
38	C₄²×D₅₉	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic, complemented
40	#14273658	C₂²	1	1	yes	characteristic, complemented
41	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic, complemented
42	#14273652	C₂²	1	1	yes	characteristic, complemented
43	#14273650	C₂²	1	1	yes	characteristic, complemented
44	#14273649	C₂²	1	1	yes	characteristic, complemented
45	#14273647	C₂²	1	1	yes	characteristic
46	#14273647	C₂²	1	1	yes	characteristic
47	#14273647	C₂²	1	1	yes	characteristic, complemented
48	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic, complemented
51	#14273637	C₂²	1	1	yes	characteristic, complemented
52	#125×C₅₉	C₂²	1	1	yes	characteristic, complemented
53	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic, complemented
55	#106×C₅₉	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic, complemented
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	C₂³×D₅₉	—	2	1	no	—
60	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂²×Dic₅₉	—	2	1	no	—
64	D₄×D₅₉	—	2	2	no	—
65	D₄×D₅₉	—	2	2	no	—
66	C₂×D₂₃₆	—	2	1	no	—
67	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
68	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	—	4	4	no	—
74	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
76	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
77	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₅₉	—	2	1	no	—
81	C₄⋊C₄×C₅₉	—	2	1	no	—
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₅₉	—	2	1	no	—
88	#11745670	—	2	2	no	—
89	#11745669	—	2	2	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	(C₂×D₅₉)⋊C₄	—	2	1	no	—
93	(C₂×D₅₉)⋊C₄	—	2	1	no	—
94	(C₂×D₅₉)⋊C₄	—	2	1	no	—
95	(C₂×D₅₉)⋊C₄	—	2	1	no	—
96	(C₂×D₅₉)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₅₉	—	2	1	no	—
98	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	D₄	1	1	yes	characteristic
103	C₄×Dic₅₉	D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
107	D₄₇₂	—	2	2	no	—
108	#11745660	—	2	2	no	—
109	C₂³×C₅₉	—	2	1	no	—
110	C₂²×D₅₉	—	2	4	no	—
111	C₂²×D₅₉	—	2	1	no	—
112	C₂²×D₅₉	—	2	1	no	—
113	C₂²×D₅₉	—	4	2	no	—
114	C₂²×D₅₉	—	4	2	no	—
115	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
116	D₄×C₅₉	—	2	1	no	—
117	D₄×C₅₉	#41	1	2	yes	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	4	2	no	—
122	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₅₉	—	2	4	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	4	2	no	—
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
132	D₂₃₆	—	2	1	no	—
133	D₂₃₆	—	2	1	no	—
134	D₂₃₆	—	2	1	no	—
135	D₂₃₆	—	4	2	no	—
136	D₂₃₆	#41	1	2	yes	—
137	C₄×D₅₉	—	2	2	no	—
138	C₄×D₅₉	—	2	1	no	—
139	C₄×D₅₉	—	2	1	no	—
140	C₄×D₅₉	—	4	1	no	—
141	C₄×D₅₉	—	4	2	no	—
142	C₄×D₅₉	—	4	2	no	—
143	C₄×D₅₉	—	4	2	no	—
144	C₄×D₅₉	—	4	2	no	—
145	Dic₁₁₈	—	2	1	no	—
146	Dic₁₁₈	—	2	1	no	—
147	C₈×C₅₉	—	2	1	no	—
148	C₈×C₅₉	—	2	1	no	—
149	C₅₉⋊C₈	—	2	1	no	—
150	C₅₉⋊C₈	—	2	1	no	—
151	C₂²×C₅₉	—	2	2	no	—
152	C₂²×C₅₉	—	4	1	no	—
153	C₂²×C₅₉	—	4	1	no	—
154	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₅₉	—	2	1	no	—
156	C₂×D₅₉	—	2	1	no	—
157	C₂×D₅₉	—	2	2	no	—
158	C₂×D₅₉	—	4	8	no	—
159	C₂×D₅₉	—	4	1	no	—
160	C₂×D₅₉	—	4	1	no	—
161	C₂×D₅₉	—	4	1	no	—
162	C₂×D₅₉	—	4	1	no	—
163	C₂×D₅₉	—	4	1	no	—
164	C₂×D₅₉	AGL(1,ℤ/8)	1	2	yes	—
165	C₄×C₅₉	—	2	1	no	—
166	C₄×C₅₉	—	2	1	no	—
167	C₄×C₅₉	—	4	1	no	—
168	C₄×C₅₉	—	4	1	no	—
169	C₄×C₅₉	—	4	1	no	—
170	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
171	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
172	Dic₅₉	—	2	1	no	—
173	Dic₅₉	—	2	1	no	—
174	Dic₅₉	—	2	1	no	—
175	Dic₅₉	—	4	1	no	—
176	Dic₅₉	—	4	1	no	—
177	#3114	—	59	1	no	maximal, Sylow
178	C₂×C₅₉	—	4	2	no	—
179	C₂×C₅₉	#525	1	1	yes	characteristic
180	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
181	C₂×C₅₉	#575	1	1	yes	characteristic
182	D₅₉	—	2	2	no	—
183	D₅₉	—	4	2	no	—
184	D₅₉	—	8	1	no	—
185	#529	—	59	1	no	—
186	C₂×C₄×D₄	—	59	1	no	—
187	#483	—	59	1	no	—
188	#480	—	59	1	no	—
189	#478	—	59	1	no	—
190	#478	D₅₉	1	1	yes	characteristic, complemented
191	#463	—	59	1	no	—
192	#463	—	59	1	no	—
193	#450	—	59	1	no	—
194	#447	—	59	1	no	—
195	#442	—	59	1	no	—
196	#442	—	59	1	no	—
197	#440	—	59	1	no	—
198	#440	—	59	1	no	—
199	#432	—	59	1	no	—
200	C₅₉	#3114	1	1	yes	characteristic, complemented, Sylow
201	C₂²×D₄	—	59	1	no	—
202	C₂³×C₄	—	118	1	no	—
203	C₂×SD₁₆	—	59	1	no	—
204	C₂×SD₁₆	—	59	1	no	—
205	C₂×D₈	—	59	1	no	—
206	C₂×D₈	—	59	1	no	—
207	#126	—	118	1	no	—
208	#125	C₂×D₅₉	1	1	yes	characteristic, complemented
209	#124	—	59	1	no	—
210	#123	—	118	1	no	—
211	C₂²⋊Q₈	—	59	1	no	—
212	#121	—	59	1	no	—
213	C₄×Q₈	—	59	1	no	—
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	59	1	no	—
216	C₄×D₄	—	59	1	no	—
217	C₄×D₄	—	59	1	no	—
218	C₄×D₄	—	118	1	no	—
219	C₄×D₄	—	118	1	no	—
220	C₄×D₄	—	118	1	no	—
221	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
222	#117	—	118	1	no	—
223	C₂×C₄⋊C₄	—	59	1	no	—
224	C₂×C₂²⋊C₄	—	118	1	no	—
225	C₂×C₄²	—	59	1	no	—
226	#107	—	59	1	no	—
227	#107	C₂×D₅₉	1	1	yes	characteristic, complemented
228	#106	—	59	1	no	—
229	#106	C₂×D₅₉	1	1	yes	characteristic, complemented
230	C₄⋊C₈	—	59	1	no	—
231	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic, complemented
232	#103	—	59	1	no	—
233	#103	—	59	1	no	—
234	#102	—	59	1	no	—
235	#102	—	59	1	no	—
236	#102	—	59	1	no	—
237	#102	—	59	1	no	—
238	#102	C₂×D₅₉	1	1	yes	characteristic
239	#102	C₂×D₅₉	1	1	yes	characteristic
240	C₂²⋊C₈	—	59	1	no	—
241	C₂²⋊C₈	—	59	1	no	—
242	#97	—	59	1	no	—
243	#97	—	59	1	no	—
244	C₂⁴	—	118	1	no	—
245	C₂×Q₈	—	59	1	no	—
246	C₂×D₄	—	59	1	no	—
247	C₂×D₄	—	59	1	no	—
248	C₂×D₄	—	118	1	no	—
249	C₂×D₄	—	118	1	no	—
250	C₂×D₄	—	118	2	no	—
251	C₂×D₄	—	118	2	no	—
252	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	59	1	no	—
255	C₂²×C₄	—	59	1	no	—
256	C₂²×C₄	—	59	1	no	—
257	C₂²×C₄	—	118	1	no	—
258	C₂²×C₄	—	118	1	no	—
259	C₂²×C₄	—	118	1	no	—
260	C₂²×C₄	—	118	1	no	—
261	C₂²×C₄	—	118	1	no	—
262	C₂²×C₄	—	236	4	no	—
263	SD₁₆	—	118	2	no	—
264	SD₁₆	—	118	2	no	—
265	D₈	—	118	2	no	—
266	D₈	—	118	2	no	—
267	C₂×C₈	—	59	1	no	—
268	C₂×C₈	—	59	1	no	—
269	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	59	1	no	—
274	C₄⋊C₄	—	118	1	no	—
275	C₄⋊C₄	—	118	1	no	—
276	C₄⋊C₄	—	118	1	no	—
277	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
278	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
279	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	118	1	no	—
282	C₂²⋊C₄	—	118	1	no	—
283	C₂²⋊C₄	—	118	1	no	—
284	C₂²⋊C₄	—	118	1	no	—
285	C₂²⋊C₄	—	118	1	no	—
286	C₂²⋊C₄	—	118	1	no	—
287	C₂²⋊C₄	—	118	1	no	—
288	C₂²⋊C₄	—	118	1	no	—
289	C₄²	—	59	1	no	—
290	C₄²	—	59	1	no	—
291	C₄²	—	59	1	no	—
292	C₄²	—	118	1	no	—
293	C₄²	C₂²×D₅₉	1	1	yes	characteristic
294	C₂³	—	2	1	no	—
295	C₂³	—	59	1	no	—
296	C₂³	—	118	4	no	—
297	C₂³	—	118	1	no	—
298	C₂³	—	118	1	no	—
299	C₂³	—	236	2	no	—
300	C₂³	—	236	2	no	—
301	Q₈	—	118	1	no	—
302	Q₈	—	118	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	59	2	no	—
305	D₄	—	118	4	no	—
306	D₄	—	118	1	no	—
307	D₄	—	118	1	no	—
308	D₄	—	118	1	no	—
309	D₄	—	236	2	no	—
310	D₄	#11745689	1	2	yes	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	59	1	no	—
316	C₂×C₄	—	59	1	no	—
317	C₂×C₄	—	59	1	no	—
318	C₂×C₄	—	118	2	no	—
319	C₂×C₄	—	118	1	no	—
320	C₂×C₄	—	118	1	no	—
321	C₂×C₄	—	118	1	no	—
322	C₂×C₄	—	118	1	no	—
323	C₂×C₄	—	236	1	no	—
324	C₂×C₄	—	236	2	no	—
325	C₂×C₄	—	236	2	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	—	236	2	no	—
328	C₂×C₄	—	236	2	no	—
329	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
330	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
331	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
332	C₈	—	2	1	no	—
333	C₈	—	2	1	no	—
334	C₈	—	118	1	no	—
335	C₈	—	118	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	4	1	no	—
339	C₂²	—	59	2	no	—
340	C₂²	—	118	1	no	—
341	C₂²	—	118	1	no	—
342	C₂²	—	118	2	no	—
343	C₂²	—	236	8	no	—
344	C₂²	—	236	1	no	—
345	C₂²	—	236	1	no	—
346	C₂²	—	236	1	no	—
347	C₂²	—	236	1	no	—
348	C₂²	—	236	1	no	—
349	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	118	1	no	—
356	C₄	—	118	1	no	—
357	C₄	—	118	1	no	—
358	C₄	—	236	1	no	—
359	C₄	—	236	1	no	—
360	C₄	#41×D₅₉	1	1	yes	characteristic
361	C₄	C₂×#11745689	1	1	yes	characteristic
362	C₂	—	4	2	no	—
363	C₂	—	118	2	no	—
364	C₂	—	472	1	no	—
365	C₂	—	236	2	no	—
366	C₂	#16837915	1	1	yes	characteristic
367	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
368	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×#478	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#478	D₅₉
hypocenter	C₅₉	#3114
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #16837937	C₂	semidirect
2. C₄×D₄×D₅₉	C₂	semidirect
3. #16837734	C₂	semidirect
4. #16837717	C₂	semidirect
5. #16837699	C₂	semidirect
6. #16837618	C₂	semidirect
7. #16837617	C₂	semidirect
8. #16837554	C₂	semidirect
10. #16837513	C₂	semidirect
11. #16837511	C₂	semidirect
12. #16837501	C₂	semidirect
13. #16837499	C₂	semidirect
14. #16837458	C₂	semidirect
15. C₅₉×#478	C₂	semidirect
23. #14273715	C₂²	semidirect
27. #14273692	C₂²	semidirect
32. #14273689	C₂²	semidirect
39. #14273660	C₂²	semidirect
40. #14273658	C₂²	semidirect
41. C₄⋊(C₅₉⋊C₈)	C₂²	semidirect
42. #14273652	C₂²	semidirect
43. #14273650	C₂²	semidirect
44. #14273649	C₂²	semidirect
47. #14273647	C₂²	semidirect
50. #14273637	C₂²	semidirect
51. #14273637	C₂²	semidirect
52. #125×C₅₉	C₂²	semidirect
54. #107×C₅₉	C₂²	semidirect
55. #106×C₅₉	C₂²	semidirect
56. C₄⋊C₈×C₅₉	C₂²	semidirect
190. #478	D₅₉	semidirect
200. C₅₉	#3114	semidirect
208. #125	C₂×D₅₉	semidirect
227. #107	C₂×D₅₉	semidirect
229. #106	C₂×D₅₉	semidirect
231. C₄⋊C₈	C₂×D₅₉	semidirect