Subgroups — #21432974

Order 7552.

484 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14273748	C₂	1	1	yes	maximal, characteristic
2	#16837884	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837881	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837512	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837511	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837507	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837506	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745660	C₂	1	1	yes	maximal, characteristic, complemented
9	SD₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837489	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837402	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837400	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837356	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×SD₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745689	C₂²	1	1	yes	characteristic
17	C₂×#11745687	C₂²	1	1	yes	characteristic
18	#14273748	C₄	1	8	yes	complemented
19	C₂×#11745672	C₂²	1	1	yes	characteristic
20	C₂×#11745660	C₂²	1	1	yes	characteristic
21	C₂×#11745669	C₂²	1	1	yes	characteristic
22	C₂×#11745659	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
24	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
26	#14273692	C₂²	1	1	yes	characteristic
27	C₄×D₂₃₆	—	2	1	no	—
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	C₄×C₂²⋊D₅₉	—	2	1	no	—
33	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
34	#14273684	—	2	1	no	—
35	#14273684	C₂²	1	1	yes	characteristic
36	#14273683	—	2	1	no	—
37	C₂×Dic₅₉⋊C₄	—	2	1	no	—
38	C₄⋊C₄×D₅₉	—	2	1	no	—
39	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
40	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
41	#14273659	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	#14273658	C₂²	1	1	yes	characteristic
44	#14273652	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	#14273648	C₂²	1	1	yes	characteristic
49	#14273647	C₂²	1	1	yes	characteristic
50	#14273638	C₂²	1	1	yes	characteristic
51	#14273637	C₂²	1	1	yes	characteristic
52	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	—	2	1	no	—
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂²×Dic₅₉	—	2	1	no	—
65	C₂²×Dic₅₉	—	2	1	no	—
66	#11745689	—	2	2	no	—
67	#11745689	C₂×C₄	1	4	yes	complemented
68	#11745687	—	2	2	no	—
69	#11745687	C₂×C₄	1	4	yes	complemented
70	C₂×D₂₃₆	—	2	1	no	—
71	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	—	2	1	no	—
74	C₂×C₄×D₅₉	—	2	1	no	—
75	C₂×C₄×D₅₉	—	2	1	no	—
76	C₂×C₄×D₅₉	—	2	1	no	—
77	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
79	SD₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
80	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	—	2	1	no	—
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	—	2	1	no	—
87	C₄²×C₅₉	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₅₉	—	2	1	no	—
89	#11745672	C₂×C₄	1	4	yes	complemented
90	#11745669	C₂×C₄	1	4	yes	complemented
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	(C₂×D₅₉)⋊C₄	—	2	1	no	—
93	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
94	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
95	C₄⋊Dic₅₉	—	2	1	no	—
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
103	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	—	2	1	no	—
106	C₄×Dic₅₉	—	2	1	no	—
107	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
110	#11745660	C₂×C₄	1	4	yes	complemented
111	#11745659	—	2	2	no	—
112	C₈×D₅₉	C₂×C₄	1	4	yes	complemented
113	C₂³×C₅₉	—	2	1	no	—
114	C₂²×D₅₉	—	2	1	no	—
115	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
116	Q₈×C₅₉	—	2	1	no	—
117	Q₈×C₅₉	C₂²×C₄	1	2	yes	—
118	D₄×C₅₉	—	2	1	no	—
119	D₄×C₅₉	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	8	1	no	—
124	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₅₉	—	2	4	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	—	2	4	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	8	1	no	—
133	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
136	D₂₃₆	—	2	4	no	—
137	D₂₃₆	—	2	1	no	—
138	D₂₃₆	C₂²×C₄	1	2	yes	—
139	C₄×D₅₉	—	2	4	no	—
140	C₄×D₅₉	—	8	1	no	—
141	C₄×D₅₉	—	8	1	no	—
142	C₄×D₅₉	—	8	1	no	—
143	C₄×D₅₉	—	8	1	no	—
144	C₄×D₅₉	C₂²×C₄	1	2	yes	—
145	C₄×D₅₉	#41	1	2	yes	—
146	Dic₁₁₈	—	2	1	no	—
147	Dic₁₁₈	C₂²×C₄	1	2	yes	—
148	C₈×C₅₉	—	2	1	no	—
149	C₈×C₅₉	C₂²×C₄	1	2	yes	—
150	C₅₉⋊C₈	—	2	1	no	—
151	C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
152	C₂²×C₅₉	—	2	2	no	—
153	C₂²×C₅₉	—	4	1	no	—
154	C₂²×C₅₉	—	4	1	no	—
155	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
156	C₂×D₅₉	—	2	1	no	—
157	C₂×D₅₉	—	2	1	no	—
158	C₂×D₅₉	—	2	2	no	—
159	C₂×D₅₉	—	4	1	no	—
160	C₂×D₅₉	—	4	1	no	—
161	C₂×D₅₉	C₄×D₄	1	2	yes	—
162	C₄×C₅₉	—	2	2	no	—
163	C₄×C₅₉	—	2	1	no	—
164	C₄×C₅₉	—	2	1	no	—
165	C₄×C₅₉	—	4	1	no	—
166	C₄×C₅₉	—	4	1	no	—
167	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
168	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
169	Dic₅₉	—	2	2	no	—
170	Dic₅₉	—	2	1	no	—
171	Dic₅₉	—	2	1	no	—
172	Dic₅₉	—	4	1	no	—
173	Dic₅₉	—	4	1	no	—
174	Dic₅₉	C₄×D₄	1	2	yes	—
175	#2952	—	59	1	no	maximal, Sylow
176	C₂×C₅₉	—	4	2	no	—
177	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
178	C₂×C₅₉	#577	1	1	yes	characteristic
179	C₂×C₅₉	#577	1	1	yes	characteristic
180	D₅₉	—	2	2	no	—
181	D₅₉	—	4	2	no	—
182	C₂×#135	—	59	1	no	—
183	#520	—	59	1	no	—
184	#520	—	59	1	no	—
185	#442	—	59	1	no	—
186	#442	—	59	1	no	—
187	#441	—	59	1	no	—
188	#441	—	59	1	no	—
189	C₄×SD₁₆	—	59	1	no	—
190	C₄×SD₁₆	—	59	1	no	—
191	C₄×SD₁₆	—	59	1	no	—
192	C₄×SD₁₆	D₅₉	1	1	yes	characteristic, complemented
193	C₂×#106	—	59	1	no	—
194	#417	—	59	1	no	—
195	#417	—	59	1	no	—
196	#405	—	59	1	no	—
197	C₅₉	#2952	1	1	yes	characteristic, complemented, Sylow
198	C₂×#41	—	59	1	no	—
199	C₂×#41	—	59	1	no	—
200	#135	—	59	8	no	—
201	C₂×Q₁₆	—	59	1	no	—
202	C₂×SD₁₆	—	59	1	no	—
203	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
204	C₂×D₈	—	59	1	no	—
205	C₂×M₁₆	—	59	1	no	—
206	C₂²×C₈	—	59	1	no	—
207	C₄×Q₈	—	59	1	no	—
208	C₄×Q₈	—	59	1	no	—
209	C₄×Q₈	—	59	1	no	—
210	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
211	C₄×D₄	—	59	1	no	—
212	C₄×D₄	—	59	1	no	—
213	C₄×D₄	—	59	1	no	—
214	C₄×D₄	—	118	1	no	—
215	C₄×D₄	—	118	1	no	—
216	C₄×D₄	—	118	1	no	—
217	C₄×D₄	—	118	1	no	—
218	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
219	#117	—	59	1	no	—
220	#117	—	118	1	no	—
221	#117	—	118	1	no	—
222	C₂×C₄⋊C₄	—	59	1	no	—
223	C₂×C₄⋊C₄	—	59	1	no	—
224	C₂×C₄⋊C₄	—	118	1	no	—
225	C₂×C₄⋊C₄	—	118	1	no	—
226	#106	—	59	1	no	—
227	#106	—	59	1	no	—
228	#106	—	59	1	no	—
229	#106	C₂×D₅₉	1	1	yes	characteristic
230	#103	—	59	1	no	—
231	#103	—	59	1	no	—
232	#103	—	59	1	no	—
233	#103	C₂×D₅₉	1	1	yes	characteristic
234	#102	—	59	1	no	—
235	#102	—	59	1	no	—
236	#102	—	59	1	no	—
237	#102	C₂×D₅₉	1	1	yes	characteristic
238	#97	—	59	1	no	—
239	#97	—	59	1	no	—
240	C₄×C₈	—	59	1	no	—
241	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
242	#41	—	59	4	no	—
243	#41	—	59	4	no	—
244	#41	—	118	2	no	—
245	#41	—	118	2	no	—
246	C₂×Q₈	—	59	1	no	—
247	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
248	C₂×D₄	—	59	1	no	—
249	C₂×D₄	—	118	1	no	—
250	C₂×D₄	—	118	1	no	—
251	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	59	1	no	—
254	C₂²×C₄	—	118	1	no	—
255	C₂²×C₄	—	118	1	no	—
256	C₂²×C₄	—	118	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	Q₁₆	—	59	4	no	—
262	SD₁₆	—	59	4	no	—
263	SD₁₆	C₄×D₅₉	1	4	yes	complemented
264	D₈	—	59	4	no	—
265	M₁₆	—	118	2	no	—
266	C₂×C₈	—	59	1	no	—
267	C₂×C₈	—	59	4	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	—	59	1	no	—
273	C₄⋊C₄	—	59	1	no	—
274	C₄⋊C₄	—	59	1	no	—
275	C₄⋊C₄	—	59	1	no	—
276	C₄⋊C₄	—	118	1	no	—
277	C₄⋊C₄	—	118	1	no	—
278	C₄⋊C₄	—	118	1	no	—
279	C₄⋊C₄	—	118	1	no	—
280	C₄⋊C₄	—	118	1	no	—
281	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
282	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
283	C₂²⋊C₄	—	2	1	no	—
284	C₂²⋊C₄	—	59	1	no	—
285	C₂²⋊C₄	—	59	1	no	—
286	C₂²⋊C₄	—	118	1	no	—
287	C₂²⋊C₄	—	118	1	no	—
288	C₂²⋊C₄	—	118	1	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	59	1	no	—
291	C₄²	—	118	1	no	—
292	C₄²	—	118	1	no	—
293	C₄²	—	118	1	no	—
294	C₄²	C₂²×D₅₉	1	1	yes	characteristic
295	C₂³	—	2	1	no	—
296	C₂³	—	59	1	no	—
297	C₂³	—	118	1	no	—
298	Q₈	—	2	1	no	—
299	Q₈	—	59	2	no	—
300	Q₈	—	118	1	no	—
301	Q₈	C₂×C₄×D₅₉	1	2	yes	—
302	D₄	—	2	1	no	—
303	D₄	—	59	2	no	—
304	D₄	—	118	4	no	—
305	D₄	—	118	4	no	—
306	D₄	—	118	1	no	—
307	D₄	C₂×C₄×D₅₉	1	2	yes	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	8	1	no	—
312	C₂×C₄	—	59	1	no	—
313	C₂×C₄	—	59	2	no	—
314	C₂×C₄	—	59	2	no	—
315	C₂×C₄	—	59	1	no	—
316	C₂×C₄	—	59	1	no	—
317	C₂×C₄	—	118	4	no	—
318	C₂×C₄	—	118	4	no	—
319	C₂×C₄	—	118	1	no	—
320	C₂×C₄	—	118	1	no	—
321	C₂×C₄	—	118	1	no	—
322	C₂×C₄	—	472	1	no	—
323	C₂×C₄	—	472	1	no	—
324	C₂×C₄	—	472	1	no	—
325	C₂×C₄	—	472	1	no	—
326	C₂×C₄	—	472	1	no	—
327	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
328	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
329	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	59	2	no	—
332	C₈	—	118	1	no	—
333	C₈	C₂×C₄×D₅₉	1	2	yes	—
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	59	2	no	—
338	C₂²	—	118	1	no	—
339	C₂²	—	118	1	no	—
340	C₂²	—	118	2	no	—
341	C₂²	—	236	1	no	—
342	C₂²	—	236	1	no	—
343	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	2	no	—
347	C₄	—	4	1	no	—
348	C₄	—	4	1	no	—
349	C₄	—	59	2	no	—
350	C₄	—	118	1	no	—
351	C₄	—	118	1	no	—
352	C₄	—	118	2	no	—
353	C₄	—	236	1	no	—
354	C₄	—	236	1	no	—
355	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
356	C₄	#41×D₅₉	1	1	yes	characteristic
357	C₂	—	4	2	no	—
358	C₂	—	118	2	no	—
359	C₂	—	236	2	no	—
360	C₂	C₄×D₄×D₅₉	1	1	yes	characteristic
361	C₂	#16838208	1	1	yes	characteristic
362	C₂	#16838209	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₄×SD₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	C₄×SD₁₆	D₅₉
hypocenter	C₅₉	#2952
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. #16837884	C₂	semidirect
3. #16837881	C₂	semidirect
4. #16837512	C₂	semidirect
5. #16837511	C₂	semidirect
6. #16837507	C₂	semidirect
7. #16837506	C₂	semidirect
8. C₄×#11745660	C₂	semidirect
9. SD₁₆×Dic₅₉	C₂	semidirect
10. #16837489	C₂	semidirect
11. #106×D₅₉	C₂	semidirect
12. #16837402	C₂	semidirect
13. #16837400	C₂	semidirect
14. #16837356	C₂	semidirect
15. C₄×SD₁₆×C₅₉	C₂	semidirect
18. #14273748	C₄	semidirect
67. #11745689	C₂×C₄	semidirect
69. #11745687	C₂×C₄	semidirect
79. SD₁₆×C₅₉	C₂×C₄	semidirect
89. #11745672	C₂×C₄	semidirect
90. #11745669	C₂×C₄	semidirect
110. #11745660	C₂×C₄	semidirect
112. C₈×D₅₉	C₂×C₄	semidirect
192. C₄×SD₁₆	D₅₉	semidirect
197. C₅₉	#2952	semidirect
263. SD₁₆	C₄×D₅₉	semidirect