Subgroups — #21432952

Order 7552.

492 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×SD₁₆×D₅₉	C₂	1	1	yes	maximal, characteristic
2	#16837878	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837874	C₂	1	1	yes	maximal, characteristic
4	#16837504	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837503	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837501	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#11745671	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#11745670	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837487	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837439	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837409	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837405	C₂	1	1	yes	maximal, characteristic
14	#16837359	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
17	C₂×D₄×D₅₉	C₂²	1	1	yes	characteristic
18	C₂×#11745660	C₂²	1	1	yes	characteristic
19	C₂×#11745670	C₂²	1	1	yes	characteristic
20	C₂×#11745671	C₂²	1	1	yes	characteristic
21	SD₁₆×D₅₉	C₄	1	8	yes	complemented
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, complemented
25	C₄×Dic₁₁₈	—	2	1	no	—
26	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic, complemented
27	#14273692	—	2	1	no	—
28	#14273692	C₂²	1	1	yes	characteristic, complemented
29	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
30	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
31	#14273689	C₂²	1	1	yes	characteristic
32	C₄×C₂²⋊D₅₉	—	2	1	no	—
33	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
34	#14273685	—	2	1	no	—
35	#14273685	C₂²	1	1	yes	characteristic
36	#14273685	C₂²	1	1	yes	characteristic
37	#14273684	—	2	1	no	—
38	#14273684	C₂²	1	1	yes	characteristic
39	C₂×(C₂×D₅₉)⋊C₄	—	2	1	no	—
40	C₂²⋊C₄×D₅₉	—	2	1	no	—
41	#14273661	C₂²	1	1	yes	characteristic, complemented
42	#14273658	C₂²	1	1	yes	characteristic, complemented
43	#14273658	C₂²	1	1	yes	characteristic, complemented
44	#14273652	C₂²	1	1	yes	characteristic, complemented
45	#14273651	C₂²	1	1	yes	characteristic, complemented
46	#14273650	C₂²	1	1	yes	characteristic, complemented
47	#14273649	C₂²	1	1	yes	characteristic
48	#14273648	C₂²	1	1	yes	characteristic
49	#14273647	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic, complemented
51	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
52	C₈×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
53	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic, complemented
55	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
56	#107×C₅₉	C₂²	1	1	yes	characteristic, complemented
57	#103×C₅₉	C₂²	1	1	yes	characteristic, complemented
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	#97×C₅₉	C₂²	1	1	yes	characteristic, complemented
60	C₂³×D₅₉	—	2	1	no	—
61	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	—	2	1	no	—
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	C₂²×Dic₅₉	—	2	1	no	—
66	Q₈×D₅₉	—	2	2	no	—
67	Q₈×D₅₉	C₂×C₄	1	4	yes	complemented
68	D₄×D₅₉	—	2	2	no	—
69	D₄×D₅₉	C₂×C₄	1	4	yes	—
70	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	—	2	1	no	—
74	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
75	C₂×Dic₁₁₈	—	2	1	no	—
76	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
77	SD₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
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	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₅₉	—	2	1	no	—
84	C₄²×C₅₉	—	2	1	no	—
85	C₄²×C₅₉	C₂³	1	1	yes	characteristic
86	C₂²⋊Dic₅₉	—	2	1	no	—
87	#11745671	C₂×C₄	1	4	yes	complemented
88	#11745670	C₂×C₄	1	4	yes	complemented
89	(C₂×D₅₉)⋊C₄	—	2	1	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	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	D₄	1	1	yes	characteristic
101	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
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	—	2	4	no	—
116	C₂²×D₅₉	—	8	1	no	—
117	C₂²×D₅₉	—	8	1	no	—
118	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
119	Q₈×C₅₉	—	2	1	no	—
120	Q₈×C₅₉	C₂²×C₄	1	2	yes	—
121	D₄×C₅₉	—	2	1	no	—
122	D₄×C₅₉	C₂²×C₄	1	2	yes	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	2	1	no	—
126	C₂×C₄×C₅₉	—	8	1	no	—
127	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
130	C₂²⋊D₅₉	—	2	4	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	2	1	no	—
133	C₂×Dic₅₉	—	2	1	no	—
134	C₂×Dic₅₉	—	8	1	no	—
135	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
138	D₂₃₆	—	2	1	no	—
139	D₂₃₆	C₂²×C₄	1	2	yes	—
140	C₄×D₅₉	—	2	4	no	—
141	C₄×D₅₉	—	8	1	no	—
142	C₄×D₅₉	—	8	1	no	—
143	C₄×D₅₉	C₂²×C₄	1	2	yes	—
144	C₄×D₅₉	#41	1	2	yes	—
145	Dic₁₁₈	—	2	1	no	—
146	Dic₁₁₈	—	2	4	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	—	8	4	no	—
162	C₂×D₅₉	C₄×D₄	1	2	yes	—
163	C₄×C₅₉	—	2	2	no	—
164	C₄×C₅₉	—	2	1	no	—
165	C₄×C₅₉	—	2	1	no	—
166	C₄×C₅₉	—	4	1	no	—
167	C₄×C₅₉	—	4	1	no	—
168	C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
169	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
170	Dic₅₉	—	2	2	no	—
171	Dic₅₉	—	2	1	no	—
172	Dic₅₉	—	2	1	no	—
173	Dic₅₉	—	4	1	no	—
174	Dic₅₉	—	4	1	no	—
175	Dic₅₉	C₄×D₄	1	2	yes	—
176	#2949	—	59	1	no	maximal, Sylow
177	C₂×C₅₉	—	4	2	no	—
178	C₂×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
179	C₂×C₅₉	#577	1	1	yes	characteristic
180	C₂×C₅₉	#577	1	1	yes	characteristic
181	D₅₉	—	2	2	no	—
182	D₅₉	—	4	2	no	—
183	C₂²×SD₁₆	—	59	1	no	—
184	#519	—	59	1	no	—
185	#518	—	59	1	no	—
186	#440	—	59	1	no	—
187	#440	—	59	1	no	—
188	#440	—	59	1	no	—
189	#440	D₅₉	1	1	yes	characteristic, complemented
190	C₄×SD₁₆	—	59	1	no	—
191	C₄×SD₁₆	—	59	1	no	—
192	C₄×SD₁₆	—	59	1	no	—
193	C₄×SD₁₆	—	59	1	no	—
194	#427	—	59	1	no	—
195	#419	—	59	1	no	—
196	#418	—	59	1	no	—
197	#405	—	59	1	no	—
198	C₅₉	#2949	1	1	yes	characteristic, complemented, Sylow
199	C₂²×Q₈	—	59	1	no	—
200	C₂²×D₄	—	59	1	no	—
201	C₂×SD₁₆	—	59	1	no	—
202	C₂×SD₁₆	—	59	1	no	—
203	C₂×SD₁₆	—	59	8	no	—
204	C₂×SD₁₆	—	59	1	no	—
205	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
206	C₂×M₁₆	—	59	1	no	—
207	C₂²×C₈	—	59	1	no	—
208	C₄×Q₈	—	59	1	no	—
209	C₄×Q₈	—	59	1	no	—
210	C₄×Q₈	—	59	1	no	—
211	C₄×Q₈	—	118	1	no	—
212	C₄×Q₈	—	118	1	no	—
213	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic, complemented
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	59	1	no	—
216	C₄×D₄	—	59	1	no	—
217	C₄×D₄	—	118	1	no	—
218	C₄×D₄	—	118	1	no	—
219	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
220	#117	—	59	1	no	—
221	#117	—	59	1	no	—
222	#117	—	59	1	no	—
223	#117	—	118	1	no	—
224	#117	—	118	1	no	—
225	C₂×C₂²⋊C₄	—	118	1	no	—
226	C₂×C₂²⋊C₄	—	118	1	no	—
227	#107	—	59	1	no	—
228	#107	C₂×D₅₉	1	1	yes	characteristic, complemented
229	#106	—	59	1	no	—
230	#106	—	59	1	no	—
231	#103	—	59	1	no	—
232	#103	—	59	1	no	—
233	#103	—	59	1	no	—
234	#103	C₂×D₅₉	1	1	yes	characteristic, complemented
235	#102	—	59	1	no	—
236	#102	—	59	1	no	—
237	#102	—	59	1	no	—
238	#102	C₂×D₅₉	1	1	yes	characteristic
239	#97	—	59	1	no	—
240	#97	C₂×D₅₉	1	1	yes	characteristic, complemented
241	C₄×C₈	—	59	1	no	—
242	C₄×C₈	—	59	1	no	—
243	C₂⁴	—	118	1	no	—
244	C₂×Q₈	—	59	4	no	—
245	C₂×Q₈	—	59	1	no	—
246	C₂×Q₈	—	118	1	no	—
247	C₂×Q₈	—	118	2	no	—
248	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
249	C₂×D₄	—	59	4	no	—
250	C₂×D₄	—	59	1	no	—
251	C₂×D₄	—	118	2	no	—
252	C₂×D₄	—	118	1	no	—
253	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
254	C₂²×C₄	—	2	1	no	—
255	C₂²×C₄	—	59	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	SD₁₆	—	59	4	no	—
261	SD₁₆	—	59	4	no	—
262	SD₁₆	—	59	4	no	—
263	SD₁₆	C₄×D₅₉	1	4	yes	complemented
264	M₁₆	—	118	2	no	—
265	C₂×C₈	—	59	1	no	—
266	C₂×C₈	—	59	4	no	—
267	C₂×C₈	—	59	1	no	—
268	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
269	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	59	1	no	—
272	C₄⋊C₄	—	59	1	no	—
273	C₄⋊C₄	—	59	1	no	—
274	C₄⋊C₄	—	59	1	no	—
275	C₄⋊C₄	—	118	1	no	—
276	C₄⋊C₄	—	118	1	no	—
277	C₄⋊C₄	—	118	1	no	—
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	—	59	1	no	—
282	C₂²⋊C₄	—	59	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	C₄²	—	2	1	no	—
289	C₄²	—	59	1	no	—
290	C₄²	—	118	1	no	—
291	C₄²	—	118	1	no	—
292	C₄²	—	118	1	no	—
293	C₄²	—	118	1	no	—
294	C₄²	—	118	1	no	—
295	C₄²	C₂²×D₅₉	1	1	yes	characteristic
296	C₂³	—	2	1	no	—
297	C₂³	—	59	1	no	—
298	C₂³	—	118	4	no	—
299	C₂³	—	118	1	no	—
300	C₂³	—	472	1	no	—
301	C₂³	—	472	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	59	2	no	—
304	Q₈	—	118	4	no	—
305	Q₈	—	118	1	no	—
306	Q₈	C₂×C₄×D₅₉	1	2	yes	—
307	D₄	—	2	1	no	—
308	D₄	—	59	2	no	—
309	D₄	—	118	4	no	—
310	D₄	—	118	1	no	—
311	D₄	C₂×C₄×D₅₉	1	2	yes	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	8	1	no	—
316	C₂×C₄	—	59	1	no	—
317	C₂×C₄	—	59	2	no	—
318	C₂×C₄	—	59	2	no	—
319	C₂×C₄	—	59	1	no	—
320	C₂×C₄	—	59	1	no	—
321	C₂×C₄	—	118	4	no	—
322	C₂×C₄	—	118	1	no	—
323	C₂×C₄	—	118	1	no	—
324	C₂×C₄	—	118	1	no	—
325	C₂×C₄	—	472	1	no	—
326	C₂×C₄	—	472	1	no	—
327	C₂×C₄	—	472	1	no	—
328	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
329	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	59	2	no	—
333	C₈	—	118	1	no	—
334	C₈	C₂×C₄×D₅₉	1	2	yes	—
335	C₂²	—	2	2	no	—
336	C₂²	—	4	1	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	59	2	no	—
339	C₂²	—	118	1	no	—
340	C₂²	—	118	1	no	—
341	C₂²	—	118	2	no	—
342	C₂²	—	472	4	no	—
343	C₂²	—	236	1	no	—
344	C₂²	—	236	1	no	—
345	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	2	2	no	—
349	C₄	—	4	1	no	—
350	C₄	—	4	1	no	—
351	C₄	—	59	2	no	—
352	C₄	—	118	1	no	—
353	C₄	—	118	1	no	—
354	C₄	—	118	2	no	—
355	C₄	—	236	1	no	—
356	C₄	—	236	1	no	—
357	C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
358	C₄	#41×D₅₉	1	1	yes	characteristic
359	C₂	—	4	2	no	—
360	C₂	—	118	2	no	—
361	C₂	—	236	2	no	—
362	C₂	C₄×D₄×D₅₉	1	1	yes	characteristic
363	C₂	#16838211	1	1	yes	characteristic
364	C₂	#16838212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#440	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#440	D₅₉
hypocenter	C₅₉	#2949
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #16837878	C₂	semidirect
4. #16837504	C₂	semidirect
5. #16837503	C₂	semidirect
6. #16837501	C₂	semidirect
7. #16837493	C₂	semidirect
8. C₄×#11745671	C₂	semidirect
9. C₄×#11745670	C₂	semidirect
10. #16837487	C₂	semidirect
11. #16837439	C₂	semidirect
12. #16837409	C₂	semidirect
14. #16837359	C₂	semidirect
15. C₅₉×#440	C₂	semidirect
21. SD₁₆×D₅₉	C₄	semidirect
24. Q₈×Dic₅₉	C₂²	semidirect
26. C₄×Dic₁₁₈	C₂²	semidirect
28. #14273692	C₂²	semidirect
41. #14273661	C₂²	semidirect
42. #14273658	C₂²	semidirect
43. #14273658	C₂²	semidirect
44. #14273652	C₂²	semidirect
45. #14273651	C₂²	semidirect
46. #14273650	C₂²	semidirect
50. #14273637	C₂²	semidirect
51. C₄×C₅₉⋊C₈	C₂²	semidirect
52. C₈×Dic₅₉	C₂²	semidirect
54. C₄×Q₈×C₅₉	C₂²	semidirect
56. #107×C₅₉	C₂²	semidirect
57. #103×C₅₉	C₂²	semidirect
59. #97×C₅₉	C₂²	semidirect
67. Q₈×D₅₉	C₂×C₄	semidirect
77. SD₁₆×C₅₉	C₂×C₄	semidirect
87. #11745671	C₂×C₄	semidirect
88. #11745670	C₂×C₄	semidirect
110. #11745660	C₂×C₄	semidirect
112. C₈×D₅₉	C₂×C₄	semidirect
189. #440	D₅₉	semidirect
198. C₅₉	#2949	semidirect
213. C₄×Q₈	C₂×D₅₉	semidirect
228. #107	C₂×D₅₉	semidirect
234. #103	C₂×D₅₉	semidirect
240. #97	C₂×D₅₉	semidirect
263. SD₁₆	C₄×D₅₉	semidirect