Subgroups — #21434333

Order 7552.

418 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838005	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837877	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837727	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837723	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837704	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837700	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837595	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837595	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837536	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837514	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837511	C₂	1	1	yes	maximal, characteristic, complemented
12	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837482	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837460	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#447	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745669	C₂²	1	1	yes	characteristic
17	C₂×#11745669	C₂²	1	1	yes	characteristic
18	#14273723	C₂²	1	1	yes	characteristic, complemented
19	#14273718	C₂²	1	1	yes	characteristic, complemented
20	#14273711	—	2	1	no	—
21	#14273709	—	2	1	no	—
22	#14273701	C₂²	1	1	yes	characteristic
23	#14273699	—	2	1	no	—
24	#14273698	—	2	1	no	—
25	D₄×Dic₅₉	—	2	1	no	—
26	D₄×Dic₅₉	—	2	1	no	—
27	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
28	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
29	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
30	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
32	#14273689	C₂²	1	1	yes	characteristic, complemented
33	#14273686	C₂²	1	1	yes	characteristic
34	#14273686	C₂²	1	1	yes	characteristic
35	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
36	C₂×C₂²⋊Dic₅₉	—	2	1	no	—
37	#14273661	C₂²	1	1	yes	characteristic, complemented
38	#14273660	C₂²	1	1	yes	characteristic, complemented
39	#14273659	C₂²	1	1	yes	characteristic, complemented
40	#14273658	C₂²	1	1	yes	characteristic, complemented
41	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic, complemented
43	#14273649	C₂²	1	1	yes	characteristic, complemented
44	#14273649	C₂²	1	1	yes	characteristic, complemented
45	#14273648	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic, complemented
48	#14273648	C₂²	1	1	yes	characteristic, complemented
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic, complemented
52	C₂²×D₄×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
55	#121×C₅₉	C₂²	1	1	yes	characteristic
56	#102×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂⁴×C₅₉	—	2	1	no	—
60	C₂×D₄×C₅₉	—	2	2	no	—
61	C₂×D₄×C₅₉	—	2	2	no	—
62	C₂×D₄×C₅₉	—	2	1	no	—
63	C₂×D₄×C₅₉	—	2	1	no	—
64	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
66	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₅₉	—	2	1	no	—
69	C₂×C₂²⋊D₅₉	—	2	1	no	—
70	C₂×C₂²⋊D₅₉	—	2	1	no	—
71	C₂²×Dic₅₉	—	2	1	no	—
72	C₂²×Dic₅₉	—	2	1	no	—
73	C₂²×Dic₅₉	—	2	1	no	—
74	C₂²×Dic₅₉	—	2	1	no	—
75	C₂²×Dic₅₉	—	2	1	no	—
76	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₅₉	—	2	1	no	—
78	D₈×C₅₉	—	2	2	no	—
79	D₈×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	C₂²⋊Dic₅₉	—	2	1	no	—
87	C₂²⋊Dic₅₉	—	2	1	no	—
88	C₂²⋊Dic₅₉	—	2	1	no	—
89	C₂²⋊Dic₅₉	—	2	1	no	—
90	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
91	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
92	#11745669	—	2	2	no	—
93	#11745669	—	2	2	no	—
94	(C₂×D₅₉)⋊C₄	—	2	1	no	—
95	(C₂×D₅₉)⋊C₄	—	2	1	no	—
96	C₄⋊Dic₅₉	—	2	1	no	—
97	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
98	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
99	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
100	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
102	Dic₅₉⋊C₄	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
105	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₅₉	—	2	4	no	—
109	C₂³×C₅₉	—	2	1	no	—
110	C₂³×C₅₉	—	2	1	no	—
111	C₂³×C₅₉	—	2	1	no	—
112	C₂³×C₅₉	—	8	1	no	—
113	C₂³×C₅₉	—	8	1	no	—
114	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×D₅₉	—	2	1	no	—
116	D₄×C₅₉	—	2	4	no	—
117	D₄×C₅₉	—	2	1	no	—
118	D₄×C₅₉	—	2	1	no	—
119	D₄×C₅₉	—	2	1	no	—
120	D₄×C₅₉	—	2	1	no	—
121	D₄×C₅₉	—	8	1	no	—
122	D₄×C₅₉	#41	1	2	yes	—
123	D₄×C₅₉	#41	1	2	yes	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	2	1	no	—
126	C₂×C₄×C₅₉	—	2	1	no	—
127	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
129	C₂²⋊D₅₉	—	8	1	no	—
130	C₂²⋊D₅₉	—	8	1	no	—
131	C₂²⋊D₅₉	—	8	1	no	—
132	C₂×Dic₅₉	—	2	1	no	—
133	C₂×Dic₅₉	—	2	1	no	—
134	C₂×Dic₅₉	—	2	1	no	—
135	C₂×Dic₅₉	—	8	1	no	—
136	C₂×Dic₅₉	—	8	1	no	—
137	C₂×Dic₅₉	—	8	1	no	—
138	C₂×Dic₅₉	—	8	1	no	—
139	C₂×Dic₅₉	—	8	1	no	—
140	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
144	D₂₃₆	—	2	1	no	—
145	D₂₃₆	—	2	1	no	—
146	C₄×D₅₉	—	8	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	1	no	—
152	C₂²×C₅₉	—	2	2	no	—
153	C₂²×C₅₉	—	2	2	no	—
154	C₂²×C₅₉	—	2	1	no	—
155	C₂²×C₅₉	—	2	1	no	—
156	C₂²×C₅₉	—	4	1	no	—
157	C₂²×C₅₉	—	4	1	no	—
158	C₂²×C₅₉	—	4	1	no	—
159	C₂²×C₅₉	—	4	1	no	—
160	C₂²×C₅₉	—	4	1	no	—
161	C₂²×C₅₉	—	4	1	no	—
162	C₂²×C₅₉	—	4	1	no	—
163	C₂²×C₅₉	—	8	4	no	—
164	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
165	C₂×D₅₉	—	4	1	no	—
166	C₂×D₅₉	—	4	1	no	—
167	C₂×D₅₉	—	4	1	no	—
168	C₄×C₅₉	—	2	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	—	2	1	no	—
176	Dic₅₉	—	4	1	no	—
177	Dic₅₉	—	4	1	no	—
178	Dic₅₉	—	4	1	no	—
179	#3122	—	59	1	no	maximal, Sylow
180	C₂×C₅₉	—	4	1	no	—
181	C₂×C₅₉	—	4	2	no	—
182	C₂×C₅₉	—	4	2	no	—
183	C₂×C₅₉	—	8	1	no	—
184	C₂×C₅₉	#525	1	1	yes	characteristic
185	C₂×C₅₉	#576	1	1	yes	characteristic
186	C₂×C₅₉	#576	1	1	yes	characteristic
187	D₅₉	—	8	1	no	—
188	#540	—	59	1	no	—
189	#518	—	59	1	no	—
190	#482	—	59	1	no	—
191	#482	—	59	1	no	—
192	#478	—	59	1	no	—
193	#478	—	59	1	no	—
194	#459	—	59	1	no	—
195	#459	—	59	1	no	—
196	#447	—	59	1	no	—
197	#447	D₅₉	1	1	yes	characteristic, complemented
198	#442	—	59	1	no	—
199	#442	—	59	1	no	—
200	C₄×D₈	—	59	1	no	—
201	C₄×D₈	—	59	1	no	—
202	#433	—	59	1	no	—
203	C₅₉	#3122	1	1	yes	characteristic, complemented, Sylow
204	C₂²×D₄	C₂×D₅₉	1	1	yes	characteristic
205	C₂×D₈	—	59	1	no	—
206	C₂×D₈	—	59	1	no	—
207	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
208	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
209	#127	—	59	1	no	—
210	#125	—	59	1	no	—
211	#123	—	118	1	no	—
212	#123	—	118	1	no	—
213	#121	—	59	1	no	—
214	#121	—	118	1	no	—
215	#121	—	118	1	no	—
216	#121	C₂×D₅₉	1	1	yes	characteristic
217	C₄×D₄	—	59	1	no	—
218	C₄×D₄	—	59	1	no	—
219	C₄×D₄	—	59	1	no	—
220	C₄×D₄	—	59	1	no	—
221	C₄×D₄	—	59	1	no	—
222	C₄×D₄	—	59	1	no	—
223	C₄×D₄	—	118	1	no	—
224	C₄×D₄	—	118	1	no	—
225	#117	—	59	1	no	—
226	#117	—	59	1	no	—
227	C₂×C₂²⋊C₄	—	118	1	no	—
228	C₂×C₂²⋊C₄	—	118	1	no	—
229	#107	—	59	1	no	—
230	#107	—	59	1	no	—
231	#106	—	59	1	no	—
232	#106	—	59	1	no	—
233	C₄⋊C₈	—	59	1	no	—
234	C₄⋊C₈	—	59	1	no	—
235	#102	—	59	1	no	—
236	#102	—	59	1	no	—
237	#102	—	59	1	no	—
238	#102	—	59	1	no	—
239	#102	—	59	1	no	—
240	#102	—	59	1	no	—
241	#102	C₂×D₅₉	1	1	yes	characteristic
242	#102	C₂×D₅₉	1	1	yes	characteristic
243	C₂²⋊C₈	—	59	1	no	—
244	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
245	#97	—	59	1	no	—
246	C₄×C₈	—	59	1	no	—
247	C₂⁴	—	2	1	no	—
248	C₂×D₄	—	2	2	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	2	1	no	—
252	C₂×D₄	—	59	1	no	—
253	C₂×D₄	—	118	1	no	—
254	C₂×D₄	—	118	1	no	—
255	C₂×D₄	—	118	1	no	—
256	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
257	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
258	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
259	C₂²×C₄	—	118	1	no	—
260	C₂²×C₄	—	118	1	no	—
261	C₂²×C₄	—	118	1	no	—
262	C₂²×C₄	—	118	1	no	—
263	C₂²×C₄	—	118	1	no	—
264	C₂²×C₄	—	118	1	no	—
265	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
266	D₈	—	2	2	no	—
267	D₈	—	2	2	no	—
268	D₈	—	118	2	no	—
269	D₈	—	118	2	no	—
270	C₂×C₈	—	59	1	no	—
271	C₂×C₈	—	59	1	no	—
272	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
273	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
274	C₄⋊C₄	—	59	1	no	—
275	C₄⋊C₄	—	59	1	no	—
276	C₄⋊C₄	—	59	1	no	—
277	C₄⋊C₄	—	59	1	no	—
278	C₄⋊C₄	—	59	1	no	—
279	C₄⋊C₄	—	118	1	no	—
280	C₄⋊C₄	—	118	1	no	—
281	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
282	C₂²⋊C₄	—	2	1	no	—
283	C₂²⋊C₄	—	59	1	no	—
284	C₂²⋊C₄	—	59	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	C₂²⋊C₄	—	118	1	no	—
290	C₂²⋊C₄	—	118	1	no	—
291	C₂²⋊C₄	—	118	1	no	—
292	C₂²⋊C₄	—	118	1	no	—
293	C₄²	—	59	1	no	—
294	C₄²	—	59	1	no	—
295	C₄²	—	118	1	no	—
296	C₂³	—	2	4	no	—
297	C₂³	—	2	1	no	—
298	C₂³	—	2	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	8	1	no	—
301	C₂³	—	8	1	no	—
302	C₂³	—	118	1	no	—
303	C₂³	D₄×D₅₉	1	1	yes	characteristic
304	D₄	—	2	4	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	D₄	—	2	1	no	—
308	D₄	—	2	1	no	—
309	D₄	—	8	1	no	—
310	D₄	—	118	1	no	—
311	D₄	—	118	1	no	—
312	D₄	—	472	1	no	—
313	D₄	—	472	1	no	—
314	D₄	—	472	1	no	—
315	D₄	#11745687	1	2	yes	—
316	D₄	#11745687	1	2	yes	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	59	1	no	—
321	C₂×C₄	—	59	1	no	—
322	C₂×C₄	—	59	1	no	—
323	C₂×C₄	—	59	1	no	—
324	C₂×C₄	—	118	1	no	—
325	C₂×C₄	—	118	1	no	—
326	C₂×C₄	—	118	1	no	—
327	C₂×C₄	—	472	1	no	—
328	C₂×C₄	—	472	1	no	—
329	C₂×C₄	—	472	1	no	—
330	C₂×C₄	—	472	1	no	—
331	C₂×C₄	—	472	1	no	—
332	C₂×C₄	—	472	1	no	—
333	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
334	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₈	—	118	1	no	—
338	C₈	—	118	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	2	no	—
341	C₂²	—	2	2	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	2	1	no	—
344	C₂²	—	4	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	4	1	no	—
349	C₂²	—	4	1	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	8	4	no	—
352	C₂²	—	236	1	no	—
353	C₂²	—	236	1	no	—
354	C₂²	—	236	1	no	—
355	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
356	C₄	—	2	1	no	—
357	C₄	—	4	1	no	—
358	C₄	—	118	1	no	—
359	C₄	—	118	1	no	—
360	C₄	—	118	1	no	—
361	C₄	—	118	1	no	—
362	C₄	—	236	1	no	—
363	C₄	—	236	1	no	—
364	C₄	—	236	1	no	—
365	C₄	C₂×#11745687	1	1	yes	characteristic
366	C₄	C₂×#11745687	1	1	yes	characteristic
367	C₂	—	4	1	no	—
368	C₂	—	4	2	no	—
369	C₂	—	4	2	no	—
370	C₂	—	8	1	no	—
371	C₂	—	472	1	no	—
372	C₂	#16837908	1	1	yes	characteristic
373	C₂	#16838203	1	1	yes	characteristic
374	C₂	#16838206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#447	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#447	D₅₉
hypocenter	C₅₉	#3122
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #447

Complements

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

N	H ≅ G/N	type
1. #16838005	C₂	semidirect
2. #16837877	C₂	semidirect
3. #16837727	C₂	semidirect
4. #16837723	C₂	semidirect
5. #16837704	C₂	semidirect
6. #16837700	C₂	semidirect
7. #16837595	C₂	semidirect
8. #16837595	C₂	semidirect
9. #16837536	C₂	semidirect
10. #16837514	C₂	semidirect
11. #16837511	C₂	semidirect
12. D₈×Dic₅₉	C₂	semidirect
13. #16837482	C₂	semidirect
14. #16837460	C₂	semidirect
15. C₅₉×#447	C₂	semidirect
18. #14273723	C₂²	semidirect
19. #14273718	C₂²	semidirect
31. D₄×Dic₅₉	C₂²	semidirect
32. #14273689	C₂²	semidirect
37. #14273661	C₂²	semidirect
38. #14273660	C₂²	semidirect
39. #14273659	C₂²	semidirect
40. #14273658	C₂²	semidirect
41. Dic₅₉⋊C₈	C₂²	semidirect
42. Dic₅₉⋊C₈	C₂²	semidirect
43. #14273649	C₂²	semidirect
44. #14273649	C₂²	semidirect
47. #14273648	C₂²	semidirect
48. #14273648	C₂²	semidirect
50. #14273637	C₂²	semidirect
51. C₈×Dic₅₉	C₂²	semidirect
197. #447	D₅₉	semidirect
203. C₅₉	#3122	semidirect