Subgroups — #28559381

Order 15104.

592 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434669	C₂	1	1	yes	maximal, characteristic
2	#21433244	C₂	1	2	yes	maximal, complemented
3	#21423001	C₂	1	2	yes	maximal, complemented
4	#21422993	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422980	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422906	C₂	1	2	yes	maximal, complemented
7	C₈×D₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
8	#21422853	C₂	1	1	yes	maximal, characteristic, complemented
9	#21422560	C₂	1	2	yes	maximal, complemented
10	#21422378	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837939	C₂²	1	2	yes	—
13	#16837803	C₄	1	1	yes	characteristic
14	#16837787	C₄	1	1	yes	characteristic
15	#16837758	C₄	1	2	yes	—
16	#16837640	C₄	1	1	yes	characteristic
17	#16837639	C₄	1	1	yes	characteristic
18	#16837634	C₄	1	2	yes	—
19	#16837533	C₂²	1	2	yes	—
20	C₄×#11745671	C₂²	1	2	yes	—
21	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
22	#16837481	—	2	2	no	—
23	#16837479	C₂²	1	2	yes	—
24	#16837472	—	2	2	no	—
25	C₈×D₂₃₆	C₂²	1	2	yes	—
26	#16837461	—	2	2	no	—
27	#16837453	C₂²	1	1	yes	characteristic
28	#16837453	C₂²	1	1	yes	characteristic
29	#16837427	—	2	2	no	—
30	#16837424	C₂²	1	2	yes	—
31	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
32	#16837155	C₂²	1	1	yes	characteristic
33	#16837154	C₂²	1	2	yes	—
34	Q₈⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
35	#16837131	C₂²	1	2	yes	—
36	#16837129	C₂²	1	2	yes	—
37	#16837124	C₂²	1	1	yes	characteristic
38	#16837124	C₂²	1	1	yes	characteristic
39	C₈×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₅₉	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
42	C₅₉×#335	C₂²	1	1	yes	characteristic
43	C₅₉×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₅₉	C₂²	1	1	yes	characteristic
45	C₂×#11745671	C₂×C₄	1	2	yes	—
46	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
47	C₂×#11745659	—	2	2	no	—
48	C₂×#11745659	—	2	2	no	—
49	C₂×C₈×D₅₉	—	2	2	no	—
50	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₅₉	C₂×C₄	1	1	yes	characteristic
52	#14273722	—	2	2	no	—
53	#14273718	C₂×C₄	1	2	yes	—
54	#14273716	C₂×C₄	1	2	yes	—
55	#14273712	—	2	2	no	—
56	#14273703	C₂×C₄	1	2	yes	—
57	#14273702	C₂×C₄	1	2	yes	—
58	C₄×D₂₃₆	—	2	2	no	—
59	C₄×D₂₃₆	C₂³	1	2	yes	—
60	#14273684	—	2	2	no	—
61	C₄²×D₅₉	C₂³	1	1	yes	characteristic
62	#14273661	C₂×C₄	1	1	yes	characteristic
63	#14273658	C₂×C₄	1	2	yes	—
64	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
65	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
66	Dic₅₉⋊C₈	—	2	2	no	—
67	Dic₅₉⋊C₈	—	2	1	no	—
68	Dic₅₉⋊C₈	D₄	1	1	yes	characteristic
69	Dic₅₉⋊C₈	D₄	1	1	yes	characteristic
70	#14273651	C₂×C₄	1	2	yes	—
71	#14273649	C₂×C₄	1	2	yes	—
72	#14273647	C₂×C₄	1	2	yes	—
73	(C₂×D₅₉)⋊C₈	—	2	1	no	—
74	(C₂×D₅₉)⋊C₈	—	2	2	no	—
75	(C₂×D₅₉)⋊C₈	—	2	2	no	—
76	(C₂×D₅₉)⋊C₈	D₄	1	1	yes	characteristic
77	(C₂×D₅₉)⋊C₈	D₄	1	1	yes	characteristic
78	#14273638	—	2	2	no	—
79	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
80	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
82	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
83	C₈×Dic₅₉	C₂×C₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
86	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₅₉	—	2	2	no	—
88	C₄⋊C₈×C₅₉	C₂³	1	2	yes	—
89	#103×C₅₉	C₂×C₄	1	2	yes	—
90	C₄×C₈×C₅₉	—	2	2	no	—
91	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
95	C₂×D₂₃₆	—	2	2	no	—
96	C₂×D₂₃₆	C₂²×C₄	1	2	yes	—
97	C₂×C₄×D₅₉	—	2	2	no	—
98	C₂×C₄×D₅₉	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×D₅₉	#41	1	1	yes	characteristic
101	Q₁₆×C₅₉	—	2	2	no	—
102	C₂×C₈×C₅₉	—	2	2	no	—
103	C₂×C₈×C₅₉	—	2	1	no	—
104	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₄⋊C₄×C₅₉	—	2	2	no	—
109	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
110	C₄²×C₅₉	—	2	2	no	—
111	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
112	#11745671	—	2	4	no	—
113	(C₂×D₅₉)⋊C₄	—	2	2	no	—
114	(C₂×D₅₉)⋊C₄	—	2	2	no	—
115	(C₂×D₅₉)⋊C₄	—	2	2	no	—
116	(C₂×D₅₉)⋊C₄	—	2	2	no	—
117	(C₂×D₅₉)⋊C₄	—	2	2	no	—
118	C₄⋊Dic₅₉	—	2	2	no	—
119	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
120	Dic₅₉⋊C₄	—	2	2	no	—
121	Dic₅₉⋊C₄	—	2	2	no	—
122	C₄×Dic₅₉	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₅₉	#41	1	1	yes	characteristic
125	C₂×C₅₉⋊C₈	—	2	2	no	—
126	C₂×C₅₉⋊C₈	—	2	1	no	—
127	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₅₉⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
131	D₄₇₂	—	2	2	no	—
132	#11745659	—	8	2	no	—
133	#11745659	—	8	2	no	—
134	C₈×D₅₉	—	2	2	no	—
135	C₈×D₅₉	—	2	2	no	—
136	C₈×D₅₉	—	8	2	no	—
137	C₂²×D₅₉	—	2	2	no	—
138	C₂²×D₅₉	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₅₉	—	2	2	no	—
140	Q₈×C₅₉	—	2	2	no	—
141	C₂×C₄×C₅₉	—	2	2	no	—
142	C₂×C₄×C₅₉	—	2	2	no	—
143	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₅₉	—	2	2	no	—
147	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
148	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₅₉	C₄×D₄	1	1	yes	characteristic
150	D₂₃₆	—	2	2	no	—
151	D₂₃₆	—	2	2	no	—
152	D₂₃₆	—	8	2	no	—
153	C₄×D₅₉	—	2	1	no	—
154	C₄×D₅₉	—	2	1	no	—
155	C₄×D₅₉	—	2	1	no	—
156	C₄×D₅₉	—	2	1	no	—
157	C₄×D₅₉	—	4	1	no	—
158	C₄×D₅₉	—	4	2	no	—
159	C₄×D₅₉	—	4	2	no	—
160	C₈×C₅₉	—	2	1	no	—
161	C₈×C₅₉	—	2	1	no	—
162	C₈×C₅₉	—	2	1	no	—
163	C₈×C₅₉	—	4	1	no	—
164	C₈×C₅₉	—	4	2	no	—
165	C₈×C₅₉	#131	1	2	yes	—
166	C₅₉⋊C₈	—	2	1	no	—
167	C₅₉⋊C₈	—	2	1	no	—
168	C₅₉⋊C₈	—	2	1	no	—
169	C₅₉⋊C₈	—	4	1	no	—
170	C₅₉⋊C₈	—	4	2	no	—
171	C₅₉⋊C₈	#131	1	2	yes	—
172	#17973	—	59	1	no	maximal, Sylow
173	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
174	C₂×D₅₉	—	2	1	no	—
175	C₂×D₅₉	—	2	1	no	—
176	C₂×D₅₉	—	2	1	no	—
177	C₂×D₅₉	—	4	2	no	—
178	C₂×D₅₉	—	4	2	no	—
179	C₂×D₅₉	—	4	2	no	—
180	C₄×C₅₉	—	2	1	no	—
181	C₄×C₅₉	—	4	2	no	—
182	C₄×C₅₉	—	4	2	no	—
183	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
184	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
185	C₄×C₅₉	#576	1	1	yes	characteristic
186	C₄×C₅₉	#578	1	1	yes	characteristic
187	Dic₅₉	—	2	1	no	—
188	Dic₅₉	—	2	1	no	—
189	Dic₅₉	—	2	1	no	—
190	Dic₅₉	—	4	2	no	—
191	#3168	—	59	1	no	—
192	#2993	—	59	2	no	—
193	#1592	—	59	2	no	—
194	#1591	—	59	1	no	—
195	#1589	—	59	1	no	—
196	SD₁₆⋊C₈	—	59	2	no	—
197	C₈×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
198	C₈×D₈	—	59	1	no	—
199	#1567	—	59	1	no	—
200	#1495	—	59	2	no	—
201	#1450	—	59	1	no	—
202	C₂×C₅₉	#2931	1	1	yes	characteristic
203	C₂×C₅₉	#2951	1	1	yes	characteristic
204	C₂×C₅₉	#2957	1	1	yes	characteristic
205	D₅₉	—	4	1	no	—
206	D₅₉	—	8	2	no	—
207	#529	—	59	2	no	—
208	#499	—	59	1	no	—
209	#495	—	59	1	no	—
210	#488	—	59	2	no	—
211	#467	—	59	1	no	—
212	#466	—	59	1	no	—
213	#465	—	59	2	no	—
214	#446	—	59	2	no	—
215	C₈×Q₈	C₂×D₅₉	1	2	yes	—
216	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
217	C₄×SD₁₆	—	59	2	no	—
218	C₄×D₈	—	59	1	no	—
219	#436	—	59	2	no	—
220	#436	—	118	2	no	—
221	#435	—	118	2	no	—
222	C₈×D₄	—	59	2	no	—
223	#433	—	118	2	no	—
224	#431	—	59	1	no	—
225	#431	—	59	1	no	—
226	#424	—	118	2	no	—
227	#423	—	59	2	no	—
228	C₂×C₄×C₈	—	59	1	no	—
229	#335	—	59	2	no	—
230	#335	—	59	1	no	—
231	#335	C₂×D₅₉	1	1	yes	characteristic
232	Q₈⋊C₈	—	59	2	no	—
233	Q₈⋊C₈	C₂×D₅₉	1	2	yes	—
234	D₄⋊C₈	—	59	2	no	—
235	D₄⋊C₈	—	59	2	no	—
236	#322	—	59	1	no	—
237	#322	—	59	1	no	—
238	C₈²	—	59	1	no	—
239	C₈²	C₂×D₅₉	1	1	yes	characteristic
240	C₅₉	#17973	1	1	yes	characteristic, complemented, Sylow
241	C₂×Q₁₆	C₄×D₅₉	1	1	yes	characteristic
242	C₂×SD₁₆	—	59	2	no	—
243	C₂×D₈	—	59	1	no	—
244	C₂×M₁₆	—	118	2	no	—
245	C₂×M₁₆	—	118	2	no	—
246	C₂²×C₈	—	59	1	no	—
247	C₂²×C₈	—	59	1	no	—
248	C₂²×C₈	—	118	2	no	—
249	#126	—	118	2	no	—
250	#125	—	59	2	no	—
251	#124	—	59	2	no	—
252	#123	—	118	2	no	—
253	C₂²⋊Q₈	—	59	2	no	—
254	#121	—	59	2	no	—
255	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
256	C₄×D₄	—	59	2	no	—
257	C₄×D₄	—	118	2	no	—
258	#117	—	118	2	no	—
259	C₂×C₄²	—	59	1	no	—
260	#107	—	59	1	no	—
261	#107	C₄×D₅₉	1	1	yes	characteristic
262	#106	—	59	2	no	—
263	C₄⋊C₈	—	2	2	no	—
264	C₄⋊C₈	—	59	1	no	—
265	C₄⋊C₈	—	59	1	no	—
266	C₄⋊C₈	—	59	2	no	—
267	C₄⋊C₈	—	118	2	no	—
268	C₄⋊C₈	—	118	2	no	—
269	C₄⋊C₈	—	118	1	no	—
270	C₄⋊C₈	C₂²×D₅₉	1	2	yes	—
271	#103	—	59	2	no	—
272	#103	C₄×D₅₉	1	2	yes	—
273	#102	—	59	2	no	—
274	#102	—	59	2	no	—
275	C₂²⋊C₈	—	59	1	no	—
276	C₂²⋊C₈	—	59	1	no	—
277	C₂²⋊C₈	—	118	2	no	—
278	C₂²⋊C₈	—	118	1	no	—
279	C₂²⋊C₈	—	118	2	no	—
280	#97	—	118	2	no	—
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	—	59	1	no	—
283	C₄×C₈	—	59	1	no	—
284	C₄×C₈	—	59	1	no	—
285	C₄×C₈	—	59	1	no	—
286	C₄×C₈	—	59	1	no	—
287	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
290	C₂×Q₈	C₂×C₄×D₅₉	1	2	yes	—
291	C₂×D₄	—	59	2	no	—
292	C₂×D₄	—	118	2	no	—
293	C₂²×C₄	—	59	1	no	—
294	C₂²×C₄	—	59	1	no	—
295	C₂²×C₄	—	59	1	no	—
296	C₂²×C₄	—	118	2	no	—
297	Q₁₆	—	2	2	no	—
298	SD₁₆	—	118	4	no	—
299	D₈	—	118	2	no	—
300	M₁₆	—	472	2	no	—
301	M₁₆	—	472	2	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	59	1	no	—
305	C₂×C₈	—	59	1	no	—
306	C₂×C₈	—	59	1	no	—
307	C₂×C₈	—	59	1	no	—
308	C₂×C₈	—	118	2	no	—
309	C₂×C₈	—	118	2	no	—
310	C₂×C₈	—	118	2	no	—
311	C₂×C₈	—	118	1	no	—
312	C₂×C₈	—	472	2	no	—
313	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
314	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
315	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
316	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	59	2	no	—
319	C₄⋊C₄	—	118	2	no	—
320	C₄⋊C₄	—	118	2	no	—
321	C₄⋊C₄	—	118	2	no	—
322	C₄⋊C₄	C₂×C₄×D₅₉	1	2	yes	—
323	C₂²⋊C₄	—	118	2	no	—
324	C₂²⋊C₄	—	118	2	no	—
325	C₂²⋊C₄	—	118	2	no	—
326	C₂²⋊C₄	—	118	2	no	—
327	C₂²⋊C₄	—	118	2	no	—
328	C₄²	—	2	2	no	—
329	C₄²	—	59	1	no	—
330	C₄²	—	59	1	no	—
331	C₄²	—	59	1	no	—
332	C₄²	C₂³×D₅₉	1	1	yes	characteristic
333	C₂³	—	59	1	no	—
334	C₂³	—	118	2	no	—
335	Q₈	—	2	2	no	—
336	Q₈	—	2	2	no	—
337	D₄	—	118	2	no	—
338	D₄	—	118	2	no	—
339	D₄	—	472	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	2	no	—
342	C₂×C₄	—	59	1	no	—
343	C₂×C₄	—	59	1	no	—
344	C₂×C₄	—	59	1	no	—
345	C₂×C₄	—	118	1	no	—
346	C₂×C₄	—	118	1	no	—
347	C₂×C₄	—	118	1	no	—
348	C₂×C₄	—	118	1	no	—
349	C₂×C₄	—	118	2	no	—
350	C₂×C₄	—	236	1	no	—
351	C₂×C₄	—	236	2	no	—
352	C₂×C₄	—	236	2	no	—
353	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
355	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	2	no	—
361	C₈	—	59	2	no	—
362	C₈	—	118	1	no	—
363	C₈	—	118	1	no	—
364	C₈	—	118	1	no	—
365	C₈	—	236	1	no	—
366	C₈	—	236	2	no	—
367	C₈	#14273733	1	2	yes	—
368	C₂²	—	118	1	no	—
369	C₂²	—	118	1	no	—
370	C₂²	—	118	1	no	—
371	C₂²	—	236	2	no	—
372	C₂²	—	236	2	no	—
373	C₂²	—	236	2	no	—
374	C₂²	C₄×D₄×D₅₉	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	4	2	no	—
377	C₄	—	4	2	no	—
378	C₄	—	118	1	no	—
379	C₄	—	118	1	no	—
380	C₄	—	118	1	no	—
381	C₄	—	236	2	no	—
382	C₄	#131×D₅₉	1	1	yes	characteristic
383	C₄	C₂×#14273733	1	1	yes	characteristic
384	C₄	#16838204	1	1	yes	characteristic
385	C₄	#16838215	1	1	yes	characteristic
386	C₂	—	472	2	no	—
387	C₂	—	236	1	no	—
388	C₂	#21432803	1	1	yes	characteristic
389	C₂	#21432971	1	1	yes	characteristic
390	C₂	#21433008	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₈×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	#16837639	C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₈×Q₁₆	D₅₉
hypocenter	C₅₉	#17973
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #21433244	C₂	semidirect
3. #21423001	C₂	semidirect
4. #21422993	C₂	semidirect
5. #21422980	C₂	semidirect
6. #21422906	C₂	semidirect
7. C₈×D₄₇₂	C₂	semidirect
8. #21422853	C₂	semidirect
9. #21422560	C₂	semidirect
10. #21422378	C₂	semidirect
11. C₈×Q₁₆×C₅₉	C₂	semidirect
197. C₈×Q₁₆	D₅₉	semidirect
240. C₅₉	#17973	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#21434669	2
#21433244	2
#21423001	2
#21422993	2
#21422980	2
#21422906	2
C₈×D₄₇₂	2
#21422853	2
#21422560	2
#21422378	2
C₈×Q₁₆×C₅₉	2
#17973	59