Subgroups — #28593125

Order 15104.

738 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432747	C₂	1	2	yes	maximal, complemented
2	#21424990	C₂	1	1	yes	maximal, characteristic
3	#21424985	C₂	1	1	yes	maximal, characteristic
4	#21424850	C₂	1	1	yes	maximal, characteristic, complemented
5	#21424845	C₂	1	1	yes	maximal, characteristic, complemented
6	#21424809	C₂	1	2	yes	maximal, complemented
7	#21424802	C₂	1	2	yes	maximal, complemented
8	#21424002	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#14273654	C₂	1	2	yes	maximal, complemented
10	#97×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₉×#2943	C₂	1	1	yes	maximal, characteristic
12	#16837463	—	2	4	no	—
13	#16837459	—	2	4	no	—
14	C₂×#14273663	C₂²	1	1	yes	characteristic
15	C₂×#14273663	C₂²	1	1	yes	characteristic
16	#16837429	C₂²	1	2	yes	—
17	#16837429	C₂²	1	2	yes	—
18	C₂×#14273654	C₂²	1	2	yes	—
19	C₂×#14273654	C₂²	1	2	yes	—
20	#16837369	C₂²	1	2	yes	—
21	#16837369	C₂²	1	2	yes	—
22	C₄×#11745664	C₂²	1	2	yes	—
23	C₂×#14273638	C₂²	1	1	yes	characteristic
24	C₂×#14273637	C₂²	1	1	yes	characteristic
25	C₂×#14273637	C₂²	1	1	yes	characteristic
26	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
27	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
28	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
29	#16837167	C₂²	1	2	yes	—
30	#16837161	C₂²	1	2	yes	—
31	#16837161	C₂²	1	2	yes	—
32	C₂×#135×C₅₉	C₂²	1	1	yes	characteristic
33	C₄×#41×C₅₉	C₂²	1	2	yes	—
34	C₅₉×#442	C₄	1	2	yes	complemented
35	C₅₉×#441	C₄	1	2	yes	complemented
36	C₅₉×#440	C₄	1	4	yes	complemented
37	C₅₉×#427	C₂²	1	1	yes	characteristic
38	C₅₉×#416	C₂²	1	2	yes	—
39	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
40	C₂×#11745664	C₂³	1	2	yes	—
41	C₂×#11745664	C₂³	1	2	yes	—
42	C₂²×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
44	C₂×C₄×Dic₅₉	—	2	2	no	—
45	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
46	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
47	#14273663	—	2	2	no	—
48	#14273663	—	2	2	no	—
49	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
50	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
51	#14273654	—	2	4	no	—
52	#14273654	—	2	4	no	—
53	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
54	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
55	#14273638	—	2	2	no	—
56	#14273638	D₄	1	4	yes	—
57	#14273637	—	2	2	no	—
58	#14273637	—	2	2	no	—
59	C₄×C₅₉⋊C₈	—	2	2	no	—
60	C₈×Dic₅₉	—	2	2	no	—
61	C₈×Dic₅₉	—	2	2	no	—
62	C₂×#41×C₅₉	C₂³	1	2	yes	—
63	#135×C₅₉	—	2	2	no	—
64	#135×C₅₉	—	2	2	no	—
65	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
66	C₂×SD₁₆×C₅₉	C₂×C₄	1	2	yes	—
67	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₄×Q₈×C₅₉	C₂×C₄	1	4	yes	—
71	C₄×D₄×C₅₉	—	2	4	no	—
72	C₄×D₄×C₅₉	C₂×C₄	1	4	yes	—
73	#117×C₅₉	—	2	2	no	—
74	#117×C₅₉	C₂³	1	2	yes	—
75	C₂×C₄²×C₅₉	—	2	2	no	—
76	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
77	#107×C₅₉	C₂×C₄	1	2	yes	—
78	#106×C₅₉	C₂×C₄	1	2	yes	—
79	#103×C₅₉	C₂×C₄	1	4	yes	—
80	#102×C₅₉	C₂×C₄	1	4	yes	—
81	#97×C₅₉	C₂×C₄	1	2	yes	—
82	#97×C₅₉	C₂×C₄	1	2	yes	—
83	#41×C₅₉	—	2	2	no	—
84	#41×C₅₉	—	2	2	no	—
85	#41×C₅₉	—	2	2	no	—
86	#41×C₅₉	—	2	2	no	—
87	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
88	C₂×D₄×C₅₉	—	2	2	no	—
89	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
90	C₂²×C₄×C₅₉	—	2	2	no	—
91	C₂²×C₄×C₅₉	—	2	4	no	—
92	C₂²×C₄×C₅₉	#41	1	2	yes	—
93	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
94	C₂²×Dic₅₉	—	2	1	no	—
95	C₂²×Dic₅₉	—	2	1	no	—
96	Q₁₆×C₅₉	—	2	2	no	—
97	SD₁₆×C₅₉	—	2	4	no	—
98	D₈×C₅₉	—	2	2	no	—
99	C₂×C₈×C₅₉	—	2	1	no	—
100	C₂×C₈×C₅₉	—	2	1	no	—
101	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
102	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₉	#41	1	2	yes	—
105	C₂×C₈×C₅₉	#41	1	2	yes	—
106	C₄⋊C₄×C₅₉	—	2	4	no	—
107	C₄⋊C₄×C₅₉	C₂²×C₄	1	4	yes	—
108	C₂²⋊C₄×C₅₉	—	2	2	no	—
109	C₂²⋊C₄×C₅₉	—	2	4	no	—
110	C₄²×C₅₉	—	2	4	no	—
111	C₄²×C₅₉	—	2	2	no	—
112	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
113	C₄²×C₅₉	C₂×D₄	1	2	yes	—
114	C₄×Dic₅₉	—	2	1	no	—
115	C₄×Dic₅₉	—	2	1	no	—
116	C₄×Dic₅₉	—	4	4	no	—
117	C₄×Dic₅₉	#41	1	2	yes	—
118	C₄×Dic₅₉	#41	1	2	yes	—
119	#11745664	—	2	2	no	—
120	#11745664	—	2	2	no	—
121	#11745664	—	2	2	no	—
122	#11745664	—	2	2	no	—
123	C₂×C₅₉⋊C₈	—	2	1	no	—
124	C₂×C₅₉⋊C₈	—	2	1	no	—
125	C₂×C₅₉⋊C₈	—	2	1	no	—
126	C₂×C₅₉⋊C₈	—	2	1	no	—
127	C₂×C₅₉⋊C₈	—	2	2	no	—
128	C₂×C₅₉⋊C₈	—	2	2	no	—
129	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
130	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
131	C₂³×C₅₉	—	2	2	no	—
132	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
133	Q₈×C₅₉	—	2	2	no	—
134	Q₈×C₅₉	—	2	2	no	—
135	D₄×C₅₉	—	2	2	no	—
136	D₄×C₅₉	—	2	2	no	—
137	D₄×C₅₉	—	4	2	no	—
138	D₄×C₅₉	—	4	2	no	—
139	C₂×C₄×C₅₉	—	2	4	no	—
140	C₂×C₄×C₅₉	—	2	2	no	—
141	C₂×C₄×C₅₉	—	4	2	no	—
142	C₂×C₄×C₅₉	—	4	2	no	—
143	C₂×C₄×C₅₉	—	4	2	no	—
144	C₂×C₄×C₅₉	—	8	4	no	—
145	C₂×C₄×C₅₉	C₄×D₄	1	4	yes	—
146	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
147	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
152	C₂×Dic₅₉	—	2	4	no	—
153	C₂×Dic₅₉	—	2	2	no	—
154	C₂×Dic₅₉	—	2	2	no	—
155	C₂×Dic₅₉	—	4	2	no	—
156	C₈×C₅₉	—	2	2	no	—
157	C₈×C₅₉	#117	1	2	yes	—
158	C₈×C₅₉	#117	1	2	yes	—
159	C₅₉⋊C₈	—	2	2	no	—
160	C₅₉⋊C₈	—	2	2	no	—
161	C₅₉⋊C₈	—	4	2	no	—
162	C₅₉⋊C₈	—	4	2	no	—
163	#21297	—	59	1	no	maximal, Sylow
164	C₂²×C₅₉	—	2	2	no	—
165	C₂²×C₅₉	—	4	4	no	—
166	C₂²×C₅₉	—	4	2	no	—
167	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
168	C₂²×C₅₉	#529	1	1	yes	characteristic
169	C₂²×C₅₉	#529	1	1	yes	characteristic
170	C₄×C₅₉	—	2	4	no	—
171	C₄×C₅₉	—	4	4	no	—
172	C₄×C₅₉	—	4	2	no	—
173	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
174	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
175	C₄×C₅₉	#532	1	1	yes	characteristic
176	C₄×C₅₉	#532	1	1	yes	characteristic
177	Dic₅₉	—	2	4	no	—
178	Dic₅₉	—	4	2	no	—
179	#2943	D₅₉	1	1	yes	characteristic
180	#2923	—	59	2	no	—
181	#1975	—	59	1	no	—
182	#1974	—	59	1	no	—
183	#1946	—	59	1	no	—
184	#1945	—	59	1	no	—
185	#1938	—	59	2	no	—
186	#1937	—	59	2	no	—
187	#1778	—	59	1	no	—
188	C₄×#104	—	59	2	no	—
189	C₄×#97	—	59	1	no	—
190	C₂×C₅₉	—	2	1	no	—
191	C₂×C₅₉	—	2	1	no	—
192	C₂×C₅₉	—	8	2	no	—
193	C₂×C₅₉	#2293	1	1	yes	characteristic
194	C₂×C₅₉	#2956	1	2	yes	—
195	C₂×#135	C₂×D₅₉	1	1	yes	characteristic
196	C₄×#41	C₂×D₅₉	1	2	yes	—
197	#442	Dic₅₉	1	2	yes	complemented
198	#441	Dic₅₉	1	2	yes	complemented
199	#440	Dic₅₉	1	4	yes	complemented
200	#433	—	118	4	no	—
201	#432	—	118	4	no	—
202	C₂×#108	—	59	1	no	—
203	C₂×#108	—	59	1	no	—
204	#427	C₂×D₅₉	1	1	yes	characteristic
205	#424	—	59	2	no	—
206	#424	—	59	2	no	—
207	C₂×#104	—	59	2	no	—
208	C₂×#104	—	59	2	no	—
209	#416	C₂×D₅₉	1	2	yes	—
210	#408	—	59	2	no	—
211	#408	—	59	2	no	—
212	C₄×M₁₆	—	59	2	no	—
213	C₂×#97	—	59	1	no	—
214	C₂×#97	—	59	1	no	—
215	C₂×#97	—	59	1	no	—
216	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
217	C₂×C₄×C₈	—	59	1	no	—
218	C₂×C₄×C₈	—	59	1	no	—
219	C₄³	—	59	1	no	—
220	#341	—	59	2	no	—
221	#339	—	59	2	no	—
222	#339	—	59	2	no	—
223	C₅₉	#21297	1	1	yes	characteristic, complemented, Sylow
224	C₂×#41	C₂²×D₅₉	1	2	yes	—
225	#135	—	2	2	no	—
226	#135	—	2	2	no	—
227	C₂×Q₁₆	C₂×Dic₅₉	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×Dic₅₉	1	2	yes	—
229	C₂×D₈	C₂×Dic₅₉	1	1	yes	characteristic
230	C₂×M₁₆	—	59	2	no	—
231	C₂×M₁₆	—	59	2	no	—
232	C₂²×C₈	—	59	1	no	—
233	C₂²×C₈	—	59	1	no	—
234	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₄×Q₈	C₂×Dic₅₉	1	4	yes	—
237	C₄×D₄	—	2	4	no	—
238	C₄×D₄	C₂×Dic₅₉	1	4	yes	—
239	#117	—	2	2	no	—
240	#117	C₂²×D₅₉	1	2	yes	—
241	C₂×C₄²	—	2	2	no	—
242	C₂×C₄²	—	59	1	no	—
243	C₂×C₄²	—	59	1	no	—
244	C₂×C₄²	—	118	2	no	—
245	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
246	#108	—	118	2	no	—
247	#108	—	118	2	no	—
248	#107	C₂×Dic₅₉	1	2	yes	—
249	#106	C₂×Dic₅₉	1	2	yes	—
250	C₄⋊C₈	—	118	4	no	—
251	C₄⋊C₈	—	118	4	no	—
252	#104	—	118	4	no	—
253	#104	—	118	4	no	—
254	#103	C₂×Dic₅₉	1	4	yes	—
255	#102	C₂×Dic₅₉	1	4	yes	—
256	C₂²⋊C₈	—	118	4	no	—
257	C₂²⋊C₈	—	118	4	no	—
258	#97	—	59	4	no	—
259	#97	—	118	2	no	—
260	#97	—	118	2	no	—
261	#97	—	118	2	no	—
262	#97	C₂×Dic₅₉	1	2	yes	—
263	#97	C₂×Dic₅₉	1	2	yes	—
264	C₄×C₈	—	118	2	no	—
265	C₄×C₈	—	118	2	no	—
266	C₄×C₈	—	118	2	no	—
267	#41	—	2	2	no	—
268	#41	—	2	2	no	—
269	#41	—	2	2	no	—
270	#41	—	2	2	no	—
271	C₂×Q₈	C₂²×Dic₅₉	1	2	yes	—
272	C₂×D₄	—	2	2	no	—
273	C₂×D₄	C₂²×Dic₅₉	1	2	yes	—
274	C₂²×C₄	—	2	2	no	—
275	C₂²×C₄	—	2	4	no	—
276	C₂²×C₄	—	118	1	no	—
277	C₂²×C₄	—	118	1	no	—
278	C₂²×C₄	#11745687	1	2	yes	—
279	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
280	Q₁₆	—	2	2	no	—
281	SD₁₆	—	2	4	no	—
282	D₈	—	2	2	no	—
283	M₁₆	—	118	2	no	—
284	M₁₆	—	118	2	no	—
285	M₁₆	—	118	2	no	—
286	M₁₆	—	118	2	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	2	1	no	—
289	C₂×C₈	—	59	2	no	—
290	C₂×C₈	—	59	2	no	—
291	C₂×C₈	—	118	1	no	—
292	C₂×C₈	—	118	1	no	—
293	C₂×C₈	—	118	1	no	—
294	C₂×C₈	—	118	1	no	—
295	C₂×C₈	—	118	2	no	—
296	C₂×C₈	—	118	2	no	—
297	C₂×C₈	#11745685	1	2	yes	—
298	C₂×C₈	#11745685	1	2	yes	—
299	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
300	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
301	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
302	C₄⋊C₄	—	2	4	no	—
303	C₄⋊C₄	C₂²×Dic₅₉	1	4	yes	—
304	C₂²⋊C₄	—	2	2	no	—
305	C₂²⋊C₄	—	2	4	no	—
306	C₄²	—	2	4	no	—
307	C₄²	—	2	2	no	—
308	C₄²	—	59	2	no	—
309	C₄²	—	59	2	no	—
310	C₄²	—	118	1	no	—
311	C₄²	—	118	1	no	—
312	C₄²	—	236	4	no	—
313	C₄²	D₄×D₅₉	1	2	yes	—
314	C₄²	C₂²×Dic₅₉	1	2	yes	—
315	C₂³	—	2	2	no	—
316	C₂³	C₂×#11745687	1	1	yes	characteristic
317	Q₈	—	2	2	no	—
318	Q₈	—	2	2	no	—
319	D₄	—	2	2	no	—
320	D₄	—	2	2	no	—
321	D₄	—	4	2	no	—
322	D₄	—	4	2	no	—
323	C₂×C₄	—	2	4	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	8	4	no	—
329	C₂×C₄	—	118	4	no	—
330	C₂×C₄	—	118	2	no	—
331	C₂×C₄	—	118	2	no	—
332	C₂×C₄	—	236	2	no	—
333	C₂×C₄	D₄×Dic₅₉	1	4	yes	—
334	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
335	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
336	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
337	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
338	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
339	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
340	C₈	—	2	2	no	—
341	C₈	—	118	2	no	—
342	C₈	—	118	2	no	—
343	C₈	—	236	2	no	—
344	C₈	—	236	2	no	—
345	C₈	#14273686	1	2	yes	—
346	C₈	#14273686	1	2	yes	—
347	C₂²	—	2	2	no	—
348	C₂²	—	4	2	no	—
349	C₂²	—	4	4	no	—
350	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
351	C₂²	#16837934	1	1	yes	characteristic
352	C₂²	#16837934	1	1	yes	characteristic
353	C₄	—	2	4	no	—
354	C₄	—	4	2	no	—
355	C₄	—	4	4	no	—
356	C₄	—	118	4	no	—
357	C₄	—	236	2	no	—
358	C₄	C₂×#14273686	1	1	yes	characteristic
359	C₄	#41×Dic₅₉	1	1	yes	characteristic
360	C₄	#16837949	1	1	yes	characteristic
361	C₄	#16837949	1	1	yes	characteristic
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	8	2	no	—
365	C₂	#21426561	1	1	yes	characteristic
366	C₂	#21433006	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#2943	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂×#135×C₅₉	C₂²
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	#2943	D₅₉
hypocenter	C₅₉	#21297
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2943

Complements

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

N	H ≅ G/N	type
1. #21432747	C₂	semidirect
4. #21424850	C₂	semidirect
5. #21424845	C₂	semidirect
6. #21424809	C₂	semidirect
7. #21424802	C₂	semidirect
8. #21424002	C₂	semidirect
9. C₄×#14273654	C₂	semidirect
10. #97×Dic₅₉	C₂	semidirect
34. C₅₉×#442	C₄	semidirect
35. C₅₉×#441	C₄	semidirect
36. C₅₉×#440	C₄	semidirect
197. #442	Dic₅₉	semidirect
198. #441	Dic₅₉	semidirect
199. #440	Dic₅₉	semidirect
223. C₅₉	#21297	semidirect

Maximal subgroups

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

subgroup	index
#21432747	2
#21424990	2
#21424985	2
#21424850	2
#21424845	2
#21424809	2
#21424802	2
#21424002	2
C₄×#14273654	2
#97×Dic₅₉	2
C₅₉×#2943	2
#21297	59