Subgroups — #28743784

Order 15104.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434666	C₂	1	1	yes	maximal, characteristic
2	#21426298	C₂	1	1	yes	maximal, characteristic, complemented
3	#21426281	C₂	1	1	yes	maximal, characteristic
4	#21426260	C₂	1	2	yes	maximal, complemented
5	#21426158	C₂	1	2	yes	maximal, complemented
6	#21426030	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426013	C₂	1	1	yes	maximal, characteristic
8	#21425965	C₂	1	2	yes	maximal, complemented
9	#21425958	C₂	1	2	yes	maximal, complemented
10	#21425899	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₄×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837936	—	2	1	no	—
13	C₂×#14273673	C₂²	1	2	yes	—
14	C₂×#14273670	C₂²	1	2	yes	—
15	#16837804	C₂²	1	1	yes	characteristic
16	#16837786	C₂²	1	1	yes	characteristic
17	#16837757	—	2	1	no	—
18	#16837643	C₂²	1	1	yes	characteristic
19	#16837638	C₂²	1	1	yes	characteristic
20	#16837631	—	2	1	no	—
21	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
22	C₄×#11745660	—	2	1	no	—
23	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
24	#16837229	C₂²	1	2	yes	—
25	#16837227	C₂²	1	2	yes	—
26	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
27	#16837206	C₂²	1	2	yes	—
28	#16837206	C₂²	1	2	yes	—
29	#16837205	C₂²	1	2	yes	—
30	#16837202	C₂²	1	2	yes	—
31	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
32	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
33	#16837178	C₂²	1	2	yes	—
34	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
36	C₄×Q₁₆×C₅₉	—	2	1	no	—
37	C₄×Q₁₆×C₅₉	—	2	1	no	—
38	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
39	C₄×Q₁₆×C₅₉	C₂²	1	2	yes	—
40	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
41	C₂×#103×C₅₉	—	2	1	no	—
42	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
43	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
44	C₂×#11745660	—	2	1	no	—
45	C₂×D₄₇₂	C₂³	1	1	yes	characteristic
46	#14273721	—	4	1	no	—
47	#14273719	—	2	1	no	—
48	#14273715	—	2	1	no	—
49	#14273710	—	4	1	no	—
50	#14273706	—	2	1	no	—
51	#14273701	—	2	1	no	—
52	C₄×Dic₁₁₈	—	2	1	no	—
53	C₄×D₂₃₆	—	2	1	no	—
54	C₄×C₂²⋊D₅₉	—	4	1	no	—
55	#14273684	—	4	1	no	—
56	#14273673	—	2	4	no	—
57	#14273670	—	2	4	no	—
58	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
59	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
60	#14273661	C₂³	1	1	yes	characteristic
61	#14273661	C₂³	1	1	yes	characteristic
62	#14273659	—	2	1	no	—
63	#14273652	—	2	1	no	—
64	#14273652	—	2	1	no	—
65	#14273649	—	2	1	no	—
66	#14273649	—	2	1	no	—
67	C₂²×Q₈×C₅₉	—	2	1	no	—
68	C₂×Q₁₆×C₅₉	—	2	4	no	—
69	C₂×Q₁₆×C₅₉	—	2	1	no	—
70	C₂×Q₁₆×C₅₉	C₂³	1	2	yes	—
71	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
72	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄×Q₈×C₅₉	—	2	2	no	—
74	C₄×Q₈×C₅₉	—	2	2	no	—
75	C₄×Q₈×C₅₉	—	4	1	no	—
76	C₄×Q₈×C₅₉	—	4	1	no	—
77	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₅₉	—	4	1	no	—
79	C₂×C₄²×C₅₉	—	4	1	no	—
80	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
81	#107×C₅₉	—	2	1	no	—
82	#107×C₅₉	C₂³	1	2	yes	—
83	#103×C₅₉	—	2	2	no	—
84	#103×C₅₉	—	2	2	no	—
85	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
87	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₅₉	—	2	4	no	—
90	C₂×Q₈×C₅₉	—	2	2	no	—
91	C₂×Q₈×C₅₉	—	4	2	no	—
92	C₂×Q₈×C₅₉	—	4	1	no	—
93	C₂²×C₄×C₅₉	—	4	1	no	—
94	C₂²×C₄×C₅₉	—	4	1	no	—
95	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
96	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
97	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₂²⋊D₅₉	—	4	1	no	—
99	C₂×D₂₃₆	—	2	1	no	—
100	C₂×C₄×D₅₉	—	4	1	no	—
101	C₂×Dic₁₁₈	—	2	1	no	—
102	Q₁₆×C₅₉	—	2	4	no	—
103	Q₁₆×C₅₉	—	2	2	no	—
104	Q₁₆×C₅₉	#41	1	4	yes	—
105	C₂×C₈×C₅₉	—	2	2	no	—
106	C₂×C₈×C₅₉	—	4	1	no	—
107	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
111	C₄⋊C₄×C₅₉	—	2	2	no	—
112	C₄⋊C₄×C₅₉	—	4	1	no	—
113	C₄⋊C₄×C₅₉	—	4	2	no	—
114	C₄⋊C₄×C₅₉	—	4	2	no	—
115	C₄²×C₅₉	—	4	2	no	—
116	C₄²×C₅₉	—	4	2	no	—
117	C₄²×C₅₉	D₈	1	1	yes	characteristic
118	C₄²×C₅₉	D₈	1	1	yes	characteristic
119	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂²⋊Dic₅₉	—	4	1	no	—
122	(C₂×D₅₉)⋊C₄	—	4	1	no	—
123	(C₂×D₅₉)⋊C₄	—	4	1	no	—
124	(C₂×D₅₉)⋊C₄	—	4	1	no	—
125	(C₂×D₅₉)⋊C₄	—	4	1	no	—
126	C₄⋊Dic₅₉	—	2	1	no	—
127	C₄⋊Dic₅₉	—	2	1	no	—
128	Dic₅₉⋊C₄	—	4	1	no	—
129	Dic₅₉⋊C₄	—	4	1	no	—
130	Dic₅₉⋊C₄	—	4	1	no	—
131	Dic₅₉⋊C₄	—	4	1	no	—
132	C₄×Dic₅₉	—	4	1	no	—
133	Dic₂₃₆	—	2	2	no	—
134	D₄₇₂	—	2	2	no	—
135	#11745660	—	4	2	no	—
136	C₅₉⋊C₁₆	—	2	2	no	—
137	C₅₉⋊C₁₆	—	2	2	no	—
138	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₅₉	—	4	1	no	—
140	Q₈×C₅₉	—	2	4	no	—
141	Q₈×C₅₉	—	4	4	no	—
142	Q₈×C₅₉	—	4	2	no	—
143	C₂×C₄×C₅₉	—	2	2	no	—
144	C₂×C₄×C₅₉	—	2	1	no	—
145	C₂×C₄×C₅₉	—	2	1	no	—
146	C₂×C₄×C₅₉	—	4	1	no	—
147	C₂×C₄×C₅₉	—	4	4	no	—
148	C₂×C₄×C₅₉	—	4	2	no	—
149	C₂×C₄×C₅₉	—	4	2	no	—
150	C₂×C₄×C₅₉	—	8	2	no	—
151	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
157	C₂²⋊D₅₉	—	8	2	no	—
158	C₂×Dic₅₉	—	4	1	no	—
159	C₂×Dic₅₉	—	4	1	no	—
160	C₂×Dic₅₉	—	4	1	no	—
161	D₂₃₆	—	4	1	no	—
162	D₂₃₆	—	4	1	no	—
163	C₄×D₅₉	—	8	2	no	—
164	Dic₁₁₈	—	4	1	no	—
165	Dic₁₁₈	—	4	1	no	—
166	C₈×C₅₉	—	2	1	no	—
167	C₈×C₅₉	—	2	1	no	—
168	C₈×C₅₉	—	2	1	no	—
169	C₈×C₅₉	C₂×#41	1	2	yes	—
170	#33677	—	59	1	no	maximal, Sylow
171	C₂²×C₅₉	—	2	1	no	—
172	C₂²×C₅₉	—	2	1	no	—
173	C₂²×C₅₉	#510	1	2	yes	—
174	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₅₉	—	8	1	no	—
176	C₂×D₅₉	—	8	1	no	—
177	C₂×D₅₉	—	8	1	no	—
178	C₄×C₅₉	—	2	1	no	—
179	C₄×C₅₉	—	2	1	no	—
180	C₄×C₅₉	—	2	1	no	—
181	C₄×C₅₉	—	2	1	no	—
182	C₄×C₅₉	—	2	1	no	—
183	C₄×C₅₉	—	4	4	no	—
184	C₄×C₅₉	—	8	2	no	—
185	C₄×C₅₉	#525	1	1	yes	characteristic
186	C₄×C₅₉	#575	1	1	yes	characteristic
187	Dic₅₉	—	8	1	no	—
188	Dic₅₉	—	8	1	no	—
189	Dic₅₉	—	8	1	no	—
190	#3168	—	59	1	no	—
191	C₂×C₄×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
192	#2236	—	59	1	no	—
193	#2233	—	59	1	no	—
194	#2229	—	59	2	no	—
195	#2211	—	59	2	no	—
196	#2188	—	59	1	no	—
197	#2185	—	59	1	no	—
198	#2176	—	59	2	no	—
199	#2175	—	59	2	no	—
200	#2163	—	59	1	no	—
201	C₂×C₅₉	—	2	2	no	—
202	C₂×C₅₉	#3100	1	1	yes	characteristic
203	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
204	C₂×C₅₉	#3414	1	1	yes	characteristic
205	D₅₉	—	16	1	no	—
206	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
207	#529	—	118	1	no	—
208	C₂×C₄×Q₈	—	2	1	no	—
209	C₂×Q₃₂	—	59	2	no	—
210	C₂×SD₃₂	—	59	2	no	—
211	#499	—	59	1	no	—
212	#495	—	59	1	no	—
213	#488	—	118	1	no	—
214	#467	—	59	1	no	—
215	#466	—	59	1	no	—
216	#465	—	118	1	no	—
217	C₄×Q₁₆	—	2	1	no	—
218	C₄×Q₁₆	—	2	1	no	—
219	C₄×Q₁₆	—	59	1	no	—
220	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
221	C₄×Q₁₆	C₂×D₅₉	1	2	yes	—
222	C₄×SD₁₆	—	118	1	no	—
223	C₄×D₈	—	59	1	no	—
224	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
225	C₂×#103	—	2	1	no	—
226	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
227	#367	—	59	2	no	—
228	#366	—	59	2	no	—
229	C₄⋊C₁₆	—	59	2	no	—
230	#358	—	59	2	no	—
231	#358	—	59	2	no	—
232	#358	—	59	2	no	—
233	#357	—	59	2	no	—
234	C₂²⋊C₁₆	—	59	1	no	—
235	C₂²⋊C₁₆	—	59	1	no	—
236	#346	—	59	2	no	—
237	C₅₉	#33677	1	1	yes	characteristic, complemented, Sylow
238	C₂²×Q₈	—	2	1	no	—
239	C₂×Q₁₆	—	2	4	no	—
240	C₂×Q₁₆	—	2	1	no	—
241	C₂×Q₁₆	—	59	1	no	—
242	C₂×Q₁₆	C₂²×D₅₉	1	2	yes	—
243	C₂×SD₁₆	—	118	1	no	—
244	C₂×D₈	—	59	1	no	—
245	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
246	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
247	#126	—	236	1	no	—
248	#125	—	118	1	no	—
249	#124	—	118	1	no	—
250	#123	—	236	1	no	—
251	C₂²⋊Q₈	—	118	1	no	—
252	#121	—	118	1	no	—
253	C₄×Q₈	—	2	2	no	—
254	C₄×Q₈	—	2	2	no	—
255	C₄×Q₈	—	4	1	no	—
256	C₄×Q₈	—	4	1	no	—
257	C₄×Q₈	—	118	1	no	—
258	C₄×D₄	—	118	1	no	—
259	C₄×D₄	—	236	1	no	—
260	#117	—	236	1	no	—
261	C₂×C₄⋊C₄	—	2	1	no	—
262	C₂×C₄⋊C₄	—	4	1	no	—
263	C₂×C₄²	—	4	1	no	—
264	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
265	Q₃₂	—	118	4	no	—
266	SD₃₂	—	118	4	no	—
267	C₂×C₁₆	—	59	2	no	—
268	C₂×C₁₆	—	59	2	no	—
269	#107	—	2	1	no	—
270	#107	—	59	1	no	—
271	#107	—	59	1	no	—
272	#107	C₂²×D₅₉	1	2	yes	—
273	#106	—	118	1	no	—
274	#103	—	2	2	no	—
275	#103	—	2	2	no	—
276	#103	—	118	1	no	—
277	#103	—	118	1	no	—
278	#102	—	118	1	no	—
279	#102	—	118	1	no	—
280	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
281	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
284	C₂×Q₈	—	2	4	no	—
285	C₂×Q₈	—	2	2	no	—
286	C₂×Q₈	—	4	2	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	—	118	1	no	—
289	C₂×D₄	—	118	1	no	—
290	C₂×D₄	—	236	1	no	—
291	C₂²×C₄	—	4	1	no	—
292	C₂²×C₄	—	4	1	no	—
293	C₂²×C₄	—	236	1	no	—
294	C₂²×C₄	#11745669	1	1	yes	characteristic
295	C₂²×C₄	#11745669	1	1	yes	characteristic
296	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
297	Q₁₆	—	2	4	no	—
298	Q₁₆	—	2	2	no	—
299	Q₁₆	—	118	2	no	—
300	Q₁₆	#11745685	1	4	yes	—
301	SD₁₆	—	236	2	no	—
302	D₈	—	118	2	no	—
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	4	1	no	—
305	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	4	1	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	118	1	no	—
314	C₄⋊C₄	—	118	1	no	—
315	C₄⋊C₄	—	236	1	no	—
316	C₄⋊C₄	—	236	1	no	—
317	C₄⋊C₄	—	236	1	no	—
318	C₄⋊C₄	—	236	1	no	—
319	C₂²⋊C₄	—	236	1	no	—
320	C₂²⋊C₄	—	236	1	no	—
321	C₂²⋊C₄	—	236	1	no	—
322	C₂²⋊C₄	—	236	1	no	—
323	C₂²⋊C₄	—	236	1	no	—
324	C₄²	—	4	2	no	—
325	C₄²	—	4	2	no	—
326	C₄²	—	236	1	no	—
327	C₄²	#11745669	1	1	yes	characteristic
328	C₄²	#11745669	1	1	yes	characteristic
329	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
330	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
331	C₁₆	—	118	2	no	—
332	C₁₆	—	118	2	no	—
333	C₂³	—	236	1	no	—
334	C₂³	C₂×#11745669	1	1	yes	characteristic
335	Q₈	—	2	4	no	—
336	Q₈	—	4	4	no	—
337	Q₈	—	4	2	no	—
338	Q₈	—	236	1	no	—
339	Q₈	—	236	1	no	—
340	D₄	—	472	2	no	—
341	D₄	—	236	1	no	—
342	D₄	—	236	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	4	no	—
348	C₂×C₄	—	4	2	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	8	2	no	—
351	C₂×C₄	—	472	2	no	—
352	C₂×C₄	—	236	1	no	—
353	C₂×C₄	—	236	1	no	—
354	C₂×C₄	—	236	1	no	—
355	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
356	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
357	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
358	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
359	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
360	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
361	C₈	—	2	1	no	—
362	C₈	—	2	1	no	—
363	C₈	—	2	1	no	—
364	C₈	C₂×#11745685	1	2	yes	—
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	472	1	no	—
368	C₂²	—	472	1	no	—
369	C₂²	—	472	1	no	—
370	C₂²	#16837843	1	2	yes	—
371	C₂²	C₂²×#11745669	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	4	no	—
378	C₄	—	8	2	no	—
379	C₄	—	472	1	no	—
380	C₄	—	472	1	no	—
381	C₄	—	472	1	no	—
382	C₄	#16837913	1	1	yes	characteristic
383	C₄	#16838200	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	944	1	no	—
386	C₂	#21434096	1	1	yes	characteristic
387	C₂	C₂×#16837843	1	1	yes	characteristic
388	C₂	#21437230	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₄×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	#16837638	C₂²
hypercenter	C₂×C₄×Q₁₆	D₅₉
hypocenter	C₅₉	#33677
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #21426298	C₂	semidirect
4. #21426260	C₂	semidirect
5. #21426158	C₂	semidirect
6. #21426030	C₂	semidirect
8. #21425965	C₂	semidirect
9. #21425958	C₂	semidirect
10. #21425899	C₂	semidirect
11. C₂×C₄×Q₁₆×C₅₉	C₂	semidirect
191. C₂×C₄×Q₁₆	D₅₉	semidirect
237. C₅₉	#33677	semidirect

Maximal subgroups

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

subgroup	index
#21434666	2
#21426298	2
#21426281	2
#21426260	2
#21426158	2
#21426030	2
#21426013	2
#21425965	2
#21425958	2
#21425899	2
C₂×C₄×Q₁₆×C₅₉	2
#33677	59