Subgroups — #28488019

Order 15104.

510 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837750	C₂	1	1	yes	maximal, characteristic
2	#21425023	C₂	1	1	yes	maximal, characteristic
3	#21423965	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837154	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16837151	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#16837145	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1987	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273724	C₂²	1	1	yes	characteristic
9	#16837750	—	2	2	no	—
10	#16837750	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#14273647	—	2	1	no	—
14	C₂×#14273647	—	2	1	no	—
15	C₂×#14273647	—	2	2	no	—
16	C₂×#14273638	—	2	1	no	—
17	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	#16837157	—	2	1	no	—
19	#16837154	—	2	2	no	—
20	#16837151	—	2	2	no	—
21	#16837145	—	2	2	no	—
22	C₂×C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#340	—	2	2	no	—
26	C₂²×D₂₃₆	—	2	1	no	—
27	C₂²×D₂₃₆	—	2	1	no	—
28	C₂²×D₂₃₆	—	2	1	no	—
29	C₂²×C₅₉⋊C₈	—	2	1	no	—
30	C₂²×C₅₉⋊C₈	—	2	1	no	—
31	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
32	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	#14273724	—	2	1	no	—
34	#14273724	—	2	1	no	—
35	#14273724	—	2	1	no	—
36	#14273724	C₂×C₄	1	2	yes	—
37	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
38	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
39	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
40	C₄⋊(C₅₉⋊C₈)	D₄	1	2	yes	—
41	#14273647	—	2	4	no	—
42	#14273647	—	2	4	no	—
43	#14273647	—	4	4	no	—
44	#14273638	—	4	2	no	—
45	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
46	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
47	C₂²×Q₈×C₅₉	—	2	1	no	—
48	C₂²×C₈×C₅₉	—	2	1	no	—
49	C₄⋊Q₈×C₅₉	—	2	1	no	—
50	C₄⋊Q₈×C₅₉	—	2	1	no	—
51	C₄⋊Q₈×C₅₉	—	2	1	no	—
52	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₅₉	D₄	1	2	yes	—
56	C₂×C₄²×C₅₉	—	2	1	no	—
57	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₅₉	—	2	1	no	—
59	C₄⋊C₈×C₅₉	—	2	1	no	—
60	#95×C₅₉	—	2	1	no	—
61	C₂³×D₅₉	—	4	1	no	—
62	C₂×Q₈×C₅₉	—	2	2	no	—
63	C₂×Q₈×C₅₉	—	4	1	no	—
64	C₂×Q₈×C₅₉	—	8	2	no	—
65	C₂²×C₄×C₅₉	—	2	2	no	—
66	C₂²×C₄×C₅₉	—	4	1	no	—
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
70	C₂×D₂₃₆	—	2	2	no	—
71	C₂×D₂₃₆	—	2	2	no	—
72	C₂×D₂₃₆	—	2	2	no	—
73	C₂×D₂₃₆	—	4	1	no	—
74	C₂×D₂₃₆	—	4	1	no	—
75	C₂×D₂₃₆	—	4	2	no	—
76	C₂×D₂₃₆	—	4	1	no	—
77	C₂×D₂₃₆	—	4	1	no	—
78	C₂×D₂₃₆	—	4	2	no	—
79	C₂×D₂₃₆	—	4	1	no	—
80	C₂×D₂₃₆	—	4	1	no	—
81	C₂×D₂₃₆	—	4	1	no	—
82	C₂×D₂₃₆	—	8	2	no	—
83	C₂×C₈×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	—	4	2	no	—
85	C₄⋊C₄×C₅₉	—	2	2	no	—
86	C₄⋊C₄×C₅₉	—	2	2	no	—
87	C₄⋊C₄×C₅₉	—	2	2	no	—
88	C₄⋊C₄×C₅₉	—	2	2	no	—
89	C₄⋊C₄×C₅₉	—	4	1	no	—
90	C₄⋊C₄×C₅₉	—	4	1	no	—
91	C₄²×C₅₉	—	4	1	no	—
92	C₄²×C₅₉	—	4	1	no	—
93	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
95	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₉	#41	1	1	yes	characteristic
97	C₂×C₅₉⋊C₈	—	2	1	no	—
98	C₂×C₅₉⋊C₈	—	2	1	no	—
99	C₂×C₅₉⋊C₈	—	2	1	no	—
100	C₂×C₅₉⋊C₈	—	2	1	no	—
101	C₂×C₅₉⋊C₈	—	2	2	no	—
102	C₂×C₅₉⋊C₈	—	4	1	no	—
103	C₂×C₅₉⋊C₈	—	4	1	no	—
104	C₂×C₅₉⋊C₈	—	4	1	no	—
105	C₂×C₅₉⋊C₈	—	4	2	no	—
106	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
107	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
108	C₂³×C₅₉	#121	1	1	yes	characteristic
109	C₂²×D₅₉	—	4	2	no	—
110	C₂²×D₅₉	—	8	1	no	—
111	C₂²×D₅₉	—	8	1	no	—
112	C₂²×D₅₉	—	8	2	no	—
113	C₂²×D₅₉	—	8	2	no	—
114	Q₈×C₅₉	—	8	2	no	—
115	Q₈×C₅₉	—	8	2	no	—
116	C₂×C₄×C₅₉	—	2	2	no	—
117	C₂×C₄×C₅₉	—	4	2	no	—
118	C₂×C₄×C₅₉	—	4	2	no	—
119	C₂×C₄×C₅₉	—	4	2	no	—
120	C₂×C₄×C₅₉	—	4	2	no	—
121	C₂×C₄×C₅₉	—	8	2	no	—
122	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
126	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
127	C₂×C₄×C₅₉	#127	1	1	yes	characteristic
128	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
129	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
130	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
131	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₅₉	#137	1	2	yes	—
133	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
134	D₂₃₆	—	4	2	no	—
135	D₂₃₆	—	4	2	no	—
136	D₂₃₆	—	4	4	no	—
137	D₂₃₆	—	8	2	no	—
138	D₂₃₆	—	8	2	no	—
139	D₂₃₆	—	8	1	no	—
140	D₂₃₆	—	8	1	no	—
141	D₂₃₆	—	8	1	no	—
142	D₂₃₆	—	8	1	no	—
143	C₈×C₅₉	—	8	1	no	—
144	C₅₉⋊C₈	—	2	2	no	—
145	C₅₉⋊C₈	—	2	2	no	—
146	C₅₉⋊C₈	—	4	2	no	—
147	C₅₉⋊C₈	—	8	1	no	—
148	#9747	—	59	1	no	maximal, Sylow
149	C₂²×C₅₉	#390	1	1	yes	characteristic
150	C₂²×C₅₉	#463	1	1	yes	characteristic
151	C₂²×C₅₉	#464	1	1	yes	characteristic
152	C₂²×C₅₉	#468	1	2	yes	—
153	C₂²×C₅₉	#472	1	2	yes	—
154	C₂×D₅₉	—	8	2	no	—
155	C₂×D₅₉	—	8	2	no	—
156	C₂×D₅₉	—	8	2	no	—
157	C₂×D₅₉	—	16	2	no	—
158	C₂×D₅₉	—	16	2	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	2	1	no	—
161	C₄×C₅₉	—	8	2	no	—
162	C₄×C₅₉	—	8	2	no	—
163	C₄×C₅₉	#416	1	1	yes	characteristic
164	C₄×C₅₉	#417	1	1	yes	characteristic
165	C₄×C₅₉	#440	1	2	yes	—
166	C₄×C₅₉	#463	1	1	yes	characteristic
167	C₄×C₅₉	#464	1	1	yes	characteristic
168	C₄×C₅₉	#496	1	1	yes	characteristic
169	C₄×C₅₉	#497	1	1	yes	characteristic
170	C₂×#486	—	59	1	no	—
171	#1987	D₅₉	1	1	yes	characteristic, complemented
172	#1982	—	59	1	no	—
173	#1770	—	59	1	no	—
174	C₂×#335	—	59	1	no	—
175	C₂×#333	—	59	1	no	—
176	C₂×#331	—	59	1	no	—
177	C₂×C₅₉	#1697	1	2	yes	—
178	C₂×C₅₉	#1698	1	2	yes	—
179	C₂×C₅₉	#1973	1	1	yes	characteristic
180	C₂×C₅₉	#1978	1	1	yes	characteristic
181	C₂×C₅₉	#1983	1	1	yes	characteristic
182	D₅₉	—	16	2	no	—
183	C₂×C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
184	C₂×#127	—	59	1	no	—
185	#486	—	59	4	no	—
186	#486	—	118	2	no	—
187	C₂×C₄⋊C₈	—	59	1	no	—
188	C₂×C₄⋊C₈	—	59	1	no	—
189	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
190	C₂×#102	—	118	1	no	—
191	C₂×#102	—	118	1	no	—
192	C₂×#102	—	118	2	no	—
193	C₂×#97	—	118	1	no	—
194	C₂×C₄×C₈	—	59	1	no	—
195	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
196	#340	—	2	2	no	—
197	#336	—	118	1	no	—
198	#335	—	118	2	no	—
199	#333	—	118	2	no	—
200	#331	—	118	2	no	—
201	C₅₉	#9747	1	1	yes	characteristic, complemented, Sylow
202	C₂²×Q₈	—	2	1	no	—
203	C₂²×D₄	—	118	1	no	—
204	C₂²×D₄	—	118	1	no	—
205	C₂²×D₄	—	118	1	no	—
206	C₂²×C₈	—	2	1	no	—
207	C₂²×C₈	—	59	1	no	—
208	C₂²×C₈	—	59	1	no	—
209	C₂²×C₈	—	118	1	no	—
210	C₂²×C₈	—	118	1	no	—
211	C₄⋊Q₈	—	2	1	no	—
212	C₄⋊Q₈	—	2	1	no	—
213	C₄⋊Q₈	—	2	1	no	—
214	C₄⋊Q₈	C₄×D₅₉	1	2	yes	—
215	#127	—	59	2	no	—
216	#127	—	118	1	no	—
217	#127	—	118	1	no	—
218	#127	—	118	1	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	C₂²⋊D₅₉	1	2	yes	—
222	C₂×C₄²	—	2	1	no	—
223	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
224	C₄⋊C₈	—	2	1	no	—
225	C₄⋊C₈	—	2	1	no	—
226	C₄⋊C₈	—	59	2	no	—
227	C₄⋊C₈	—	118	1	no	—
228	C₄⋊C₈	—	118	1	no	—
229	C₄⋊C₈	—	118	1	no	—
230	#102	—	118	4	no	—
231	#102	—	118	4	no	—
232	#102	—	236	4	no	—
233	#97	—	236	2	no	—
234	C₄×C₈	—	59	2	no	—
235	C₄×C₈	—	59	2	no	—
236	#95	—	2	1	no	—
237	C₂⁴	—	236	1	no	—
238	C₂×Q₈	—	2	2	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	8	2	no	—
241	C₂×D₄	—	118	2	no	—
242	C₂×D₄	—	118	2	no	—
243	C₂×D₄	—	118	2	no	—
244	C₂×D₄	—	472	2	no	—
245	C₂×D₄	—	236	1	no	—
246	C₂×D₄	—	236	1	no	—
247	C₂×D₄	—	236	1	no	—
248	C₂×D₄	—	236	1	no	—
249	C₂×D₄	—	236	1	no	—
250	C₂×D₄	—	236	2	no	—
251	C₂×D₄	—	236	1	no	—
252	C₂×D₄	—	236	2	no	—
253	C₂×D₄	—	236	1	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	1	no	—
256	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
257	C₂²×C₄	#11745687	1	1	yes	characteristic
258	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	2	no	—
261	C₂×C₈	—	59	2	no	—
262	C₂×C₈	—	59	2	no	—
263	C₂×C₈	—	118	1	no	—
264	C₂×C₈	—	118	1	no	—
265	C₂×C₈	—	118	1	no	—
266	C₂×C₈	—	118	1	no	—
267	C₂×C₈	—	118	2	no	—
268	C₂×C₈	—	236	1	no	—
269	C₂×C₈	—	236	2	no	—
270	C₂×C₈	—	236	1	no	—
271	C₂×C₈	—	236	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
281	C₄²	#11745685	1	1	yes	characteristic
282	C₄²	D₄×D₅₉	1	1	yes	characteristic
283	C₄²	D₄×D₅₉	1	1	yes	characteristic
284	C₂³	—	472	2	no	—
285	C₂³	—	472	2	no	—
286	C₂³	—	472	1	no	—
287	C₂³	—	472	1	no	—
288	C₂³	—	236	2	no	—
289	C₂³	#14273699	1	1	yes	characteristic
290	Q₈	—	8	2	no	—
291	Q₈	—	8	2	no	—
292	D₄	—	472	2	no	—
293	D₄	—	472	2	no	—
294	D₄	—	472	1	no	—
295	D₄	—	472	1	no	—
296	D₄	—	472	1	no	—
297	D₄	—	472	1	no	—
298	D₄	—	236	2	no	—
299	D₄	—	236	2	no	—
300	D₄	—	236	4	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	2	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	2	no	—
307	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
310	C₂×C₄	#14273700	1	1	yes	characteristic
311	C₂×C₄	#14273713	1	1	yes	characteristic
312	C₂×C₄	#14273723	1	1	yes	characteristic
313	C₂×C₄	#14273748	1	1	yes	characteristic
314	C₂×C₄	#14273750	1	1	yes	characteristic
315	C₂×C₄	#14273752	1	2	yes	—
316	C₂×C₄	#14273754	1	1	yes	characteristic
317	C₂×C₄	#14273758	1	2	yes	—
318	C₂×C₄	#14273759	1	1	yes	characteristic
319	C₈	—	8	1	no	—
320	C₈	—	118	2	no	—
321	C₈	—	118	2	no	—
322	C₈	—	472	1	no	—
323	C₈	—	236	2	no	—
324	C₂²	—	944	2	no	—
325	C₂²	—	944	2	no	—
326	C₂²	—	472	2	no	—
327	C₂²	—	472	2	no	—
328	C₂²	—	472	2	no	—
329	C₂²	#16837302	1	1	yes	characteristic
330	C₂²	#16837619	1	1	yes	characteristic
331	C₂²	#16837626	1	1	yes	characteristic
332	C₂²	#16837650	1	2	yes	—
333	C₂²	#16837670	1	2	yes	—
334	C₄	—	2	1	no	—
335	C₄	—	2	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	2	no	—
338	C₄	#16837397	1	1	yes	characteristic
339	C₄	#16837402	1	1	yes	characteristic
340	C₄	#16837500	1	2	yes	—
341	C₄	#16837620	1	1	yes	characteristic
342	C₄	#16837627	1	1	yes	characteristic
343	C₄	#16837792	1	1	yes	characteristic
344	C₄	#16837794	1	1	yes	characteristic
345	C₂	—	944	2	no	—
346	C₂	#21423631	1	2	yes	—
347	C₂	#21423638	1	2	yes	—
348	C₂	#21424980	1	1	yes	characteristic
349	C₂	#21425005	1	1	yes	characteristic
350	C₂	#21425025	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273699
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1987	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂×#14273724	C₂²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1987	D₅₉
hypocenter	C₅₉	#9747
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1987

Complements

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

N	H ≅ G/N	type
3. #21423965	C₂	semidirect
4. C₂×#16837154	C₂	semidirect
5. C₂×#16837151	C₂	semidirect
7. C₅₉×#1987	C₂	semidirect
10. #16837750	C₄	semidirect
171. #1987	D₅₉	semidirect
201. C₅₉	#9747	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837750	2
#21425023	2
#21423965	2
C₂×#16837154	2
C₂×#16837151	2
C₂×#16837145	2
C₅₉×#1987	2
#9747	59