Subgroups — #28481802

Order 15104.

494 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	C₂×#16837479	C₂	1	1	yes	maximal, characteristic
2	#21424223	C₂	1	1	yes	maximal, characteristic, complemented
3	#21424097	C₂	1	1	yes	maximal, characteristic
4	#21424093	C₂	1	1	yes	maximal, characteristic
5	C₂×#16837152	C₂	1	1	yes	maximal, characteristic, complemented
6	#21421980	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1296	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11745659	—	2	1	no	—
9	C₂×C₄×D₂₃₆	C₂²	1	1	yes	characteristic
10	#16837479	—	2	2	no	—
11	#16837479	C₄	1	4	yes	complemented
12	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×#14273649	—	2	1	no	—
14	C₂×#14273647	—	2	1	no	—
15	C₂×(C₂×D₅₉)⋊C₈	—	2	1	no	—
16	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	#16837272	C₂²	1	1	yes	characteristic
18	#16837165	—	2	1	no	—
19	#16837164	—	2	1	no	—
20	#16837164	—	2	1	no	—
21	#16837164	—	2	1	no	—
22	#16837152	—	2	2	no	—
23	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#384	C₂²	1	1	yes	characteristic
26	C₂²×D₂₃₆	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₅₉	—	2	1	no	—
28	C₂×#11745659	—	2	4	no	—
29	C₂×#11745659	—	4	2	no	—
30	C₂×#11745659	—	4	2	no	—
31	C₂²×C₅₉⋊C₈	—	2	1	no	—
32	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
34	C₄×D₂₃₆	—	2	1	no	—
35	C₄×D₂₃₆	—	2	1	no	—
36	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
37	C₄×D₂₃₆	Q₈	1	2	yes	—
38	C₂×C₄⋊Dic₅₉	—	2	1	no	—
39	C₂×C₄⋊Dic₅₉	—	2	1	no	—
40	C₂×C₄⋊Dic₅₉	—	2	1	no	—
41	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
42	C₂×(C₂×D₅₉)⋊C₄	—	2	1	no	—
43	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
44	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
45	#14273649	—	4	2	no	—
46	#14273647	—	4	2	no	—
47	(C₂×D₅₉)⋊C₈	—	2	4	no	—
48	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
49	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
50	C₂²×C₈×C₅₉	—	2	1	no	—
51	C₂²×C₈×C₅₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
53	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	—	2	1	no	—
57	C₄⋊C₈×C₅₉	—	2	1	no	—
58	C₄⋊C₈×C₅₉	—	2	1	no	—
59	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
60	C₂³×D₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	—	4	1	no	—
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	4	1	no	—
67	C₂×D₂₃₆	—	2	1	no	—
68	C₂×D₂₃₆	—	2	1	no	—
69	C₂×D₂₃₆	—	2	1	no	—
70	C₂×D₂₃₆	—	2	1	no	—
71	C₂×D₂₃₆	—	2	1	no	—
72	C₂×D₂₃₆	C₄⋊C₄	1	2	yes	—
73	C₂×C₄×D₅₉	—	2	2	no	—
74	C₂×C₄×D₅₉	—	2	2	no	—
75	C₂×C₄×D₅₉	—	4	1	no	—
76	C₂×C₄×D₅₉	—	4	1	no	—
77	C₂×C₄×D₅₉	—	4	1	no	—
78	C₂×C₄×D₅₉	—	4	1	no	—
79	C₂×C₈×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	—	4	1	no	—
81	C₂×C₈×C₅₉	—	4	1	no	—
82	C₂×C₈×C₅₉	—	4	1	no	—
83	C₂×C₈×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	—	4	1	no	—
85	C₄⋊C₄×C₅₉	—	4	1	no	—
86	C₄⋊C₄×C₅₉	—	4	1	no	—
87	C₄⋊C₄×C₅₉	—	4	1	no	—
88	C₄⋊C₄×C₅₉	—	4	1	no	—
89	C₄⋊C₄×C₅₉	—	8	1	no	—
90	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
91	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
93	C₄²×C₅₉	#41	1	1	yes	characteristic
94	(C₂×D₅₉)⋊C₄	—	2	2	no	—
95	(C₂×D₅₉)⋊C₄	—	2	2	no	—
96	C₄⋊Dic₅₉	—	2	1	no	—
97	C₄⋊Dic₅₉	—	4	1	no	—
98	C₄⋊Dic₅₉	—	4	1	no	—
99	C₄⋊Dic₅₉	—	4	1	no	—
100	C₄⋊Dic₅₉	—	4	1	no	—
101	C₄⋊Dic₅₉	—	8	1	no	—
102	C₄⋊Dic₅₉	C₄⋊C₄	1	2	yes	—
103	C₂×C₅₉⋊C₈	—	2	1	no	—
104	C₂×C₅₉⋊C₈	—	2	1	no	—
105	C₂×C₅₉⋊C₈	—	2	1	no	—
106	C₂×C₅₉⋊C₈	—	2	1	no	—
107	C₂×C₅₉⋊C₈	—	4	1	no	—
108	C₂×C₅₉⋊C₈	—	4	1	no	—
109	C₂×C₅₉⋊C₈	—	4	1	no	—
110	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
111	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
112	#11745659	—	4	8	no	—
113	C₂³×C₅₉	#121	1	1	yes	characteristic
114	C₂²×D₅₉	—	2	2	no	—
115	C₂²×D₅₉	—	4	1	no	—
116	C₂²×D₅₉	—	4	1	no	—
117	C₂²×D₅₉	—	4	1	no	—
118	C₂²×D₅₉	—	4	1	no	—
119	C₂²×D₅₉	—	4	1	no	—
120	C₂²×D₅₉	—	4	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	8	1	no	—
124	C₂×C₄×C₅₉	—	8	1	no	—
125	C₂×C₄×C₅₉	—	8	1	no	—
126	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
129	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
133	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
134	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
135	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
136	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
137	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
139	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
140	C₂×Dic₅₉	—	2	2	no	—
141	C₂×Dic₅₉	—	4	1	no	—
142	C₂×Dic₅₉	—	4	1	no	—
143	C₂×Dic₅₉	—	8	1	no	—
144	C₂×Dic₅₉	—	8	1	no	—
145	C₂×Dic₅₉	—	8	1	no	—
146	D₂₃₆	—	2	2	no	—
147	D₂₃₆	—	2	2	no	—
148	D₂₃₆	—	4	1	no	—
149	D₂₃₆	—	4	1	no	—
150	C₄×D₅₉	—	4	2	no	—
151	C₄×D₅₉	—	4	2	no	—
152	C₄×D₅₉	—	4	2	no	—
153	C₄×D₅₉	—	4	2	no	—
154	C₈×C₅₉	—	4	2	no	—
155	C₈×C₅₉	—	8	1	no	—
156	C₅₉⋊C₈	—	2	2	no	—
157	C₅₉⋊C₈	—	2	2	no	—
158	C₅₉⋊C₈	—	8	1	no	—
159	#8656	—	59	1	no	maximal, Sylow
160	C₂²×C₅₉	#380	1	1	yes	characteristic
161	C₂²×C₅₉	#463	1	1	yes	characteristic
162	C₂²×C₅₉	#463	1	1	yes	characteristic
163	C₂²×C₅₉	#469	1	1	yes	characteristic
164	C₂²×C₅₉	#470	1	1	yes	characteristic
165	C₂²×C₅₉	#471	1	1	yes	characteristic
166	C₂²×C₅₉	#471	1	1	yes	characteristic
167	C₂×D₅₉	—	4	2	no	—
168	C₂×D₅₉	—	4	2	no	—
169	C₂×D₅₉	—	4	2	no	—
170	C₂×D₅₉	—	8	1	no	—
171	C₂×D₅₉	—	8	1	no	—
172	C₂×D₅₉	—	8	1	no	—
173	C₂×D₅₉	—	8	1	no	—
174	C₄×C₅₉	—	2	1	no	—
175	C₄×C₅₉	—	2	1	no	—
176	C₄×C₅₉	—	16	1	no	—
177	C₄×C₅₉	#416	1	1	yes	characteristic
178	C₄×C₅₉	#418	1	1	yes	characteristic
179	C₄×C₅₉	#428	1	1	yes	characteristic
180	C₄×C₅₉	#428	1	1	yes	characteristic
181	C₄×C₅₉	#468	1	1	yes	characteristic
182	C₄×C₅₉	#468	1	1	yes	characteristic
183	C₄×C₅₉	#478	1	1	yes	characteristic
184	C₄×C₅₉	#478	1	1	yes	characteristic
185	Dic₅₉	—	4	2	no	—
186	Dic₅₉	—	16	1	no	—
187	C₂×#436	—	59	1	no	—
188	#1834	—	59	1	no	—
189	#1802	—	59	1	no	—
190	#1802	—	59	1	no	—
191	C₂×#334	—	59	1	no	—
192	#1296	—	59	1	no	—
193	#1296	D₅₉	1	1	yes	characteristic, complemented
194	C₂×C₅₉	#1701	1	1	yes	characteristic
195	C₂×C₅₉	#1701	1	1	yes	characteristic
196	C₂×C₅₉	#1703	1	1	yes	characteristic
197	C₂×C₅₉	#1703	1	1	yes	characteristic
198	C₂×C₅₉	#1804	1	1	yes	characteristic
199	C₂×C₅₉	#1809	1	1	yes	characteristic
200	C₂×C₅₉	#1815	1	1	yes	characteristic
201	D₅₉	—	8	2	no	—
202	C₂²×M₁₆	—	118	1	no	—
203	C₂×C₄×D₄	—	59	1	no	—
204	#436	—	59	4	no	—
205	#436	—	118	2	no	—
206	C₂×C₄⋊C₈	—	59	1	no	—
207	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
208	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
209	C₂×#102	—	118	1	no	—
210	C₂×#102	—	118	1	no	—
211	C₂×C₂²⋊C₈	—	118	1	no	—
212	C₂×C₄×C₈	—	59	1	no	—
213	#384	—	59	1	no	—
214	#384	C₂×D₅₉	1	1	yes	characteristic
215	#340	—	118	1	no	—
216	#340	—	118	1	no	—
217	#340	—	118	1	no	—
218	#340	—	118	1	no	—
219	#334	—	118	2	no	—
220	C₅₉	#8656	1	1	yes	characteristic, complemented, Sylow
221	C₂²×D₄	—	59	1	no	—
222	C₂³×C₄	—	118	1	no	—
223	C₂×M₁₆	—	118	4	no	—
224	C₂×M₁₆	—	236	2	no	—
225	C₂×M₁₆	—	236	2	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	—	2	1	no	—
228	C₂²×C₈	—	59	1	no	—
229	C₂²×C₈	—	59	1	no	—
230	C₂²×C₈	—	118	1	no	—
231	C₄×D₄	—	59	2	no	—
232	C₄×D₄	—	59	2	no	—
233	C₄×D₄	—	118	1	no	—
234	C₄×D₄	—	118	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	—	2	1	no	—
238	C₂×C₄⋊C₄	—	59	1	no	—
239	C₂×C₄⋊C₄	—	118	1	no	—
240	C₂×C₄⋊C₄	—	118	1	no	—
241	C₂×C₄⋊C₄	—	118	1	no	—
242	C₂×C₂²⋊C₄	—	118	1	no	—
243	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
244	C₄⋊C₈	—	2	1	no	—
245	C₄⋊C₈	—	2	1	no	—
246	C₄⋊C₈	—	2	1	no	—
247	C₄⋊C₈	—	118	1	no	—
248	C₄⋊C₈	—	118	1	no	—
249	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
250	#102	—	236	2	no	—
251	#102	—	236	2	no	—
252	C₂²⋊C₈	—	118	4	no	—
253	C₄×C₈	—	59	2	no	—
254	C₄×C₈	—	59	2	no	—
255	C₂⁴	—	118	1	no	—
256	C₂×D₄	—	59	2	no	—
257	C₂×D₄	—	118	1	no	—
258	C₂×D₄	—	118	1	no	—
259	C₂×D₄	—	118	1	no	—
260	C₂×D₄	—	118	1	no	—
261	C₂×D₄	—	118	1	no	—
262	C₂²×C₄	—	4	1	no	—
263	C₂²×C₄	—	118	1	no	—
264	C₂²×C₄	—	118	2	no	—
265	C₂²×C₄	—	118	2	no	—
266	C₂²×C₄	—	236	1	no	—
267	C₂²×C₄	—	236	1	no	—
268	C₂²×C₄	—	236	1	no	—
269	C₂²×C₄	—	236	1	no	—
270	C₂²×C₄	—	236	1	no	—
271	C₂²×C₄	#11745685	1	1	yes	characteristic
272	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
273	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
274	M₁₆	—	236	8	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	4	1	no	—
277	C₂×C₈	—	4	1	no	—
278	C₂×C₈	—	4	1	no	—
279	C₂×C₈	—	4	1	no	—
280	C₂×C₈	—	4	1	no	—
281	C₂×C₈	—	59	2	no	—
282	C₂×C₈	—	59	2	no	—
283	C₂×C₈	—	118	1	no	—
284	C₂×C₈	—	118	1	no	—
285	C₂×C₈	—	118	1	no	—
286	C₂×C₈	—	118	1	no	—
287	C₂×C₈	—	236	1	no	—
288	C₂×C₈	—	236	1	no	—
289	C₂×C₈	—	236	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	—	8	1	no	—
295	C₄⋊C₄	—	59	2	no	—
296	C₄⋊C₄	—	118	1	no	—
297	C₄⋊C₄	—	472	1	no	—
298	C₄⋊C₄	—	236	1	no	—
299	C₄⋊C₄	—	236	1	no	—
300	C₄⋊C₄	—	236	1	no	—
301	C₄⋊C₄	—	236	1	no	—
302	C₂²⋊C₄	—	118	2	no	—
303	C₂²⋊C₄	—	118	2	no	—
304	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
305	C₄²	#11745685	1	1	yes	characteristic
306	C₄²	D₄×D₅₉	1	1	yes	characteristic
307	C₄²	Q₈×D₅₉	1	1	yes	characteristic
308	C₂³	—	118	2	no	—
309	C₂³	—	236	1	no	—
310	C₂³	—	236	1	no	—
311	C₂³	—	236	1	no	—
312	C₂³	—	236	1	no	—
313	C₂³	—	236	1	no	—
314	C₂³	—	236	1	no	—
315	C₂³	#14273700	1	1	yes	characteristic
316	D₄	—	118	2	no	—
317	D₄	—	118	2	no	—
318	D₄	—	236	1	no	—
319	D₄	—	236	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	8	1	no	—
323	C₂×C₄	—	8	1	no	—
324	C₂×C₄	—	8	1	no	—
325	C₂×C₄	—	118	2	no	—
326	C₂×C₄	—	472	1	no	—
327	C₂×C₄	—	472	1	no	—
328	C₂×C₄	—	472	1	no	—
329	C₂×C₄	—	236	1	no	—
330	C₂×C₄	—	236	1	no	—
331	C₂×C₄	—	236	2	no	—
332	C₂×C₄	—	236	2	no	—
333	C₂×C₄	—	236	2	no	—
334	C₂×C₄	—	236	2	no	—
335	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
336	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
337	C₂×C₄	#14273684	1	1	yes	characteristic
338	C₂×C₄	#14273700	1	1	yes	characteristic
339	C₂×C₄	#14273705	1	1	yes	characteristic
340	C₂×C₄	#14273705	1	1	yes	characteristic
341	C₂×C₄	#14273748	1	1	yes	characteristic
342	C₂×C₄	#14273750	1	1	yes	characteristic
343	C₂×C₄	#14273751	1	1	yes	characteristic
344	C₂×C₄	#14273751	1	1	yes	characteristic
345	C₂×C₄	#14273754	1	1	yes	characteristic
346	C₂×C₄	#14273754	1	1	yes	characteristic
347	C₂×C₄	#14273756	1	1	yes	characteristic
348	C₂×C₄	#14273759	1	1	yes	characteristic
349	C₈	—	4	2	no	—
350	C₈	—	8	1	no	—
351	C₈	—	118	2	no	—
352	C₈	—	118	2	no	—
353	C₈	—	472	1	no	—
354	C₂²	—	472	1	no	—
355	C₂²	—	472	1	no	—
356	C₂²	—	472	1	no	—
357	C₂²	—	472	1	no	—
358	C₂²	—	236	2	no	—
359	C₂²	—	236	2	no	—
360	C₂²	—	236	2	no	—
361	C₂²	#16837261	1	1	yes	characteristic
362	C₂²	#16837616	1	1	yes	characteristic
363	C₂²	#16837620	1	1	yes	characteristic
364	C₂²	#16837654	1	1	yes	characteristic
365	C₂²	#16837659	1	1	yes	characteristic
366	C₂²	#16837664	1	1	yes	characteristic
367	C₂²	#16837668	1	1	yes	characteristic
368	C₄	—	2	1	no	—
369	C₄	—	2	1	no	—
370	C₄	—	16	1	no	—
371	C₄	—	944	1	no	—
372	C₄	—	236	2	no	—
373	C₄	#16837397	1	1	yes	characteristic
374	C₄	#16837405	1	1	yes	characteristic
375	C₄	#16837441	1	1	yes	characteristic
376	C₄	#16837443	1	1	yes	characteristic
377	C₄	#16837647	1	1	yes	characteristic
378	C₄	#16837651	1	1	yes	characteristic
379	C₄	#16837701	1	1	yes	characteristic
380	C₄	#16837705	1	1	yes	characteristic
381	C₂	—	472	2	no	—
382	C₂	#21423656	1	1	yes	characteristic
383	C₂	#21423660	1	1	yes	characteristic
384	C₂	#21423670	1	1	yes	characteristic
385	C₂	#21423674	1	1	yes	characteristic
386	C₂	#21424105	1	1	yes	characteristic
387	C₂	#21424128	1	1	yes	characteristic
388	C₂	#21424154	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273700
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1296	C₂
socle	C₂³×C₅₉	#121
prime radical	C₂²×D₂₃₆	D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1296	D₅₉
hypocenter	C₅₉	#8656
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
2. #21424223	C₂	semidirect
5. C₂×#16837152	C₂	semidirect
6. #21421980	C₂	semidirect
7. C₅₉×#1296	C₂	semidirect
11. #16837479	C₄	semidirect
193. #1296	D₅₉	semidirect
220. C₅₉	#8656	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837479	2
#21424223	2
#21424097	2
#21424093	2
C₂×#16837152	2
#21421980	2
C₅₉×#1296	2
#8656	59