Subgroups — #28572482

Order 15104.

690 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433889	C₂	1	2	yes	maximal
2	C₂×C₈×D₂₃₆	C₂	1	1	yes	maximal, characteristic
3	#21423544	C₂	1	4	yes	maximal, complemented
4	#21423492	C₂	1	2	yes	maximal, complemented
5	#21423458	C₂	1	2	yes	maximal, complemented
6	#21422855	C₂	1	1	yes	maximal, characteristic, complemented
7	#21422786	C₂	1	2	yes	maximal, complemented
8	C₅₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₈×D₅₉	—	2	1	no	—
10	C₂×C₄×D₂₃₆	C₂²	1	1	yes	characteristic
11	#16837699	—	2	4	no	—
12	#16837689	C₂²	1	4	yes	—
13	#16837679	C₂²	1	4	yes	—
14	C₈×D₂₃₆	—	2	1	no	—
15	C₈×D₂₃₆	—	2	1	no	—
16	C₈×D₂₃₆	C₂²	1	2	yes	—
17	C₈×D₂₃₆	C₂²	1	2	yes	—
18	#16837445	—	2	2	no	—
19	#16837426	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
21	#16837406	—	2	2	no	—
22	C₂×(C₂×D₅₉)⋊C₈	—	2	1	no	—
23	#16837155	C₂²	1	2	yes	complemented
24	#16837155	C₂²	1	2	yes	complemented
25	#16837151	C₂²	1	4	yes	complemented
26	#16837148	C₂²	1	4	yes	complemented
27	C₅₉×#533	C₂²	1	2	yes	—
28	C₅₉×#500	C₂²	1	1	yes	characteristic, complemented
29	C₅₉×#494	C₂²	1	1	yes	characteristic, complemented
30	C₅₉×#487	C₂²	1	2	yes	complemented
31	C₅₉×#467	—	2	1	no	—
32	C₅₉×#467	—	2	1	no	—
33	C₄×Q₁₆×C₅₉	—	2	1	no	—
34	C₄×Q₁₆×C₅₉	—	2	1	no	—
35	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
36	C₂²×D₂₃₆	D₄	1	1	yes	characteristic
37	C₂²×C₄×D₅₉	—	2	1	no	—
38	C₂×#11745664	—	2	2	no	—
39	C₂²×C₅₉⋊C₈	—	2	1	no	—
40	C₂×C₈×D₅₉	—	2	2	no	—
41	C₂×C₈×D₅₉	—	2	2	no	—
42	C₂×C₈×D₅₉	—	4	4	no	—
43	C₄×D₂₃₆	—	2	1	no	—
44	C₄×D₂₃₆	—	2	1	no	—
45	C₄×D₂₃₆	D₄	1	2	yes	—
46	C₄×D₂₃₆	C₂³	1	2	yes	—
47	C₂×C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
48	C₂×(C₂×D₅₉)⋊C₄	—	2	1	no	—
49	#14273660	—	2	4	no	—
50	#14273658	—	2	4	no	—
51	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
52	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
53	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
54	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
55	#14273647	—	2	4	no	—
56	#14273647	—	2	4	no	—
57	(C₂×D₅₉)⋊C₈	—	2	2	no	—
58	(C₂×D₅₉)⋊C₈	—	2	2	no	—
59	C₂×Q₁₆×C₅₉	—	2	1	no	—
60	C₂×Q₁₆×C₅₉	—	2	1	no	—
61	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
63	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
64	C₄⋊Q₈×C₅₉	C₂³	1	2	yes	—
65	#125×C₅₉	—	2	2	no	—
66	C₂²⋊Q₈×C₅₉	—	2	2	no	—
67	C₂²⋊Q₈×C₅₉	—	2	2	no	—
68	C₄×Q₈×C₅₉	—	2	2	no	—
69	C₄×Q₈×C₅₉	—	2	2	no	—
70	#117×C₅₉	—	2	2	no	—
71	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
72	#107×C₅₉	—	2	1	no	—
73	#107×C₅₉	—	2	1	no	—
74	#103×C₅₉	—	2	2	no	—
75	#103×C₅₉	—	2	2	no	—
76	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
77	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
78	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
80	C₂³×D₅₉	—	2	1	no	—
81	C₂×Q₈×C₅₉	—	2	2	no	—
82	C₂×Q₈×C₅₉	—	2	2	no	—
83	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
86	C₂²×Dic₅₉	—	2	1	no	—
87	C₂×D₂₃₆	—	2	1	no	—
88	C₂×D₂₃₆	—	2	2	no	—
89	C₂×D₂₃₆	Q₁₆	1	4	yes	complemented
90	C₂×D₂₃₆	C₂×D₄	1	2	yes	—
91	C₂×C₄×D₅₉	—	2	2	no	—
92	C₂×C₄×D₅₉	—	2	2	no	—
93	C₂×C₄×D₅₉	—	4	2	no	—
94	C₂×C₄×D₅₉	—	4	2	no	—
95	Q₁₆×C₅₉	—	4	2	no	—
96	Q₁₆×C₅₉	—	4	2	no	—
97	C₂×C₈×C₅₉	—	2	2	no	—
98	C₂×C₈×C₅₉	—	4	1	no	—
99	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
103	C₄⋊C₄×C₅₉	—	2	2	no	—
104	C₄⋊C₄×C₅₉	—	2	2	no	—
105	C₄⋊C₄×C₅₉	—	2	2	no	—
106	C₄⋊C₄×C₅₉	—	2	2	no	—
107	C₄⋊C₄×C₅₉	—	4	2	no	—
108	C₄⋊C₄×C₅₉	—	4	2	no	—
109	C₂²⋊C₄×C₅₉	—	4	2	no	—
110	C₄²×C₅₉	—	4	2	no	—
111	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₅₉	#41	1	1	yes	characteristic
114	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
115	(C₂×D₅₉)⋊C₄	—	2	2	no	—
116	(C₂×D₅₉)⋊C₄	—	2	2	no	—
117	C₄⋊Dic₅₉	—	2	1	no	—
118	C₄⋊Dic₅₉	C₂×D₄	1	2	yes	—
119	#11745664	—	4	4	no	—
120	C₂×C₅₉⋊C₈	—	2	2	no	—
121	C₂×C₅₉⋊C₈	—	2	4	no	—
122	C₂×C₅₉⋊C₈	—	4	2	no	—
123	C₈×D₅₉	—	4	4	no	—
124	C₈×D₅₉	—	4	4	no	—
125	C₂³×C₅₉	#121	1	1	yes	characteristic
126	C₂²×D₅₉	—	2	2	no	—
127	C₂²×D₅₉	—	2	4	no	—
128	C₂²×D₅₉	—	4	2	no	—
129	C₂²×D₅₉	—	4	2	no	—
130	Q₈×C₅₉	—	4	2	no	—
131	Q₈×C₅₉	—	4	2	no	—
132	Q₈×C₅₉	—	8	2	no	—
133	C₂×C₄×C₅₉	—	2	1	no	—
134	C₂×C₄×C₅₉	—	2	1	no	—
135	C₂×C₄×C₅₉	—	4	1	no	—
136	C₂×C₄×C₅₉	—	4	2	no	—
137	C₂×C₄×C₅₉	—	4	2	no	—
138	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
139	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
140	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
141	C₂×C₄×C₅₉	C₂×Q₁₆	1	2	yes	—
142	C₂×C₄×C₅₉	#135	1	2	yes	—
143	C₂×C₄×C₅₉	C₂²×D₄	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	—	4	2	no	—
148	D₂₃₆	—	2	2	no	—
149	D₂₃₆	—	2	4	no	—
150	D₂₃₆	C₂×Q₁₆	1	4	yes	—
151	C₄×D₅₉	—	4	4	no	—
152	C₄×D₅₉	—	4	2	no	—
153	C₄×D₅₉	—	4	2	no	—
154	C₈×C₅₉	—	2	1	no	—
155	C₈×C₅₉	—	2	1	no	—
156	C₈×C₅₉	—	2	1	no	—
157	C₈×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
158	C₅₉⋊C₈	—	4	4	no	—
159	C₅₉⋊C₈	—	4	2	no	—
160	#19256	—	59	1	no	maximal, Sylow
161	C₂²×C₅₉	—	2	1	no	—
162	C₂²×C₅₉	—	2	1	no	—
163	C₂²×C₅₉	#467	1	2	yes	—
164	C₂²×C₅₉	C₂×#121	1	1	yes	characteristic
165	C₂×D₅₉	—	2	4	no	—
166	C₂×D₅₉	—	4	8	no	—
167	C₂×D₅₉	—	4	2	no	—
168	C₂×D₅₉	—	4	2	no	—
169	C₄×C₅₉	—	2	1	no	—
170	C₄×C₅₉	—	2	1	no	—
171	C₄×C₅₉	—	2	1	no	—
172	C₄×C₅₉	—	8	2	no	—
173	C₄×C₅₉	—	8	2	no	—
174	C₄×C₅₉	#467	1	2	yes	—
175	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
176	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
177	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₄×C₅₉	#575	1	1	yes	characteristic
179	Dic₅₉	—	4	2	no	—
180	#3170	D₅₉	1	1	yes	characteristic, complemented
181	#3077	—	59	2	no	—
182	C₂×C₈×D₄	—	59	1	no	—
183	#1683	—	59	4	no	—
184	#1675	—	59	2	no	—
185	#1669	—	59	2	no	—
186	#1567	—	59	1	no	—
187	#1550	—	59	2	no	—
188	C₂×C₅₉	—	2	2	no	—
189	C₂×C₅₉	#3036	1	1	yes	characteristic
190	C₂×C₅₉	C₂×#467	1	1	yes	characteristic
191	C₂×C₅₉	#3067	1	1	yes	characteristic
192	D₅₉	—	4	4	no	—
193	C₂³×C₈	—	118	1	no	—
194	#533	C₂×D₅₉	1	2	yes	—
195	C₂×C₄×D₄	—	59	1	no	—
196	#500	C₂×D₅₉	1	1	yes	characteristic, complemented
197	#494	C₂×D₅₉	1	1	yes	characteristic, complemented
198	#487	C₂×D₅₉	1	2	yes	complemented
199	#478	—	118	4	no	—
200	#476	—	59	4	no	—
201	#474	—	59	4	no	—
202	#467	—	2	1	no	—
203	#467	—	2	1	no	—
204	C₄×Q₁₆	—	2	1	no	—
205	C₄×Q₁₆	—	2	1	no	—
206	C₈×D₄	—	59	2	no	—
207	C₈×D₄	—	59	2	no	—
208	C₈×D₄	—	118	1	no	—
209	C₈×D₄	—	118	1	no	—
210	#428	—	118	2	no	—
211	#423	—	59	2	no	—
212	C₂×C₄⋊C₈	—	59	1	no	—
213	#418	—	118	2	no	—
214	C₂×C₂²⋊C₈	—	118	1	no	—
215	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
216	#335	—	59	2	no	—
217	#335	—	59	2	no	—
218	#333	—	59	4	no	—
219	#332	—	59	4	no	—
220	C₅₉	#19256	1	1	yes	characteristic, complemented, Sylow
221	C₂²×D₄	—	59	1	no	—
222	C₂³×C₄	—	118	1	no	—
223	C₂×Q₁₆	—	2	1	no	—
224	C₂×Q₁₆	—	2	1	no	—
225	C₂×M₁₆	—	118	2	no	—
226	C₂²×C₈	—	118	1	no	—
227	C₂²×C₈	—	118	2	no	—
228	C₂²×C₈	—	118	2	no	—
229	C₂²×C₈	—	236	4	no	—
230	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
231	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
232	C₄⋊Q₈	C₂²×D₅₉	1	2	yes	—
233	C₄⋊Q₈	C₂²×D₅₉	1	2	yes	—
234	#125	—	2	2	no	—
235	C₂²⋊Q₈	—	2	2	no	—
236	C₂²⋊Q₈	—	2	2	no	—
237	C₄×Q₈	—	2	2	no	—
238	C₄×Q₈	—	2	2	no	—
239	C₄×D₄	—	59	2	no	—
240	C₄×D₄	—	59	2	no	—
241	C₄×D₄	—	118	1	no	—
242	C₄×D₄	—	118	1	no	—
243	#117	—	2	2	no	—
244	C₂×C₄⋊C₄	—	59	1	no	—
245	C₂×C₂²⋊C₄	—	118	1	no	—
246	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
247	#107	—	2	1	no	—
248	#107	—	2	1	no	—
249	#107	—	118	4	no	—
250	#106	—	118	4	no	—
251	C₄⋊C₈	—	59	2	no	—
252	C₄⋊C₈	—	59	4	no	—
253	C₄⋊C₈	—	118	2	no	—
254	C₄⋊C₈	—	118	1	no	—
255	#103	—	2	2	no	—
256	#103	—	2	2	no	—
257	#102	—	118	4	no	—
258	#102	—	118	4	no	—
259	C₂²⋊C₈	—	118	2	no	—
260	C₂²⋊C₈	—	118	2	no	—
261	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
262	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
263	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
264	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
265	C₂⁴	—	118	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	2	2	no	—
268	C₂×D₄	—	59	4	no	—
269	C₂×D₄	—	59	2	no	—
270	C₂×D₄	—	118	1	no	—
271	C₂×D₄	—	118	2	no	—
272	C₂²×C₄	—	118	1	no	—
273	C₂²×C₄	—	118	2	no	—
274	C₂²×C₄	—	118	2	no	—
275	C₂²×C₄	—	236	2	no	—
276	C₂²×C₄	—	236	2	no	—
277	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
278	C₂²×C₄	#11745687	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
280	Q₁₆	—	4	2	no	—
281	Q₁₆	—	4	2	no	—
282	M₁₆	—	236	4	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	118	4	no	—
286	C₂×C₈	—	118	2	no	—
287	C₂×C₈	—	236	4	no	—
288	C₂×C₈	—	236	2	no	—
289	C₂×C₈	—	236	4	no	—
290	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
292	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
293	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
294	C₄⋊C₄	—	2	2	no	—
295	C₄⋊C₄	—	2	2	no	—
296	C₄⋊C₄	—	2	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	4	2	no	—
299	C₄⋊C₄	—	4	2	no	—
300	C₄⋊C₄	—	59	2	no	—
301	C₄⋊C₄	—	118	1	no	—
302	C₂²⋊C₄	—	4	2	no	—
303	C₂²⋊C₄	—	118	2	no	—
304	C₂²⋊C₄	—	118	2	no	—
305	C₄²	—	4	2	no	—
306	C₄²	D₄×D₅₉	1	1	yes	characteristic
307	C₄²	#11745687	1	1	yes	characteristic
308	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
309	C₄²	C₂³×D₅₉	1	1	yes	characteristic
310	C₂³	—	118	2	no	—
311	C₂³	—	118	4	no	—
312	C₂³	—	236	2	no	—
313	C₂³	—	236	2	no	—
314	C₂³	#14273698	1	1	yes	characteristic
315	Q₈	—	4	2	no	—
316	Q₈	—	4	2	no	—
317	Q₈	—	8	2	no	—
318	D₄	—	59	4	no	—
319	D₄	—	118	4	no	—
320	D₄	—	118	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	118	2	no	—
327	C₂×C₄	—	236	4	no	—
328	C₂×C₄	—	236	2	no	—
329	C₂×C₄	—	236	2	no	—
330	C₂×C₄	—	236	2	no	—
331	C₂×C₄	#14273698	1	1	yes	characteristic
332	C₂×C₄	#14273698	1	1	yes	characteristic
333	C₂×C₄	#14273698	1	1	yes	characteristic
334	C₂×C₄	Q₁₆×D₅₉	1	2	yes	—
335	C₂×C₄	#14273750	1	2	yes	—
336	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
337	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
338	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
339	C₈	—	2	1	no	—
340	C₈	—	2	1	no	—
341	C₈	—	2	1	no	—
342	C₈	—	236	4	no	—
343	C₈	—	236	2	no	—
344	C₈	#14273752	1	2	yes	—
345	C₂²	—	2	1	no	—
346	C₂²	—	2	1	no	—
347	C₂²	—	118	4	no	—
348	C₂²	—	236	8	no	—
349	C₂²	—	236	2	no	—
350	C₂²	—	236	2	no	—
351	C₂²	#16837640	1	2	yes	—
352	C₂²	C₂×#14273698	1	1	yes	characteristic
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	2	1	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	—	236	2	no	—
359	C₄	#16837640	1	2	yes	—
360	C₄	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
361	C₄	C₂×#14273750	1	1	yes	characteristic
362	C₄	C₂×#14273752	1	1	yes	characteristic
363	C₄	#16838200	1	1	yes	characteristic
364	C₂	—	2	2	no	—
365	C₂	—	236	4	no	—
366	C₂	#21433556	1	1	yes	characteristic
367	C₂	C₂×#16837640	1	1	yes	characteristic
368	C₂	#21433812	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14273698
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#3170	C₂
socle	C₂²×C₅₉	C₂×#121
prime radical	C₂²×D₂₃₆	D₄
hypercenter	#3170	D₅₉
hypocenter	C₅₉	#19256
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3170

Complements

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

N	H ≅ G/N	type
3. #21423544	C₂	semidirect
4. #21423492	C₂	semidirect
5. #21423458	C₂	semidirect
6. #21422855	C₂	semidirect
7. #21422786	C₂	semidirect
8. C₅₉×#3170	C₂	semidirect
23. #16837155	C₂²	semidirect
24. #16837155	C₂²	semidirect
25. #16837151	C₂²	semidirect
26. #16837148	C₂²	semidirect
28. C₅₉×#500	C₂²	semidirect
29. C₅₉×#494	C₂²	semidirect
30. C₅₉×#487	C₂²	semidirect
89. C₂×D₂₃₆	Q₁₆	semidirect
180. #3170	D₅₉	semidirect
196. #500	C₂×D₅₉	semidirect
197. #494	C₂×D₅₉	semidirect
198. #487	C₂×D₅₉	semidirect
220. C₅₉	#19256	semidirect

Maximal subgroups

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

subgroup	index
#21433889	2
C₂×C₈×D₂₃₆	2
#21423544	2
#21423492	2
#21423458	2
#21422855	2
#21422786	2
C₅₉×#3170	2
#19256	59