Subgroups — #28560120

Order 15104.

656 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433110	C₂	1	2	yes	maximal, complemented
2	#21422993	C₂	1	2	yes	maximal, complemented
3	#21422933	C₂	1	4	yes	maximal, complemented
4	D₈×C₅₉⋊C₈	C₂	1	2	yes	maximal, complemented
5	#21422859	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422530	C₂	1	2	yes	maximal, complemented
7	C₈×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₅₉	C₂	1	1	yes	maximal, characteristic
9	#16837533	C₂²	1	4	yes	—
10	#16837474	—	2	4	no	—
11	D₄×C₅₉⋊C₈	C₂²	1	4	yes	—
12	#16837459	—	2	2	no	—
13	#16837459	C₂²	1	2	yes	—
14	#16837455	C₂²	1	1	yes	characteristic
15	C₄×#11745664	C₂²	1	1	yes	characteristic
16	C₂×#14273638	—	2	2	no	—
17	#16837155	C₂²	1	2	yes	—
18	#16837153	C₂²	1	2	yes	—
19	Q₈⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
20	D₄⋊(C₅₉⋊C₈)	C₂²	1	4	yes	—
21	#16837124	C₂²	1	2	yes	—
22	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
23	C₂×#135×C₅₉	C₄	1	1	yes	characteristic
24	C₄×#41×C₅₉	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₉	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₅₉	C₄	1	2	yes	—
28	C₄×SD₁₆×C₅₉	C₂²	1	2	yes	—
29	C₄×D₈×C₅₉	C₄	1	1	yes	characteristic
30	C₄×D₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₅₉×#427	C₄	1	1	yes	characteristic
32	C₅₉×#416	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
34	C₂×#11745664	D₄	1	2	yes	—
35	C₂²×C₅₉⋊C₈	—	2	4	no	—
36	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
37	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
38	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
39	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
40	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
41	C₂²⋊(C₅₉⋊C₈)	—	2	4	no	—
42	#14273638	—	2	2	no	—
43	#14273638	—	2	4	no	—
44	#14273638	D₄	1	1	yes	characteristic
45	#14273638	D₄	1	1	yes	characteristic
46	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
47	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
48	C₂×#41×C₅₉	C₂×C₄	1	2	yes	—
49	#135×C₅₉	—	2	4	no	—
50	C₂×Q₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₅₉	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₅₉	C₂³	1	2	yes	—
57	C₄×D₄×C₅₉	—	2	2	no	—
58	C₄×D₄×C₅₉	—	2	2	no	—
59	C₄×D₄×C₅₉	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₅₉	C₂³	1	2	yes	—
61	#117×C₅₉	—	2	2	no	—
62	#117×C₅₉	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₅₉	—	2	2	no	—
64	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
65	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
66	#107×C₅₉	C₂×C₄	1	1	yes	characteristic
67	#106×C₅₉	C₂×C₄	1	2	yes	—
68	#103×C₅₉	C₂×C₄	1	2	yes	—
69	#103×C₅₉	C₂×C₄	1	2	yes	—
70	#102×C₅₉	C₂×C₄	1	2	yes	—
71	#102×C₅₉	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	#41×C₅₉	—	2	4	no	—
77	#41×C₅₉	—	4	2	no	—
78	C₂×Q₈×C₅₉	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₅₉	—	2	2	no	—
80	C₂×D₄×C₅₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₉	—	2	2	no	—
82	C₂²×C₄×C₅₉	—	2	2	no	—
83	C₂²×C₄×C₅₉	—	2	2	no	—
84	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
87	Q₁₆×C₅₉	—	2	2	no	—
88	SD₁₆×C₅₉	—	2	4	no	—
89	D₈×C₅₉	M₁₆	1	4	yes	complemented
90	C₂×C₈×C₅₉	—	2	2	no	—
91	C₂×C₈×C₅₉	—	4	1	no	—
92	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₅₉	—	2	2	no	—
97	C₄⋊C₄×C₅₉	—	2	2	no	—
98	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
99	C₄⋊C₄×C₅₉	C₂²×C₄	1	2	yes	—
100	C₂²⋊C₄×C₅₉	—	2	2	no	—
101	C₂²⋊C₄×C₅₉	—	2	2	no	—
102	C₂²⋊C₄×C₅₉	—	2	2	no	—
103	C₄²×C₅₉	—	2	2	no	—
104	C₄²×C₅₉	—	2	2	no	—
105	C₄²×C₅₉	—	2	2	no	—
106	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₉	#41	1	1	yes	characteristic
109	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
110	#11745664	—	2	4	no	—
111	C₂×C₅₉⋊C₈	—	2	4	no	—
112	C₂×C₅₉⋊C₈	—	2	2	no	—
113	C₂×C₅₉⋊C₈	—	4	8	no	—
114	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
115	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
116	C₂³×C₅₉	—	2	2	no	—
117	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₅₉	—	2	2	no	—
119	Q₈×C₅₉	—	2	2	no	—
120	D₄×C₅₉	—	2	2	no	—
121	D₄×C₅₉	—	4	4	no	—
122	D₄×C₅₉	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	—	2	1	no	—
125	C₂×C₄×C₅₉	—	2	2	no	—
126	C₂×C₄×C₅₉	—	2	2	no	—
127	C₂×C₄×C₅₉	—	2	2	no	—
128	C₂×C₄×C₅₉	—	2	2	no	—
129	C₂×C₄×C₅₉	—	4	1	no	—
130	C₂×C₄×C₅₉	—	4	4	no	—
131	C₂×C₄×C₅₉	—	4	4	no	—
132	C₂×C₄×C₅₉	—	8	2	no	—
133	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
143	C₅₉⋊C₈	—	2	2	no	—
144	C₅₉⋊C₈	—	4	2	no	—
145	C₅₉⋊C₈	—	4	4	no	—
146	C₅₉⋊C₈	#135	1	4	yes	complemented
147	#18035	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	2	4	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	—	2	1	no	—
152	C₂²×C₅₉	—	4	2	no	—
153	C₂²×C₅₉	—	4	2	no	—
154	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	4	2	no	—
160	C₄×C₅₉	—	4	2	no	—
161	C₄×C₅₉	—	4	2	no	—
162	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
163	C₄×C₅₉	#568	1	1	yes	characteristic
164	C₄×C₅₉	C₂×#135	1	2	yes	—
165	#2970	—	59	2	no	—
166	C₄×#135	D₅₉	1	1	yes	characteristic
167	#1591	—	59	2	no	—
168	#1582	—	59	4	no	—
169	C₈×D₈	—	59	2	no	—
170	#1568	—	59	1	no	—
171	#1488	—	59	2	no	—
172	C₈×M₁₆	—	59	1	no	—
173	C₂×C₅₉	—	4	1	no	—
174	C₂×C₅₉	—	4	4	no	—
175	C₂×C₅₉	D₄×M₁₆	1	1	yes	characteristic
176	C₂×C₅₉	C₄×#135	1	1	yes	characteristic
177	C₂×C₅₉	#2957	1	1	yes	characteristic
178	C₂×#135	Dic₅₉	1	1	yes	characteristic
179	C₄×#41	C₂×D₅₉	1	2	yes	—
180	#446	—	59	4	no	—
181	C₄×Q₁₆	Dic₅₉	1	1	yes	characteristic
182	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
183	C₄×SD₁₆	Dic₅₉	1	2	yes	—
184	C₄×SD₁₆	C₂×D₅₉	1	2	yes	—
185	C₄×D₈	Dic₅₉	1	1	yes	characteristic
186	C₄×D₈	C₂×D₅₉	1	1	yes	characteristic
187	#435	—	118	4	no	—
188	C₈×D₄	—	59	4	no	—
189	#432	—	59	2	no	—
190	#432	—	118	2	no	—
191	#431	—	59	1	no	—
192	#427	Dic₅₉	1	1	yes	characteristic
193	#416	Dic₅₉	1	2	yes	—
194	C₄×M₁₆	—	59	1	no	—
195	C₂×#97	—	118	2	no	—
196	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
197	#335	—	59	2	no	—
198	#334	—	59	2	no	—
199	Q₈⋊C₈	—	59	4	no	—
200	D₄⋊C₈	—	59	4	no	—
201	#322	—	59	2	no	—
202	C₈²	—	59	2	no	—
203	C₅₉	#18035	1	1	yes	characteristic, complemented, Sylow
204	C₂×#41	C₂×Dic₅₉	1	2	yes	—
205	#135	—	2	4	no	—
206	C₂×Q₁₆	C₂×Dic₅₉	1	1	yes	characteristic
207	C₂×SD₁₆	C₂×Dic₅₉	1	2	yes	—
208	C₂×D₈	C₂×Dic₅₉	1	1	yes	characteristic
209	C₂×M₁₆	—	59	2	no	—
210	C₂²×C₈	—	118	4	no	—
211	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
212	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
213	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
214	C₄×Q₈	C₂²×D₅₉	1	2	yes	—
215	C₄×D₄	—	2	2	no	—
216	C₄×D₄	—	2	2	no	—
217	C₄×D₄	C₂×Dic₅₉	1	2	yes	—
218	C₄×D₄	C₂²×D₅₉	1	2	yes	—
219	#117	—	2	2	no	—
220	#117	C₂×Dic₅₉	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
223	#107	C₂×Dic₅₉	1	1	yes	characteristic
224	#107	C₂×Dic₅₉	1	1	yes	characteristic
225	#106	C₂×Dic₅₉	1	2	yes	—
226	C₄⋊C₈	—	59	4	no	—
227	C₄⋊C₈	—	118	4	no	—
228	C₄⋊C₈	—	118	1	no	—
229	#103	C₂×Dic₅₉	1	2	yes	—
230	#103	C₂×Dic₅₉	1	2	yes	—
231	#102	C₂×Dic₅₉	1	2	yes	—
232	#102	C₂×Dic₅₉	1	2	yes	—
233	C₂²⋊C₈	—	118	4	no	—
234	C₂²⋊C₈	—	118	2	no	—
235	C₂²⋊C₈	—	118	1	no	—
236	#97	—	59	1	no	—
237	#97	—	59	1	no	—
238	#97	—	118	4	no	—
239	#97	—	118	2	no	—
240	C₄×C₈	—	59	2	no	—
241	C₄×C₈	—	59	2	no	—
242	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
243	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
246	#41	—	2	4	no	—
247	#41	—	4	2	no	—
248	C₂×Q₈	C₂²×Dic₅₉	1	2	yes	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	C₂²×Dic₅₉	1	2	yes	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
255	C₂²×C₄	#11745687	1	1	yes	characteristic
256	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
257	Q₁₆	—	2	2	no	—
258	SD₁₆	—	2	4	no	—
259	D₈	#11745664	1	4	yes	complemented
260	M₁₆	—	118	4	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	59	2	no	—
264	C₂×C₈	—	59	2	no	—
265	C₂×C₈	—	118	4	no	—
266	C₂×C₈	—	118	2	no	—
267	C₂×C₈	—	236	8	no	—
268	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
269	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	C₂²×Dic₅₉	1	2	yes	—
275	C₄⋊C₄	C₂²×Dic₅₉	1	2	yes	—
276	C₂²⋊C₄	—	2	2	no	—
277	C₂²⋊C₄	—	2	2	no	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	D₄×D₅₉	1	1	yes	characteristic
283	C₄²	#11745687	1	1	yes	characteristic
284	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
285	C₄²	C₂³×D₅₉	1	1	yes	characteristic
286	C₂³	—	2	2	no	—
287	C₂³	D₄×Dic₅₉	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	D₄	—	2	2	no	—
291	D₄	—	4	4	no	—
292	D₄	C₂×#11745664	1	4	yes	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₅₉	1	1	yes	characteristic
306	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
308	C₂×C₄	C₂×#11745687	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	59	4	no	—
313	C₈	—	118	2	no	—
314	C₈	—	236	2	no	—
315	C₈	—	236	4	no	—
316	C₈	C₂×#11745664	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	4	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	4	2	no	—
322	C₂²	—	4	2	no	—
323	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	2	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	C₂²×#11745664	1	1	yes	characteristic
332	C₄	#16838169	1	1	yes	characteristic
333	C₄	#135×D₅₉	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	4	4	no	—
336	C₂	D₄×#11745664	1	1	yes	characteristic
337	C₂	#135×Dic₅₉	1	1	yes	characteristic
338	C₂	#21433013	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₄×#135×C₅₉	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂×#135×C₅₉	C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	C₄×#135	D₅₉
hypocenter	C₅₉	#18035
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #21433110	C₂	semidirect
2. #21422993	C₂	semidirect
3. #21422933	C₂	semidirect
4. D₈×C₅₉⋊C₈	C₂	semidirect
5. #21422859	C₂	semidirect
6. #21422530	C₂	semidirect
7. C₈×#11745664	C₂	semidirect
89. D₈×C₅₉	M₁₆	semidirect
146. C₅₉⋊C₈	#135	semidirect
203. C₅₉	#18035	semidirect
259. D₈	#11745664	semidirect

Maximal subgroups

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

subgroup	index
#21433110	2
#21422993	2
#21422933	2
D₈×C₅₉⋊C₈	2
#21422859	2
#21422530	2
C₈×#11745664	2
C₄×#135×C₅₉	2
#18035	59