Subgroups — #28503726

Order 15104.

514 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837357	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
3	C₂×#16837177	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	#21422173	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422173	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₅₉×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
8	C₂²×#11745664	C₄	1	1	yes	characteristic
9	C₂×#14273686	C₄	1	1	yes	characteristic
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
12	#16837357	—	2	2	no	—
13	#16837357	—	2	2	no	—
14	C₂×#14273637	C₄	1	1	yes	characteristic
15	C₂×#14273637	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂²⋊(C₅₉⋊C₁₆)	C₄	1	4	yes	complemented
19	#16837177	—	2	2	no	—
20	C₁₆×Dic₅₉	—	2	2	no	—
21	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
23	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×C₂²⋊C₁₆	C₄	1	4	yes	complemented
25	C₂×#11745664	—	2	2	no	—
26	C₂×#11745664	—	2	2	no	—
27	C₂×#11745664	C₈	1	4	yes	—
28	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
30	#14273686	—	2	2	no	—
31	#14273686	—	2	2	no	—
32	C₂×C₄⋊Dic₅₉	C₈	1	2	yes	—
33	C₂×C₂²⋊Dic₅₉	C₈	1	2	yes	—
34	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
36	C₂×C₅₉⋊C₁₆	—	2	1	no	—
37	C₂×C₅₉⋊C₁₆	—	2	1	no	—
38	C₂×C₅₉⋊C₁₆	—	2	1	no	—
39	C₂×C₅₉⋊C₁₆	—	2	1	no	—
40	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
42	#14273637	—	2	2	no	—
43	#14273637	—	2	2	no	—
44	C₈×Dic₅₉	—	2	1	no	—
45	C₈×Dic₅₉	—	2	1	no	—
46	C₈×Dic₅₉	D₄	1	2	yes	—
47	C₈×Dic₅₉	D₄	1	2	yes	—
48	C₂³×C₄×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	—	2	1	no	—
50	C₂²×C₈×C₅₉	—	2	1	no	—
51	C₂²×C₈×C₅₉	—	2	1	no	—
52	C₂²×C₈×C₅₉	—	2	1	no	—
53	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	C₂×C₁₆×C₅₉	—	2	1	no	—
58	C₂×C₁₆×C₅₉	—	2	1	no	—
59	C₂×C₁₆×C₅₉	—	2	1	no	—
60	C₂×C₁₆×C₅₉	—	2	1	no	—
61	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
62	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
63	C₂⁴×C₅₉	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	—	2	1	no	—
65	C₂²×C₄×C₅₉	—	2	1	no	—
66	C₂²×C₄×C₅₉	—	2	1	no	—
67	C₂²×C₄×C₅₉	—	2	1	no	—
68	C₂²×C₄×C₅₉	C₄²	1	2	yes	—
69	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
71	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
72	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
73	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
74	C₂×C₈×C₅₉	—	2	1	no	—
75	C₂×C₈×C₅₉	—	2	1	no	—
76	C₂×C₈×C₅₉	—	2	2	no	—
77	C₂×C₈×C₅₉	—	2	2	no	—
78	C₂×C₈×C₅₉	—	2	2	no	—
79	C₂×C₈×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	C₄²	1	2	yes	—
81	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
86	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
87	C₂²⋊Dic₅₉	—	2	4	no	—
88	C₄⋊Dic₅₉	—	2	4	no	—
89	C₄×Dic₅₉	—	2	1	no	—
90	C₄×Dic₅₉	—	2	1	no	—
91	C₄×Dic₅₉	C₂²⋊C₄	1	2	yes	—
92	C₄×Dic₅₉	M₁₆	1	2	yes	—
93	#11745664	—	2	8	no	—
94	C₂×C₅₉⋊C₈	—	2	1	no	—
95	C₂×C₅₉⋊C₈	—	2	1	no	—
96	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
97	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
98	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
99	C₂×C₅₉⋊C₈	M₁₆	1	2	yes	—
100	C₁₆×C₅₉	—	2	2	no	—
101	C₁₆×C₅₉	—	2	2	no	—
102	C₅₉⋊C₁₆	—	2	2	no	—
103	C₅₉⋊C₁₆	—	2	2	no	—
104	C₂³×C₅₉	—	2	1	no	—
105	C₂³×C₅₉	—	2	1	no	—
106	C₂³×C₅₉	—	2	2	no	—
107	C₂³×C₅₉	—	2	2	no	—
108	C₂³×C₅₉	C₄×C₈	1	2	yes	—
109	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	2	no	—
113	C₂×C₄×C₅₉	—	2	2	no	—
114	C₂×C₄×C₅₉	—	2	2	no	—
115	C₂×C₄×C₅₉	—	2	2	no	—
116	C₂×C₄×C₅₉	C₄×C₈	1	2	yes	—
117	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
123	C₂×Dic₅₉	—	2	2	no	—
124	C₂×Dic₅₉	—	2	2	no	—
125	C₂×Dic₅₉	—	2	2	no	—
126	C₂×Dic₅₉	—	2	2	no	—
127	C₂×Dic₅₉	—	2	2	no	—
128	C₂×Dic₅₉	C₂²⋊C₈	1	4	yes	—
129	C₈×C₅₉	—	2	2	no	—
130	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
131	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
132	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
133	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
134	C₅₉⋊C₈	—	2	2	no	—
135	C₅₉⋊C₈	C₂²⋊C₈	1	4	yes	—
136	#12600	—	59	1	no	maximal, Sylow
137	C₂²×C₅₉	—	2	2	no	—
138	C₂²×C₅₉	—	2	2	no	—
139	C₂²×C₅₉	—	2	2	no	—
140	C₂²×C₅₉	—	4	2	no	—
141	C₂²×C₅₉	—	4	2	no	—
142	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
143	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₂²×C₅₉	#431	1	1	yes	characteristic
145	C₂²×C₅₉	#504	1	2	yes	—
146	C₂²×C₅₉	#504	1	2	yes	—
147	C₄×C₅₉	—	2	2	no	—
148	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
149	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
150	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
151	C₄×C₅₉	#431	1	1	yes	characteristic
152	Dic₅₉	—	2	4	no	—
153	Dic₅₉	—	4	2	no	—
154	C₂×#405	—	59	1	no	—
155	C₂×C₂²⋊C₁₆	—	59	1	no	—
156	C₂×C₂²⋊C₁₆	D₅₉	1	1	yes	characteristic
157	C₂×#346	—	59	1	no	—
158	C₂×C₄×C₁₆	—	59	1	no	—
159	#1374	—	59	1	no	—
160	#1374	—	59	1	no	—
161	C₂×C₅₉	—	4	2	no	—
162	C₂×C₅₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
163	C₂×C₅₉	#2108	1	1	yes	characteristic
164	C₂×C₅₉	#2108	1	1	yes	characteristic
165	C₂×C₅₉	#2113	1	2	yes	—
166	C₂×C₅₉	#2164	1	2	yes	—
167	C₂²×M₁₆	—	59	1	no	—
168	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
169	C₂×#117	—	59	1	no	—
170	C₂²×C₁₆	—	59	1	no	—
171	C₂²×C₁₆	—	59	1	no	—
172	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
173	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
174	#405	—	118	2	no	—
175	#405	—	118	2	no	—
176	C₂×#97	—	59	1	no	—
177	C₂×#97	—	59	1	no	—
178	C₂×C₄×C₈	—	59	1	no	—
179	C₂×C₄×C₈	—	59	1	no	—
180	C₂²⋊C₁₆	—	59	4	no	—
181	C₂²⋊C₁₆	Dic₅₉	1	4	yes	complemented
182	#346	—	118	2	no	—
183	C₄×C₁₆	—	118	2	no	—
184	C₅₉	#12600	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₄×D₅₉	1	1	yes	characteristic
186	C₂×M₁₆	—	59	4	no	—
187	C₂×M₁₆	—	118	2	no	—
188	C₂×M₁₆	—	118	2	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	2	1	no	—
192	C₂²×C₈	—	2	1	no	—
193	C₂²×C₈	—	59	1	no	—
194	C₂²×C₈	—	59	1	no	—
195	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
196	C₂²×C₈	C₄×D₅₉	1	2	yes	—
197	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
198	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
199	#117	—	118	2	no	—
200	#117	—	118	2	no	—
201	C₂×C₄⋊C₄	—	59	2	no	—
202	C₂×C₂²⋊C₄	—	59	2	no	—
203	C₂×C₄²	—	59	1	no	—
204	C₂×C₄²	—	59	1	no	—
205	C₂×C₁₆	—	2	1	no	—
206	C₂×C₁₆	—	2	1	no	—
207	C₂×C₁₆	—	2	1	no	—
208	C₂×C₁₆	—	2	1	no	—
209	C₂×C₁₆	—	59	2	no	—
210	C₂×C₁₆	—	59	2	no	—
211	C₂×C₁₆	—	118	1	no	—
212	C₂×C₁₆	—	118	1	no	—
213	C₂×C₁₆	—	118	1	no	—
214	C₂×C₁₆	—	118	1	no	—
215	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
216	C₂×C₁₆	C₂×Dic₅₉	1	2	yes	—
217	#97	—	118	2	no	—
218	#97	—	118	2	no	—
219	C₄×C₈	—	59	2	no	—
220	C₄×C₈	—	59	2	no	—
221	C₄×C₈	—	118	1	no	—
222	C₄×C₈	—	118	1	no	—
223	C₂⁴	C₈×D₅₉	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	2	1	no	—
228	C₂²×C₄	—	59	2	no	—
229	C₂²×C₄	—	59	2	no	—
230	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
231	C₂²×C₄	C₈×D₅₉	1	2	yes	—
232	C₂²×C₄	C₄×Dic₅₉	1	2	yes	—
233	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
234	M₁₆	—	118	8	no	—
235	C₂×C₈	—	2	1	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	59	2	no	—
242	C₂×C₈	—	59	2	no	—
243	C₂×C₈	—	59	2	no	—
244	C₂×C₈	—	59	2	no	—
245	C₂×C₈	—	118	1	no	—
246	C₂×C₈	—	118	1	no	—
247	C₂×C₈	C₄×Dic₅₉	1	2	yes	—
248	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
249	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
250	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
251	C₂×C₈	#11745687	1	1	yes	characteristic
252	C₂×C₈	#11745687	1	1	yes	characteristic
253	C₂×C₈	C₂²×Dic₅₉	1	1	yes	characteristic
254	C₄⋊C₄	—	118	4	no	—
255	C₂²⋊C₄	—	118	4	no	—
256	C₄²	—	59	2	no	—
257	C₄²	—	59	2	no	—
258	C₄²	—	118	1	no	—
259	C₄²	—	118	1	no	—
260	C₁₆	—	2	2	no	—
261	C₁₆	—	2	2	no	—
262	C₁₆	—	118	2	no	—
263	C₁₆	—	118	2	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	2	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₈×Dic₅₉	1	2	yes	—
269	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	59	4	no	—
277	C₂×C₄	—	118	2	no	—
278	C₂×C₄	—	118	2	no	—
279	C₂×C₄	—	118	2	no	—
280	C₂×C₄	—	118	2	no	—
281	C₂×C₄	—	118	2	no	—
282	C₂×C₄	C₈×Dic₅₉	1	2	yes	—
283	C₂×C₄	C₂×C₄×Dic₅₉	1	1	yes	characteristic
284	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
285	C₂×C₄	#14273683	1	1	yes	characteristic
286	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
287	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
288	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
289	C₈	—	2	2	no	—
290	C₈	—	59	4	no	—
291	C₈	—	118	2	no	—
292	C₈	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
293	C₈	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
294	C₈	D₄×Dic₅₉	1	1	yes	characteristic
295	C₈	D₄×Dic₅₉	1	1	yes	characteristic
296	C₂²	—	2	2	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	4	2	no	—
301	C₂²	C₂×C₈×Dic₅₉	1	1	yes	characteristic
302	C₂²	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
303	C₂²	#16837452	1	1	yes	characteristic
304	C₂²	#16837818	1	2	yes	—
305	C₂²	#16837820	1	2	yes	—
306	C₄	—	2	2	no	—
307	C₄	—	118	4	no	—
308	C₄	—	236	2	no	—
309	C₄	C₂²⋊C₄×Dic₅₉	1	1	yes	characteristic
310	C₄	M₁₆×Dic₅₉	1	1	yes	characteristic
311	C₄	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
312	C₄	#16837452	1	1	yes	characteristic
313	C₂	—	4	2	no	—
314	C₂	C₂²⋊C₈×Dic₅₉	1	1	yes	characteristic
315	C₂	#21425692	1	1	yes	characteristic
316	C₂	#21425695	1	1	yes	characteristic
317	C₂	#21425708	1	2	yes	—
318	C₂	#21425900	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₉
commutator	C₂²×C₅₉	C₂×C₄×C₈
Fitting	C₂×C₅₉×C₂²⋊C₁₆	C₂
socle	C₂³×C₅₉	C₂²×C₈
prime radical	C₂⁴×C₅₉	C₂×C₈
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₂²⋊C₁₆	D₅₉
hypocenter	C₅₉	#12600
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₂²⋊C₁₆

Complements

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

N	H ≅ G/N	type
3. C₂×#16837177	C₂	semidirect
4. C₂×C₁₆×Dic₅₉	C₂	semidirect
5. #21422173	C₂	semidirect
6. #21422173	C₂	semidirect
18. C₂²⋊(C₅₉⋊C₁₆)	C₄	semidirect
24. C₅₉×C₂²⋊C₁₆	C₄	semidirect
181. C₂²⋊C₁₆	Dic₅₉	semidirect
184. C₅₉	#12600	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837357	2
C₂×C₂²⋊(C₅₉⋊C₁₆)	2
C₂×#16837177	2
C₂×C₁₆×Dic₅₉	2
#21422173	2
C₂×C₅₉×C₂²⋊C₁₆	2
#12600	59