Subgroups — #28741682

Order 15104.

706 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837820	C₂	1	2	yes	maximal
2	C₄×#14273733	C₂	1	1	yes	maximal, characteristic
3	#21425856	C₂	1	2	yes	maximal, complemented
4	#21425853	C₂	1	4	yes	maximal, complemented
5	#21425850	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄⋊C₁₆×D₅₉	C₂	1	2	yes	maximal, complemented
8	C₅₉×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273733	C₄	1	1	yes	characteristic
10	C₂×#14273733	C₂²	1	1	yes	characteristic
11	C₄×#11745685	C₄	1	1	yes	characteristic
12	#16837820	—	2	8	no	—
13	C₂×#14273666	C₂²	1	4	yes	—
14	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
15	C₂×C₁₆×D₅₉	C₂²	1	4	yes	—
16	#16837357	C₄	1	1	yes	characteristic
17	#16837357	C₂²	1	1	yes	characteristic
18	#16837356	C₄	1	2	yes	—
19	#16837356	C₂²	1	2	yes	—
20	C₄×#11745664	C₄	1	1	yes	characteristic
21	C₄×#11745659	C₄	1	2	yes	—
22	C₄×C₈×D₅₉	C₂²	1	2	yes	—
23	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
24	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
25	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
26	Dic₅₉⋊C₁₆	C₂²	1	4	yes	—
27	Dic₅₉⋊C₁₆	C₂²	1	4	yes	—
28	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
30	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
32	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
33	#14273733	—	2	4	no	—
34	#14273733	D₄	1	4	yes	—
35	#14273733	Q₈	1	4	yes	—
36	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
37	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
38	C₂×#11745659	C₂×C₄	1	2	yes	—
39	C₂×#11745659	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
41	C₂×C₈×D₅₉	C₂³	1	2	yes	—
42	C₄×Dic₁₁₈	C₈	1	1	yes	characteristic
43	C₄×Dic₁₁₈	C₈	1	1	yes	characteristic
44	C₄×D₂₃₆	C₈	1	1	yes	characteristic
45	C₄×D₂₃₆	C₈	1	1	yes	characteristic
46	C₄×C₂²⋊D₅₉	C₈	1	2	yes	—
47	C₄×C₂²⋊D₅₉	C₈	1	2	yes	—
48	#14273686	C₂×C₄	1	1	yes	characteristic
49	#14273684	C₂×C₄	1	2	yes	—
50	C₄²×D₅₉	C₂×C₄	1	2	yes	—
51	#14273666	—	2	8	no	—
52	C₂×C₅₉⋊C₁₆	—	2	4	no	—
53	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
54	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
55	C₁₆×D₅₉	—	2	8	no	—
56	#14273638	C₂×C₄	1	2	yes	—
57	#14273637	C₂×C₄	1	2	yes	—
58	#14273637	C₂×C₄	1	2	yes	—
59	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
60	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
61	C₈×Dic₅₉	C₂³	1	2	yes	—
62	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
65	M₃₂×C₅₉	—	2	4	no	—
66	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
67	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
69	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
72	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
73	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₅₉	C₂×C₈	1	2	yes	—
75	#11745685	C₄⋊C₄	1	4	yes	—
76	#11745685	M₁₆	1	4	yes	—
77	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
78	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
79	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
80	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
81	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
82	C₂×C₈×C₅₉	—	2	2	no	—
83	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
88	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
89	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
90	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
91	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
93	C₂²⋊Dic₅₉	C₂×C₈	1	2	yes	—
94	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
95	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
96	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
97	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
98	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
99	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
100	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
101	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
102	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
103	#11745664	—	2	2	no	—
104	#11745664	C₄⋊C₄	1	4	yes	—
105	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
106	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
107	#11745659	—	2	4	no	—
108	#11745659	C₄⋊C₄	1	8	yes	—
109	C₈×D₅₉	—	2	4	no	—
110	C₈×D₅₉	C₂×D₄	1	4	yes	—
111	C₈×D₅₉	C₂×Q₈	1	4	yes	—
112	C₁₆×C₅₉	—	2	4	no	—
113	C₅₉⋊C₁₆	—	2	2	no	—
114	C₅₉⋊C₁₆	—	2	2	no	—
115	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
116	C₂²×D₅₉	C₂²×C₈	1	2	yes	—
117	C₂×C₄×C₅₉	—	2	2	no	—
118	C₂×C₄×C₅₉	—	4	1	no	—
119	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
126	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
127	C₂²⋊D₅₉	C₄⋊C₈	1	8	yes	—
128	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
129	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
130	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
131	D₂₃₆	C₄⋊C₈	1	4	yes	—
132	C₄×D₅₉	—	2	4	no	—
133	C₄×D₅₉	—	4	2	no	—
134	C₄×D₅₉	C₂×C₄⋊C₄	1	4	yes	—
135	C₄×D₅₉	C₂×M₁₆	1	4	yes	—
136	Dic₁₁₈	C₄⋊C₈	1	4	yes	—
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₈×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
140	C₈×C₅₉	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
142	C₅₉⋊C₈	—	2	2	no	—
143	C₅₉⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
144	#33398	—	59	1	no	maximal, Sylow
145	C₂²×C₅₉	—	2	1	no	—
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
148	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
149	C₂×D₅₉	—	2	2	no	—
150	C₂×D₅₉	—	2	2	no	—
151	C₂×D₅₉	C₂×C₄⋊C₈	1	4	yes	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	C₂×C₄⋊C₈	1	2	yes	—
156	C₄×C₅₉	#504	1	2	yes	complemented
157	C₄×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
159	Dic₅₉	—	2	2	no	—
160	Dic₅₉	—	2	2	no	—
161	Dic₅₉	C₂×C₄⋊C₈	1	4	yes	—
162	C₂×#504	—	59	2	no	—
163	C₄×#131	—	59	1	no	—
164	#2151	—	59	2	no	—
165	#2151	—	59	4	no	—
166	#2150	—	59	2	no	—
167	#2150	D₅₉	1	1	yes	characteristic, complemented
168	C₂×C₄⋊C₁₆	—	59	1	no	—
169	C₂×C₄⋊C₁₆	—	59	2	no	—
170	C₂×C₅₉	—	2	2	no	—
171	C₂×C₅₉	C₂²×C₄⋊C₈	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#504	1	1	yes	characteristic
173	C₂×C₅₉	#3407	1	1	yes	characteristic
174	D₅₉	—	2	4	no	—
175	C₂×#131	—	59	1	no	—
176	C₂×#131	—	59	1	no	—
177	C₄×#41	—	59	1	no	—
178	#504	—	118	8	no	—
179	C₂×M₃₂	—	59	4	no	—
180	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
181	C₂²×C₁₆	—	59	2	no	—
182	C₂²×C₁₆	—	59	4	no	—
183	#405	—	59	1	no	—
184	#405	—	59	1	no	—
185	#405	—	59	2	no	—
186	#405	—	59	2	no	—
187	C₄×M₁₆	—	59	1	no	—
188	C₄×M₁₆	—	59	2	no	—
189	C₂×C₄×C₈	—	59	2	no	—
190	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
191	C₄⋊C₁₆	—	59	4	no	—
192	C₄⋊C₁₆	—	59	4	no	—
193	C₄⋊C₁₆	—	59	1	no	—
194	C₄⋊C₁₆	—	59	1	no	—
195	C₄⋊C₁₆	—	59	2	no	—
196	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
197	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
198	C₅₉	#33398	1	1	yes	characteristic, complemented, Sylow
199	C₂×#41	—	59	1	no	—
200	#131	—	59	4	no	—
201	#131	—	59	4	no	—
202	#131	—	118	4	no	—
203	C₂×M₁₆	—	59	1	no	—
204	C₂×M₁₆	—	59	1	no	—
205	C₂×M₁₆	—	59	2	no	—
206	C₂×M₁₆	—	59	2	no	—
207	C₂²×C₈	—	59	2	no	—
208	C₂²×C₈	—	59	2	no	—
209	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
210	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
211	C₄×Q₈	—	59	1	no	—
212	C₄×Q₈	—	59	1	no	—
213	C₄×D₄	—	59	2	no	—
214	C₄×D₄	—	59	2	no	—
215	C₄×D₄	—	59	1	no	—
216	C₄×D₄	—	59	1	no	—
217	#117	—	59	1	no	—
218	#117	—	59	2	no	—
219	C₂×C₄²	—	59	2	no	—
220	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
221	M₃₂	—	2	4	no	—
222	M₃₂	—	118	8	no	—
223	C₂×C₁₆	—	59	2	no	—
224	C₂×C₁₆	—	59	2	no	—
225	C₂×C₁₆	—	118	8	no	—
226	C₂×C₁₆	—	118	4	no	—
227	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
228	#97	—	59	2	no	—
229	#97	—	59	2	no	—
230	#97	—	59	2	no	—
231	C₄×C₈	—	59	2	no	—
232	C₄×C₈	—	59	2	no	—
233	C₄×C₈	—	59	2	no	—
234	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
235	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
238	#41	—	59	4	no	—
239	#41	—	59	4	no	—
240	C₂×Q₈	—	59	1	no	—
241	C₂×D₄	—	59	2	no	—
242	C₂×D₄	—	59	1	no	—
243	C₂²×C₄	—	59	2	no	—
244	C₂²×C₄	—	59	2	no	—
245	C₂²×C₄	—	59	2	no	—
246	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
247	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
248	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
249	M₁₆	—	59	4	no	—
250	M₁₆	—	59	8	no	—
251	M₁₆	—	118	4	no	—
252	M₁₆	—	118	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	59	2	no	—
255	C₂×C₈	—	59	4	no	—
256	C₂×C₈	—	59	4	no	—
257	C₂×C₈	—	59	2	no	—
258	C₂×C₈	—	118	4	no	—
259	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
260	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
261	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
262	C₂×C₈	D₄×D₅₉	1	2	yes	—
263	C₂×C₈	Q₈×D₅₉	1	2	yes	—
264	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
265	C₄⋊C₄	—	59	2	no	—
266	C₄⋊C₄	—	59	2	no	—
267	C₄⋊C₄	—	59	1	no	—
268	C₄⋊C₄	—	59	1	no	—
269	C₂²⋊C₄	—	59	2	no	—
270	C₂²⋊C₄	—	59	2	no	—
271	C₂²⋊C₄	—	59	2	no	—
272	C₄²	—	59	2	no	—
273	C₄²	—	59	2	no	—
274	C₄²	—	59	2	no	—
275	C₄²	C₈×D₅₉	1	1	yes	characteristic
276	C₄²	C₈×D₅₉	1	1	yes	characteristic
277	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
278	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
279	C₁₆	—	2	4	no	—
280	C₁₆	—	118	2	no	—
281	C₁₆	—	118	2	no	—
282	C₂³	—	59	2	no	—
283	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
284	Q₈	—	59	4	no	—
285	D₄	—	59	8	no	—
286	D₄	—	59	4	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	59	2	no	—
290	C₂×C₄	—	59	4	no	—
291	C₂×C₄	—	59	4	no	—
292	C₂×C₄	—	59	2	no	—
293	C₂×C₄	—	59	2	no	—
294	C₂×C₄	—	118	4	no	—
295	C₂×C₄	—	236	2	no	—
296	C₂×C₄	C₄⋊C₄×D₅₉	1	2	yes	—
297	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
298	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
299	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
300	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
301	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
302	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
303	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	59	4	no	—
307	C₈	—	118	2	no	—
308	C₈	C₄⋊C₄×D₅₉	1	2	yes	—
309	C₈	C₂×D₄×D₅₉	1	1	yes	characteristic
310	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	59	4	no	—
314	C₂²	—	118	2	no	—
315	C₂²	—	118	2	no	—
316	C₂²	C₄⋊C₈×D₅₉	1	2	yes	—
317	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	59	4	no	—
322	C₄	—	118	2	no	—
323	C₄	—	118	2	no	—
324	C₄	C₄⋊C₈×D₅₉	1	2	yes	—
325	C₄	#16837820	1	2	yes	complemented
326	C₄	C₂×C₄⋊C₄×D₅₉	1	1	yes	characteristic
327	C₄	C₂×M₁₆×D₅₉	1	1	yes	characteristic
328	C₂	—	2	2	no	—
329	C₂	—	118	4	no	—
330	C₂	C₂×C₄⋊C₈×D₅₉	1	1	yes	characteristic
331	C₂	C₂×#16837820	1	1	yes	characteristic
332	C₂	#21437198	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	C₅₉×#2150	C₂
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2150	D₅₉
hypocenter	C₅₉	#33398
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ #2150

Complements

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

N	H ≅ G/N	type
3. #21425856	C₂	semidirect
4. #21425853	C₂	semidirect
5. #21425850	C₂	semidirect
6. C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
7. C₄⋊C₁₆×D₅₉	C₂	semidirect
8. C₅₉×#2150	C₂	semidirect
156. C₄×C₅₉	#504	semidirect
167. #2150	D₅₉	semidirect
198. C₅₉	#33398	semidirect
325. C₄	#16837820	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837820	2
C₄×#14273733	2
#21425856	2
#21425853	2
#21425850	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
C₄⋊C₁₆×D₅₉	2
C₅₉×#2150	2
#33398	59