Subgroups — #28505611

Order 15104.

466 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437198	C₂	1	1	yes	maximal, characteristic, complemented
2	#21432815	C₂	1	1	yes	maximal, characteristic
3	#21425725	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425723	C₂	1	1	yes	maximal, characteristic, complemented
5	#21425716	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422176	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1375	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	C₂×#14273733	C₂²	1	1	yes	characteristic
10	#16837952	C₄	1	1	yes	characteristic
11	#16837820	—	2	2	no	—
12	#16837818	—	2	2	no	—
13	C₂×#14273667	C₂²	1	1	yes	characteristic
14	C₂×#14273667	C₂²	1	1	yes	characteristic
15	M₃₂×D₅₉	—	2	2	no	—
16	#16837475	—	2	2	no	—
17	#16837475	—	2	2	no	—
18	#16837426	C₄	1	1	yes	characteristic
19	#16837368	C₄	1	1	yes	characteristic
20	#16837368	C₄	1	1	yes	characteristic
21	#16837187	—	2	2	no	—
22	#16837185	—	2	2	no	—
23	#16837183	—	2	2	no	—
24	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
28	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
29	#14273733	—	2	1	no	—
30	#14273733	—	2	1	no	—
31	#14273733	—	2	1	no	—
32	#14273733	—	2	1	no	—
33	#14273733	—	2	1	no	—
34	#14273733	—	2	1	no	—
35	#14273733	—	2	1	no	—
36	#14273733	—	2	1	no	—
37	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
38	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
39	C₂×#11745659	—	2	1	no	—
40	C₂×#11745659	D₄	1	2	yes	—
41	C₂×C₈×D₅₉	—	2	1	no	—
42	C₂×C₈×D₅₉	D₄	1	2	yes	—
43	#14273727	C₈	1	1	yes	characteristic
44	#14273724	C₈	1	1	yes	characteristic
45	#14273715	C₈	1	1	yes	characteristic
46	#14273715	C₈	1	1	yes	characteristic
47	#14273701	—	2	2	no	—
48	C₄×D₂₃₆	—	2	2	no	—
49	#14273667	—	2	1	no	—
50	#14273667	—	2	1	no	—
51	#14273667	—	2	1	no	—
52	#14273667	—	2	1	no	—
53	#14273666	—	2	4	no	—
54	C₂×C₅₉⋊C₁₆	—	2	1	no	—
55	C₂×C₅₉⋊C₁₆	D₄	1	2	yes	complemented
56	C₁₆×D₅₉	—	2	4	no	—
57	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
58	(C₂×D₅₉)⋊C₈	—	2	2	no	—
59	(C₂×D₅₉)⋊C₈	—	2	2	no	—
60	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
63	M₃₂×C₅₉	—	2	1	no	—
64	M₃₂×C₅₉	—	2	1	no	—
65	M₃₂×C₅₉	—	2	1	no	—
66	M₃₂×C₅₉	—	2	1	no	—
67	C₂×C₁₆×C₅₉	—	2	1	no	—
68	C₂×C₁₆×C₅₉	D₄	1	2	yes	complemented
69	#97×C₅₉	—	2	1	no	—
70	#97×C₅₉	—	2	1	no	—
71	C₂²×C₄×C₅₉	—	2	1	no	—
72	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₂²⋊D₅₉	—	2	1	no	—
74	C₂×C₂²⋊D₅₉	M₁₆	1	2	yes	—
75	#11745685	—	2	1	no	—
76	#11745685	—	2	1	no	—
77	#11745685	—	2	1	no	—
78	#11745685	—	2	1	no	—
79	#11745685	—	2	1	no	—
80	#11745685	—	2	1	no	—
81	#11745685	—	2	1	no	—
82	#11745685	—	2	1	no	—
83	C₂×D₂₃₆	—	2	2	no	—
84	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
85	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
86	C₂×C₄×D₅₉	—	2	1	no	—
87	C₂×C₄×D₅₉	C₂²⋊C₄	1	2	yes	—
88	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
89	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
90	C₂×C₈×C₅₉	—	2	1	no	—
91	C₂×C₈×C₅₉	—	2	1	no	—
92	C₂×C₈×C₅₉	—	2	1	no	—
93	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
98	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
99	C₄²×C₅₉	—	2	1	no	—
100	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
101	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
102	(C₂×D₅₉)⋊C₄	—	2	2	no	—
103	(C₂×D₅₉)⋊C₄	—	2	2	no	—
104	C₄⋊Dic₅₉	—	2	2	no	—
105	#11745664	—	2	1	no	—
106	#11745664	—	2	1	no	—
107	#11745664	—	2	1	no	—
108	#11745664	—	2	1	no	—
109	C₂×C₅₉⋊C₈	—	2	1	no	—
110	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
111	#11745659	—	2	2	no	—
112	#11745659	—	2	2	no	—
113	#11745659	—	4	1	no	—
114	#11745659	—	4	1	no	—
115	C₈×D₅₉	—	2	2	no	—
116	C₈×D₅₉	—	2	2	no	—
117	C₈×D₅₉	—	4	1	no	—
118	C₈×D₅₉	—	4	1	no	—
119	C₁₆×C₅₉	—	2	2	no	—
120	C₁₆×C₅₉	—	4	1	no	—
121	C₅₉⋊C₁₆	—	2	2	no	—
122	C₅₉⋊C₁₆	—	4	1	no	—
123	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
124	C₂²×D₅₉	—	2	1	no	—
125	C₂²×D₅₉	C₂²⋊C₈	1	2	yes	—
126	C₂×C₄×C₅₉	—	2	1	no	—
127	C₂×C₄×C₅₉	—	2	1	no	—
128	C₂×C₄×C₅₉	—	8	1	no	—
129	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
134	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
135	C₂²⋊D₅₉	—	2	2	no	—
136	C₂²⋊D₅₉	—	2	2	no	—
137	C₂²⋊D₅₉	—	4	1	no	—
138	C₂²⋊D₅₉	—	4	1	no	—
139	C₂×Dic₅₉	—	2	1	no	—
140	C₂×Dic₅₉	C₂²⋊C₈	1	2	yes	—
141	D₂₃₆	—	2	1	no	—
142	D₂₃₆	—	2	1	no	—
143	D₂₃₆	—	2	1	no	—
144	D₂₃₆	—	2	1	no	—
145	D₂₃₆	—	8	2	no	—
146	C₄×D₅₉	—	2	2	no	—
147	C₄×D₅₉	—	2	2	no	—
148	C₄×D₅₉	—	4	1	no	—
149	C₄×D₅₉	—	4	1	no	—
150	Dic₁₁₈	—	2	1	no	—
151	Dic₁₁₈	—	2	1	no	—
152	Dic₁₁₈	—	2	1	no	—
153	Dic₁₁₈	—	2	1	no	—
154	C₈×C₅₉	—	4	1	no	—
155	C₈×C₅₉	#120	1	1	yes	characteristic
156	C₈×C₅₉	#121	1	1	yes	characteristic
157	C₈×C₅₉	#121	1	1	yes	characteristic
158	C₈×C₅₉	#123	1	1	yes	characteristic
159	C₅₉⋊C₈	—	2	2	no	—
160	C₅₉⋊C₈	—	4	1	no	—
161	#12988	—	59	1	no	maximal, Sylow
162	C₂²×C₅₉	—	4	1	no	—
163	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
164	C₂²×C₅₉	#435	1	1	yes	characteristic
165	C₂²×C₅₉	#435	1	1	yes	characteristic
166	C₂×D₅₉	—	2	2	no	—
167	C₂×D₅₉	—	4	2	no	—
168	C₂×D₅₉	—	4	2	no	—
169	C₂×D₅₉	—	4	1	no	—
170	C₄×C₅₉	—	4	1	no	—
171	C₄×C₅₉	—	4	1	no	—
172	C₄×C₅₉	#386	1	1	yes	characteristic
173	C₄×C₅₉	#408	1	1	yes	characteristic
174	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
175	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
176	Dic₅₉	—	2	2	no	—
177	Dic₅₉	—	4	1	no	—
178	#3407	—	59	1	no	—
179	#2932	—	59	1	no	—
180	#2117	—	59	1	no	—
181	#2117	—	59	1	no	—
182	#2115	—	59	1	no	—
183	#1375	—	59	1	no	—
184	#1375	D₅₉	1	1	yes	characteristic, complemented
185	C₂×C₅₉	—	2	1	no	—
186	C₂×C₅₉	—	2	1	no	—
187	C₂×C₅₉	—	2	1	no	—
188	C₂×C₅₉	#1869	1	1	yes	characteristic
189	D₅₉	—	4	2	no	—
190	D₅₉	—	8	1	no	—
191	C₂×#131	—	59	1	no	—
192	C₂×#131	—	59	1	no	—
193	#532	—	59	1	no	—
194	#504	—	118	2	no	—
195	#504	—	118	2	no	—
196	C₂×M₃₂	—	59	1	no	—
197	C₂×M₃₂	—	59	1	no	—
198	C₂×M₃₂	—	118	2	no	—
199	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
200	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
201	#435	—	118	2	no	—
202	#435	—	118	2	no	—
203	#423	—	59	1	no	—
204	#408	—	59	1	no	—
205	#408	—	59	1	no	—
206	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
207	#350	—	118	2	no	—
208	#350	—	118	2	no	—
209	#349	—	118	2	no	—
210	C₅₉	#12988	1	1	yes	characteristic, complemented, Sylow
211	C₂×#41	—	59	1	no	—
212	C₂×#41	—	59	1	no	—
213	#131	—	118	1	no	—
214	#131	—	118	1	no	—
215	#131	—	118	1	no	—
216	#131	—	118	1	no	—
217	#131	—	118	1	no	—
218	#131	—	118	1	no	—
219	#131	—	118	1	no	—
220	#131	—	118	1	no	—
221	C₂×M₁₆	—	59	1	no	—
222	C₂×M₁₆	—	59	2	no	—
223	C₂×M₁₆	—	59	1	no	—
224	C₂×M₁₆	—	118	1	no	—
225	C₂²×C₈	—	59	2	no	—
226	C₂²×C₈	—	118	1	no	—
227	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
228	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
229	C₄⋊Q₈	—	59	1	no	—
230	#127	—	59	1	no	—
231	#124	—	59	1	no	—
232	#124	—	59	1	no	—
233	#121	—	118	2	no	—
234	C₄×D₄	—	118	2	no	—
235	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
236	M₃₂	—	2	1	no	—
237	M₃₂	—	2	1	no	—
238	M₃₂	—	2	1	no	—
239	M₃₂	—	2	1	no	—
240	M₃₂	—	118	4	no	—
241	M₃₂	—	118	1	no	—
242	M₃₂	—	118	1	no	—
243	M₃₂	—	118	1	no	—
244	M₃₂	—	118	1	no	—
245	C₂×C₁₆	—	2	1	no	—
246	C₂×C₁₆	—	59	2	no	—
247	C₂×C₁₆	—	118	4	no	—
248	C₂×C₁₆	—	118	1	no	—
249	C₂×C₁₆	D₂₃₆	1	2	yes	complemented
250	C₄⋊C₈	—	118	2	no	—
251	C₂²⋊C₈	—	118	2	no	—
252	C₂²⋊C₈	—	118	2	no	—
253	#97	—	2	1	no	—
254	#97	—	2	1	no	—
255	#41	—	118	1	no	—
256	#41	—	118	1	no	—
257	#41	—	118	1	no	—
258	#41	—	118	1	no	—
259	#41	—	118	1	no	—
260	#41	—	118	1	no	—
261	#41	—	118	1	no	—
262	#41	—	118	1	no	—
263	C₂×Q₈	—	59	1	no	—
264	C₂×Q₈	—	59	1	no	—
265	C₂×D₄	—	59	2	no	—
266	C₂×D₄	—	59	1	no	—
267	C₂×D₄	—	59	1	no	—
268	C₂×D₄	—	118	2	no	—
269	C₂×D₄	—	118	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	59	2	no	—
272	C₂²×C₄	—	118	1	no	—
273	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
274	M₁₆	—	118	1	no	—
275	M₁₆	—	118	1	no	—
276	M₁₆	—	118	2	no	—
277	M₁₆	—	118	2	no	—
278	M₁₆	—	118	1	no	—
279	M₁₆	—	118	1	no	—
280	M₁₆	—	236	1	no	—
281	M₁₆	—	236	1	no	—
282	C₂×C₈	—	2	1	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	2	1	no	—
285	C₂×C₈	—	59	2	no	—
286	C₂×C₈	—	118	2	no	—
287	C₂×C₈	—	118	2	no	—
288	C₂×C₈	—	118	1	no	—
289	C₂×C₈	—	236	1	no	—
290	C₂×C₈	—	236	1	no	—
291	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
292	C₂×C₈	#11745685	1	1	yes	characteristic
293	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
294	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
295	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
296	C₂×C₈	#11745689	1	1	yes	characteristic
297	C₄⋊C₄	—	118	2	no	—
298	C₂²⋊C₄	—	118	2	no	—
299	C₂²⋊C₄	—	118	2	no	—
300	C₄²	—	2	1	no	—
301	C₄²	C₈×D₅₉	1	1	yes	characteristic
302	C₄²	C₈×D₅₉	1	1	yes	characteristic
303	C₁₆	—	2	2	no	—
304	C₁₆	—	4	1	no	—
305	C₁₆	—	118	2	no	—
306	C₁₆	—	236	1	no	—
307	C₂³	—	59	2	no	—
308	C₂³	—	118	1	no	—
309	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
310	Q₈	—	118	1	no	—
311	Q₈	—	118	1	no	—
312	Q₈	—	118	1	no	—
313	Q₈	—	118	1	no	—
314	D₄	—	118	2	no	—
315	D₄	—	118	2	no	—
316	D₄	—	118	1	no	—
317	D₄	—	118	1	no	—
318	D₄	—	118	1	no	—
319	D₄	—	118	1	no	—
320	D₄	—	472	2	no	—
321	D₄	—	236	1	no	—
322	D₄	—	236	1	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	8	1	no	—
326	C₂×C₄	—	59	2	no	—
327	C₂×C₄	—	118	2	no	—
328	C₂×C₄	—	118	2	no	—
329	C₂×C₄	—	118	1	no	—
330	C₂×C₄	—	236	1	no	—
331	C₂×C₄	—	236	1	no	—
332	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
333	C₂×C₄	#14273689	1	1	yes	characteristic
334	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
335	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
336	C₂×C₄	#14273733	1	1	yes	characteristic
337	C₂×C₄	#14273735	1	1	yes	characteristic
338	C₈	—	4	1	no	—
339	C₈	—	118	2	no	—
340	C₈	—	236	1	no	—
341	C₈	#14273697	1	1	yes	characteristic
342	C₈	#14273700	1	1	yes	characteristic
343	C₈	#14273702	1	1	yes	characteristic
344	C₈	#14273712	1	1	yes	characteristic
345	C₂²	—	4	1	no	—
346	C₂²	—	118	2	no	—
347	C₂²	—	236	2	no	—
348	C₂²	—	236	2	no	—
349	C₂²	—	236	1	no	—
350	C₂²	C₂²⋊C₈×D₅₉	1	1	yes	characteristic
351	C₂²	#16837472	1	1	yes	characteristic
352	C₂²	#16837475	1	1	yes	characteristic
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	118	2	no	—
356	C₄	—	236	1	no	—
357	C₄	#16837280	1	1	yes	characteristic
358	C₄	#16837367	1	1	yes	characteristic
359	C₄	#16837467	1	1	yes	characteristic
360	C₄	C₈×D₂₃₆	1	1	yes	characteristic
361	C₂	—	2	1	no	—
362	C₂	—	2	1	no	—
363	C₂	—	2	1	no	—
364	C₂	—	472	1	no	—
365	C₂	—	236	2	no	—
366	C₂	#21424387	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837280
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1375	C₂
socle	C₂×C₅₉	#1869
prime radical	#16837952	C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1375	D₅₉
hypocenter	C₅₉	#12988
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1375

Complements

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

N	H ≅ G/N	type
1. #21437198	C₂	semidirect
3. #21425725	C₂	semidirect
4. #21425723	C₂	semidirect
5. #21425716	C₂	semidirect
6. #21422176	C₂	semidirect
7. C₅₉×#1375	C₂	semidirect
55. C₂×C₅₉⋊C₁₆	D₄	semidirect
68. C₂×C₁₆×C₅₉	D₄	semidirect
184. #1375	D₅₉	semidirect
210. C₅₉	#12988	semidirect
249. C₂×C₁₆	D₂₃₆	semidirect

Maximal subgroups

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

subgroup	index
#21437198	2
#21432815	2
#21425725	2
#21425723	2
#21425716	2
#21422176	2
C₅₉×#1375	2
#12988	59