Subgroups — #28744212

Order 15104.

514 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434609	C₂	1	1	yes	maximal, characteristic
2	#500×D₅₉	C₂	1	1	yes	maximal, characteristic
3	#21426383	C₂	1	1	yes	maximal, characteristic, complemented
4	#21426382	C₂	1	1	yes	maximal, characteristic, complemented
5	#21426379	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426377	C₂	1	2	yes	maximal, complemented
7	#21426370	C₂	1	2	yes	maximal, complemented
8	#21425883	C₂	1	2	yes	maximal
9	#21425881	C₂	1	2	yes	maximal
10	#21425689	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#2255	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837958	—	2	1	no	—
13	C₄⋊Q₈×D₅₉	—	2	1	no	—
14	C₂×#14273666	C₂²	1	2	yes	—
15	#16837807	C₂²	1	1	yes	characteristic
16	#16837806	—	2	1	no	—
17	#16837806	C₂²	1	1	yes	characteristic
18	#16837806	C₂²	1	2	yes	—
19	#16837806	C₂²	1	1	yes	characteristic
20	#16837805	—	2	1	no	—
21	#16837804	C₂²	1	1	yes	characteristic
22	#16837801	—	2	1	no	—
23	#16837799	—	2	1	no	—
24	#16837439	C₂²	1	1	yes	characteristic
25	#16837439	C₂²	1	1	yes	characteristic
26	#16837437	—	2	1	no	—
27	#107×D₅₉	—	2	1	no	—
28	#107×D₅₉	C₂²	1	2	yes	—
29	C₄×C₈×D₅₉	C₂²	1	1	yes	characteristic
30	#16837230	C₂²	1	2	yes	—
31	#16837229	C₂²	1	2	yes	—
32	#16837229	C₂²	1	2	yes	—
33	#16837228	C₂²	1	2	yes	—
34	#16837227	C₂²	1	2	yes	—
35	#16837227	C₂²	1	2	yes	—
36	#16837178	C₂²	1	1	yes	characteristic
37	C₁₆×Dic₅₉	C₂²	1	2	yes	—
38	C₅₉×#500	C₂²	1	1	yes	characteristic
39	C₅₉×#499	C₂²	1	1	yes	characteristic
40	C₅₉×#367	C₂²	1	2	yes	—
41	C₅₉×#366	C₂²	1	2	yes	—
42	C₅₉×#346	C₂²	1	1	yes	characteristic
43	C₂×Q₈×D₅₉	—	4	1	no	—
44	C₂×C₈×D₅₉	D₄	1	1	yes	characteristic
45	C₂×C₈×D₅₉	C₂³	1	1	yes	characteristic
46	#14273727	—	2	1	no	—
47	#14273726	—	2	2	no	—
48	#14273726	—	2	1	no	—
49	#14273726	—	2	1	no	—
50	#14273726	—	4	1	no	—
51	#14273725	—	4	1	no	—
52	#14273719	—	2	1	no	—
53	#14273705	—	4	1	no	—
54	#14273705	—	4	1	no	—
55	#14273692	—	4	1	no	—
56	#14273692	—	4	1	no	—
57	#14273685	—	2	1	no	—
58	#14273685	—	2	1	no	—
59	C₄⋊C₄×D₅₉	—	2	2	no	—
60	C₄⋊C₄×D₅₉	—	4	1	no	—
61	C₄²×D₅₉	D₄	1	1	yes	characteristic
62	#14273666	Q₈	1	8	yes	complemented
63	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
64	#14273661	—	2	1	no	—
65	#14273661	C₂³	1	2	yes	—
66	#14273661	C₂³	1	1	yes	characteristic
67	#14273661	C₂³	1	1	yes	characteristic
68	#14273660	—	2	2	no	—
69	#14273660	—	2	2	no	—
70	#14273660	—	2	1	no	—
71	#14273660	—	2	1	no	—
72	#14273659	—	2	1	no	—
73	#14273658	—	2	1	no	—
74	#14273658	—	2	1	no	—
75	C₄×C₅₉⋊C₈	D₄	1	1	yes	characteristic
76	C₈×Dic₅₉	D₄	1	1	yes	characteristic
77	C₈×Dic₅₉	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₅₉	—	2	1	no	—
79	#125×C₅₉	—	2	1	no	—
80	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
81	#107×C₅₉	—	2	1	no	—
82	#107×C₅₉	C₂³	1	1	yes	characteristic
83	#107×C₅₉	C₂³	1	2	yes	—
84	#107×C₅₉	C₂³	1	1	yes	characteristic
85	#106×C₅₉	—	2	1	no	—
86	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₅₉	—	4	1	no	—
88	Q₈×D₅₉	—	8	4	no	—
89	C₂×C₄×D₅₉	—	4	2	no	—
90	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
91	C₂×C₄×D₅₉	D₈	1	1	yes	characteristic
92	C₂×C₄×D₅₉	C₂×D₄	1	1	yes	characteristic
93	C₂×Dic₁₁₈	—	4	1	no	—
94	C₂×Dic₁₁₈	—	4	1	no	—
95	C₂×Dic₁₁₈	—	4	1	no	—
96	C₂×Dic₁₁₈	—	4	2	no	—
97	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
99	C₄⋊C₄×C₅₉	—	2	1	no	—
100	C₄⋊C₄×C₅₉	—	2	1	no	—
101	C₄⋊C₄×C₅₉	—	2	2	no	—
102	C₄⋊C₄×C₅₉	—	4	1	no	—
103	C₄⋊C₄×C₅₉	—	4	1	no	—
104	C₄⋊C₄×C₅₉	—	4	1	no	—
105	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
106	(C₂×D₅₉)⋊C₄	—	4	1	no	—
107	(C₂×D₅₉)⋊C₄	—	4	1	no	—
108	C₄⋊Dic₅₉	—	2	1	no	—
109	C₄⋊Dic₅₉	—	2	1	no	—
110	C₄⋊Dic₅₉	—	2	2	no	—
111	C₄⋊Dic₅₉	—	4	1	no	—
112	Dic₅₉⋊C₄	—	4	1	no	—
113	Dic₅₉⋊C₄	—	4	1	no	—
114	Dic₅₉⋊C₄	—	4	2	no	—
115	Dic₅₉⋊C₄	—	4	2	no	—
116	Dic₅₉⋊C₄	—	4	1	no	—
117	Dic₅₉⋊C₄	—	4	1	no	—
118	C₄×Dic₅₉	—	4	1	no	—
119	C₄×Dic₅₉	—	4	1	no	—
120	C₄×Dic₅₉	D₈	1	1	yes	characteristic
121	C₄×Dic₅₉	D₈	1	1	yes	characteristic
122	C₄×Dic₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
124	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
125	C₈×D₅₉	Q₁₆	1	4	yes	—
126	C₈×D₅₉	C₂×Q₈	1	4	yes	—
127	C₁₆×C₅₉	C₂×Q₈	1	4	yes	complemented
128	C₅₉⋊C₁₆	C₂×Q₈	1	4	yes	complemented
129	C₂²×D₅₉	C₂×D₈	1	1	yes	characteristic
130	Q₈×C₅₉	—	8	2	no	—
131	C₂×C₄×C₅₉	—	4	1	no	—
132	C₂×C₄×C₅₉	—	4	1	no	—
133	C₂×C₄×C₅₉	—	4	2	no	—
134	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×Dic₅₉	—	4	1	no	—
138	C₂×Dic₅₉	—	4	1	no	—
139	C₂×Dic₅₉	—	4	2	no	—
140	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
141	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
142	C₂×Dic₅₉	C₂×D₈	1	1	yes	characteristic
143	C₄×D₅₉	—	2	2	no	—
144	C₄×D₅₉	—	2	2	no	—
145	C₄×D₅₉	—	8	4	no	—
146	C₄×D₅₉	C₄⋊Q₈	1	2	yes	—
147	C₄×D₅₉	C₂×Q₁₆	1	2	yes	—
148	Dic₁₁₈	—	8	2	no	—
149	Dic₁₁₈	—	8	2	no	—
150	Dic₁₁₈	—	8	4	no	—
151	Dic₁₁₈	—	8	2	no	—
152	C₈×C₅₉	C₂×Q₁₆	1	2	yes	—
153	C₈×C₅₉	C₂²×Q₈	1	2	yes	—
154	C₅₉⋊C₈	C₄⋊Q₈	1	2	yes	—
155	C₅₉⋊C₈	C₂×Q₁₆	1	2	yes	—
156	#33731	—	59	1	no	maximal, Sylow
157	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₅₉	—	2	1	no	—
159	C₂×D₅₉	—	2	1	no	—
160	C₂×D₅₉	#500	1	2	yes	—
161	C₄×C₅₉	—	2	1	no	—
162	C₄×C₅₉	—	2	1	no	—
163	C₄×C₅₉	—	8	2	no	—
164	C₄×C₅₉	—	8	1	no	—
165	C₄×C₅₉	—	8	1	no	—
166	C₄×C₅₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
167	C₄×C₅₉	C₂²×Q₁₆	1	1	yes	characteristic
168	Dic₅₉	—	8	2	no	—
169	Dic₅₉	—	8	1	no	—
170	Dic₅₉	—	8	1	no	—
171	Dic₅₉	#500	1	2	yes	—
172	Dic₅₉	#509	1	2	yes	—
173	Dic₅₉	#510	1	2	yes	—
174	#3160	—	59	1	no	—
175	C₂×#500	—	59	1	no	—
176	#2255	—	59	1	no	—
177	#2255	—	59	1	no	—
178	#2255	—	59	1	no	—
179	#2255	D₅₉	1	1	yes	characteristic, complemented
180	#2254	—	59	2	no	—
181	#2252	—	59	2	no	—
182	#2159	—	59	2	no	—
183	#2159	—	59	2	no	—
184	C₄×M₃₂	—	59	1	no	—
185	C₂×C₅₉	C₂×#500	1	1	yes	characteristic
186	C₂×C₅₉	C₂×#509	1	1	yes	characteristic
187	C₂×C₅₉	C₂×#510	1	1	yes	characteristic
188	D₅₉	—	2	2	no	—
189	#533	—	118	1	no	—
190	C₂×C₄⋊Q₈	—	118	1	no	—
191	C₂×M₃₂	—	59	2	no	—
192	#500	—	59	2	no	—
193	#500	—	59	1	no	—
194	#500	—	59	1	no	—
195	#500	—	59	1	no	—
196	#500	—	118	1	no	—
197	#500	—	118	1	no	—
198	#500	C₂×D₅₉	1	1	yes	characteristic
199	#499	—	59	1	no	—
200	#499	—	118	1	no	—
201	#499	C₂×D₅₉	1	1	yes	characteristic
202	#498	—	118	1	no	—
203	#427	—	59	1	no	—
204	#427	—	59	1	no	—
205	#427	—	118	1	no	—
206	C₂×#107	—	59	2	no	—
207	C₂×#107	—	118	1	no	—
208	C₂×C₄×C₈	—	59	1	no	—
209	#367	—	59	2	no	—
210	#367	—	59	2	no	—
211	#367	—	59	2	no	—
212	#367	C₂×D₅₉	1	2	yes	—
213	#366	—	59	2	no	—
214	#366	—	59	2	no	—
215	#366	—	59	2	no	—
216	#366	C₂×D₅₉	1	2	yes	—
217	#346	—	59	1	no	—
218	#346	C₂×D₅₉	1	1	yes	characteristic
219	C₄×C₁₆	—	59	2	no	—
220	C₅₉	#33731	1	1	yes	characteristic, complemented, Sylow
221	C₂²×Q₈	—	236	1	no	—
222	C₂²×C₈	—	59	1	no	—
223	C₂²×C₈	—	59	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	118	2	no	—
226	C₄⋊Q₈	—	118	1	no	—
227	C₄⋊Q₈	—	118	1	no	—
228	C₄⋊Q₈	—	118	1	no	—
229	C₄⋊Q₈	—	236	1	no	—
230	C₄⋊Q₈	—	236	1	no	—
231	#125	—	2	1	no	—
232	#125	—	118	1	no	—
233	C₂²⋊Q₈	—	236	1	no	—
234	C₂²⋊Q₈	—	236	1	no	—
235	C₄×Q₈	—	236	1	no	—
236	C₄×Q₈	—	236	1	no	—
237	#117	—	118	1	no	—
238	#117	—	118	1	no	—
239	C₂×C₄⋊C₄	—	118	2	no	—
240	C₂×C₄⋊C₄	—	236	1	no	—
241	C₂×C₄²	—	59	1	no	—
242	M₃₂	—	59	8	no	—
243	C₂×C₁₆	—	59	2	no	—
244	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
245	#107	—	2	1	no	—
246	#107	—	59	1	no	—
247	#107	—	59	2	no	—
248	#107	—	59	1	no	—
249	#107	—	118	2	no	—
250	#107	—	118	2	no	—
251	#107	—	118	1	no	—
252	#107	—	118	1	no	—
253	#107	—	118	1	no	—
254	#107	C₂²×D₅₉	1	1	yes	characteristic
255	#107	C₂²×D₅₉	1	2	yes	—
256	#107	C₂²×D₅₉	1	1	yes	characteristic
257	#106	—	2	1	no	—
258	#106	—	118	1	no	—
259	#106	—	118	1	no	—
260	#106	—	118	1	no	—
261	C₄×C₈	—	59	1	no	—
262	C₄×C₈	—	59	1	no	—
263	C₄×C₈	—	59	1	no	—
264	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
265	C₂×Q₈	—	4	1	no	—
266	C₂×Q₈	—	472	4	no	—
267	C₂×Q₈	—	236	1	no	—
268	C₂×Q₈	—	236	1	no	—
269	C₂×Q₈	—	236	2	no	—
270	C₂×Q₈	—	236	1	no	—
271	C₂²×C₄	—	59	1	no	—
272	C₂²×C₄	—	59	1	no	—
273	C₂²×C₄	—	59	1	no	—
274	C₂²×C₄	—	236	2	no	—
275	C₂×C₈	—	59	1	no	—
276	C₂×C₈	—	59	1	no	—
277	C₂×C₈	—	59	4	no	—
278	C₂×C₈	—	59	4	no	—
279	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
280	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	—	2	1	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	118	1	no	—
288	C₄⋊C₄	—	118	1	no	—
289	C₄⋊C₄	—	118	2	no	—
290	C₄⋊C₄	—	236	2	no	—
291	C₄⋊C₄	—	236	2	no	—
292	C₄⋊C₄	—	236	1	no	—
293	C₄⋊C₄	—	236	1	no	—
294	C₄⋊C₄	—	236	1	no	—
295	C₄⋊C₄	—	236	1	no	—
296	C₄⋊C₄	—	236	1	no	—
297	C₂²⋊C₄	—	236	1	no	—
298	C₂²⋊C₄	—	236	1	no	—
299	C₄²	—	59	1	no	—
300	C₄²	—	59	1	no	—
301	C₄²	—	59	1	no	—
302	C₄²	—	236	1	no	—
303	C₄²	—	236	1	no	—
304	C₄²	D₄×D₅₉	1	1	yes	characteristic
305	C₁₆	—	59	4	no	—
306	C₁₆	Q₈×D₅₉	1	4	yes	complemented
307	C₂³	—	59	1	no	—
308	Q₈	—	8	2	no	—
309	Q₈	—	472	2	no	—
310	Q₈	—	472	4	no	—
311	Q₈	—	472	2	no	—
312	Q₈	—	472	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	59	1	no	—
317	C₂×C₄	—	59	1	no	—
318	C₂×C₄	—	59	1	no	—
319	C₂×C₄	—	59	2	no	—
320	C₂×C₄	—	59	2	no	—
321	C₂×C₄	—	118	2	no	—
322	C₂×C₄	—	118	2	no	—
323	C₂×C₄	—	472	4	no	—
324	C₂×C₄	—	236	1	no	—
325	C₂×C₄	—	236	1	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
328	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
330	C₈	—	59	2	no	—
331	C₈	—	59	2	no	—
332	C₈	Q₁₆×D₅₉	1	2	yes	—
333	C₈	C₂×Q₈×D₅₉	1	2	yes	—
334	C₂²	—	59	2	no	—
335	C₂²	—	118	1	no	—
336	C₂²	—	118	1	no	—
337	C₂²	C₂×D₈×D₅₉	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	8	2	no	—
341	C₄	—	8	1	no	—
342	C₄	—	8	1	no	—
343	C₄	—	59	2	no	—
344	C₄	—	59	2	no	—
345	C₄	—	59	2	no	—
346	C₄	—	472	2	no	—
347	C₄	—	472	1	no	—
348	C₄	—	472	1	no	—
349	C₄	C₄⋊Q₈×D₅₉	1	1	yes	characteristic
350	C₄	C₂×Q₁₆×D₅₉	1	1	yes	characteristic
351	C₂	—	118	2	no	—
352	C₂	#500×D₅₉	1	1	yes	characteristic
353	C₂	#509×D₅₉	1	1	yes	characteristic
354	C₂	#510×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₅₉
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2255	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂²×D₅₉	C₂×D₈
hypercenter	#2255	D₅₉
hypocenter	C₅₉	#33731
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2255

Complements

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

N	H ≅ G/N	type
3. #21426383	C₂	semidirect
4. #21426382	C₂	semidirect
5. #21426379	C₂	semidirect
6. #21426377	C₂	semidirect
7. #21426370	C₂	semidirect
11. C₅₉×#2255	C₂	semidirect
62. #14273666	Q₈	semidirect
127. C₁₆×C₅₉	C₂×Q₈	semidirect
128. C₅₉⋊C₁₆	C₂×Q₈	semidirect
179. #2255	D₅₉	semidirect
220. C₅₉	#33731	semidirect
306. C₁₆	Q₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21434609	2
#500×D₅₉	2
#21426383	2
#21426382	2
#21426379	2
#21426377	2
#21426370	2
#21425883	2
#21425881	2
#21425689	2
C₅₉×#2255	2
#33731	59