Subgroups — #28504729

Order 15104.

510 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837484	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₅₉⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837204	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837202	C₂	1	1	yes	maximal, characteristic
5	#21424254	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422187	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1380	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₇₂	C₂²	1	1	yes	characteristic
9	C₂×#14273689	—	2	1	no	—
10	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
11	#16837484	—	2	2	no	—
12	#16837484	C₄	1	4	yes	complemented
13	C₂×#14273661	C₂²	1	1	yes	characteristic
14	C₂×#14273660	—	2	1	no	—
15	C₂×#14273647	—	2	1	no	—
16	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
17	#16837271	—	2	1	no	—
18	Dic₅₉⋊C₁₆	—	2	2	no	—
19	#16837204	—	2	2	no	—
20	#16837202	—	2	2	no	—
21	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
22	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×#107×C₅₉	C₂²	1	1	yes	characteristic
24	C₂²×D₂₃₆	—	2	1	no	—
25	C₂²×C₄×D₅₉	—	4	1	no	—
26	C₂×D₄₇₂	—	2	1	no	—
27	C₂×D₄₇₂	—	2	1	no	—
28	C₂×D₄₇₂	—	2	1	no	—
29	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
30	C₂×D₄₇₂	D₄	1	2	yes	—
31	C₂×D₄₇₂	Q₈	1	2	yes	—
32	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
33	#14273689	—	2	4	no	—
34	#14273689	—	4	2	no	—
35	C₂×C₄⋊Dic₅₉	—	2	1	no	—
36	C₂×Dic₅₉⋊C₄	—	4	1	no	—
37	C₂×Dic₅₉⋊C₄	—	4	1	no	—
38	C₂×(C₂×D₅₉)⋊C₄	—	4	1	no	—
39	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	—	2	1	no	—
42	C₂×C₅₉⋊C₁₆	—	2	1	no	—
43	#14273661	—	2	1	no	—
44	#14273661	—	2	1	no	—
45	#14273660	—	4	2	no	—
46	#14273647	—	2	4	no	—
47	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
48	C₈×Dic₅₉	D₄	1	2	yes	—
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
51	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
52	C₂×C₁₆×C₅₉	—	2	1	no	—
53	C₂×C₁₆×C₅₉	—	2	1	no	—
54	C₂×C₁₆×C₅₉	—	2	1	no	—
55	#107×C₅₉	—	2	1	no	—
56	#107×C₅₉	—	2	1	no	—
57	#107×C₅₉	—	2	1	no	—
58	#107×C₅₉	C₂×C₄	1	2	yes	—
59	C₂³×D₅₉	—	4	1	no	—
60	C₂²×C₄×C₅₉	—	4	1	no	—
61	C₂²×C₄×C₅₉	—	4	1	no	—
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₅₉	—	4	1	no	—
64	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
65	C₂²×Dic₅₉	D₈	1	1	yes	characteristic
66	C₂×D₂₃₆	—	2	2	no	—
67	C₂×D₂₃₆	—	2	2	no	—
68	C₂×D₂₃₆	—	2	2	no	—
69	C₂×D₂₃₆	—	4	1	no	—
70	C₂×D₂₃₆	—	4	1	no	—
71	C₂×D₂₃₆	—	4	1	no	—
72	C₂×C₄×D₅₉	—	4	4	no	—
73	C₂×C₄×D₅₉	—	8	2	no	—
74	C₂×C₄×D₅₉	—	8	2	no	—
75	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
79	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
80	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	—	2	2	no	—
82	C₄⋊C₄×C₅₉	—	4	1	no	—
83	C₄⋊C₄×C₅₉	—	4	1	no	—
84	C₄⋊C₄×C₅₉	—	4	1	no	—
85	(C₂×D₅₉)⋊C₄	—	4	4	no	—
86	C₄⋊Dic₅₉	—	4	1	no	—
87	C₄⋊Dic₅₉	—	4	1	no	—
88	Dic₅₉⋊C₄	—	8	2	no	—
89	Dic₅₉⋊C₄	—	8	2	no	—
90	C₄×Dic₅₉	C₂²⋊C₄	1	2	yes	—
91	C₄×Dic₅₉	SD₁₆	1	2	yes	—
92	C₂×C₅₉⋊C₈	—	2	1	no	—
93	C₂×C₅₉⋊C₈	—	2	1	no	—
94	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
95	D₄₇₂	—	2	2	no	—
96	D₄₇₂	—	2	2	no	—
97	D₄₇₂	—	2	2	no	—
98	D₄₇₂	C₄⋊C₄	1	4	yes	complemented
99	C₁₆×C₅₉	—	4	1	no	—
100	C₅₉⋊C₁₆	—	4	1	no	—
101	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
102	C₂²×D₅₉	—	4	2	no	—
103	C₂²×D₅₉	—	4	2	no	—
104	C₂²×D₅₉	—	4	2	no	—
105	C₂²×D₅₉	—	8	1	no	—
106	C₂²×D₅₉	—	8	1	no	—
107	C₂²×D₅₉	—	8	1	no	—
108	C₂²×D₅₉	—	8	1	no	—
109	C₂×C₄×C₅₉	—	4	2	no	—
110	C₂×C₄×C₅₉	—	8	1	no	—
111	C₂×C₄×C₅₉	—	8	1	no	—
112	C₂×C₄×C₅₉	—	8	1	no	—
113	C₂×C₄×C₅₉	—	8	1	no	—
114	C₂×C₄×C₅₉	—	8	1	no	—
115	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	8	1	no	—
124	C₂×Dic₅₉	—	8	1	no	—
125	C₂×Dic₅₉	—	8	1	no	—
126	C₂×Dic₅₉	#102	1	2	yes	—
127	C₂×Dic₅₉	#102	1	2	yes	—
128	C₂×Dic₅₉	D₁₆	1	2	yes	—
129	C₂×Dic₅₉	SD₃₂	1	2	yes	—
130	D₂₃₆	—	2	4	no	—
131	D₂₃₆	—	4	2	no	—
132	D₂₃₆	—	4	2	no	—
133	D₂₃₆	—	4	2	no	—
134	C₄×D₅₉	—	8	8	no	—
135	C₈×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
136	C₈×C₅₉	#117	1	1	yes	characteristic
137	C₈×C₅₉	#121	1	1	yes	characteristic
138	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
139	C₅₉⋊C₈	—	2	2	no	—
140	#12811	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
142	C₂²×C₅₉	#459	1	1	yes	characteristic
143	C₂²×C₅₉	#474	1	1	yes	characteristic
144	C₂²×C₅₉	C₂×D₁₆	1	1	yes	characteristic
145	C₂²×C₅₉	C₂×SD₃₂	1	1	yes	characteristic
146	C₂²×C₅₉	#509	1	1	yes	characteristic
147	C₂²×C₅₉	#510	1	1	yes	characteristic
148	C₂×D₅₉	—	4	4	no	—
149	C₂×D₅₉	—	8	2	no	—
150	C₂×D₅₉	—	8	2	no	—
151	C₂×D₅₉	—	8	2	no	—
152	C₂×D₅₉	—	8	2	no	—
153	C₂×D₅₉	—	8	2	no	—
154	C₂×D₅₉	—	8	2	no	—
155	C₄×C₅₉	—	8	2	no	—
156	C₄×C₅₉	—	16	1	no	—
157	C₄×C₅₉	#380	1	1	yes	characteristic
158	C₄×C₅₉	#417	1	1	yes	characteristic
159	C₄×C₅₉	#461	1	1	yes	characteristic
160	C₄×C₅₉	#476	1	1	yes	characteristic
161	Dic₅₉	—	2	2	no	—
162	Dic₅₉	—	16	1	no	—
163	Dic₅₉	#357	1	4	yes	complemented
164	C₂×C₄×D₈	—	59	1	no	—
165	C₂×C₄⋊C₁₆	—	59	1	no	—
166	C₂×#357	—	59	1	no	—
167	C₂×#357	—	59	1	no	—
168	#1845	—	59	1	no	—
169	#1380	—	59	1	no	—
170	#1380	D₅₉	1	1	yes	characteristic, complemented
171	C₂×C₅₉	#1802	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#357	1	1	yes	characteristic
173	C₂×C₅₉	#2139	1	1	yes	characteristic
174	C₂×C₅₉	#2206	1	1	yes	characteristic
175	C₂×C₅₉	#2208	1	1	yes	characteristic
176	C₂×C₅₉	#2224	1	1	yes	characteristic
177	C₂×C₅₉	#2226	1	1	yes	characteristic
178	D₅₉	—	8	4	no	—
179	C₂²×D₈	—	59	1	no	—
180	C₂×C₄×D₄	—	118	1	no	—
181	C₂²×C₁₆	—	59	1	no	—
182	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
183	C₄×D₈	—	59	4	no	—
184	C₄×D₈	—	118	2	no	—
185	C₂×#107	—	59	1	no	—
186	C₂×#107	—	118	1	no	—
187	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
188	C₂×#107	C₂×D₅₉	1	1	yes	characteristic
189	C₂×#102	—	118	1	no	—
190	C₂×C₄×C₈	—	59	1	no	—
191	#384	—	118	1	no	—
192	C₄⋊C₁₆	—	118	2	no	—
193	#357	—	118	2	no	—
194	#357	—	118	2	no	—
195	C₅₉	#12811	1	1	yes	characteristic, complemented, Sylow
196	C₂²×D₄	—	118	1	no	—
197	C₂³×C₄	—	236	1	no	—
198	C₂×D₈	—	59	2	no	—
199	C₂×D₈	—	59	2	no	—
200	C₂×D₈	—	59	2	no	—
201	C₂×D₈	—	118	1	no	—
202	C₂×D₈	—	118	1	no	—
203	C₂×D₈	—	118	1	no	—
204	C₂²×C₈	—	59	1	no	—
205	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
206	C₄×D₄	—	118	4	no	—
207	C₄×D₄	—	236	2	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	—	2	1	no	—
210	C₂×C₄⋊C₄	—	118	1	no	—
211	C₂×C₄⋊C₄	—	236	1	no	—
212	C₂×C₄⋊C₄	—	236	1	no	—
213	C₂×C₂²⋊C₄	—	236	1	no	—
214	C₂×C₄²	—	59	1	no	—
215	C₂×C₁₆	—	2	1	no	—
216	C₂×C₁₆	—	2	1	no	—
217	C₂×C₁₆	—	2	1	no	—
218	C₂×C₁₆	—	118	1	no	—
219	C₂×C₁₆	—	118	1	no	—
220	C₂×C₁₆	—	118	1	no	—
221	#107	—	2	1	no	—
222	#107	—	2	1	no	—
223	#107	—	2	1	no	—
224	#107	—	118	1	no	—
225	#107	—	118	1	no	—
226	#107	—	236	2	no	—
227	#107	C₄×D₅₉	1	2	yes	—
228	#102	—	118	4	no	—
229	C₄×C₈	—	59	2	no	—
230	C₄×C₈	—	59	2	no	—
231	C₂⁴	—	236	1	no	—
232	C₂×D₄	—	118	2	no	—
233	C₂×D₄	—	118	2	no	—
234	C₂×D₄	—	118	2	no	—
235	C₂×D₄	—	236	1	no	—
236	C₂×D₄	—	236	1	no	—
237	C₂×D₄	—	236	1	no	—
238	C₂²×C₄	—	4	1	no	—
239	C₂²×C₄	—	4	1	no	—
240	C₂²×C₄	—	59	1	no	—
241	C₂²×C₄	—	59	1	no	—
242	C₂²×C₄	—	472	2	no	—
243	C₂²×C₄	—	472	2	no	—
244	C₂²×C₄	—	236	4	no	—
245	C₂²×C₄	—	236	1	no	—
246	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
247	D₈	—	59	4	no	—
248	D₈	—	118	2	no	—
249	D₈	—	118	2	no	—
250	D₈	—	118	2	no	—
251	C₂×C₈	—	59	2	no	—
252	C₂×C₈	—	118	1	no	—
253	C₂×C₈	—	118	1	no	—
254	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
255	C₂×C₈	#11745685	1	1	yes	characteristic
256	C₂×C₈	#11745685	1	1	yes	characteristic
257	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
258	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
259	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	472	2	no	—
265	C₄⋊C₄	—	472	2	no	—
266	C₄⋊C₄	—	236	1	no	—
267	C₄⋊C₄	—	236	1	no	—
268	C₂²⋊C₄	—	236	4	no	—
269	C₄²	—	59	2	no	—
270	C₄²	—	59	2	no	—
271	C₁₆	—	4	1	no	—
272	C₁₆	—	236	1	no	—
273	C₂³	—	472	1	no	—
274	C₂³	—	472	1	no	—
275	C₂³	—	472	1	no	—
276	C₂³	—	472	1	no	—
277	C₂³	—	236	2	no	—
278	C₂³	—	236	2	no	—
279	C₂³	—	236	2	no	—
280	C₂³	D₈×D₅₉	1	1	yes	characteristic
281	D₄	—	118	4	no	—
282	D₄	—	236	2	no	—
283	D₄	—	236	2	no	—
284	D₄	—	236	2	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	8	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	8	1	no	—
289	C₂×C₄	—	8	1	no	—
290	C₂×C₄	—	8	1	no	—
291	C₂×C₄	—	59	2	no	—
292	C₂×C₄	—	59	2	no	—
293	C₂×C₄	—	59	2	no	—
294	C₂×C₄	—	59	2	no	—
295	C₂×C₄	—	118	1	no	—
296	C₂×C₄	—	118	1	no	—
297	C₂×C₄	—	472	8	no	—
298	C₂×C₄	—	472	1	no	—
299	C₂×C₄	—	472	1	no	—
300	C₂×C₄	—	472	1	no	—
301	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
302	C₂×C₄	#14273700	1	1	yes	characteristic
303	C₂×C₄	#14273705	1	1	yes	characteristic
304	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
305	C₂×C₄	#14273754	1	1	yes	characteristic
306	C₂×C₄	#14273759	1	1	yes	characteristic
307	C₈	—	118	2	no	—
308	C₈	C₄⋊C₄×D₅₉	1	1	yes	characteristic
309	C₈	#14273684	1	1	yes	characteristic
310	C₈	#14273700	1	1	yes	characteristic
311	C₈	#14273705	1	1	yes	characteristic
312	C₂²	—	472	2	no	—
313	C₂²	—	472	2	no	—
314	C₂²	—	472	2	no	—
315	C₂²	—	472	2	no	—
316	C₂²	—	472	2	no	—
317	C₂²	—	472	2	no	—
318	C₂²	—	236	4	no	—
319	C₂²	#102×D₅₉	1	1	yes	characteristic
320	C₂²	#16837596	1	1	yes	characteristic
321	C₂²	#16837681	1	1	yes	characteristic
322	C₂²	D₁₆×D₅₉	1	1	yes	characteristic
323	C₂²	SD₃₂×D₅₉	1	1	yes	characteristic
324	C₂²	#16837839	1	1	yes	characteristic
325	C₂²	#16837844	1	1	yes	characteristic
326	C₄	—	8	2	no	—
327	C₄	—	16	1	no	—
328	C₄	—	59	4	no	—
329	C₄	—	118	2	no	—
330	C₄	—	944	1	no	—
331	C₄	#16837261	1	1	yes	characteristic
332	C₄	#16837402	1	1	yes	characteristic
333	C₄	#16837606	1	1	yes	characteristic
334	C₄	#16837691	1	1	yes	characteristic
335	C₂	—	472	4	no	—
336	C₂	#21424095	1	1	yes	characteristic
337	C₂	#357×D₅₉	1	1	yes	characteristic
338	C₂	#21425807	1	1	yes	characteristic
339	C₂	#21426130	1	1	yes	characteristic
340	C₂	#21426140	1	1	yes	characteristic
341	C₂	#21426232	1	1	yes	characteristic
342	C₂	#21426242	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	D₈×D₅₉
commutator	C₂×C₈×C₅₉	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1380	C₂
socle	C₂³×C₅₉	C₂×D₈
prime radical	C₂²×D₄₇₂	C₂²
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1380	D₅₉
hypocenter	C₅₉	#12811
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1380

Complements

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

N	H ≅ G/N	type
2. C₂×Dic₅₉⋊C₁₆	C₂	semidirect
5. #21424254	C₂	semidirect
6. #21422187	C₂	semidirect
7. C₅₉×#1380	C₂	semidirect
12. #16837484	C₄	semidirect
98. D₄₇₂	C₄⋊C₄	semidirect
163. Dic₅₉	#357	semidirect
170. #1380	D₅₉	semidirect
195. C₅₉	#12811	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837484	2
C₂×Dic₅₉⋊C₁₆	2
C₂×#16837204	2
C₂×#16837202	2
#21424254	2
#21422187	2
C₅₉×#1380	2
#12811	59