Subgroups — #28580217

Order 15104.

606 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21436964	C₂	1	1	yes	maximal, characteristic, complemented
2	#21436444	C₂	1	1	yes	maximal, characteristic
3	#21436414	C₂	1	2	yes	maximal, complemented
4	#21423706	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423672	C₂	1	2	yes	maximal, complemented
6	#21423495	C₂	1	1	yes	maximal, characteristic, complemented
7	#21423490	C₂	1	1	yes	maximal, characteristic, complemented
8	#21423312	C₂	1	2	yes	maximal, complemented
9	#21423249	C₂	1	2	yes	maximal, complemented
10	#21422982	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838112	C₂²	1	2	yes	—
13	#16838056	C₂²	1	2	yes	—
14	#16837807	C₂²	1	1	yes	characteristic
15	#16837807	C₂²	1	1	yes	characteristic
16	#16837804	—	2	1	no	—
17	#16837753	C₂²	1	2	yes	—
18	#16837753	C₂²	1	2	yes	—
19	#16837734	—	2	2	no	—
20	#16837716	—	2	2	no	—
21	#16837703	—	2	2	no	—
22	#16837693	C₂²	1	2	yes	—
23	#16837692	C₂²	1	2	yes	—
24	#16837683	C₂²	1	2	yes	—
25	#16837659	—	2	2	no	—
26	#16837638	—	2	1	no	—
27	#16837612	C₂²	1	2	yes	—
28	#16837561	—	2	2	no	—
29	C₄×D₄₇₂	—	2	1	no	—
30	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
31	#16837481	C₂²	1	2	yes	—
32	C₂×#14273661	—	2	1	no	—
33	#16837400	—	2	2	no	—
34	#16837155	C₂²	1	1	yes	characteristic
35	#16837150	C₂²	1	2	yes	—
36	#16837147	C₂²	1	2	yes	—
37	#16837131	C₂²	1	2	yes	—
38	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₄×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
40	C₈×D₄×C₅₉	C₂²	1	1	yes	characteristic
41	C₅₉×#334	C₂²	1	1	yes	characteristic
42	C₅₉×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₅₉×D₄⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₅₉	C₂²	1	1	yes	characteristic
45	C₂×#11745687	—	2	2	no	—
46	C₂×#11745672	—	2	2	no	—
47	C₂×D₄₇₂	D₄	1	1	yes	characteristic
48	C₂×#11745659	—	2	2	no	—
49	#14273727	C₂³	1	2	yes	—
50	#14273727	C₂³	1	2	yes	—
51	#14273719	—	2	2	no	—
52	#14273711	—	4	2	no	—
53	#14273707	—	2	2	no	—
54	#14273705	—	4	2	no	—
55	#14273703	—	2	2	no	—
56	#14273701	—	2	2	no	—
57	C₄×D₂₃₆	—	2	2	no	—
58	C₄×D₂₃₆	C₂³	1	2	yes	—
59	C₂×C₄⋊Dic₅₉	—	2	2	no	—
60	C₄⋊C₄×D₅₉	—	2	2	no	—
61	#14273661	—	2	1	no	—
62	#14273661	—	2	1	no	—
63	#14273661	—	2	1	no	—
64	#14273661	—	2	1	no	—
65	#14273661	—	2	2	no	—
66	#14273661	D₄	1	1	yes	characteristic
67	#14273659	—	2	2	no	—
68	#14273658	—	2	2	no	—
69	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
70	#14273652	—	2	2	no	—
71	#14273652	—	2	2	no	—
72	#14273652	—	4	2	no	—
73	#14273650	—	2	2	no	—
74	#14273649	—	2	2	no	—
75	#14273649	—	2	2	no	—
76	#14273649	D₄	1	2	yes	—
77	#14273648	—	2	2	no	—
78	(C₂×D₅₉)⋊C₈	—	2	2	no	—
79	C₂×SD₁₆×C₅₉	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₅₉	—	2	1	no	—
81	C₄×Q₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₄×D₄×C₅₉	C₂³	1	1	yes	characteristic
83	#106×C₅₉	D₄	1	1	yes	characteristic
84	C₄⋊C₈×C₅₉	—	2	1	no	—
85	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
86	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
87	#103×C₅₉	D₄	1	1	yes	characteristic
88	#102×C₅₉	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₅₉	—	2	1	no	—
90	C₄×C₈×C₅₉	—	2	1	no	—
91	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
95	C₂×D₄×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₂²×C₄×C₅₉	—	2	1	no	—
97	C₂×C₂²⋊D₅₉	—	4	2	no	—
98	C₂²×Dic₅₉	—	4	2	no	—
99	#11745687	—	4	4	no	—
100	#11745687	—	8	2	no	—
101	C₂×D₂₃₆	C₂×D₄	1	2	yes	—
102	C₂×C₄×D₅₉	—	2	2	no	—
103	C₂×C₄×D₅₉	—	4	2	no	—
104	C₂×Dic₁₁₈	—	2	2	no	—
105	C₂×Dic₁₁₈	—	2	2	no	—
106	SD₁₆×C₅₉	—	2	2	no	—
107	C₂×C₈×C₅₉	—	2	1	no	—
108	C₂×C₈×C₅₉	—	2	1	no	—
109	C₂×C₈×C₅₉	—	2	1	no	—
110	C₂×C₈×C₅₉	—	4	2	no	—
111	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₅₉	—	2	1	no	—
116	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
117	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₂²⋊C₄×C₅₉	—	2	1	no	—
119	C₄²×C₅₉	—	2	1	no	—
120	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
121	#11745672	—	8	2	no	—
122	(C₂×D₅₉)⋊C₄	—	2	2	no	—
123	(C₂×D₅₉)⋊C₄	—	4	2	no	—
124	(C₂×D₅₉)⋊C₄	—	4	2	no	—
125	C₄⋊Dic₅₉	—	2	2	no	—
126	C₄⋊Dic₅₉	—	2	2	no	—
127	C₄⋊Dic₅₉	—	2	2	no	—
128	C₄⋊Dic₅₉	—	2	2	no	—
129	C₄⋊Dic₅₉	—	4	2	no	—
130	C₄⋊Dic₅₉	C₂×D₄	1	2	yes	—
131	Dic₅₉⋊C₄	—	4	2	no	—
132	Dic₅₉⋊C₄	—	4	2	no	—
133	C₂×C₅₉⋊C₈	—	2	2	no	—
134	D₄₇₂	—	2	2	no	—
135	#11745659	—	8	2	no	—
136	C₂³×C₅₉	—	2	1	no	—
137	C₂²×D₅₉	—	2	2	no	—
138	Q₈×C₅₉	—	2	1	no	—
139	Q₈×C₅₉	—	2	1	no	—
140	D₄×C₅₉	—	2	1	no	—
141	D₄×C₅₉	#135	1	2	yes	—
142	C₂×C₄×C₅₉	—	2	1	no	—
143	C₂×C₄×C₅₉	—	2	1	no	—
144	C₂×C₄×C₅₉	—	2	1	no	—
145	C₂×C₄×C₅₉	—	4	1	no	—
146	C₂×C₄×C₅₉	—	4	1	no	—
147	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
149	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂²⋊D₅₉	—	8	4	no	—
151	C₂×Dic₅₉	—	2	2	no	—
152	C₂×Dic₅₉	—	4	2	no	—
153	C₂×Dic₅₉	—	4	2	no	—
154	C₂×Dic₅₉	—	8	4	no	—
155	D₂₃₆	—	2	2	no	—
156	D₂₃₆	—	2	2	no	—
157	C₄×D₅₉	—	4	2	no	—
158	C₄×D₅₉	—	4	2	no	—
159	C₄×D₅₉	—	16	2	no	—
160	Dic₁₁₈	—	4	2	no	—
161	Dic₁₁₈	—	4	2	no	—
162	Dic₁₁₈	—	8	2	no	—
163	C₈×C₅₉	—	2	1	no	—
164	C₈×C₅₉	—	2	1	no	—
165	C₈×C₅₉	—	2	1	no	—
166	C₈×C₅₉	—	4	1	no	—
167	C₈×C₅₉	—	4	1	no	—
168	C₈×C₅₉	—	4	1	no	—
169	C₈×C₅₉	#137	1	2	yes	—
170	C₅₉⋊C₈	—	4	2	no	—
171	#19903	—	59	1	no	maximal, Sylow
172	C₂²×C₅₉	—	2	2	no	—
173	C₂²×C₅₉	—	4	1	no	—
174	C₂²×C₅₉	—	4	1	no	—
175	C₂²×C₅₉	D₄²	1	1	yes	characteristic
176	C₂×D₅₉	—	4	2	no	—
177	C₂×D₅₉	—	4	2	no	—
178	C₂×D₅₉	—	4	2	no	—
179	C₄×C₅₉	—	2	1	no	—
180	C₄×C₅₉	—	4	1	no	—
181	C₄×C₅₉	—	4	1	no	—
182	C₄×C₅₉	—	4	1	no	—
183	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
184	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
185	C₄×C₅₉	#576	1	1	yes	characteristic
186	C₄×C₅₉	#578	1	1	yes	characteristic
187	Dic₅₉	—	4	2	no	—
188	Dic₅₉	—	8	2	no	—
189	Dic₅₉	—	8	2	no	—
190	#3380	—	59	1	no	—
191	#3334	—	59	1	no	—
192	#3331	—	59	2	no	—
193	#1710	—	59	1	no	—
194	#1703	—	59	2	no	—
195	#1675	—	59	1	no	—
196	#1674	—	59	1	no	—
197	#1644	—	59	2	no	—
198	#1635	—	59	2	no	—
199	#1589	—	59	1	no	—
200	C₈×SD₁₆	D₅₉	1	1	yes	characteristic, complemented
201	C₂×C₅₉	—	4	2	no	—
202	C₂×C₅₉	#3276	1	1	yes	characteristic
203	C₂×C₅₉	#3280	1	1	yes	characteristic
204	C₂×C₅₉	#3290	1	1	yes	characteristic
205	D₅₉	—	8	2	no	—
206	#554	—	59	2	no	—
207	#547	—	59	2	no	—
208	#500	—	59	1	no	—
209	#500	—	59	1	no	—
210	#499	—	118	1	no	—
211	#487	—	59	2	no	—
212	#487	—	59	2	no	—
213	#483	—	118	2	no	—
214	#480	—	118	2	no	—
215	#478	—	118	2	no	—
216	#476	—	59	2	no	—
217	#476	—	59	2	no	—
218	#474	—	59	2	no	—
219	#470	—	118	2	no	—
220	#466	—	118	1	no	—
221	#462	—	59	2	no	—
222	#452	—	118	2	no	—
223	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
224	C₄×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
225	C₄×D₈	—	59	1	no	—
226	C₄×D₈	—	118	1	no	—
227	#436	—	59	2	no	—
228	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
229	C₂×#107	—	118	1	no	—
230	#417	—	118	2	no	—
231	#335	—	59	1	no	—
232	#334	C₂×D₅₉	1	1	yes	characteristic
233	#333	—	59	2	no	—
234	#332	—	59	2	no	—
235	Q₈⋊C₈	C₂×D₅₉	1	1	yes	characteristic
236	D₄⋊C₈	—	59	2	no	—
237	D₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
238	C₈²	C₂×D₅₉	1	1	yes	characteristic
239	C₅₉	#19903	1	1	yes	characteristic, complemented, Sylow
240	C₂×#41	—	118	2	no	—
241	C₂×Q₁₆	—	118	2	no	—
242	C₂×SD₁₆	D₂₃₆	1	1	yes	characteristic
243	C₂×D₈	—	59	1	no	—
244	C₂×M₁₆	—	118	2	no	—
245	C₂²×C₈	—	2	1	no	—
246	C₄⋊Q₈	—	59	2	no	—
247	C₄⋊Q₈	—	59	2	no	—
248	#125	—	118	2	no	—
249	#123	—	236	2	no	—
250	C₂²⋊Q₈	—	118	2	no	—
251	C₂²⋊Q₈	—	118	2	no	—
252	C₂²⋊Q₈	—	236	2	no	—
253	#121	—	118	2	no	—
254	C₄×Q₈	C₂²×D₅₉	1	1	yes	characteristic
255	C₄×D₄	—	59	2	no	—
256	C₄×D₄	—	118	2	no	—
257	C₄×D₄	C₂²×D₅₉	1	1	yes	characteristic
258	C₂×C₄⋊C₄	—	118	2	no	—
259	C₂×C₄⋊C₄	—	118	2	no	—
260	#107	—	59	1	no	—
261	#107	—	118	2	no	—
262	#107	—	118	1	no	—
263	#107	—	118	1	no	—
264	#107	—	118	1	no	—
265	#107	—	118	1	no	—
266	#106	—	118	2	no	—
267	#106	—	118	2	no	—
268	#106	D₂₃₆	1	1	yes	characteristic
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	—	59	2	no	—
271	C₄⋊C₈	C₂²×D₅₉	1	1	yes	characteristic
272	C₄⋊C₈	C₂²×D₅₉	1	1	yes	characteristic
273	#103	—	118	2	no	—
274	#103	—	118	2	no	—
275	#103	—	118	2	no	—
276	#103	—	236	2	no	—
277	#103	D₂₃₆	1	1	yes	characteristic
278	#102	—	59	2	no	—
279	#102	—	118	2	no	—
280	#102	—	118	2	no	—
281	#102	—	118	2	no	—
282	#102	D₂₃₆	1	1	yes	characteristic
283	C₂²⋊C₈	—	2	1	no	—
284	C₂²⋊C₈	—	118	2	no	—
285	C₄×C₈	—	2	1	no	—
286	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
287	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
288	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
289	#41	—	472	2	no	—
290	#41	—	236	4	no	—
291	C₂×Q₈	—	118	2	no	—
292	C₂×Q₈	—	118	2	no	—
293	C₂×Q₈	C₂×D₂₃₆	1	1	yes	characteristic
294	C₂×D₄	—	59	2	no	—
295	C₂×D₄	—	236	2	no	—
296	C₂×D₄	C₂×D₂₃₆	1	1	yes	characteristic
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	118	2	no	—
299	C₂²×C₄	—	236	2	no	—
300	C₂²×C₄	—	236	2	no	—
301	Q₁₆	—	472	2	no	—
302	SD₁₆	—	2	2	no	—
303	D₈	—	118	2	no	—
304	M₁₆	—	472	2	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	2	1	no	—
308	C₂×C₈	—	4	2	no	—
309	C₂×C₈	—	118	2	no	—
310	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
311	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
312	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
313	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
314	C₄⋊C₄	—	2	1	no	—
315	C₄⋊C₄	—	59	2	no	—
316	C₄⋊C₄	—	118	2	no	—
317	C₄⋊C₄	—	118	2	no	—
318	C₄⋊C₄	—	118	2	no	—
319	C₄⋊C₄	—	118	2	no	—
320	C₄⋊C₄	—	236	2	no	—
321	C₄⋊C₄	—	236	2	no	—
322	C₄⋊C₄	—	236	2	no	—
323	C₄⋊C₄	C₂×D₂₃₆	1	1	yes	characteristic
324	C₄⋊C₄	C₂×D₂₃₆	1	1	yes	characteristic
325	C₂²⋊C₄	—	2	1	no	—
326	C₂²⋊C₄	—	118	2	no	—
327	C₂²⋊C₄	—	236	2	no	—
328	C₂²⋊C₄	—	236	2	no	—
329	C₄²	—	2	1	no	—
330	C₄²	C₂³×D₅₉	1	1	yes	characteristic
331	C₂³	—	2	1	no	—
332	C₂³	—	118	2	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	2	1	no	—
335	Q₈	—	472	2	no	—
336	Q₈	—	236	2	no	—
337	Q₈	—	236	2	no	—
338	D₄	—	2	1	no	—
339	D₄	—	118	2	no	—
340	D₄	—	118	2	no	—
341	D₄	—	472	4	no	—
342	D₄	#14273749	1	2	yes	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	4	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	118	2	no	—
349	C₂×C₄	—	944	2	no	—
350	C₂×C₄	—	472	4	no	—
351	C₂×C₄	—	236	2	no	—
352	C₂×C₄	—	236	2	no	—
353	C₂×C₄	—	236	2	no	—
354	C₂×C₄	—	236	2	no	—
355	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
356	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
357	C₂×C₄	#14273773	1	1	yes	characteristic
358	C₈	—	2	1	no	—
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	1	no	—
363	C₈	—	4	1	no	—
364	C₈	—	236	2	no	—
365	C₈	#14273756	1	2	yes	—
366	C₂²	—	2	2	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	236	2	no	—
370	C₂²	—	236	2	no	—
371	C₂²	—	236	2	no	—
372	C₂²	D₄×D₂₃₆	1	1	yes	characteristic
373	C₄	—	2	1	no	—
374	C₄	—	4	1	no	—
375	C₄	—	4	1	no	—
376	C₄	—	4	1	no	—
377	C₄	—	472	2	no	—
378	C₄	—	472	2	no	—
379	C₄	—	236	2	no	—
380	C₄	C₂×#14273749	1	1	yes	characteristic
381	C₄	C₂×#14273756	1	1	yes	characteristic
382	C₄	#16838207	1	1	yes	characteristic
383	C₄	#16838216	1	1	yes	characteristic
384	C₂	—	4	2	no	—
385	C₂	—	472	2	no	—
386	C₂	#21435743	1	1	yes	characteristic
387	C₂	#21435783	1	1	yes	characteristic
388	C₂	#21435883	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃₆
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₈×SD₁₆×C₅₉	C₂
socle	C₂²×C₅₉	D₄²
prime radical	#21436444	C₂
hypercenter	C₈×SD₁₆	D₅₉
hypocenter	C₅₉	#19903
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×SD₁₆

Complements

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

N	H ≅ G/N	type
1. #21436964	C₂	semidirect
3. #21436414	C₂	semidirect
4. #21423706	C₂	semidirect
5. #21423672	C₂	semidirect
6. #21423495	C₂	semidirect
7. #21423490	C₂	semidirect
8. #21423312	C₂	semidirect
9. #21423249	C₂	semidirect
10. #21422982	C₂	semidirect
11. C₈×SD₁₆×C₅₉	C₂	semidirect
200. C₈×SD₁₆	D₅₉	semidirect
239. C₅₉	#19903	semidirect

Maximal subgroups

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

subgroup	index
#21436964	2
#21436444	2
#21436414	2
#21423706	2
#21423672	2
#21423495	2
#21423490	2
#21423312	2
#21423249	2
#21422982	2
C₈×SD₁₆×C₅₉	2
#19903	59