Subgroups — #28580291

Order 15104.

638 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437071	C₂	1	1	yes	maximal, characteristic
2	#21436964	C₂	1	1	yes	maximal, characteristic, complemented
3	#21436436	C₂	1	2	yes	maximal, complemented
4	#21423688	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423630	C₂	1	2	yes	maximal, complemented
6	#21423500	C₂	1	1	yes	maximal, characteristic, complemented
7	#21423456	C₂	1	1	yes	maximal, characteristic, complemented
8	#21423225	C₂	1	2	yes	maximal, complemented
9	#21423171	C₂	1	2	yes	maximal, complemented
10	#21422982	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838112	C₂²	1	2	yes	—
13	#16838074	C₂²	1	2	yes	—
14	#16837807	C₂²	1	1	yes	characteristic
15	#16837804	—	2	1	no	—
16	#16837779	—	2	1	no	—
17	#16837779	C₂²	1	1	yes	characteristic
18	#16837775	C₂²	1	2	yes	—
19	#16837749	C₂²	1	2	yes	—
20	#16837717	—	2	2	no	—
21	#16837703	—	2	2	no	—
22	#16837697	C₂²	1	2	yes	—
23	#16837693	C₂²	1	2	yes	—
24	#16837651	—	2	2	no	—
25	#16837602	C₂²	1	2	yes	—
26	#16837598	—	2	2	no	—
27	#16837598	C₂²	1	2	yes	—
28	#16837563	—	2	2	no	—
29	C₄×D₄₇₂	—	2	1	no	—
30	C₄×D₄₇₂	—	2	1	no	—
31	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
32	#16837481	C₂²	1	2	yes	—
33	#16837402	—	2	2	no	—
34	#16837155	C₂²	1	1	yes	characteristic
35	#16837150	C₂²	1	2	yes	—
36	#16837145	C₂²	1	2	yes	—
37	#16837131	C₂²	1	2	yes	—
38	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
39	C₈×Q₈×C₅₉	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
41	C₅₉×#335	C₂²	1	1	yes	characteristic
42	C₅₉×Q₈⋊C₈	C₂²	1	1	yes	characteristic
43	C₅₉×Q₈⋊C₈	C₂²	1	1	yes	characteristic
44	C₈²×C₅₉	C₂²	1	1	yes	characteristic
45	C₂×#11745689	—	2	2	no	—
46	C₂×#11745660	—	4	2	no	—
47	C₂×#11745671	—	2	2	no	—
48	C₂×D₄₇₂	—	2	1	no	—
49	C₂×D₄₇₂	—	2	2	no	—
50	C₂×D₄₇₂	—	2	1	no	—
51	C₂×D₄₇₂	D₄	1	1	yes	characteristic
52	C₂×#11745659	—	2	2	no	—
53	#14273727	C₂³	1	2	yes	—
54	#14273724	—	2	2	no	—
55	#14273724	C₂³	1	2	yes	—
56	#14273719	—	2	2	no	—
57	#14273705	—	4	2	no	—
58	#14273703	—	2	2	no	—
59	#14273702	—	2	2	no	—
60	#14273702	—	4	2	no	—
61	C₄×D₂₃₆	—	2	2	no	—
62	C₄×D₂₃₆	—	2	2	no	—
63	C₄×D₂₃₆	C₂³	1	2	yes	—
64	C₄⋊C₄×D₅₉	—	2	2	no	—
65	#14273661	—	2	1	no	—
66	#14273661	—	2	1	no	—
67	#14273661	D₄	1	1	yes	characteristic
68	#14273660	—	2	2	no	—
69	#14273659	—	2	2	no	—
70	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
71	#14273651	—	2	2	no	—
72	#14273649	—	2	2	no	—
73	#14273649	—	2	2	no	—
74	#14273649	—	2	2	no	—
75	#14273649	—	2	2	no	—
76	#14273649	D₄	1	2	yes	—
77	#14273647	—	2	2	no	—
78	(C₂×D₅₉)⋊C₈	—	2	2	no	—
79	C₂×Q₁₆×C₅₉	D₄	1	1	yes	characteristic
80	C₄×Q₈×C₅₉	C₂³	1	1	yes	characteristic
81	C₄×Q₈×C₅₉	C₂³	1	1	yes	characteristic
82	#107×C₅₉	D₄	1	1	yes	characteristic
83	C₄⋊C₈×C₅₉	—	2	1	no	—
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	#103×C₅₉	D₄	1	1	yes	characteristic
89	C₄×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×Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
96	#11745689	—	4	4	no	—
97	#11745689	—	8	2	no	—
98	C₂×D₂₃₆	—	2	2	no	—
99	C₂×D₂₃₆	—	2	2	no	—
100	C₂×D₂₃₆	—	2	2	no	—
101	C₂×D₂₃₆	—	4	2	no	—
102	C₂×D₂₃₆	—	4	2	no	—
103	C₂×D₂₃₆	C₂×D₄	1	2	yes	—
104	C₂×C₄×D₅₉	—	2	2	no	—
105	C₂×C₄×D₅₉	—	4	2	no	—
106	C₂×C₄×D₅₉	—	4	2	no	—
107	C₂×Dic₁₁₈	—	2	2	no	—
108	Q₁₆×C₅₉	—	2	2	no	—
109	C₂×C₈×C₅₉	—	2	1	no	—
110	C₂×C₈×C₅₉	—	2	1	no	—
111	C₂×C₈×C₅₉	—	2	1	no	—
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	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₄⋊C₄×C₅₉	—	2	1	no	—
117	C₄⋊C₄×C₅₉	—	2	1	no	—
118	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅₉	—	2	1	no	—
121	C₄²×C₅₉	—	2	1	no	—
122	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
123	#11745671	—	8	2	no	—
124	(C₂×D₅₉)⋊C₄	—	2	2	no	—
125	(C₂×D₅₉)⋊C₄	—	4	2	no	—
126	(C₂×D₅₉)⋊C₄	—	4	2	no	—
127	C₄⋊Dic₅₉	—	2	2	no	—
128	C₄⋊Dic₅₉	—	2	2	no	—
129	C₄⋊Dic₅₉	C₂×D₄	1	2	yes	—
130	Dic₅₉⋊C₄	—	4	2	no	—
131	Dic₅₉⋊C₄	—	4	2	no	—
132	C₂×C₅₉⋊C₈	—	2	2	no	—
133	D₄₇₂	—	2	2	no	—
134	D₄₇₂	—	4	2	no	—
135	D₄₇₂	—	8	1	no	—
136	D₄₇₂	—	8	2	no	—
137	#11745660	—	8	4	no	—
138	#11745659	—	8	2	no	—
139	C₂²×D₅₉	—	2	2	no	—
140	C₂²×D₅₉	—	4	2	no	—
141	Q₈×C₅₉	—	2	1	no	—
142	Q₈×C₅₉	—	2	1	no	—
143	Q₈×C₅₉	—	2	1	no	—
144	Q₈×C₅₉	#135	1	2	yes	—
145	C₂×C₄×C₅₉	—	2	1	no	—
146	C₂×C₄×C₅₉	—	2	1	no	—
147	C₂×C₄×C₅₉	—	2	1	no	—
148	C₂×C₄×C₅₉	—	2	1	no	—
149	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×Dic₅₉	—	2	2	no	—
153	C₂×Dic₅₉	—	4	2	no	—
154	D₂₃₆	—	2	2	no	—
155	D₂₃₆	—	2	2	no	—
156	D₂₃₆	—	4	2	no	—
157	D₂₃₆	—	4	2	no	—
158	D₂₃₆	—	4	2	no	—
159	D₂₃₆	—	4	2	no	—
160	D₂₃₆	—	8	4	no	—
161	D₂₃₆	—	8	2	no	—
162	D₂₃₆	—	16	2	no	—
163	C₄×D₅₉	—	4	2	no	—
164	C₄×D₅₉	—	4	2	no	—
165	C₄×D₅₉	—	8	4	no	—
166	C₄×D₅₉	—	16	2	no	—
167	Dic₁₁₈	—	8	2	no	—
168	C₈×C₅₉	—	2	2	no	—
169	C₈×C₅₉	—	2	1	no	—
170	C₈×C₅₉	—	2	1	no	—
171	C₈×C₅₉	—	2	1	no	—
172	C₈×C₅₉	—	4	1	no	—
173	C₈×C₅₉	—	4	1	no	—
174	C₈×C₅₉	#137	1	2	yes	—
175	C₅₉⋊C₈	—	4	2	no	—
176	#19909	—	59	1	no	maximal, Sylow
177	C₂²×C₅₉	D₄²	1	1	yes	characteristic
178	C₂×D₅₉	—	4	2	no	—
179	C₂×D₅₉	—	4	2	no	—
180	C₂×D₅₉	—	4	2	no	—
181	C₂×D₅₉	—	8	2	no	—
182	C₂×D₅₉	—	8	2	no	—
183	C₂×D₅₉	—	8	2	no	—
184	C₄×C₅₉	—	2	2	no	—
185	C₄×C₅₉	—	2	1	no	—
186	C₄×C₅₉	—	4	1	no	—
187	C₄×C₅₉	—	4	1	no	—
188	C₄×C₅₉	—	4	1	no	—
189	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
190	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
191	C₄×C₅₉	#576	1	1	yes	characteristic
192	C₄×C₅₉	#577	1	1	yes	characteristic
193	Dic₅₉	—	4	2	no	—
194	Dic₅₉	—	8	2	no	—
195	#3391	—	59	1	no	—
196	#3380	—	59	1	no	—
197	#3333	—	59	2	no	—
198	#1706	—	59	1	no	—
199	#1697	—	59	2	no	—
200	#1676	—	59	1	no	—
201	#1668	—	59	1	no	—
202	#1631	—	59	2	no	—
203	#1622	—	59	2	no	—
204	#1589	—	59	1	no	—
205	C₈×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
206	C₂×C₅₉	#3278	1	1	yes	characteristic
207	C₂×C₅₉	#3280	1	1	yes	characteristic
208	C₂×C₅₉	#3292	1	1	yes	characteristic
209	D₅₉	—	8	2	no	—
210	D₅₉	—	16	2	no	—
211	#554	—	59	2	no	—
212	#550	—	59	2	no	—
213	#500	—	59	1	no	—
214	#499	—	118	1	no	—
215	#493	—	59	1	no	—
216	#493	—	118	1	no	—
217	#492	—	59	2	no	—
218	#486	—	59	2	no	—
219	#480	—	118	2	no	—
220	#478	—	118	2	no	—
221	#477	—	59	2	no	—
222	#476	—	59	2	no	—
223	#468	—	118	2	no	—
224	#460	—	59	2	no	—
225	#459	—	59	2	no	—
226	#459	—	118	2	no	—
227	#452	—	118	2	no	—
228	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
229	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
230	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
231	C₄×D₈	—	59	1	no	—
232	C₄×D₈	—	118	1	no	—
233	C₄×D₈	—	118	1	no	—
234	#436	—	59	2	no	—
235	#417	—	118	2	no	—
236	#335	—	59	1	no	—
237	#335	C₂×D₅₉	1	1	yes	characteristic
238	#333	—	59	2	no	—
239	#331	—	59	2	no	—
240	Q₈⋊C₈	C₂×D₅₉	1	1	yes	characteristic
241	Q₈⋊C₈	C₂×D₅₉	1	1	yes	characteristic
242	D₄⋊C₈	—	59	2	no	—
243	C₈²	C₂×D₅₉	1	1	yes	characteristic
244	C₅₉	#19909	1	1	yes	characteristic, complemented, Sylow
245	C₂×#41	—	118	2	no	—
246	C₂×Q₁₆	D₂₃₆	1	1	yes	characteristic
247	C₂×SD₁₆	—	118	2	no	—
248	C₂×SD₁₆	—	236	2	no	—
249	C₂×D₈	—	59	1	no	—
250	C₂×D₈	—	118	2	no	—
251	C₂×D₈	—	118	1	no	—
252	C₂×D₈	—	118	1	no	—
253	C₂×M₁₆	—	118	2	no	—
254	C₄⋊Q₈	—	59	2	no	—
255	#127	—	59	2	no	—
256	#127	—	118	2	no	—
257	#125	—	118	2	no	—
258	C₂²⋊Q₈	—	118	2	no	—
259	C₂²⋊Q₈	—	236	2	no	—
260	#121	—	118	2	no	—
261	#121	—	236	2	no	—
262	C₄×Q₈	C₂²×D₅₉	1	1	yes	characteristic
263	C₄×Q₈	C₂²×D₅₉	1	1	yes	characteristic
264	C₄×D₄	—	59	2	no	—
265	C₄×D₄	—	118	2	no	—
266	C₄×D₄	—	118	2	no	—
267	C₂×C₄⋊C₄	—	118	2	no	—
268	#107	—	59	1	no	—
269	#107	—	118	1	no	—
270	#107	—	118	1	no	—
271	#107	—	118	2	no	—
272	#107	D₂₃₆	1	1	yes	characteristic
273	#106	—	118	2	no	—
274	C₄⋊C₈	—	2	1	no	—
275	C₄⋊C₈	—	2	1	no	—
276	C₄⋊C₈	—	59	2	no	—
277	C₄⋊C₈	C₂²×D₅₉	1	1	yes	characteristic
278	C₄⋊C₈	C₂²×D₅₉	1	1	yes	characteristic
279	#103	—	118	2	no	—
280	#103	D₂₃₆	1	1	yes	characteristic
281	#103	D₂₃₆	1	1	yes	characteristic
282	#102	—	59	2	no	—
283	#102	—	118	2	no	—
284	#102	—	118	2	no	—
285	#102	—	118	2	no	—
286	#102	—	118	2	no	—
287	#102	—	118	2	no	—
288	C₂²⋊C₈	—	118	2	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	2	1	no	—
291	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
292	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
293	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
294	#41	—	472	2	no	—
295	#41	—	236	4	no	—
296	C₂×Q₈	—	118	2	no	—
297	C₂×Q₈	C₂×D₂₃₆	1	1	yes	characteristic
298	C₂×Q₈	C₂×D₂₃₆	1	1	yes	characteristic
299	C₂×D₄	—	59	2	no	—
300	C₂×D₄	—	118	2	no	—
301	C₂×D₄	—	118	2	no	—
302	C₂×D₄	—	118	2	no	—
303	C₂×D₄	—	236	2	no	—
304	C₂×D₄	—	236	2	no	—
305	C₂²×C₄	—	118	2	no	—
306	C₂²×C₄	—	236	2	no	—
307	C₂²×C₄	—	236	2	no	—
308	Q₁₆	—	2	2	no	—
309	SD₁₆	—	472	4	no	—
310	SD₁₆	—	472	2	no	—
311	D₈	—	118	2	no	—
312	D₈	—	472	1	no	—
313	D₈	—	472	2	no	—
314	D₈	—	236	2	no	—
315	M₁₆	—	472	2	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	1	no	—
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	118	2	no	—
320	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
321	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
322	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
323	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
324	C₄⋊C₄	—	2	1	no	—
325	C₄⋊C₄	—	2	1	no	—
326	C₄⋊C₄	—	59	2	no	—
327	C₄⋊C₄	—	118	2	no	—
328	C₄⋊C₄	—	118	2	no	—
329	C₄⋊C₄	—	236	2	no	—
330	C₄⋊C₄	—	236	2	no	—
331	C₄⋊C₄	C₂×D₂₃₆	1	1	yes	characteristic
332	C₄⋊C₄	C₂×D₂₃₆	1	1	yes	characteristic
333	C₂²⋊C₄	—	118	2	no	—
334	C₂²⋊C₄	—	236	2	no	—
335	C₂²⋊C₄	—	236	2	no	—
336	C₄²	—	2	1	no	—
337	C₄²	—	2	1	no	—
338	C₄²	C₂³×D₅₉	1	1	yes	characteristic
339	C₂³	—	118	2	no	—
340	C₂³	—	236	2	no	—
341	Q₈	—	2	1	no	—
342	Q₈	—	2	1	no	—
343	Q₈	—	2	1	no	—
344	Q₈	—	472	2	no	—
345	Q₈	#14273749	1	2	yes	—
346	D₄	—	118	2	no	—
347	D₄	—	118	2	no	—
348	D₄	—	944	2	no	—
349	D₄	—	472	2	no	—
350	D₄	—	472	4	no	—
351	D₄	—	236	2	no	—
352	D₄	—	236	2	no	—
353	D₄	—	236	2	no	—
354	D₄	—	236	2	no	—
355	C₂×C₄	—	2	1	no	—
356	C₂×C₄	—	2	1	no	—
357	C₂×C₄	—	2	1	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	118	2	no	—
360	C₂×C₄	—	944	2	no	—
361	C₂×C₄	—	472	4	no	—
362	C₂×C₄	—	236	2	no	—
363	C₂×C₄	—	236	2	no	—
364	C₂×C₄	—	236	2	no	—
365	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
366	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
367	C₂×C₄	#14273773	1	1	yes	characteristic
368	C₈	—	2	2	no	—
369	C₈	—	2	1	no	—
370	C₈	—	2	1	no	—
371	C₈	—	2	1	no	—
372	C₈	—	4	1	no	—
373	C₈	—	4	1	no	—
374	C₈	—	236	2	no	—
375	C₈	#14273759	1	2	yes	—
376	C₂²	—	472	2	no	—
377	C₂²	—	472	2	no	—
378	C₂²	—	472	2	no	—
379	C₂²	—	236	2	no	—
380	C₂²	—	236	2	no	—
381	C₂²	—	236	2	no	—
382	C₂²	D₄×D₂₃₆	1	1	yes	characteristic
383	C₄	—	2	2	no	—
384	C₄	—	2	1	no	—
385	C₄	—	4	1	no	—
386	C₄	—	4	1	no	—
387	C₄	—	4	1	no	—
388	C₄	—	472	2	no	—
389	C₄	—	236	2	no	—
390	C₄	C₂×#14273749	1	1	yes	characteristic
391	C₄	C₂×#14273759	1	1	yes	characteristic
392	C₄	#16838207	1	1	yes	characteristic
393	C₄	#16838209	1	1	yes	characteristic
394	C₂	—	944	2	no	—
395	C₂	—	472	2	no	—
396	C₂	#21435763	1	1	yes	characteristic
397	C₂	#21435783	1	1	yes	characteristic
398	C₂	#21435896	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₂₃₆
commutator, cosocle	C₄²×C₅₉	C₂⁴
Frattini	C₄²	C₂³×D₅₉
Fitting	C₈×Q₁₆×C₅₉	C₂
socle	C₂²×C₅₉	D₄²
prime radical	#21437071	C₂
hypercenter	C₈×Q₁₆	D₅₉
hypocenter	C₅₉	#19909
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
2. #21436964	C₂	semidirect
3. #21436436	C₂	semidirect
4. #21423688	C₂	semidirect
5. #21423630	C₂	semidirect
6. #21423500	C₂	semidirect
7. #21423456	C₂	semidirect
8. #21423225	C₂	semidirect
9. #21423171	C₂	semidirect
10. #21422982	C₂	semidirect
11. C₈×Q₁₆×C₅₉	C₂	semidirect
205. C₈×Q₁₆	D₅₉	semidirect
244. C₅₉	#19909	semidirect

Maximal subgroups

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

subgroup	index
#21437071	2
#21436964	2
#21436436	2
#21423688	2
#21423630	2
#21423500	2
#21423456	2
#21423225	2
#21423171	2
#21422982	2
C₈×Q₁₆×C₅₉	2
#19909	59