Subgroups — #28580269

Order 15104.

602 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437085	C₂	1	1	yes	maximal, characteristic
2	Q₈×D₄₇₂	C₂	1	1	yes	maximal, characteristic
3	#21436381	C₂	1	2	yes	maximal, complemented
4	#21423693	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423644	C₂	1	2	yes	maximal, complemented
6	#21423490	C₂	1	1	yes	maximal, characteristic, complemented
7	#21423466	C₂	1	1	yes	maximal, characteristic, complemented
8	#21423368	C₂	1	2	yes	maximal, complemented
9	#21423362	C₂	1	2	yes	maximal, complemented
10	#21422982	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#16838125	C₂²	1	2	yes	—
13	Q₈×D₂₃₆	C₂²	1	2	yes	—
14	#16837807	—	2	1	no	—
15	#16837807	C₂²	1	1	yes	characteristic
16	#16837800	C₂²	1	2	yes	—
17	#16837786	—	2	1	no	—
18	#16837782	C₂²	1	1	yes	characteristic
19	#16837753	C₂²	1	2	yes	—
20	#16837745	—	2	2	no	—
21	#16837687	C₂²	1	2	yes	—
22	#16837683	—	2	2	no	—
23	#16837683	C₂²	1	2	yes	—
24	#16837659	—	2	2	no	—
25	#16837625	—	2	2	no	—
26	#16837612	C₂²	1	2	yes	—
27	#16837608	C₂²	1	2	yes	—
28	#16837556	—	2	2	no	—
29	C₄×Dic₂₃₆	—	2	1	no	—
30	C₄×D₄₇₂	—	2	1	no	—
31	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
32	#16837481	C₂²	1	2	yes	—
33	#16837409	—	2	2	no	—
34	#16837155	C₂²	1	1	yes	characteristic
35	#16837149	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	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×Q₈×D₅₉	—	2	2	no	—
46	C₂×Dic₂₃₆	—	2	1	no	—
47	C₂×Dic₂₃₆	—	2	1	no	—
48	C₂×#11745672	—	2	2	no	—
49	C₂×#11745660	—	2	2	no	—
50	C₂×D₄₇₂	D₄	1	1	yes	characteristic
51	C₂×#11745659	—	2	2	no	—
52	#14273727	—	2	2	no	—
53	#14273727	C₂³	1	2	yes	—
54	#14273727	C₂³	1	2	yes	—
55	#14273722	—	4	2	no	—
56	#14273715	—	2	2	no	—
57	#14273707	—	2	2	no	—
58	#14273707	—	4	2	no	—
59	#14273703	—	2	2	no	—
60	C₄×Dic₁₁₈	—	2	2	no	—
61	C₄×D₂₃₆	—	2	2	no	—
62	C₄×D₂₃₆	C₂³	1	2	yes	—
63	#14273685	—	2	2	no	—
64	#14273661	—	2	1	no	—
65	#14273661	—	2	1	no	—
66	#14273661	—	2	2	no	—
67	#14273661	D₄	1	1	yes	characteristic
68	#14273659	—	4	2	no	—
69	#14273658	—	2	2	no	—
70	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
71	#14273652	—	2	2	no	—
72	#14273652	—	2	2	no	—
73	#14273652	—	2	2	no	—
74	#14273651	—	2	2	no	—
75	#14273650	—	2	2	no	—
76	#14273649	—	2	2	no	—
77	#14273649	D₄	1	2	yes	—
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	Q₈×D₅₉	—	4	4	no	—
97	Q₈×D₅₉	—	8	2	no	—
98	C₂×D₂₃₆	C₂×D₄	1	2	yes	—
99	C₂×C₄×D₅₉	—	2	2	no	—
100	C₂×C₄×D₅₉	—	4	2	no	—
101	C₂×Dic₁₁₈	—	2	2	no	—
102	C₂×Dic₁₁₈	—	2	2	no	—
103	C₂×Dic₁₁₈	—	4	2	no	—
104	Q₁₆×C₅₉	—	2	2	no	—
105	C₂×C₈×C₅₉	—	2	1	no	—
106	C₂×C₈×C₅₉	—	2	1	no	—
107	C₂×C₈×C₅₉	—	2	1	no	—
108	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₄⋊C₄×C₅₉	—	2	1	no	—
113	C₄⋊C₄×C₅₉	—	2	1	no	—
114	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₅₉	—	2	1	no	—
117	C₄²×C₅₉	—	2	1	no	—
118	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
119	#11745672	—	8	2	no	—
120	(C₂×D₅₉)⋊C₄	—	2	2	no	—
121	(C₂×D₅₉)⋊C₄	—	4	2	no	—
122	(C₂×D₅₉)⋊C₄	—	4	2	no	—
123	(C₂×D₅₉)⋊C₄	—	4	2	no	—
124	C₄⋊Dic₅₉	—	2	2	no	—
125	C₄⋊Dic₅₉	—	2	2	no	—
126	C₄⋊Dic₅₉	—	2	2	no	—
127	C₄⋊Dic₅₉	—	2	2	no	—
128	C₄⋊Dic₅₉	—	4	2	no	—
129	C₄⋊Dic₅₉	C₂×D₄	1	2	yes	—
130	Dic₅₉⋊C₄	—	4	2	no	—
131	C₄×Dic₅₉	—	4	2	no	—
132	C₂×C₅₉⋊C₈	—	2	2	no	—
133	Dic₂₃₆	—	4	2	no	—
134	Dic₂₃₆	—	8	1	no	—
135	D₄₇₂	—	2	2	no	—
136	#11745660	—	8	2	no	—
137	#11745659	—	8	2	no	—
138	C₂²×D₅₉	—	2	2	no	—
139	Q₈×C₅₉	—	2	1	no	—
140	Q₈×C₅₉	—	2	1	no	—
141	Q₈×C₅₉	—	2	1	no	—
142	Q₈×C₅₉	#135	1	2	yes	—
143	C₂×C₄×C₅₉	—	2	1	no	—
144	C₂×C₄×C₅₉	—	2	1	no	—
145	C₂×C₄×C₅₉	—	2	1	no	—
146	C₂×C₄×C₅₉	—	2	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	C₂×Dic₅₉	—	2	2	no	—
151	C₂×Dic₅₉	—	4	2	no	—
152	C₂×Dic₅₉	—	4	2	no	—
153	D₂₃₆	—	2	2	no	—
154	D₂₃₆	—	2	2	no	—
155	C₄×D₅₉	—	4	2	no	—
156	C₄×D₅₉	—	4	2	no	—
157	C₄×D₅₉	—	8	4	no	—
158	Dic₁₁₈	—	4	2	no	—
159	Dic₁₁₈	—	4	2	no	—
160	Dic₁₁₈	—	4	2	no	—
161	Dic₁₁₈	—	4	2	no	—
162	Dic₁₁₈	—	8	4	no	—
163	C₈×C₅₉	—	2	2	no	—
164	C₈×C₅₉	—	2	1	no	—
165	C₈×C₅₉	—	2	1	no	—
166	C₈×C₅₉	—	2	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	#19907	—	59	1	no	maximal, Sylow
172	C₂²×C₅₉	D₄²	1	1	yes	characteristic
173	C₂×D₅₉	—	4	2	no	—
174	C₂×D₅₉	—	4	2	no	—
175	C₂×D₅₉	—	4	2	no	—
176	C₄×C₅₉	—	2	2	no	—
177	C₄×C₅₉	—	2	1	no	—
178	C₄×C₅₉	—	4	1	no	—
179	C₄×C₅₉	—	4	1	no	—
180	C₄×C₅₉	—	4	1	no	—
181	C₄×C₅₉	C₂×#135	1	1	yes	characteristic
182	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
183	C₄×C₅₉	#576	1	1	yes	characteristic
184	C₄×C₅₉	#577	1	1	yes	characteristic
185	Dic₅₉	—	4	2	no	—
186	Dic₅₉	—	8	2	no	—
187	Dic₅₉	—	8	2	no	—
188	#3393	—	59	1	no	—
189	Q₈×D₈	—	59	1	no	—
190	#3328	—	59	2	no	—
191	#1707	—	59	1	no	—
192	#1699	—	59	2	no	—
193	#1674	—	59	1	no	—
194	#1670	—	59	1	no	—
195	#1652	—	59	2	no	—
196	#1651	—	59	2	no	—
197	#1589	—	59	1	no	—
198	C₈×Q₁₆	D₅₉	1	1	yes	characteristic, complemented
199	C₂×C₅₉	#3278	1	1	yes	characteristic
200	C₂×C₅₉	#3280	1	1	yes	characteristic
201	C₂×C₅₉	#3293	1	1	yes	characteristic
202	D₅₉	—	8	2	no	—
203	#556	—	59	2	no	—
204	D₄×Q₈	—	59	2	no	—
205	#500	—	59	1	no	—
206	#500	—	118	1	no	—
207	#498	—	59	2	no	—
208	#495	—	118	1	no	—
209	#494	—	59	1	no	—
210	#487	—	59	2	no	—
211	#485	—	118	2	no	—
212	#475	—	59	2	no	—
213	#474	—	59	2	no	—
214	#474	—	118	2	no	—
215	#470	—	118	2	no	—
216	#464	—	118	2	no	—
217	#462	—	59	2	no	—
218	#461	—	59	2	no	—
219	#451	—	118	2	no	—
220	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
221	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
222	C₄×Q₁₆	—	118	1	no	—
223	C₄×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
224	C₄×D₈	—	59	1	no	—
225	C₄×D₈	—	118	1	no	—
226	#436	—	59	2	no	—
227	#419	—	118	2	no	—
228	#335	—	59	1	no	—
229	#335	C₂×D₅₉	1	1	yes	characteristic
230	#332	—	59	2	no	—
231	#332	—	59	2	no	—
232	Q₈⋊C₈	C₂×D₅₉	1	1	yes	characteristic
233	Q₈⋊C₈	C₂×D₅₉	1	1	yes	characteristic
234	D₄⋊C₈	—	59	2	no	—
235	C₈²	C₂×D₅₉	1	1	yes	characteristic
236	C₅₉	#19907	1	1	yes	characteristic, complemented, Sylow
237	C₂²×Q₈	—	118	2	no	—
238	C₂×Q₁₆	—	118	2	no	—
239	C₂×Q₁₆	—	118	1	no	—
240	C₂×Q₁₆	—	118	1	no	—
241	C₂×Q₁₆	D₂₃₆	1	1	yes	characteristic
242	C₂×SD₁₆	—	118	2	no	—
243	C₂×D₈	—	59	1	no	—
244	C₂×M₁₆	—	118	2	no	—
245	C₄⋊Q₈	—	59	2	no	—
246	C₄⋊Q₈	—	59	2	no	—
247	C₄⋊Q₈	—	118	2	no	—
248	#126	—	236	2	no	—
249	#124	—	118	2	no	—
250	C₂²⋊Q₈	—	118	2	no	—
251	C₂²⋊Q₈	—	118	2	no	—
252	C₂²⋊Q₈	—	236	2	no	—
253	C₄×Q₈	—	118	2	no	—
254	C₄×Q₈	C₂²×D₅₉	1	1	yes	characteristic
255	C₄×Q₈	C₂²×D₅₉	1	1	yes	characteristic
256	C₄×D₄	—	59	2	no	—
257	C₄×D₄	—	118	2	no	—
258	#117	—	118	2	no	—
259	#107	—	59	1	no	—
260	#107	—	118	2	no	—
261	#107	—	118	1	no	—
262	#107	—	118	1	no	—
263	#107	D₂₃₆	1	1	yes	characteristic
264	#106	—	118	2	no	—
265	#106	—	236	2	no	—
266	C₄⋊C₈	—	2	1	no	—
267	C₄⋊C₈	—	2	1	no	—
268	C₄⋊C₈	—	59	2	no	—
269	C₄⋊C₈	C₂²×D₅₉	1	1	yes	characteristic
270	C₄⋊C₈	C₂²×D₅₉	1	1	yes	characteristic
271	#103	—	118	2	no	—
272	#103	—	118	2	no	—
273	#103	—	118	2	no	—
274	#103	—	118	2	no	—
275	#103	—	118	2	no	—
276	#103	D₂₃₆	1	1	yes	characteristic
277	#103	D₂₃₆	1	1	yes	characteristic
278	#102	—	59	2	no	—
279	#102	—	118	2	no	—
280	C₂²⋊C₈	—	118	2	no	—
281	C₄×C₈	—	2	1	no	—
282	C₄×C₈	—	2	1	no	—
283	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
286	C₂×Q₈	—	118	2	no	—
287	C₂×Q₈	—	118	2	no	—
288	C₂×Q₈	—	472	2	no	—
289	C₂×Q₈	—	236	2	no	—
290	C₂×Q₈	—	236	4	no	—
291	C₂×Q₈	C₂×D₂₃₆	1	1	yes	characteristic
292	C₂×Q₈	C₂×D₂₃₆	1	1	yes	characteristic
293	C₂×D₄	—	59	2	no	—
294	C₂²×C₄	—	118	2	no	—
295	C₂²×C₄	—	236	2	no	—
296	Q₁₆	—	2	2	no	—
297	Q₁₆	—	472	1	no	—
298	Q₁₆	—	472	2	no	—
299	Q₁₆	—	236	2	no	—
300	SD₁₆	—	472	2	no	—
301	D₈	—	118	2	no	—
302	M₁₆	—	472	2	no	—
303	C₂×C₈	—	2	1	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	118	2	no	—
307	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
308	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
309	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
310	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
311	C₄⋊C₄	—	2	1	no	—
312	C₄⋊C₄	—	2	1	no	—
313	C₄⋊C₄	—	59	2	no	—
314	C₄⋊C₄	—	118	2	no	—
315	C₄⋊C₄	—	118	2	no	—
316	C₄⋊C₄	—	118	2	no	—
317	C₄⋊C₄	—	118	2	no	—
318	C₄⋊C₄	—	236	2	no	—
319	C₄⋊C₄	—	236	2	no	—
320	C₄⋊C₄	C₂×D₂₃₆	1	1	yes	characteristic
321	C₄⋊C₄	C₂×D₂₃₆	1	1	yes	characteristic
322	C₂²⋊C₄	—	118	2	no	—
323	C₂²⋊C₄	—	236	2	no	—
324	C₂²⋊C₄	—	236	2	no	—
325	C₂²⋊C₄	—	236	2	no	—
326	C₄²	—	2	1	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	236	2	no	—
329	C₄²	C₂³×D₅₉	1	1	yes	characteristic
330	C₂³	—	118	2	no	—
331	Q₈	—	2	1	no	—
332	Q₈	—	2	1	no	—
333	Q₈	—	2	1	no	—
334	Q₈	—	472	4	no	—
335	Q₈	—	236	2	no	—
336	Q₈	—	236	2	no	—
337	Q₈	—	236	2	no	—
338	Q₈	—	236	2	no	—
339	Q₈	#14273749	1	2	yes	—
340	D₄	—	118	2	no	—
341	D₄	—	118	2	no	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	118	2	no	—
347	C₂×C₄	—	472	4	no	—
348	C₂×C₄	—	236	2	no	—
349	C₂×C₄	—	236	2	no	—
350	C₂×C₄	—	236	2	no	—
351	C₂×C₄	—	236	2	no	—
352	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
353	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
354	C₂×C₄	#14273773	1	1	yes	characteristic
355	C₈	—	2	2	no	—
356	C₈	—	2	1	no	—
357	C₈	—	2	1	no	—
358	C₈	—	2	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	1	no	—
361	C₈	—	236	2	no	—
362	C₈	#14273759	1	2	yes	—
363	C₂²	—	236	2	no	—
364	C₂²	—	236	2	no	—
365	C₂²	—	236	2	no	—
366	C₂²	D₄×D₂₃₆	1	1	yes	characteristic
367	C₄	—	2	2	no	—
368	C₄	—	2	1	no	—
369	C₄	—	4	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	472	2	no	—
373	C₄	—	472	2	no	—
374	C₄	—	236	2	no	—
375	C₄	C₂×#14273749	1	1	yes	characteristic
376	C₄	C₂×#14273759	1	1	yes	characteristic
377	C₄	#16838207	1	1	yes	characteristic
378	C₄	#16838209	1	1	yes	characteristic
379	C₂	—	472	2	no	—
380	C₂	#21435763	1	1	yes	characteristic
381	C₂	#21435783	1	1	yes	characteristic
382	C₂	#21435901	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	C₂×D₄₇₂	D₄
hypercenter	C₈×Q₁₆	D₅₉
hypocenter	C₅₉	#19907
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
3. #21436381	C₂	semidirect
4. #21423693	C₂	semidirect
5. #21423644	C₂	semidirect
6. #21423490	C₂	semidirect
7. #21423466	C₂	semidirect
8. #21423368	C₂	semidirect
9. #21423362	C₂	semidirect
11. C₈×Q₁₆×C₅₉	C₂	semidirect
198. C₈×Q₁₆	D₅₉	semidirect
236. C₅₉	#19907	semidirect

Maximal subgroups

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

subgroup	index
#21437085	2
Q₈×D₄₇₂	2
#21436381	2
#21423693	2
#21423644	2
#21423490	2
#21423466	2
#21423368	2
#21423362	2
#21422982	2
C₈×Q₁₆×C₅₉	2
#19907	59