Subgroups — #28580003

Order 15104.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×D₄₇₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#21436444	C₂	1	1	yes	maximal, characteristic
3	#21435743	C₂	1	2	yes	maximal, complemented
4	#21423714	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423586	C₂	1	2	yes	maximal, complemented
6	#21423490	C₂	1	1	yes	maximal, characteristic, complemented
7	#21423485	C₂	1	1	yes	maximal, characteristic, complemented
8	#21423347	C₂	1	2	yes	maximal, complemented
9	#21423333	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	C₂×#14273756	—	2	2	no	—
13	Q₈×D₂₃₆	C₂²	1	2	yes	—
14	#16838056	C₂²	1	2	yes	—
15	#16837807	—	2	1	no	—
16	#16837807	C₂²	1	1	yes	characteristic
17	#16837807	C₂²	1	1	yes	characteristic
18	#16837807	C₂²	1	2	yes	—
19	#16837782	C₂²	1	2	yes	—
20	#16837716	—	2	2	no	—
21	#16837683	—	2	2	no	—
22	#16837683	C₂²	1	2	yes	—
23	#16837683	C₂²	1	2	yes	—
24	#16837661	—	2	2	no	—
25	#16837638	—	2	1	no	—
26	#16837612	C₂²	1	2	yes	—
27	#16837607	C₂²	1	2	yes	—
28	#16837565	—	2	2	no	—
29	#16837559	—	2	2	no	—
30	C₄×D₄₇₂	—	2	1	no	—
31	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
32	#16837481	C₂²	1	2	yes	—
33	C₂×#14273661	—	2	1	no	—
34	#16837155	C₂²	1	1	yes	characteristic
35	#16837147	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×Q₈×D₅₉	—	2	2	no	—
47	#14273756	—	4	8	no	—
48	C₂×Dic₂₃₆	—	2	2	no	—
49	C₂×Dic₂₃₆	—	2	2	no	—
50	C₂×#11745672	—	2	2	no	—
51	C₂×#11745670	—	2	2	no	—
52	C₂×D₄₇₂	D₄	1	1	yes	characteristic
53	C₂×#11745659	—	2	2	no	—
54	#14273727	—	2	2	no	—
55	#14273727	C₂³	1	2	yes	—
56	#14273727	C₂³	1	2	yes	—
57	#14273711	—	4	2	no	—
58	#14273707	—	2	2	no	—
59	#14273707	—	2	2	no	—
60	#14273707	—	4	2	no	—
61	#14273701	—	2	2	no	—
62	C₄×D₂₃₆	—	2	2	no	—
63	C₄×D₂₃₆	C₂³	1	2	yes	—
64	C₂×C₄⋊Dic₅₉	—	2	2	no	—
65	#14273661	—	2	1	no	—
66	#14273661	—	2	1	no	—
67	#14273661	—	2	1	no	—
68	#14273661	—	2	1	no	—
69	#14273661	—	2	2	no	—
70	#14273661	—	2	2	no	—
71	#14273661	—	4	2	no	—
72	#14273661	D₄	1	1	yes	characteristic
73	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
74	#14273650	—	2	2	no	—
75	#14273650	—	2	2	no	—
76	#14273649	—	2	2	no	—
77	#14273649	—	2	2	no	—
78	#14273649	D₄	1	2	yes	—
79	(C₂×D₅₉)⋊C₈	—	2	2	no	—
80	C₂×SD₁₆×C₅₉	D₄	1	1	yes	characteristic
81	C₂²×C₈×C₅₉	—	2	1	no	—
82	C₄×Q₈×C₅₉	C₂³	1	1	yes	characteristic
83	C₄×D₄×C₅₉	C₂³	1	1	yes	characteristic
84	#106×C₅₉	D₄	1	1	yes	characteristic
85	C₄⋊C₈×C₅₉	—	2	1	no	—
86	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
87	C₄⋊C₈×C₅₉	C₂³	1	1	yes	characteristic
88	#103×C₅₉	D₄	1	1	yes	characteristic
89	#102×C₅₉	D₄	1	1	yes	characteristic
90	C₂²⋊C₈×C₅₉	—	2	1	no	—
91	C₄×C₈×C₅₉	—	2	1	no	—
92	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₂×D₄×C₅₉	C₂×D₄	1	1	yes	characteristic
97	C₂²×C₄×C₅₉	—	2	1	no	—
98	C₂×C₂²⋊D₅₉	—	4	2	no	—
99	C₂²×Dic₅₉	—	4	2	no	—
100	Q₈×D₅₉	—	4	4	no	—
101	Q₈×D₅₉	—	8	2	no	—
102	#11745687	—	4	4	no	—
103	#11745687	—	8	2	no	—
104	C₂×D₂₃₆	C₂×D₄	1	2	yes	—
105	C₂×C₄×D₅₉	—	2	2	no	—
106	C₂×C₄×D₅₉	—	4	2	no	—
107	C₂×Dic₁₁₈	—	2	2	no	—
108	C₂×Dic₁₁₈	—	2	2	no	—
109	C₂×Dic₁₁₈	—	4	2	no	—
110	SD₁₆×C₅₉	D₈	1	4	yes	complemented
111	C₂×C₈×C₅₉	—	2	1	no	—
112	C₂×C₈×C₅₉	—	2	1	no	—
113	C₂×C₈×C₅₉	—	2	1	no	—
114	C₂×C₈×C₅₉	—	4	2	no	—
115	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₅₉	—	2	1	no	—
120	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂²⋊C₄×C₅₉	—	2	1	no	—
123	C₄²×C₅₉	—	2	1	no	—
124	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
125	#11745672	—	4	4	no	—
126	#11745670	—	4	4	no	—
127	(C₂×D₅₉)⋊C₄	—	2	2	no	—
128	(C₂×D₅₉)⋊C₄	—	4	2	no	—
129	(C₂×D₅₉)⋊C₄	—	4	2	no	—
130	C₄⋊Dic₅₉	—	2	2	no	—
131	C₄⋊Dic₅₉	—	2	2	no	—
132	C₄⋊Dic₅₉	—	2	2	no	—
133	C₄⋊Dic₅₉	—	2	2	no	—
134	C₄⋊Dic₅₉	—	4	2	no	—
135	C₄⋊Dic₅₉	—	4	2	no	—
136	C₄⋊Dic₅₉	C₂×D₄	1	2	yes	—
137	C₂×C₅₉⋊C₈	—	2	2	no	—
138	Dic₂₃₆	—	4	4	no	—
139	Dic₂₃₆	—	8	2	no	—
140	D₄₇₂	—	2	2	no	—
141	#11745659	—	4	4	no	—
142	C₂³×C₅₉	—	2	1	no	—
143	C₂²×D₅₉	—	2	2	no	—
144	Q₈×C₅₉	—	2	1	no	—
145	Q₈×C₅₉	C₂×D₈	1	2	yes	—
146	D₄×C₅₉	—	2	1	no	—
147	D₄×C₅₉	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₅₉	—	2	1	no	—
149	C₂×C₄×C₅₉	—	2	1	no	—
150	C₂×C₄×C₅₉	—	2	1	no	—
151	C₂×C₄×C₅₉	—	4	1	no	—
152	C₂×C₄×C₅₉	—	4	1	no	—
153	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂²⋊D₅₉	—	8	4	no	—
157	C₂×Dic₅₉	—	2	2	no	—
158	C₂×Dic₅₉	—	4	2	no	—
159	C₂×Dic₅₉	—	4	2	no	—
160	C₂×Dic₅₉	—	8	4	no	—
161	D₂₃₆	—	2	2	no	—
162	D₂₃₆	—	2	2	no	—
163	C₄×D₅₉	—	4	2	no	—
164	C₄×D₅₉	—	4	2	no	—
165	C₄×D₅₉	—	8	4	no	—
166	Dic₁₁₈	—	4	2	no	—
167	Dic₁₁₈	—	4	2	no	—
168	Dic₁₁₈	—	4	2	no	—
169	Dic₁₁₈	—	4	2	no	—
170	Dic₁₁₈	—	8	4	no	—
171	C₈×C₅₉	—	2	2	no	—
172	C₈×C₅₉	—	2	1	no	—
173	C₈×C₅₉	—	2	1	no	—
174	C₈×C₅₉	—	4	1	no	—
175	C₈×C₅₉	—	4	1	no	—
176	C₈×C₅₉	C₂×D₈	1	2	yes	—
177	C₈×C₅₉	#137	1	2	yes	complemented
178	C₅₉⋊C₈	—	4	2	no	—
179	#19887	—	59	1	no	maximal, Sylow
180	C₂²×C₅₉	—	2	2	no	—
181	C₂²×C₅₉	—	4	1	no	—
182	C₂²×C₅₉	—	4	1	no	—
183	C₂²×C₅₉	D₄²	1	1	yes	characteristic
184	C₂×D₅₉	—	4	2	no	—
185	C₂×D₅₉	—	4	2	no	—
186	C₂×D₅₉	—	4	2	no	—
187	C₄×C₅₉	—	2	2	no	—
188	C₄×C₅₉	—	2	1	no	—
189	C₄×C₅₉	—	4	1	no	—
190	C₄×C₅₉	—	4	1	no	—
191	C₄×C₅₉	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
193	C₄×C₅₉	#576	1	1	yes	characteristic
194	C₄×C₅₉	#578	1	1	yes	characteristic
195	Dic₅₉	—	4	2	no	—
196	Dic₅₉	—	8	2	no	—
197	Dic₅₉	—	8	2	no	—
198	Q₈×D₈	—	59	1	no	—
199	#3334	—	59	1	no	—
200	#3276	—	59	2	no	—
201	#1713	—	59	1	no	—
202	#1689	—	59	2	no	—
203	#1674	—	59	1	no	—
204	#1673	—	59	1	no	—
205	#1649	—	59	2	no	—
206	#1647	—	59	2	no	—
207	#1589	—	59	1	no	—
208	C₈×SD₁₆	D₅₉	1	1	yes	characteristic, complemented
209	C₂×C₅₉	—	4	2	no	—
210	C₂×C₅₉	#3276	1	1	yes	characteristic
211	C₂×C₅₉	D₄×D₈	1	1	yes	characteristic
212	C₂×C₅₉	#3293	1	1	yes	characteristic
213	D₅₉	—	8	2	no	—
214	C₂×#137	—	118	2	no	—
215	D₄×Q₈	—	59	2	no	—
216	#547	—	59	2	no	—
217	#500	—	59	1	no	—
218	#500	—	59	1	no	—
219	#500	—	59	2	no	—
220	#500	—	118	1	no	—
221	#494	—	59	2	no	—
222	#480	—	118	2	no	—
223	#474	—	59	2	no	—
224	#474	—	59	2	no	—
225	#474	—	118	2	no	—
226	#470	—	118	2	no	—
227	#466	—	118	1	no	—
228	#462	—	59	2	no	—
229	#461	—	59	2	no	—
230	#452	—	118	2	no	—
231	#451	—	118	2	no	—
232	C₈×Q₈	C₂×D₅₉	1	1	yes	characteristic
233	C₄×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
234	C₄×D₈	—	59	1	no	—
235	C₄×D₈	—	118	1	no	—
236	#436	—	59	2	no	—
237	C₈×D₄	C₂×D₅₉	1	1	yes	characteristic
238	C₂×#107	—	118	1	no	—
239	#335	—	59	1	no	—
240	#334	C₂×D₅₉	1	1	yes	characteristic
241	#332	—	59	2	no	—
242	#332	—	59	2	no	—
243	Q₈⋊C₈	C₂×D₅₉	1	1	yes	characteristic
244	D₄⋊C₈	—	59	2	no	—
245	D₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
246	C₈²	C₂×D₅₉	1	1	yes	characteristic
247	C₅₉	#19887	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	118	2	no	—
249	C₂²×Q₈	—	118	2	no	—
250	#137	—	236	8	no	—
251	C₂×Q₁₆	—	118	2	no	—
252	C₂×Q₁₆	—	118	2	no	—
253	C₂×Q₁₆	—	118	2	no	—
254	C₂×SD₁₆	—	118	2	no	—
255	C₂×SD₁₆	D₂₃₆	1	1	yes	characteristic
256	C₂×D₈	—	59	1	no	—
257	C₂×M₁₆	—	118	2	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₄⋊Q₈	—	59	2	no	—
260	C₄⋊Q₈	—	59	2	no	—
261	C₄⋊Q₈	—	118	2	no	—
262	#123	—	236	2	no	—
263	C₂²⋊Q₈	—	118	2	no	—
264	C₂²⋊Q₈	—	118	2	no	—
265	C₂²⋊Q₈	—	236	2	no	—
266	#121	—	118	2	no	—
267	C₄×Q₈	C₂²×D₅₉	1	1	yes	characteristic
268	C₄×D₄	—	59	2	no	—
269	C₄×D₄	—	118	2	no	—
270	C₄×D₄	C₂²×D₅₉	1	1	yes	characteristic
271	C₂×C₄⋊C₄	—	118	2	no	—
272	#107	—	59	1	no	—
273	#107	—	118	2	no	—
274	#107	—	118	2	no	—
275	#107	—	118	1	no	—
276	#107	—	118	1	no	—
277	#107	—	118	1	no	—
278	#107	—	118	1	no	—
279	#107	—	236	2	no	—
280	#106	D₂₃₆	1	1	yes	characteristic
281	C₄⋊C₈	—	2	1	no	—
282	C₄⋊C₈	—	59	2	no	—
283	C₄⋊C₈	C₂²×D₅₉	1	1	yes	characteristic
284	C₄⋊C₈	C₂²×D₅₉	1	1	yes	characteristic
285	#103	—	118	2	no	—
286	#103	—	118	2	no	—
287	#103	D₂₃₆	1	1	yes	characteristic
288	#102	—	59	2	no	—
289	#102	—	118	2	no	—
290	#102	—	118	2	no	—
291	#102	D₂₃₆	1	1	yes	characteristic
292	C₂²⋊C₈	—	2	1	no	—
293	C₂²⋊C₈	—	118	2	no	—
294	C₄×C₈	—	2	1	no	—
295	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
296	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
297	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
298	#41	—	472	2	no	—
299	#41	—	236	4	no	—
300	C₂×Q₈	—	118	2	no	—
301	C₂×Q₈	—	118	2	no	—
302	C₂×Q₈	—	472	2	no	—
303	C₂×Q₈	—	236	2	no	—
304	C₂×Q₈	—	236	4	no	—
305	C₂×Q₈	C₂×D₂₃₆	1	1	yes	characteristic
306	C₂×D₄	—	59	2	no	—
307	C₂×D₄	—	236	2	no	—
308	C₂×D₄	C₂×D₂₃₆	1	1	yes	characteristic
309	C₂²×C₄	—	2	1	no	—
310	C₂²×C₄	—	118	2	no	—
311	C₂²×C₄	—	236	2	no	—
312	C₂²×C₄	—	236	2	no	—
313	Q₁₆	—	472	2	no	—
314	Q₁₆	—	236	4	no	—
315	Q₁₆	—	236	4	no	—
316	SD₁₆	—	236	4	no	—
317	SD₁₆	D₄₇₂	1	4	yes	complemented
318	D₈	—	118	2	no	—
319	M₁₆	—	236	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	1	no	—
323	C₂×C₈	—	4	2	no	—
324	C₂×C₈	—	118	2	no	—
325	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
326	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
327	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
328	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	59	2	no	—
331	C₄⋊C₄	—	118	2	no	—
332	C₄⋊C₄	—	118	2	no	—
333	C₄⋊C₄	—	118	2	no	—
334	C₄⋊C₄	—	118	2	no	—
335	C₄⋊C₄	—	236	2	no	—
336	C₄⋊C₄	—	236	2	no	—
337	C₄⋊C₄	C₂×D₂₃₆	1	1	yes	characteristic
338	C₄⋊C₄	C₂×D₂₃₆	1	1	yes	characteristic
339	C₂²⋊C₄	—	2	1	no	—
340	C₂²⋊C₄	—	118	2	no	—
341	C₂²⋊C₄	—	236	2	no	—
342	C₂²⋊C₄	—	236	2	no	—
343	C₄²	—	2	1	no	—
344	C₄²	C₂³×D₅₉	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	118	2	no	—
347	Q₈	—	2	1	no	—
348	Q₈	—	472	4	no	—
349	Q₈	—	236	2	no	—
350	Q₈	—	236	2	no	—
351	Q₈	—	236	2	no	—
352	Q₈	—	236	2	no	—
353	Q₈	C₂×D₄₇₂	1	2	yes	—
354	D₄	—	2	1	no	—
355	D₄	—	118	2	no	—
356	D₄	—	118	2	no	—
357	D₄	—	472	4	no	—
358	D₄	C₂×D₄₇₂	1	2	yes	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	1	no	—
361	C₂×C₄	—	2	1	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	1	no	—
364	C₂×C₄	—	118	2	no	—
365	C₂×C₄	—	472	4	no	—
366	C₂×C₄	—	472	4	no	—
367	C₂×C₄	—	236	2	no	—
368	C₂×C₄	—	236	2	no	—
369	C₂×C₄	—	236	2	no	—
370	C₂×C₄	—	236	2	no	—
371	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
372	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
373	C₂×C₄	#14273773	1	1	yes	characteristic
374	C₈	—	2	2	no	—
375	C₈	—	2	1	no	—
376	C₈	—	2	1	no	—
377	C₈	—	4	1	no	—
378	C₈	—	4	1	no	—
379	C₈	—	236	2	no	—
380	C₈	C₂×D₄₇₂	1	2	yes	—
381	C₈	#14273756	1	2	yes	complemented
382	C₂²	—	2	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	236	2	no	—
386	C₂²	—	236	2	no	—
387	C₂²	—	236	2	no	—
388	C₂²	D₄×D₂₃₆	1	1	yes	characteristic
389	C₄	—	2	2	no	—
390	C₄	—	2	1	no	—
391	C₄	—	4	1	no	—
392	C₄	—	4	1	no	—
393	C₄	—	472	2	no	—
394	C₄	—	472	2	no	—
395	C₄	—	236	2	no	—
396	C₄	C₂²×D₄₇₂	1	1	yes	characteristic
397	C₄	C₂×#14273756	1	1	yes	characteristic
398	C₄	#16838207	1	1	yes	characteristic
399	C₄	#16838216	1	1	yes	characteristic
400	C₂	—	4	2	no	—
401	C₂	—	472	2	no	—
402	C₂	#21435743	1	1	yes	characteristic
403	C₂	D₄×D₄₇₂	1	1	yes	characteristic
404	C₂	#21435901	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	#19887
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. Q₈×D₄₇₂	C₂	semidirect
3. #21435743	C₂	semidirect
4. #21423714	C₂	semidirect
5. #21423586	C₂	semidirect
6. #21423490	C₂	semidirect
7. #21423485	C₂	semidirect
8. #21423347	C₂	semidirect
9. #21423333	C₂	semidirect
10. #21422982	C₂	semidirect
11. C₈×SD₁₆×C₅₉	C₂	semidirect
110. SD₁₆×C₅₉	D₈	semidirect
177. C₈×C₅₉	#137	semidirect
208. C₈×SD₁₆	D₅₉	semidirect
247. C₅₉	#19887	semidirect
317. SD₁₆	D₄₇₂	semidirect
381. C₈	#14273756	semidirect

Maximal subgroups

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

subgroup	index
Q₈×D₄₇₂	2
#21436444	2
#21435743	2
#21423714	2
#21423586	2
#21423490	2
#21423485	2
#21423347	2
#21423333	2
#21422982	2
C₈×SD₁₆×C₅₉	2
#19887	59