Subgroups — #28742138

Order 15104.

652 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	#21437232	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837523	C₂	1	1	yes	maximal, characteristic, complemented
3	#21432896	C₂	1	1	yes	maximal, characteristic, complemented
4	#21425993	C₂	1	4	yes	maximal, complemented
5	#21425986	C₂	1	4	yes	maximal, complemented
6	#21425843	C₂	1	1	yes	maximal, characteristic, complemented
7	#21425831	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#16837179	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#2155	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#14273749	C₂²	1	1	yes	characteristic
11	C₂×#14273749	C₂²	1	1	yes	characteristic
12	C₂×#14273735	—	2	1	no	—
13	#16837870	—	2	1	no	—
14	#16837845	C₄	1	2	yes	complemented
15	#16837845	C₂²	1	2	yes	—
16	#16837840	C₄	1	2	yes	complemented
17	#16837840	C₂²	1	2	yes	—
18	#16837834	C₄	1	4	yes	complemented
19	C₂×#14273667	C₂²	1	1	yes	characteristic
20	#16837523	—	2	2	no	—
21	#16837523	C₂²	1	4	yes	—
22	#16837513	C₂²	1	2	yes	complemented
23	#16837508	C₂²	1	2	yes	complemented
24	#16837501	—	2	2	no	—
25	C₂×#14273653	—	2	1	no	—
26	#16837396	—	2	1	no	—
27	C₂×#14273637	C₂²	1	1	yes	characteristic
28	#16837217	C₂²	1	2	yes	complemented
29	#16837214	C₂²	1	2	yes	complemented
30	#16837211	C₂²	1	4	yes	—
31	#16837179	C₂²	1	4	yes	complemented
32	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
33	C₂×#108×C₅₉	C₂²	1	1	yes	characteristic
34	C₅₉×#427	C₂²	1	1	yes	characteristic
35	C₅₉×#365	C₂²	1	4	yes	complemented
36	C₂×#11745685	—	2	1	no	—
37	C₂×#11745685	—	2	1	no	—
38	#14273749	—	2	1	no	—
39	#14273749	—	2	1	no	—
40	#14273749	—	2	1	no	—
41	#14273749	—	2	1	no	—
42	#14273749	C₂×C₄	1	2	yes	—
43	#14273749	C₂×C₄	1	2	yes	—
44	#14273749	C₂×C₄	1	2	yes	complemented
45	#14273749	C₂³	1	2	yes	—
46	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic, complemented
47	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
48	C₂×#11745660	—	2	1	no	—
49	C₂×#11745660	—	2	1	no	—
50	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic, complemented
51	C₂×D₄₇₂	C₂³	1	1	yes	characteristic
52	#14273735	—	2	4	no	—
53	#14273735	—	4	2	no	—
54	C₂×#11745659	—	4	1	no	—
55	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
56	C₂×C₈×D₅₉	—	4	1	no	—
57	#14273692	—	2	2	no	—
58	#14273689	—	2	2	no	—
59	(C₂²⋊D₅₉)⋊C₄	—	4	2	no	—
60	#14273684	—	4	1	no	—
61	C₂×C₄×Dic₅₉	D₄	1	1	yes	characteristic
62	C₄²×D₅₉	—	4	1	no	—
63	Dic₄₇₂	C₂×C₄	1	4	yes	complemented
64	#14273672	C₂×C₄	1	4	yes	complemented
65	#14273672	C₂×C₄	1	4	yes	complemented
66	D₉₄₄	C₂×C₄	1	4	yes	complemented
67	#14273667	—	2	1	no	—
68	#14273667	C₂³	1	2	yes	—
69	C₂×C₅₉⋊C₁₆	—	2	1	no	—
70	#14273653	—	2	4	no	—
71	#14273650	—	2	2	no	—
72	#14273647	—	2	2	no	—
73	#14273637	D₄	1	2	yes	—
74	#14273637	C₂³	1	2	yes	—
75	C₂×M₁₆×C₅₉	—	2	1	no	—
76	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
77	#117×C₅₉	—	2	1	no	—
78	M₃₂×C₅₉	C₂×C₄	1	2	yes	complemented
79	M₃₂×C₅₉	C₂³	1	2	yes	—
80	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	complemented
81	#108×C₅₉	—	2	1	no	—
82	#108×C₅₉	C₂³	1	2	yes	—
83	#107×C₅₉	—	2	1	no	—
84	#106×C₅₉	C₂³	1	2	yes	—
85	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
86	C₂×C₂²⋊D₅₉	—	4	1	no	—
87	C₂×C₂²⋊D₅₉	—	4	1	no	—
88	C₂²×Dic₅₉	—	2	1	no	—
89	#11745685	—	2	2	no	—
90	#11745685	—	2	2	no	—
91	#11745685	—	2	2	no	—
92	#11745685	—	2	2	no	—
93	#11745685	—	4	1	no	—
94	#11745685	—	4	1	no	—
95	#11745685	—	4	1	no	—
96	#11745685	—	4	1	no	—
97	C₂×D₂₃₆	—	2	1	no	—
98	C₂×D₂₃₆	—	2	1	no	—
99	C₂×C₄×D₅₉	—	4	1	no	—
100	C₂×C₄×D₅₉	—	4	1	no	—
101	C₂×C₄×D₅₉	—	4	2	no	—
102	C₂×Dic₁₁₈	—	2	1	no	—
103	C₂×Dic₁₁₈	—	2	1	no	—
104	M₁₆×C₅₉	—	2	2	no	—
105	M₁₆×C₅₉	—	4	1	no	—
106	C₂×C₈×C₅₉	—	4	1	no	—
107	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
111	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
112	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₅₉	—	2	2	no	—
114	C₂²⋊C₄×C₅₉	—	4	1	no	—
115	C₄²×C₅₉	—	4	1	no	—
116	(C₂×D₅₉)⋊C₄	—	4	2	no	—
117	Dic₅₉⋊C₄	—	4	2	no	—
118	C₄×Dic₅₉	—	2	1	no	—
119	C₄×Dic₅₉	—	4	2	no	—
120	C₄×Dic₅₉	C₂×D₄	1	2	yes	—
121	C₂×C₅₉⋊C₈	—	2	1	no	—
122	C₂×C₅₉⋊C₈	—	2	1	no	—
123	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
124	Dic₂₃₆	—	2	1	no	—
125	Dic₂₃₆	—	2	1	no	—
126	Dic₂₃₆	C₂²×C₄	1	2	yes	—
127	Dic₂₃₆	C₂²×C₄	1	2	yes	—
128	D₄₇₂	—	2	1	no	—
129	D₄₇₂	—	2	1	no	—
130	D₄₇₂	C₂²×C₄	1	2	yes	—
131	D₄₇₂	C₂²×C₄	1	2	yes	—
132	#11745660	—	2	2	no	—
133	#11745660	—	2	2	no	—
134	#11745660	—	2	2	no	—
135	#11745660	—	2	2	no	—
136	#11745659	—	4	4	no	—
137	C₈×D₅₉	—	4	4	no	—
138	C₁₆×C₅₉	C₂²×C₄	1	4	yes	—
139	C₅₉⋊C₁₆	—	2	2	no	—
140	C₂³×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
141	C₂²×D₅₉	—	4	1	no	—
142	C₂²×D₅₉	—	4	1	no	—
143	C₂×C₄×C₅₉	—	4	2	no	—
144	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
147	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
148	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₅₉	—	4	2	no	—
151	C₂²⋊D₅₉	—	4	2	no	—
152	C₂²⋊D₅₉	—	4	2	no	—
153	C₂²⋊D₅₉	—	4	2	no	—
154	C₂×Dic₅₉	—	2	2	no	—
155	C₂×Dic₅₉	—	4	1	no	—
156	C₂×Dic₅₉	—	4	1	no	—
157	C₂×Dic₅₉	—	8	1	no	—
158	D₂₃₆	—	2	2	no	—
159	D₂₃₆	—	2	2	no	—
160	D₂₃₆	—	4	1	no	—
161	D₂₃₆	—	4	1	no	—
162	C₄×D₅₉	—	4	2	no	—
163	C₄×D₅₉	—	4	2	no	—
164	C₄×D₅₉	—	4	2	no	—
165	C₄×D₅₉	—	4	2	no	—
166	C₄×D₅₉	—	8	4	no	—
167	Dic₁₁₈	—	2	2	no	—
168	Dic₁₁₈	—	2	2	no	—
169	Dic₁₁₈	—	4	1	no	—
170	Dic₁₁₈	—	4	1	no	—
171	C₈×C₅₉	—	4	2	no	—
172	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
173	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
174	C₈×C₅₉	C₂³×C₄	1	1	yes	characteristic
175	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
176	C₅₉⋊C₈	—	2	2	no	—
177	#33468	—	59	1	no	maximal, Sylow
178	C₂²×C₅₉	—	4	1	no	—
179	C₂²×C₅₉	#442	1	1	yes	characteristic
180	C₂²×C₅₉	#442	1	1	yes	characteristic
181	C₂²×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×D₅₉	—	4	2	no	—
183	C₂×D₅₉	—	4	2	no	—
184	C₂×D₅₉	—	8	2	no	—
185	C₂×D₅₉	—	8	2	no	—
186	C₄×C₅₉	—	8	2	no	—
187	C₄×C₅₉	#442	1	1	yes	characteristic
188	C₄×C₅₉	#442	1	1	yes	characteristic
189	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
190	C₄×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
191	Dic₅₉	—	4	2	no	—
192	Dic₅₉	—	4	2	no	—
193	Dic₅₉	—	4	2	no	—
194	#3414	—	59	1	no	—
195	C₂×#444	—	59	1	no	—
196	#2943	—	59	1	no	—
197	AGL(1,ℤ/16)	—	59	4	no	—
198	#2180	—	59	4	no	—
199	#2155	D₅₉	1	1	yes	characteristic, complemented
200	#2148	—	59	1	no	—
201	#2145	—	59	1	no	—
202	C₂×#347	—	59	1	no	—
203	C₂×C₅₉	—	2	1	no	—
204	C₂×C₅₉	—	2	1	no	—
205	C₂×C₅₉	—	2	1	no	—
206	C₂×C₅₉	C₂×#442	1	1	yes	characteristic
207	D₅₉	—	8	2	no	—
208	D₅₉	—	8	2	no	—
209	C₂×#135	—	59	1	no	—
210	C₂×#135	—	59	1	no	—
211	C₂×#131	—	118	1	no	—
212	C₄×#41	—	118	1	no	—
213	#510	—	59	2	no	—
214	#510	—	59	2	no	—
215	#509	—	59	2	no	—
216	#509	—	59	2	no	—
217	#508	—	59	4	no	—
218	C₂×M₃₂	—	59	1	no	—
219	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
220	#444	—	59	4	no	—
221	#444	—	118	2	no	—
222	#442	—	59	2	no	—
223	#441	—	59	2	no	—
224	#440	—	118	2	no	—
225	C₂×#108	C₂×D₅₉	1	1	yes	characteristic
226	#427	C₂×D₅₉	1	1	yes	characteristic
227	C₂×#104	—	118	1	no	—
228	#416	—	118	1	no	—
229	C₂×#97	—	59	1	no	—
230	#365	C₂×D₅₉	1	4	yes	complemented
231	#362	—	59	2	no	—
232	#361	—	59	2	no	—
233	#360	—	59	4	no	—
234	#347	—	59	4	no	—
235	C₅₉	#33468	1	1	yes	characteristic, complemented, Sylow
236	C₂×#41	—	118	1	no	—
237	C₂×#41	—	118	1	no	—
238	#135	—	59	2	no	—
239	#135	—	59	2	no	—
240	#135	—	59	2	no	—
241	#135	—	59	2	no	—
242	#135	—	118	1	no	—
243	#135	—	118	1	no	—
244	#135	—	118	1	no	—
245	#135	—	118	1	no	—
246	C₂×Q₁₆	—	59	1	no	—
247	C₂×Q₁₆	—	59	1	no	—
248	C₂×SD₁₆	—	118	1	no	—
249	C₂×SD₁₆	—	118	1	no	—
250	C₂×D₈	—	59	1	no	—
251	C₂×D₈	—	59	1	no	—
252	#131	—	118	4	no	—
253	#131	—	236	2	no	—
254	C₂×M₁₆	—	2	1	no	—
255	C₂×M₁₆	—	236	1	no	—
256	C₂²×C₈	—	59	1	no	—
257	C₂²×C₈	—	236	1	no	—
258	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
259	C₄×Q₈	—	118	2	no	—
260	C₄×D₄	—	118	2	no	—
261	C₄×D₄	—	236	2	no	—
262	#117	—	2	1	no	—
263	#117	—	236	1	no	—
264	C₂×C₄²	—	59	1	no	—
265	C₂×C₄²	—	236	1	no	—
266	Q₃₂	—	59	4	no	—
267	SD₃₂	—	59	4	no	—
268	SD₃₂	—	59	4	no	—
269	D₁₆	—	59	4	no	—
270	M₃₂	—	59	2	no	—
271	M₃₂	—	118	1	no	—
272	M₃₂	C₄×D₅₉	1	2	yes	complemented
273	M₃₂	C₂²×D₅₉	1	2	yes	—
274	C₂×C₁₆	—	118	1	no	—
275	C₂×C₁₆	C₄×D₅₉	1	2	yes	complemented
276	#108	—	2	1	no	—
277	#108	C₂²×D₅₉	1	2	yes	—
278	#107	—	2	1	no	—
279	#106	C₂²×D₅₉	1	2	yes	—
280	#104	—	118	4	no	—
281	#103	—	118	2	no	—
282	#102	—	118	2	no	—
283	#97	—	59	2	no	—
284	#97	—	59	2	no	—
285	#41	—	118	2	no	—
286	#41	—	118	2	no	—
287	#41	—	118	2	no	—
288	#41	—	118	2	no	—
289	#41	—	236	1	no	—
290	#41	—	236	1	no	—
291	#41	—	236	1	no	—
292	#41	—	236	1	no	—
293	C₂×Q₈	—	118	1	no	—
294	C₂×Q₈	—	118	1	no	—
295	C₂×D₄	—	118	1	no	—
296	C₂×D₄	—	118	1	no	—
297	C₂×D₄	—	236	1	no	—
298	C₂×D₄	—	236	1	no	—
299	C₂²×C₄	—	118	1	no	—
300	C₂²×C₄	—	236	2	no	—
301	C₂²×C₄	—	236	1	no	—
302	C₂²×C₄	—	236	1	no	—
303	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
304	Q₁₆	—	59	2	no	—
305	Q₁₆	—	59	2	no	—
306	Q₁₆	—	118	1	no	—
307	Q₁₆	—	118	1	no	—
308	SD₁₆	—	118	2	no	—
309	SD₁₆	—	118	2	no	—
310	SD₁₆	—	118	2	no	—
311	SD₁₆	—	118	2	no	—
312	D₈	—	59	2	no	—
313	D₈	—	59	2	no	—
314	D₈	—	118	1	no	—
315	D₈	—	118	1	no	—
316	M₁₆	—	2	2	no	—
317	M₁₆	—	4	1	no	—
318	M₁₆	—	236	4	no	—
319	C₂×C₈	—	4	1	no	—
320	C₂×C₈	—	59	2	no	—
321	C₂×C₈	—	118	1	no	—
322	C₂×C₈	—	118	1	no	—
323	C₂×C₈	—	236	4	no	—
324	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
325	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
326	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
327	C₂×C₈	#11745689	1	1	yes	characteristic
328	C₂×C₈	#11745689	1	1	yes	characteristic
329	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	236	2	no	—
332	C₂²⋊C₄	—	4	1	no	—
333	C₂²⋊C₄	—	236	2	no	—
334	C₄²	—	4	1	no	—
335	C₄²	—	59	2	no	—
336	C₄²	—	118	1	no	—
337	C₄²	—	236	2	no	—
338	C₁₆	—	118	2	no	—
339	C₁₆	C₂×C₄×D₅₉	1	4	yes	—
340	C₂³	—	236	1	no	—
341	C₂³	—	236	1	no	—
342	C₂³	#14273754	1	1	yes	characteristic
343	Q₈	—	118	2	no	—
344	Q₈	—	118	2	no	—
345	Q₈	—	236	1	no	—
346	Q₈	—	236	1	no	—
347	D₄	—	118	2	no	—
348	D₄	—	118	2	no	—
349	D₄	—	236	2	no	—
350	D₄	—	236	2	no	—
351	D₄	—	236	2	no	—
352	D₄	—	236	2	no	—
353	D₄	—	236	1	no	—
354	D₄	—	236	1	no	—
355	C₂×C₄	—	4	2	no	—
356	C₂×C₄	—	118	2	no	—
357	C₂×C₄	—	472	4	no	—
358	C₂×C₄	—	472	1	no	—
359	C₂×C₄	—	236	1	no	—
360	C₂×C₄	—	236	1	no	—
361	C₂×C₄	—	236	2	no	—
362	C₂×C₄	—	236	2	no	—
363	C₂×C₄	—	236	2	no	—
364	C₂×C₄	—	236	2	no	—
365	C₂×C₄	#14273689	1	1	yes	characteristic
366	C₂×C₄	#14273689	1	1	yes	characteristic
367	C₂×C₄	#14273754	1	1	yes	characteristic
368	C₂×C₄	#14273754	1	1	yes	characteristic
369	C₂×C₄	#14273754	1	1	yes	characteristic
370	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
371	C₈	—	4	2	no	—
372	C₈	—	118	2	no	—
373	C₈	#14273689	1	1	yes	characteristic
374	C₈	#14273689	1	1	yes	characteristic
375	C₈	C₂²×C₄×D₅₉	1	1	yes	characteristic
376	C₈	C₂×#11745689	1	1	yes	characteristic
377	C₂²	—	4	1	no	—
378	C₂²	—	472	2	no	—
379	C₂²	—	472	2	no	—
380	C₂²	—	236	2	no	—
381	C₂²	—	236	2	no	—
382	C₂²	#16837513	1	1	yes	characteristic
383	C₂²	#16837513	1	1	yes	characteristic
384	C₂²	C₂×#14273754	1	1	yes	characteristic
385	C₄	—	8	2	no	—
386	C₄	—	236	2	no	—
387	C₄	—	236	2	no	—
388	C₄	—	236	2	no	—
389	C₄	#16837513	1	1	yes	characteristic
390	C₄	#16837513	1	1	yes	characteristic
391	C₄	C₂×#14273689	1	1	yes	characteristic
392	C₄	C₂×#14273754	1	1	yes	characteristic
393	C₂	—	2	1	no	—
394	C₂	—	2	1	no	—
395	C₂	—	2	1	no	—
396	C₂	—	472	2	no	—
397	C₂	—	472	2	no	—
398	C₂	C₂×#16837513	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837513
commutator	C₈×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#2155	C₂
socle	C₂×C₅₉	C₂×#442
prime radical	#21437232	C₂
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	#2155	D₅₉
hypocenter	C₅₉	#33468
LCS	G ▷ C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2155

Complements

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

N	H ≅ G/N	type
2. C₂×#16837523	C₂	semidirect
3. #21432896	C₂	semidirect
4. #21425993	C₂	semidirect
5. #21425986	C₂	semidirect
6. #21425843	C₂	semidirect
7. #21425831	C₂	semidirect
8. C₂×#16837179	C₂	semidirect
9. C₅₉×#2155	C₂	semidirect
14. #16837845	C₄	semidirect
16. #16837840	C₄	semidirect
18. #16837834	C₄	semidirect
22. #16837513	C₂²	semidirect
23. #16837508	C₂²	semidirect
28. #16837217	C₂²	semidirect
29. #16837214	C₂²	semidirect
31. #16837179	C₂²	semidirect
35. C₅₉×#365	C₂²	semidirect
44. #14273749	C₂×C₄	semidirect
46. C₂×Dic₂₃₆	C₂×C₄	semidirect
50. C₂×D₄₇₂	C₂×C₄	semidirect
63. Dic₄₇₂	C₂×C₄	semidirect
64. #14273672	C₂×C₄	semidirect
65. #14273672	C₂×C₄	semidirect
66. D₉₄₄	C₂×C₄	semidirect
78. M₃₂×C₅₉	C₂×C₄	semidirect
80. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
199. #2155	D₅₉	semidirect
230. #365	C₂×D₅₉	semidirect
235. C₅₉	#33468	semidirect
272. M₃₂	C₄×D₅₉	semidirect
275. C₂×C₁₆	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21437232	2
C₂×#16837523	2
#21432896	2
#21425993	2
#21425986	2
#21425843	2
#21425831	2
C₂×#16837179	2
C₅₉×#2155	2
#33468	59