Subgroups — #28633819

Order 15104.

620 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	C₂×#16837674	C₂	1	1	yes	maximal, characteristic
2	#21425564	C₂	1	2	yes	maximal, complemented
3	#21425374	C₂	1	2	yes	maximal
4	#21425012	C₂	1	1	yes	maximal, characteristic, complemented
5	#21424886	C₂	1	1	yes	maximal, characteristic, complemented
6	#21424590	C₂	1	2	yes	maximal
7	#21424219	C₂	1	1	yes	maximal, characteristic, complemented
8	#21424210	C₂	1	2	yes	maximal, complemented
9	#21424147	C₂	1	1	yes	maximal, characteristic
10	#21423897	C₂	1	1	yes	maximal, characteristic
11	C₅₉×#2950	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14273758	C₂²	1	2	yes	—
13	C₂×#14273733	C₂²	1	1	yes	characteristic
14	#16837674	—	2	2	no	—
15	#16837674	—	2	2	no	—
16	#16837450	C₂²	1	2	yes	—
17	C₂×#14273663	C₂²	1	1	yes	characteristic
18	C₂×#14273663	C₂²	1	1	yes	characteristic
19	#16837420	C₂²	1	2	yes	—
20	#16837411	C₂²	1	2	yes	—
21	#16837384	C₂²	1	1	yes	characteristic
22	C₂×#14273646	C₂²	1	2	yes	—
23	C₂×#14273646	C₂²	1	2	yes	—
24	#16837375	C₂²	1	1	yes	characteristic
25	#16837357	C₂²	1	1	yes	characteristic
26	#16837174	C₂²	1	1	yes	characteristic
27	#16837172	C₂²	1	2	yes	—
28	#16837172	C₂²	1	1	yes	characteristic
29	#16837172	C₂²	1	2	yes	—
30	#16837172	C₂²	1	2	yes	—
31	#16837158	C₂²	1	2	yes	—
32	C₂²×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
33	C₅₉×#519	C₂²	1	2	yes	—
34	C₅₉×#441	—	2	2	no	—
35	C₄×Q₁₆×C₅₉	—	2	2	no	—
36	C₅₉×#427	C₂²	1	1	yes	characteristic
37	C₅₉×#419	C₂²	1	2	yes	—
38	C₅₉×#405	C₂²	1	1	yes	characteristic
39	C₂×#11745685	C₂³	1	1	yes	characteristic
40	#14273758	—	2	4	no	—
41	#14273758	—	4	2	no	—
42	C₂×#11745672	—	2	2	no	—
43	C₂×#11745671	—	2	2	no	—
44	#14273733	—	2	1	no	—
45	#14273733	—	2	1	no	—
46	#14273733	—	2	1	no	—
47	#14273733	—	2	1	no	—
48	C₂×#11745664	C₂³	1	1	yes	characteristic
49	C₂×#11745664	C₂³	1	1	yes	characteristic
50	C₂×#11745664	C₂³	1	2	yes	—
51	C₂×#11745664	C₂³	1	2	yes	—
52	C₂×#11745659	—	2	1	no	—
53	C₂×C₈×D₅₉	—	2	1	no	—
54	#14273686	C₂³	1	1	yes	characteristic
55	#14273663	—	2	1	no	—
56	#14273663	—	2	1	no	—
57	#14273663	—	2	1	no	—
58	#14273663	—	2	1	no	—
59	#14273662	—	4	2	no	—
60	#14273655	—	4	2	no	—
61	#14273651	—	4	2	no	—
62	#14273646	—	2	4	no	—
63	#14273646	—	4	2	no	—
64	#14273645	—	2	1	no	—
65	#14273643	—	2	1	no	—
66	#14273641	—	2	2	no	—
67	#14273637	—	2	1	no	—
68	C₈×Dic₅₉	—	2	1	no	—
69	C₂²×Q₈×C₅₉	C₂³	1	2	yes	—
70	C₂×Q₁₆×C₅₉	—	2	1	no	—
71	C₂×Q₁₆×C₅₉	—	2	1	no	—
72	C₂×Q₁₆×C₅₉	—	2	1	no	—
73	C₂×Q₁₆×C₅₉	—	2	1	no	—
74	C₂×Q₁₆×C₅₉	D₄	1	2	yes	—
75	C₂×Q₁₆×C₅₉	D₄	1	2	yes	—
76	C₂×M₁₆×C₅₉	C₂³	1	1	yes	characteristic
77	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₅₉	—	2	4	no	—
79	C₄×Q₈×C₅₉	—	4	2	no	—
80	#117×C₅₉	—	2	4	no	—
81	#117×C₅₉	C₂³	1	2	yes	—
82	#117×C₅₉	C₂³	1	1	yes	characteristic
83	#107×C₅₉	—	2	1	no	—
84	#106×C₅₉	—	2	1	no	—
85	#103×C₅₉	—	2	4	no	—
86	#97×C₅₉	—	2	1	no	—
87	C₄×C₈×C₅₉	—	2	1	no	—
88	C₂×Q₈×C₅₉	—	2	2	no	—
89	C₂×Q₈×C₅₉	—	2	2	no	—
90	C₂×Q₈×C₅₉	—	2	2	no	—
91	C₂×Q₈×C₅₉	—	2	2	no	—
92	C₂×Q₈×C₅₉	—	2	2	no	—
93	C₂×Q₈×C₅₉	C₂×D₄	1	4	yes	—
94	C₂²×C₄×C₅₉	—	2	2	no	—
95	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
96	C₂×C₂²⋊D₅₉	—	2	1	no	—
97	#11745685	—	2	1	no	—
98	#11745685	—	2	1	no	—
99	#11745685	—	2	1	no	—
100	#11745685	—	2	1	no	—
101	C₂×D₂₃₆	#41	1	1	yes	characteristic
102	C₂×C₄×D₅₉	—	2	1	no	—
103	C₂×Dic₁₁₈	#41	1	1	yes	characteristic
104	Q₁₆×C₅₉	—	2	2	no	—
105	Q₁₆×C₅₉	—	2	2	no	—
106	Q₁₆×C₅₉	—	2	2	no	—
107	Q₁₆×C₅₉	—	2	2	no	—
108	M₁₆×C₅₉	—	2	1	no	—
109	M₁₆×C₅₉	—	2	1	no	—
110	C₂×C₈×C₅₉	—	2	1	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	#41	1	1	yes	characteristic
114	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
115	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
116	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
117	C₄⋊C₄×C₅₉	—	2	1	no	—
118	C₄⋊C₄×C₅₉	—	2	2	no	—
119	C₄⋊C₄×C₅₉	—	4	4	no	—
120	C₂²⋊C₄×C₅₉	—	2	2	no	—
121	C₂²⋊C₄×C₅₉	#41	1	2	yes	—
122	C₄²×C₅₉	—	2	1	no	—
123	C₄²×C₅₉	—	2	2	no	—
124	C₄²×C₅₉	—	4	4	no	—
125	C₂²⋊Dic₅₉	—	2	1	no	—
126	#11745672	—	4	4	no	—
127	#11745671	—	4	4	no	—
128	C₄⋊Dic₅₉	—	2	1	no	—
129	C₄×Dic₅₉	—	2	1	no	—
130	#11745664	—	2	1	no	—
131	#11745664	—	2	2	no	—
132	#11745664	—	2	1	no	—
133	#11745664	—	2	2	no	—
134	#11745664	—	2	1	no	—
135	#11745664	—	2	1	no	—
136	#11745664	—	2	2	no	—
137	#11745664	—	2	2	no	—
138	C₂×C₅₉⋊C₈	—	2	1	no	—
139	C₂×C₅₉⋊C₈	—	2	1	no	—
140	C₂×C₅₉⋊C₈	—	2	2	no	—
141	C₂×C₅₉⋊C₈	—	2	2	no	—
142	#11745659	—	4	1	no	—
143	#11745659	—	4	1	no	—
144	C₈×D₅₉	—	4	1	no	—
145	C₈×D₅₉	—	4	1	no	—
146	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
147	C₂²×D₅₉	—	2	1	no	—
148	Q₈×C₅₉	—	2	4	no	—
149	Q₈×C₅₉	—	2	4	no	—
150	Q₈×C₅₉	—	4	2	no	—
151	Q₈×C₅₉	—	4	2	no	—
152	C₂×C₄×C₅₉	—	2	2	no	—
153	C₂×C₄×C₅₉	—	2	4	no	—
154	C₂×C₄×C₅₉	—	4	2	no	—
155	C₂×C₄×C₅₉	—	4	2	no	—
156	C₂×C₄×C₅₉	—	4	2	no	—
157	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
158	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
160	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
161	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂²⋊D₅₉	—	4	1	no	—
164	C₂²⋊D₅₉	—	4	1	no	—
165	C₂×Dic₅₉	—	2	1	no	—
166	C₂×Dic₅₉	—	4	1	no	—
167	D₂₃₆	—	2	1	no	—
168	D₂₃₆	—	2	1	no	—
169	C₄×D₅₉	—	4	1	no	—
170	C₄×D₅₉	—	4	1	no	—
171	Dic₁₁₈	—	2	1	no	—
172	Dic₁₁₈	—	2	1	no	—
173	C₈×C₅₉	—	4	1	no	—
174	C₈×C₅₉	#123	1	1	yes	characteristic
175	C₈×C₅₉	#123	1	1	yes	characteristic
176	C₈×C₅₉	#123	1	1	yes	characteristic
177	C₈×C₅₉	#123	1	1	yes	characteristic
178	C₅₉⋊C₈	—	4	1	no	—
179	C₅₉⋊C₈	—	4	2	no	—
180	C₅₉⋊C₈	—	4	2	no	—
181	C₅₉⋊C₈	—	4	1	no	—
182	#24989	—	59	1	no	maximal, Sylow
183	C₂²×C₅₉	—	4	1	no	—
184	C₂²×C₅₉	#529	1	1	yes	characteristic
185	C₂²×C₅₉	#546	1	1	yes	characteristic
186	C₂²×C₅₉	#555	1	1	yes	characteristic
187	C₂×D₅₉	—	4	1	no	—
188	C₂×D₅₉	—	8	1	no	—
189	C₄×C₅₉	—	4	2	no	—
190	C₄×C₅₉	—	4	4	no	—
191	C₄×C₅₉	—	8	2	no	—
192	C₄×C₅₉	C₂×#123	1	1	yes	characteristic
193	C₄×C₅₉	#540	1	1	yes	characteristic
194	C₄×C₅₉	#546	1	1	yes	characteristic
195	C₄×C₅₉	#553	1	1	yes	characteristic
196	Dic₅₉	—	4	1	no	—
197	Dic₅₉	—	8	1	no	—
198	C₂×#473	—	59	1	no	—
199	#2950	D₅₉	1	1	yes	characteristic, complemented
200	#2082	—	59	2	no	—
201	#2049	—	59	2	no	—
202	#1979	—	59	1	no	—
203	#1953	—	59	1	no	—
204	#1905	—	59	2	no	—
205	#1833	—	59	1	no	—
206	#1830	—	59	2	no	—
207	#1814	—	59	1	no	—
208	#1757	—	59	1	no	—
209	C₂×C₅₉	—	2	1	no	—
210	C₂×C₅₉	—	2	1	no	—
211	C₂×C₅₉	—	2	1	no	—
212	C₂×C₅₉	#2420	1	1	yes	characteristic
213	D₅₉	—	8	1	no	—
214	C₂×#137	—	59	2	no	—
215	C₂²×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
216	C₂×#131	—	59	1	no	—
217	#519	C₂×D₅₉	1	2	yes	—
218	#473	—	118	2	no	—
219	#473	—	118	2	no	—
220	#441	—	2	2	no	—
221	C₄×Q₁₆	—	2	2	no	—
222	#430	—	59	2	no	—
223	C₂×#108	—	59	1	no	—
224	C₂×#108	—	59	1	no	—
225	#427	C₂×D₅₉	1	1	yes	characteristic
226	#421	—	59	2	no	—
227	#419	—	59	2	no	—
228	#419	C₂×D₅₉	1	2	yes	—
229	#413	—	59	1	no	—
230	C₂×#101	—	59	2	no	—
231	C₂×#101	—	59	2	no	—
232	#410	—	59	1	no	—
233	#405	—	59	1	no	—
234	#405	C₂×D₅₉	1	1	yes	characteristic
235	#344	—	59	1	no	—
236	#344	—	59	2	no	—
237	#344	—	59	2	no	—
238	#344	—	59	1	no	—
239	#344	—	59	2	no	—
240	H(3,ℤ/4)	—	59	2	no	—
241	C₅₉	#24989	1	1	yes	characteristic, complemented, Sylow
242	C₂×#41	—	59	1	no	—
243	C₂²×Q₈	C₂²×D₅₉	1	2	yes	—
244	#137	—	118	4	no	—
245	#137	—	236	2	no	—
246	C₂×Q₁₆	—	2	1	no	—
247	C₂×Q₁₆	—	2	1	no	—
248	C₂×Q₁₆	—	2	1	no	—
249	C₂×Q₁₆	—	2	1	no	—
250	C₂×Q₁₆	—	118	2	no	—
251	C₂×Q₁₆	C₂²⋊D₅₉	1	2	yes	—
252	C₂×Q₁₆	C₂²⋊D₅₉	1	2	yes	—
253	C₂×SD₁₆	—	118	2	no	—
254	#131	—	118	1	no	—
255	#131	—	118	1	no	—
256	#131	—	118	1	no	—
257	#131	—	118	1	no	—
258	C₂×M₁₆	—	59	1	no	—
259	C₂×M₁₆	—	59	2	no	—
260	C₂×M₁₆	—	59	1	no	—
261	C₂×M₁₆	—	59	2	no	—
262	C₂×M₁₆	—	118	1	no	—
263	C₂×M₁₆	C₂²×D₅₉	1	1	yes	characteristic
264	C₂²×C₈	—	118	1	no	—
265	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
266	C₄×Q₈	—	2	4	no	—
267	C₄×Q₈	—	4	2	no	—
268	#117	—	2	4	no	—
269	#117	—	59	1	no	—
270	#117	C₂²×D₅₉	1	2	yes	—
271	#117	C₂²×D₅₉	1	1	yes	characteristic
272	#108	—	118	1	no	—
273	#108	—	118	1	no	—
274	#108	—	118	1	no	—
275	#108	—	118	1	no	—
276	#108	—	236	2	no	—
277	#107	—	2	1	no	—
278	#106	—	2	1	no	—
279	#104	—	236	2	no	—
280	#103	—	2	4	no	—
281	#103	—	236	2	no	—
282	#101	—	118	1	no	—
283	#101	—	118	4	no	—
284	#101	—	236	2	no	—
285	#100	—	118	1	no	—
286	#99	—	118	2	no	—
287	#97	—	2	1	no	—
288	#97	—	118	1	no	—
289	C₄×C₈	—	2	1	no	—
290	C₄×C₈	—	118	1	no	—
291	#41	—	118	1	no	—
292	#41	—	118	1	no	—
293	#41	—	118	1	no	—
294	#41	—	118	1	no	—
295	C₂×Q₈	—	2	2	no	—
296	C₂×Q₈	—	2	2	no	—
297	C₂×Q₈	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	59	1	no	—
301	C₂×Q₈	C₂×C₂²⋊D₅₉	1	4	yes	—
302	C₂×D₄	—	59	1	no	—
303	C₂×D₄	—	118	1	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	—	118	1	no	—
306	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
307	Q₁₆	—	2	2	no	—
308	Q₁₆	—	2	2	no	—
309	Q₁₆	—	2	2	no	—
310	Q₁₆	—	2	2	no	—
311	Q₁₆	—	236	4	no	—
312	SD₁₆	—	236	4	no	—
313	M₁₆	—	2	1	no	—
314	M₁₆	—	2	1	no	—
315	M₁₆	—	118	1	no	—
316	M₁₆	—	118	2	no	—
317	M₁₆	—	118	1	no	—
318	M₁₆	—	118	2	no	—
319	M₁₆	—	118	1	no	—
320	M₁₆	—	118	1	no	—
321	M₁₆	—	118	2	no	—
322	M₁₆	—	118	2	no	—
323	M₁₆	—	236	1	no	—
324	M₁₆	—	236	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	118	2	no	—
327	C₂×C₈	—	118	1	no	—
328	C₂×C₈	—	118	1	no	—
329	C₂×C₈	—	118	2	no	—
330	C₂×C₈	—	236	1	no	—
331	C₂×C₈	—	236	1	no	—
332	C₂×C₈	#11745685	1	1	yes	characteristic
333	C₂×C₈	#11745685	1	1	yes	characteristic
334	C₂×C₈	#11745685	1	1	yes	characteristic
335	C₂×C₈	#11745685	1	1	yes	characteristic
336	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
337	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	4	4	no	—
341	C₄⋊C₄	—	118	1	no	—
342	C₂²⋊C₄	—	2	2	no	—
343	C₂²⋊C₄	—	118	1	no	—
344	C₂²⋊C₄	#11745687	1	2	yes	—
345	C₄²	—	2	1	no	—
346	C₄²	—	2	2	no	—
347	C₄²	—	4	4	no	—
348	C₄²	—	118	1	no	—
349	C₂³	—	118	1	no	—
350	C₂³	C₂×#11745687	1	1	yes	characteristic
351	Q₈	—	2	4	no	—
352	Q₈	—	2	4	no	—
353	Q₈	—	4	2	no	—
354	Q₈	—	4	2	no	—
355	Q₈	—	118	1	no	—
356	Q₈	—	118	1	no	—
357	D₄	—	118	1	no	—
358	D₄	—	118	1	no	—
359	D₄	—	236	1	no	—
360	D₄	—	236	1	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	4	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	2	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	118	1	no	—
367	C₂×C₄	—	236	1	no	—
368	C₂×C₄	—	236	1	no	—
369	C₂×C₄	—	236	1	no	—
370	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
371	C₂×C₄	#41×D₅₉	1	1	yes	characteristic
372	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
373	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
374	C₂×C₄	#14273772	1	1	yes	characteristic
375	C₂×C₄	#14273773	1	1	yes	characteristic
376	C₈	—	4	1	no	—
377	C₈	—	236	1	no	—
378	C₈	—	236	2	no	—
379	C₈	—	236	2	no	—
380	C₈	—	236	1	no	—
381	C₈	#14273710	1	1	yes	characteristic
382	C₈	#14273710	1	1	yes	characteristic
383	C₈	#14273710	1	1	yes	characteristic
384	C₈	#14273710	1	1	yes	characteristic
385	C₂²	—	4	1	no	—
386	C₂²	—	472	1	no	—
387	C₂²	—	236	1	no	—
388	C₂²	#16837934	1	1	yes	characteristic
389	C₂²	#16838046	1	1	yes	characteristic
390	C₂²	#16838117	1	1	yes	characteristic
391	C₄	—	4	2	no	—
392	C₄	—	4	4	no	—
393	C₄	—	8	2	no	—
394	C₄	—	472	1	no	—
395	C₄	—	236	1	no	—
396	C₄	C₂×#14273710	1	1	yes	characteristic
397	C₄	#16838003	1	1	yes	characteristic
398	C₄	#16838040	1	1	yes	characteristic
399	C₄	#16838102	1	1	yes	characteristic
400	C₂	—	2	1	no	—
401	C₂	—	2	1	no	—
402	C₂	—	2	1	no	—
403	C₂	—	472	1	no	—
404	C₂	#21427582	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#21427582
commutator, cosocle	C₂²×C₄×C₅₉	C₂⁴
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting	C₅₉×#2950	C₂
socle	C₂×C₅₉	#2420
prime radical	C₂×#11745685	C₂³
hypercenter	#2950	D₅₉
hypocenter	C₅₉	#24989
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2950

Complements

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

N	H ≅ G/N	type
2. #21425564	C₂	semidirect
4. #21425012	C₂	semidirect
5. #21424886	C₂	semidirect
7. #21424219	C₂	semidirect
8. #21424210	C₂	semidirect
11. C₅₉×#2950	C₂	semidirect
199. #2950	D₅₉	semidirect
241. C₅₉	#24989	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837674	2
#21425564	2
#21425374	2
#21425012	2
#21424886	2
#21424590	2
#21424219	2
#21424210	2
#21424147	2
#21423897	2
C₅₉×#2950	2
#24989	59