Subgroups — #28743337

Order 15104.

546 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21434610	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#14273749	C₂	1	1	yes	maximal, characteristic
3	#21426261	C₂	1	2	yes	maximal, complemented
4	#21426254	C₂	1	2	yes	maximal, complemented
5	#21426249	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426244	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426239	C₂	1	1	yes	maximal, characteristic, complemented
8	#21426234	C₂	1	1	yes	maximal, characteristic, complemented
9	#21425880	C₂	1	2	yes	maximal, complemented
10	#21425804	C₂	1	2	yes	maximal
11	C₂×C₅₉×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14273749	C₂²	1	1	yes	characteristic
13	#16837959	—	2	1	no	—
14	C₄×#11745685	—	2	1	no	—
15	#16837807	C₂²	1	1	yes	characteristic
16	#16837807	C₂²	1	1	yes	characteristic
17	#16837804	—	2	1	no	—
18	#16837804	C₂²	1	2	yes	—
19	#16837800	—	2	1	no	—
20	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
22	C₄×#11745660	—	2	1	no	—
23	C₄×#11745660	—	2	1	no	—
24	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
25	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
26	#16837440	—	2	1	no	—
27	#16837440	C₂²	1	1	yes	characteristic
28	#16837440	C₂²	1	2	yes	—
29	#16837399	—	2	1	no	—
30	#16837230	C₂²	1	2	yes	—
31	#16837230	C₂²	1	2	yes	—
32	#16837228	C₂²	1	2	yes	—
33	#16837228	C₂²	1	2	yes	—
34	#16837207	C₂²	1	2	yes	—
35	#16837207	C₂²	1	2	yes	—
36	#16837204	C₂²	1	2	yes	—
37	#16837204	C₂²	1	2	yes	—
38	C₂²×C₁₆×C₅₉	C₂²	1	2	yes	—
39	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
40	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
41	C₅₉×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₅₉×C₄⋊C₁₆	C₂²	1	2	yes	—
43	C₂×#11745685	—	2	1	no	—
44	#14273749	—	2	2	no	—
45	#14273749	Q₈	1	4	yes	complemented
46	C₂×Dic₂₃₆	C₂³	1	1	yes	characteristic
47	C₂×#11745660	—	2	1	no	—
48	C₂×D₄₇₂	C₂³	1	1	yes	characteristic
49	#14273727	—	2	1	no	—
50	#14273727	—	2	1	no	—
51	#14273719	—	2	2	no	—
52	#14273706	—	4	1	no	—
53	#14273706	—	4	1	no	—
54	C₄×Dic₁₁₈	—	2	1	no	—
55	C₄×Dic₁₁₈	—	2	1	no	—
56	C₄×Dic₁₁₈	—	4	1	no	—
57	C₄×Dic₁₁₈	—	4	1	no	—
58	C₄×D₂₃₆	—	2	1	no	—
59	C₄×D₂₃₆	—	2	1	no	—
60	C₄×C₂²⋊D₅₉	—	4	1	no	—
61	C₄×C₂²⋊D₅₉	—	4	1	no	—
62	#14273686	—	2	1	no	—
63	#14273686	—	2	2	no	—
64	#14273684	—	4	1	no	—
65	C₄²×D₅₉	—	4	1	no	—
66	#14273661	—	2	1	no	—
67	#14273661	—	2	1	no	—
68	#14273661	C₂³	1	2	yes	—
69	#14273661	C₂³	1	2	yes	—
70	#14273661	C₂³	1	1	yes	characteristic
71	#14273661	C₂³	1	1	yes	characteristic
72	#14273659	—	2	2	no	—
73	#14273659	—	2	1	no	—
74	#14273659	—	2	2	no	—
75	#14273652	—	2	1	no	—
76	#14273652	—	2	1	no	—
77	#14273649	—	2	1	no	—
78	#14273649	—	2	1	no	—
79	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
80	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
81	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
82	C₂×C₁₆×C₅₉	—	2	4	no	—
83	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
84	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
85	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
86	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
87	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
88	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
89	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
90	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
92	C₂×C₂²⋊D₅₉	—	4	1	no	—
93	#11745685	—	2	4	no	—
94	#11745685	—	4	2	no	—
95	C₂×D₂₃₆	—	2	1	no	—
96	C₂×C₄×D₅₉	—	4	1	no	—
97	C₂×C₄×D₅₉	—	4	1	no	—
98	C₂×C₄×D₅₉	—	4	1	no	—
99	C₂×Dic₁₁₈	—	2	1	no	—
100	C₂×Dic₁₁₈	—	4	1	no	—
101	C₂×Dic₁₁₈	—	4	1	no	—
102	C₂×C₈×C₅₉	—	2	2	no	—
103	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₅₉	C₂×Q₈	1	2	yes	—
107	C₂×C₈×C₅₉	#41	1	2	yes	—
108	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
109	C₄²×C₅₉	D₈	1	1	yes	characteristic
110	C₄²×C₅₉	D₈	1	1	yes	characteristic
111	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
113	C₂²⋊Dic₅₉	—	4	1	no	—
114	C₂²⋊Dic₅₉	—	4	2	no	—
115	(C₂×D₅₉)⋊C₄	—	4	1	no	—
116	(C₂×D₅₉)⋊C₄	—	4	1	no	—
117	C₄⋊Dic₅₉	—	2	1	no	—
118	C₄⋊Dic₅₉	—	2	1	no	—
119	C₄⋊Dic₅₉	—	2	2	no	—
120	C₄⋊Dic₅₉	—	2	2	no	—
121	C₄⋊Dic₅₉	—	4	1	no	—
122	C₄⋊Dic₅₉	—	4	1	no	—
123	Dic₅₉⋊C₄	—	4	1	no	—
124	Dic₅₉⋊C₄	—	4	1	no	—
125	Dic₅₉⋊C₄	—	4	2	no	—
126	Dic₅₉⋊C₄	—	4	2	no	—
127	C₄×Dic₅₉	—	4	1	no	—
128	C₄×Dic₅₉	—	4	1	no	—
129	C₄×Dic₅₉	—	4	1	no	—
130	C₄×Dic₅₉	—	4	2	no	—
131	Dic₂₃₆	—	2	1	no	—
132	Dic₂₃₆	C₂×Q₈	1	2	yes	—
133	D₄₇₂	—	2	1	no	—
134	D₄₇₂	C₂×Q₈	1	2	yes	—
135	#11745660	—	2	4	no	—
136	C₁₆×C₅₉	—	2	2	no	—
137	C₁₆×C₅₉	—	2	2	no	—
138	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₅₉	—	4	1	no	—
140	C₂×C₄×C₅₉	—	2	2	no	—
141	C₂×C₄×C₅₉	—	4	1	no	—
142	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₅₉	#137	1	2	yes	—
149	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
150	C₂²⋊D₅₉	—	4	4	no	—
151	C₂×Dic₅₉	—	4	1	no	—
152	C₂×Dic₅₉	—	4	1	no	—
153	C₂×Dic₅₉	—	4	1	no	—
154	C₂×Dic₅₉	—	4	2	no	—
155	C₂×Dic₅₉	—	4	2	no	—
156	D₂₃₆	—	2	2	no	—
157	D₂₃₆	—	4	1	no	—
158	C₄×D₅₉	—	4	4	no	—
159	C₄×D₅₉	—	8	2	no	—
160	C₄×D₅₉	—	8	2	no	—
161	Dic₁₁₈	—	2	2	no	—
162	Dic₁₁₈	—	4	1	no	—
163	Dic₁₁₈	—	8	2	no	—
164	Dic₁₁₈	—	8	2	no	—
165	C₈×C₅₉	—	2	1	no	—
166	C₈×C₅₉	—	2	1	no	—
167	C₈×C₅₉	C₂²⋊Q₈	1	2	yes	—
168	C₈×C₅₉	C₂²×Q₈	1	1	yes	characteristic
169	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
170	#33626	—	59	1	no	maximal, Sylow
171	C₂²×C₅₉	—	2	1	no	—
172	C₂²×C₅₉	—	2	1	no	—
173	C₂²×C₅₉	#474	1	2	yes	—
174	C₂²×C₅₉	C₂²×D₈	1	1	yes	characteristic
175	C₂×D₅₉	—	4	2	no	—
176	C₂×D₅₉	—	8	1	no	—
177	C₂×D₅₉	—	8	1	no	—
178	C₄×C₅₉	—	2	1	no	—
179	C₄×C₅₉	—	2	1	no	—
180	C₄×C₅₉	—	2	1	no	—
181	C₄×C₅₉	#474	1	2	yes	—
182	C₄×C₅₉	#508	1	2	yes	—
183	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
184	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
185	Dic₅₉	—	4	2	no	—
186	Dic₅₉	—	8	2	no	—
187	Dic₅₉	—	8	1	no	—
188	Dic₅₉	—	8	2	no	—
189	Dic₅₉	—	8	1	no	—
190	#3160	—	59	1	no	—
191	C₄×#135	—	59	1	no	—
192	#2229	—	59	2	no	—
193	#2228	—	59	2	no	—
194	#2227	—	59	1	no	—
195	#2226	—	59	1	no	—
196	#2225	—	59	1	no	—
197	#2224	—	59	1	no	—
198	#2158	—	59	2	no	—
199	C₂×C₄⋊C₁₆	D₅₉	1	1	yes	characteristic, complemented
200	#2138	—	59	2	no	—
201	C₂×C₅₉	—	2	2	no	—
202	C₂×C₅₉	C₂×#474	1	1	yes	characteristic
203	C₂×C₅₉	C₂×#508	1	1	yes	characteristic
204	C₂×C₅₉	#3414	1	1	yes	characteristic
205	D₅₉	—	8	2	no	—
206	C₂×#135	—	59	1	no	—
207	#533	—	118	1	no	—
208	C₄×#41	—	118	1	no	—
209	C₂²×C₁₆	C₂×D₅₉	1	2	yes	—
210	#500	—	59	1	no	—
211	#500	—	59	1	no	—
212	#499	—	59	2	no	—
213	#499	—	118	1	no	—
214	#498	—	118	1	no	—
215	C₄×Q₁₆	—	59	1	no	—
216	C₄×Q₁₆	—	59	1	no	—
217	C₄×SD₁₆	—	118	1	no	—
218	C₄×SD₁₆	—	118	1	no	—
219	C₄×D₈	—	59	1	no	—
220	C₄×D₈	—	59	1	no	—
221	#427	—	59	2	no	—
222	#427	—	59	1	no	—
223	#427	—	118	1	no	—
224	#416	—	118	1	no	—
225	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
226	#367	—	59	2	no	—
227	#367	—	59	2	no	—
228	#366	—	59	2	no	—
229	#366	—	59	2	no	—
230	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
231	C₄⋊C₁₆	C₂×D₅₉	1	1	yes	characteristic
232	C₄⋊C₁₆	C₂×D₅₉	1	2	yes	—
233	#358	—	59	2	no	—
234	#358	—	59	2	no	—
235	#357	—	59	2	no	—
236	#357	—	59	2	no	—
237	C₅₉	#33626	1	1	yes	characteristic, complemented, Sylow
238	C₂×#41	—	118	1	no	—
239	#135	—	59	4	no	—
240	#135	—	118	2	no	—
241	C₂×Q₁₆	—	59	1	no	—
242	C₂×SD₁₆	—	118	1	no	—
243	C₂×D₈	—	59	1	no	—
244	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
246	C₄⋊Q₈	—	118	1	no	—
247	C₄⋊Q₈	—	118	1	no	—
248	#125	—	118	2	no	—
249	C₂²⋊Q₈	—	236	1	no	—
250	C₂²⋊Q₈	—	236	1	no	—
251	C₄×Q₈	—	118	1	no	—
252	C₄×Q₈	—	118	1	no	—
253	C₄×Q₈	—	236	1	no	—
254	C₄×Q₈	—	236	1	no	—
255	C₄×D₄	—	118	1	no	—
256	C₄×D₄	—	118	1	no	—
257	C₄×D₄	—	236	1	no	—
258	C₄×D₄	—	236	1	no	—
259	#117	—	118	1	no	—
260	#117	—	118	2	no	—
261	#117	—	236	1	no	—
262	C₂×C₄²	—	236	1	no	—
263	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
264	C₂×C₁₆	—	2	4	no	—
265	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
266	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
267	#107	—	59	2	no	—
268	#107	—	59	2	no	—
269	#107	—	59	1	no	—
270	#107	—	59	1	no	—
271	#107	—	118	1	no	—
272	#107	—	118	1	no	—
273	#106	—	118	1	no	—
274	#106	—	118	2	no	—
275	#106	—	118	2	no	—
276	#103	—	118	1	no	—
277	#103	—	118	1	no	—
278	#102	—	118	1	no	—
279	#102	—	118	1	no	—
280	C₄×C₈	D₂₃₆	1	1	yes	characteristic
281	C₄×C₈	D₂₃₆	1	1	yes	characteristic
282	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
283	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
284	#41	—	118	4	no	—
285	#41	—	236	2	no	—
286	C₂×Q₈	—	118	1	no	—
287	C₂×Q₈	—	236	1	no	—
288	C₂×Q₈	—	236	1	no	—
289	C₂×D₄	—	118	1	no	—
290	C₂×D₄	—	236	1	no	—
291	C₂²×C₄	—	236	1	no	—
292	C₂²×C₄	—	236	1	no	—
293	C₂²×C₄	—	236	1	no	—
294	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
295	C₂²×C₄	D₄₇₂	1	1	yes	characteristic
296	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
297	Q₁₆	—	59	2	no	—
298	Q₁₆	—	118	1	no	—
299	SD₁₆	—	118	4	no	—
300	D₈	—	59	2	no	—
301	D₈	—	118	1	no	—
302	C₂×C₈	—	2	2	no	—
303	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
304	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
305	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
306	C₂×C₈	#11745687	1	2	yes	—
307	C₂×C₈	Q₈×D₅₉	1	2	yes	—
308	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
309	C₄⋊C₄	—	118	1	no	—
310	C₄⋊C₄	—	118	1	no	—
311	C₄⋊C₄	—	118	2	no	—
312	C₄⋊C₄	—	118	2	no	—
313	C₄⋊C₄	—	236	1	no	—
314	C₄⋊C₄	—	236	1	no	—
315	C₄⋊C₄	—	236	2	no	—
316	C₄⋊C₄	—	236	2	no	—
317	C₄⋊C₄	—	236	1	no	—
318	C₄⋊C₄	—	236	1	no	—
319	C₂²⋊C₄	—	236	1	no	—
320	C₂²⋊C₄	—	236	2	no	—
321	C₂²⋊C₄	—	236	1	no	—
322	C₂²⋊C₄	—	236	1	no	—
323	C₄²	—	236	1	no	—
324	C₄²	—	236	1	no	—
325	C₄²	—	236	1	no	—
326	C₄²	—	236	2	no	—
327	C₄²	D₄₇₂	1	1	yes	characteristic
328	C₄²	D₄₇₂	1	1	yes	characteristic
329	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
330	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
331	C₁₆	—	2	2	no	—
332	C₁₆	—	2	2	no	—
333	C₂³	—	236	1	no	—
334	C₂³	C₂×D₄₇₂	1	1	yes	characteristic
335	Q₈	—	118	2	no	—
336	Q₈	—	472	2	no	—
337	Q₈	—	472	2	no	—
338	Q₈	—	236	1	no	—
339	D₄	—	118	2	no	—
340	D₄	—	236	4	no	—
341	D₄	—	236	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	—	472	2	no	—
345	C₂×C₄	—	472	2	no	—
346	C₂×C₄	—	236	1	no	—
347	C₂×C₄	—	236	1	no	—
348	C₂×C₄	—	236	2	no	—
349	C₂×C₄	—	236	2	no	—
350	C₂×C₄	—	236	1	no	—
351	C₂×C₄	—	236	4	no	—
352	C₂×C₄	#14273707	1	2	yes	—
353	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
355	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
356	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
357	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
358	C₂×C₄	#14273760	1	2	yes	—
359	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	#14273707	1	2	yes	—
363	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
364	C₈	C₂×#11745687	1	1	yes	characteristic
365	C₂²	—	2	1	no	—
366	C₂²	—	2	1	no	—
367	C₂²	—	472	1	no	—
368	C₂²	—	472	1	no	—
369	C₂²	—	236	2	no	—
370	C₂²	#16837683	1	2	yes	—
371	C₂²	C₂²×D₄₇₂	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	472	1	no	—
376	C₄	—	472	2	no	—
377	C₄	—	472	1	no	—
378	C₄	—	472	2	no	—
379	C₄	—	236	2	no	—
380	C₄	#16837683	1	2	yes	—
381	C₄	#16837834	1	2	yes	—
382	C₄	C₂×#14273707	1	1	yes	characteristic
383	C₄	C₂×#14273760	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	472	2	no	—
386	C₂	C₂×#16837683	1	1	yes	characteristic
387	C₂	C₂×#16837834	1	1	yes	characteristic
388	C₂	#21437232	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄₇₂
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×C₄⋊C₁₆	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×#14273749	C₂²
hypercenter	C₂×C₄⋊C₁₆	D₅₉
hypocenter	C₅₉	#33626
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄⋊C₁₆

Complements

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

N	H ≅ G/N	type
1. #21434610	C₂	semidirect
3. #21426261	C₂	semidirect
4. #21426254	C₂	semidirect
5. #21426249	C₂	semidirect
6. #21426244	C₂	semidirect
7. #21426239	C₂	semidirect
8. #21426234	C₂	semidirect
9. #21425880	C₂	semidirect
11. C₂×C₅₉×C₄⋊C₁₆	C₂	semidirect
45. #14273749	Q₈	semidirect
199. C₂×C₄⋊C₁₆	D₅₉	semidirect
237. C₅₉	#33626	semidirect

Maximal subgroups

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

subgroup	index
#21434610	2
C₄×#14273749	2
#21426261	2
#21426254	2
#21426249	2
#21426244	2
#21426239	2
#21426234	2
#21425880	2
#21425804	2
C₂×C₅₉×C₄⋊C₁₆	2
#33626	59