Subgroups — #28743329

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	C₄×#14273749	C₂	1	1	yes	maximal, characteristic
2	#21426257	C₂	1	2	yes	maximal, complemented
3	#21426250	C₂	1	2	yes	maximal, complemented
4	#21426245	C₂	1	1	yes	maximal, characteristic, complemented
5	#21426240	C₂	1	1	yes	maximal, characteristic, complemented
6	#21426235	C₂	1	1	yes	maximal, characteristic, complemented
7	#21426230	C₂	1	1	yes	maximal, characteristic, complemented
8	#21425878	C₂	1	2	yes	maximal, complemented
9	C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
10	#21425801	C₂	1	2	yes	maximal
11	C₅₉×#3160	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#14273749	C₂²	1	1	yes	characteristic
13	C₄×#11745685	—	2	1	no	—
14	C₂²×C₅₉⋊C₁₆	C₂²	1	2	yes	—
15	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
16	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
17	C₄×#11745660	—	2	1	no	—
18	C₄×#11745660	—	2	1	no	—
19	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
20	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
21	#16837440	C₂²	1	1	yes	characteristic
22	#16837399	—	2	1	no	—
23	#16837229	C₂²	1	2	yes	—
24	#16837229	C₂²	1	2	yes	—
25	#16837227	C₂²	1	2	yes	—
26	#16837227	C₂²	1	2	yes	—
27	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
28	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	1	yes	characteristic
29	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
30	#16837205	C₂²	1	2	yes	—
31	#16837205	C₂²	1	2	yes	—
32	#16837202	C₂²	1	2	yes	—
33	#16837202	C₂²	1	2	yes	—
34	C₅₉×#533	—	2	1	no	—
35	C₅₉×#500	C₂²	1	1	yes	characteristic
36	C₅₉×#500	C₂²	1	1	yes	characteristic
37	C₅₉×#499	—	2	1	no	—
38	C₅₉×#499	C₂²	1	2	yes	—
39	C₅₉×#498	—	2	1	no	—
40	C₅₉×#427	—	2	1	no	—
41	C₅₉×#427	C₂²	1	2	yes	—
42	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
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	C₄×Dic₁₁₈	—	2	1	no	—
50	C₄×Dic₁₁₈	—	2	1	no	—
51	C₄×D₂₃₆	—	2	1	no	—
52	C₄×D₂₃₆	—	2	1	no	—
53	C₄×C₂²⋊D₅₉	—	4	1	no	—
54	C₄×C₂²⋊D₅₉	—	4	1	no	—
55	#14273686	—	2	1	no	—
56	#14273684	—	4	1	no	—
57	C₄²×D₅₉	—	4	1	no	—
58	C₂×C₅₉⋊C₁₆	—	2	4	no	—
59	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
60	C₂×C₅₉⋊C₁₆	C₂³	1	2	yes	—
61	#14273661	C₂³	1	1	yes	characteristic
62	#14273661	C₂³	1	1	yes	characteristic
63	#14273659	—	2	1	no	—
64	#14273652	—	2	1	no	—
65	#14273652	—	2	1	no	—
66	#14273649	—	2	1	no	—
67	#14273649	—	2	1	no	—
68	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
69	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₄⋊Q₈×C₅₉	—	2	1	no	—
71	C₄⋊Q₈×C₅₉	—	2	1	no	—
72	#125×C₅₉	—	2	2	no	—
73	C₂²⋊Q₈×C₅₉	—	4	1	no	—
74	C₂²⋊Q₈×C₅₉	—	4	1	no	—
75	C₄×Q₈×C₅₉	—	4	1	no	—
76	C₄×Q₈×C₅₉	—	4	1	no	—
77	#117×C₅₉	—	2	2	no	—
78	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
79	#107×C₅₉	—	2	1	no	—
80	#107×C₅₉	—	2	1	no	—
81	#107×C₅₉	C₂³	1	2	yes	—
82	#107×C₅₉	C₂³	1	2	yes	—
83	#106×C₅₉	—	2	2	no	—
84	#106×C₅₉	—	2	2	no	—
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	C₂×Q₈×C₅₉	—	4	1	no	—
90	C₂×Q₈×C₅₉	—	4	1	no	—
91	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₅₉	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₂²⋊D₅₉	—	4	1	no	—
95	#11745685	—	2	4	no	—
96	#11745685	—	4	2	no	—
97	C₂×D₂₃₆	—	2	1	no	—
98	C₂×C₄×D₅₉	—	4	1	no	—
99	C₂×C₄×D₅₉	—	4	1	no	—
100	C₂×C₄×D₅₉	—	4	1	no	—
101	C₂×Dic₁₁₈	—	2	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	C₄⋊C₄×C₅₉	—	2	2	no	—
110	C₄⋊C₄×C₅₉	—	2	2	no	—
111	C₄⋊C₄×C₅₉	—	4	2	no	—
112	C₄⋊C₄×C₅₉	—	4	2	no	—
113	C₄⋊C₄×C₅₉	—	4	1	no	—
114	C₄⋊C₄×C₅₉	—	4	1	no	—
115	C₂²⋊C₄×C₅₉	—	4	2	no	—
116	C₄²×C₅₉	—	4	2	no	—
117	C₄²×C₅₉	D₈	1	1	yes	characteristic
118	C₄²×C₅₉	D₈	1	1	yes	characteristic
119	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂²⋊Dic₅₉	—	4	1	no	—
122	(C₂×D₅₉)⋊C₄	—	4	1	no	—
123	(C₂×D₅₉)⋊C₄	—	4	1	no	—
124	C₄⋊Dic₅₉	—	2	1	no	—
125	C₄⋊Dic₅₉	—	2	1	no	—
126	Dic₅₉⋊C₄	—	4	1	no	—
127	Dic₅₉⋊C₄	—	4	1	no	—
128	C₄×Dic₅₉	—	4	1	no	—
129	C₄×Dic₅₉	—	4	1	no	—
130	C₄×Dic₅₉	—	4	1	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	C₅₉⋊C₁₆	—	2	2	no	—
137	C₅₉⋊C₁₆	—	2	2	no	—
138	C₂³×C₅₉	C₂×D₈	1	1	yes	characteristic
139	C₂²×D₅₉	—	4	1	no	—
140	Q₈×C₅₉	—	8	2	no	—
141	Q₈×C₅₉	—	8	2	no	—
142	C₂×C₄×C₅₉	—	2	2	no	—
143	C₂×C₄×C₅₉	—	4	1	no	—
144	C₂×C₄×C₅₉	—	4	2	no	—
145	C₂×C₄×C₅₉	—	4	2	no	—
146	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
147	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₅₉	#137	1	2	yes	—
153	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
154	C₂²⋊D₅₉	—	4	4	no	—
155	C₂×Dic₅₉	—	4	1	no	—
156	C₂×Dic₅₉	—	4	1	no	—
157	C₂×Dic₅₉	—	4	1	no	—
158	D₂₃₆	—	2	2	no	—
159	D₂₃₆	—	4	1	no	—
160	C₄×D₅₉	—	4	4	no	—
161	C₄×D₅₉	—	8	2	no	—
162	C₄×D₅₉	—	8	2	no	—
163	Dic₁₁₈	—	2	2	no	—
164	Dic₁₁₈	—	4	1	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	—	8	2	no	—
182	C₄×C₅₉	—	8	2	no	—
183	C₄×C₅₉	#474	1	2	yes	—
184	C₄×C₅₉	#508	1	2	yes	—
185	C₄×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
186	C₄×C₅₉	C₂×#137	1	1	yes	characteristic
187	Dic₅₉	—	4	2	no	—
188	Dic₅₉	—	8	1	no	—
189	Dic₅₉	—	8	1	no	—
190	#3160	D₅₉	1	1	yes	characteristic, complemented
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	—	59	1	no	—
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	—	2	1	no	—
208	C₄×#41	—	118	1	no	—
209	C₂²×C₁₆	—	59	2	no	—
210	#500	C₂×D₅₉	1	1	yes	characteristic
211	#500	C₂×D₅₉	1	1	yes	characteristic
212	#499	—	2	1	no	—
213	#499	C₂×D₅₉	1	2	yes	—
214	#498	—	2	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	—	2	1	no	—
222	#427	—	59	1	no	—
223	#427	C₂×D₅₉	1	2	yes	—
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	—	59	1	no	—
231	C₄⋊C₁₆	—	59	1	no	—
232	C₄⋊C₁₆	—	59	2	no	—
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	C₂²⋊D₅₉	1	1	yes	characteristic
245	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
246	C₄⋊Q₈	—	2	1	no	—
247	C₄⋊Q₈	—	2	1	no	—
248	#125	—	2	2	no	—
249	C₂²⋊Q₈	—	4	1	no	—
250	C₂²⋊Q₈	—	4	1	no	—
251	C₄×Q₈	—	4	1	no	—
252	C₄×Q₈	—	4	1	no	—
253	C₄×Q₈	—	118	1	no	—
254	C₄×Q₈	—	118	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	—	2	2	no	—
260	#117	—	118	1	no	—
261	#117	—	236	1	no	—
262	C₂×C₄²	—	236	1	no	—
263	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
264	C₂×C₁₆	—	59	2	no	—
265	C₂×C₁₆	—	59	2	no	—
266	C₂×C₁₆	—	118	4	no	—
267	#107	—	2	1	no	—
268	#107	—	2	1	no	—
269	#107	—	59	1	no	—
270	#107	—	59	1	no	—
271	#107	C₂²×D₅₉	1	2	yes	—
272	#107	C₂²×D₅₉	1	2	yes	—
273	#106	—	2	2	no	—
274	#106	—	2	2	no	—
275	#106	—	118	1	no	—
276	#103	—	118	1	no	—
277	#103	—	118	1	no	—
278	#102	—	118	1	no	—
279	#102	—	118	1	no	—
280	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
281	C₄×C₈	C₂²⋊D₅₉	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	—	4	1	no	—
287	C₂×Q₈	—	4	1	no	—
288	C₂×Q₈	—	118	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	#11745669	1	1	yes	characteristic
295	C₂²×C₄	#11745669	1	1	yes	characteristic
296	C₂²×C₄	C₂×C₂²⋊D₅₉	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	Q₈×D₅₉	1	2	yes	—
304	C₂×C₈	#11745689	1	2	yes	—
305	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
306	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
307	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
308	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
309	C₄⋊C₄	—	2	2	no	—
310	C₄⋊C₄	—	2	2	no	—
311	C₄⋊C₄	—	4	2	no	—
312	C₄⋊C₄	—	4	2	no	—
313	C₄⋊C₄	—	4	1	no	—
314	C₄⋊C₄	—	4	1	no	—
315	C₄⋊C₄	—	118	1	no	—
316	C₄⋊C₄	—	118	1	no	—
317	C₄⋊C₄	—	236	1	no	—
318	C₄⋊C₄	—	236	1	no	—
319	C₂²⋊C₄	—	4	2	no	—
320	C₂²⋊C₄	—	236	1	no	—
321	C₂²⋊C₄	—	236	1	no	—
322	C₂²⋊C₄	—	236	1	no	—
323	C₄²	—	4	2	no	—
324	C₄²	—	236	1	no	—
325	C₄²	—	236	1	no	—
326	C₄²	—	236	1	no	—
327	C₄²	#11745669	1	1	yes	characteristic
328	C₄²	#11745669	1	1	yes	characteristic
329	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
330	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
331	C₁₆	—	118	2	no	—
332	C₁₆	—	118	2	no	—
333	C₂³	—	236	1	no	—
334	C₂³	C₂×#11745669	1	1	yes	characteristic
335	Q₈	—	8	2	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	118	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	—	4	2	no	—
345	C₂×C₄	—	4	2	no	—
346	C₂×C₄	—	472	2	no	—
347	C₂×C₄	—	472	2	no	—
348	C₂×C₄	—	236	1	no	—
349	C₂×C₄	—	236	1	no	—
350	C₂×C₄	—	236	1	no	—
351	C₂×C₄	—	236	4	no	—
352	C₂×C₄	#14273703	1	2	yes	—
353	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
354	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
355	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
356	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
357	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
358	C₂×C₄	#14273758	1	2	yes	—
359	C₂×C₄	C₂²×C₂²⋊D₅₉	1	1	yes	characteristic
360	C₈	—	2	1	no	—
361	C₈	—	2	1	no	—
362	C₈	#14273703	1	2	yes	—
363	C₈	C₂×Q₈×D₅₉	1	1	yes	characteristic
364	C₈	C₂×#11745689	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	#16837679	1	2	yes	—
371	C₂²	C₂²×#11745669	1	1	yes	characteristic
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	2	1	no	—
375	C₄	—	8	2	no	—
376	C₄	—	8	2	no	—
377	C₄	—	472	1	no	—
378	C₄	—	472	1	no	—
379	C₄	—	236	2	no	—
380	C₄	#16837679	1	2	yes	—
381	C₄	#16837832	1	2	yes	—
382	C₄	C₂×#14273703	1	1	yes	characteristic
383	C₄	C₂×#14273758	1	1	yes	characteristic
384	C₂	—	2	2	no	—
385	C₂	—	472	2	no	—
386	C₂	C₂×#16837679	1	1	yes	characteristic
387	C₂	C₂×#16837832	1	1	yes	characteristic
388	C₂	#21437230	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11745669
commutator, cosocle	C₂×C₈×C₅₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₅₉×#3160	C₂
socle	C₂²×C₅₉	C₂²×D₈
prime radical	C₂×#14273749	C₂²
hypercenter	#3160	D₅₉
hypocenter	C₅₉	#33626
LCS	G ▷ C₂×C₈×C₅₉ ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #21426257	C₂	semidirect
3. #21426250	C₂	semidirect
4. #21426245	C₂	semidirect
5. #21426240	C₂	semidirect
6. #21426235	C₂	semidirect
7. #21426230	C₂	semidirect
8. #21425878	C₂	semidirect
9. C₂×C₄⋊(C₅₉⋊C₁₆)	C₂	semidirect
11. C₅₉×#3160	C₂	semidirect
45. #14273749	Q₈	semidirect
190. #3160	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
C₄×#14273749	2
#21426257	2
#21426250	2
#21426245	2
#21426240	2
#21426235	2
#21426230	2
#21425878	2
C₂×C₄⋊(C₅₉⋊C₁₆)	2
#21425801	2
C₅₉×#3160	2
#33626	59