Subgroups — #28481855

Order 15104.

454 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11745685	C₂	1	1	yes	maximal, characteristic
2	#21424113	C₂	1	2	yes	maximal
3	C₄×#14273661	C₂	1	1	yes	maximal, characteristic, complemented
4	#21421997	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1567	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11745685	C₂²	1	1	yes	characteristic
7	C₈×Dic₁₁₈	C₄	1	1	yes	characteristic, complemented
8	C₈×Dic₁₁₈	C₄	1	1	yes	characteristic, complemented
9	C₈×D₂₃₆	C₄	1	1	yes	characteristic, complemented
10	C₈×D₂₃₆	C₄	1	1	yes	characteristic, complemented
11	C₈×C₂²⋊D₅₉	—	2	1	no	—
12	C₈×C₂²⋊D₅₉	—	2	1	no	—
13	#16837455	C₂²	1	1	yes	characteristic
14	#16837453	—	2	1	no	—
15	C₂×#14273661	—	2	1	no	—
16	C₂×#14273653	—	2	2	no	—
17	C₄×C₈×D₅₉	—	2	1	no	—
18	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
19	#16837165	—	2	1	no	—
20	#16837163	—	2	2	no	—
21	C₅₉×#423	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#335	—	2	1	no	—
24	C₅₉×#335	—	2	1	no	—
25	C₂×#11745685	D₄	1	1	yes	characteristic
26	C₂×C₈×D₅₉	—	2	1	no	—
27	C₂×C₈×D₅₉	—	2	1	no	—
28	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
29	C₄×Dic₁₁₈	Q₈	1	1	yes	characteristic
30	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
31	C₄×D₂₃₆	Q₈	1	1	yes	characteristic
32	C₄×C₂²⋊D₅₉	—	2	1	no	—
33	C₄×C₂²⋊D₅₉	—	2	1	no	—
34	#14273686	D₄	1	1	yes	characteristic
35	#14273684	—	2	1	no	—
36	C₂×C₄⋊Dic₅₉	—	2	2	no	—
37	C₂×C₄×Dic₅₉	—	2	2	no	—
38	C₄²×D₅₉	—	2	1	no	—
39	#14273661	—	2	4	no	—
40	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
41	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
42	Dic₅₉⋊C₈	—	2	1	no	—
43	Dic₅₉⋊C₈	—	2	1	no	—
44	#14273653	—	4	4	no	—
45	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
46	(C₂×D₅₉)⋊C₈	—	2	1	no	—
47	(C₂×D₅₉)⋊C₈	—	2	1	no	—
48	C₄×C₅₉⋊C₈	—	2	1	no	—
49	C₈×Dic₅₉	—	2	1	no	—
50	C₈×Dic₅₉	—	2	1	no	—
51	C₅₉⋊#95	—	2	2	no	—
52	C₂×M₁₆×C₅₉	—	2	2	no	—
53	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	—	2	2	no	—
57	C₄⋊C₈×C₅₉	—	2	2	no	—
58	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
60	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
65	C₂×C₂²⋊D₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	4	2	no	—
67	#11745685	—	2	1	no	—
68	#11745685	—	2	1	no	—
69	#11745685	D₈	1	2	yes	—
70	#11745685	Q₁₆	1	2	yes	—
71	C₂×D₂₃₆	C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	—	2	1	no	—
74	C₂×C₄×D₅₉	—	2	1	no	—
75	C₂×Dic₁₁₈	C₄⋊C₄	1	1	yes	characteristic
76	M₁₆×C₅₉	—	4	2	no	—
77	M₁₆×C₅₉	—	4	2	no	—
78	C₂×C₈×C₅₉	—	2	2	no	—
79	C₂×C₈×C₅₉	—	4	1	no	—
80	C₂×C₈×C₅₉	—	4	2	no	—
81	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
85	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
86	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
88	C₄²×C₅₉	#41	1	1	yes	characteristic
89	C₂²⋊Dic₅₉	—	2	1	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	C₄⋊Dic₅₉	—	2	4	no	—
93	C₄⋊Dic₅₉	—	4	2	no	—
94	C₄⋊Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
95	C₄⋊Dic₅₉	C₄⋊C₄	1	1	yes	characteristic
96	Dic₅₉⋊C₄	—	2	1	no	—
97	Dic₅₉⋊C₄	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	4	2	no	—
102	C₄×Dic₅₉	—	4	2	no	—
103	C₂×C₅₉⋊C₈	—	2	1	no	—
104	C₂×C₅₉⋊C₈	—	2	1	no	—
105	C₈×D₅₉	—	4	2	no	—
106	C₈×D₅₉	—	4	2	no	—
107	C₂³×C₅₉	#121	1	1	yes	characteristic
108	C₂²×D₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	4	1	no	—
112	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
122	C₂²⋊D₅₉	—	2	2	no	—
123	C₂²⋊D₅₉	—	2	2	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	4	4	no	—
128	C₂×Dic₅₉	—	8	2	no	—
129	C₂×Dic₅₉	—	8	2	no	—
130	D₂₃₆	—	2	1	no	—
131	D₂₃₆	#107	1	2	yes	complemented
132	C₄×D₅₉	—	2	2	no	—
133	C₄×D₅₉	—	2	2	no	—
134	C₄×D₅₉	—	4	2	no	—
135	C₄×D₅₉	—	4	1	no	—
136	C₄×D₅₉	—	4	1	no	—
137	Dic₁₁₈	—	2	1	no	—
138	Dic₁₁₈	#107	1	2	yes	complemented
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	—	8	2	no	—
143	C₈×C₅₉	#104	1	2	yes	complemented
144	C₅₉⋊C₈	—	4	1	no	—
145	C₅₉⋊C₈	—	4	1	no	—
146	#8667	—	59	1	no	maximal, Sylow
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	#380	1	1	yes	characteristic
150	C₂²×C₅₉	#466	1	1	yes	characteristic
151	C₂²×C₅₉	#467	1	1	yes	characteristic
152	C₂×D₅₉	—	2	2	no	—
153	C₂×D₅₉	—	4	1	no	—
154	C₂×D₅₉	—	4	1	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	C₂×#104	1	1	yes	characteristic
159	C₄×C₅₉	#421	1	1	yes	characteristic
160	C₄×C₅₉	C₂×#107	1	1	yes	characteristic
161	C₄×C₅₉	#427	1	1	yes	characteristic
162	C₄×C₅₉	#466	1	1	yes	characteristic
163	C₄×C₅₉	#467	1	1	yes	characteristic
164	Dic₅₉	—	2	2	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	—	8	4	no	—
168	C₈×#41	—	59	1	no	—
169	#1806	—	59	2	no	—
170	C₄×#107	—	59	1	no	—
171	#1567	D₅₉	1	1	yes	characteristic, complemented
172	#1303	—	59	2	no	—
173	C₂×C₅₉	—	2	1	no	—
174	C₂×C₅₉	—	2	1	no	—
175	C₂×C₅₉	#1806	1	1	yes	characteristic
176	C₂×C₅₉	#1808	1	1	yes	characteristic
177	C₂×C₅₉	#1814	1	1	yes	characteristic
178	D₅₉	—	4	2	no	—
179	C₄×#41	—	59	1	no	—
180	C₈×Q₈	—	59	1	no	—
181	C₈×Q₈	—	59	1	no	—
182	C₈×D₄	—	59	1	no	—
183	C₈×D₄	—	59	1	no	—
184	C₈×D₄	—	118	1	no	—
185	C₈×D₄	—	118	1	no	—
186	#431	—	59	1	no	—
187	#431	—	118	1	no	—
188	C₂×#107	—	118	1	no	—
189	#423	C₂×D₅₉	1	2	yes	—
190	C₂×#104	—	118	2	no	—
191	C₂×C₄×C₈	—	118	1	no	—
192	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
193	C₄×C₄⋊C₄	—	59	2	no	—
194	#340	—	118	1	no	—
195	#339	—	118	2	no	—
196	#335	—	2	1	no	—
197	#335	—	2	1	no	—
198	C₅₉	#8667	1	1	yes	characteristic, complemented, Sylow
199	C₂×#41	—	59	1	no	—
200	C₂×M₁₆	—	2	2	no	—
201	C₂²×C₈	—	118	1	no	—
202	C₂²×C₈	—	118	1	no	—
203	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
204	C₂²×C₈	D₂₃₆	1	1	yes	characteristic
205	C₄×Q₈	—	59	1	no	—
206	C₄×Q₈	—	59	1	no	—
207	C₄×D₄	—	59	1	no	—
208	C₄×D₄	—	59	1	no	—
209	C₄×D₄	—	118	1	no	—
210	C₄×D₄	—	118	1	no	—
211	#117	—	59	1	no	—
212	#117	—	118	1	no	—
213	C₂×C₄⋊C₄	—	118	2	no	—
214	C₂×C₄²	—	118	1	no	—
215	C₂×C₄²	—	118	2	no	—
216	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
217	#107	—	118	4	no	—
218	C₄⋊C₈	—	2	2	no	—
219	C₄⋊C₈	—	2	2	no	—
220	C₄⋊C₈	—	59	1	no	—
221	C₄⋊C₈	—	59	1	no	—
222	C₄⋊C₈	—	118	1	no	—
223	C₄⋊C₈	—	118	1	no	—
224	#104	—	236	4	no	—
225	C₂²⋊C₈	—	118	1	no	—
226	C₂²⋊C₈	—	118	1	no	—
227	C₂²⋊C₈	—	118	1	no	—
228	C₄×C₈	—	118	1	no	—
229	C₄×C₈	—	118	1	no	—
230	C₄×C₈	—	118	1	no	—
231	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
232	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
233	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
234	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
235	#95	—	118	2	no	—
236	#41	—	59	2	no	—
237	#41	—	59	2	no	—
238	#41	—	118	1	no	—
239	#41	—	118	1	no	—
240	C₂×Q₈	—	59	1	no	—
241	C₂×D₄	—	59	1	no	—
242	C₂×D₄	—	118	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	C₂²×C₄	—	118	1	no	—
245	C₂²×C₄	—	118	1	no	—
246	C₂²×C₄	—	236	2	no	—
247	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
248	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
249	C₂²×C₄	#11745689	1	1	yes	characteristic
250	M₁₆	—	4	2	no	—
251	M₁₆	—	4	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	2	no	—
255	C₂×C₈	—	118	1	no	—
256	C₂×C₈	—	118	1	no	—
257	C₂×C₈	—	236	2	no	—
258	C₂×C₈	—	236	2	no	—
259	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
260	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
261	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
262	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
263	C₄⋊C₄	—	59	1	no	—
264	C₄⋊C₄	—	59	1	no	—
265	C₄⋊C₄	—	118	4	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	—	118	1	no	—
268	C₄⋊C₄	—	236	2	no	—
269	C₂²⋊C₄	—	118	1	no	—
270	C₂²⋊C₄	—	118	1	no	—
271	C₂²⋊C₄	—	118	1	no	—
272	C₄²	—	118	1	no	—
273	C₄²	—	118	1	no	—
274	C₄²	—	118	1	no	—
275	C₄²	—	236	2	no	—
276	C₄²	—	236	2	no	—
277	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
278	C₄²	#11745687	1	1	yes	characteristic
279	C₄²	Q₈×D₅₉	1	1	yes	characteristic
280	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
281	C₂³	—	118	1	no	—
282	C₂³	#14273702	1	1	yes	characteristic
283	Q₈	—	59	2	no	—
284	Q₈	—	118	1	no	—
285	D₄	—	59	2	no	—
286	D₄	—	118	2	no	—
287	D₄	—	118	2	no	—
288	D₄	—	118	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	118	1	no	—
293	C₂×C₄	—	118	2	no	—
294	C₂×C₄	—	118	2	no	—
295	C₂×C₄	—	118	1	no	—
296	C₂×C₄	—	118	1	no	—
297	C₂×C₄	—	472	2	no	—
298	C₂×C₄	—	472	2	no	—
299	C₂×C₄	—	236	2	no	—
300	C₂×C₄	—	236	4	no	—
301	C₂×C₄	—	236	1	no	—
302	C₂×C₄	—	236	1	no	—
303	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
304	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
305	C₂×C₄	#14273685	1	1	yes	characteristic
306	C₂×C₄	#14273698	1	1	yes	characteristic
307	C₂×C₄	#14273703	1	1	yes	characteristic
308	C₂×C₄	#14273707	1	1	yes	characteristic
309	C₂×C₄	D₈×D₅₉	1	1	yes	characteristic
310	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
311	C₂×C₄	#14273746	1	1	yes	characteristic
312	C₂×C₄	#14273750	1	1	yes	characteristic
313	C₈	—	2	1	no	—
314	C₈	—	2	1	no	—
315	C₈	—	2	1	no	—
316	C₈	—	8	2	no	—
317	C₈	—	236	1	no	—
318	C₈	—	236	1	no	—
319	C₈	#14273653	1	2	yes	complemented
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	118	2	no	—
323	C₂²	—	236	1	no	—
324	C₂²	—	236	1	no	—
325	C₂²	#16837259	1	1	yes	characteristic
326	C₂²	#16837639	1	1	yes	characteristic
327	C₂²	#16837644	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	118	2	no	—
332	C₄	—	472	4	no	—
333	C₄	—	236	1	no	—
334	C₄	—	236	1	no	—
335	C₄	C₂×#14273653	1	1	yes	characteristic
336	C₄	#16837418	1	1	yes	characteristic
337	C₄	#107×D₅₉	1	1	yes	characteristic
338	C₄	#16837439	1	1	yes	characteristic
339	C₄	#16837635	1	1	yes	characteristic
340	C₄	#16837640	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	236	2	no	—
344	C₂	#21424113	1	1	yes	characteristic
345	C₂	#21424122	1	1	yes	characteristic
346	C₂	#21424151	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14273698
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1567	C₂
socle	C₂²×C₅₉	#380
prime radical	C₂×#11745685	D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1567	D₅₉
hypocenter	C₅₉	#8667
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1567

Complements

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

N	H ≅ G/N	type
3. C₄×#14273661	C₂	semidirect
4. #21421997	C₂	semidirect
5. C₅₉×#1567	C₂	semidirect
7. C₈×Dic₁₁₈	C₄	semidirect
8. C₈×Dic₁₁₈	C₄	semidirect
9. C₈×D₂₃₆	C₄	semidirect
10. C₈×D₂₃₆	C₄	semidirect
131. D₂₃₆	#107	semidirect
138. Dic₁₁₈	#107	semidirect
143. C₈×C₅₉	#104	semidirect
171. #1567	D₅₉	semidirect
198. C₅₉	#8667	semidirect
319. C₈	#14273653	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11745685	2
#21424113	2
C₄×#14273661	2
#21421997	2
C₅₉×#1567	2
#8667	59