Subgroups — #28560118

Order 15104.

656 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433108	C₂	1	2	yes	maximal, complemented
2	C₄×#14273749	C₂	1	1	yes	maximal, characteristic
3	#21422992	C₂	1	2	yes	maximal, complemented
4	#21422931	C₂	1	4	yes	maximal, complemented
5	#21422890	C₂	1	2	yes	maximal, complemented
6	#21422528	C₂	1	2	yes	maximal, complemented
7	C₈×#11745664	C₂	1	1	yes	maximal, characteristic, complemented
8	C₅₉×#1568	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#14273749	C₄	1	1	yes	characteristic
10	C₄×#11745685	C₂²	1	2	yes	—
11	#16837529	C₂²	1	4	yes	—
12	C₄×Dic₂₃₆	C₄	1	1	yes	characteristic
13	C₄×Dic₂₃₆	C₂²	1	1	yes	characteristic
14	C₄×#11745660	C₄	1	2	yes	—
15	C₄×#11745660	C₂²	1	2	yes	—
16	C₄×D₄₇₂	C₄	1	1	yes	characteristic
17	C₄×D₄₇₂	C₂²	1	1	yes	characteristic
18	#16837476	—	2	4	no	—
19	#16837467	C₂²	1	4	yes	—
20	#16837457	—	2	2	no	—
21	#16837455	C₂²	1	1	yes	characteristic
22	#16837440	C₄	1	1	yes	characteristic
23	#16837399	C₄	1	2	yes	—
24	C₄×#11745664	C₂²	1	1	yes	characteristic
25	#97×D₅₉	—	2	2	no	—
26	#16837132	C₂²	1	4	yes	—
27	#16837129	C₂²	1	4	yes	—
28	#16837124	C₂²	1	2	yes	—
29	C₈×C₅₉⋊C₈	C₂²	1	2	yes	—
30	C₅₉×#432	C₂²	1	2	yes	—
31	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
32	C₅₉×#335	C₂²	1	2	yes	—
33	C₅₉×#334	C₂²	1	2	yes	—
34	C₂×#11745685	C₂×C₄	1	2	yes	—
35	#14273749	—	2	4	no	—
36	C₂×Dic₂₃₆	C₂×C₄	1	1	yes	characteristic
37	C₂×#11745660	C₂×C₄	1	2	yes	—
38	C₂×D₄₇₂	C₂×C₄	1	1	yes	characteristic
39	C₂×#11745664	D₄	1	2	yes	—
40	C₂×C₈×D₅₉	—	2	4	no	—
41	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
42	C₄×Dic₁₁₈	C₂³	1	2	yes	—
43	C₄×D₂₃₆	C₂×C₄	1	2	yes	—
44	C₄×D₂₃₆	C₂³	1	2	yes	—
45	C₄×C₂²⋊D₅₉	—	2	2	no	—
46	C₄×C₂²⋊D₅₉	—	2	2	no	—
47	#14273686	C₂×C₄	1	2	yes	—
48	#14273684	—	2	2	no	—
49	C₄²×D₅₉	—	2	2	no	—
50	#14273661	C₂×C₄	1	1	yes	characteristic
51	#14273661	C₂×C₄	1	1	yes	characteristic
52	#14273659	C₂×C₄	1	2	yes	—
53	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
54	Dic₅₉⋊C₈	—	2	4	no	—
55	#14273652	C₂×C₄	1	2	yes	—
56	#14273652	C₂×C₄	1	2	yes	—
57	#14273649	C₂×C₄	1	2	yes	—
58	#14273649	C₂×C₄	1	2	yes	—
59	C₂²⋊(C₅₉⋊C₈)	—	2	1	no	—
60	(C₂×D₅₉)⋊C₈	—	2	4	no	—
61	#14273638	D₄	1	1	yes	characteristic
62	#14273638	D₄	1	1	yes	characteristic
63	#14273637	—	2	4	no	—
64	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
65	C₄×C₅₉⋊C₈	C₂³	1	2	yes	—
66	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₅₉	C₂³	1	4	yes	—
70	C₂²⋊C₈×C₅₉	—	2	2	no	—
71	#97×C₅₉	—	2	2	no	—
72	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
79	C₂×C₂²⋊D₅₉	—	2	2	no	—
80	#11745685	—	2	4	no	—
81	#11745685	—	4	2	no	—
82	C₂×D₂₃₆	C₂²×C₄	1	2	yes	—
83	C₂×C₄×D₅₉	—	2	2	no	—
84	C₂×C₄×D₅₉	—	2	2	no	—
85	C₂×C₄×D₅₉	—	2	2	no	—
86	C₂×Dic₁₁₈	C₂²×C₄	1	2	yes	—
87	C₂×C₈×C₅₉	—	2	2	no	—
88	C₂×C₈×C₅₉	—	2	4	no	—
89	C₂×C₈×C₅₉	—	4	1	no	—
90	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₅₉	#41	1	1	yes	characteristic
97	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
98	C₂²⋊Dic₅₉	—	2	2	no	—
99	(C₂×D₅₉)⋊C₄	—	2	2	no	—
100	(C₂×D₅₉)⋊C₄	—	2	2	no	—
101	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
102	C₄⋊Dic₅₉	C₂²×C₄	1	2	yes	—
103	Dic₅₉⋊C₄	—	2	2	no	—
104	Dic₅₉⋊C₄	—	2	2	no	—
105	C₄×Dic₅₉	—	2	2	no	—
106	C₄×Dic₅₉	—	2	2	no	—
107	C₄×Dic₅₉	—	2	2	no	—
108	#11745664	—	2	4	no	—
109	C₂×C₅₉⋊C₈	—	2	2	no	—
110	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
111	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
112	Dic₂₃₆	—	2	2	no	—
113	D₄₇₂	M₁₆	1	4	yes	complemented
114	#11745660	—	2	4	no	—
115	C₈×D₅₉	—	4	8	no	—
116	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₅₉	—	2	2	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	2	no	—
121	C₂×C₄×C₅₉	—	4	1	no	—
122	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
128	C₂²⋊D₅₉	—	4	4	no	—
129	C₂×Dic₅₉	—	2	2	no	—
130	C₂×Dic₅₉	—	2	2	no	—
131	C₂×Dic₅₉	—	2	2	no	—
132	D₂₃₆	—	2	2	no	—
133	D₂₃₆	C₂×M₁₆	1	4	yes	—
134	C₄×D₅₉	—	4	4	no	—
135	C₄×D₅₉	—	4	4	no	—
136	C₄×D₅₉	—	8	2	no	—
137	Dic₁₁₈	—	2	2	no	—
138	Dic₁₁₈	—	2	2	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₈×C₅₉	—	4	4	no	—
143	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
144	C₅₉⋊C₈	—	2	2	no	—
145	C₅₉⋊C₈	—	4	2	no	—
146	C₅₉⋊C₈	#135	1	4	yes	complemented
147	#18035	—	59	1	no	maximal, Sylow
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂×D₅₉	—	2	4	no	—
153	C₂×D₅₉	—	4	2	no	—
154	C₂×D₅₉	—	4	2	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	2	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
160	C₄×C₅₉	#568	1	1	yes	characteristic
161	C₄×C₅₉	C₂×#135	1	2	yes	—
162	Dic₅₉	—	4	2	no	—
163	Dic₅₉	—	4	2	no	—
164	Dic₅₉	—	4	2	no	—
165	#2970	—	59	2	no	—
166	C₄×#135	—	59	1	no	—
167	#1591	—	59	2	no	—
168	#1582	—	59	4	no	—
169	C₈×D₈	—	59	2	no	—
170	#1568	D₅₉	1	1	yes	characteristic, complemented
171	#1488	—	59	2	no	—
172	C₈×M₁₆	—	59	1	no	—
173	C₂×C₅₉	—	4	1	no	—
174	C₂×C₅₉	D₄×M₁₆	1	1	yes	characteristic
175	C₂×C₅₉	C₄×#135	1	1	yes	characteristic
176	C₂×C₅₉	#2957	1	1	yes	characteristic
177	D₅₉	—	4	4	no	—
178	C₂×#135	—	59	1	no	—
179	C₄×#41	—	59	2	no	—
180	#446	—	59	4	no	—
181	C₄×Q₁₆	—	59	1	no	—
182	C₄×Q₁₆	—	59	1	no	—
183	C₄×SD₁₆	—	59	2	no	—
184	C₄×SD₁₆	—	59	2	no	—
185	C₄×D₈	—	59	1	no	—
186	C₄×D₈	—	59	1	no	—
187	#435	—	118	4	no	—
188	C₈×D₄	—	59	4	no	—
189	#432	—	118	2	no	—
190	#432	C₂×D₅₉	1	2	yes	—
191	#431	—	59	1	no	—
192	#427	—	59	1	no	—
193	#416	—	59	2	no	—
194	C₄×M₁₆	—	59	1	no	—
195	C₂×#97	—	118	2	no	—
196	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
197	#335	C₂×D₅₉	1	2	yes	—
198	#334	C₂×D₅₉	1	2	yes	—
199	Q₈⋊C₈	—	59	4	no	—
200	D₄⋊C₈	—	59	4	no	—
201	#322	—	59	2	no	—
202	C₈²	—	59	2	no	—
203	C₅₉	#18035	1	1	yes	characteristic, complemented, Sylow
204	C₂×#41	—	59	2	no	—
205	#135	—	118	4	no	—
206	C₂×Q₁₆	—	59	1	no	—
207	C₂×SD₁₆	—	59	2	no	—
208	C₂×D₈	—	59	1	no	—
209	C₂×M₁₆	—	59	2	no	—
210	C₂²×C₈	—	118	4	no	—
211	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
212	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
213	C₄×Q₈	—	59	2	no	—
214	C₄×Q₈	—	59	2	no	—
215	C₄×D₄	—	59	2	no	—
216	C₄×D₄	—	59	2	no	—
217	C₄×D₄	—	118	2	no	—
218	C₄×D₄	—	118	2	no	—
219	#117	—	59	2	no	—
220	#117	—	118	2	no	—
221	C₂×C₄²	—	118	2	no	—
222	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
223	#107	—	59	1	no	—
224	#107	—	59	1	no	—
225	#106	—	59	2	no	—
226	C₄⋊C₈	—	118	4	no	—
227	C₄⋊C₈	—	118	1	no	—
228	C₄⋊C₈	C₂²×D₅₉	1	4	yes	—
229	#103	—	59	2	no	—
230	#103	—	59	2	no	—
231	#102	—	59	2	no	—
232	#102	—	59	2	no	—
233	C₂²⋊C₈	—	2	2	no	—
234	C₂²⋊C₈	—	118	4	no	—
235	C₂²⋊C₈	—	118	1	no	—
236	#97	—	2	2	no	—
237	#97	—	59	1	no	—
238	#97	—	59	1	no	—
239	#97	—	118	4	no	—
240	C₄×C₈	—	59	2	no	—
241	C₄×C₈	—	59	2	no	—
242	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
243	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
244	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
246	#41	—	118	4	no	—
247	#41	—	236	2	no	—
248	C₂×Q₈	—	59	2	no	—
249	C₂×D₄	—	59	2	no	—
250	C₂×D₄	—	118	2	no	—
251	C₂²×C₄	—	118	2	no	—
252	C₂²×C₄	—	118	2	no	—
253	C₂²×C₄	—	118	2	no	—
254	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
255	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
256	C₂²×C₄	#11745689	1	1	yes	characteristic
257	Q₁₆	—	118	2	no	—
258	SD₁₆	—	118	4	no	—
259	D₈	—	59	4	no	—
260	M₁₆	—	118	4	no	—
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	59	2	no	—
265	C₂×C₈	—	59	2	no	—
266	C₂×C₈	—	118	2	no	—
267	C₂×C₈	—	236	8	no	—
268	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
269	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
271	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
272	C₄⋊C₄	—	59	2	no	—
273	C₄⋊C₄	—	59	2	no	—
274	C₄⋊C₄	—	118	2	no	—
275	C₄⋊C₄	—	118	2	no	—
276	C₂²⋊C₄	—	118	2	no	—
277	C₂²⋊C₄	—	118	2	no	—
278	C₂²⋊C₄	—	118	2	no	—
279	C₄²	—	118	2	no	—
280	C₄²	—	118	2	no	—
281	C₄²	—	118	2	no	—
282	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
283	C₄²	D₄×D₅₉	1	1	yes	characteristic
284	C₄²	#11745689	1	1	yes	characteristic
285	C₄²	C₂³×D₅₉	1	1	yes	characteristic
286	C₂³	—	118	2	no	—
287	C₂³	#14273689	1	1	yes	characteristic
288	Q₈	—	118	2	no	—
289	Q₈	—	118	2	no	—
290	D₄	—	59	4	no	—
291	D₄	—	118	2	no	—
292	D₄	—	236	4	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	118	2	no	—
298	C₂×C₄	—	118	2	no	—
299	C₂×C₄	—	118	2	no	—
300	C₂×C₄	—	472	2	no	—
301	C₂×C₄	—	236	4	no	—
302	C₂×C₄	—	236	4	no	—
303	C₂×C₄	#14273689	1	1	yes	characteristic
304	C₂×C₄	#14273689	1	1	yes	characteristic
305	C₂×C₄	#14273689	1	1	yes	characteristic
306	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
307	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
308	C₂×C₄	C₂×#11745689	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	4	4	no	—
313	C₈	—	59	4	no	—
314	C₈	—	118	2	no	—
315	C₈	—	236	2	no	—
316	C₈	M₁₆×D₅₉	1	2	yes	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	118	4	no	—
321	C₂²	—	236	2	no	—
322	C₂²	—	236	2	no	—
323	C₂²	C₂×#14273689	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	1	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	236	2	no	—
329	C₄	—	236	2	no	—
330	C₄	—	236	2	no	—
331	C₄	C₂×M₁₆×D₅₉	1	1	yes	characteristic
332	C₄	#16838168	1	1	yes	characteristic
333	C₄	#135×D₅₉	1	2	yes	—
334	C₂	—	4	1	no	—
335	C₂	—	236	4	no	—
336	C₂	#21432831	1	1	yes	characteristic
337	C₂	#21432867	1	1	yes	characteristic
338	C₂	#21433012	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#1568	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂×#14273749	C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1568	D₅₉
hypocenter	C₅₉	#18035
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1568

Complements

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

N	H ≅ G/N	type
1. #21433108	C₂	semidirect
3. #21422992	C₂	semidirect
4. #21422931	C₂	semidirect
5. #21422890	C₂	semidirect
6. #21422528	C₂	semidirect
7. C₈×#11745664	C₂	semidirect
8. C₅₉×#1568	C₂	semidirect
113. D₄₇₂	M₁₆	semidirect
146. C₅₉⋊C₈	#135	semidirect
170. #1568	D₅₉	semidirect
203. C₅₉	#18035	semidirect

Maximal subgroups

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

subgroup	index
#21433108	2
C₄×#14273749	2
#21422992	2
#21422931	2
#21422890	2
#21422528	2
C₈×#11745664	2
C₅₉×#1568	2
#18035	59