Subgroups — #28741833

Order 15104.

528 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14273733	C₂	1	1	yes	maximal, characteristic
2	#21425911	C₂	1	4	yes	maximal, complemented
3	#21425907	C₂	1	4	yes	maximal, complemented
4	C₂×#16837223	C₂	1	1	yes	maximal, characteristic, complemented
5	#21425716	C₂	1	2	yes	maximal
6	#21425700	C₂	1	2	yes	maximal, complemented
7	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₄	1	1	yes	characteristic
9	C₂×#14273733	C₂²	1	1	yes	characteristic
10	C₄×#11745685	C₄	1	1	yes	characteristic
11	C₂×#14273667	C₂²	1	2	yes	—
12	#16837357	C₄	1	1	yes	characteristic
13	#16837357	C₂²	1	1	yes	characteristic
14	#16837356	C₄	1	2	yes	—
15	#16837356	C₂²	1	2	yes	—
16	C₄×#11745664	C₄	1	1	yes	characteristic
17	C₄×#11745659	C₂²	1	2	yes	—
18	C₄×C₈×D₅₉	C₄	1	2	yes	—
19	#16837223	C₂²	1	2	yes	—
20	#16837223	C₂²	1	2	yes	—
21	#16837183	C₂²	1	8	yes	—
22	#16837179	C₂²	1	8	yes	—
23	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
25	C₅₉×#364	C₂²	1	2	yes	—
26	C₅₉×#364	C₂²	1	2	yes	—
27	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
28	#14273733	—	2	2	no	—
29	#14273733	—	2	2	no	—
30	#14273733	—	2	1	no	—
31	#14273733	—	2	1	no	—
32	#14273733	—	2	1	no	—
33	#14273733	—	2	1	no	—
34	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
35	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
36	C₂×#11745659	C₂×C₄	1	2	yes	—
37	C₂×#11745659	C₂³	1	2	yes	—
38	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
40	C₄×Dic₁₁₈	—	2	1	no	—
41	C₄×D₂₃₆	—	2	1	no	—
42	C₄×C₂²⋊D₅₉	—	2	2	no	—
43	#14273686	C₂×C₄	1	1	yes	characteristic
44	#14273684	C₂×C₄	1	2	yes	—
45	C₄²×D₅₉	C₂×C₄	1	2	yes	—
46	#14273667	—	2	2	no	—
47	#14273667	C₂³	1	4	yes	—
48	C₂×C₅₉⋊C₁₆	—	2	2	no	—
49	#14273638	C₂×C₄	1	2	yes	—
50	#14273637	C₂×C₄	1	2	yes	—
51	#14273637	C₂³	1	2	yes	—
52	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
53	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
54	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
55	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₅₉	—	2	2	no	—
59	M₃₂×C₅₉	C₂³	1	4	yes	—
60	C₂×C₁₆×C₅₉	—	2	2	no	—
61	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	—	2	1	no	—
66	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂×C₂²⋊D₅₉	M₁₆	1	2	yes	—
68	#11745685	—	2	1	no	—
69	#11745685	—	2	1	no	—
70	#11745685	—	2	1	no	—
71	#11745685	—	2	1	no	—
72	C₂×D₂₃₆	M₁₆	1	1	yes	characteristic
73	C₂×C₄×D₅₉	—	2	2	no	—
74	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
75	C₂×Dic₁₁₈	M₁₆	1	1	yes	characteristic
76	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
77	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
78	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
82	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
83	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
84	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
85	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
86	C₄²×C₅₉	—	2	1	no	—
87	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
88	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₅₉	—	2	1	no	—
90	(C₂×D₅₉)⋊C₄	—	2	2	no	—
91	C₄⋊Dic₅₉	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	2	no	—
93	C₄×Dic₅₉	—	2	2	no	—
94	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
95	#11745664	—	2	1	no	—
96	#11745664	—	2	1	no	—
97	#11745664	—	2	1	no	—
98	#11745664	—	2	1	no	—
99	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
100	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
101	#11745659	—	2	4	no	—
102	#11745659	—	2	2	no	—
103	#11745659	—	2	2	no	—
104	C₈×D₅₉	—	2	2	no	—
105	C₈×D₅₉	—	2	2	no	—
106	C₈×D₅₉	—	2	4	no	—
107	C₁₆×C₅₉	—	2	4	no	—
108	C₅₉⋊C₁₆	—	2	4	no	—
109	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
110	C₂²×D₅₉	C₂×M₁₆	1	2	yes	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	4	2	no	—
114	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
120	C₂²⋊D₅₉	—	2	2	no	—
121	C₂²⋊D₅₉	—	2	2	no	—
122	C₂×Dic₅₉	—	2	2	no	—
123	C₂×Dic₅₉	C₂×M₁₆	1	2	yes	—
124	D₂₃₆	—	2	1	no	—
125	D₂₃₆	—	2	1	no	—
126	C₄×D₅₉	—	2	2	no	—
127	C₄×D₅₉	—	2	2	no	—
128	C₄×D₅₉	—	4	4	no	—
129	Dic₁₁₈	—	2	1	no	—
130	Dic₁₁₈	—	2	1	no	—
131	C₈×C₅₉	#117	1	1	yes	characteristic
132	C₈×C₅₉	#117	1	1	yes	characteristic
133	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
134	C₈×C₅₉	C₂×M₁₆	1	2	yes	—
135	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
136	C₈×C₅₉	C₂×#41	1	1	yes	characteristic
137	C₅₉⋊C₈	—	2	2	no	—
138	C₅₉⋊C₈	—	2	2	no	—
139	#33428	—	59	1	no	maximal, Sylow
140	C₂²×C₅₉	—	4	1	no	—
141	C₂²×C₅₉	#432	1	1	yes	characteristic
142	C₂²×C₅₉	#432	1	1	yes	characteristic
143	C₂²×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
144	C₂×D₅₉	—	2	2	no	—
145	C₂×D₅₉	—	4	2	no	—
146	C₄×C₅₉	—	2	2	no	—
147	C₄×C₅₉	—	4	1	no	—
148	C₄×C₅₉	#432	1	1	yes	characteristic
149	C₄×C₅₉	#432	1	1	yes	characteristic
150	C₄×C₅₉	C₂×#117	1	1	yes	characteristic
151	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
152	Dic₅₉	—	2	2	no	—
153	Dic₅₉	—	4	2	no	—
154	C₄×#131	—	59	1	no	—
155	#2166	—	59	4	no	—
156	#2165	—	59	4	no	—
157	C₂×#364	—	59	1	no	—
158	C₂×#364	D₅₉	1	1	yes	characteristic, complemented
159	#2115	—	59	2	no	—
160	#2110	—	59	2	no	—
161	C₂×C₅₉	—	2	1	no	—
162	C₂×C₅₉	—	2	1	no	—
163	C₂×C₅₉	—	2	1	no	—
164	C₂×C₅₉	C₂×#432	1	1	yes	characteristic
165	D₅₉	—	4	2	no	—
166	C₂×#131	—	59	1	no	—
167	C₂×#131	—	59	1	no	—
168	C₄×#41	—	59	1	no	—
169	C₂×M₃₂	—	59	2	no	—
170	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
171	#405	—	59	1	no	—
172	#405	—	59	1	no	—
173	#405	—	59	2	no	—
174	#405	—	59	2	no	—
175	C₄×M₁₆	—	59	1	no	—
176	C₄×M₁₆	—	59	2	no	—
177	C₂×C₄×C₈	—	59	2	no	—
178	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
179	#364	—	59	2	no	—
180	#364	—	59	2	no	—
181	#364	C₂×D₅₉	1	2	yes	—
182	#364	C₂×D₅₉	1	2	yes	—
183	#349	—	59	8	no	—
184	#347	—	59	8	no	—
185	C₅₉	#33428	1	1	yes	characteristic, complemented, Sylow
186	C₂×#41	—	59	1	no	—
187	#131	—	118	2	no	—
188	#131	—	118	2	no	—
189	#131	—	118	1	no	—
190	#131	—	118	1	no	—
191	#131	—	118	1	no	—
192	#131	—	118	1	no	—
193	C₂×M₁₆	—	59	1	no	—
194	C₂×M₁₆	—	59	1	no	—
195	C₂×M₁₆	—	59	2	no	—
196	C₂×M₁₆	—	59	2	no	—
197	C₂²×C₈	—	59	2	no	—
198	C₂²×C₈	—	59	2	no	—
199	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
200	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
201	C₄×Q₈	—	118	1	no	—
202	C₄×D₄	—	118	1	no	—
203	C₄×D₄	—	118	2	no	—
204	#117	—	59	1	no	—
205	#117	—	59	2	no	—
206	C₂×C₄²	—	59	2	no	—
207	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
208	M₃₂	—	2	2	no	—
209	M₃₂	—	59	4	no	—
210	M₃₂	—	118	2	no	—
211	M₃₂	C₂²×D₅₉	1	4	yes	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	—	118	2	no	—
214	#97	—	59	2	no	—
215	#97	—	59	2	no	—
216	#97	—	59	2	no	—
217	C₄×C₈	—	59	2	no	—
218	C₄×C₈	—	59	2	no	—
219	C₄×C₈	—	59	2	no	—
220	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
221	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
222	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
223	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
224	#41	—	118	1	no	—
225	#41	—	118	1	no	—
226	#41	—	118	1	no	—
227	#41	—	118	1	no	—
228	C₂×Q₈	—	59	1	no	—
229	C₂×D₄	—	59	2	no	—
230	C₂×D₄	—	59	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	59	2	no	—
233	C₂²×C₄	—	118	2	no	—
234	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
235	M₁₆	—	118	1	no	—
236	M₁₆	—	118	1	no	—
237	M₁₆	—	118	1	no	—
238	M₁₆	—	118	1	no	—
239	M₁₆	—	118	4	no	—
240	M₁₆	—	118	2	no	—
241	M₁₆	—	118	2	no	—
242	C₂×C₈	—	59	2	no	—
243	C₂×C₈	—	59	2	no	—
244	C₂×C₈	—	118	2	no	—
245	C₂×C₈	—	118	2	no	—
246	C₂×C₈	—	118	4	no	—
247	C₂×C₈	#11745659	1	2	yes	—
248	C₂×C₈	#11745659	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
250	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
252	C₂×C₈	#11745687	1	1	yes	characteristic
253	C₂×C₈	#11745687	1	1	yes	characteristic
254	C₂×C₈	#11745689	1	1	yes	characteristic
255	C₂×C₈	#11745689	1	1	yes	characteristic
256	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	118	2	no	—
258	C₄⋊C₄	—	118	1	no	—
259	C₂²⋊C₄	—	118	2	no	—
260	C₂²⋊C₄	—	118	1	no	—
261	C₄²	—	2	1	no	—
262	C₄²	—	59	2	no	—
263	C₄²	—	118	2	no	—
264	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
265	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
266	C₁₆	—	2	4	no	—
267	C₁₆	—	118	4	no	—
268	C₂³	—	59	2	no	—
269	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
270	Q₈	—	118	1	no	—
271	Q₈	—	118	1	no	—
272	D₄	—	118	2	no	—
273	D₄	—	118	2	no	—
274	D₄	—	118	1	no	—
275	D₄	—	118	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	59	2	no	—
280	C₂×C₄	—	118	2	no	—
281	C₂×C₄	—	118	2	no	—
282	C₂×C₄	—	118	2	no	—
283	C₂×C₄	—	236	4	no	—
284	C₂×C₄	#14273685	1	1	yes	characteristic
285	C₂×C₄	#14273685	1	1	yes	characteristic
286	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
287	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
288	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
289	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
290	C₈	—	118	2	no	—
291	C₈	—	118	2	no	—
292	C₈	#14273685	1	1	yes	characteristic
293	C₈	#14273685	1	1	yes	characteristic
294	C₈	C₂×#11745659	1	2	yes	—
295	C₈	C₂×#11745659	1	2	yes	—
296	C₈	C₂×#11745687	1	1	yes	characteristic
297	C₈	C₂×#11745689	1	1	yes	characteristic
298	C₂²	—	4	1	no	—
299	C₂²	—	118	2	no	—
300	C₂²	—	236	2	no	—
301	C₂²	#16837458	1	1	yes	characteristic
302	C₂²	#16837458	1	1	yes	characteristic
303	C₂²	C₂×M₁₆×D₅₉	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	118	2	no	—
307	C₄	—	236	2	no	—
308	C₄	#16837458	1	1	yes	characteristic
309	C₄	#16837458	1	1	yes	characteristic
310	C₄	C₂×#14273685	1	1	yes	characteristic
311	C₄	C₂²×#11745659	1	1	yes	characteristic
312	C₂	—	2	1	no	—
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	236	2	no	—
316	C₂	C₂×#16837458	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273685
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂×C₅₉	C₂×#432
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#33428
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. #21425911	C₂	semidirect
3. #21425907	C₂	semidirect
4. C₂×#16837223	C₂	semidirect
6. #21425700	C₂	semidirect
7. C₂×C₅₉×#364	C₂	semidirect
158. C₂×#364	D₅₉	semidirect
185. C₅₉	#33428	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273733	2
#21425911	2
#21425907	2
C₂×#16837223	2
#21425716	2
#21425700	2
C₂×C₅₉×#364	2
#33428	59