Subgroups — #28741348

Order 15104.

716 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437198	C₂	1	2	yes	maximal
2	#21432521	C₂	1	1	yes	maximal, characteristic
3	#21425700	C₂	1	2	yes	maximal, complemented
4	#21425699	C₂	1	4	yes	maximal, complemented
5	C₂×#16837180	C₂	1	1	yes	maximal, characteristic, complemented
6	#347×D₅₉	C₂	1	4	yes	maximal, complemented
7	C₂×C₅₉×#347	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	#16837820	C₄	1	8	yes	complemented
12	#16837818	C₄	1	8	yes	complemented
13	C₂×#14273667	C₂²	1	2	yes	—
14	M₃₂×D₅₉	C₂²	1	8	yes	—
15	#16837359	C₄	1	2	yes	—
16	#16837359	C₂²	1	2	yes	—
17	#16837357	C₄	1	1	yes	characteristic
18	#16837357	C₂²	1	1	yes	characteristic
19	C₄×#11745664	C₄	1	1	yes	characteristic
20	#16837354	C₄	1	2	yes	—
21	#97×D₅₉	C₂²	1	2	yes	—
22	#16837180	C₂²	1	2	yes	—
23	#16837180	C₂²	1	2	yes	—
24	#16837179	C₂²	1	8	yes	—
25	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
26	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#347	C₂²	1	2	yes	—
28	C₅₉×#347	C₂²	1	2	yes	—
29	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
30	#14273733	—	2	2	no	—
31	#14273733	—	2	2	no	—
32	#14273733	C₂×C₄	1	4	yes	—
33	#14273733	C₂×C₄	1	4	yes	—
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	C₂×C₄	1	2	yes	—
38	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₈×D₅₉	C₂³	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	C₂×C₄	1	4	yes	complemented
47	#14273667	C₂³	1	4	yes	—
48	#14273666	C₂×C₄	1	16	yes	complemented
49	C₂×C₅₉⋊C₁₆	C₂×C₄	1	4	yes	complemented
50	C₁₆×D₅₉	C₂×C₄	1	16	yes	complemented
51	#14273638	C₂×C₄	1	2	yes	—
52	#14273637	C₂×C₄	1	2	yes	—
53	#14273637	C₂³	1	2	yes	—
54	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
55	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
56	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
60	M₃₂×C₅₉	C₂×C₄	1	4	yes	complemented
61	M₃₂×C₅₉	C₂³	1	4	yes	—
62	C₂×C₁₆×C₅₉	C₂×C₄	1	4	yes	complemented
63	#97×C₅₉	C₂×C₄	1	1	yes	characteristic
64	#97×C₅₉	C₂×C₄	1	1	yes	characteristic
65	#97×C₅₉	C₂³	1	1	yes	characteristic
66	#97×C₅₉	C₂³	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	—	2	1	no	—
68	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₂²⋊D₅₉	M₁₆	1	2	yes	—
70	#11745685	C₄²	1	2	yes	—
71	#11745685	C₄²	1	2	yes	—
72	#11745685	M₁₆	1	2	yes	—
73	#11745685	M₁₆	1	2	yes	—
74	C₂×D₂₃₆	M₁₆	1	1	yes	characteristic
75	C₂×C₄×D₅₉	—	2	2	no	—
76	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
77	C₂×Dic₁₁₈	M₁₆	1	1	yes	characteristic
78	C₂×C₈×C₅₉	—	2	1	no	—
79	C₂×C₈×C₅₉	—	2	1	no	—
80	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
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	2	yes	—
84	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
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	C₄²	1	2	yes	—
98	#11745664	C₄²	1	2	yes	—
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	C₄²	1	8	yes	—
103	C₈×D₅₉	—	2	4	no	—
104	C₈×D₅₉	C₂²×C₄	1	8	yes	—
105	C₁₆×C₅₉	C₂²×C₄	1	8	yes	—
106	C₅₉⋊C₁₆	C₂²×C₄	1	8	yes	—
107	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
108	C₂²×D₅₉	C₂×M₁₆	1	2	yes	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	8	1	no	—
112	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
118	C₂²⋊D₅₉	#97	1	4	yes	—
119	C₂²⋊D₅₉	#97	1	4	yes	—
120	C₂×Dic₅₉	—	2	2	no	—
121	C₂×Dic₅₉	C₂×M₁₆	1	2	yes	—
122	D₂₃₆	#97	1	2	yes	—
123	D₂₃₆	#97	1	2	yes	—
124	C₄×D₅₉	—	8	2	no	—
125	C₄×D₅₉	C₂×C₄²	1	4	yes	—
126	C₄×D₅₉	C₂×M₁₆	1	4	yes	—
127	Dic₁₁₈	#97	1	2	yes	—
128	Dic₁₁₈	#97	1	2	yes	—
129	C₈×C₅₉	—	2	1	no	—
130	C₈×C₅₉	—	2	1	no	—
131	C₈×C₅₉	C₂×C₄²	1	2	yes	—
132	C₈×C₅₉	C₂³×C₄	1	2	yes	—
133	C₅₉⋊C₈	—	2	2	no	—
134	C₅₉⋊C₈	C₂×C₄²	1	4	yes	—
135	#33329	—	59	1	no	maximal, Sylow
136	C₂²×C₅₉	—	4	1	no	—
137	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
138	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
139	C₂²×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
140	C₂×D₅₉	—	4	2	no	—
141	C₂×D₅₉	C₂×#97	1	4	yes	—
142	C₄×C₅₉	—	4	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
145	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
146	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
147	C₄×C₅₉	C₂²×M₁₆	1	1	yes	characteristic
148	Dic₅₉	—	4	2	no	—
149	Dic₅₉	C₂×#97	1	4	yes	—
150	#3407	—	59	2	no	—
151	#2875	—	59	1	no	—
152	#2110	—	59	2	no	—
153	#2110	—	59	4	no	—
154	C₂×#347	—	59	1	no	—
155	C₂×#347	—	59	4	no	—
156	C₂×#347	D₅₉	1	1	yes	characteristic, complemented
157	C₂×C₅₉	—	2	1	no	—
158	C₂×C₅₉	—	2	1	no	—
159	C₂×C₅₉	—	2	1	no	—
160	C₂×C₅₉	C₂²×#97	1	1	yes	characteristic
161	D₅₉	—	2	4	no	—
162	C₂×#131	—	59	1	no	—
163	C₂×#131	—	59	1	no	—
164	C₄×#41	—	59	1	no	—
165	#504	—	59	8	no	—
166	#504	—	59	8	no	—
167	C₂×M₃₂	—	59	2	no	—
168	C₂×M₃₂	—	59	8	no	—
169	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
170	#405	—	59	1	no	—
171	#405	—	59	1	no	—
172	#405	—	59	2	no	—
173	#405	—	59	2	no	—
174	C₄×M₁₆	—	59	1	no	—
175	C₄×M₁₆	—	59	2	no	—
176	C₂×#97	—	59	2	no	—
177	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
178	#347	—	59	8	no	—
179	#347	—	59	2	no	—
180	#347	—	59	2	no	—
181	#347	C₂×D₅₉	1	2	yes	—
182	#347	C₂×D₅₉	1	2	yes	—
183	C₅₉	#33329	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	59	1	no	—
185	#131	—	59	4	no	—
186	#131	—	59	4	no	—
187	#131	—	118	2	no	—
188	#131	—	118	2	no	—
189	C₂×M₁₆	—	59	1	no	—
190	C₂×M₁₆	—	59	1	no	—
191	C₂×M₁₆	—	59	2	no	—
192	C₂×M₁₆	—	59	2	no	—
193	C₂²×C₈	—	59	2	no	—
194	C₂²×C₈	—	59	2	no	—
195	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
196	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
197	C₄×Q₈	—	118	1	no	—
198	C₄×D₄	—	118	1	no	—
199	C₄×D₄	—	118	2	no	—
200	#117	—	59	1	no	—
201	#117	—	59	2	no	—
202	C₂×C₄²	—	59	2	no	—
203	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
204	M₃₂	—	59	4	no	—
205	M₃₂	—	59	4	no	—
206	M₃₂	—	59	16	no	—
207	M₃₂	C₄×D₅₉	1	4	yes	complemented
208	M₃₂	C₂²×D₅₉	1	4	yes	—
209	C₂×C₁₆	—	59	4	no	—
210	C₂×C₁₆	—	59	16	no	—
211	C₂×C₁₆	C₄×D₅₉	1	4	yes	complemented
212	#97	—	59	2	no	—
213	#97	—	59	2	no	—
214	#97	—	59	2	no	—
215	#97	C₄×D₅₉	1	1	yes	characteristic
216	#97	C₄×D₅₉	1	1	yes	characteristic
217	#97	C₂²×D₅₉	1	1	yes	characteristic
218	#97	C₂²×D₅₉	1	1	yes	characteristic
219	C₄×C₈	—	59	2	no	—
220	C₄×C₈	—	59	2	no	—
221	C₄×C₈	—	59	2	no	—
222	#41	—	59	2	no	—
223	#41	—	59	2	no	—
224	#41	—	59	2	no	—
225	#41	—	59	2	no	—
226	C₂×Q₈	—	59	1	no	—
227	C₂×D₄	—	59	2	no	—
228	C₂×D₄	—	59	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	59	2	no	—
231	C₂²×C₄	—	118	2	no	—
232	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
233	M₁₆	—	59	2	no	—
234	M₁₆	—	59	2	no	—
235	M₁₆	—	59	8	no	—
236	M₁₆	—	118	4	no	—
237	M₁₆	—	118	1	no	—
238	M₁₆	—	118	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	59	2	no	—
242	C₂×C₈	—	59	8	no	—
243	C₂×C₈	—	59	2	no	—
244	C₂×C₈	—	118	4	no	—
245	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
246	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
247	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
248	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₅₉	1	2	yes	—
250	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
251	C₄⋊C₄	—	118	2	no	—
252	C₄⋊C₄	—	118	1	no	—
253	C₂²⋊C₄	—	118	2	no	—
254	C₂²⋊C₄	—	118	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	59	2	no	—
257	C₄²	—	118	2	no	—
258	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
259	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
260	C₁₆	—	59	8	no	—
261	C₁₆	C₂×C₄×D₅₉	1	8	yes	—
262	C₂³	—	59	2	no	—
263	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
264	Q₈	—	59	2	no	—
265	Q₈	—	59	2	no	—
266	D₄	—	59	4	no	—
267	D₄	—	59	4	no	—
268	D₄	—	59	2	no	—
269	D₄	—	59	2	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	59	2	no	—
274	C₂×C₄	—	59	4	no	—
275	C₂×C₄	—	59	4	no	—
276	C₂×C₄	—	118	2	no	—
277	C₂×C₄	—	472	2	no	—
278	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
279	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
280	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
281	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
282	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
283	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	59	4	no	—
287	C₈	—	118	2	no	—
288	C₈	C₄²×D₅₉	1	2	yes	—
289	C₈	C₂²×C₄×D₅₉	1	2	yes	—
290	C₂²	—	4	1	no	—
291	C₂²	—	59	4	no	—
292	C₂²	—	236	2	no	—
293	C₂²	#97×D₅₉	1	1	yes	characteristic
294	C₂²	#97×D₅₉	1	1	yes	characteristic
295	C₂²	C₂×M₁₆×D₅₉	1	1	yes	characteristic
296	C₄	—	4	1	no	—
297	C₄	—	4	1	no	—
298	C₄	—	59	4	no	—
299	C₄	—	236	2	no	—
300	C₄	#97×D₅₉	1	1	yes	characteristic
301	C₄	#97×D₅₉	1	1	yes	characteristic
302	C₄	C₂×C₄²×D₅₉	1	1	yes	characteristic
303	C₄	C₂×M₁₆×D₅₉	1	1	yes	characteristic
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	1	no	—
307	C₂	—	118	4	no	—
308	C₂	C₂×#97×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×C₄²×D₅₉
commutator	C₄×C₅₉	C₂²×C₄²
Frattini	C₂×C₈	C₂³×D₅₉
Fitting	C₂×C₅₉×#347	C₂
socle	C₂×C₅₉	C₂²×#97
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₂×#347	D₅₉
hypocenter	C₅₉	#33329
LCS	G ▷ C₄×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂×#347

Complements

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

N	H ≅ G/N	type
3. #21425700	C₂	semidirect
4. #21425699	C₂	semidirect
5. C₂×#16837180	C₂	semidirect
6. #347×D₅₉	C₂	semidirect
7. C₂×C₅₉×#347	C₂	semidirect
11. #16837820	C₄	semidirect
12. #16837818	C₄	semidirect
46. #14273667	C₂×C₄	semidirect
48. #14273666	C₂×C₄	semidirect
49. C₂×C₅₉⋊C₁₆	C₂×C₄	semidirect
50. C₁₆×D₅₉	C₂×C₄	semidirect
60. M₃₂×C₅₉	C₂×C₄	semidirect
62. C₂×C₁₆×C₅₉	C₂×C₄	semidirect
156. C₂×#347	D₅₉	semidirect
183. C₅₉	#33329	semidirect
207. M₃₂	C₄×D₅₉	semidirect
211. C₂×C₁₆	C₄×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21437198	2
#21432521	2
#21425700	2
#21425699	2
C₂×#16837180	2
#347×D₅₉	2
C₂×C₅₉×#347	2
#33329	59