Subgroups — #28503813

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21437198	C₂	1	1	yes	maximal, characteristic
2	#21425725	C₂	1	1	yes	maximal, characteristic
3	#21425700	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16837180	C₂	1	1	yes	maximal, characteristic, complemented
5	#21423940	C₂	1	1	yes	maximal, characteristic
6	#21422183	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#1377	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	#16837820	C₄	1	4	yes	complemented
10	#16837818	—	2	2	no	—
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	C₂×#14273667	C₂²	1	1	yes	characteristic
13	M₃₂×D₅₉	—	2	2	no	—
14	C₂×#14273655	C₄	1	1	yes	characteristic
15	#16837399	C₄	1	1	yes	characteristic
16	#16837357	C₂²	1	1	yes	characteristic
17	#16837187	C₄	1	4	yes	complemented
18	#16837180	—	2	2	no	—
19	#16837179	—	2	2	no	—
20	#16837167	C₄	1	1	yes	characteristic
21	#16837162	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
24	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
25	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
26	#14273733	—	2	1	no	—
27	#14273733	—	2	1	no	—
28	#14273733	C₂×C₄	1	2	yes	—
29	#14273733	C₂×C₄	1	2	yes	—
30	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
32	C₂×#11745659	—	2	1	no	—
33	C₂×C₈×D₅₉	—	2	1	no	—
34	#14273686	C₂×C₄	1	1	yes	characteristic
35	#14273667	—	2	1	no	—
36	#14273667	—	2	1	no	—
37	#14273667	D₄	1	2	yes	—
38	#14273667	D₄	1	2	yes	—
39	#14273666	—	2	4	no	—
40	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
41	C₂×C₅₉⋊C₁₆	C₂×C₄	1	2	yes	—
42	C₁₆×D₅₉	—	2	4	no	—
43	#14273655	—	2	2	no	—
44	#14273652	—	2	1	no	—
45	#14273649	—	2	1	no	—
46	#14273637	—	2	1	no	—
47	C₈×Dic₅₉	—	2	1	no	—
48	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
51	M₃₂×C₅₉	—	2	1	no	—
52	M₃₂×C₅₉	—	2	1	no	—
53	M₃₂×C₅₉	C₂×C₄	1	2	yes	—
54	M₃₂×C₅₉	C₂×C₄	1	2	yes	—
55	C₂×C₁₆×C₅₉	—	2	1	no	—
56	C₂×C₁₆×C₅₉	—	2	1	no	—
57	#97×C₅₉	—	2	1	no	—
58	#97×C₅₉	—	2	1	no	—
59	C₂²×C₄×C₅₉	—	2	1	no	—
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₉	—	2	1	no	—
62	#11745685	—	2	1	no	—
63	#11745685	—	2	1	no	—
64	#11745685	C₄²	1	2	yes	—
65	#11745685	M₁₆	1	2	yes	—
66	C₂×D₂₃₆	M₁₆	1	1	yes	characteristic
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×Dic₁₁₈	M₁₆	1	1	yes	characteristic
69	C₂×C₈×C₅₉	—	2	1	no	—
70	C₂×C₈×C₅₉	—	2	1	no	—
71	C₂×C₈×C₅₉	—	2	1	no	—
72	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
77	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
78	C₄²×C₅₉	—	2	1	no	—
79	C₄²×C₅₉	—	2	1	no	—
80	C₂²⋊Dic₅₉	—	2	1	no	—
81	C₄⋊Dic₅₉	—	2	1	no	—
82	C₄×Dic₅₉	—	2	1	no	—
83	#11745664	—	2	1	no	—
84	#11745664	—	2	1	no	—
85	#11745664	—	2	1	no	—
86	#11745664	C₄²	1	2	yes	—
87	C₂×C₅₉⋊C₈	—	2	1	no	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	#11745659	—	2	2	no	—
90	#11745659	—	2	2	no	—
91	C₈×D₅₉	—	2	2	no	—
92	C₈×D₅₉	—	2	2	no	—
93	C₁₆×C₅₉	—	2	2	no	—
94	C₁₆×C₅₉	—	2	2	no	—
95	C₅₉⋊C₁₆	—	2	2	no	—
96	C₅₉⋊C₁₆	C₂²⋊C₄	1	4	yes	—
97	C₂³×C₅₉	#131	1	1	yes	characteristic
98	C₂²×D₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	4	1	no	—
100	C₂×C₄×C₅₉	—	8	1	no	—
101	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
107	C₂²⋊D₅₉	—	2	2	no	—
108	C₂²⋊D₅₉	—	2	2	no	—
109	C₂×Dic₅₉	—	2	1	no	—
110	C₂×Dic₅₉	—	4	1	no	—
111	D₂₃₆	—	2	1	no	—
112	D₂₃₆	#97	1	2	yes	—
113	C₄×D₅₉	—	2	2	no	—
114	C₄×D₅₉	—	2	2	no	—
115	Dic₁₁₈	—	2	1	no	—
116	Dic₁₁₈	#97	1	2	yes	—
117	C₈×C₅₉	—	4	1	no	—
118	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
119	C₈×C₅₉	#117	1	1	yes	characteristic
120	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
121	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
122	C₅₉⋊C₈	—	2	2	no	—
123	C₅₉⋊C₈	—	4	1	no	—
124	#12618	—	59	1	no	maximal, Sylow
125	C₂²×C₅₉	—	4	1	no	—
126	C₂²×C₅₉	#405	1	1	yes	characteristic
127	C₂²×C₅₉	#435	1	1	yes	characteristic
128	C₂²×C₅₉	#435	1	1	yes	characteristic
129	C₂×D₅₉	—	2	2	no	—
130	C₂×D₅₉	—	8	1	no	—
131	C₄×C₅₉	—	8	1	no	—
132	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
133	C₄×C₅₉	C₂×#97	1	1	yes	characteristic
134	C₄×C₅₉	#435	1	1	yes	characteristic
135	C₄×C₅₉	#435	1	1	yes	characteristic
136	Dic₅₉	—	2	2	no	—
137	Dic₅₉	—	8	1	no	—
138	#3407	—	59	1	no	—
139	#2117	—	59	1	no	—
140	#2110	—	59	1	no	—
141	C₂×#347	—	59	1	no	—
142	#1765	—	59	1	no	—
143	#1378	—	59	1	no	—
144	#1377	D₅₉	1	1	yes	characteristic, complemented
145	C₂×C₅₉	—	2	1	no	—
146	C₂×C₅₉	—	2	1	no	—
147	C₂×C₅₉	—	2	1	no	—
148	C₂×C₅₉	#1752	1	1	yes	characteristic
149	D₅₉	—	4	2	no	—
150	C₂×#131	—	59	1	no	—
151	#504	—	59	4	no	—
152	#504	—	118	2	no	—
153	C₂×M₃₂	—	59	1	no	—
154	C₂×M₃₂	—	59	1	no	—
155	C₂×M₃₂	—	118	2	no	—
156	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
157	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
158	C₂×#104	—	59	1	no	—
159	#416	—	59	1	no	—
160	#405	—	59	1	no	—
161	C₂×#97	C₂×D₅₉	1	1	yes	characteristic
162	#350	—	59	4	no	—
163	#347	—	118	2	no	—
164	#347	—	118	2	no	—
165	#341	—	59	1	no	—
166	#339	—	59	1	no	—
167	C₅₉	#12618	1	1	yes	characteristic, complemented, Sylow
168	C₂×#41	—	59	1	no	—
169	#131	—	59	2	no	—
170	#131	—	59	2	no	—
171	#131	—	118	1	no	—
172	#131	—	118	1	no	—
173	C₂×M₁₆	—	59	1	no	—
174	C₂×M₁₆	—	59	1	no	—
175	C₂×M₁₆	—	118	1	no	—
176	C₂²×C₈	—	118	1	no	—
177	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
178	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
179	#117	—	59	1	no	—
180	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
181	M₃₂	—	2	1	no	—
182	M₃₂	—	2	1	no	—
183	M₃₂	—	59	2	no	—
184	M₃₂	—	59	2	no	—
185	M₃₂	—	118	4	no	—
186	M₃₂	—	118	1	no	—
187	M₃₂	—	118	1	no	—
188	M₃₂	C₄×D₅₉	1	2	yes	—
189	M₃₂	C₄×D₅₉	1	2	yes	—
190	C₂×C₁₆	—	2	1	no	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	59	2	no	—
193	C₂×C₁₆	—	59	2	no	—
194	C₂×C₁₆	—	118	4	no	—
195	#104	—	118	2	no	—
196	#103	—	118	1	no	—
197	#102	—	118	1	no	—
198	#97	—	2	1	no	—
199	#97	—	2	1	no	—
200	#97	—	118	1	no	—
201	C₄×C₈	—	118	1	no	—
202	#41	—	59	2	no	—
203	#41	—	59	2	no	—
204	#41	—	118	1	no	—
205	#41	—	118	1	no	—
206	C₂×Q₈	—	59	1	no	—
207	C₂×D₄	—	59	1	no	—
208	C₂×D₄	—	118	1	no	—
209	C₂²×C₄	—	2	1	no	—
210	C₂²×C₄	—	118	1	no	—
211	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
212	M₁₆	—	59	2	no	—
213	M₁₆	—	118	1	no	—
214	M₁₆	—	118	2	no	—
215	M₁₆	—	118	2	no	—
216	M₁₆	—	118	1	no	—
217	M₁₆	—	118	1	no	—
218	C₂×C₈	—	2	1	no	—
219	C₂×C₈	—	2	1	no	—
220	C₂×C₈	—	2	1	no	—
221	C₂×C₈	—	118	1	no	—
222	C₂×C₈	—	118	2	no	—
223	C₂×C₈	—	118	2	no	—
224	C₂×C₈	—	118	1	no	—
225	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
227	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
228	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
229	C₂×C₈	#11745687	1	1	yes	characteristic
230	C₂×C₈	#11745689	1	1	yes	characteristic
231	C₄⋊C₄	—	118	1	no	—
232	C₂²⋊C₄	—	118	1	no	—
233	C₄²	—	2	1	no	—
234	C₄²	—	2	1	no	—
235	C₄²	—	118	1	no	—
236	C₁₆	—	2	2	no	—
237	C₁₆	—	2	2	no	—
238	C₁₆	—	59	4	no	—
239	C₁₆	—	118	2	no	—
240	C₂³	—	118	1	no	—
241	C₂³	#14273733	1	1	yes	characteristic
242	Q₈	—	59	2	no	—
243	Q₈	—	118	1	no	—
244	D₄	—	59	2	no	—
245	D₄	—	118	2	no	—
246	D₄	—	118	2	no	—
247	D₄	—	118	1	no	—
248	C₂×C₄	—	4	1	no	—
249	C₂×C₄	—	8	1	no	—
250	C₂×C₄	—	118	1	no	—
251	C₂×C₄	—	118	2	no	—
252	C₂×C₄	—	118	2	no	—
253	C₂×C₄	—	236	1	no	—
254	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
255	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
256	C₂×C₄	#14273689	1	1	yes	characteristic
257	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
258	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
259	C₂×C₄	#14273733	1	1	yes	characteristic
260	C₈	—	4	1	no	—
261	C₈	—	118	2	no	—
262	C₈	—	236	1	no	—
263	C₈	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
264	C₈	#14273685	1	1	yes	characteristic
265	C₈	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
266	C₈	#14273689	1	1	yes	characteristic
267	C₂²	—	4	1	no	—
268	C₂²	—	118	2	no	—
269	C₂²	—	472	1	no	—
270	C₂²	#16837356	1	1	yes	characteristic
271	C₂²	#16837472	1	1	yes	characteristic
272	C₂²	#16837476	1	1	yes	characteristic
273	C₄	—	8	1	no	—
274	C₄	—	118	2	no	—
275	C₄	—	472	1	no	—
276	C₄	#16837249	1	1	yes	characteristic
277	C₄	#97×D₅₉	1	1	yes	characteristic
278	C₄	#16837472	1	1	yes	characteristic
279	C₄	#16837476	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	2	1	no	—
283	C₂	—	236	2	no	—
284	C₂	#21423871	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#16837249
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#1377	C₂
socle	C₂×C₅₉	#1752
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#1377	D₅₉
hypocenter	C₅₉	#12618
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #1377

Complements

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

N	H ≅ G/N	type
3. #21425700	C₂	semidirect
4. C₂×#16837180	C₂	semidirect
6. #21422183	C₂	semidirect
7. C₅₉×#1377	C₂	semidirect
9. #16837820	C₄	semidirect
17. #16837187	C₄	semidirect
144. #1377	D₅₉	semidirect
167. C₅₉	#12618	semidirect

Maximal subgroups

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

subgroup	index
#21437198	2
#21425725	2
#21425700	2
C₂×#16837180	2
#21423940	2
#21422183	2
C₅₉×#1377	2
#12618	59