Subgroups — #28497076

Order 15104.

602 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432660	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837174	C₂	1	2	yes	maximal
3	#21423829	C₂	1	1	yes	maximal, characteristic
4	#21423792	C₂	1	2	yes	maximal
5	C₅₉×#1729	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11745664	C₂²	1	2	yes	—
7	C₂×#14273686	C₂²	1	2	yes	—
8	#16837429	—	2	4	no	—
9	#16837426	—	2	2	no	—
10	#16837174	—	2	4	no	—
11	#16837169	C₄	1	4	yes	—
12	#16837156	—	2	4	no	—
13	C₂²×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
14	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₅₉	C₂²	1	1	yes	characteristic
16	C₅₉×#336	—	2	2	no	—
17	C₂×#11745664	—	2	2	no	—
18	C₂×#11745664	—	2	2	no	—
19	C₂×#11745664	—	2	2	no	—
20	C₂×#11745664	—	2	2	no	—
21	C₂×#11745664	—	2	2	no	—
22	C₂×#11745664	—	2	2	no	—
23	C₂²×C₅₉⋊C₈	D₄	1	4	yes	—
24	#14273686	—	2	2	no	—
25	#14273686	—	2	2	no	—
26	#14273686	—	2	2	no	—
27	#14273686	—	2	2	no	—
28	C₂×C₄⋊Dic₅₉	—	2	2	no	—
29	C₂×C₂²⋊Dic₅₉	C₂×C₄	1	4	yes	—
30	C₂×C₄×Dic₅₉	—	2	2	no	—
31	C₄⋊(C₅₉⋊C₈)	—	2	8	no	—
32	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
33	C₂×M₁₆×C₅₉	—	2	1	no	—
34	C₂×M₁₆×C₅₉	—	2	2	no	—
35	C₂×M₁₆×C₅₉	—	2	2	no	—
36	C₂×M₁₆×C₅₉	D₄	1	2	yes	—
37	C₂²×C₈×C₅₉	—	2	1	no	—
38	C₂²×C₈×C₅₉	—	2	1	no	—
39	C₂²×C₈×C₅₉	—	2	2	no	—
40	C₂²×C₈×C₅₉	—	2	1	no	—
41	C₂²×C₈×C₅₉	—	2	2	no	—
42	C₂²×C₈×C₅₉	Q₈	1	2	yes	—
43	#117×C₅₉	—	2	2	no	—
44	#117×C₅₉	—	2	2	no	—
45	C₂×C₄⋊C₄×C₅₉	Q₈	1	2	yes	—
46	C₂×C₂²⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
47	C₂×C₄²×C₅₉	D₄	1	2	yes	—
48	C₂⁴×C₅₉	C₂²×C₄	1	1	yes	characteristic
49	C₂²×C₄×C₅₉	C₄⋊C₄	1	4	yes	—
50	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
51	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	#41	1	2	yes	—
57	C₂²×C₄×C₅₉	#41	1	2	yes	—
58	C₂²×C₄×C₅₉	#41	1	2	yes	—
59	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
60	C₂²×Dic₅₉	—	2	4	no	—
61	M₁₆×C₅₉	—	2	4	no	—
62	M₁₆×C₅₉	—	4	2	no	—
63	C₂×C₈×C₅₉	—	2	2	no	—
64	C₂×C₈×C₅₉	—	2	4	no	—
65	C₂×C₈×C₅₉	—	2	2	no	—
66	C₂×C₈×C₅₉	—	4	1	no	—
67	C₂×C₈×C₅₉	—	4	1	no	—
68	C₂×C₈×C₅₉	—	4	2	no	—
69	C₂×C₈×C₅₉	—	4	1	no	—
70	C₂×C₈×C₅₉	—	4	1	no	—
71	C₄⋊C₄×C₅₉	—	2	4	no	—
72	C₂²⋊C₄×C₅₉	—	2	4	no	—
73	C₄²×C₅₉	—	2	2	no	—
74	C₄²×C₅₉	—	2	2	no	—
75	C₂²⋊Dic₅₉	—	2	4	no	—
76	C₂²⋊Dic₅₉	—	2	4	no	—
77	C₄⋊Dic₅₉	—	4	2	no	—
78	C₄⋊Dic₅₉	—	4	2	no	—
79	C₄×Dic₅₉	—	4	2	no	—
80	C₄×Dic₅₉	—	4	2	no	—
81	#11745664	—	4	2	no	—
82	#11745664	—	4	2	no	—
83	#11745664	—	4	2	no	—
84	#11745664	—	4	2	no	—
85	C₂×C₅₉⋊C₈	—	2	4	no	—
86	C₂×C₅₉⋊C₈	—	2	4	no	—
87	C₂×C₅₉⋊C₈	—	2	4	no	—
88	C₂³×C₅₉	—	4	2	no	—
89	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
90	C₂³×C₅₉	#117	1	2	yes	—
91	C₂³×C₅₉	C₄×D₄	1	2	yes	—
92	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
93	C₂³×C₅₉	C₄×Q₈	1	1	yes	characteristic
94	C₂×C₄×C₅₉	—	2	4	no	—
95	C₂×C₄×C₅₉	—	4	4	no	—
96	C₂×C₄×C₅₉	#120	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₉	#121	1	2	yes	—
99	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
100	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
101	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#123	1	2	yes	—
104	C₂×C₄×C₅₉	#123	1	2	yes	—
105	C₂×C₄×C₅₉	#124	1	2	yes	—
106	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#125	1	2	yes	—
109	C₂×C₄×C₅₉	#125	1	2	yes	—
110	C₂×C₄×C₅₉	#125	1	1	yes	characteristic
111	C₂×C₄×C₅₉	#126	1	2	yes	—
112	C₂×C₄×C₅₉	#126	1	2	yes	—
113	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×Dic₅₉	—	4	8	no	—
115	C₂×Dic₅₉	—	4	4	no	—
116	C₈×C₅₉	—	4	2	no	—
117	C₈×C₅₉	—	4	2	no	—
118	C₅₉⋊C₈	—	4	4	no	—
119	#11288	—	59	1	no	maximal, Sylow
120	C₂²×C₅₉	—	2	4	no	—
121	C₂²×C₅₉	—	2	2	no	—
122	C₂²×C₅₉	—	4	2	no	—
123	C₂²×C₅₉	—	4	1	no	—
124	C₂²×C₅₉	—	4	1	no	—
125	C₂²×C₅₉	#382	1	1	yes	characteristic
126	C₂²×C₅₉	#385	1	1	yes	characteristic
127	C₂²×C₅₉	#387	1	2	yes	—
128	C₂²×C₅₉	#388	1	2	yes	—
129	C₂²×C₅₉	#389	1	1	yes	characteristic
130	C₄×C₅₉	—	4	4	no	—
131	C₄×C₅₉	#393	1	1	yes	characteristic
132	C₄×C₅₉	#394	1	1	yes	characteristic
133	C₄×C₅₉	#397	1	1	yes	characteristic
134	C₄×C₅₉	#398	1	2	yes	—
135	C₄×C₅₉	#400	1	1	yes	characteristic
136	C₄×C₅₉	#400	1	2	yes	—
137	Dic₅₉	—	8	4	no	—
138	#2905	—	59	1	no	—
139	C₂×#344	—	59	2	no	—
140	#1740	—	59	1	no	—
141	#1729	—	59	2	no	—
142	#1729	D₅₉	1	1	yes	characteristic
143	C₂×C₅₉	—	2	2	no	—
144	C₂×C₅₉	—	2	1	no	—
145	C₂×C₅₉	—	2	2	no	—
146	C₂×C₅₉	—	2	1	no	—
147	C₂×C₅₉	#1442	1	1	yes	characteristic
148	C₂×C₅₉	#1868	1	2	yes	—
149	C₂²×M₁₆	—	59	2	no	—
150	C₂²×M₁₆	C₂×D₅₉	1	1	yes	characteristic
151	C₂³×C₈	C₂×D₅₉	1	1	yes	characteristic
152	C₂×#117	—	59	2	no	—
153	C₂×#117	C₂×D₅₉	1	1	yes	characteristic
154	#424	—	118	4	no	—
155	#423	—	118	2	no	—
156	#344	—	118	4	no	—
157	#342	—	59	4	no	—
158	#336	—	2	2	no	—
159	#336	—	118	4	no	—
160	C₅₉	#11288	1	1	yes	characteristic, complemented, Sylow
161	C₂³×C₄	C₂²×D₅₉	1	1	yes	characteristic
162	C₂×M₁₆	—	2	1	no	—
163	C₂×M₁₆	—	2	2	no	—
164	C₂×M₁₆	—	2	2	no	—
165	C₂×M₁₆	—	118	2	no	—
166	C₂×M₁₆	—	118	2	no	—
167	C₂×M₁₆	—	118	2	no	—
168	C₂×M₁₆	—	118	2	no	—
169	C₂×M₁₆	—	118	2	no	—
170	C₂×M₁₆	—	118	2	no	—
171	C₂×M₁₆	D₂₃₆	1	2	yes	—
172	C₂²×C₈	—	2	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	2	2	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₂²×C₈	—	2	1	no	—
177	C₂²×C₈	—	59	4	no	—
178	C₂²×C₈	Dic₁₁₈	1	2	yes	—
179	#117	—	2	2	no	—
180	#117	—	2	2	no	—
181	#117	—	118	2	no	—
182	#117	—	118	2	no	—
183	#117	—	118	2	no	—
184	#117	—	118	2	no	—
185	C₂×C₄⋊C₄	—	118	2	no	—
186	C₂×C₄⋊C₄	Dic₁₁₈	1	2	yes	—
187	C₂×C₂²⋊C₄	—	59	4	no	—
188	C₂×C₂²⋊C₄	C₄×D₅₉	1	2	yes	—
189	C₂×C₄²	—	118	2	no	—
190	C₂×C₄²	C₂²⋊D₅₉	1	2	yes	—
191	C₄⋊C₈	—	118	8	no	—
192	C₂⁴	C₂×C₄×D₅₉	1	1	yes	characteristic
193	C₂²×C₄	—	118	4	no	—
194	C₂²×C₄	Dic₅₉⋊C₄	1	4	yes	—
195	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
196	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
198	C₂²×C₄	#11745685	1	1	yes	characteristic
199	C₂²×C₄	#11745685	1	1	yes	characteristic
200	C₂²×C₄	D₄×D₅₉	1	2	yes	—
201	C₂²×C₄	#11745687	1	2	yes	—
202	C₂²×C₄	#11745687	1	2	yes	—
203	C₂²×C₄	#11745689	1	2	yes	—
204	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
205	M₁₆	—	2	4	no	—
206	M₁₆	—	4	2	no	—
207	M₁₆	—	236	2	no	—
208	M₁₆	—	236	2	no	—
209	M₁₆	—	236	2	no	—
210	M₁₆	—	236	2	no	—
211	C₂×C₈	—	2	2	no	—
212	C₂×C₈	—	2	4	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	4	1	no	—
215	C₂×C₈	—	4	1	no	—
216	C₂×C₈	—	4	2	no	—
217	C₂×C₈	—	4	1	no	—
218	C₂×C₈	—	4	1	no	—
219	C₂×C₈	—	118	4	no	—
220	C₂×C₈	—	118	4	no	—
221	C₂×C₈	—	118	4	no	—
222	C₄⋊C₄	—	2	4	no	—
223	C₄⋊C₄	—	236	2	no	—
224	C₄⋊C₄	—	236	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	118	4	no	—
227	C₂²⋊C₄	—	118	4	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	2	2	no	—
230	C₄²	—	236	2	no	—
231	C₄²	—	236	2	no	—
232	C₂³	—	4	2	no	—
233	C₂³	C₂×Dic₅₉⋊C₄	1	1	yes	characteristic
234	C₂³	#14273685	1	2	yes	—
235	C₂³	(C₂²⋊D₅₉)⋊C₄	1	2	yes	—
236	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
237	C₂³	C₄×Dic₁₁₈	1	1	yes	characteristic
238	C₂×C₄	—	2	4	no	—
239	C₂×C₄	—	4	4	no	—
240	C₂×C₄	—	236	8	no	—
241	C₂×C₄	—	236	4	no	—
242	C₂×C₄	#14273695	1	1	yes	characteristic
243	C₂×C₄	#14273701	1	1	yes	characteristic
244	C₂×C₄	#14273702	1	2	yes	—
245	C₂×C₄	#14273704	1	2	yes	—
246	C₂×C₄	#14273704	1	2	yes	—
247	C₂×C₄	#14273706	1	1	yes	characteristic
248	C₂×C₄	#14273708	1	1	yes	characteristic
249	C₂×C₄	#14273711	1	2	yes	—
250	C₂×C₄	#14273712	1	2	yes	—
251	C₂×C₄	#14273713	1	2	yes	—
252	C₂×C₄	#14273714	1	1	yes	characteristic
253	C₂×C₄	#14273716	1	1	yes	characteristic
254	C₂×C₄	#14273718	1	2	yes	—
255	C₂×C₄	#14273718	1	2	yes	—
256	C₂×C₄	#14273719	1	1	yes	characteristic
257	C₂×C₄	#14273720	1	2	yes	—
258	C₂×C₄	#14273722	1	2	yes	—
259	C₂×C₄	#14273727	1	1	yes	characteristic
260	C₈	—	4	2	no	—
261	C₈	—	4	2	no	—
262	C₈	—	236	4	no	—
263	C₂²	—	2	2	no	—
264	C₂²	—	2	4	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	#16837267	1	1	yes	characteristic
269	C₂²	#16837274	1	1	yes	characteristic
270	C₂²	#16837289	1	2	yes	—
271	C₂²	#16837294	1	2	yes	—
272	C₂²	#16837300	1	1	yes	characteristic
273	C₄	—	4	4	no	—
274	C₄	—	472	4	no	—
275	C₄	#16837314	1	1	yes	characteristic
276	C₄	#16837319	1	1	yes	characteristic
277	C₄	#16837327	1	1	yes	characteristic
278	C₄	#16837334	1	2	yes	—
279	C₄	#16837341	1	1	yes	characteristic
280	C₄	#16837344	1	2	yes	—
281	C₂	—	2	2	no	—
282	C₂	—	2	1	no	—
283	C₂	—	2	2	no	—
284	C₂	—	2	1	no	—
285	C₂	#21422352	1	1	yes	characteristic
286	C₂	#21424381	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14273695
commutator, prime radical	C₂⁴×C₅₉	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₅₉
Fitting	C₅₉×#1729	C₂
socle	C₂²×C₅₉	#385
cosocle	C₂³×C₄×C₅₉	C₂³
hypercenter	#1729	D₅₉
hypocenter	C₅₉	#11288
LCS	G ▷ C₂⁴×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1729

Complements

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

N	H ≅ G/N	type
160. C₅₉	#11288	semidirect

Maximal subgroups

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

subgroup	index
#21432660	2
C₂×#16837174	2
#21423829	2
#21423792	2
C₅₉×#1729	2
#11288	59