Subgroups — #28479515

Order 15104.

478 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837529	C₂	1	1	yes	maximal, characteristic
2	#21423965	C₂	1	1	yes	maximal, characteristic
3	C₄×#14273650	C₂	1	1	yes	maximal, characteristic
4	C₂×#16837134	C₂	1	1	yes	maximal, characteristic
5	C₂×C₈×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
6	#21421930	C₂	1	1	yes	maximal, characteristic
7	C₅₉×#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
9	#16837529	—	2	2	no	—
10	#16837529	C₄	1	4	yes	complemented
11	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×Dic₅₉⋊C₈	—	2	1	no	—
13	C₂×#14273650	C₄	1	2	yes	—
14	C₂×#14273638	—	2	1	no	—
15	C₂×#14273637	—	2	1	no	—
16	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
18	#16837164	C₄	1	2	yes	—
19	#16837157	—	2	1	no	—
20	#16837134	C₄	1	4	yes	complemented
21	C₈×C₅₉⋊C₈	—	2	2	no	—
22	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₅₉⋊C₈	—	2	1	no	—
27	C₂²×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
29	C₂²×C₅₉⋊C₈	D₄	1	1	yes	characteristic
30	C₄×Dic₁₁₈	—	2	1	no	—
31	C₄×Dic₁₁₈	—	2	1	no	—
32	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
33	C₄×Dic₁₁₈	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂×Dic₅₉⋊C₄	—	2	1	no	—
36	C₂×C₄×Dic₅₉	—	2	1	no	—
37	C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
38	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
39	Dic₅₉⋊C₈	—	2	4	no	—
40	#14273650	—	2	4	no	—
41	#14273638	—	4	2	no	—
42	#14273637	—	2	4	no	—
43	C₄×C₅₉⋊C₈	—	2	1	no	—
44	C₄×C₅₉⋊C₈	—	2	1	no	—
45	C₄×C₅₉⋊C₈	C₂×C₄	1	2	yes	—
46	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
47	C₂²×C₈×C₅₉	—	2	1	no	—
48	C₂²×C₈×C₅₉	—	2	1	no	—
49	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₅₉	—	2	1	no	—
51	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
52	C₄⋊C₈×C₅₉	—	2	1	no	—
53	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₅₉	—	2	1	no	—
55	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
56	#95×C₅₉	—	2	1	no	—
57	C₂²×C₄×C₅₉	—	2	2	no	—
58	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
61	C₂²×Dic₅₉	—	2	1	no	—
62	C₂²×Dic₅₉	—	2	1	no	—
63	C₂×Dic₁₁₈	—	2	2	no	—
64	C₂×Dic₁₁₈	—	2	2	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	C₄²	1	2	yes	—
67	C₂×C₈×C₅₉	—	2	4	no	—
68	C₂×C₈×C₅₉	—	2	2	no	—
69	C₂×C₈×C₅₉	—	2	2	no	—
70	C₂×C₈×C₅₉	—	4	1	no	—
71	C₂×C₈×C₅₉	—	4	1	no	—
72	C₄⋊C₄×C₅₉	—	2	2	no	—
73	C₄⋊C₄×C₅₉	—	2	2	no	—
74	C₄²×C₅₉	—	4	1	no	—
75	C₄²×C₅₉	—	4	1	no	—
76	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₉	#41	1	1	yes	characteristic
80	C₄⋊Dic₅₉	—	2	1	no	—
81	C₄⋊Dic₅₉	C₄²	1	2	yes	—
82	Dic₅₉⋊C₄	—	2	2	no	—
83	Dic₅₉⋊C₄	—	2	2	no	—
84	C₄×Dic₅₉	—	2	2	no	—
85	C₄×Dic₅₉	—	2	2	no	—
86	C₂×C₅₉⋊C₈	—	2	2	no	—
87	C₂×C₅₉⋊C₈	—	2	2	no	—
88	C₂×C₅₉⋊C₈	—	2	1	no	—
89	C₂×C₅₉⋊C₈	—	2	1	no	—
90	C₂×C₅₉⋊C₈	—	2	2	no	—
91	C₂×C₅₉⋊C₈	—	4	1	no	—
92	C₂×C₅₉⋊C₈	—	4	1	no	—
93	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
94	C₂×C₅₉⋊C₈	C₂²⋊C₄	1	2	yes	—
95	C₂×C₅₉⋊C₈	M₁₆	1	4	yes	—
96	C₂×C₅₉⋊C₈	SD₁₆	1	2	yes	—
97	C₂×C₅₉⋊C₈	Q₁₆	1	2	yes	—
98	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	—	2	2	no	—
100	C₂×C₄×C₅₉	—	4	2	no	—
101	C₂×C₄×C₅₉	—	4	2	no	—
102	C₂×C₄×C₅₉	—	4	2	no	—
103	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×M₁₆	1	2	yes	—
110	C₂×C₄×C₅₉	#131	1	2	yes	—
111	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
114	C₂×C₄×C₅₉	#135	1	1	yes	characteristic
115	C₂×Dic₅₉	—	2	2	no	—
116	C₂×Dic₅₉	—	2	2	no	—
117	C₂×Dic₅₉	—	4	2	no	—
118	C₂×Dic₅₉	—	4	2	no	—
119	Dic₁₁₈	—	2	2	no	—
120	Dic₁₁₈	—	2	2	no	—
121	Dic₁₁₈	—	4	2	no	—
122	C₈×C₅₉	—	2	4	no	—
123	C₈×C₅₉	—	4	2	no	—
124	C₅₉⋊C₈	—	2	2	no	—
125	C₅₉⋊C₈	—	2	4	no	—
126	C₅₉⋊C₈	—	4	2	no	—
127	C₅₉⋊C₈	#103	1	4	yes	complemented
128	#8207	—	59	1	no	maximal, Sylow
129	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
130	C₂²×C₅₉	#436	1	2	yes	—
131	C₂²×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
132	C₂²×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
133	C₂²×C₅₉	#443	1	2	yes	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	8	2	no	—
137	C₄×C₅₉	C₄×M₁₆	1	1	yes	characteristic
138	C₄×C₅₉	#405	1	1	yes	characteristic
139	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
140	C₄×C₅₉	#416	1	1	yes	characteristic
141	C₄×C₅₉	#435	1	2	yes	—
142	C₄×C₅₉	C₄×SD₁₆	1	1	yes	characteristic
143	C₄×C₅₉	C₄×Q₁₆	1	1	yes	characteristic
144	Dic₅₉	—	4	2	no	—
145	Dic₅₉	—	4	2	no	—
146	C₂×#446	—	59	1	no	—
147	#1770	—	59	1	no	—
148	C₄×#103	—	59	1	no	—
149	C₂×Q₈⋊C₈	—	59	1	no	—
150	C₂×C₈²	—	59	1	no	—
151	#1279	D₅₉	1	1	yes	characteristic
152	#1277	—	59	1	no	—
153	C₂×C₅₉	#1590	1	2	yes	—
154	C₂×C₅₉	#1591	1	2	yes	—
155	C₂×C₅₉	#1753	1	1	yes	characteristic
156	C₂×C₅₉	C₄×#103	1	1	yes	characteristic
157	C₂×C₅₉	#1764	1	1	yes	characteristic
158	C₂×C₄×Q₈	—	59	1	no	—
159	#446	—	59	4	no	—
160	#446	—	118	2	no	—
161	C₂×C₄⋊C₈	—	59	1	no	—
162	C₂×C₄⋊C₈	—	118	1	no	—
163	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
164	C₂×#103	—	59	2	no	—
165	C₂×#97	—	118	1	no	—
166	C₂×#97	—	118	1	no	—
167	C₂×C₄×C₈	—	59	1	no	—
168	C₂×C₄×C₈	—	59	1	no	—
169	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
170	C₄×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
171	#340	—	59	2	no	—
172	#336	—	118	1	no	—
173	Q₈⋊C₈	—	59	4	no	—
174	C₈²	—	118	2	no	—
175	C₅₉	#8207	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	59	1	no	—
177	C₂²×C₈	—	2	1	no	—
178	C₂²×C₈	—	2	1	no	—
179	C₂²×C₈	—	59	1	no	—
180	C₂²×C₈	—	59	2	no	—
181	C₂²×C₈	—	59	1	no	—
182	C₂²×C₈	—	118	1	no	—
183	C₄×Q₈	—	59	2	no	—
184	C₄×Q₈	—	59	2	no	—
185	C₄×Q₈	—	118	1	no	—
186	C₄×Q₈	—	118	1	no	—
187	C₂×C₄⋊C₄	—	59	1	no	—
188	C₂×C₄⋊C₄	—	118	1	no	—
189	C₂×C₄⋊C₄	C₄×D₅₉	1	2	yes	—
190	C₂×C₄²	—	2	1	no	—
191	C₂×C₄²	—	118	1	no	—
192	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
193	C₄⋊C₈	—	2	1	no	—
194	C₄⋊C₈	—	59	2	no	—
195	C₄⋊C₈	—	118	4	no	—
196	C₄⋊C₈	—	118	1	no	—
197	C₄⋊C₈	C₄×D₅₉	1	2	yes	—
198	#103	—	118	4	no	—
199	#97	—	118	4	no	—
200	#97	—	236	2	no	—
201	C₄×C₈	—	2	1	no	—
202	C₄×C₈	—	59	2	no	—
203	C₄×C₈	—	59	2	no	—
204	C₄×C₈	—	118	1	no	—
205	C₄×C₈	—	118	1	no	—
206	C₄×C₈	C₄×D₅₉	1	2	yes	—
207	#95	—	2	1	no	—
208	C₂×Q₈	—	59	2	no	—
209	C₂×Q₈	—	118	1	no	—
210	C₂×Q₈	—	118	2	no	—
211	C₂×Q₈	—	118	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	118	1	no	—
214	C₂²×C₄	—	118	1	no	—
215	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
216	C₂²×C₄	D₄×D₅₉	1	1	yes	characteristic
217	C₂²×C₄	#11745687	1	1	yes	characteristic
218	C₂×C₈	—	2	4	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	4	1	no	—
222	C₂×C₈	—	4	1	no	—
223	C₂×C₈	—	59	2	no	—
224	C₂×C₈	—	59	2	no	—
225	C₂×C₈	—	59	2	no	—
226	C₂×C₈	—	59	2	no	—
227	C₂×C₈	—	59	4	no	—
228	C₂×C₈	—	118	2	no	—
229	C₂×C₈	—	118	2	no	—
230	C₂×C₈	—	118	1	no	—
231	C₂×C₈	—	118	1	no	—
232	C₂×C₈	—	118	2	no	—
233	C₂×C₈	—	236	1	no	—
234	C₂×C₈	—	236	1	no	—
235	C₄⋊C₄	—	2	2	no	—
236	C₄⋊C₄	—	2	2	no	—
237	C₄⋊C₄	—	59	2	no	—
238	C₄⋊C₄	—	118	2	no	—
239	C₄⋊C₄	—	118	2	no	—
240	C₄⋊C₄	—	118	1	no	—
241	C₄²	—	4	1	no	—
242	C₄²	—	4	1	no	—
243	C₄²	—	118	2	no	—
244	C₄²	—	118	2	no	—
245	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
246	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
247	C₄²	D₄×D₅₉	1	1	yes	characteristic
248	C₄²	#11745689	1	1	yes	characteristic
249	C₂³	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
250	Q₈	—	118	2	no	—
251	Q₈	—	118	2	no	—
252	Q₈	—	236	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	2	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	2	no	—
257	C₂×C₄	—	118	2	no	—
258	C₂×C₄	—	118	2	no	—
259	C₂×C₄	—	236	2	no	—
260	C₂×C₄	—	236	2	no	—
261	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
262	C₂×C₄	C₂²⋊C₄×D₅₉	1	1	yes	characteristic
263	C₂×C₄	#14273685	1	1	yes	characteristic
264	C₂×C₄	(C₂²⋊D₅₉)⋊C₄	1	1	yes	characteristic
265	C₂×C₄	#14273689	1	1	yes	characteristic
266	C₂×C₄	#14273689	1	1	yes	characteristic
267	C₂×C₄	M₁₆×D₅₉	1	2	yes	—
268	C₂×C₄	#14273733	1	2	yes	—
269	C₂×C₄	SD₁₆×D₅₉	1	1	yes	characteristic
270	C₂×C₄	Q₁₆×D₅₉	1	1	yes	characteristic
271	C₂×C₄	#14273746	1	1	yes	characteristic
272	C₂×C₄	#14273748	1	1	yes	characteristic
273	C₈	—	2	4	no	—
274	C₈	—	4	2	no	—
275	C₈	—	59	4	no	—
276	C₈	—	118	2	no	—
277	C₈	—	118	4	no	—
278	C₈	—	236	2	no	—
279	C₂²	#16837249	1	1	yes	characteristic
280	C₂²	#16837477	1	2	yes	—
281	C₂²	#16837493	1	1	yes	characteristic
282	C₂²	#16837494	1	1	yes	characteristic
283	C₂²	#16837516	1	2	yes	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	8	2	no	—
287	C₄	—	236	2	no	—
288	C₄	—	236	2	no	—
289	C₄	#16837354	1	1	yes	characteristic
290	C₄	#16837356	1	1	yes	characteristic
291	C₄	#103×D₅₉	1	1	yes	characteristic
292	C₄	#16837395	1	1	yes	characteristic
293	C₄	#16837472	1	2	yes	—
294	C₄	#16837489	1	1	yes	characteristic
295	C₄	#16837496	1	1	yes	characteristic
296	C₂	#21422989	1	2	yes	—
297	C₂	#21422990	1	2	yes	—
298	C₂	#21423878	1	1	yes	characteristic
299	C₂	#21423915	1	1	yes	characteristic
300	C₂	#21423930	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1279	C₂
socle, prime radical	C₂³×C₅₉	C₄×D₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1279	D₅₉
hypocenter	C₅₉	#8207
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1279

Complements

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

N	H ≅ G/N	type
10. #16837529	C₄	semidirect
20. #16837134	C₄	semidirect
127. C₅₉⋊C₈	#103	semidirect
175. C₅₉	#8207	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837529	2
#21423965	2
C₄×#14273650	2
C₂×#16837134	2
C₂×C₈×C₅₉⋊C₈	2
#21421930	2
C₅₉×#1279	2
#8207	59