Subgroups — #28477106

Order 15104.

384 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21432478	C₂	1	1	yes	maximal, characteristic
2	#21424239	C₂	1	1	yes	maximal, characteristic
3	#21421921	C₂	1	2	yes	maximal
4	#21421918	C₂	1	2	yes	maximal
5	C₄×C₄⋊C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
6	#16837308	C₄	1	1	yes	characteristic
7	#16837308	C₄	1	1	yes	characteristic
8	#16837308	C₂²	1	1	yes	characteristic
9	#16837300	C₄	1	1	yes	characteristic
10	#16837300	C₄	1	1	yes	characteristic
11	#16837300	C₂²	1	1	yes	characteristic
12	#16837286	—	2	1	no	—
13	#16837286	—	2	1	no	—
14	#16837286	—	2	1	no	—
15	#16837270	—	2	1	no	—
16	#16837157	—	2	1	no	—
17	#16837157	—	2	1	no	—
18	#16837157	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
20	C₂×C₄×C₈×C₅₉	—	2	1	no	—
21	C₄³×C₅₉	C₂²	1	1	yes	characteristic
22	C₅₉×#336	C₂²	1	2	yes	—
23	C₂²×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₅₉⋊C₈	—	2	2	no	—
25	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄⋊Dic₅₉	C₂×C₄	1	1	yes	characteristic
28	C₂×Dic₅₉⋊C₄	—	2	1	no	—
29	C₂×Dic₅₉⋊C₄	—	2	1	no	—
30	C₂×Dic₅₉⋊C₄	—	2	1	no	—
31	C₅₉⋊#95	—	2	1	no	—
32	C₅₉⋊#95	—	2	1	no	—
33	C₅₉⋊#95	—	2	1	no	—
34	C₅₉⋊#95	—	2	1	no	—
35	C₅₉⋊#95	—	2	1	no	—
36	C₅₉⋊#95	—	2	1	no	—
37	C₅₉⋊#95	—	2	1	no	—
38	C₅₉⋊#95	—	2	1	no	—
39	C₅₉⋊#95	—	2	1	no	—
40	C₂²×C₈×C₅₉	—	2	2	no	—
41	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
48	C₄⋊C₈×C₅₉	—	2	4	no	—
49	C₄×C₈×C₅₉	—	4	2	no	—
50	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₉	—	2	1	no	—
58	C₂²×Dic₅₉	—	2	1	no	—
59	C₂²×Dic₅₉	—	2	1	no	—
60	C₂²×Dic₅₉	—	2	1	no	—
61	C₂×Dic₁₁₈	—	2	2	no	—
62	C₂×Dic₁₁₈	—	4	1	no	—
63	C₂×Dic₁₁₈	—	4	1	no	—
64	C₂×C₈×C₅₉	—	2	4	no	—
65	C₂×C₈×C₅₉	—	4	2	no	—
66	C₂×C₈×C₅₉	—	4	2	no	—
67	C₄²×C₅₉	—	2	2	no	—
68	C₄²×C₅₉	—	2	2	no	—
69	C₄²×C₅₉	—	2	2	no	—
70	C₄²×C₅₉	—	4	1	no	—
71	C₄²×C₅₉	—	4	1	no	—
72	C₄²×C₅₉	—	4	1	no	—
73	C₄²×C₅₉	SD₁₆	1	2	yes	—
74	C₄²×C₅₉	Q₁₆	1	2	yes	—
75	C₄⋊Dic₅₉	—	2	2	no	—
76	C₄⋊Dic₅₉	—	4	1	no	—
77	C₄⋊Dic₅₉	—	4	1	no	—
78	Dic₅₉⋊C₄	—	4	2	no	—
79	Dic₅₉⋊C₄	—	4	2	no	—
80	Dic₅₉⋊C₄	—	4	2	no	—
81	C₂×C₅₉⋊C₈	—	4	2	no	—
82	C₂×C₅₉⋊C₈	—	4	2	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₉	—	2	1	no	—
86	C₂×C₄×C₅₉	—	2	1	no	—
87	C₂×C₄×C₅₉	—	2	1	no	—
88	C₂×C₄×C₅₉	—	2	1	no	—
89	C₂×C₄×C₅₉	—	2	1	no	—
90	C₂×C₄×C₅₉	—	2	1	no	—
91	C₂×C₄×C₅₉	—	2	1	no	—
92	C₂×C₄×C₅₉	—	2	1	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	1	no	—
96	C₂×C₄×C₅₉	—	2	1	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	#103	1	2	yes	—
101	C₂×C₄×C₅₉	#103	1	2	yes	—
102	C₂×C₄×C₅₉	#104	1	2	yes	—
103	C₂×C₄×C₅₉	#131	1	2	yes	—
104	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#135	1	2	yes	—
107	C₂×Dic₅₉	—	4	1	no	—
108	C₂×Dic₅₉	—	4	1	no	—
109	C₂×Dic₅₉	—	4	1	no	—
110	C₂×Dic₅₉	—	4	1	no	—
111	C₂×Dic₅₉	—	4	1	no	—
112	C₂×Dic₅₉	—	4	1	no	—
113	C₂×Dic₅₉	—	4	1	no	—
114	C₂×Dic₅₉	—	4	1	no	—
115	C₂×Dic₅₉	—	4	1	no	—
116	C₂×Dic₅₉	—	4	1	no	—
117	C₂×Dic₅₉	—	4	1	no	—
118	C₂×Dic₅₉	—	4	1	no	—
119	Dic₁₁₈	—	8	2	no	—
120	C₈×C₅₉	—	4	4	no	—
121	C₅₉⋊C₈	—	8	2	no	—
122	#7673	—	59	1	no	maximal, Sylow
123	C₂²×C₅₉	#408	1	1	yes	characteristic
124	C₂²×C₅₉	#410	1	1	yes	characteristic
125	C₂²×C₅₉	#413	1	1	yes	characteristic
126	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
127	C₂²×C₅₉	#416	1	1	yes	characteristic
128	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
129	C₂²×C₅₉	#421	1	1	yes	characteristic
130	C₄×C₅₉	—	2	2	no	—
131	C₄×C₅₉	—	2	2	no	—
132	C₄×C₅₉	—	2	2	no	—
133	C₄×C₅₉	—	2	2	no	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	2	2	no	—
137	C₄×C₅₉	—	4	1	no	—
138	Dic₅₉	—	8	1	no	—
139	Dic₅₉	—	8	1	no	—
140	Dic₅₉	—	8	1	no	—
141	Dic₅₉	—	8	1	no	—
142	#2863	—	59	1	no	—
143	#1839	—	59	1	no	—
144	C₄×C₄⋊C₈	D₅₉	1	1	yes	characteristic
145	#1272	—	59	2	no	—
146	#1271	—	59	2	no	—
147	C₂×C₅₉	#1473	1	1	yes	characteristic
148	C₂×C₅₉	#1481	1	1	yes	characteristic
149	C₂×C₅₉	#1494	1	1	yes	characteristic
150	C₂×C₅₉	#1503	1	1	yes	characteristic
151	C₂×C₅₉	#1521	1	1	yes	characteristic
152	C₂×C₅₉	#1529	1	1	yes	characteristic
153	C₂×C₅₉	#1542	1	1	yes	characteristic
154	C₂×C₄⋊C₈	—	2	1	no	—
155	C₂×C₄×C₈	—	2	1	no	—
156	#391	—	59	1	no	—
157	#391	—	59	1	no	—
158	#391	—	59	1	no	—
159	#389	—	59	1	no	—
160	#389	—	59	1	no	—
161	#389	—	59	1	no	—
162	#387	—	118	1	no	—
163	#387	—	118	1	no	—
164	#387	—	118	1	no	—
165	#383	—	118	1	no	—
166	C₄³	C₂×D₅₉	1	1	yes	characteristic
167	#336	—	59	2	no	—
168	#336	—	118	1	no	—
169	#336	—	118	1	no	—
170	#336	C₂×D₅₉	1	2	yes	—
171	C₅₉	#7673	1	1	yes	characteristic, complemented, Sylow
172	C₂²×Q₈	—	59	1	no	—
173	C₂²×C₈	—	2	2	no	—
174	C₂²×C₈	—	118	2	no	—
175	C₂×C₄⋊C₄	—	59	1	no	—
176	C₂×C₄⋊C₄	—	59	1	no	—
177	C₂×C₄⋊C₄	—	59	1	no	—
178	C₂×C₄⋊C₄	—	118	1	no	—
179	C₂×C₄⋊C₄	—	118	1	no	—
180	C₂×C₄⋊C₄	—	118	1	no	—
181	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
182	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
183	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
184	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
185	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
186	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
188	C₄⋊C₈	—	2	4	no	—
189	C₄×C₈	—	4	2	no	—
190	#95	—	118	1	no	—
191	#95	—	118	1	no	—
192	#95	—	118	1	no	—
193	#95	—	118	1	no	—
194	#95	—	118	1	no	—
195	#95	—	118	1	no	—
196	#95	—	118	1	no	—
197	#95	—	118	1	no	—
198	#95	—	118	1	no	—
199	C₂×Q₈	—	118	2	no	—
200	C₂×Q₈	—	236	1	no	—
201	C₂×Q₈	—	236	1	no	—
202	C₂²×C₄	—	118	1	no	—
203	C₂²×C₄	—	118	1	no	—
204	C₂²×C₄	—	118	1	no	—
205	C₂²×C₄	—	118	1	no	—
206	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	4	2	no	—
215	C₂×C₈	—	4	2	no	—
216	C₂×C₈	—	236	2	no	—
217	C₂×C₈	—	236	2	no	—
218	C₂×C₈	—	236	2	no	—
219	C₄⋊C₄	—	118	2	no	—
220	C₄⋊C₄	—	236	1	no	—
221	C₄⋊C₄	—	236	1	no	—
222	C₄⋊C₄	—	236	2	no	—
223	C₄⋊C₄	—	236	2	no	—
224	C₄⋊C₄	—	236	2	no	—
225	C₄²	—	2	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	4	1	no	—
229	C₄²	—	4	1	no	—
230	C₄²	—	4	1	no	—
231	C₄²	#11745670	1	2	yes	—
232	C₄²	#11745672	1	2	yes	—
233	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
234	Q₈	—	472	2	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	1	no	—
238	C₂×C₄	—	2	1	no	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	236	1	no	—
251	C₂×C₄	—	236	1	no	—
252	C₂×C₄	—	236	1	no	—
253	C₂×C₄	—	236	1	no	—
254	C₂×C₄	—	236	1	no	—
255	C₂×C₄	—	236	1	no	—
256	C₂×C₄	—	236	1	no	—
257	C₂×C₄	—	236	1	no	—
258	C₂×C₄	—	236	1	no	—
259	C₂×C₄	—	236	1	no	—
260	C₂×C₄	—	236	1	no	—
261	C₂×C₄	—	236	1	no	—
262	C₂×C₄	#14273650	1	2	yes	—
263	C₂×C₄	#14273650	1	2	yes	—
264	C₂×C₄	#14273655	1	2	yes	—
265	C₂×C₄	#14273733	1	2	yes	—
266	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
267	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
268	C₂×C₄	#14273749	1	2	yes	—
269	C₈	—	4	4	no	—
270	C₈	—	472	2	no	—
271	C₂²	#16837368	1	1	yes	characteristic
272	C₂²	#16837375	1	1	yes	characteristic
273	C₂²	#16837384	1	1	yes	characteristic
274	C₂²	C₂×#14273650	1	1	yes	characteristic
275	C₂²	#16837399	1	1	yes	characteristic
276	C₂²	C₂×#14273655	1	1	yes	characteristic
277	C₂²	#16837418	1	1	yes	characteristic
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	2	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	2	2	no	—
285	C₄	—	4	1	no	—
286	C₄	—	472	1	no	—
287	C₄	—	472	1	no	—
288	C₄	—	472	1	no	—
289	C₄	—	472	1	no	—
290	C₂	#21422463	1	1	yes	characteristic
291	C₂	#21422498	1	1	yes	characteristic
292	C₂	#21422557	1	1	yes	characteristic
293	C₂	#21422594	1	1	yes	characteristic
294	C₂	#21422674	1	1	yes	characteristic
295	C₂	#21422701	1	1	yes	characteristic
296	C₂	#21422754	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×(C₂×D₅₉)⋊C₄
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊C₈	D₅₉
hypocenter	C₅₉	#7673
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
171. C₅₉	#7673	semidirect

Maximal subgroups

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

subgroup	index
#21432478	2
#21424239	2
#21421921	2
#21421918	2
C₄×C₄⋊C₈×C₅₉	2
#7673	59