Subgroups — #28481723

Order 15104.

404 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21422855	C₂	1	1	yes	maximal, characteristic, complemented
2	#21422852	C₂	1	1	yes	maximal, characteristic, complemented
3	#21422831	C₂	1	2	yes	maximal
4	C₅₉⋊#1309	C₂	1	2	yes	maximal, complemented
5	C₅₉×#2931	C₂	1	1	yes	maximal, characteristic
6	#16837426	C₂²	1	2	yes	—
7	#16837426	C₂²	1	2	yes	—
8	#16837155	—	2	1	no	—
9	#16837155	—	2	1	no	—
10	#16837153	—	2	1	no	—
11	#16837153	—	2	1	no	—
12	#16837151	—	2	2	no	—
13	#16837146	—	2	2	no	—
14	C₂×#131×C₅₉	C₄	1	2	yes	—
15	C₅₉×#532	C₂²	1	1	yes	characteristic
16	C₅₉×#436	—	2	1	no	—
17	C₅₉×#436	—	2	1	no	—
18	C₈×D₄×C₅₉	—	2	1	no	—
19	C₈×D₄×C₅₉	—	2	1	no	—
20	C₅₉×#423	C₂²	1	1	yes	characteristic
21	C₅₉×#408	C₄	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×#11745664	—	2	2	no	—
24	C₂×#11745664	—	2	2	no	—
25	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
28	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
29	C₂×#41×C₅₉	C₂×C₄	1	2	yes	—
30	#131×C₅₉	—	4	4	no	—
31	C₂×M₁₆×C₅₉	—	2	2	no	—
32	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
33	C₂²×C₈×C₅₉	—	2	2	no	—
34	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Q₈×C₅₉	D₄	1	1	yes	characteristic
37	#127×C₅₉	D₄	1	1	yes	characteristic
38	#124×C₅₉	Q₈	1	2	yes	—
39	#121×C₅₉	—	2	1	no	—
40	#121×C₅₉	—	2	1	no	—
41	C₄×D₄×C₅₉	—	2	1	no	—
42	C₄×D₄×C₅₉	—	2	1	no	—
43	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
44	C₄⋊C₈×C₅₉	—	2	1	no	—
45	C₄⋊C₈×C₅₉	—	2	1	no	—
46	C₂²⋊C₈×C₅₉	—	2	2	no	—
47	C₂²⋊C₈×C₅₉	—	2	2	no	—
48	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
51	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
52	#41×C₅₉	—	4	2	no	—
53	#41×C₅₉	—	4	2	no	—
54	C₂×Q₈×C₅₉	C₄⋊C₄	1	2	yes	—
55	C₂×D₄×C₅₉	—	2	2	no	—
56	C₂×D₄×C₅₉	—	2	1	no	—
57	C₂×D₄×C₅₉	—	2	1	no	—
58	C₂×D₄×C₅₉	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₉	—	2	2	no	—
60	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
63	M₁₆×C₅₉	—	4	4	no	—
64	M₁₆×C₅₉	—	4	2	no	—
65	C₂×C₈×C₅₉	—	2	2	no	—
66	C₂×C₈×C₅₉	—	2	2	no	—
67	C₂×C₈×C₅₉	—	4	1	no	—
68	C₂×C₈×C₅₉	—	4	4	no	—
69	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	—	2	1	no	—
75	C₂²⋊C₄×C₅₉	—	2	2	no	—
76	C₂²⋊C₄×C₅₉	—	2	2	no	—
77	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
78	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
81	#11745664	—	8	2	no	—
82	#11745664	—	8	2	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂×C₅₉⋊C₈	—	4	2	no	—
85	C₂³×C₅₉	—	2	2	no	—
86	C₂³×C₅₉	#117	1	1	yes	characteristic
87	Q₈×C₅₉	—	4	2	no	—
88	D₄×C₅₉	—	4	2	no	—
89	D₄×C₅₉	—	4	4	no	—
90	D₄×C₅₉	—	4	2	no	—
91	D₄×C₅₉	—	4	1	no	—
92	D₄×C₅₉	—	4	1	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	2	no	—
96	C₂×C₄×C₅₉	—	2	1	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	—	4	2	no	—
100	C₂×C₄×C₅₉	—	4	2	no	—
101	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
104	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
107	C₈×C₅₉	—	2	1	no	—
108	C₈×C₅₉	—	2	1	no	—
109	C₈×C₅₉	—	2	1	no	—
110	C₈×C₅₉	—	4	2	no	—
111	C₈×C₅₉	#100	1	2	yes	—
112	C₅₉⋊C₈	—	8	2	no	—
113	C₅₉⋊C₈	—	8	2	no	—
114	#8640	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	—	2	1	no	—
116	C₂²×C₅₉	—	2	1	no	—
117	C₂²×C₅₉	—	2	1	no	—
118	C₂²×C₅₉	—	4	2	no	—
119	C₂²×C₅₉	—	4	2	no	—
120	C₂²×C₅₉	—	4	2	no	—
121	C₂²×C₅₉	#380	1	1	yes	characteristic
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	—	2	1	no	—
124	C₄×C₅₉	—	2	1	no	—
125	C₄×C₅₉	—	4	2	no	—
126	C₄×C₅₉	C₂×#100	1	1	yes	characteristic
127	C₄×C₅₉	#413	1	1	yes	characteristic
128	C₄×C₅₉	#471	1	2	yes	—
129	C₄×C₅₉	#472	1	1	yes	characteristic
130	C₄×C₅₉	#473	1	1	yes	characteristic
131	#2931	D₅₉	1	1	yes	characteristic
132	#1567	—	59	1	no	—
133	#1566	—	59	1	no	—
134	#1560	—	59	2	no	—
135	#1309	—	59	2	no	—
136	C₂×C₅₉	—	4	1	no	—
137	C₂×C₅₉	—	8	2	no	—
138	C₂×C₅₉	#1800	1	1	yes	characteristic
139	C₂×C₅₉	#1815	1	1	yes	characteristic
140	C₂×C₅₉	#1816	1	1	yes	characteristic
141	C₂×#131	Dic₅₉	1	2	yes	—
142	#532	C₂×D₅₉	1	1	yes	characteristic
143	#436	—	2	1	no	—
144	#436	—	2	1	no	—
145	C₈×D₄	—	2	1	no	—
146	C₈×D₄	—	2	1	no	—
147	#423	—	59	2	no	—
148	#423	—	59	2	no	—
149	#423	C₂×D₅₉	1	1	yes	characteristic
150	#408	Dic₅₉	1	2	yes	—
151	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
152	#335	—	118	1	no	—
153	#335	—	118	1	no	—
154	#334	—	118	1	no	—
155	#334	—	118	1	no	—
156	#333	—	118	2	no	—
157	#331	—	118	2	no	—
158	C₅₉	#8640	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	C₂×Dic₅₉	1	2	yes	—
160	#131	—	4	4	no	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂×M₁₆	—	118	2	no	—
163	C₂×M₁₆	—	118	2	no	—
164	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
165	C₂²×C₈	—	2	2	no	—
166	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
167	C₂²×C₈	C₂×Dic₅₉	1	1	yes	characteristic
168	C₄⋊Q₈	D₂₃₆	1	1	yes	characteristic
169	#127	D₂₃₆	1	1	yes	characteristic
170	#124	Dic₁₁₈	1	2	yes	—
171	#121	—	2	1	no	—
172	#121	—	2	1	no	—
173	C₄×D₄	—	2	1	no	—
174	C₄×D₄	—	2	1	no	—
175	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
176	C₄⋊C₈	—	2	1	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	—	118	2	no	—
179	C₄⋊C₈	—	118	2	no	—
180	C₄⋊C₈	—	118	2	no	—
181	C₄⋊C₈	—	118	2	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₂²⋊C₈	—	2	2	no	—
184	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
185	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
186	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
187	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
188	#41	—	4	2	no	—
189	#41	—	4	2	no	—
190	C₂×Q₈	C₄⋊Dic₅₉	1	2	yes	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	1	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	C₄⋊Dic₅₉	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	#11745685	1	1	yes	characteristic
197	C₂²×C₄	#11745685	1	1	yes	characteristic
198	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
199	M₁₆	—	4	4	no	—
200	M₁₆	—	4	2	no	—
201	M₁₆	—	472	2	no	—
202	M₁₆	—	472	2	no	—
203	C₂×C₈	—	2	2	no	—
204	C₂×C₈	—	2	2	no	—
205	C₂×C₈	—	4	1	no	—
206	C₂×C₈	—	4	4	no	—
207	C₂×C₈	—	236	2	no	—
208	C₂×C₈	—	236	2	no	—
209	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
210	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
211	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
212	C₂×C₈	C₂²⋊Dic₅₉	1	1	yes	characteristic
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	—	2	1	no	—
215	C₂²⋊C₄	—	2	2	no	—
216	C₂²⋊C₄	—	2	2	no	—
217	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
218	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
219	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
220	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
221	C₂³	—	2	2	no	—
222	C₂³	#14273686	1	1	yes	characteristic
223	Q₈	—	4	2	no	—
224	D₄	—	4	2	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	2	no	—
227	D₄	—	4	1	no	—
228	D₄	—	4	1	no	—
229	C₂×C₄	—	2	1	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	1	no	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
238	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
239	C₂×C₄	#14273701	1	1	yes	characteristic
240	C₂×C₄	#14273701	1	1	yes	characteristic
241	C₂×C₄	#14273706	1	1	yes	characteristic
242	C₂×C₄	#14273706	1	1	yes	characteristic
243	C₈	—	2	1	no	—
244	C₈	—	2	1	no	—
245	C₈	—	2	1	no	—
246	C₈	—	4	2	no	—
247	C₈	—	472	2	no	—
248	C₈	—	472	2	no	—
249	C₈	#14273644	1	2	yes	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	#16837262	1	1	yes	characteristic
257	C₄	—	2	1	no	—
258	C₄	—	2	1	no	—
259	C₄	—	2	1	no	—
260	C₄	—	4	2	no	—
261	C₄	C₂×#14273644	1	1	yes	characteristic
262	C₄	#16837386	1	1	yes	characteristic
263	C₄	#16837665	1	2	yes	—
264	C₄	#16837672	1	1	yes	characteristic
265	C₄	#16837677	1	1	yes	characteristic
266	C₂	—	4	1	no	—
267	C₂	—	8	2	no	—
268	C₂	#21424088	1	1	yes	characteristic
269	C₂	#21424155	1	1	yes	characteristic
270	C₂	#21424160	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#16837262
commutator	C₂²×C₄×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#2931	C₂
socle	C₂²×C₅₉	#380
prime radical	C₅₉×#532	C₂²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#2931	D₅₉
hypocenter	C₅₉	#8640
LCS	G ▷ C₂²×C₄×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2931

Complements

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

N	H ≅ G/N	type
1. #21422855	C₂	semidirect
2. #21422852	C₂	semidirect
4. C₅₉⋊#1309	C₂	semidirect
158. C₅₉	#8640	semidirect

Maximal subgroups

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

subgroup	index
#21422855	2
#21422852	2
#21422831	2
C₅₉⋊#1309	2
C₅₉×#2931	2
#8640	59