Subgroups — #28597304

Order 15104.

1046 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837519	C₂	1	4	yes	maximal, complemented
2	#21432808	C₂	1	2	yes	maximal, complemented
3	#21424985	C₂	1	4	yes	maximal, complemented
4	#21424925	C₂	1	2	yes	maximal, complemented
5	#21424268	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×C₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#3147	C₂	1	1	yes	maximal, characteristic
8	C₂×#14273734	C₂²	1	4	yes	—
9	#16837519	—	2	16	no	—
10	#16837480	—	2	4	no	—
11	D₄×C₅₉⋊C₈	—	2	4	no	—
12	C₂×#14273663	C₂²	1	4	yes	—
13	#16837426	C₂²	1	2	yes	—
14	C₂×#14273654	C₂²	1	8	yes	—
15	#16837369	C₂²	1	4	yes	—
16	C₂×C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
18	C₂×C₈×Dic₅₉	C₂²	1	2	yes	—
19	C₄²×Dic₅₉	C₂²	1	1	yes	characteristic
20	C₂×#135×C₅₉	C₂²	1	4	yes	—
21	C₅₉×#532	C₂²	1	2	yes	—
22	C₅₉×#495	C₄	1	4	yes	complemented
23	C₅₉×#494	C₄	1	1	yes	characteristic, complemented
24	C₅₉×#493	C₄	1	1	yes	characteristic, complemented
25	C₅₉×#492	C₄	1	2	yes	complemented
26	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
27	#14273734	—	2	16	no	—
28	C₂×#11745664	—	2	4	no	—
29	C₂×#11745664	C₂³	1	4	yes	—
30	C₂²×C₅₉⋊C₈	—	2	4	no	—
31	C₂²×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
32	C₂²×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
33	C₂×C₄×Dic₅₉	—	2	2	no	—
34	C₂×C₄×Dic₅₉	C₂³	1	2	yes	—
35	#14273663	—	2	8	no	—
36	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
37	#14273654	—	2	16	no	—
38	C₂²⋊(C₅₉⋊C₈)	—	2	8	no	—
39	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
40	C₄×C₅₉⋊C₈	D₄	1	2	yes	—
41	C₈×Dic₅₉	—	2	4	no	—
42	C₈×Dic₅₉	D₄	1	8	yes	complemented
43	C₂×#41×C₅₉	C₂³	1	4	yes	—
44	#135×C₅₉	—	2	8	no	—
45	#135×C₅₉	—	2	8	no	—
46	C₂×Q₁₆×C₅₉	C₂×C₄	1	4	yes	—
47	C₂×SD₁₆×C₅₉	C₂×C₄	1	8	yes	—
48	C₂×D₈×C₅₉	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₅₉	C₂³	1	2	yes	—
50	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
51	#127×C₅₉	C₂×C₄	1	2	yes	—
52	#124×C₅₉	C₂×C₄	1	4	yes	—
53	#121×C₅₉	—	2	4	no	—
54	C₄×D₄×C₅₉	—	2	4	no	—
55	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
56	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
59	#41×C₅₉	—	2	8	no	—
60	#41×C₅₉	—	2	8	no	—
61	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
62	C₂×D₄×C₅₉	—	2	4	no	—
63	C₂×D₄×C₅₉	—	2	4	no	—
64	C₂×D₄×C₅₉	C₂²×C₄	1	4	yes	—
65	C₂²×C₄×C₅₉	—	2	4	no	—
66	C₂²×C₄×C₅₉	#41	1	2	yes	—
67	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
68	C₂²×Dic₅₉	—	2	2	no	—
69	Q₁₆×C₅₉	—	2	8	no	—
70	SD₁₆×C₅₉	—	2	16	no	—
71	D₈×C₅₉	—	2	8	no	—
72	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
74	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
75	C₂×C₈×C₅₉	#41	1	4	yes	—
76	C₄⋊C₄×C₅₉	—	2	4	no	—
77	C₂²⋊C₄×C₅₉	—	2	8	no	—
78	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₉	C₂×D₄	1	2	yes	—
81	C₄×Dic₅₉	—	2	8	no	—
82	C₄×Dic₅₉	—	2	2	no	—
83	C₄×Dic₅₉	C₂×D₄	1	4	yes	—
84	#11745664	—	2	8	no	—
85	#11745664	—	4	8	no	—
86	C₂×C₅₉⋊C₈	—	2	2	no	—
87	C₂×C₅₉⋊C₈	—	2	4	no	—
88	C₂×C₅₉⋊C₈	—	4	8	no	—
89	C₂×C₅₉⋊C₈	C₂×D₄	1	2	yes	—
90	C₂×C₅₉⋊C₈	C₂×D₄	1	4	yes	—
91	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₅₉⋊C₈	C₂×D₄	1	1	yes	characteristic
93	C₂³×C₅₉	—	2	4	no	—
94	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
95	Q₈×C₅₉	—	2	8	no	—
96	D₄×C₅₉	—	2	8	no	—
97	D₄×C₅₉	—	4	8	no	—
98	D₄×C₅₉	—	4	8	no	—
99	C₂×C₄×C₅₉	—	2	4	no	—
100	C₂×C₄×C₅₉	—	2	4	no	—
101	C₂×C₄×C₅₉	—	4	4	no	—
102	C₂×C₄×C₅₉	—	4	4	no	—
103	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
104	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂²×D₄	1	2	yes	—
107	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×#41	1	2	yes	—
109	C₂×Dic₅₉	—	2	8	no	—
110	C₂×Dic₅₉	—	2	4	no	—
111	C₈×C₅₉	C₄×D₄	1	4	yes	—
112	C₈×C₅₉	C₄×D₄	1	4	yes	—
113	C₅₉⋊C₈	—	2	2	no	—
114	C₅₉⋊C₈	—	4	4	no	—
115	C₅₉⋊C₈	#127	1	2	yes	—
116	C₅₉⋊C₈	#127	1	2	yes	—
117	#21705	—	59	1	no	maximal, Sylow
118	C₂²×C₅₉	—	2	2	no	—
119	C₂²×C₅₉	—	4	4	no	—
120	C₂²×C₅₉	—	4	8	no	—
121	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
122	C₂²×C₅₉	#532	1	2	yes	—
123	C₄×C₅₉	—	2	2	no	—
124	C₄×C₅₉	—	4	4	no	—
125	C₄×C₅₉	#443	1	4	yes	complemented
126	C₄×C₅₉	C₂×C₄×D₄	1	2	yes	—
127	C₄×C₅₉	C₂×#127	1	1	yes	characteristic
128	C₄×C₅₉	#532	1	1	yes	characteristic
129	Dic₅₉	—	2	8	no	—
130	#3147	D₅₉	1	1	yes	characteristic
131	C₂×#443	—	59	4	no	—
132	#2931	—	59	2	no	—
133	#1974	—	59	4	no	—
134	#1962	—	59	2	no	—
135	#1848	—	59	1	no	—
136	C₄²×C₈	—	59	1	no	—
137	C₂×C₅₉	—	2	2	no	—
138	C₂×C₅₉	—	8	4	no	—
139	C₂×C₅₉	C₄×#127	1	1	yes	characteristic
140	C₂×C₅₉	C₂×#443	1	2	yes	—
141	C₂×#135	C₂×D₅₉	1	4	yes	—
142	C₂×#131	—	59	4	no	—
143	#532	C₂×D₅₉	1	2	yes	—
144	#495	Dic₅₉	1	4	yes	complemented
145	#494	Dic₅₉	1	1	yes	characteristic, complemented
146	#493	Dic₅₉	1	1	yes	characteristic, complemented
147	#492	Dic₅₉	1	2	yes	complemented
148	#443	—	118	16	no	—
149	#436	—	118	4	no	—
150	C₈×D₄	—	118	4	no	—
151	C₂×#108	—	59	4	no	—
152	#423	—	59	2	no	—
153	C₂×#104	—	59	8	no	—
154	#408	—	59	4	no	—
155	C₂×C₄×C₈	—	59	2	no	—
156	C₂×C₄×C₈	—	59	1	no	—
157	C₂×C₄×C₈	—	59	2	no	—
158	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
159	C₄³	—	59	1	no	—
160	C₅₉	#21705	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	C₂²×D₅₉	1	4	yes	—
162	#135	—	2	8	no	—
163	#135	—	2	8	no	—
164	C₂×Q₁₆	C₂×Dic₅₉	1	4	yes	—
165	C₂×SD₁₆	C₂×Dic₅₉	1	8	yes	—
166	C₂×D₈	C₂×Dic₅₉	1	4	yes	—
167	#131	—	118	16	no	—
168	C₂×M₁₆	—	59	4	no	—
169	C₂×M₁₆	—	118	4	no	—
170	C₂²×C₈	—	59	1	no	—
171	C₂²×C₈	—	59	1	no	—
172	C₂²×C₈	—	118	4	no	—
173	C₂²×C₈	C₂²×D₅₉	1	2	yes	—
174	C₄⋊Q₈	C₂×Dic₅₉	1	2	yes	—
175	#127	C₂×Dic₅₉	1	2	yes	—
176	#124	C₂×Dic₅₉	1	4	yes	—
177	#121	—	2	4	no	—
178	C₄×D₄	—	2	4	no	—
179	C₂×C₄²	—	59	2	no	—
180	C₂×C₄²	—	118	2	no	—
181	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
182	#108	—	118	8	no	—
183	C₄⋊C₈	—	118	4	no	—
184	#104	—	118	16	no	—
185	C₂²⋊C₈	—	118	8	no	—
186	C₄×C₈	—	59	8	no	—
187	C₄×C₈	—	59	2	no	—
188	C₄×C₈	—	59	2	no	—
189	C₄×C₈	—	118	4	no	—
190	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
191	C₄×C₈	C₂×Dic₅₉	1	1	yes	characteristic
192	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
193	#41	—	2	8	no	—
194	#41	—	2	8	no	—
195	C₂×Q₈	C₂²×Dic₅₉	1	4	yes	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	C₂²×Dic₅₉	1	4	yes	—
199	C₂²×C₄	—	2	4	no	—
200	C₂²×C₄	—	118	2	no	—
201	C₂²×C₄	#11745687	1	2	yes	—
202	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
203	Q₁₆	—	2	8	no	—
204	SD₁₆	—	2	16	no	—
205	D₈	—	2	8	no	—
206	M₁₆	—	118	8	no	—
207	M₁₆	—	236	8	no	—
208	C₂×C₈	—	59	4	no	—
209	C₂×C₈	—	59	1	no	—
210	C₂×C₈	—	59	1	no	—
211	C₂×C₈	—	59	2	no	—
212	C₂×C₈	—	118	2	no	—
213	C₂×C₈	—	118	4	no	—
214	C₂×C₈	—	236	8	no	—
215	C₂×C₈	D₄×D₅₉	1	4	yes	—
216	C₂×C₈	#11745687	1	4	yes	—
217	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
218	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
219	C₄⋊C₄	—	2	4	no	—
220	C₂²⋊C₄	—	2	8	no	—
221	C₄²	—	59	4	no	—
222	C₄²	—	118	8	no	—
223	C₄²	—	118	2	no	—
224	C₄²	D₄×D₅₉	1	2	yes	—
225	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
226	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
227	C₂³	—	2	4	no	—
228	C₂³	C₂×#11745687	1	1	yes	characteristic
229	Q₈	—	2	8	no	—
230	D₄	—	2	8	no	—
231	D₄	—	4	8	no	—
232	D₄	—	4	8	no	—
233	C₂×C₄	—	2	4	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	118	8	no	—
238	C₂×C₄	—	118	4	no	—
239	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
240	C₂×C₄	D₄×Dic₅₉	1	2	yes	—
241	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×D₅₉	1	2	yes	—
243	C₂×C₄	C₂×D₄×D₅₉	1	1	yes	characteristic
244	C₂×C₄	C₂×#11745687	1	2	yes	—
245	C₈	—	59	2	no	—
246	C₈	—	59	2	no	—
247	C₈	—	118	2	no	—
248	C₈	—	236	4	no	—
249	C₈	D₄×Dic₅₉	1	4	yes	—
250	C₈	D₄×Dic₅₉	1	4	yes	—
251	C₂²	—	2	2	no	—
252	C₂²	—	4	8	no	—
253	C₂²	—	4	4	no	—
254	C₂²	C₂×D₄×Dic₅₉	1	1	yes	characteristic
255	C₂²	#16837950	1	2	yes	—
256	C₄	—	2	2	no	—
257	C₄	—	4	4	no	—
258	C₄	—	118	8	no	—
259	C₄	#16837519	1	4	yes	complemented
260	C₄	C₂×D₄×Dic₅₉	1	2	yes	—
261	C₄	#127×D₅₉	1	1	yes	characteristic
262	C₄	#16837950	1	1	yes	characteristic
263	C₂	—	2	2	no	—
264	C₂	—	8	4	no	—
265	C₂	#127×Dic₅₉	1	1	yes	characteristic
266	C₂	C₂×#16837519	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₉
Fitting, prime radical	C₅₉×#3147	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
cosocle	C₂²×C₄×C₅₉	C₂⁴
hypercenter	#3147	D₅₉
hypocenter	C₅₉	#21705
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#16837519	C₂	semidirect
2. #21432808	C₂	semidirect
3. #21424985	C₂	semidirect
4. #21424925	C₂	semidirect
5. #21424268	C₂	semidirect
6. C₄×C₈×Dic₅₉	C₂	semidirect
22. C₅₉×#495	C₄	semidirect
23. C₅₉×#494	C₄	semidirect
24. C₅₉×#493	C₄	semidirect
25. C₅₉×#492	C₄	semidirect
42. C₈×Dic₅₉	D₄	semidirect
125. C₄×C₅₉	#443	semidirect
144. #495	Dic₅₉	semidirect
145. #494	Dic₅₉	semidirect
146. #493	Dic₅₉	semidirect
147. #492	Dic₅₉	semidirect
160. C₅₉	#21705	semidirect
259. C₄	#16837519	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837519	2
#21432808	2
#21424985	2
#21424925	2
#21424268	2
C₄×C₈×Dic₅₉	2
C₅₉×#3147	2
#21705	59