Subgroups — #568×Dic₅₉ #29452103

Order 15104.

3250 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	C₂×#16838169	C₂	1	1	yes	maximal, characteristic
2	C₂×#41×Dic₅₉	C₂	1	1	yes	maximal, characteristic
3	#21432522	C₂	1	8	yes	maximal, complemented
4	#131×Dic₅₉	C₂	1	8	yes	maximal, complemented
5	#21432514	C₂	1	6	yes	maximal, complemented
6	C₂×M₁₆×Dic₅₉	C₂	1	6	yes	maximal, complemented
7	C₂×C₅₉×#568	C₂	1	1	yes	maximal, characteristic
8	#16838169	C₄	1	16	yes	complemented
9	C₂×#14273734	C₂²	1	8	yes	—
10	C₂²×#11745664	C₂²	1	6	yes	—
11	#41×Dic₅₉	C₄	1	8	yes	—
12	#41×Dic₅₉	C₂²	1	4	yes	—
13	#41×Dic₅₉	C₂²	1	4	yes	—
14	C₂×Q₈×Dic₅₉	C₄	1	2	yes	—
15	C₂×D₄×Dic₅₉	C₄	1	6	yes	—
16	C₂×#14273686	C₂²	1	3	yes	—
17	C₂²×C₄×Dic₅₉	C₂²	1	3	yes	—
18	#16837360	C₂²	1	24	yes	complemented
19	#16837357	C₂²	1	24	yes	complemented
20	M₁₆×Dic₅₉	C₂²	1	4	yes	—
21	M₁₆×Dic₅₉	C₂²	1	12	yes	—
22	M₁₆×Dic₅₉	C₂²	1	12	yes	complemented
23	M₁₆×Dic₅₉	C₂²	1	12	yes	complemented
24	C₂×#14273637	C₂²	1	12	yes	complemented
25	C₂×C₈×Dic₅₉	C₂²	1	12	yes	complemented
26	C₂²×#41×C₅₉	C₂²	1	1	yes	characteristic
27	C₅₉×#568	C₄	1	16	yes	complemented
28	C₂×#131×C₅₉	C₂²	1	8	yes	—
29	C₂²×M₁₆×C₅₉	C₂²	1	6	yes	—
30	C₂³×Dic₅₉	C₂×C₄	1	6	yes	—
31	#14273734	C₂×C₄	1	64	yes	complemented
32	C₂×#11745664	C₂×C₄	1	48	yes	complemented
33	C₂×#11745664	C₂³	1	4	yes	—
34	C₂×#11745664	C₂³	1	12	yes	—
35	C₂²×C₅₉⋊C₈	C₂³	1	12	yes	—
36	Q₈×Dic₅₉	C₂×C₄	1	8	yes	—
37	Q₈×Dic₅₉	C₂×C₄	1	8	yes	—
38	D₄×Dic₅₉	C₂×C₄	1	24	yes	—
39	D₄×Dic₅₉	C₂×C₄	1	24	yes	—
40	#14273686	C₂×C₄	1	12	yes	—
41	#14273686	C₂³	1	12	yes	—
42	C₂×C₄⋊Dic₅₉	C₂×C₄	1	6	yes	—
43	C₂×C₂²⋊Dic₅₉	C₂×C₄	1	6	yes	—
44	C₂×C₄×Dic₅₉	C₂×C₄	1	6	yes	—
45	C₂×C₄×Dic₅₉	C₂×C₄	1	12	yes	—
46	C₂×C₄×Dic₅₉	C₂³	1	1	yes	characteristic
47	C₂×C₄×Dic₅₉	C₂³	1	3	yes	—
48	C₂×C₄×Dic₅₉	C₂³	1	6	yes	—
49	C₂×C₄×Dic₅₉	C₂³	1	6	yes	—
50	#14273637	C₂³	1	8	yes	—
51	#14273637	C₂³	1	24	yes	—
52	C₈×Dic₅₉	C₂³	1	8	yes	—
53	C₈×Dic₅₉	C₂³	1	24	yes	—
54	C₂×#41×C₅₉	C₂×C₄	1	8	yes	—
55	C₂×#41×C₅₉	C₂×C₄	1	4	yes	—
56	C₂×#41×C₅₉	C₂×C₄	1	8	yes	—
57	C₂×#41×C₅₉	C₂³	1	4	yes	—
58	C₂²×Q₈×C₅₉	C₂×C₄	1	1	yes	characteristic
59	C₂²×D₄×C₅₉	C₂×C₄	1	3	yes	—
60	C₂³×C₄×C₅₉	C₂³	1	3	yes	—
61	#131×C₅₉	C₂×C₄	1	64	yes	complemented
62	C₂×M₁₆×C₅₉	C₂×C₄	1	48	yes	complemented
63	C₂×M₁₆×C₅₉	C₂³	1	4	yes	—
64	C₂×M₁₆×C₅₉	C₂³	1	12	yes	—
65	C₂²×C₈×C₅₉	C₂³	1	12	yes	—
66	C₂⁴×C₅₉	C₂²×C₄	1	3	yes	—
67	#41×C₅₉	C₄²	1	32	yes	—
68	#41×C₅₉	C₂²×C₄	1	16	yes	—
69	#41×C₅₉	C₂²×C₄	1	16	yes	—
70	C₂×Q₈×C₅₉	C₄²	1	8	yes	—
71	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
72	C₂×D₄×C₅₉	C₄²	1	24	yes	—
73	C₂×D₄×C₅₉	C₂²×C₄	1	12	yes	—
74	C₂²×C₄×C₅₉	C₂²×C₄	1	3	yes	—
75	C₂²×C₄×C₅₉	C₂²×C₄	1	6	yes	—
76	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
77	C₂²×C₄×C₅₉	C₂²×C₄	1	12	yes	—
78	C₂²×C₄×C₅₉	C₂⁴	1	6	yes	—
79	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
80	C₂²×Dic₅₉	—	4	12	no	—
81	C₂²×Dic₅₉	C₂²×C₄	1	2	yes	—
82	C₂²×Dic₅₉	C₂²×C₄	1	12	yes	—
83	C₂²×Dic₅₉	C₂²×C₄	1	6	yes	—
84	C₂²×Dic₅₉	C₂²×C₄	1	12	yes	—
85	M₁₆×C₅₉	C₂²×C₄	1	16	yes	—
86	M₁₆×C₅₉	C₂²×C₄	1	48	yes	complemented
87	C₂×C₈×C₅₉	C₂²×C₄	1	48	yes	complemented
88	C₂×C₈×C₅₉	C₂⁴	1	8	yes	—
89	C₂²⋊Dic₅₉	C₂²×C₄	1	24	yes	—
90	C₄⋊Dic₅₉	C₂²×C₄	1	24	yes	—
91	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
92	C₄×Dic₅₉	C₂²×C₄	1	6	yes	—
93	C₄×Dic₅₉	C₂²×C₄	1	12	yes	—
94	C₄×Dic₅₉	C₂²×C₄	1	12	yes	—
95	C₄×Dic₅₉	C₂⁴	1	1	yes	characteristic
96	C₄×Dic₅₉	C₂⁴	1	6	yes	—
97	C₄×Dic₅₉	C₂⁴	1	1	yes	characteristic
98	#11745664	C₂²×C₄	1	16	yes	—
99	#11745664	C₂²×C₄	1	48	yes	complemented
100	C₂×C₅₉⋊C₈	C₂²×C₄	1	48	yes	complemented
101	C₂×C₅₉⋊C₈	C₂⁴	1	8	yes	—
102	C₂³×C₅₉	—	4	6	no	—
103	C₂³×C₅₉	C₂×C₄²	1	12	yes	—
104	C₂³×C₅₉	C₂³×C₄	1	6	yes	—
105	C₂³×C₅₉	C₂³×C₄	1	1	yes	characteristic
106	Q₈×C₅₉	C₂×C₄²	1	16	yes	—
107	D₄×C₅₉	C₂×C₄²	1	48	yes	—
108	C₂×C₄×C₅₉	C₂×C₄²	1	12	yes	—
109	C₂×C₄×C₅₉	C₂×C₄²	1	24	yes	—
110	C₂×C₄×C₅₉	C₂³×C₄	1	12	yes	—
111	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
112	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂³×C₄	1	6	yes	—
114	C₂×C₄×C₅₉	C₂³×C₄	1	12	yes	—
115	C₂×C₄×C₅₉	C₂³×C₄	1	2	yes	—
116	C₂×C₄×C₅₉	C₂⁵	1	1	yes	characteristic
117	C₂×Dic₅₉	—	2	4	no	—
118	C₂×Dic₅₉	—	2	24	no	—
119	C₂×Dic₅₉	—	4	12	no	—
120	C₂×Dic₅₉	—	4	6	no	—
121	C₂×Dic₅₉	C₂³×C₄	1	2	yes	—
122	C₂×Dic₅₉	C₂³×C₄	1	12	yes	—
123	C₂×Dic₅₉	C₂³×C₄	1	2	yes	—
124	C₈×C₅₉	C₂³×C₄	1	16	yes	—
125	C₅₉⋊C₈	C₂³×C₄	1	16	yes	—
126	C₄×#568	—	59	1	no	maximal, Sylow
127	C₂²×C₅₉	—	2	2	no	—
128	C₂²×C₅₉	—	2	12	no	—
129	C₂²×C₅₉	—	4	12	no	—
130	C₂²×C₅₉	C₂²×C₄²	1	2	yes	—
131	C₂²×C₅₉	C₂²×C₄²	1	12	yes	—
132	C₂²×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
133	C₄×C₅₉	C₂²×C₄²	1	2	yes	—
134	C₄×C₅₉	C₂²×C₄²	1	12	yes	—
135	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
136	C₄×C₅₉	C₂⁴×C₄	1	1	yes	characteristic
137	Dic₅₉	—	2	2	no	—
138	Dic₅₉	—	2	12	no	—
139	Dic₅₉	#568	1	4	yes	direct product factor
140	C₂×#568	—	59	1	no	—
141	C₂×#568	D₅₉	1	1	yes	characteristic
142	C₂×C₄×#41	—	59	1	no	—
143	#2875	—	59	8	no	—
144	C₄×#131	—	59	8	no	—
145	#2873	—	59	6	no	—
146	C₂×C₄×M₁₆	—	59	6	no	—
147	C₂×C₅₉	—	2	2	no	—
148	C₂×C₅₉	—	2	12	no	—
149	C₂×C₅₉	C₂³×C₄²	1	1	yes	characteristic
150	C₂×C₅₉	C₂×#568	1	2	yes	—
151	C₂²×#41	C₂×D₅₉	1	1	yes	characteristic
152	#568	—	59	16	no	—
153	#568	Dic₅₉	1	16	yes	direct product factor
154	C₂×#131	—	59	8	no	—
155	C₂×#131	C₂×D₅₉	1	8	yes	—
156	C₂²×M₁₆	—	59	6	no	—
157	C₂²×M₁₆	C₂×D₅₉	1	6	yes	—
158	C₄×#41	—	59	8	no	—
159	C₄×#41	—	59	4	no	—
160	C₄×#41	—	59	4	no	—
161	C₂×C₄×Q₈	—	59	2	no	—
162	C₂×C₄×D₄	—	59	6	no	—
163	C₂×#117	—	59	3	no	—
164	C₂²×C₄²	—	59	3	no	—
165	#405	—	59	24	no	—
166	#405	—	59	24	no	—
167	C₄×M₁₆	—	59	4	no	—
168	C₄×M₁₆	—	59	12	no	—
169	C₄×M₁₆	—	59	12	no	—
170	C₄×M₁₆	—	59	12	no	—
171	C₂×#97	—	59	12	no	—
172	C₂×C₄×C₈	—	59	12	no	—
173	C₅₉	C₄×#568	1	1	yes	characteristic, complemented, Sylow
174	C₂×#41	C₄×D₅₉	1	8	yes	—
175	C₂×#41	C₄×D₅₉	1	4	yes	—
176	C₂×#41	C₂×Dic₅₉	1	8	yes	—
177	C₂×#41	C₂²×D₅₉	1	4	yes	—
178	C₂²×Q₈	C₄×D₅₉	1	1	yes	characteristic
179	C₂²×D₄	C₄×D₅₉	1	3	yes	—
180	C₂³×C₄	—	59	6	no	—
181	C₂³×C₄	C₂²×D₅₉	1	3	yes	—
182	#131	—	59	64	no	—
183	#131	C₂×Dic₅₉	1	64	yes	complemented
184	C₂×M₁₆	—	59	48	no	—
185	C₂×M₁₆	—	59	4	no	—
186	C₂×M₁₆	—	59	12	no	—
187	C₂×M₁₆	C₂×Dic₅₉	1	48	yes	complemented
188	C₂×M₁₆	C₂²×D₅₉	1	4	yes	—
189	C₂×M₁₆	C₂²×D₅₉	1	12	yes	—
190	C₂²×C₈	—	59	12	no	—
191	C₂²×C₈	C₂²×D₅₉	1	12	yes	—
192	C₄×Q₈	—	59	8	no	—
193	C₄×Q₈	—	59	8	no	—
194	C₄×D₄	—	59	24	no	—
195	C₄×D₄	—	59	24	no	—
196	#117	—	59	12	no	—
197	#117	—	59	12	no	—
198	C₂×C₄⋊C₄	—	59	6	no	—
199	C₂×C₂²⋊C₄	—	59	6	no	—
200	C₂×C₄²	—	59	6	no	—
201	C₂×C₄²	—	59	12	no	—
202	C₂×C₄²	—	59	3	no	—
203	C₂×C₄²	—	59	6	no	—
204	C₂×C₄²	—	59	6	no	—
205	C₂×C₄²	—	59	1	no	—
206	#97	—	59	8	no	—
207	#97	—	59	24	no	—
208	C₄×C₈	—	59	8	no	—
209	C₄×C₈	—	59	24	no	—
210	C₂⁴	C₂×C₄×D₅₉	1	3	yes	—
211	#41	C₄×Dic₅₉	1	32	yes	—
212	#41	C₂×C₄×D₅₉	1	16	yes	—
213	#41	C₂²×Dic₅₉	1	16	yes	—
214	C₂×Q₈	C₄×Dic₅₉	1	8	yes	—
215	C₂×Q₈	C₂×C₄×D₅₉	1	4	yes	—
216	C₂×D₄	C₄×Dic₅₉	1	24	yes	—
217	C₂×D₄	C₂×C₄×D₅₉	1	12	yes	—
218	C₂²×C₄	—	59	2	no	—
219	C₂²×C₄	—	59	12	no	—
220	C₂²×C₄	—	59	6	no	—
221	C₂²×C₄	—	59	12	no	—
222	C₂²×C₄	—	236	12	no	—
223	C₂²×C₄	C₂×C₄×D₅₉	1	3	yes	—
224	C₂²×C₄	C₂×C₄×D₅₉	1	6	yes	—
225	C₂²×C₄	C₂×C₄×D₅₉	1	12	yes	—
226	C₂²×C₄	C₂²×Dic₅₉	1	12	yes	—
227	C₂²×C₄	C₂³×D₅₉	1	1	yes	characteristic
228	C₂²×C₄	C₂³×D₅₉	1	6	yes	—
229	M₁₆	—	59	16	no	—
230	M₁₆	—	59	48	no	—
231	M₁₆	C₂²×Dic₅₉	1	16	yes	—
232	M₁₆	C₂²×Dic₅₉	1	48	yes	complemented
233	C₂×C₈	—	59	48	no	—
234	C₂×C₈	—	59	8	no	—
235	C₂×C₈	C₂²×Dic₅₉	1	48	yes	complemented
236	C₂×C₈	C₂³×D₅₉	1	8	yes	—
237	C₄⋊C₄	—	59	24	no	—
238	C₂²⋊C₄	—	59	24	no	—
239	C₄²	—	59	2	no	—
240	C₄²	—	59	6	no	—
241	C₄²	—	59	12	no	—
242	C₄²	—	59	12	no	—
243	C₄²	—	59	1	no	—
244	C₄²	—	59	6	no	—
245	C₄²	—	59	1	no	—
246	C₂³	—	4	6	no	—
247	C₂³	C₂×C₄×Dic₅₉	1	12	yes	—
248	C₂³	C₂²×C₄×D₅₉	1	6	yes	—
249	C₂³	C₂²×C₄×D₅₉	1	1	yes	characteristic
250	Q₈	C₂×C₄×Dic₅₉	1	16	yes	—
251	D₄	C₂×C₄×Dic₅₉	1	48	yes	—
252	C₂×C₄	—	59	2	no	—
253	C₂×C₄	—	59	2	no	—
254	C₂×C₄	—	59	12	no	—
255	C₂×C₄	—	118	24	no	—
256	C₂×C₄	—	118	4	no	—
257	C₂×C₄	—	236	12	no	—
258	C₂×C₄	—	236	6	no	—
259	C₂×C₄	C₂×C₄×Dic₅₉	1	12	yes	—
260	C₂×C₄	C₂×C₄×Dic₅₉	1	24	yes	—
261	C₂×C₄	C₂²×C₄×D₅₉	1	12	yes	—
262	C₂×C₄	C₂²×C₄×D₅₉	1	2	yes	—
263	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
264	C₂×C₄	C₂²×C₄×D₅₉	1	6	yes	—
265	C₂×C₄	C₂³×Dic₅₉	1	12	yes	—
266	C₂×C₄	C₂³×Dic₅₉	1	2	yes	—
267	C₂×C₄	C₂⁴×D₅₉	1	1	yes	characteristic
268	C₈	—	59	16	no	—
269	C₈	C₂³×Dic₅₉	1	16	yes	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	12	no	—
272	C₂²	—	4	12	no	—
273	C₂²	C₂²×C₄×Dic₅₉	1	2	yes	—
274	C₂²	C₂²×C₄×Dic₅₉	1	12	yes	—
275	C₂²	C₂³×C₄×D₅₉	1	1	yes	characteristic
276	C₄	—	59	4	no	—
277	C₄	—	118	2	no	—
278	C₄	—	118	12	no	—
279	C₄	C₂²×C₄×Dic₅₉	1	2	yes	—
280	C₄	C₂²×C₄×Dic₅₉	1	12	yes	—
281	C₄	C₂³×C₄×D₅₉	1	1	yes	characteristic
282	C₄	C₂⁴×Dic₅₉	1	1	yes	characteristic
283	C₂	—	2	2	no	—
284	C₂	—	2	12	no	—
285	C₂	C₂³×C₄×Dic₅₉	1	1	yes	characteristic
286	C₂	#568×D₅₉	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂⁴×D₅₉
commutator	C₂×C₅₉	C₂³×C₄²
Fitting	C₂×C₅₉×#568	C₂
socle	C₂²×C₅₉	C₂⁴×C₄
prime radical	C₂²×#41×C₅₉	C₂²
cosocle	C₂×C₄×C₅₉	C₂⁵
hypercenter	C₂×#568	D₅₉
hypocenter	C₅₉	C₄×#568
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#568

Complements

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

N	H ≅ G/N	type
3. #21432522	C₂	semidirect
4. #131×Dic₅₉	C₂	semidirect
5. #21432514	C₂	semidirect
6. C₂×M₁₆×Dic₅₉	C₂	semidirect
8. #16838169	C₄	semidirect
18. #16837360	C₂²	semidirect
19. #16837357	C₂²	semidirect
22. M₁₆×Dic₅₉	C₂²	semidirect
23. M₁₆×Dic₅₉	C₂²	semidirect
24. C₂×#14273637	C₂²	semidirect
25. C₂×C₈×Dic₅₉	C₂²	semidirect
27. C₅₉×#568	C₄	semidirect
31. #14273734	C₂×C₄	semidirect
32. C₂×#11745664	C₂×C₄	semidirect
61. #131×C₅₉	C₂×C₄	semidirect
62. C₂×M₁₆×C₅₉	C₂×C₄	semidirect
86. M₁₆×C₅₉	C₂²×C₄	semidirect
87. C₂×C₈×C₅₉	C₂²×C₄	semidirect
99. #11745664	C₂²×C₄	semidirect
100. C₂×C₅₉⋊C₈	C₂²×C₄	semidirect
139. Dic₅₉	#568	direct (N × H)
153. #568	Dic₅₉	direct (N × H)
173. C₅₉	C₄×#568	semidirect
183. #131	C₂×Dic₅₉	semidirect
187. C₂×M₁₆	C₂×Dic₅₉	semidirect
232. M₁₆	C₂²×Dic₅₉	semidirect
235. C₂×C₈	C₂²×Dic₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16838169	2
C₂×#41×Dic₅₉	2
#21432522	2
#131×Dic₅₉	2
#21432514	2
C₂×M₁₆×Dic₅₉	2
C₂×C₅₉×#568	2
C₄×#568	59

Subgroups — #568×Dic59 #29452103

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #568×Dic₅₉ #29452103