Subgroups — C₂³×#14273709 #29495336

Order 15104.

15150 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×C₂²⋊D₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂⁴×C₄×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×#14273709	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₄⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×Dic₅₉⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×C₂²⋊Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×(C₂×D₅₉)⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×C₂²⋊C₄×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁵×D₅₉	C₂²	1	1	yes	characteristic
10	C₂³×C₂²⋊D₅₉	—	2	16	no	—
11	C₂³×C₂²⋊D₅₉	C₂²	1	28	yes	—
12	C₂⁴×Dic₅₉	C₂²	1	1	yes	characteristic
13	C₂⁴×Dic₅₉	C₂²	1	1	yes	characteristic
14	C₂⁴×Dic₅₉	C₂²	1	1	yes	characteristic
15	C₂³×C₄×D₅₉	—	2	16	no	—
16	C₂³×C₄×D₅₉	C₂²	1	28	yes	complemented
17	C₂×#14273709	C₂²	1	448	yes	direct product factor
18	C₂²×C₄⋊Dic₅₉	C₂²	1	28	yes	complemented
19	C₂²×Dic₅₉⋊C₄	C₂²	1	28	yes	complemented
20	C₂²×C₂²⋊Dic₅₉	C₂²	1	28	yes	complemented
21	C₂²×(C₂×D₅₉)⋊C₄	C₂²	1	28	yes	complemented
22	C₂⁶×C₅₉	C₂²	1	1	yes	characteristic
23	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
24	C₂⁴×C₄×C₅₉	C₂²	1	1	yes	characteristic
25	C₂²×C₂²⋊C₄×C₅₉	C₂²	1	28	yes	complemented
26	C₂⁴×D₅₉	—	2	8	no	—
27	C₂⁴×D₅₉	—	2	8	no	—
28	C₂⁴×D₅₉	D₄	1	16	yes	—
29	C₂⁴×D₅₉	C₂³	1	14	yes	—
30	C₂²×C₂²⋊D₅₉	—	2	224	no	—
31	C₂²×C₂²⋊D₅₉	C₂³	1	112	yes	—
32	C₂³×Dic₅₉	—	2	8	no	—
33	C₂³×Dic₅₉	—	2	8	no	—
34	C₂³×Dic₅₉	—	2	8	no	—
35	C₂³×Dic₅₉	C₂³	1	14	yes	—
36	C₂³×Dic₅₉	C₂³	1	14	yes	—
37	C₂³×Dic₅₉	C₂³	1	14	yes	—
38	C₂²×C₄×D₅₉	—	2	224	no	—
39	C₂²×C₄×D₅₉	C₂³	1	112	yes	complemented
40	#14273709	C₂³	1	512	yes	direct product factor
41	C₂×C₄⋊Dic₅₉	C₂³	1	112	yes	complemented
42	C₂×Dic₅₉⋊C₄	C₂³	1	112	yes	complemented
43	C₂×C₂²⋊Dic₅₉	C₂³	1	112	yes	complemented
44	C₂×(C₂×D₅₉)⋊C₄	C₂³	1	112	yes	complemented
45	C₂⁵×C₅₉	—	2	8	no	—
46	C₂⁵×C₅₉	—	2	8	no	—
47	C₂⁵×C₅₉	—	2	8	no	—
48	C₂⁵×C₅₉	C₂³	1	14	yes	—
49	C₂⁵×C₅₉	C₂³	1	1	yes	characteristic
50	C₂³×C₄×C₅₉	—	2	8	no	—
51	C₂³×C₄×C₅₉	—	2	8	no	—
52	C₂³×C₄×C₅₉	C₂³	1	14	yes	—
53	C₂³×C₄×C₅₉	C₂³	1	14	yes	—
54	C₂×C₂²⋊C₄×C₅₉	C₂³	1	112	yes	complemented
55	C₂⁴×C₅₉	—	2	56	no	—
56	C₂⁴×C₅₉	—	2	56	no	—
57	C₂⁴×C₅₉	—	2	56	no	—
58	C₂⁴×C₅₉	—	4	64	no	—
59	C₂⁴×C₅₉	C₂×D₄	1	8	yes	—
60	C₂⁴×C₅₉	#41	1	8	yes	—
61	C₂⁴×C₅₉	#41	1	8	yes	—
62	C₂⁴×C₅₉	C₂⁴	1	28	yes	—
63	C₂⁴×C₅₉	C₂⁴	1	7	yes	—
64	C₂³×D₅₉	—	2	128	no	—
65	C₂³×D₅₉	—	2	56	no	—
66	C₂³×D₅₉	—	2	56	no	—
67	C₂³×D₅₉	C₂×D₄	1	112	yes	—
68	C₂³×D₅₉	C₂⁴	1	28	yes	—
69	C₂²×C₄×C₅₉	—	2	56	no	—
70	C₂²×C₄×C₅₉	—	2	56	no	—
71	C₂²×C₄×C₅₉	C₂⁴	1	28	yes	—
72	C₂²×C₄×C₅₉	C₂⁴	1	28	yes	—
73	C₂×C₂²⋊D₅₉	—	2	448	no	—
74	C₂×C₂²⋊D₅₉	C₂⁴	1	64	yes	—
75	C₂²×Dic₅₉	—	2	56	no	—
76	C₂²×Dic₅₉	—	2	56	no	—
77	C₂²×Dic₅₉	—	2	56	no	—
78	C₂²×Dic₅₉	C₂⁴	1	28	yes	—
79	C₂²×Dic₅₉	C₂⁴	1	28	yes	—
80	C₂²×Dic₅₉	C₂⁴	1	28	yes	—
81	C₂×C₄×D₅₉	—	2	448	no	—
82	C₂×C₄×D₅₉	C₂⁴	1	64	yes	complemented
83	C₂²⋊C₄×C₅₉	C₂⁴	1	64	yes	complemented
84	C₂²⋊Dic₅₉	C₂⁴	1	64	yes	complemented
85	(C₂×D₅₉)⋊C₄	C₂⁴	1	64	yes	complemented
86	C₄⋊Dic₅₉	C₂⁴	1	64	yes	complemented
87	Dic₅₉⋊C₄	C₂⁴	1	64	yes	complemented
88	C₂³×C₅₉	—	2	56	no	—
89	C₂³×C₅₉	—	2	56	no	—
90	C₂³×C₅₉	—	2	56	no	—
91	C₂³×C₅₉	—	4	224	no	—
92	C₂³×C₅₉	#123	1	64	yes	complemented
93	C₂³×C₅₉	C₂²×D₄	1	28	yes	—
94	C₂³×C₅₉	C₂×#41	1	28	yes	—
95	C₂³×C₅₉	C₂×#41	1	28	yes	—
96	C₂³×C₅₉	C₂⁵	1	8	yes	—
97	C₂³×C₅₉	C₂⁵	1	7	yes	—
98	C₂²×D₅₉	—	2	448	no	—
99	C₂²×D₅₉	—	2	56	no	—
100	C₂²×D₅₉	—	2	56	no	—
101	C₂²×D₅₉	C₂²×D₄	1	112	yes	—
102	C₂²×D₅₉	C₂⁵	1	8	yes	—
103	C₂×C₄×C₅₉	—	2	56	no	—
104	C₂×C₄×C₅₉	—	2	56	no	—
105	C₂×C₄×C₅₉	C₂⁵	1	8	yes	—
106	C₂×C₄×C₅₉	C₂⁵	1	8	yes	—
107	C₂²⋊D₅₉	—	2	128	no	—
108	C₂×Dic₅₉	—	2	56	no	—
109	C₂×Dic₅₉	—	2	56	no	—
110	C₂×Dic₅₉	—	2	56	no	—
111	C₂×Dic₅₉	C₂⁵	1	8	yes	—
112	C₂×Dic₅₉	C₂⁵	1	8	yes	—
113	C₂×Dic₅₉	C₂⁵	1	8	yes	—
114	C₄×D₅₉	—	2	128	no	—
115	C₂³×#123	—	59	1	no	maximal, Sylow
116	C₂²×C₅₉	—	2	8	no	—
117	C₂²×C₅₉	—	2	8	no	—
118	C₂²×C₅₉	—	2	8	no	—
119	C₂²×C₅₉	—	4	112	no	—
120	C₂²×C₅₉	C₂×#123	1	112	yes	complemented
121	C₂²×C₅₉	C₂³×D₄	1	14	yes	—
122	C₂²×C₅₉	C₂²×#41	1	14	yes	—
123	C₂²×C₅₉	C₂²×#41	1	14	yes	—
124	C₂²×C₅₉	C₂⁶	1	1	yes	characteristic
125	C₂×D₅₉	—	2	8	no	—
126	C₂×D₅₉	—	2	8	no	—
127	C₂×D₅₉	—	2	224	no	—
128	C₂×D₅₉	C₂³×D₄	1	16	yes	—
129	C₄×C₅₉	—	2	8	no	—
130	C₄×C₅₉	—	2	8	no	—
131	Dic₅₉	—	2	8	no	—
132	Dic₅₉	—	2	8	no	—
133	Dic₅₉	—	2	8	no	—
134	C₂⁴×D₄	—	59	1	no	—
135	C₂⁵×C₄	—	59	1	no	—
136	C₂²×#123	—	59	56	no	—
137	C₂³×C₄⋊C₄	—	59	1	no	—
138	C₂³×C₄⋊C₄	—	59	1	no	—
139	C₂³×C₂²⋊C₄	—	59	1	no	—
140	C₂³×C₂²⋊C₄	—	59	1	no	—
141	C₂³×C₂²⋊C₄	D₅₉	1	1	yes	characteristic, complemented
142	C₂×C₅₉	—	4	8	no	—
143	C₂×C₅₉	C₂²×#123	1	28	yes	complemented
144	C₂×C₅₉	C₂⁴×D₄	1	1	yes	characteristic
145	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
146	C₂×C₅₉	C₂³×#41	1	1	yes	characteristic
147	D₅₉	—	2	16	no	—
148	C₂⁶	—	59	1	no	—
149	C₂⁶	C₂×D₅₉	1	1	yes	characteristic
150	C₂³×D₄	—	59	28	no	—
151	C₂³×D₄	—	118	16	no	—
152	C₂⁴×C₄	—	59	1	no	—
153	C₂⁴×C₄	—	59	1	no	—
154	C₂⁴×C₄	—	59	1	no	—
155	C₂⁴×C₄	—	59	28	no	—
156	C₂⁴×C₄	—	118	16	no	—
157	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
158	C₂⁴×C₄	C₂×D₅₉	1	1	yes	characteristic
159	C₂×#123	—	59	448	no	—
160	C₂²×C₄⋊C₄	—	59	28	no	—
161	C₂²×C₄⋊C₄	—	59	28	no	—
162	C₂²×C₂²⋊C₄	—	59	28	no	—
163	C₂²×C₂²⋊C₄	—	59	28	no	—
164	C₂²×C₂²⋊C₄	C₂×D₅₉	1	28	yes	complemented
165	C₅₉	C₂³×#123	1	1	yes	characteristic, complemented, Sylow
166	C₂⁵	—	2	8	no	—
167	C₂⁵	—	2	8	no	—
168	C₂⁵	—	2	8	no	—
169	C₂⁵	—	59	16	no	—
170	C₂⁵	—	59	14	no	—
171	C₂⁵	—	118	8	no	—
172	C₂⁵	—	118	8	no	—
173	C₂⁵	C₂²×D₅₉	1	14	yes	—
174	C₂⁵	C₂²×D₅₉	1	1	yes	characteristic
175	C₂²×D₄	—	59	112	no	—
176	C₂²×D₄	—	118	224	no	—
177	C₂³×C₄	—	2	8	no	—
178	C₂³×C₄	—	2	8	no	—
179	C₂³×C₄	—	59	14	no	—
180	C₂³×C₄	—	59	14	no	—
181	C₂³×C₄	—	59	14	no	—
182	C₂³×C₄	—	59	112	no	—
183	C₂³×C₄	—	118	224	no	—
184	C₂³×C₄	—	118	8	no	—
185	C₂³×C₄	—	118	8	no	—
186	C₂³×C₄	—	118	8	no	—
187	C₂³×C₄	C₂²×D₅₉	1	14	yes	—
188	C₂³×C₄	C₂²×D₅₉	1	14	yes	—
189	#123	—	59	512	no	—
190	C₂×C₄⋊C₄	—	59	112	no	—
191	C₂×C₄⋊C₄	—	59	112	no	—
192	C₂×C₂²⋊C₄	—	59	112	no	—
193	C₂×C₂²⋊C₄	—	59	112	no	—
194	C₂×C₂²⋊C₄	C₂²×D₅₉	1	112	yes	complemented
195	C₂⁴	—	2	56	no	—
196	C₂⁴	—	2	56	no	—
197	C₂⁴	—	2	56	no	—
198	C₂⁴	—	4	64	no	—
199	C₂⁴	—	59	112	no	—
200	C₂⁴	—	59	28	no	—
201	C₂⁴	—	118	128	no	—
202	C₂⁴	—	118	56	no	—
203	C₂⁴	—	118	56	no	—
204	C₂⁴	#11745685	1	8	yes	—
205	C₂⁴	D₄×D₅₉	1	8	yes	—
206	C₂⁴	#11745687	1	8	yes	—
207	C₂⁴	C₂³×D₅₉	1	28	yes	—
208	C₂⁴	C₂³×D₅₉	1	7	yes	—
209	C₂×D₄	—	59	64	no	—
210	C₂×D₄	—	118	448	no	—
211	C₂²×C₄	—	2	56	no	—
212	C₂²×C₄	—	2	56	no	—
213	C₂²×C₄	—	59	28	no	—
214	C₂²×C₄	—	59	28	no	—
215	C₂²×C₄	—	59	28	no	—
216	C₂²×C₄	—	59	64	no	—
217	C₂²×C₄	—	118	448	no	—
218	C₂²×C₄	—	118	56	no	—
219	C₂²×C₄	—	118	56	no	—
220	C₂²×C₄	—	118	56	no	—
221	C₂²×C₄	C₂³×D₅₉	1	28	yes	—
222	C₂²×C₄	C₂³×D₅₉	1	28	yes	—
223	C₄⋊C₄	—	59	64	no	—
224	C₄⋊C₄	—	59	64	no	—
225	C₂²⋊C₄	—	59	64	no	—
226	C₂²⋊C₄	—	59	64	no	—
227	C₂²⋊C₄	C₂³×D₅₉	1	64	yes	complemented
228	C₂³	—	2	56	no	—
229	C₂³	—	2	56	no	—
230	C₂³	—	2	56	no	—
231	C₂³	—	4	224	no	—
232	C₂³	—	59	112	no	—
233	C₂³	—	59	8	no	—
234	C₂³	—	118	448	no	—
235	C₂³	—	118	56	no	—
236	C₂³	—	118	56	no	—
237	C₂³	#14273709	1	64	yes	direct product factor
238	C₂³	C₂×D₄×D₅₉	1	28	yes	—
239	C₂³	C₂×#11745687	1	28	yes	—
240	C₂³	C₂×#11745685	1	28	yes	—
241	C₂³	C₂⁴×D₅₉	1	8	yes	—
242	C₂³	C₂⁴×D₅₉	1	7	yes	—
243	D₄	—	118	128	no	—
244	C₂×C₄	—	2	56	no	—
245	C₂×C₄	—	2	56	no	—
246	C₂×C₄	—	59	8	no	—
247	C₂×C₄	—	59	8	no	—
248	C₂×C₄	—	59	8	no	—
249	C₂×C₄	—	118	128	no	—
250	C₂×C₄	—	118	56	no	—
251	C₂×C₄	—	118	56	no	—
252	C₂×C₄	—	118	56	no	—
253	C₂×C₄	C₂⁴×D₅₉	1	8	yes	—
254	C₂×C₄	C₂⁴×D₅₉	1	8	yes	—
255	C₂²	—	2	8	no	—
256	C₂²	—	2	8	no	—
257	C₂²	—	2	8	no	—
258	C₂²	—	4	112	no	—
259	C₂²	—	59	16	no	—
260	C₂²	—	118	224	no	—
261	C₂²	—	118	8	no	—
262	C₂²	—	118	8	no	—
263	C₂²	C₂×#14273709	1	112	yes	direct product factor
264	C₂²	C₂²×D₄×D₅₉	1	14	yes	—
265	C₂²	C₂²×#11745687	1	14	yes	—
266	C₂²	C₂²×#11745685	1	14	yes	—
267	C₂²	C₂⁵×D₅₉	1	1	yes	characteristic
268	C₄	—	2	8	no	—
269	C₄	—	2	8	no	—
270	C₄	—	118	8	no	—
271	C₄	—	118	8	no	—
272	C₄	—	118	8	no	—
273	C₂	—	4	8	no	—
274	C₂	—	118	16	no	—
275	C₂	C₂²×#14273709	1	28	yes	direct product factor
276	C₂	C₂³×D₄×D₅₉	1	1	yes	characteristic
277	C₂	C₂³×#11745687	1	1	yes	characteristic
278	C₂	C₂³×#11745685	1	1	yes	characteristic

	subgroup	G/H
center	C₂⁵	C₂²×D₅₉
commutator, cosocle	C₂²×C₅₉	C₂⁶
Frattini	C₂²	C₂⁵×D₅₉
Fitting	C₂³×C₂²⋊C₄×C₅₉	C₂
socle	C₂⁵×C₅₉	C₂³
prime radical	C₂⁴×C₂²⋊D₅₉	C₂
hypercenter	C₂³×C₂²⋊C₄	D₅₉
hypocenter	C₅₉	C₂³×#123
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
2. C₂⁴×C₄×D₅₉	C₂	semidirect
3. C₂²×#14273709	C₂	direct (N × H)
4. C₂³×C₄⋊Dic₅₉	C₂	semidirect
5. C₂³×Dic₅₉⋊C₄	C₂	semidirect
6. C₂³×C₂²⋊Dic₅₉	C₂	semidirect
7. C₂³×(C₂×D₅₉)⋊C₄	C₂	semidirect
8. C₂³×C₂²⋊C₄×C₅₉	C₂	semidirect
16. C₂³×C₄×D₅₉	C₂²	semidirect
17. C₂×#14273709	C₂²	direct (N × H)
18. C₂²×C₄⋊Dic₅₉	C₂²	semidirect
19. C₂²×Dic₅₉⋊C₄	C₂²	semidirect
20. C₂²×C₂²⋊Dic₅₉	C₂²	semidirect
21. C₂²×(C₂×D₅₉)⋊C₄	C₂²	semidirect
25. C₂²×C₂²⋊C₄×C₅₉	C₂²	semidirect
39. C₂²×C₄×D₅₉	C₂³	semidirect
40. #14273709	C₂³	direct (N × H)
41. C₂×C₄⋊Dic₅₉	C₂³	semidirect
42. C₂×Dic₅₉⋊C₄	C₂³	semidirect
43. C₂×C₂²⋊Dic₅₉	C₂³	semidirect
44. C₂×(C₂×D₅₉)⋊C₄	C₂³	semidirect
54. C₂×C₂²⋊C₄×C₅₉	C₂³	semidirect
82. C₂×C₄×D₅₉	C₂⁴	semidirect
83. C₂²⋊C₄×C₅₉	C₂⁴	semidirect
84. C₂²⋊Dic₅₉	C₂⁴	semidirect
85. (C₂×D₅₉)⋊C₄	C₂⁴	semidirect
86. C₄⋊Dic₅₉	C₂⁴	semidirect
87. Dic₅₉⋊C₄	C₂⁴	semidirect
92. C₂³×C₅₉	#123	semidirect
120. C₂²×C₅₉	C₂×#123	semidirect
141. C₂³×C₂²⋊C₄	D₅₉	semidirect
143. C₂×C₅₉	C₂²×#123	semidirect
164. C₂²×C₂²⋊C₄	C₂×D₅₉	semidirect
165. C₅₉	C₂³×#123	semidirect
194. C₂×C₂²⋊C₄	C₂²×D₅₉	semidirect
227. C₂²⋊C₄	C₂³×D₅₉	semidirect
237. C₂³	#14273709	direct (N × H)
263. C₂²	C₂×#14273709	direct (N × H)
275. C₂	C₂²×#14273709	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×C₂²⋊D₅₉	2
C₂⁴×C₄×D₅₉	2
C₂²×#14273709	2
C₂³×C₄⋊Dic₅₉	2
C₂³×Dic₅₉⋊C₄	2
C₂³×C₂²⋊Dic₅₉	2
C₂³×(C₂×D₅₉)⋊C₄	2
C₂³×C₂²⋊C₄×C₅₉	2
C₂³×#123	59

Subgroups — C23×#14273709 #29495336

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14273709 #29495336