Subgroups — #28558148

Order 15104.

782 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	#21433119	C₂	1	2	yes	maximal, complemented
2	C₂×C₄×D₄₇₂	C₂	1	1	yes	maximal, characteristic
3	#21422921	C₂	1	4	yes	maximal, complemented
4	C₈×D₄₇₂	C₂	1	4	yes	maximal, complemented
5	#21422844	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422488	C₂	1	2	yes	maximal, complemented
7	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×D₄₇₂	C₄	1	1	yes	characteristic
9	C₂×C₄×D₂₃₆	C₂²	1	2	yes	—
10	C₄×D₄₇₂	C₄	1	2	yes	—
11	C₄×D₄₇₂	C₄	1	2	yes	—
12	C₄×D₄₇₂	C₂²	1	2	yes	—
13	C₄×D₄₇₂	C₂²	1	2	yes	—
14	C₈×D₂₃₆	C₂²	1	8	yes	complemented
15	#16837467	—	2	4	no	—
16	#16837455	C₂²	1	2	yes	—
17	C₂×#14273661	C₄	1	1	yes	characteristic
18	C₂×#14273649	C₄	1	2	yes	—
19	C₂²⋊C₈×D₅₉	—	2	4	no	—
20	#16837155	C₂²	1	2	yes	complemented
21	#16837153	C₂²	1	2	yes	complemented
22	#16837131	C₂²	1	8	yes	complemented
23	C₅₉×#431	C₂²	1	1	yes	characteristic
24	C₅₉×#431	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
26	C₅₉×#322	C₂²	1	2	yes	complemented
27	C₈²×C₅₉	C₂²	1	2	yes	complemented
28	C₂²×D₂₃₆	C₂×C₄	1	2	yes	—
29	C₂²×C₄×D₅₉	—	2	2	no	—
30	C₂×D₄₇₂	C₈	1	8	yes	complemented
31	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
32	C₂×D₄₇₂	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₉	—	2	8	no	—
34	C₄×D₂₃₆	—	2	2	no	—
35	C₄×D₂₃₆	—	2	2	no	—
36	C₄×D₂₃₆	C₂×C₄	1	4	yes	—
37	C₄×D₂₃₆	C₂³	1	4	yes	—
38	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
39	C₂×(C₂×D₅₉)⋊C₄	—	2	2	no	—
40	#14273661	C₂×C₄	1	2	yes	—
41	#14273661	C₂×C₄	1	2	yes	—
42	C₄⋊(C₅₉⋊C₈)	C₂³	1	4	yes	—
43	#14273649	C₂×C₄	1	4	yes	—
44	#14273649	C₂×C₄	1	4	yes	—
45	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
46	(C₂×D₅₉)⋊C₈	—	2	8	no	—
47	C₄×C₅₉⋊C₈	—	2	2	no	—
48	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
51	C₄⋊C₈×C₅₉	—	2	1	no	—
52	C₄⋊C₈×C₅₉	D₄	1	2	yes	—
53	C₂²⋊C₈×C₅₉	—	2	1	no	—
54	C₂²⋊C₈×C₅₉	D₄	1	2	yes	—
55	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
57	C₄×C₈×C₅₉	C₂³	1	2	yes	—
58	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₄×C₈×C₅₉	C₂³	1	2	yes	—
61	C₂³×D₅₉	—	2	2	no	—
62	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
65	C₂²×Dic₅₉	—	2	2	no	—
66	C₂×D₂₃₆	—	2	2	no	—
67	C₂×D₂₃₆	—	2	4	no	—
68	C₂×D₂₃₆	C₂×C₈	1	8	yes	complemented
69	C₂×D₂₃₆	C₂²×C₄	1	4	yes	—
70	C₂×C₄×D₅₉	—	2	4	no	—
71	C₂×C₄×D₅₉	—	2	4	no	—
72	C₂×C₄×D₅₉	—	8	4	no	—
73	C₂×C₈×C₅₉	—	2	2	no	—
74	C₂×C₈×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	complemented
76	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
79	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂×D₄	1	4	yes	—
81	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₉	#41	1	1	yes	characteristic
84	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
85	(C₂×D₅₉)⋊C₄	—	2	4	no	—
86	(C₂×D₅₉)⋊C₄	—	2	4	no	—
87	C₄⋊Dic₅₉	—	2	2	no	—
88	C₄⋊Dic₅₉	C₂²×C₄	1	4	yes	—
89	C₂×C₅₉⋊C₈	—	2	4	no	—
90	D₄₇₂	C₂×C₈	1	8	yes	complemented
91	D₄₇₂	C₂×C₈	1	8	yes	complemented
92	C₈×D₅₉	—	8	8	no	—
93	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
94	C₂²×D₅₉	—	2	4	no	—
95	C₂²×D₅₉	—	2	8	no	—
96	C₂²×D₅₉	—	8	2	no	—
97	C₂²×D₅₉	—	8	2	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	—	4	1	no	—
100	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅₉	#131	1	2	yes	—
105	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
108	C₂×Dic₅₉	—	2	4	no	—
109	C₂×Dic₅₉	—	8	2	no	—
110	D₂₃₆	—	2	4	no	—
111	D₂₃₆	—	2	8	no	—
112	D₂₃₆	C₂²×C₈	1	8	yes	—
113	C₄×D₅₉	—	4	8	no	—
114	C₄×D₅₉	—	8	4	no	—
115	C₈×C₅₉	—	2	1	no	—
116	C₈×C₅₉	—	2	1	no	—
117	C₈×C₅₉	—	2	4	no	—
118	C₈×C₅₉	—	4	2	no	—
119	C₈×C₅₉	C₂²×C₈	1	2	yes	—
120	C₈×C₅₉	C₂²×C₈	1	2	yes	—
121	C₅₉⋊C₈	—	4	4	no	—
122	#17854	—	59	1	no	maximal, Sylow
123	C₂²×C₅₉	—	2	1	no	—
124	C₂²×C₅₉	—	2	1	no	—
125	C₂²×C₅₉	C₈×D₄	1	2	yes	—
126	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
127	C₂×D₅₉	—	2	8	no	—
128	C₂×D₅₉	—	4	4	no	—
129	C₂×D₅₉	—	4	4	no	—
130	C₂×D₅₉	—	8	8	no	—
131	C₄×C₅₉	—	2	1	no	—
132	C₄×C₅₉	—	2	1	no	—
133	C₄×C₅₉	—	2	1	no	—
134	C₄×C₅₉	C₈×D₄	1	2	yes	—
135	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
136	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
137	C₄×C₅₉	#576	1	2	yes	—
138	Dic₅₉	—	4	4	no	—
139	#2972	—	59	2	no	—
140	C₂×C₄×D₈	—	59	1	no	—
141	#1580	—	59	4	no	—
142	C₈×D₈	—	59	4	no	—
143	#1564	—	59	1	no	—
144	#1478	—	59	2	no	—
145	#1450	D₅₉	1	1	yes	characteristic, complemented
146	C₂×C₅₉	—	2	2	no	—
147	C₂×C₅₉	C₂×C₈×D₄	1	1	yes	characteristic
148	C₂×C₅₉	#2948	1	1	yes	characteristic
149	C₂×C₅₉	#2957	1	1	yes	characteristic
150	D₅₉	—	4	8	no	—
151	C₂²×D₈	—	59	1	no	—
152	C₂×C₄×D₄	—	59	2	no	—
153	C₄×D₈	—	59	2	no	—
154	C₄×D₈	—	59	2	no	—
155	C₄×D₈	—	59	2	no	—
156	C₄×D₈	—	59	2	no	—
157	C₈×D₄	—	59	8	no	—
158	C₈×D₄	—	118	4	no	—
159	#431	—	59	2	no	—
160	#431	C₂×D₅₉	1	1	yes	characteristic
161	#431	C₂×D₅₉	1	1	yes	characteristic
162	C₂×#107	—	59	1	no	—
163	C₂×#102	—	59	2	no	—
164	C₂×C₂²⋊C₈	—	118	4	no	—
165	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
166	#335	—	59	2	no	—
167	#334	—	59	2	no	—
168	D₄⋊C₈	—	59	8	no	—
169	#322	C₂×D₅₉	1	2	yes	complemented
170	C₈²	C₂×D₅₉	1	2	yes	complemented
171	C₅₉	#17854	1	1	yes	characteristic, complemented, Sylow
172	C₂²×D₄	—	59	2	no	—
173	C₂³×C₄	—	118	2	no	—
174	C₂×D₈	—	59	8	no	—
175	C₂×D₈	—	59	2	no	—
176	C₂×D₈	—	59	2	no	—
177	C₂²×C₈	—	118	8	no	—
178	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
179	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
180	C₄×D₄	—	59	4	no	—
181	C₄×D₄	—	59	4	no	—
182	C₄×D₄	—	118	2	no	—
183	C₄×D₄	—	118	2	no	—
184	C₂×C₄⋊C₄	—	59	2	no	—
185	C₂×C₂²⋊C₄	—	118	2	no	—
186	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
187	#107	—	59	2	no	—
188	#107	—	59	2	no	—
189	C₄⋊C₈	—	2	1	no	—
190	C₄⋊C₈	—	59	4	no	—
191	C₄⋊C₈	D₂₃₆	1	2	yes	—
192	#102	—	59	4	no	—
193	#102	—	59	4	no	—
194	C₂²⋊C₈	—	2	1	no	—
195	C₂²⋊C₈	—	118	2	no	—
196	C₂²⋊C₈	—	118	8	no	—
197	C₂²⋊C₈	D₂₃₆	1	2	yes	—
198	C₄×C₈	—	118	2	no	—
199	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
200	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₅₉	1	2	yes	—
202	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₅₉	1	2	yes	—
205	C₂⁴	—	118	2	no	—
206	C₂×D₄	—	59	8	no	—
207	C₂×D₄	—	59	4	no	—
208	C₂×D₄	—	118	2	no	—
209	C₂×D₄	—	118	4	no	—
210	C₂²×C₄	—	118	2	no	—
211	C₂²×C₄	—	118	4	no	—
212	C₂²×C₄	—	118	4	no	—
213	C₂²×C₄	—	472	4	no	—
214	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
215	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
216	C₂²×C₄	#11745685	1	1	yes	characteristic
217	D₈	—	59	8	no	—
218	D₈	—	59	8	no	—
219	C₂×C₈	—	2	2	no	—
220	C₂×C₈	—	2	2	no	—
221	C₂×C₈	—	118	4	no	—
222	C₂×C₈	—	472	8	no	—
223	C₂×C₈	C₈×D₅₉	1	4	yes	complemented
224	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
226	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
228	C₂×C₈	C₂×D₂₃₆	1	4	yes	—
229	C₄⋊C₄	—	59	4	no	—
230	C₄⋊C₄	—	118	2	no	—
231	C₂²⋊C₄	—	118	4	no	—
232	C₂²⋊C₄	—	118	4	no	—
233	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
234	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
235	C₄²	#11745685	1	1	yes	characteristic
236	C₄²	C₂³×D₅₉	1	1	yes	characteristic
237	C₂³	—	118	8	no	—
238	C₂³	—	118	4	no	—
239	C₂³	—	472	2	no	—
240	C₂³	—	472	2	no	—
241	C₂³	C₄×D₂₃₆	1	1	yes	characteristic
242	D₄	—	59	8	no	—
243	D₄	—	118	4	no	—
244	D₄	—	118	8	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	4	1	no	—
247	C₂×C₄	—	118	4	no	—
248	C₂×C₄	—	472	4	no	—
249	C₂×C₄	—	472	2	no	—
250	C₂×C₄	—	236	8	no	—
251	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
252	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
253	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
254	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
255	C₂×C₄	#14273733	1	2	yes	—
256	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
257	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
258	C₂×C₄	C₂×#11745685	1	1	yes	characteristic
259	C₈	—	2	1	no	—
260	C₈	—	2	1	no	—
261	C₈	—	2	4	no	—
262	C₈	—	4	2	no	—
263	C₈	—	236	4	no	—
264	C₈	C₂×C₈×D₅₉	1	2	yes	—
265	C₈	C₂×C₈×D₅₉	1	2	yes	—
266	C₂²	—	2	1	no	—
267	C₂²	—	2	1	no	—
268	C₂²	—	118	8	no	—
269	C₂²	—	472	8	no	—
270	C₂²	—	236	4	no	—
271	C₂²	—	236	4	no	—
272	C₂²	C₈×D₂₃₆	1	2	yes	—
273	C₂²	C₂×C₄×D₂₃₆	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	236	4	no	—
278	C₄	C₈×D₂₃₆	1	2	yes	—
279	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
280	C₄	C₂×#14273733	1	1	yes	characteristic
281	C₄	#16838207	1	2	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	236	8	no	—
284	C₂	C₂×C₈×D₂₃₆	1	1	yes	characteristic
285	C₂	#21432949	1	1	yes	characteristic
286	C₂	#21433014	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃₆
commutator	C₄×C₅₉	C₂³×C₈
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₂²×D₄₇₂	C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#17854
LCS	G ▷ C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ #431 ◁ #1450

Complements

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

N	H ≅ G/N	type
1. #21433119	C₂	semidirect
3. #21422921	C₂	semidirect
4. C₈×D₄₇₂	C₂	semidirect
5. #21422844	C₂	semidirect
6. #21422488	C₂	semidirect
7. C₅₉×#1450	C₂	semidirect
14. C₈×D₂₃₆	C₂²	semidirect
20. #16837155	C₂²	semidirect
21. #16837153	C₂²	semidirect
22. #16837131	C₂²	semidirect
26. C₅₉×#322	C₂²	semidirect
27. C₈²×C₅₉	C₂²	semidirect
30. C₂×D₄₇₂	C₈	semidirect
68. C₂×D₂₃₆	C₂×C₈	semidirect
75. C₂×C₈×C₅₉	C₂×C₈	semidirect
90. D₄₇₂	C₂×C₈	semidirect
91. D₄₇₂	C₂×C₈	semidirect
145. #1450	D₅₉	semidirect
169. #322	C₂×D₅₉	semidirect
170. C₈²	C₂×D₅₉	semidirect
171. C₅₉	#17854	semidirect
223. C₂×C₈	C₈×D₅₉	semidirect

Maximal subgroups

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

subgroup	index
#21433119	2
C₂×C₄×D₄₇₂	2
#21422921	2
C₈×D₄₇₂	2
#21422844	2
#21422488	2
C₅₉×#1450	2
#17854	59