Subgroups — #28741328

Order 15104.

826 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₃₂×D₅₉	C₂	1	2	yes	maximal
2	C₂×#16837356	C₂	1	1	yes	maximal, characteristic
3	#21425694	C₂	1	4	yes	maximal, complemented
4	#21425691	C₂	1	4	yes	maximal, complemented
5	C₄×#14273667	C₂	1	1	yes	maximal, characteristic, complemented
6	M₃₂×Dic₅₉	C₂	1	2	yes	maximal, complemented
7	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11745659	C₄	1	1	yes	characteristic
9	C₂²×C₈×D₅₉	C₂²	1	1	yes	characteristic
10	C₂×#14273684	C₄	1	1	yes	characteristic
11	C₂×#14273667	C₂²	1	2	yes	—
12	C₂×#14273666	C₂²	1	4	yes	—
13	M₃₂×D₅₉	C₄	1	16	yes	complemented
14	C₂×C₁₆×D₅₉	C₂²	1	4	yes	—
15	#16837356	C₄	1	2	yes	—
16	#16837356	C₄	1	2	yes	—
17	#16837356	C₂²	1	2	yes	—
18	#16837356	C₂²	1	2	yes	—
19	C₂×#14273638	C₄	1	1	yes	characteristic
20	C₂×#14273637	C₄	1	1	yes	characteristic
21	C₂×C₈×Dic₅₉	C₂²	1	1	yes	characteristic
22	#16837178	C₂²	1	2	yes	complemented
23	#16837177	C₂²	1	4	yes	complemented
24	C₄×C₅₉⋊C₁₆	C₂²	1	2	yes	complemented
25	C₁₆×Dic₅₉	C₂²	1	4	yes	complemented
26	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#346	C₂²	1	2	yes	complemented
29	C₄×C₁₆×C₅₉	C₂²	1	2	yes	complemented
30	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
31	C₂×#11745659	C₈	1	8	yes	—
32	C₂×#11745659	C₂×C₄	1	2	yes	—
33	C₂×#11745659	C₂×C₄	1	2	yes	—
34	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
36	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
37	C₂×C₈×D₅₉	C₂×C₄	1	8	yes	—
38	C₂×C₈×D₅₉	C₂³	1	2	yes	—
39	#14273684	C₈	1	4	yes	—
40	#14273684	C₂×C₄	1	2	yes	—
41	#14273684	C₂×C₄	1	2	yes	—
42	C₂×Dic₅₉⋊C₄	C₈	1	2	yes	—
43	C₂×(C₂×D₅₉)⋊C₄	C₈	1	2	yes	—
44	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
45	#14273667	C₂×C₄	1	8	yes	—
46	#14273666	C₂×C₄	1	16	yes	complemented
47	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
48	C₁₆×D₅₉	C₂×C₄	1	16	yes	complemented
49	#14273638	C₂×C₄	1	2	yes	—
50	#14273638	C₂×C₄	1	2	yes	—
51	#14273637	C₂×C₄	1	2	yes	—
52	#14273637	C₂×C₄	1	2	yes	—
53	C₈×Dic₅₉	C₂×C₄	1	2	yes	—
54	C₈×Dic₅₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
58	M₃₂×C₅₉	C₂×C₄	1	8	yes	—
59	C₂×C₁₆×C₅₉	C₂³	1	4	yes	—
60	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
61	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₂³×D₅₉	C₂×C₈	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂×C₈	1	2	yes	—
66	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×Dic₅₉	C₂×C₈	1	2	yes	—
68	C₂²×Dic₅₉	C₂×C₈	1	1	yes	characteristic
69	C₂×C₄×D₅₉	C₄²	1	4	yes	—
70	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₅₉	C₂×C₈	1	4	yes	—
72	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
73	C₂×C₈×C₅₉	C₂×C₈	1	4	yes	—
74	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
76	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂²×C₄	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂⁴	1	1	yes	characteristic
79	C₄²×C₅₉	C₂×C₈	1	2	yes	—
80	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
81	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	(C₂×D₅₉)⋊C₄	C₂×C₈	1	4	yes	—
83	(C₂×D₅₉)⋊C₄	C₂×C₈	1	4	yes	—
84	Dic₅₉⋊C₄	C₂×C₈	1	4	yes	—
85	Dic₅₉⋊C₄	C₂×C₈	1	4	yes	—
86	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
87	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
88	C₂×C₅₉⋊C₈	C₄²	1	4	yes	—
89	C₂×C₅₉⋊C₈	C₂×C₈	1	4	yes	—
90	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
91	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
92	#11745659	C₂×C₈	1	8	yes	—
93	#11745659	C₂×C₈	1	8	yes	—
94	C₈×D₅₉	C₄²	1	8	yes	—
95	C₈×D₅₉	C₂²×C₄	1	4	yes	—
96	C₈×D₅₉	C₂²×C₄	1	4	yes	—
97	C₁₆×C₅₉	C₂²×C₄	1	8	yes	—
98	C₅₉⋊C₁₆	C₂²×C₄	1	8	yes	—
99	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
100	C₂²×D₅₉	—	4	2	no	—
101	C₂²×D₅₉	C₄×C₈	1	4	yes	—
102	C₂²×D₅₉	C₂²×C₈	1	2	yes	—
103	C₂×C₄×C₅₉	—	4	2	no	—
104	C₂×C₄×C₅₉	C₂×C₄²	1	2	yes	—
105	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
107	C₂×C₄×C₅₉	C₂²×C₈	1	2	yes	—
108	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
110	C₂×Dic₅₉	—	4	2	no	—
111	C₂×Dic₅₉	C₄×C₈	1	4	yes	—
112	C₂×Dic₅₉	C₂²×C₈	1	4	yes	—
113	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
114	C₄×D₅₉	C₄×C₈	1	8	yes	—
115	C₄×D₅₉	C₂×C₄²	1	4	yes	—
116	C₄×D₅₉	C₂²×C₈	1	4	yes	—
117	C₈×C₅₉	C₂×C₄²	1	2	yes	—
118	C₈×C₅₉	C₂²×C₈	1	2	yes	—
119	C₈×C₅₉	C₂²×C₈	1	2	yes	—
120	C₈×C₅₉	C₂³×C₄	1	2	yes	—
121	C₅₉⋊C₈	C₂×C₄²	1	4	yes	—
122	C₅₉⋊C₈	C₂²×C₈	1	4	yes	—
123	#33324	—	59	1	no	maximal, Sylow
124	C₂²×C₅₉	—	2	2	no	—
125	C₂²×C₅₉	C₂×C₄×C₈	1	2	yes	—
126	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
127	C₂×D₅₉	—	2	4	no	—
128	C₂×D₅₉	—	4	4	no	—
129	C₂×D₅₉	C₂×C₄×C₈	1	4	yes	—
130	C₄×C₅₉	—	2	2	no	—
131	C₄×C₅₉	—	2	2	no	—
132	C₄×C₅₉	C₂×C₄×C₈	1	2	yes	—
133	C₄×C₅₉	C₂²×C₄²	1	1	yes	characteristic
134	C₄×C₅₉	C₂³×C₈	1	1	yes	characteristic
135	Dic₅₉	—	2	4	no	—
136	Dic₅₉	C₂×C₄×C₈	1	4	yes	—
137	C₂²×M₃₂	—	59	2	no	—
138	C₂×#405	—	59	1	no	—
139	#2108	—	59	4	no	—
140	#2108	—	59	4	no	—
141	C₄×M₃₂	—	59	2	no	—
142	C₄×M₃₂	—	59	1	no	—
143	C₄×M₃₂	D₅₉	1	1	yes	characteristic, complemented
144	C₂×C₅₉	—	2	2	no	—
145	C₂×C₅₉	C₂²×C₄×C₈	1	1	yes	characteristic
146	C₂×C₅₉	#3407	1	2	yes	—
147	D₅₉	—	2	4	no	—
148	C₂²×M₁₆	—	59	1	no	—
149	C₂³×C₈	—	59	1	no	—
150	C₂×#117	—	59	1	no	—
151	C₂×M₃₂	—	59	2	no	—
152	C₂×M₃₂	—	59	16	no	—
153	C₂×M₃₂	—	59	4	no	—
154	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
155	C₂²×C₁₆	—	59	4	no	—
156	#405	—	59	2	no	—
157	#405	—	59	2	no	—
158	#405	—	59	2	no	—
159	#405	—	59	2	no	—
160	C₂×#97	—	59	1	no	—
161	C₂×#97	—	59	1	no	—
162	C₂×C₄×C₈	—	59	1	no	—
163	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
164	#346	—	59	4	no	—
165	#346	—	59	2	no	—
166	#346	C₂×D₅₉	1	2	yes	complemented
167	C₄×C₁₆	—	59	4	no	—
168	C₄×C₁₆	—	59	2	no	—
169	C₄×C₁₆	C₂×D₅₉	1	2	yes	complemented
170	C₅₉	#33324	1	1	yes	characteristic, complemented, Sylow
171	C₂³×C₄	—	59	1	no	—
172	C₂×M₁₆	—	59	8	no	—
173	C₂×M₁₆	—	59	2	no	—
174	C₂×M₁₆	—	59	2	no	—
175	C₂²×C₈	—	59	1	no	—
176	C₂²×C₈	—	59	1	no	—
177	C₂²×C₈	—	59	2	no	—
178	C₂²×C₈	—	59	8	no	—
179	C₂²×C₈	—	59	2	no	—
180	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
181	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
182	#117	—	59	4	no	—
183	#117	—	59	2	no	—
184	#117	—	59	2	no	—
185	C₂×C₄⋊C₄	—	59	2	no	—
186	C₂×C₂²⋊C₄	—	59	2	no	—
187	C₂×C₄²	—	59	1	no	—
188	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
189	M₃₂	—	59	8	no	—
190	M₃₂	—	59	16	no	—
191	M₃₂	C₄×D₅₉	1	8	yes	—
192	C₂×C₁₆	—	59	4	no	—
193	C₂×C₁₆	—	59	16	no	—
194	C₂×C₁₆	C₂²×D₅₉	1	4	yes	—
195	#97	—	59	2	no	—
196	#97	—	59	2	no	—
197	#97	—	59	2	no	—
198	#97	—	59	2	no	—
199	C₄×C₈	—	59	2	no	—
200	C₄×C₈	—	59	2	no	—
201	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
202	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
203	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
204	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
205	C₂⁴	—	59	1	no	—
206	C₂²×C₄	—	59	1	no	—
207	C₂²×C₄	—	59	4	no	—
208	C₂²×C₄	—	59	2	no	—
209	C₂²×C₄	—	59	4	no	—
210	C₂²×C₄	—	59	2	no	—
211	C₂²×C₄	—	59	2	no	—
212	C₂²×C₄	C₈×D₅₉	1	2	yes	—
213	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
214	M₁₆	—	59	8	no	—
215	M₁₆	—	59	8	no	—
216	C₂×C₈	—	59	4	no	—
217	C₂×C₈	—	59	4	no	—
218	C₂×C₈	—	59	2	no	—
219	C₂×C₈	—	59	2	no	—
220	C₂×C₈	—	59	8	no	—
221	C₂×C₈	—	59	4	no	—
222	C₂×C₈	—	59	4	no	—
223	C₂×C₈	C₈×D₅₉	1	4	yes	—
224	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
225	C₂×C₈	C₂×C₄×D₅₉	1	4	yes	—
226	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
227	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
228	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
229	C₄⋊C₄	—	59	4	no	—
230	C₄⋊C₄	—	59	4	no	—
231	C₂²⋊C₄	—	59	4	no	—
232	C₂²⋊C₄	—	59	4	no	—
233	C₄²	—	59	2	no	—
234	C₄²	—	59	2	no	—
235	C₄²	C₈×D₅₉	1	2	yes	—
236	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
237	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
238	C₁₆	—	59	8	no	—
239	C₁₆	C₂×C₄×D₅₉	1	8	yes	—
240	C₂³	—	59	2	no	—
241	C₂³	—	59	4	no	—
242	C₂³	—	236	2	no	—
243	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	59	2	no	—
246	C₂×C₄	—	59	4	no	—
247	C₂×C₄	—	59	4	no	—
248	C₂×C₄	—	59	4	no	—
249	C₂×C₄	—	59	8	no	—
250	C₂×C₄	—	59	4	no	—
251	C₂×C₄	—	236	2	no	—
252	C₂×C₄	C₄²×D₅₉	1	2	yes	—
253	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
254	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
255	C₂×C₄	C₂×C₈×D₅₉	1	2	yes	—
256	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
258	C₈	—	59	4	no	—
259	C₈	—	59	4	no	—
260	C₈	C₄²×D₅₉	1	2	yes	—
261	C₈	C₂×C₈×D₅₉	1	2	yes	—
262	C₈	C₂×C₈×D₅₉	1	2	yes	—
263	C₈	C₂²×C₄×D₅₉	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	59	4	no	—
266	C₂²	—	118	4	no	—
267	C₂²	—	236	4	no	—
268	C₂²	C₄×C₈×D₅₉	1	2	yes	—
269	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	59	4	no	—
273	C₄	—	118	4	no	—
274	C₄	C₄×C₈×D₅₉	1	2	yes	—
275	C₄	C₂×C₄²×D₅₉	1	1	yes	characteristic
276	C₄	C₂²×C₈×D₅₉	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	118	4	no	—
279	C₂	C₂×C₄×C₈×D₅₉	1	1	yes	characteristic
280	C₂	#21437198	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator	C₂×C₅₉	C₂²×C₄×C₈
Fitting	C₄×M₃₂×C₅₉	C₂
socle	C₂²×C₅₉	C₂³×C₈
prime radical	C₂³×D₅₉	C₂×C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#33324
LCS	G ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
3. #21425694	C₂	semidirect
4. #21425691	C₂	semidirect
5. C₄×#14273667	C₂	semidirect
6. M₃₂×Dic₅₉	C₂	semidirect
7. C₄×M₃₂×C₅₉	C₂	semidirect
13. M₃₂×D₅₉	C₄	semidirect
22. #16837178	C₂²	semidirect
23. #16837177	C₂²	semidirect
24. C₄×C₅₉⋊C₁₆	C₂²	semidirect
25. C₁₆×Dic₅₉	C₂²	semidirect
28. C₅₉×#346	C₂²	semidirect
29. C₄×C₁₆×C₅₉	C₂²	semidirect
46. #14273666	C₂×C₄	semidirect
48. C₁₆×D₅₉	C₂×C₄	semidirect
143. C₄×M₃₂	D₅₉	semidirect
166. #346	C₂×D₅₉	semidirect
169. C₄×C₁₆	C₂×D₅₉	semidirect
170. C₅₉	#33324	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₃₂×D₅₉	2
C₂×#16837356	2
#21425694	2
#21425691	2
C₄×#14273667	2
M₃₂×Dic₅₉	2
C₄×M₃₂×C₅₉	2
#33324	59