Subgroups — #28742264

Order 15104.

552 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈×#11745685	C₂	1	1	yes	maximal, characteristic
2	#21426004	C₂	1	2	yes	maximal, complemented
3	#21425919	C₂	1	4	yes	maximal, complemented
4	C₁₆×D₂₃₆	C₂	1	2	yes	maximal, complemented
5	#21425899	C₂	1	1	yes	maximal, characteristic, complemented
6	#21425897	C₂	1	2	yes	maximal, complemented
7	#346×D₅₉	C₂	1	2	yes	maximal, complemented
8	C₄×M₃₂×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#11745685	C₄	1	1	yes	characteristic
10	C₂×C₁₆×D₅₉	C₂²	1	4	yes	—
11	C₈×Dic₁₁₈	C₄	1	1	yes	characteristic
12	C₈×Dic₁₁₈	C₂²	1	1	yes	characteristic
13	C₈×D₂₃₆	C₄	1	1	yes	characteristic
14	C₈×D₂₃₆	C₂²	1	1	yes	characteristic
15	C₈×C₂²⋊D₅₉	C₄	1	2	yes	—
16	C₈×C₂²⋊D₅₉	C₂²	1	2	yes	—
17	#16837455	C₄	1	1	yes	characteristic
18	#16837453	C₄	1	2	yes	—
19	C₄×C₈×D₅₉	C₂²	1	2	yes	—
20	C₄⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
21	Dic₅₉⋊C₁₆	C₂²	1	4	yes	—
22	C₂²⋊(C₅₉⋊C₁₆)	C₂²	1	2	yes	—
23	(C₂×D₅₉)⋊C₁₆	C₂²	1	4	yes	—
24	#16837178	C₂²	1	2	yes	—
25	#16837177	C₂²	1	4	yes	—
26	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
28	C₅₉×#346	C₂²	1	1	yes	characteristic
29	C₅₉×#346	C₂²	1	1	yes	characteristic
30	C₄×C₁₆×C₅₉	C₂²	1	2	yes	—
31	C₂×#11745685	C₈	1	1	yes	characteristic
32	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
33	C₂×C₈×D₅₉	C₂³	1	2	yes	—
34	C₄×Dic₁₁₈	C₈	1	1	yes	characteristic
35	C₄×Dic₁₁₈	C₂×C₄	1	1	yes	characteristic
36	C₄×D₂₃₆	C₈	1	1	yes	characteristic
37	C₄×D₂₃₆	C₂×C₄	1	1	yes	characteristic
38	C₄×C₂²⋊D₅₉	C₈	1	2	yes	—
39	C₄×C₂²⋊D₅₉	C₂×C₄	1	2	yes	—
40	#14273686	C₈	1	1	yes	characteristic
41	#14273684	C₈	1	2	yes	—
42	C₄²×D₅₉	C₂×C₄	1	2	yes	—
43	C₂×C₅₉⋊C₁₆	C₂³	1	4	yes	—
44	C₁₆×D₅₉	—	2	8	no	—
45	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
46	C₄⋊(C₅₉⋊C₈)	C₂×C₄	1	1	yes	characteristic
47	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
48	Dic₅₉⋊C₈	C₂×C₄	1	2	yes	—
49	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	2	yes	—
50	(C₂×D₅₉)⋊C₈	C₂×C₄	1	2	yes	—
51	(C₂×D₅₉)⋊C₈	C₂×C₄	1	2	yes	—
52	C₄×C₅₉⋊C₈	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	—	2	4	no	—
59	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
60	C₂×C₁₆×C₅₉	C₂³	1	2	yes	—
61	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
62	C₄×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
64	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
65	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₉	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₅₉	C₂×C₈	1	2	yes	—
69	#11745685	—	2	4	no	—
70	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₅₉	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₅₉	—	2	2	no	—
76	C₂×C₈×C₅₉	—	2	2	no	—
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	C₂⁴	1	1	yes	characteristic
81	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×C₈	1	1	yes	characteristic
83	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
85	C₂²⋊Dic₅₉	C₂×C₈	1	2	yes	—
86	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
87	(C₂×D₅₉)⋊C₄	C₂×C₈	1	2	yes	—
88	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
89	C₄⋊Dic₅₉	C₂×C₈	1	1	yes	characteristic
90	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
91	Dic₅₉⋊C₄	C₂×C₈	1	2	yes	—
92	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
93	C₄×Dic₅₉	C₂×C₈	1	2	yes	—
94	C₄×Dic₅₉	C₂²×C₄	1	2	yes	—
95	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
96	C₂×C₅₉⋊C₈	C₂²×C₄	1	2	yes	—
97	C₈×D₅₉	—	2	4	no	—
98	C₈×D₅₉	—	2	4	no	—
99	C₁₆×C₅₉	—	2	2	no	—
100	C₁₆×C₅₉	#41	1	4	yes	complemented
101	C₅₉⋊C₁₆	—	2	4	no	—
102	C₂³×C₅₉	C₂²×C₈	1	1	yes	characteristic
103	C₂²×D₅₉	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	2	no	—
107	C₂×C₄×C₅₉	—	4	1	no	—
108	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
114	C₂²⋊D₅₉	—	2	4	no	—
115	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
116	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
117	C₂×Dic₅₉	C₂²×C₈	1	2	yes	—
118	D₂₃₆	M₃₂	1	4	yes	complemented
119	C₄×D₅₉	—	2	4	no	—
120	C₄×D₅₉	—	2	2	no	—
121	C₄×D₅₉	—	2	2	no	—
122	C₄×D₅₉	—	4	2	no	—
123	Dic₁₁₈	—	2	2	no	—
124	C₈×C₅₉	—	2	1	no	—
125	C₈×C₅₉	—	2	1	no	—
126	C₈×C₅₉	#131	1	2	yes	—
127	C₈×C₅₉	C₂×#41	1	2	yes	—
128	C₅₉⋊C₈	—	2	2	no	—
129	C₅₉⋊C₈	—	2	2	no	—
130	#33484	—	59	1	no	maximal, Sylow
131	C₂²×C₅₉	—	2	1	no	—
132	C₂²×C₅₉	—	2	1	no	—
133	C₂²×C₅₉	—	2	1	no	—
134	C₂²×C₅₉	C₂³×C₈	1	1	yes	characteristic
135	C₂×D₅₉	—	2	2	no	—
136	C₂×D₅₉	—	2	2	no	—
137	C₂×D₅₉	C₂×M₃₂	1	4	yes	—
138	C₄×C₅₉	—	2	1	no	—
139	C₄×C₅₉	—	2	1	no	—
140	C₄×C₅₉	—	2	1	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	C₂×M₃₂	1	2	yes	—
143	C₄×C₅₉	C₄×#41	1	1	yes	characteristic
144	C₄×C₅₉	C₂×#131	1	1	yes	characteristic
145	Dic₅₉	—	2	2	no	—
146	Dic₅₉	—	2	2	no	—
147	Dic₅₉	—	2	2	no	—
148	C₈×#41	—	59	1	no	—
149	#2183	—	59	2	no	—
150	#2168	—	59	4	no	—
151	D₄×C₁₆	—	59	2	no	—
152	#2163	—	59	1	no	—
153	#2163	—	59	2	no	—
154	C₄×M₃₂	D₅₉	1	1	yes	characteristic, complemented
155	C₂×#346	—	59	2	no	—
156	C₂×C₅₉	—	4	1	no	—
157	C₂×C₅₉	C₈×#41	1	1	yes	characteristic
158	C₂×C₅₉	C₂²×M₃₂	1	1	yes	characteristic
159	C₂×C₅₉	#3407	1	1	yes	characteristic
160	D₅₉	—	2	4	no	—
161	C₄×#41	—	59	1	no	—
162	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
163	C₂²×C₁₆	—	59	4	no	—
164	C₈×Q₈	—	59	1	no	—
165	C₈×Q₈	—	59	1	no	—
166	C₈×D₄	—	59	2	no	—
167	C₈×D₄	—	59	2	no	—
168	C₈×D₄	—	59	1	no	—
169	C₈×D₄	—	59	1	no	—
170	#431	—	59	1	no	—
171	#431	—	59	2	no	—
172	C₂×C₄×C₈	—	59	2	no	—
173	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
174	C₄⋊C₁₆	—	59	4	no	—
175	C₄⋊C₁₆	—	59	2	no	—
176	C₂²⋊C₁₆	—	59	2	no	—
177	C₂²⋊C₁₆	—	59	4	no	—
178	#346	—	59	4	no	—
179	#346	—	59	2	no	—
180	#346	C₂×D₅₉	1	1	yes	characteristic
181	#346	C₂×D₅₉	1	1	yes	characteristic
182	C₄×C₁₆	C₂×D₅₉	1	2	yes	—
183	C₅₉	#33484	1	1	yes	characteristic, complemented, Sylow
184	C₂×#41	—	59	1	no	—
185	C₂²×C₈	—	59	2	no	—
186	C₂²×C₈	—	59	2	no	—
187	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
188	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
189	C₄×Q₈	—	59	1	no	—
190	C₄×Q₈	—	59	1	no	—
191	C₄×D₄	—	59	2	no	—
192	C₄×D₄	—	59	2	no	—
193	C₄×D₄	—	59	1	no	—
194	C₄×D₄	—	59	1	no	—
195	#117	—	59	1	no	—
196	#117	—	59	2	no	—
197	C₂×C₄²	—	59	2	no	—
198	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
199	M₃₂	—	2	4	no	—
200	C₂×C₁₆	—	59	4	no	—
201	C₂×C₁₆	—	118	8	no	—
202	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
203	C₂×C₁₆	C₂²×D₅₉	1	2	yes	—
204	C₄⋊C₈	—	59	2	no	—
205	C₄⋊C₈	—	59	2	no	—
206	C₄⋊C₈	—	59	1	no	—
207	C₄⋊C₈	—	59	1	no	—
208	C₂²⋊C₈	—	59	2	no	—
209	C₂²⋊C₈	—	59	2	no	—
210	C₂²⋊C₈	—	59	2	no	—
211	C₄×C₈	—	59	2	no	—
212	C₄×C₈	—	59	2	no	—
213	C₄×C₈	—	59	2	no	—
214	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
215	C₄×C₈	C₄×D₅₉	1	1	yes	characteristic
216	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
217	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
218	#41	—	118	4	no	—
219	C₂×Q₈	—	59	1	no	—
220	C₂×D₄	—	59	2	no	—
221	C₂×D₄	—	59	1	no	—
222	C₂²×C₄	—	59	2	no	—
223	C₂²×C₄	—	59	2	no	—
224	C₂²×C₄	—	59	2	no	—
225	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
226	C₂²×C₄	C₈×D₅₉	1	1	yes	characteristic
227	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	59	2	no	—
231	C₂×C₈	—	59	2	no	—
232	C₂×C₈	—	118	4	no	—
233	C₂×C₈	—	118	4	no	—
234	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
235	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
236	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₅₉	1	1	yes	characteristic
238	C₄⋊C₄	—	59	2	no	—
239	C₄⋊C₄	—	59	2	no	—
240	C₄⋊C₄	—	59	1	no	—
241	C₄⋊C₄	—	59	1	no	—
242	C₂²⋊C₄	—	59	2	no	—
243	C₂²⋊C₄	—	59	2	no	—
244	C₂²⋊C₄	—	59	2	no	—
245	C₄²	—	59	2	no	—
246	C₄²	—	59	2	no	—
247	C₄²	—	59	2	no	—
248	C₄²	C₈×D₅₉	1	1	yes	characteristic
249	C₄²	C₈×D₅₉	1	1	yes	characteristic
250	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
251	C₄²	C₂×C₄×D₅₉	1	1	yes	characteristic
252	C₁₆	—	2	2	no	—
253	C₁₆	—	118	4	no	—
254	C₁₆	#11745685	1	4	yes	complemented
255	C₂³	—	59	2	no	—
256	C₂³	C₂×C₈×D₅₉	1	1	yes	characteristic
257	Q₈	—	118	2	no	—
258	D₄	—	59	4	no	—
259	D₄	—	118	4	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	59	2	no	—
265	C₂×C₄	—	59	2	no	—
266	C₂×C₄	—	59	2	no	—
267	C₂×C₄	—	118	4	no	—
268	C₂×C₄	—	118	2	no	—
269	C₂×C₄	—	118	2	no	—
270	C₂×C₄	—	236	2	no	—
271	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
272	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
273	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
274	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
276	C₂×C₄	C₂²×C₄×D₅₉	1	1	yes	characteristic
277	C₈	—	2	1	no	—
278	C₈	—	2	1	no	—
279	C₈	—	118	2	no	—
280	C₈	—	118	2	no	—
281	C₈	#14273733	1	2	yes	—
282	C₈	C₂×#11745685	1	2	yes	—
283	C₂²	—	2	1	no	—
284	C₂²	—	2	1	no	—
285	C₂²	—	2	1	no	—
286	C₂²	—	59	4	no	—
287	C₂²	—	118	2	no	—
288	C₂²	—	118	2	no	—
289	C₂²	C₂²×C₈×D₅₉	1	1	yes	characteristic
290	C₄	—	2	1	no	—
291	C₄	—	2	1	no	—
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	118	2	no	—
295	C₄	—	118	2	no	—
296	C₄	—	118	2	no	—
297	C₄	M₃₂×D₅₉	1	2	yes	—
298	C₄	C₄×#11745685	1	1	yes	characteristic
299	C₄	C₂×#14273733	1	1	yes	characteristic
300	C₂	—	4	1	no	—
301	C₂	—	118	4	no	—
302	C₂	C₈×#11745685	1	1	yes	characteristic
303	C₂	C₂×M₃₂×D₅₉	1	1	yes	characteristic
304	C₂	#21437198	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₅₉
commutator, socle	C₂²×C₅₉	C₂³×C₈
Fitting	C₄×M₃₂×C₅₉	C₂
prime radical	C₂×#11745685	C₈
cosocle	C₂×C₈×C₅₉	C₂⁴
hypercenter	C₄×M₃₂	D₅₉
hypocenter	C₅₉	#33484
LCS	G ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
2. #21426004	C₂	semidirect
3. #21425919	C₂	semidirect
4. C₁₆×D₂₃₆	C₂	semidirect
5. #21425899	C₂	semidirect
6. #21425897	C₂	semidirect
7. #346×D₅₉	C₂	semidirect
8. C₄×M₃₂×C₅₉	C₂	semidirect
100. C₁₆×C₅₉	#41	semidirect
118. D₂₃₆	M₃₂	semidirect
154. C₄×M₃₂	D₅₉	semidirect
183. C₅₉	#33484	semidirect
254. C₁₆	#11745685	semidirect

Maximal subgroups

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

subgroup	index
C₈×#11745685	2
#21426004	2
#21425919	2
C₁₆×D₂₃₆	2
#21425899	2
#21425897	2
#346×D₅₉	2
C₄×M₃₂×C₅₉	2
#33484	59