Subgroups — #28479218

Order 15104.

506 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837356	C₂	1	1	yes	maximal, characteristic
2	#21422408	C₂	1	2	yes	maximal
3	#21422379	C₂	1	1	yes	maximal, characteristic, complemented
4	#21421940	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1450	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11745659	C₄	1	1	yes	characteristic
7	C₂²×C₈×D₅₉	C₄	1	1	yes	characteristic
8	C₂×#14273684	C₂²	1	1	yes	characteristic
9	#16837455	C₂²	1	2	yes	—
10	#16837356	—	2	1	no	—
11	#16837356	—	2	1	no	—
12	#16837356	—	2	1	no	—
13	#16837356	—	2	1	no	—
14	C₂×#14273638	C₂²	1	1	yes	characteristic
15	C₂×#14273637	C₄	1	1	yes	characteristic
16	C₂×C₈×Dic₅₉	C₄	1	1	yes	characteristic
17	#16837127	C₄	1	4	yes	—
18	#16837125	C₄	1	4	yes	—
19	#16837123	—	2	1	no	—
20	C₈×C₅₉⋊C₈	—	2	1	no	—
21	C₅₉×#431	C₂²	1	2	yes	—
22	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
23	C₅₉×#322	—	2	1	no	—
24	C₈²×C₅₉	—	2	1	no	—
25	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
26	C₂×#11745659	—	2	1	no	—
27	C₂×#11745659	—	2	1	no	—
28	C₂×#11745659	—	2	2	no	—
29	C₂×#11745659	C₈	1	4	yes	complemented
30	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×D₅₉	—	2	1	no	—
33	C₂×C₈×D₅₉	—	2	1	no	—
34	C₂×C₈×D₅₉	—	2	2	no	—
35	C₂×C₈×D₅₉	C₈	1	4	yes	complemented
36	#14273684	—	2	1	no	—
37	#14273684	—	2	1	no	—
38	#14273684	—	2	2	no	—
39	C₂×Dic₅₉⋊C₄	C₂×C₄	1	2	yes	—
40	C₂×(C₂×D₅₉)⋊C₄	C₂×C₄	1	2	yes	—
41	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
42	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
43	C₂²⋊(C₅₉⋊C₈)	C₂×C₄	1	4	yes	—
44	#14273638	—	2	1	no	—
45	#14273638	—	2	1	no	—
46	#14273637	—	2	1	no	—
47	#14273637	—	2	1	no	—
48	C₄×C₅₉⋊C₈	—	2	2	no	—
49	C₈×Dic₅₉	—	2	1	no	—
50	C₈×Dic₅₉	—	2	1	no	—
51	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₅₉	—	2	2	no	—
55	C₂²⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
56	C₄×C₈×C₅₉	—	2	2	no	—
57	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
58	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
59	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
60	C₄×C₈×C₅₉	D₄	1	1	yes	characteristic
61	C₂³×D₅₉	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
64	C₂²×Dic₅₉	C₄²	1	1	yes	characteristic
65	C₂²×Dic₅₉	C₄²	1	2	yes	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
71	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
72	C₂×C₈×C₅₉	—	2	4	no	—
73	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
78	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
80	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
81	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
82	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
83	C₄²×C₅₉	#41	1	2	yes	—
84	(C₂×D₅₉)⋊C₄	—	2	2	no	—
85	(C₂×D₅₉)⋊C₄	—	2	2	no	—
86	Dic₅₉⋊C₄	—	2	2	no	—
87	Dic₅₉⋊C₄	—	2	2	no	—
88	C₄×Dic₅₉	—	2	1	no	—
89	C₄×Dic₅₉	—	2	1	no	—
90	C₂×C₅₉⋊C₈	—	2	1	no	—
91	C₂×C₅₉⋊C₈	—	2	1	no	—
92	C₂×C₅₉⋊C₈	—	2	1	no	—
93	C₂×C₅₉⋊C₈	—	2	1	no	—
94	C₂×C₅₉⋊C₈	—	2	4	no	—
95	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
96	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
97	#11745659	—	2	4	no	—
98	#11745659	—	2	4	no	—
99	C₈×D₅₉	—	2	4	no	—
100	C₈×D₅₉	—	2	4	no	—
101	C₂³×C₅₉	C₂×C₄²	1	1	yes	characteristic
102	C₂²×D₅₉	—	2	1	no	—
103	C₂²×D₅₉	—	2	1	no	—
104	C₂²×D₅₉	—	4	2	no	—
105	C₂²×D₅₉	C₄×C₈	1	2	yes	—
106	C₂×C₄×C₅₉	—	4	2	no	—
107	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₅₉	C₄×D₄	1	2	yes	—
111	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	—	2	1	no	—
117	C₂×Dic₅₉	—	2	2	no	—
118	C₂×Dic₅₉	—	4	2	no	—
119	C₂×Dic₅₉	C₄×C₈	1	2	yes	—
120	C₄×D₅₉	—	2	2	no	—
121	C₄×D₅₉	—	2	2	no	—
122	C₄×D₅₉	—	2	2	no	—
123	C₄×D₅₉	—	2	2	no	—
124	C₈×C₅₉	—	4	4	no	—
125	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
126	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
127	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
128	C₈×C₅₉	C₂²⋊C₈	1	2	yes	—
129	C₅₉⋊C₈	—	2	2	no	—
130	C₅₉⋊C₈	—	2	2	no	—
131	C₅₉⋊C₈	—	4	4	no	—
132	#8156	—	59	1	no	maximal, Sylow
133	C₂²×C₅₉	—	2	2	no	—
134	C₂²×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₂²×C₅₉	C₂×C₄×C₈	1	1	yes	characteristic
136	C₂²×C₅₉	C₄×M₁₆	1	1	yes	characteristic
137	C₂×D₅₉	—	2	2	no	—
138	C₂×D₅₉	—	4	2	no	—
139	C₂×D₅₉	—	4	2	no	—
140	C₂×D₅₉	—	4	2	no	—
141	C₄×C₅₉	—	2	2	no	—
142	C₄×C₅₉	—	2	2	no	—
143	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
144	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
145	C₄×C₅₉	#431	1	1	yes	characteristic
146	C₄×C₅₉	#431	1	1	yes	characteristic
147	Dic₅₉	—	2	2	no	—
148	Dic₅₉	—	4	2	no	—
149	C₂×#405	—	59	1	no	—
150	#1458	—	59	2	no	—
151	#1450	—	59	1	no	—
152	#1450	D₅₉	1	1	yes	characteristic, complemented
153	#1281	—	59	2	no	—
154	C₂×C₅₉	—	2	1	no	—
155	C₂×C₅₉	—	2	1	no	—
156	C₂×C₅₉	C₄×C₂²⋊C₈	1	1	yes	characteristic
157	C₂×C₅₉	#1757	1	2	yes	—
158	D₅₉	—	4	2	no	—
159	C₂²×M₁₆	—	59	1	no	—
160	C₂³×C₈	—	59	1	no	—
161	C₂×#117	—	59	1	no	—
162	#431	—	59	2	no	—
163	#431	C₂×D₅₉	1	2	yes	—
164	#405	—	118	1	no	—
165	#405	—	118	1	no	—
166	#405	—	118	1	no	—
167	#405	—	118	1	no	—
168	C₂×#97	—	59	1	no	—
169	C₂×#97	—	59	1	no	—
170	C₂×C₄×C₈	—	59	1	no	—
171	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
172	#324	—	59	4	no	—
173	#323	—	59	4	no	—
174	#322	—	2	1	no	—
175	#322	—	118	1	no	—
176	C₈²	—	2	1	no	—
177	C₈²	—	118	1	no	—
178	C₅₉	#8156	1	1	yes	characteristic, complemented, Sylow
179	C₂³×C₄	—	59	1	no	—
180	C₂×M₁₆	—	59	4	no	—
181	C₂×M₁₆	—	118	2	no	—
182	C₂×M₁₆	—	118	1	no	—
183	C₂×M₁₆	—	118	1	no	—
184	C₂²×C₈	—	59	1	no	—
185	C₂²×C₈	—	59	1	no	—
186	C₂²×C₈	—	59	4	no	—
187	C₂²×C₈	—	118	2	no	—
188	C₂²×C₈	—	118	1	no	—
189	C₂²×C₈	—	118	1	no	—
190	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
191	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
192	#117	—	118	1	no	—
193	#117	—	118	2	no	—
194	#117	—	118	1	no	—
195	C₂×C₄⋊C₄	—	59	2	no	—
196	C₂×C₂²⋊C₄	—	59	2	no	—
197	C₂×C₄²	—	59	1	no	—
198	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
199	C₄⋊C₈	—	2	2	no	—
200	C₄⋊C₈	—	118	2	no	—
201	C₂²⋊C₈	—	59	4	no	—
202	C₂²⋊C₈	C₄×D₅₉	1	4	yes	—
203	#97	—	118	1	no	—
204	#97	—	118	1	no	—
205	#97	—	118	1	no	—
206	#97	—	118	1	no	—
207	C₄×C₈	—	2	2	no	—
208	C₄×C₈	—	118	1	no	—
209	C₄×C₈	—	118	1	no	—
210	C₄×C₈	—	118	2	no	—
211	C₄×C₈	D₂₃₆	1	1	yes	characteristic
212	C₄×C₈	D₂₃₆	1	1	yes	characteristic
213	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
214	C₄×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
215	C₂⁴	—	59	1	no	—
216	C₂²×C₄	—	59	1	no	—
217	C₂²×C₄	—	59	2	no	—
218	C₂²×C₄	—	59	2	no	—
219	C₂²×C₄	—	59	2	no	—
220	C₂²×C₄	—	118	1	no	—
221	C₂²×C₄	—	118	1	no	—
222	C₂²×C₄	—	118	1	no	—
223	C₂²×C₄	—	118	1	no	—
224	C₂²×C₄	C₂×C₄×D₅₉	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
226	M₁₆	—	118	4	no	—
227	M₁₆	—	118	4	no	—
228	C₂×C₈	—	2	4	no	—
229	C₂×C₈	—	59	2	no	—
230	C₂×C₈	—	59	2	no	—
231	C₂×C₈	—	118	1	no	—
232	C₂×C₈	—	118	1	no	—
233	C₂×C₈	—	118	1	no	—
234	C₂×C₈	—	118	1	no	—
235	C₂×C₈	—	118	4	no	—
236	C₂×C₈	—	118	4	no	—
237	C₂×C₈	—	118	4	no	—
238	C₂×C₈	C₈×D₅₉	1	2	yes	—
239	C₂×C₈	C₈×D₅₉	1	2	yes	—
240	C₂×C₈	#11745659	1	2	yes	—
241	C₂×C₈	#11745659	1	2	yes	—
242	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
243	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
244	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
245	C₂×C₈	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
246	C₄⋊C₄	—	118	2	no	—
247	C₄⋊C₄	—	118	2	no	—
248	C₂²⋊C₄	—	118	2	no	—
249	C₂²⋊C₄	—	118	2	no	—
250	C₄²	—	118	1	no	—
251	C₄²	—	118	1	no	—
252	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
253	C₄²	#11745685	1	2	yes	—
254	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
255	C₂³	—	59	2	no	—
256	C₂³	—	118	1	no	—
257	C₂³	—	118	1	no	—
258	C₂³	—	236	2	no	—
259	C₂³	C₄²×D₅₉	1	1	yes	characteristic
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	—	59	2	no	—
262	C₂×C₄	—	118	1	no	—
263	C₂×C₄	—	118	1	no	—
264	C₂×C₄	—	118	2	no	—
265	C₂×C₄	—	118	2	no	—
266	C₂×C₄	—	118	2	no	—
267	C₂×C₄	—	118	2	no	—
268	C₂×C₄	—	118	2	no	—
269	C₂×C₄	—	236	2	no	—
270	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
271	C₂×C₄	#14273684	1	1	yes	characteristic
272	C₂×C₄	C₄×C₂²⋊D₅₉	1	2	yes	—
273	C₂×C₄	C₄×D₂₃₆	1	2	yes	—
274	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
275	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
276	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
277	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
278	C₈	—	4	4	no	—
279	C₈	—	118	2	no	—
280	C₈	—	118	2	no	—
281	C₈	—	236	4	no	—
282	C₈	(C₂×D₅₉)⋊C₈	1	2	yes	—
283	C₈	(C₂×D₅₉)⋊C₈	1	2	yes	—
284	C₈	(C₂×D₅₉)⋊C₈	1	2	yes	—
285	C₈	(C₂×D₅₉)⋊C₈	1	2	yes	—
286	C₂²	—	2	2	no	—
287	C₂²	—	118	2	no	—
288	C₂²	—	236	2	no	—
289	C₂²	—	236	2	no	—
290	C₂²	—	236	2	no	—
291	C₂²	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
292	C₂²	C₄×C₈×D₅₉	1	1	yes	characteristic
293	C₂²	C₄×#11745659	1	1	yes	characteristic
294	C₄	—	2	2	no	—
295	C₄	—	2	2	no	—
296	C₄	—	118	2	no	—
297	C₄	—	236	2	no	—
298	C₄	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
299	C₄	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
300	C₄	#16837453	1	1	yes	characteristic
301	C₄	#16837453	1	1	yes	characteristic
302	C₂	—	2	1	no	—
303	C₂	—	2	1	no	—
304	C₂	—	236	2	no	—
305	C₂	C₄×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
306	C₂	#21423897	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	(C₂×D₅₉)⋊C₄
commutator	C₂²×C₅₉	C₂×C₄×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₅₉×#1450	C₂
socle	C₂²×C₅₉	C₄×C₂²⋊C₄
prime radical	C₂³×D₅₉	C₄²
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	#1450	D₅₉
hypocenter	C₅₉	#8156
LCS	G ▷ C₂²×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1450

Complements

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

N	H ≅ G/N	type
3. #21422379	C₂	semidirect
4. #21421940	C₂	semidirect
5. C₅₉×#1450	C₂	semidirect
29. C₂×#11745659	C₈	semidirect
35. C₂×C₈×D₅₉	C₈	semidirect
152. #1450	D₅₉	semidirect
178. C₅₉	#8156	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837356	2
#21422408	2
#21422379	2
#21421940	2
C₅₉×#1450	2
#8156	59