Subgroups — #28504640

Order 15104.

482 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#11745659	C₂	1	1	yes	maximal, characteristic
2	C₂×#16837223	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16837183	C₂	1	2	yes	maximal
4	#21422175	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₉×#364	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11745659	C₄	1	1	yes	characteristic
7	C₂²×#11745659	C₂²	1	1	yes	characteristic
8	C₂×C₄²×D₅₉	C₄	1	1	yes	characteristic
9	C₂×#14273667	C₂²	1	2	yes	—
10	C₄×#11745659	—	2	2	no	—
11	C₄×#11745659	—	2	2	no	—
12	C₂×#14273637	C₄	1	1	yes	characteristic
13	C₂×#14273637	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₅₉⋊C₈	C₄	1	1	yes	characteristic
15	#16837223	—	2	2	no	—
16	#16837183	—	2	4	no	—
17	C₂×M₃₂×C₅₉	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
19	C₅₉×#364	—	2	2	no	—
20	C₂²×C₄×D₅₉	—	2	1	no	—
21	C₂²×C₄×D₅₉	C₂×C₄	1	1	yes	characteristic
22	C₂×#11745659	—	2	2	no	—
23	C₂×#11745659	—	2	2	no	—
24	C₂×#11745659	—	2	1	no	—
25	C₂×#11745659	—	2	1	no	—
26	C₂×#11745659	—	2	1	no	—
27	C₂×#11745659	—	2	1	no	—
28	C₂×#11745659	C₈	1	4	yes	—
29	C₂×#11745659	D₄	1	2	yes	—
30	C₂×#11745659	Q₈	1	2	yes	—
31	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₅₉⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄×Dic₅₉	—	2	1	no	—
34	C₂×C₄×Dic₅₉	C₂×C₄	1	1	yes	characteristic
35	C₄²×D₅₉	—	2	2	no	—
36	C₄²×D₅₉	—	2	2	no	—
37	#14273667	—	2	2	no	—
38	#14273667	—	2	2	no	—
39	C₂×C₅₉⋊C₁₆	—	2	2	no	—
40	#14273637	—	2	2	no	—
41	#14273637	—	2	1	no	—
42	#14273637	—	2	1	no	—
43	C₄×C₅₉⋊C₈	—	2	2	no	—
44	C₂²×C₈×C₅₉	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
47	M₃₂×C₅₉	—	2	2	no	—
48	M₃₂×C₅₉	—	2	2	no	—
49	C₂×C₁₆×C₅₉	—	2	2	no	—
50	C₄×C₈×C₅₉	D₄	1	2	yes	—
51	C₄×C₈×C₅₉	D₄	1	2	yes	—
52	C₂³×D₅₉	M₁₆	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	—	2	1	no	—
54	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₅₉	—	2	1	no	—
56	C₂²×Dic₅₉	M₁₆	1	1	yes	characteristic
57	C₂×C₄×D₅₉	—	2	1	no	—
58	C₂×C₄×D₅₉	—	2	1	no	—
59	C₂×C₄×D₅₉	—	2	1	no	—
60	C₂×C₄×D₅₉	—	2	1	no	—
61	C₂×C₄×D₅₉	—	2	4	no	—
62	C₂×C₄×D₅₉	—	4	4	no	—
63	C₂×C₄×D₅₉	C₄⋊C₄	1	2	yes	—
64	C₂×C₄×D₅₉	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
66	C₂×C₈×C₅₉	C₂×C₈	1	2	yes	—
67	C₂×C₈×C₅₉	M₁₆	1	2	yes	—
68	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂×Q₈	1	1	yes	characteristic
72	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
73	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
74	C₄²×C₅₉	—	2	1	no	—
75	C₄²×C₅₉	C₂²⋊C₄	1	2	yes	—
76	C₄×Dic₅₉	—	2	4	no	—
77	C₄×Dic₅₉	—	2	1	no	—
78	C₄×Dic₅₉	M₁₆	1	2	yes	—
79	C₂×C₅₉⋊C₈	—	2	1	no	—
80	C₂×C₅₉⋊C₈	—	2	1	no	—
81	C₂×C₅₉⋊C₈	—	2	1	no	—
82	C₂×C₅₉⋊C₈	—	2	1	no	—
83	C₂×C₅₉⋊C₈	C₄⋊C₄	1	2	yes	—
84	C₂×C₅₉⋊C₈	C₂×C₈	1	2	yes	—
85	#11745659	—	2	8	no	—
86	#11745659	—	2	2	no	—
87	#11745659	—	2	2	no	—
88	#11745659	—	2	2	no	—
89	#11745659	—	2	2	no	—
90	C₁₆×C₅₉	—	4	2	no	—
91	C₅₉⋊C₁₆	—	4	2	no	—
92	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
93	C₂²×D₅₉	—	2	1	no	—
94	C₂²×D₅₉	—	2	1	no	—
95	C₂²×D₅₉	—	4	2	no	—
96	C₂²×D₅₉	C₄⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₉	—	2	4	no	—
98	C₂×C₄×C₅₉	—	2	2	no	—
99	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
103	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
105	C₂×Dic₅₉	—	2	1	no	—
106	C₂×Dic₅₉	—	2	1	no	—
107	C₂×Dic₅₉	—	2	2	no	—
108	C₂×Dic₅₉	—	4	2	no	—
109	C₂×Dic₅₉	C₄⋊C₈	1	2	yes	—
110	C₄×D₅₉	—	2	2	no	—
111	C₄×D₅₉	—	2	2	no	—
112	C₄×D₅₉	—	2	2	no	—
113	C₄×D₅₉	—	2	2	no	—
114	C₄×D₅₉	—	4	8	no	—
115	C₈×C₅₉	C₂²⋊C₈	1	4	yes	—
116	C₈×C₅₉	#121	1	1	yes	characteristic
117	C₈×C₅₉	#121	1	1	yes	characteristic
118	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
119	C₈×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
120	C₅₉⋊C₈	—	2	2	no	—
121	C₅₉⋊C₈	—	2	2	no	—
122	#12791	—	59	1	no	maximal, Sylow
123	C₂²×C₅₉	—	2	2	no	—
124	C₂²×C₅₉	#407	1	1	yes	characteristic
125	C₂²×C₅₉	C₂×C₄⋊C₈	1	1	yes	characteristic
126	C₂²×C₅₉	#432	1	1	yes	characteristic
127	C₂×D₅₉	—	2	2	no	—
128	C₂×D₅₉	—	4	2	no	—
129	C₂×D₅₉	—	4	2	no	—
130	C₂×D₅₉	—	4	2	no	—
131	C₄×C₅₉	—	2	4	no	—
132	C₄×C₅₉	#380	1	1	yes	characteristic
133	C₄×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
134	C₄×C₅₉	#423	1	1	yes	characteristic
135	C₄×C₅₉	#431	1	1	yes	characteristic
136	Dic₅₉	—	2	2	no	—
137	Dic₅₉	—	4	2	no	—
138	C₂×C₄×M₁₆	—	59	1	no	—
139	C₂×#364	—	59	1	no	—
140	C₂×#364	D₅₉	1	1	yes	characteristic, complemented
141	C₂×#349	—	59	2	no	—
142	#1375	—	59	2	no	—
143	C₂×C₅₉	—	2	1	no	—
144	C₂×C₅₉	—	2	1	no	—
145	C₂×C₅₉	#1797	1	1	yes	characteristic
146	C₂×C₅₉	#2165	1	2	yes	—
147	D₅₉	—	4	2	no	—
148	C₂²×M₁₆	—	59	1	no	—
149	C₂²×M₁₆	—	59	1	no	—
150	C₂²×C₄²	—	59	1	no	—
151	C₂×M₃₂	—	59	2	no	—
152	C₂×M₃₂	C₂×D₅₉	1	2	yes	—
153	C₄×M₁₆	—	118	2	no	—
154	C₄×M₁₆	—	118	2	no	—
155	C₂×#97	—	59	1	no	—
156	C₂×#97	—	59	1	no	—
157	C₂×C₄×C₈	—	59	1	no	—
158	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
159	#364	—	2	2	no	—
160	#364	—	118	2	no	—
161	#349	—	118	4	no	—
162	C₅₉	#12791	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	—	59	1	no	—
164	C₂³×C₄	—	118	1	no	—
165	C₂×M₁₆	—	59	4	no	—
166	C₂×M₁₆	—	59	2	no	—
167	C₂×M₁₆	—	59	2	no	—
168	C₂×M₁₆	—	118	2	no	—
169	C₂×M₁₆	—	118	2	no	—
170	C₂×M₁₆	—	118	1	no	—
171	C₂×M₁₆	—	118	1	no	—
172	C₂×M₁₆	—	118	1	no	—
173	C₂×M₁₆	—	118	1	no	—
174	C₂²×C₈	—	59	1	no	—
175	C₂²×C₈	—	59	1	no	—
176	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
177	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
178	C₂×C₄²	—	59	1	no	—
179	C₂×C₄²	—	118	2	no	—
180	C₂×C₄²	—	118	1	no	—
181	C₂×C₄²	—	118	2	no	—
182	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
183	M₃₂	—	2	2	no	—
184	M₃₂	—	2	2	no	—
185	M₃₂	—	118	2	no	—
186	M₃₂	—	118	2	no	—
187	C₂×C₁₆	—	2	2	no	—
188	C₂×C₁₆	—	118	2	no	—
189	#97	—	118	2	no	—
190	#97	—	118	1	no	—
191	#97	—	118	1	no	—
192	C₄×C₈	—	118	2	no	—
193	C₄×C₈	D₂₃₆	1	2	yes	—
194	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
195	C₂⁴	—	59	1	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	59	1	no	—
198	C₂²×C₄	—	59	2	no	—
199	C₂²×C₄	—	59	2	no	—
200	C₂²×C₄	—	118	4	no	—
201	C₂²×C₄	—	118	1	no	—
202	C₂²×C₄	—	118	1	no	—
203	C₂²×C₄	—	118	1	no	—
204	C₂²×C₄	—	118	1	no	—
205	C₂²×C₄	—	118	1	no	—
206	C₂²×C₄	—	236	4	no	—
207	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
208	M₁₆	—	118	8	no	—
209	M₁₆	—	118	2	no	—
210	M₁₆	—	118	2	no	—
211	M₁₆	—	118	2	no	—
212	M₁₆	—	118	2	no	—
213	C₂×C₈	—	59	2	no	—
214	C₂×C₈	—	59	2	no	—
215	C₂×C₈	—	118	1	no	—
216	C₂×C₈	—	118	1	no	—
217	C₂×C₈	—	118	1	no	—
218	C₂×C₈	—	118	1	no	—
219	C₂×C₈	C₈×D₅₉	1	2	yes	—
220	C₂×C₈	#11745659	1	2	yes	—
221	C₂×C₈	(C₂×D₅₉)⋊C₄	1	2	yes	—
222	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
223	C₂×C₈	D₄×D₅₉	1	1	yes	characteristic
224	C₂×C₈	#11745687	1	1	yes	characteristic
225	C₂×C₈	Q₈×D₅₉	1	1	yes	characteristic
226	C₂×C₈	#11745689	1	1	yes	characteristic
227	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
228	C₄²	—	2	1	no	—
229	C₄²	—	59	2	no	—
230	C₄²	—	118	4	no	—
231	C₄²	—	118	1	no	—
232	C₄²	(C₂×D₅₉)⋊C₄	1	2	yes	—
233	C₁₆	—	4	2	no	—
234	C₁₆	—	236	2	no	—
235	C₂³	—	59	2	no	—
236	C₂³	—	118	1	no	—
237	C₂³	—	118	1	no	—
238	C₂³	—	236	2	no	—
239	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	59	2	no	—
243	C₂×C₄	—	118	1	no	—
244	C₂×C₄	—	118	1	no	—
245	C₂×C₄	—	118	2	no	—
246	C₂×C₄	—	118	2	no	—
247	C₂×C₄	—	118	2	no	—
248	C₂×C₄	—	118	2	no	—
249	C₂×C₄	—	118	2	no	—
250	C₂×C₄	—	236	8	no	—
251	C₂×C₄	—	236	2	no	—
252	C₂×C₄	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
253	C₂×C₄	C₄⋊C₄×D₅₉	1	1	yes	characteristic
254	C₂×C₄	#14273685	1	1	yes	characteristic
255	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
256	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
257	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
258	C₈	—	118	2	no	—
259	C₈	—	118	2	no	—
260	C₈	(C₂×D₅₉)⋊C₈	1	4	yes	—
261	C₈	#14273698	1	1	yes	characteristic
262	C₈	#14273702	1	1	yes	characteristic
263	C₈	#14273703	1	1	yes	characteristic
264	C₈	#14273707	1	1	yes	characteristic
265	C₂²	—	2	2	no	—
266	C₂²	—	118	2	no	—
267	C₂²	—	236	2	no	—
268	C₂²	—	236	2	no	—
269	C₂²	—	236	2	no	—
270	C₂²	#16837365	1	1	yes	characteristic
271	C₂²	C₄⋊C₈×D₅₉	1	1	yes	characteristic
272	C₂²	#16837458	1	1	yes	characteristic
273	C₄	—	2	4	no	—
274	C₄	—	118	2	no	—
275	C₄	—	236	2	no	—
276	C₄	#16837259	1	1	yes	characteristic
277	C₄	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
278	C₄	#16837424	1	1	yes	characteristic
279	C₄	#16837454	1	1	yes	characteristic
280	C₂	—	2	1	no	—
281	C₂	—	2	1	no	—
282	C₂	—	236	2	no	—
283	C₂	#21424073	1	1	yes	characteristic
284	C₂	#21425907	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#14273685
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₅₉×#364	C₂
socle	C₂²×C₅₉	#432
prime radical	C₂³×D₅₉	M₁₆
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×#364	D₅₉
hypocenter	C₅₉	#12791
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#364

Complements

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

N	H ≅ G/N	type
2. C₂×#16837223	C₂	semidirect
4. #21422175	C₂	semidirect
5. C₂×C₅₉×#364	C₂	semidirect
140. C₂×#364	D₅₉	semidirect
162. C₅₉	#12791	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#11745659	2
C₂×#16837223	2
C₂×#16837183	2
#21422175	2
C₂×C₅₉×#364	2
#12791	59