Subgroups — #28503810

Order 15104.

382 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16837820	C₂	1	1	yes	maximal, characteristic
2	#21425716	C₂	1	1	yes	maximal, characteristic
3	#21425700	C₂	1	1	yes	maximal, characteristic, complemented
4	#21423897	C₂	1	1	yes	maximal, characteristic
5	#21422186	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422185	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₉×#2110	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	#16837820	—	2	2	no	—
10	#16837820	—	2	2	no	—
11	C₂×#14273667	C₂²	1	1	yes	characteristic
12	C₂×#14273666	C₄	1	2	yes	complemented
13	C₂²×C₅₉⋊C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₅₉	C₄	1	2	yes	complemented
15	#16837384	C₄	1	1	yes	characteristic
16	#16837375	C₄	1	1	yes	characteristic
17	#16837357	C₂²	1	1	yes	characteristic
18	#16837183	C₄	1	4	yes	complemented
19	#16837179	—	2	2	no	—
20	#16837174	C₄	1	1	yes	characteristic
21	#16837172	C₄	1	1	yes	characteristic
22	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
23	C₂×M₃₂×C₅₉	C₂²	1	1	yes	characteristic
24	C₅₉×#405	C₂²	1	1	yes	characteristic
25	C₅₉×#347	—	2	2	no	—
26	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
27	#14273733	—	2	1	no	—
28	#14273733	—	2	1	no	—
29	#14273733	—	2	1	no	—
30	#14273733	—	2	1	no	—
31	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
32	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
33	C₂×#11745659	C₂×C₄	1	2	yes	—
34	C₂×C₈×D₅₉	C₂×C₄	1	2	yes	—
35	#14273686	C₂×C₄	1	1	yes	characteristic
36	#14273667	—	2	1	no	—
37	#14273667	C₂×C₄	1	2	yes	—
38	#14273666	—	2	4	no	—
39	C₂×C₅₉⋊C₁₆	—	2	1	no	—
40	C₂×C₅₉⋊C₁₆	—	2	1	no	—
41	C₂×C₅₉⋊C₁₆	—	2	1	no	—
42	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
43	C₂×C₅₉⋊C₁₆	C₂×C₄	1	1	yes	characteristic
44	C₁₆×D₅₉	—	2	4	no	—
45	#14273645	—	2	1	no	—
46	#14273643	—	2	1	no	—
47	#14273641	—	2	2	no	—
48	#14273637	—	2	1	no	—
49	C₈×Dic₅₉	—	2	1	no	—
50	C₂×M₁₆×C₅₉	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
53	M₃₂×C₅₉	—	2	1	no	—
54	M₃₂×C₅₉	C₂×C₄	1	2	yes	—
55	M₃₂×C₅₉	D₄	1	2	yes	—
56	M₃₂×C₅₉	D₄	1	2	yes	—
57	C₂×C₁₆×C₅₉	—	2	1	no	—
58	C₂×C₁₆×C₅₉	C₂×C₄	1	2	yes	—
59	#97×C₅₉	—	2	1	no	—
60	C₄×C₈×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	M₁₆	1	2	yes	—
63	#11745685	—	2	1	no	—
64	#11745685	—	2	1	no	—
65	#11745685	—	2	1	no	—
66	#11745685	—	2	1	no	—
67	C₂×D₂₃₆	M₁₆	1	1	yes	characteristic
68	C₂×C₄×D₅₉	C₄²	1	2	yes	—
69	C₂×Dic₁₁₈	M₁₆	1	1	yes	characteristic
70	M₁₆×C₅₉	—	2	1	no	—
71	M₁₆×C₅₉	—	2	1	no	—
72	C₂×C₈×C₅₉	—	2	1	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	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
78	C₂×C₈×C₅₉	#41	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	—	2	1	no	—
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄×Dic₅₉	—	2	1	no	—
85	#11745664	—	2	1	no	—
86	#11745664	—	2	1	no	—
87	#11745664	—	2	1	no	—
88	#11745664	—	2	1	no	—
89	C₂×C₅₉⋊C₈	—	2	1	no	—
90	C₂×C₅₉⋊C₈	C₄²	1	2	yes	—
91	#11745659	—	2	2	no	—
92	#11745659	—	2	2	no	—
93	C₈×D₅₉	—	2	2	no	—
94	C₈×D₅₉	—	2	2	no	—
95	C₁₆×C₅₉	—	2	2	no	—
96	C₁₆×C₅₉	C₂²⋊C₄	1	4	yes	—
97	C₅₉⋊C₁₆	—	2	1	no	—
98	C₅₉⋊C₁₆	—	2	1	no	—
99	C₅₉⋊C₁₆	—	2	2	no	—
100	C₂³×C₅₉	C₂×M₁₆	1	1	yes	characteristic
101	C₂²×D₅₉	#97	1	2	yes	—
102	C₂×C₄×C₅₉	—	4	1	no	—
103	C₂×C₄×C₅₉	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
107	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
108	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
109	C₂²⋊D₅₉	—	2	2	no	—
110	C₂²⋊D₅₉	—	2	2	no	—
111	C₂×Dic₅₉	—	4	1	no	—
112	C₂×Dic₅₉	#97	1	2	yes	—
113	D₂₃₆	—	2	1	no	—
114	D₂₃₆	—	2	1	no	—
115	C₄×D₅₉	—	2	2	no	—
116	C₄×D₅₉	—	2	2	no	—
117	Dic₁₁₈	—	2	1	no	—
118	Dic₁₁₈	—	2	1	no	—
119	C₈×C₅₉	—	4	1	no	—
120	C₈×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₈×C₅₉	#117	1	1	yes	characteristic
122	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
123	C₈×C₅₉	C₄×D₄	1	1	yes	characteristic
124	C₅₉⋊C₈	—	2	2	no	—
125	C₅₉⋊C₈	—	4	1	no	—
126	#12617	—	59	1	no	maximal, Sylow
127	C₂²×C₅₉	—	4	1	no	—
128	C₂²×C₅₉	C₂×#97	1	1	yes	characteristic
129	C₂²×C₅₉	#435	1	1	yes	characteristic
130	C₂²×C₅₉	#435	1	1	yes	characteristic
131	C₂×D₅₉	—	2	2	no	—
132	C₂×D₅₉	—	4	2	no	—
133	C₄×C₅₉	—	8	1	no	—
134	C₄×C₅₉	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₅₉	#405	1	1	yes	characteristic
136	C₄×C₅₉	#435	1	1	yes	characteristic
137	C₄×C₅₉	#435	1	1	yes	characteristic
138	Dic₅₉	—	2	2	no	—
139	Dic₅₉	—	8	1	no	—
140	C₂×#504	—	59	1	no	—
141	#2115	—	59	1	no	—
142	#2110	—	59	1	no	—
143	#2110	D₅₉	1	1	yes	characteristic, complemented
144	#1757	—	59	1	no	—
145	#1379	—	59	1	no	—
146	#1379	—	59	1	no	—
147	C₂×C₅₉	—	2	1	no	—
148	C₂×C₅₉	—	2	1	no	—
149	C₂×C₅₉	—	2	1	no	—
150	C₂×C₅₉	#1752	1	1	yes	characteristic
151	D₅₉	—	4	2	no	—
152	C₂×#131	—	59	1	no	—
153	#504	—	118	2	no	—
154	#504	—	118	2	no	—
155	C₂×M₃₂	—	59	2	no	—
156	C₂×M₃₂	—	59	1	no	—
157	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
158	C₂×M₃₂	C₂×D₅₉	1	1	yes	characteristic
159	C₂²×C₁₆	—	59	1	no	—
160	C₂²×C₁₆	—	59	2	no	—
161	#413	—	59	1	no	—
162	#410	—	59	1	no	—
163	#405	—	59	1	no	—
164	#405	C₂×D₅₉	1	1	yes	characteristic
165	#349	—	59	4	no	—
166	#347	—	2	2	no	—
167	#347	—	118	2	no	—
168	#344	—	59	1	no	—
169	#344	—	59	1	no	—
170	C₅₉	#12617	1	1	yes	characteristic, complemented, Sylow
171	C₂×#41	—	59	1	no	—
172	#131	—	118	1	no	—
173	#131	—	118	1	no	—
174	#131	—	118	1	no	—
175	#131	—	118	1	no	—
176	C₂×M₁₆	—	59	1	no	—
177	C₂×M₁₆	—	59	2	no	—
178	C₂×M₁₆	—	59	1	no	—
179	C₂×M₁₆	C₄×D₅₉	1	1	yes	characteristic
180	C₂²×C₈	—	59	2	no	—
181	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
182	#117	—	59	1	no	—
183	#117	C₄×D₅₉	1	1	yes	characteristic
184	M₃₂	—	2	1	no	—
185	M₃₂	—	59	2	no	—
186	M₃₂	—	118	1	no	—
187	M₃₂	—	118	4	no	—
188	M₃₂	C₄×D₅₉	1	2	yes	—
189	M₃₂	D₂₃₆	1	2	yes	—
190	M₃₂	C₂²⋊D₅₉	1	2	yes	—
191	C₂×C₁₆	—	2	1	no	—
192	C₂×C₁₆	—	59	1	no	—
193	C₂×C₁₆	—	59	1	no	—
194	C₂×C₁₆	—	118	1	no	—
195	C₂×C₁₆	—	118	1	no	—
196	C₂×C₁₆	—	118	1	no	—
197	C₂×C₁₆	—	118	4	no	—
198	C₂×C₁₆	C₄×D₅₉	1	2	yes	—
199	#101	—	118	1	no	—
200	#100	—	118	1	no	—
201	#99	—	118	2	no	—
202	#97	—	2	1	no	—
203	#97	—	118	1	no	—
204	C₄×C₈	—	2	1	no	—
205	C₄×C₈	—	118	1	no	—
206	#41	—	118	1	no	—
207	#41	—	118	1	no	—
208	#41	—	118	1	no	—
209	#41	—	118	1	no	—
210	C₂×Q₈	—	59	1	no	—
211	C₂×D₄	—	59	2	no	—
212	C₂×D₄	—	59	1	no	—
213	C₂²×C₄	—	59	2	no	—
214	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
215	M₁₆	—	2	1	no	—
216	M₁₆	—	2	1	no	—
217	M₁₆	—	118	1	no	—
218	M₁₆	—	118	1	no	—
219	M₁₆	—	118	2	no	—
220	M₁₆	—	118	2	no	—
221	M₁₆	—	118	1	no	—
222	M₁₆	—	118	1	no	—
223	C₂×C₈	—	2	1	no	—
224	C₂×C₈	—	59	2	no	—
225	C₂×C₈	—	118	2	no	—
226	C₂×C₈	—	118	2	no	—
227	C₂×C₈	—	118	1	no	—
228	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
229	C₂×C₈	C₂×C₄×D₅₉	1	1	yes	characteristic
230	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
231	C₂×C₈	#11745685	1	1	yes	characteristic
232	C₂×C₈	#11745685	1	1	yes	characteristic
233	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
234	C₄⋊C₄	—	2	1	no	—
235	C₄⋊C₄	—	118	1	no	—
236	C₂²⋊C₄	—	2	1	no	—
237	C₂²⋊C₄	—	118	1	no	—
238	C₄²	—	2	1	no	—
239	C₄²	—	118	1	no	—
240	C₁₆	—	2	2	no	—
241	C₁₆	—	118	1	no	—
242	C₁₆	—	118	1	no	—
243	C₁₆	—	118	2	no	—
244	C₁₆	(C₂×D₅₉)⋊C₄	1	4	yes	—
245	C₂³	—	59	2	no	—
246	C₂³	M₁₆×D₅₉	1	1	yes	characteristic
247	Q₈	—	118	1	no	—
248	Q₈	—	118	1	no	—
249	D₄	—	118	2	no	—
250	D₄	—	118	2	no	—
251	D₄	—	118	1	no	—
252	D₄	—	118	1	no	—
253	C₂×C₄	—	4	1	no	—
254	C₂×C₄	—	59	2	no	—
255	C₂×C₄	—	118	2	no	—
256	C₂×C₄	—	118	2	no	—
257	C₂×C₄	—	236	1	no	—
258	C₂×C₄	C₄²×D₅₉	1	1	yes	characteristic
259	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
260	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
261	C₂×C₄	M₁₆×D₅₉	1	1	yes	characteristic
262	C₂×C₄	#14273735	1	1	yes	characteristic
263	C₂×C₄	#14273735	1	1	yes	characteristic
264	C₈	—	4	1	no	—
265	C₈	—	118	2	no	—
266	C₈	—	236	1	no	—
267	C₈	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
268	C₈	#14273684	1	1	yes	characteristic
269	C₈	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
270	C₈	C₄×D₂₃₆	1	1	yes	characteristic
271	C₂²	—	4	1	no	—
272	C₂²	—	118	2	no	—
273	C₂²	—	236	2	no	—
274	C₂²	#97×D₅₉	1	1	yes	characteristic
275	C₂²	#16837471	1	1	yes	characteristic
276	C₂²	#16837475	1	1	yes	characteristic
277	C₄	—	8	1	no	—
278	C₄	—	118	2	no	—
279	C₄	—	472	1	no	—
280	C₄	C₄×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
281	C₄	#16837359	1	1	yes	characteristic
282	C₄	#16837471	1	1	yes	characteristic
283	C₄	#16837475	1	1	yes	characteristic
284	C₂	—	2	1	no	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	236	2	no	—
288	C₂	#21423873	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14273684
commutator	C₂×C₄×C₅₉	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₅₉×#2110	C₂
socle	C₂×C₅₉	#1752
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	#2110	D₅₉
hypocenter	C₅₉	#12617
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ #2110

Complements

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

N	H ≅ G/N	type
3. #21425700	C₂	semidirect
5. #21422186	C₂	semidirect
6. #21422185	C₂	semidirect
7. C₅₉×#2110	C₂	semidirect
12. C₂×#14273666	C₄	semidirect
14. C₂×C₁₆×D₅₉	C₄	semidirect
18. #16837183	C₄	semidirect
143. #2110	D₅₉	semidirect
170. C₅₉	#12617	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16837820	2
#21425716	2
#21425700	2
#21423897	2
#21422186	2
#21422185	2
C₅₉×#2110	2
#12617	59