Subgroups — #28506444

Order 15104.

332 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21425854	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16837185	C₂	1	1	yes	maximal, characteristic
3	#21425709	C₂	1	1	yes	maximal, characteristic
4	#21423933	C₂	1	1	yes	maximal, characteristic
5	#21422184	C₂	1	1	yes	maximal, characteristic, complemented
6	#21422178	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14273733	C₂²	1	1	yes	characteristic
9	C₂×#14273667	C₂²	1	1	yes	characteristic
10	C₂×#14273667	C₂²	1	1	yes	characteristic
11	C₂×#14273655	C₄	1	1	yes	characteristic
12	#16837399	C₄	1	1	yes	characteristic
13	#16837357	C₂²	1	1	yes	characteristic
14	Dic₅₉⋊C₁₆	—	2	2	no	—
15	#16837185	—	2	2	no	—
16	(C₂×D₅₉)⋊C₁₆	—	2	2	no	—
17	#16837167	C₄	1	1	yes	characteristic
18	#16837162	C₄	1	1	yes	characteristic
19	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₁₆×C₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₅₉	—	2	2	no	—
23	C₂×#11745685	C₂×C₄	1	1	yes	characteristic
24	#14273733	—	2	1	no	—
25	#14273733	—	2	1	no	—
26	#14273733	—	2	1	no	—
27	#14273733	—	2	1	no	—
28	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
29	C₂×#11745664	C₂×C₄	1	1	yes	characteristic
30	C₂×#11745659	—	2	1	no	—
31	C₂×C₈×D₅₉	—	2	1	no	—
32	#14273686	C₂×C₄	1	1	yes	characteristic
33	#14273667	—	2	1	no	—
34	#14273667	—	2	1	no	—
35	#14273667	—	2	1	no	—
36	#14273667	—	2	1	no	—
37	C₂×C₅₉⋊C₁₆	—	2	1	no	—
38	C₂×C₅₉⋊C₁₆	—	2	1	no	—
39	#14273655	—	2	2	no	—
40	#14273652	C₈	1	2	yes	—
41	#14273649	C₈	1	2	yes	—
42	#14273637	—	2	1	no	—
43	C₈×Dic₅₉	—	2	1	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	C₂×C₁₆×C₅₉	—	2	1	no	—
48	C₂×C₁₆×C₅₉	—	2	1	no	—
49	C₂×C₁₆×C₅₉	—	2	1	no	—
50	C₂×C₁₆×C₅₉	—	2	1	no	—
51	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
52	C₂×C₁₆×C₅₉	D₄	1	2	yes	—
53	C₄×C₈×C₅₉	—	2	1	no	—
54	C₄×C₈×C₅₉	—	2	1	no	—
55	C₂²×C₄×C₅₉	—	2	1	no	—
56	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₅₉	—	2	1	no	—
58	#11745685	—	2	1	no	—
59	#11745685	—	2	1	no	—
60	#11745685	—	2	1	no	—
61	#11745685	—	2	1	no	—
62	C₂×D₂₃₆	C₂×C₈	1	1	yes	characteristic
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×Dic₁₁₈	C₂×C₈	1	1	yes	characteristic
65	C₂×C₈×C₅₉	—	2	1	no	—
66	C₂×C₈×C₅₉	—	2	1	no	—
67	C₂×C₈×C₅₉	C₂×C₈	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	#41	1	1	yes	characteristic
71	C₂×C₈×C₅₉	#41	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	M₁₆	1	2	yes	—
76	C₂²⋊Dic₅₉	—	2	1	no	—
77	C₄⋊Dic₅₉	C₂×C₈	1	2	yes	—
78	C₄×Dic₅₉	—	2	1	no	—
79	#11745664	—	2	1	no	—
80	#11745664	—	2	1	no	—
81	#11745664	—	2	1	no	—
82	#11745664	—	2	1	no	—
83	C₂×C₅₉⋊C₈	—	2	1	no	—
84	C₂×C₅₉⋊C₈	—	2	1	no	—
85	#11745659	—	4	1	no	—
86	#11745659	—	4	1	no	—
87	C₈×D₅₉	—	4	1	no	—
88	C₈×D₅₉	—	4	1	no	—
89	C₁₆×C₅₉	—	2	2	no	—
90	C₁₆×C₅₉	—	2	2	no	—
91	C₅₉⋊C₁₆	—	4	1	no	—
92	C₅₉⋊C₁₆	—	4	1	no	—
93	C₂³×C₅₉	#131	1	1	yes	characteristic
94	C₂²×D₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	4	no	—
96	C₂×C₄×C₅₉	—	2	2	no	—
97	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₉	#131	1	1	yes	characteristic
103	C₂²⋊D₅₉	—	4	1	no	—
104	C₂²⋊D₅₉	—	4	1	no	—
105	C₂×Dic₅₉	—	2	2	no	—
106	C₂×Dic₅₉	—	2	1	no	—
107	D₂₃₆	—	2	1	no	—
108	D₂₃₆	—	2	1	no	—
109	C₄×D₅₉	—	4	1	no	—
110	C₄×D₅₉	—	4	1	no	—
111	Dic₁₁₈	—	2	1	no	—
112	Dic₁₁₈	—	2	1	no	—
113	C₈×C₅₉	—	2	2	no	—
114	C₈×C₅₉	#121	1	1	yes	characteristic
115	C₈×C₅₉	#123	1	1	yes	characteristic
116	C₈×C₅₉	#124	1	1	yes	characteristic
117	C₈×C₅₉	#126	1	1	yes	characteristic
118	C₅₉⋊C₈	—	4	1	no	—
119	C₅₉⋊C₈	—	4	1	no	—
120	#13124	—	59	1	no	maximal, Sylow
121	C₂²×C₅₉	—	2	2	no	—
122	C₂²×C₅₉	#433	1	1	yes	characteristic
123	C₂²×C₅₉	C₈×D₄	1	1	yes	characteristic
124	C₂²×C₅₉	#435	1	1	yes	characteristic
125	C₂×D₅₉	—	4	2	no	—
126	C₂×D₅₉	—	4	1	no	—
127	C₄×C₅₉	—	2	4	no	—
128	C₄×C₅₉	#388	1	1	yes	characteristic
129	C₄×C₅₉	#431	1	1	yes	characteristic
130	C₄×C₅₉	C₈×D₄	1	1	yes	characteristic
131	C₄×C₅₉	#436	1	1	yes	characteristic
132	Dic₅₉	—	4	2	no	—
133	Dic₅₉	—	4	1	no	—
134	#2151	—	59	1	no	—
135	C₂×#350	—	59	1	no	—
136	#2113	—	59	1	no	—
137	C₂×C₄×C₁₆	D₅₉	1	1	yes	characteristic, complemented
138	#1764	—	59	1	no	—
139	#1379	—	59	1	no	—
140	#1376	—	59	1	no	—
141	C₂×C₅₉	—	2	1	no	—
142	C₂×C₅₉	—	2	1	no	—
143	C₂×C₅₉	#1923	1	1	yes	characteristic
144	C₂×C₅₉	#2170	1	2	yes	—
145	D₅₉	—	8	1	no	—
146	C₂×#131	—	59	1	no	—
147	C₂×M₃₂	—	59	1	no	—
148	C₂×M₃₂	—	59	1	no	—
149	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
150	C₂²×C₁₆	C₂×D₅₉	1	1	yes	characteristic
151	C₂×#104	—	59	1	no	—
152	#416	—	59	1	no	—
153	#405	—	59	1	no	—
154	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
155	C₄⋊C₁₆	—	118	2	no	—
156	#350	—	118	2	no	—
157	C₂²⋊C₁₆	—	118	2	no	—
158	C₄×C₁₆	—	2	2	no	—
159	#341	—	59	1	no	—
160	#339	—	59	1	no	—
161	C₅₉	#13124	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	—	59	1	no	—
163	#131	—	118	1	no	—
164	#131	—	118	1	no	—
165	#131	—	118	1	no	—
166	#131	—	118	1	no	—
167	C₂×M₁₆	—	59	1	no	—
168	C₂×M₁₆	—	59	1	no	—
169	C₂×M₁₆	—	118	1	no	—
170	C₂²×C₈	—	118	1	no	—
171	C₂²×C₈	C₄×D₅₉	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₅₉	1	1	yes	characteristic
173	#117	—	59	1	no	—
174	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
175	M₃₂	—	118	1	no	—
176	M₃₂	—	118	1	no	—
177	M₃₂	—	118	1	no	—
178	M₃₂	—	118	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	2	1	no	—
182	C₂×C₁₆	—	2	1	no	—
183	C₂×C₁₆	—	118	1	no	—
184	C₂×C₁₆	—	118	1	no	—
185	C₂×C₁₆	D₂₃₆	1	2	yes	—
186	C₂×C₁₆	C₂²⋊D₅₉	1	2	yes	—
187	#104	—	118	2	no	—
188	#103	—	59	2	no	—
189	#102	—	59	2	no	—
190	#97	—	118	1	no	—
191	C₄×C₈	—	2	1	no	—
192	C₄×C₈	—	2	1	no	—
193	C₄×C₈	—	118	1	no	—
194	#41	—	118	1	no	—
195	#41	—	118	1	no	—
196	#41	—	118	1	no	—
197	#41	—	118	1	no	—
198	C₂×Q₈	—	59	1	no	—
199	C₂×D₄	—	59	1	no	—
200	C₂×D₄	—	118	1	no	—
201	C₂²×C₄	—	2	1	no	—
202	C₂²×C₄	—	118	1	no	—
203	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
204	M₁₆	—	118	1	no	—
205	M₁₆	—	118	1	no	—
206	M₁₆	—	118	1	no	—
207	M₁₆	—	118	1	no	—
208	M₁₆	—	236	1	no	—
209	M₁₆	—	236	1	no	—
210	C₂×C₈	—	2	1	no	—
211	C₂×C₈	—	2	1	no	—
212	C₂×C₈	—	118	1	no	—
213	C₂×C₈	—	118	1	no	—
214	C₂×C₈	—	236	1	no	—
215	C₂×C₈	—	236	1	no	—
216	C₂×C₈	C₈×D₅₉	1	2	yes	—
217	C₂×C₈	C₂×D₂₃₆	1	1	yes	characteristic
218	C₂×C₈	#11745685	1	1	yes	characteristic
219	C₂×C₈	#11745685	1	1	yes	characteristic
220	C₂×C₈	#11745685	1	1	yes	characteristic
221	C₂×C₈	#11745685	1	1	yes	characteristic
222	C₂×C₈	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
223	C₄⋊C₄	—	59	2	no	—
224	C₂²⋊C₄	—	118	1	no	—
225	C₄²	—	2	1	no	—
226	C₄²	—	118	1	no	—
227	C₄²	#11745659	1	2	yes	—
228	C₁₆	—	2	2	no	—
229	C₁₆	—	2	2	no	—
230	C₁₆	—	236	1	no	—
231	C₁₆	—	236	1	no	—
232	C₂³	—	118	1	no	—
233	C₂³	#14273733	1	1	yes	characteristic
234	Q₈	—	118	1	no	—
235	Q₈	—	118	1	no	—
236	D₄	—	118	1	no	—
237	D₄	—	118	1	no	—
238	D₄	—	236	1	no	—
239	D₄	—	236	1	no	—
240	C₂×C₄	—	2	4	no	—
241	C₂×C₄	—	2	2	no	—
242	C₂×C₄	—	118	2	no	—
243	C₂×C₄	—	118	1	no	—
244	C₂×C₄	—	236	1	no	—
245	C₂×C₄	—	236	1	no	—
246	C₂×C₄	#14273684	1	1	yes	characteristic
247	C₂×C₄	C₄×C₂²⋊D₅₉	1	1	yes	characteristic
248	C₂×C₄	C₄×D₂₃₆	1	1	yes	characteristic
249	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
250	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
251	C₂×C₄	#14273733	1	1	yes	characteristic
252	C₈	—	2	2	no	—
253	C₈	—	236	1	no	—
254	C₈	—	236	1	no	—
255	C₈	#14273701	1	1	yes	characteristic
256	C₈	#14273710	1	1	yes	characteristic
257	C₈	#14273715	1	1	yes	characteristic
258	C₈	#14273721	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	236	2	no	—
261	C₂²	—	236	1	no	—
262	C₂²	#16837461	1	1	yes	characteristic
263	C₂²	C₈×D₂₃₆	1	1	yes	characteristic
264	C₂²	#16837473	1	1	yes	characteristic
265	C₄	—	2	4	no	—
266	C₄	—	236	2	no	—
267	C₄	—	236	1	no	—
268	C₄	#16837293	1	1	yes	characteristic
269	C₄	#16837453	1	1	yes	characteristic
270	C₄	C₈×C₂²⋊D₅₉	1	1	yes	characteristic
271	C₄	#16837481	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	2	1	no	—
274	C₂	—	472	1	no	—
275	C₂	#21424702	1	1	yes	characteristic
276	C₂	#21425934	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₉
Fitting	C₂×C₄×C₁₆×C₅₉	C₂
socle	C₂²×C₅₉	C₈×D₄
prime radical	C₂×#11745685	C₂×C₄
cosocle	C₂²×C₈×C₅₉	C₂³
hypercenter	C₂×C₄×C₁₆	D₅₉
hypocenter	C₅₉	#13124
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. #21425854	C₂	semidirect
5. #21422184	C₂	semidirect
6. #21422178	C₂	semidirect
7. C₂×C₄×C₁₆×C₅₉	C₂	semidirect
137. C₂×C₄×C₁₆	D₅₉	semidirect
161. C₅₉	#13124	semidirect

Maximal subgroups

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

subgroup	index
#21425854	2
C₂×#16837185	2
#21425709	2
#21423933	2
#21422184	2
#21422178	2
C₂×C₄×C₁₆×C₅₉	2
#13124	59