Subgroups — #28481788

Order 15104.

434 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424256	C₂	1	1	yes	maximal, characteristic
2	#21424101	C₂	1	2	yes	maximal
3	C₂×#16837153	C₂	1	1	yes	maximal, characteristic
4	#21421980	C₂	1	2	yes	maximal
5	C₂×C₅₉×#446	C₂	1	1	yes	maximal, characteristic
6	C₂×#14273661	—	2	1	no	—
7	C₂×#14273661	—	2	1	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
9	C₂×#14273651	—	2	2	no	—
10	#16837272	C₂²	1	2	yes	—
11	#16837164	—	2	2	no	—
12	#16837153	—	2	2	no	—
13	C₂×C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
14	C₅₉×#446	—	2	2	no	—
15	C₅₉×#446	C₄	1	4	yes	complemented
16	C₂×C₄⋊C₈×C₅₉	—	2	1	no	—
17	C₂×C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
18	C₂×#97×C₅₉	—	2	1	no	—
19	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₅₉	—	2	2	no	—
22	C₂×C₄⋊Dic₅₉	—	2	2	no	—
23	C₂×C₄⋊Dic₅₉	—	2	2	no	—
24	#14273661	—	4	2	no	—
25	#14273661	—	4	2	no	—
26	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
27	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
28	#14273651	—	4	4	no	—
29	C₂²×Q₈×C₅₉	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₉	—	2	1	no	—
31	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₉	D₄	1	1	yes	characteristic
33	C₄×Q₈×C₅₉	—	2	1	no	—
34	C₄×Q₈×C₅₉	—	2	1	no	—
35	C₄×Q₈×C₅₉	C₂×C₄	1	2	yes	—
36	C₄×Q₈×C₅₉	Q₈	1	2	yes	—
37	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
38	C₂×C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₅₉	—	2	1	no	—
40	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₅₉	—	2	4	no	—
42	C₄⋊C₈×C₅₉	—	2	1	no	—
43	C₄⋊C₈×C₅₉	C₂×C₄	1	2	yes	—
44	#97×C₅₉	—	2	4	no	—
45	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₅₉	D₄	1	2	yes	—
47	C₂×Q₈×C₅₉	—	2	1	no	—
48	C₂×Q₈×C₅₉	—	2	2	no	—
49	C₂×Q₈×C₅₉	—	2	2	no	—
50	C₂×Q₈×C₅₉	C₄⋊C₄	1	2	yes	—
51	C₂²×C₄×C₅₉	—	2	1	no	—
52	C₂²×C₄×C₅₉	—	2	1	no	—
53	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
56	C₂²×Dic₅₉	—	4	2	no	—
57	C₂×C₈×C₅₉	—	2	1	no	—
58	C₂×C₈×C₅₉	—	2	1	no	—
59	C₂×C₈×C₅₉	—	2	2	no	—
60	C₂×C₈×C₅₉	—	4	1	no	—
61	C₂×C₈×C₅₉	—	4	1	no	—
62	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
63	C₂×C₈×C₅₉	C₂²⋊C₄	1	2	yes	—
64	C₂×C₈×C₅₉	SD₁₆	1	2	yes	—
65	C₂×C₈×C₅₉	Q₁₆	1	2	yes	—
66	C₄⋊C₄×C₅₉	—	2	2	no	—
67	C₄⋊C₄×C₅₉	—	2	1	no	—
68	C₄⋊C₄×C₅₉	—	2	2	no	—
69	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
70	C₄²×C₅₉	—	2	2	no	—
71	C₄²×C₅₉	—	2	2	no	—
72	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
75	C₄²×C₅₉	#41	1	1	yes	characteristic
76	C₄⋊Dic₅₉	—	4	2	no	—
77	C₄⋊Dic₅₉	—	4	2	no	—
78	C₄⋊Dic₅₉	—	4	2	no	—
79	C₄⋊Dic₅₉	—	4	2	no	—
80	C₄⋊Dic₅₉	—	8	2	no	—
81	C₂×C₅₉⋊C₈	—	4	2	no	—
82	C₂×C₅₉⋊C₈	—	4	2	no	—
83	C₂×C₅₉⋊C₈	—	4	2	no	—
84	C₂³×C₅₉	#121	1	1	yes	characteristic
85	Q₈×C₅₉	—	2	4	no	—
86	Q₈×C₅₉	—	4	2	no	—
87	C₂×C₄×C₅₉	—	2	1	no	—
88	C₂×C₄×C₅₉	—	2	1	no	—
89	C₂×C₄×C₅₉	—	2	2	no	—
90	C₂×C₄×C₅₉	—	2	2	no	—
91	C₂×C₄×C₅₉	—	4	1	no	—
92	C₂×C₄×C₅₉	—	4	1	no	—
93	C₂×C₄×C₅₉	—	4	1	no	—
94	C₂×C₄×C₅₉	—	4	1	no	—
95	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₉	#117	1	1	yes	characteristic
98	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
99	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄×C₅₉	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	1	yes	characteristic
104	C₂×C₄×C₅₉	AGL(1,ℤ/8)	1	2	yes	—
105	C₂×C₄×C₅₉	#137	1	1	yes	characteristic
106	C₂×C₄×C₅₉	#137	1	2	yes	—
107	C₂×Dic₅₉	—	8	2	no	—
108	C₂×Dic₅₉	—	8	2	no	—
109	C₂×Dic₅₉	—	8	2	no	—
110	C₈×C₅₉	—	2	2	no	—
111	C₈×C₅₉	—	4	2	no	—
112	C₈×C₅₉	#103	1	4	yes	complemented
113	C₅₉⋊C₈	—	8	2	no	—
114	#8653	—	59	1	no	maximal, Sylow
115	C₂²×C₅₉	#380	1	1	yes	characteristic
116	C₂²×C₅₉	#460	1	1	yes	characteristic
117	C₂²×C₅₉	#462	1	1	yes	characteristic
118	C₂²×C₅₉	#469	1	1	yes	characteristic
119	C₂²×C₅₉	#470	1	1	yes	characteristic
120	C₂²×C₅₉	#472	1	1	yes	characteristic
121	C₂²×C₅₉	#473	1	1	yes	characteristic
122	C₄×C₅₉	—	2	1	no	—
123	C₄×C₅₉	—	2	1	no	—
124	C₄×C₅₉	—	4	2	no	—
125	C₄×C₅₉	—	4	2	no	—
126	C₄×C₅₉	C₂×#103	1	1	yes	characteristic
127	C₄×C₅₉	#417	1	1	yes	characteristic
128	C₄×C₅₉	#428	1	2	yes	—
129	C₄×C₅₉	#468	1	2	yes	—
130	C₄×C₅₉	#475	1	1	yes	characteristic
131	C₄×C₅₉	#477	1	1	yes	characteristic
132	Dic₅₉	—	16	2	no	—
133	C₂×#446	D₅₉	1	1	yes	characteristic
134	#1845	—	59	1	no	—
135	#1803	—	59	2	no	—
136	C₂×#334	—	59	1	no	—
137	#1296	—	59	2	no	—
138	C₂×C₅₉	#1688	1	1	yes	characteristic
139	C₂×C₅₉	#1690	1	1	yes	characteristic
140	C₂×C₅₉	#1694	1	1	yes	characteristic
141	C₂×C₅₉	#1696	1	1	yes	characteristic
142	C₂×C₅₉	#1803	1	1	yes	characteristic
143	C₂×C₅₉	#1809	1	1	yes	characteristic
144	C₂×C₅₉	#1816	1	1	yes	characteristic
145	C₂×C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
146	#446	—	2	2	no	—
147	#446	Dic₅₉	1	4	yes	complemented
148	C₂×#107	—	118	1	no	—
149	C₂×#107	—	118	1	no	—
150	C₂×C₄⋊C₈	—	2	1	no	—
151	C₂×C₄⋊C₈	—	59	2	no	—
152	C₂×C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
153	C₂×#103	—	118	2	no	—
154	C₂×#97	—	2	1	no	—
155	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
156	#384	—	59	2	no	—
157	#340	—	118	2	no	—
158	#334	—	118	2	no	—
159	C₅₉	#8653	1	1	yes	characteristic, complemented, Sylow
160	C₂²×Q₈	D₂₃₆	1	1	yes	characteristic
161	C₂²×C₈	—	2	1	no	—
162	C₂²×C₈	—	118	2	no	—
163	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
164	C₂²×C₈	C₂²⋊D₅₉	1	1	yes	characteristic
165	C₄×Q₈	—	2	1	no	—
166	C₄×Q₈	—	2	1	no	—
167	C₄×Q₈	Dic₁₁₈	1	2	yes	—
168	C₄×Q₈	C₂×Dic₅₉	1	2	yes	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	118	2	no	—
171	C₂×C₄⋊C₄	—	118	2	no	—
172	C₂×C₄⋊C₄	—	118	2	no	—
173	C₂×C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
174	C₂×C₄²	—	2	1	no	—
175	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
176	#107	—	236	2	no	—
177	#107	—	236	2	no	—
178	C₄⋊C₈	—	2	4	no	—
179	C₄⋊C₈	—	2	1	no	—
180	C₄⋊C₈	—	118	2	no	—
181	C₄⋊C₈	—	118	2	no	—
182	C₄⋊C₈	C₂×Dic₅₉	1	2	yes	—
183	#103	—	236	4	no	—
184	#97	—	2	4	no	—
185	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
186	C₄×C₈	C₂²⋊D₅₉	1	2	yes	—
187	C₂×Q₈	—	2	1	no	—
188	C₂×Q₈	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	C₄⋊Dic₅₉	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	2	1	no	—
193	C₂²×C₄	—	236	2	no	—
194	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
195	C₂²×C₄	#11745685	1	1	yes	characteristic
196	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
197	C₂×C₈	—	2	1	no	—
198	C₂×C₈	—	2	1	no	—
199	C₂×C₈	—	2	2	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	1	no	—
202	C₂×C₈	—	236	2	no	—
203	C₂×C₈	—	236	2	no	—
204	C₂×C₈	—	236	2	no	—
205	C₂×C₈	#11745671	1	2	yes	—
206	C₂×C₈	#11745672	1	2	yes	—
207	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
208	C₂×C₈	C₂²⋊Dic₅₉	1	2	yes	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	2	no	—
212	C₄⋊C₄	—	472	2	no	—
213	C₄⋊C₄	—	236	2	no	—
214	C₄⋊C₄	—	236	2	no	—
215	C₄⋊C₄	—	236	2	no	—
216	C₄⋊C₄	—	236	2	no	—
217	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
218	C₄²	—	2	2	no	—
219	C₄²	—	2	2	no	—
220	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
221	C₄²	#11745685	1	1	yes	characteristic
222	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
223	C₄²	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
224	C₂³	#14273701	1	1	yes	characteristic
225	Q₈	—	2	4	no	—
226	Q₈	—	4	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	4	1	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	1	no	—
234	C₂×C₄	—	4	1	no	—
235	C₂×C₄	—	472	2	no	—
236	C₂×C₄	—	472	2	no	—
237	C₂×C₄	—	472	2	no	—
238	C₂×C₄	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
239	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
240	C₂×C₄	#14273686	1	1	yes	characteristic
241	C₂×C₄	#14273701	1	1	yes	characteristic
242	C₂×C₄	#14273706	1	1	yes	characteristic
243	C₂×C₄	#14273706	1	1	yes	characteristic
244	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
245	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
246	C₂×C₄	#14273752	1	1	yes	characteristic
247	C₂×C₄	#14273755	1	2	yes	—
248	C₂×C₄	#14273757	1	1	yes	characteristic
249	C₂×C₄	#14273760	1	2	yes	—
250	C₈	—	2	2	no	—
251	C₈	—	4	2	no	—
252	C₈	—	472	2	no	—
253	C₈	#14273651	1	4	yes	complemented
254	C₂²	#16837262	1	1	yes	characteristic
255	C₂²	#16837602	1	1	yes	characteristic
256	C₂²	#16837612	1	1	yes	characteristic
257	C₂²	#16837655	1	1	yes	characteristic
258	C₂²	#16837660	1	1	yes	characteristic
259	C₂²	#16837672	1	1	yes	characteristic
260	C₂²	#16837677	1	1	yes	characteristic
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	4	2	no	—
265	C₄	—	944	2	no	—
266	C₄	C₂×#14273651	1	1	yes	characteristic
267	C₄	#16837403	1	1	yes	characteristic
268	C₄	#16837444	1	2	yes	—
269	C₄	#16837648	1	2	yes	—
270	C₄	#16837687	1	1	yes	characteristic
271	C₄	#16837697	1	1	yes	characteristic
272	C₂	#21423580	1	1	yes	characteristic
273	C₂	#21423590	1	1	yes	characteristic
274	C₂	#21423614	1	1	yes	characteristic
275	C₂	#21423624	1	1	yes	characteristic
276	C₂	#21424101	1	1	yes	characteristic
277	C₂	#21424129	1	1	yes	characteristic
278	C₂	#21424160	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14273701
commutator	C₄²×C₅₉	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₂×C₅₉×#446	C₂
socle, prime radical	C₂³×C₅₉	#121
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₂×#446	D₅₉
hypocenter	C₅₉	#8653
LCS	G ▷ C₄²×C₅₉ ▷ C₂²×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#446

Complements

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

N	H ≅ G/N	type
15. C₅₉×#446	C₄	semidirect
112. C₈×C₅₉	#103	semidirect
147. #446	Dic₅₉	semidirect
159. C₅₉	#8653	semidirect
253. C₈	#14273651	semidirect

Maximal subgroups

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

subgroup	index
#21424256	2
#21424101	2
C₂×#16837153	2
#21421980	2
C₂×C₅₉×#446	2
#8653	59