Subgroups — #28488697

Order 15104.

566 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21424248	C₂	1	1	yes	maximal, characteristic
2	C₄²×C₅₉⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#21421937	C₂	1	4	yes	maximal
4	C₄×C₄⋊Q₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
6	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
9	C₂×C₄×C₅₉⋊C₈	—	2	1	no	—
10	C₂×C₄×C₅₉⋊C₈	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈×C₅₉	C₄	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₅₉	—	2	1	no	—
13	C₅₉×#391	—	2	1	no	—
14	C₅₉×#389	—	2	1	no	—
15	C₅₉×#389	C₄	1	2	yes	—
16	C₅₉×#384	C₄	1	1	yes	characteristic
17	C₄×C₄⋊C₄×C₅₉	—	2	1	no	—
18	C₄×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
19	C₄³×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×C₅₉⋊C₈	—	2	2	no	—
21	C₂²×C₅₉⋊C₈	—	2	2	no	—
22	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
23	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
24	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
25	C₄×C₅₉⋊C₈	—	2	4	no	—
26	C₄×C₅₉⋊C₈	—	2	4	no	—
27	C₄×C₅₉⋊C₈	—	2	2	no	—
28	C₄×C₅₉⋊C₈	Q₈	1	4	yes	complemented
29	C₂²×Q₈×C₅₉	—	2	1	no	—
30	C₄⋊Q₈×C₅₉	—	2	2	no	—
31	C₄⋊Q₈×C₅₉	C₈	1	4	yes	complemented
32	C₄×Q₈×C₅₉	—	4	2	no	—
33	C₄×Q₈×C₅₉	—	4	2	no	—
34	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
35	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
36	C₂×C₄⋊C₄×C₅₉	—	2	1	no	—
37	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	2	yes	—
40	C₂×C₄²×C₅₉	—	2	2	no	—
41	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
43	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
47	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
48	#95×C₅₉	—	2	2	no	—
49	#95×C₅₉	—	2	2	no	—
50	C₂×Q₈×C₅₉	—	2	4	no	—
51	C₂×Q₈×C₅₉	—	4	2	no	—
52	C₂×Q₈×C₅₉	—	4	2	no	—
53	C₂²×C₄×C₅₉	—	2	2	no	—
54	C₂²×C₄×C₅₉	—	2	2	no	—
55	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₉	—	2	4	no	—
63	C₄⋊C₄×C₅₉	—	2	2	no	—
64	C₄⋊C₄×C₅₉	—	2	4	no	—
65	C₄⋊C₄×C₅₉	—	4	4	no	—
66	C₄⋊C₄×C₅₉	—	4	2	no	—
67	C₄⋊C₄×C₅₉	—	4	2	no	—
68	C₄⋊C₄×C₅₉	C₂×C₈	1	4	yes	—
69	C₄²×C₅₉	—	2	2	no	—
70	C₄²×C₅₉	—	2	2	no	—
71	C₄²×C₅₉	—	2	2	no	—
72	C₄²×C₅₉	—	4	2	no	—
73	C₄²×C₅₉	—	4	2	no	—
74	C₄²×C₅₉	C₂×C₈	1	2	yes	—
75	C₄²×C₅₉	M₁₆	1	2	yes	—
76	C₄²×C₅₉	D₈	1	2	yes	—
77	C₄²×C₅₉	SD₁₆	1	2	yes	—
78	C₄²×C₅₉	SD₁₆	1	2	yes	—
79	C₄²×C₅₉	Q₁₆	1	2	yes	—
80	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
81	C₄²×C₅₉	#41	1	2	yes	—
82	C₂×C₅₉⋊C₈	—	2	4	no	—
83	C₂×C₅₉⋊C₈	—	2	4	no	—
84	C₂×C₅₉⋊C₈	—	2	4	no	—
85	C₂×C₅₉⋊C₈	—	4	2	no	—
86	C₂×C₅₉⋊C₈	—	4	2	no	—
87	C₂×C₅₉⋊C₈	—	4	2	no	—
88	C₂³×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₅₉	—	4	8	no	—
90	C₂×C₄×C₅₉	—	2	1	no	—
91	C₂×C₄×C₅₉	—	2	1	no	—
92	C₂×C₄×C₅₉	—	2	1	no	—
93	C₂×C₄×C₅₉	—	2	1	no	—
94	C₂×C₄×C₅₉	—	2	1	no	—
95	C₂×C₄×C₅₉	—	2	1	no	—
96	C₂×C₄×C₅₉	—	2	4	no	—
97	C₂×C₄×C₅₉	—	4	2	no	—
98	C₂×C₄×C₅₉	—	4	2	no	—
99	C₂×C₄×C₅₉	—	4	2	no	—
100	C₂×C₄×C₅₉	—	4	2	no	—
101	C₂×C₄×C₅₉	—	4	2	no	—
102	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
104	C₂×C₄×C₅₉	#102	1	2	yes	—
105	C₂×C₄×C₅₉	#102	1	2	yes	—
106	C₂×C₄×C₅₉	#103	1	2	yes	—
107	C₂×C₄×C₅₉	#103	1	2	yes	—
108	C₂×C₄×C₅₉	#104	1	2	yes	—
109	C₂×C₄×C₅₉	#104	1	2	yes	—
110	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
111	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
116	C₂×C₄×C₅₉	#131	1	2	yes	—
117	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
121	C₅₉⋊C₈	—	2	8	no	—
122	C₅₉⋊C₈	—	8	2	no	—
123	#9870	—	59	1	no	maximal, Sylow
124	C₂²×C₅₉	#391	1	1	yes	characteristic
125	C₂²×C₅₉	C₂×C₂²⋊C₈	1	1	yes	characteristic
126	C₂²×C₅₉	#408	1	1	yes	characteristic
127	C₂²×C₅₉	C₂×#102	1	1	yes	characteristic
128	C₂²×C₅₉	C₂×#103	1	1	yes	characteristic
129	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
130	C₂²×C₅₉	C₂×#104	1	1	yes	characteristic
131	C₄×C₅₉	—	2	2	no	—
132	C₄×C₅₉	—	2	2	no	—
133	C₄×C₅₉	—	2	2	no	—
134	C₄×C₅₉	—	4	4	no	—
135	C₄×C₅₉	—	8	2	no	—
136	C₄×C₅₉	D₄⋊C₈	1	4	yes	—
137	C₄×C₅₉	Q₈⋊C₈	1	4	yes	—
138	C₄×C₅₉	C₈×Q₈	1	2	yes	—
139	C₄×C₅₉	#446	1	2	yes	—
140	C₄×C₅₉	#486	1	1	yes	characteristic
141	C₄×C₅₉	#487	1	1	yes	characteristic
142	C₄×C₅₉	#498	1	1	yes	characteristic
143	C₄×C₅₉	#500	1	1	yes	characteristic
144	C₄×C₄⋊Q₈	D₅₉	1	1	yes	characteristic
145	#1842	—	59	1	no	—
146	C₄²×C₈	—	59	1	no	—
147	#1279	—	59	4	no	—
148	C₂×C₅₉	C₂×D₄⋊C₈	1	1	yes	characteristic
149	C₂×C₅₉	C₂×Q₈⋊C₈	1	1	yes	characteristic
150	C₂×C₅₉	#1492	1	1	yes	characteristic
151	C₂×C₅₉	#1493	1	1	yes	characteristic
152	C₂×C₅₉	#1984	1	1	yes	characteristic
153	C₂×C₅₉	#1990	1	1	yes	characteristic
154	C₂×C₅₉	#1994	1	1	yes	characteristic
155	C₂×C₄⋊Q₈	Dic₅₉	1	1	yes	characteristic
156	C₂×C₄×Q₈	—	2	1	no	—
157	C₂×C₄⋊C₈	—	59	2	no	—
158	C₂×C₄⋊C₈	—	118	1	no	—
159	C₂×C₄⋊C₈	—	118	1	no	—
160	C₂×C₄×C₈	—	59	2	no	—
161	C₂×C₄×C₈	—	118	1	no	—
162	C₂×C₄×C₈	—	118	1	no	—
163	#391	—	2	1	no	—
164	#389	—	2	1	no	—
165	#389	Dic₅₉	1	2	yes	—
166	#384	Dic₅₉	1	1	yes	characteristic
167	C₄×C₄⋊C₄	—	2	1	no	—
168	C₄×C₄⋊C₄	C₂×D₅₉	1	2	yes	—
169	C₄³	C₂×D₅₉	1	1	yes	characteristic
170	C₅₉	#9870	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	—	2	1	no	—
172	C₂²×C₈	—	118	2	no	—
173	C₂²×C₈	—	118	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₅₉⋊C₈	1	4	yes	complemented
176	C₄×Q₈	—	4	2	no	—
177	C₄×Q₈	—	4	2	no	—
178	C₂×C₄⋊C₄	—	2	1	no	—
179	C₂×C₄⋊C₄	—	2	1	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	1	no	—
182	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
183	C₂×C₄⋊C₄	C₂×Dic₅₉	1	2	yes	—
184	C₂×C₄²	—	2	2	no	—
185	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
186	C₂×C₄²	C₂×Dic₅₉	1	1	yes	characteristic
187	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
188	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
189	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
190	C₂×C₄²	C₂²⋊D₅₉	1	1	yes	characteristic
191	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
192	C₄⋊C₈	—	118	4	no	—
193	C₄⋊C₈	—	236	2	no	—
194	C₄⋊C₈	—	236	2	no	—
195	C₄×C₈	—	59	4	no	—
196	C₄×C₈	—	118	4	no	—
197	C₄×C₈	—	118	4	no	—
198	C₄×C₈	—	118	2	no	—
199	#95	—	2	2	no	—
200	#95	—	2	2	no	—
201	C₂×Q₈	—	2	4	no	—
202	C₂×Q₈	—	4	2	no	—
203	C₂×Q₈	—	4	2	no	—
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
207	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
208	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
209	C₂²×C₄	C₂²⋊Dic₅₉	1	1	yes	characteristic
210	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
213	C₂×C₈	—	118	4	no	—
214	C₂×C₈	—	118	4	no	—
215	C₂×C₈	—	118	4	no	—
216	C₂×C₈	—	236	2	no	—
217	C₂×C₈	—	236	2	no	—
218	C₂×C₈	—	236	2	no	—
219	C₄⋊C₄	—	2	4	no	—
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	4	no	—
222	C₄⋊C₄	—	4	4	no	—
223	C₄⋊C₄	—	4	2	no	—
224	C₄⋊C₄	—	4	2	no	—
225	C₄⋊C₄	C₂×C₅₉⋊C₈	1	4	yes	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	C₂×C₅₉⋊C₈	1	2	yes	—
232	C₄²	#11745664	1	2	yes	—
233	C₄²	#11745669	1	2	yes	—
234	C₄²	#11745670	1	2	yes	—
235	C₄²	#11745671	1	2	yes	—
236	C₄²	#11745672	1	2	yes	—
237	C₄²	Q₈×D₅₉	1	2	yes	—
238	C₄²	#11745689	1	2	yes	—
239	C₂³	C₂×C₂²⋊Dic₅₉	1	1	yes	characteristic
240	Q₈	—	4	8	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	1	no	—
244	C₂×C₄	—	2	1	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	4	2	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
254	C₂×C₄	C₂²⋊(C₅₉⋊C₈)	1	2	yes	—
255	C₂×C₄	#14273648	1	2	yes	—
256	C₂×C₄	#14273648	1	2	yes	—
257	C₂×C₄	#14273651	1	2	yes	—
258	C₂×C₄	#14273651	1	2	yes	—
259	C₂×C₄	#14273654	1	2	yes	—
260	C₂×C₄	#14273654	1	2	yes	—
261	C₂×C₄	Q₈×Dic₅₉	1	2	yes	—
262	C₂×C₄	#14273703	1	2	yes	—
263	C₂×C₄	#14273716	1	1	yes	characteristic
264	C₂×C₄	#14273725	1	1	yes	characteristic
265	C₂×C₄	C₂²×C₅₉⋊C₈	1	1	yes	characteristic
266	C₂×C₄	C₂×#11745664	1	1	yes	characteristic
267	C₂×C₄	#14273734	1	2	yes	—
268	C₂×C₄	C₂×#11745669	1	1	yes	characteristic
269	C₂×C₄	C₂×#11745671	1	1	yes	characteristic
270	C₂×C₄	C₂×#11745670	1	1	yes	characteristic
271	C₂×C₄	C₂×#11745672	1	1	yes	characteristic
272	C₈	—	118	8	no	—
273	C₈	—	472	2	no	—
274	C₂²	#16837309	1	1	yes	characteristic
275	C₂²	C₂×C₂²⋊(C₅₉⋊C₈)	1	1	yes	characteristic
276	C₂²	#16837369	1	1	yes	characteristic
277	C₂²	C₂×#14273648	1	1	yes	characteristic
278	C₂²	C₂×#14273651	1	1	yes	characteristic
279	C₂²	C₂×#14273654	1	1	yes	characteristic
280	C₂²	C₂×#14273654	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	4	no	—
285	C₄	—	8	2	no	—
286	C₄	D₄⋊(C₅₉⋊C₈)	1	4	yes	—
287	C₄	Q₈⋊(C₅₉⋊C₈)	1	4	yes	—
288	C₄	Q₈×C₅₉⋊C₈	1	2	yes	—
289	C₄	#16837533	1	2	yes	—
290	C₄	#16837750	1	1	yes	characteristic
291	C₄	#16837754	1	1	yes	characteristic
292	C₄	#16837798	1	1	yes	characteristic
293	C₄	#16837805	1	1	yes	characteristic
294	C₂	C₂×D₄⋊(C₅₉⋊C₈)	1	1	yes	characteristic
295	C₂	C₂×Q₈⋊(C₅₉⋊C₈)	1	1	yes	characteristic
296	C₂	#21422547	1	1	yes	characteristic
297	C₂	#21422551	1	1	yes	characteristic
298	C₂	#21425033	1	1	yes	characteristic
299	C₂	#21425058	1	1	yes	characteristic
300	C₂	#21425074	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₅₉
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄×C₄⋊Q₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄×C₄⋊Q₈	D₅₉
hypocenter	C₅₉	#9870
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄×C₄⋊Q₈

Complements

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

N	H ≅ G/N	type
28. C₄×C₅₉⋊C₈	Q₈	semidirect
31. C₄⋊Q₈×C₅₉	C₈	semidirect
170. C₅₉	#9870	semidirect
175. C₄⋊Q₈	C₅₉⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#21424248	2
C₄²×C₅₉⋊C₈	2
#21421937	2
C₄×C₄⋊Q₈×C₅₉	2
#9870	59