Subgroups — #28488696

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	C₄×#14273727	C₂	1	1	yes	maximal, characteristic
2	#21424248	C₂	1	1	yes	maximal, characteristic
3	#21421936	C₂	1	4	yes	maximal
4	C₄²×C₈×C₅₉	C₂	1	1	yes	maximal, characteristic
5	C₂×#14273727	C₄	1	1	yes	characteristic
6	C₂×C₄×Dic₁₁₈	—	2	1	no	—
7	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₅₉⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
10	#16837308	—	2	1	no	—
11	#16837300	—	2	1	no	—
12	#16837300	C₄	1	2	yes	—
13	#16837272	C₄	1	1	yes	characteristic
14	C₄×C₄⋊Dic₅₉	—	2	1	no	—
15	C₄×C₄⋊Dic₅₉	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₅₉	—	2	1	no	—
17	C₂×C₄×C₈×C₅₉	—	2	1	no	—
18	C₂×C₄×C₈×C₅₉	C₂²	1	2	yes	—
19	C₄³×C₅₉	C₂²	1	1	yes	characteristic
20	C₂²×Dic₁₁₈	—	2	1	no	—
21	C₂²×C₅₉⋊C₈	—	2	2	no	—
22	#14273727	—	2	2	no	—
23	#14273727	C₈	1	4	yes	complemented
24	C₄×Dic₁₁₈	—	4	2	no	—
25	C₄×Dic₁₁₈	—	4	2	no	—
26	C₂×C₄⋊Dic₅₉	—	2	1	no	—
27	C₂×C₄⋊Dic₅₉	—	2	1	no	—
28	C₂×C₄⋊Dic₅₉	—	2	1	no	—
29	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
30	C₂×C₄⋊Dic₅₉	C₂×C₄	1	2	yes	—
31	C₂×Dic₅₉⋊C₄	—	2	2	no	—
32	C₂×C₄×Dic₅₉	—	2	2	no	—
33	C₄⋊(C₅₉⋊C₈)	—	2	4	no	—
34	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
35	C₄⋊(C₅₉⋊C₈)	—	4	2	no	—
36	C₅₉⋊#95	—	2	2	no	—
37	C₅₉⋊#95	—	2	2	no	—
38	C₂²×C₈×C₅₉	—	2	2	no	—
39	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄²×C₅₉	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₉	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₅₉	D₄	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	C₂³	1	1	yes	characteristic
46	C₄×C₈×C₅₉	—	2	4	no	—
47	C₄×C₈×C₅₉	—	2	4	no	—
48	C₄×C₈×C₅₉	—	2	2	no	—
49	C₄×C₈×C₅₉	Q₈	1	4	yes	complemented
50	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
57	C₂²×Dic₅₉	—	2	2	no	—
58	C₂²×Dic₅₉	—	2	2	no	—
59	C₂×Dic₁₁₈	—	2	4	no	—
60	C₂×Dic₁₁₈	—	4	2	no	—
61	C₂×Dic₁₁₈	—	4	2	no	—
62	C₂×C₈×C₅₉	—	2	4	no	—
63	C₂×C₈×C₅₉	—	2	4	no	—
64	C₂×C₈×C₅₉	—	2	4	no	—
65	C₄²×C₅₉	—	2	2	no	—
66	C₄²×C₅₉	—	2	2	no	—
67	C₄²×C₅₉	—	2	2	no	—
68	C₄²×C₅₉	C₂×C₈	1	2	yes	—
69	C₄²×C₅₉	M₁₆	1	2	yes	—
70	C₄²×C₅₉	D₈	1	2	yes	—
71	C₄²×C₅₉	SD₁₆	1	2	yes	—
72	C₄²×C₅₉	SD₁₆	1	2	yes	—
73	C₄²×C₅₉	Q₁₆	1	2	yes	—
74	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
75	C₄²×C₅₉	#41	1	2	yes	—
76	C₄⋊Dic₅₉	—	2	4	no	—
77	C₄⋊Dic₅₉	—	2	2	no	—
78	C₄⋊Dic₅₉	—	2	4	no	—
79	C₄⋊Dic₅₉	—	4	2	no	—
80	C₄⋊Dic₅₉	—	4	2	no	—
81	C₄⋊Dic₅₉	C₂×C₈	1	4	yes	—
82	Dic₅₉⋊C₄	—	4	4	no	—
83	C₄×Dic₅₉	—	4	2	no	—
84	C₄×Dic₅₉	—	4	2	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	C₂×C₄×C₅₉	—	2	1	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	C₂²⋊C₈	1	2	yes	—
96	C₂×C₄×C₅₉	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₉	#102	1	2	yes	—
98	C₂×C₄×C₅₉	#102	1	2	yes	—
99	C₂×C₄×C₅₉	#103	1	2	yes	—
100	C₂×C₄×C₅₉	#103	1	2	yes	—
101	C₂×C₄×C₅₉	#104	1	2	yes	—
102	C₂×C₄×C₅₉	#104	1	2	yes	—
103	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
104	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
105	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
106	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
107	C₂×C₄×C₅₉	C₂²×C₈	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×M₁₆	1	1	yes	characteristic
109	C₂×C₄×C₅₉	#131	1	2	yes	—
110	C₂×C₄×C₅₉	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₉	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×Q₁₆	1	1	yes	characteristic
114	C₂×Dic₅₉	—	2	4	no	—
115	C₂×Dic₅₉	—	4	2	no	—
116	C₂×Dic₅₉	—	4	2	no	—
117	C₂×Dic₅₉	—	4	2	no	—
118	C₂×Dic₅₉	—	4	2	no	—
119	C₂×Dic₅₉	—	4	2	no	—
120	Dic₁₁₈	—	4	8	no	—
121	C₈×C₅₉	—	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	D₄⋊C₈	1	4	yes	—
135	C₄×C₅₉	Q₈⋊C₈	1	4	yes	—
136	C₄×C₅₉	C₈×Q₈	1	2	yes	—
137	C₄×C₅₉	#446	1	2	yes	—
138	C₄×C₅₉	#486	1	1	yes	characteristic
139	C₄×C₅₉	#487	1	1	yes	characteristic
140	C₄×C₅₉	#498	1	1	yes	characteristic
141	C₄×C₅₉	#500	1	1	yes	characteristic
142	Dic₅₉	—	4	4	no	—
143	Dic₅₉	—	8	2	no	—
144	C₄×C₄⋊Q₈	—	59	1	no	—
145	#1842	—	59	1	no	—
146	C₄²×C₈	D₅₉	1	1	yes	characteristic
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	—	59	1	no	—
156	C₂×C₄×Q₈	—	118	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	—	2	1	no	—
161	C₂×C₄×C₈	—	2	1	no	—
162	C₂×C₄×C₈	C₂×D₅₉	1	2	yes	—
163	#391	—	118	1	no	—
164	#389	—	59	2	no	—
165	#389	—	118	1	no	—
166	#384	—	59	1	no	—
167	C₄×C₄⋊C₄	—	59	2	no	—
168	C₄×C₄⋊C₄	—	118	1	no	—
169	C₄³	C₂×D₅₉	1	1	yes	characteristic
170	C₅₉	#9870	1	1	yes	characteristic, complemented, Sylow
171	C₂²×Q₈	—	118	1	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	118	2	no	—
174	C₄⋊Q₈	—	59	4	no	—
175	C₄⋊Q₈	—	118	2	no	—
176	C₄×Q₈	—	236	2	no	—
177	C₄×Q₈	—	236	2	no	—
178	C₂×C₄⋊C₄	—	59	2	no	—
179	C₂×C₄⋊C₄	—	59	2	no	—
180	C₂×C₄⋊C₄	—	118	1	no	—
181	C₂×C₄⋊C₄	—	118	1	no	—
182	C₂×C₄⋊C₄	—	118	2	no	—
183	C₂×C₄⋊C₄	—	118	1	no	—
184	C₂×C₄²	—	118	2	no	—
185	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
186	C₂×C₄²	C₄×D₅₉	1	1	yes	characteristic
187	C₂×C₄²	D₂₃₆	1	1	yes	characteristic
188	C₂×C₄²	D₂₃₆	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	—	2	4	no	—
196	C₄×C₈	—	2	4	no	—
197	C₄×C₈	—	2	2	no	—
198	C₄×C₈	Dic₁₁₈	1	4	yes	complemented
199	#95	—	118	2	no	—
200	#95	—	118	2	no	—
201	C₂×Q₈	—	118	4	no	—
202	C₂×Q₈	—	236	2	no	—
203	C₂×Q₈	—	236	2	no	—
204	C₂²×C₄	—	118	2	no	—
205	C₂²×C₄	—	118	2	no	—
206	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
207	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
208	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
209	C₂²×C₄	(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
210	C₂²×C₄	C₂×C₄×D₅₉	1	1	yes	characteristic
211	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
212	C₂²×C₄	C₂×C₂²⋊D₅₉	1	1	yes	characteristic
213	C₂×C₈	—	2	4	no	—
214	C₂×C₈	—	2	4	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	236	2	no	—
217	C₂×C₈	—	236	2	no	—
218	C₂×C₈	—	236	2	no	—
219	C₄⋊C₄	—	59	4	no	—
220	C₄⋊C₄	—	118	4	no	—
221	C₄⋊C₄	—	118	2	no	—
222	C₄⋊C₄	—	118	4	no	—
223	C₄⋊C₄	—	236	4	no	—
224	C₄⋊C₄	—	236	2	no	—
225	C₄⋊C₄	—	236	2	no	—
226	C₄²	—	2	2	no	—
227	C₄²	—	2	2	no	—
228	C₄²	—	2	2	no	—
229	C₄²	—	236	2	no	—
230	C₄²	—	236	2	no	—
231	C₄²	C₈×D₅₉	1	2	yes	—
232	C₄²	#11745659	1	2	yes	—
233	C₄²	#11745660	1	2	yes	—
234	C₄²	#11745660	1	2	yes	—
235	C₄²	D₄₇₂	1	2	yes	—
236	C₄²	Dic₂₃₆	1	2	yes	—
237	C₄²	C₂×Dic₁₁₈	1	2	yes	—
238	C₄²	#11745685	1	2	yes	—
239	C₂³	C₂×(C₂×D₅₉)⋊C₄	1	1	yes	characteristic
240	Q₈	—	236	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	—	118	4	no	—
248	C₂×C₄	—	236	2	no	—
249	C₂×C₄	—	236	2	no	—
250	C₂×C₄	—	236	2	no	—
251	C₂×C₄	—	236	2	no	—
252	C₂×C₄	—	236	2	no	—
253	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
254	C₂×C₄	(C₂×D₅₉)⋊C₈	1	2	yes	—
255	C₂×C₄	#14273649	1	2	yes	—
256	C₂×C₄	#14273649	1	2	yes	—
257	C₂×C₄	#14273652	1	2	yes	—
258	C₂×C₄	#14273652	1	2	yes	—
259	C₂×C₄	#14273655	1	2	yes	—
260	C₂×C₄	#14273655	1	2	yes	—
261	C₂×C₄	C₄×Dic₁₁₈	1	2	yes	—
262	C₂×C₄	#14273706	1	2	yes	—
263	C₂×C₄	#14273715	1	1	yes	characteristic
264	C₂×C₄	#14273727	1	1	yes	characteristic
265	C₂×C₄	C₂×C₈×D₅₉	1	1	yes	characteristic
266	C₂×C₄	C₂×#11745659	1	1	yes	characteristic
267	C₂×C₄	#14273733	1	2	yes	—
268	C₂×C₄	C₂×D₄₇₂	1	1	yes	characteristic
269	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
270	C₂×C₄	C₂×#11745660	1	1	yes	characteristic
271	C₂×C₄	C₂×Dic₂₃₆	1	1	yes	characteristic
272	C₈	—	2	8	no	—
273	C₈	—	472	2	no	—
274	C₂²	#16837308	1	1	yes	characteristic
275	C₂²	C₂×(C₂×D₅₉)⋊C₈	1	1	yes	characteristic
276	C₂²	#16837368	1	1	yes	characteristic
277	C₂²	C₂×#14273649	1	1	yes	characteristic
278	C₂²	C₂×#14273652	1	1	yes	characteristic
279	C₂²	C₂×#14273655	1	1	yes	characteristic
280	C₂²	C₂×#14273655	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	2	2	no	—
283	C₄	—	2	2	no	—
284	C₄	—	472	2	no	—
285	C₄	—	236	4	no	—
286	C₄	#16837131	1	4	yes	—
287	C₄	#16837134	1	4	yes	—
288	C₄	C₈×Dic₁₁₈	1	2	yes	—
289	C₄	#16837532	1	2	yes	—
290	C₄	#16837749	1	1	yes	characteristic
291	C₄	#16837753	1	1	yes	characteristic
292	C₄	#16837800	1	1	yes	characteristic
293	C₄	#16837807	1	1	yes	characteristic
294	C₂	C₂×#16837131	1	1	yes	characteristic
295	C₂	C₂×#16837134	1	1	yes	characteristic
296	C₂	#21422548	1	1	yes	characteristic
297	C₂	#21422552	1	1	yes	characteristic
298	C₂	#21425032	1	1	yes	characteristic
299	C₂	#21425057	1	1	yes	characteristic
300	C₂	#21425073	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₉
Fitting	C₄²×C₈×C₅₉	C₂
socle, prime radical	C₂³×C₅₉	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₅₉	C₂³
hypercenter	C₄²×C₈	D₅₉
hypocenter	C₅₉	#9870
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×C₈

Complements

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

N	H ≅ G/N	type
23. #14273727	C₈	semidirect
49. C₄×C₈×C₅₉	Q₈	semidirect
170. C₅₉	#9870	semidirect
198. C₄×C₈	Dic₁₁₈	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14273727	2
#21424248	2
#21421936	2
C₄²×C₈×C₅₉	2
#9870	59