Subgroups — #28557898

Order 15104.

572 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21433213	C₂	1	2	yes	maximal
2	#21422885	C₂	1	2	yes	maximal, complemented
3	#21422881	C₂	1	2	yes	maximal, complemented
4	#21422876	C₂	1	2	yes	maximal, complemented
5	#21422867	C₂	1	2	yes	maximal, complemented
6	#21422863	C₂	1	2	yes	maximal, complemented
7	#21422859	C₂	1	1	yes	maximal, characteristic, complemented
8	#21422852	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₉×#2922	C₂	1	1	yes	maximal, characteristic
10	#16837533	—	2	2	no	—
11	#16837480	—	2	2	no	—
12	#16837474	—	2	2	no	—
13	#16837474	—	2	2	no	—
14	#16837459	C₂²	1	2	yes	—
15	#16837459	C₂²	1	2	yes	—
16	#16837455	C₂²	1	2	yes	—
17	#16837426	C₂²	1	2	yes	—
18	#16837426	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₅₉⋊C₈)	C₂²	1	2	yes	—
20	#16837155	C₂²	1	2	yes	—
21	#16837155	C₂²	1	2	yes	—
22	#16837155	C₂²	1	2	yes	—
23	#16837153	C₂²	1	2	yes	—
24	#16837153	C₂²	1	1	yes	characteristic
25	#16837153	C₂²	1	1	yes	characteristic
26	#16837153	C₂²	1	1	yes	characteristic
27	#16837153	C₂²	1	1	yes	characteristic
28	#16837153	C₂²	1	2	yes	—
29	#16837153	C₂²	1	2	yes	—
30	C₄×#41×C₅₉	C₂²	1	1	yes	characteristic
31	C₅₉×#433	C₄	1	4	yes	—
32	C₅₉×#431	C₂²	1	2	yes	—
33	C₅₉×#431	C₂²	1	2	yes	—
34	C₅₉×#424	C₄	1	2	yes	—
35	C₅₉×#408	C₄	1	2	yes	—
36	C₄×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
37	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
38	C₂×#11745664	—	2	2	no	—
39	C₂×#11745664	—	2	2	no	—
40	C₂²×C₅₉⋊C₈	—	2	2	no	—
41	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
42	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
43	C₄⋊(C₅₉⋊C₈)	—	2	2	no	—
44	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
45	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
46	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
47	C₄⋊(C₅₉⋊C₈)	C₂³	1	2	yes	—
48	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
49	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
50	C₂²⋊(C₅₉⋊C₈)	—	2	2	no	—
51	#14273638	—	2	2	no	—
52	#14273638	—	2	2	no	—
53	C₄×C₅₉⋊C₈	—	2	2	no	—
54	C₂×#41×C₅₉	C₂×C₄	1	1	yes	characteristic
55	C₂×M₁₆×C₅₉	C₂×C₄	1	2	yes	—
56	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
57	C₄×Q₈×C₅₉	D₄	1	1	yes	characteristic
58	C₄×Q₈×C₅₉	Q₈	1	1	yes	characteristic
59	C₄×D₄×C₅₉	D₄	1	1	yes	characteristic
60	C₄×D₄×C₅₉	D₄	1	2	yes	—
61	C₄×D₄×C₅₉	Q₈	1	2	yes	—
62	C₄×D₄×C₅₉	Q₈	1	1	yes	characteristic
63	#117×C₅₉	C₂×C₄	1	1	yes	characteristic
64	#117×C₅₉	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₅₉	C₂³	1	2	yes	—
66	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
68	C₄⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
69	C₂²⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
70	C₂²⋊C₈×C₅₉	C₂×C₄	1	4	yes	—
71	#97×C₅₉	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₉	C₂×C₄	1	2	yes	—
73	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₉	C₂³	1	2	yes	—
76	#41×C₅₉	—	4	2	no	—
77	C₂×Q₈×C₅₉	C₄⋊C₄	1	1	yes	characteristic
78	C₂×D₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
79	C₂×D₄×C₅₉	C₄⋊C₄	1	2	yes	—
80	C₂²×C₄×C₅₉	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₅₉	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
85	C₂²×C₄×C₅₉	C₂×Q₈	1	1	yes	characteristic
86	M₁₆×C₅₉	—	4	2	no	—
87	C₂×C₈×C₅₉	—	4	2	no	—
88	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₅₉	C₂²×C₄	1	4	yes	—
91	C₄⋊C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
92	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
93	C₄⋊C₄×C₅₉	C₄⋊C₄	1	1	yes	characteristic
94	C₄⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
95	C₂²⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
96	C₂²⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
97	C₂²⋊C₄×C₅₉	C₄⋊C₄	1	2	yes	—
98	C₄²×C₅₉	C₂²×C₄	1	1	yes	characteristic
99	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
100	C₄²×C₅₉	C₂×D₄	1	2	yes	—
101	C₄²×C₅₉	C₂×D₄	1	1	yes	characteristic
102	C₄²×C₅₉	C₂×Q₈	1	2	yes	—
103	C₄²×C₅₉	C₂×Q₈	1	1	yes	characteristic
104	C₄²×C₅₉	C₂⁴	1	1	yes	characteristic
105	#11745664	—	8	2	no	—
106	#11745664	—	8	2	no	—
107	C₂×C₅₉⋊C₈	—	2	2	no	—
108	C₂×C₅₉⋊C₈	—	2	2	no	—
109	C₂×C₅₉⋊C₈	—	2	2	no	—
110	C₂×C₅₉⋊C₈	—	2	2	no	—
111	C₂×C₅₉⋊C₈	—	8	2	no	—
112	C₂³×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
113	C₂³×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
114	Q₈×C₅₉	—	4	1	no	—
115	D₄×C₅₉	—	4	1	no	—
116	D₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	—	2	2	no	—
118	C₂×C₄×C₅₉	—	2	4	no	—
119	C₂×C₄×C₅₉	—	2	2	no	—
120	C₂×C₄×C₅₉	—	2	2	no	—
121	C₂×C₄×C₅₉	—	2	2	no	—
122	C₂×C₄×C₅₉	—	4	2	no	—
123	C₂×C₄×C₅₉	—	4	1	no	—
124	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
128	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	2	yes	—
129	C₂×C₄×C₅₉	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
131	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
133	C₈×C₅₉	—	2	2	no	—
134	C₈×C₅₉	—	2	2	no	—
135	C₈×C₅₉	—	2	4	no	—
136	C₅₉⋊C₈	—	4	2	no	—
137	C₅₉⋊C₈	—	4	2	no	—
138	C₅₉⋊C₈	—	4	2	no	—
139	C₅₉⋊C₈	—	4	2	no	—
140	#17824	—	59	1	no	maximal, Sylow
141	C₂²×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	2	no	—
143	C₂²×C₅₉	—	2	2	no	—
144	C₂²×C₅₉	—	2	1	no	—
145	C₂²×C₅₉	—	2	2	no	—
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	C₂²×C₄⋊C₄	1	1	yes	characteristic
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	2	no	—
150	C₄×C₅₉	—	2	2	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	—	2	2	no	—
155	C₄×C₅₉	#568	1	2	yes	—
156	C₄×C₅₉	#577	1	2	yes	—
157	#2989	—	59	2	no	—
158	#2922	D₅₉	1	1	yes	characteristic
159	#1574	—	59	2	no	—
160	#1573	—	59	2	no	—
161	#1572	—	59	2	no	—
162	#1570	—	59	2	no	—
163	#1569	—	59	2	no	—
164	#1568	—	59	1	no	—
165	#1566	—	59	1	no	—
166	C₂×C₅₉	—	4	1	no	—
167	C₂×C₅₉	—	4	2	no	—
168	C₂×C₅₉	#2905	1	1	yes	characteristic
169	C₂×C₅₉	#2912	1	1	yes	characteristic
170	C₂×C₅₉	#2916	1	1	yes	characteristic
171	C₄×#41	C₂×D₅₉	1	1	yes	characteristic
172	#446	—	118	2	no	—
173	#436	—	118	2	no	—
174	#435	—	118	2	no	—
175	#435	—	118	2	no	—
176	#433	Dic₅₉	1	4	yes	—
177	#432	—	59	2	no	—
178	#432	—	59	2	no	—
179	#431	—	59	2	no	—
180	#431	C₂×D₅₉	1	2	yes	—
181	#431	C₂×D₅₉	1	2	yes	—
182	#424	Dic₅₉	1	2	yes	—
183	#423	—	59	2	no	—
184	#423	—	59	2	no	—
185	C₂×C₄⋊C₈	—	59	2	no	—
186	#408	Dic₅₉	1	2	yes	—
187	C₄×M₁₆	C₂×D₅₉	1	1	yes	characteristic
188	C₂×C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
189	#335	—	59	2	no	—
190	#335	—	59	2	no	—
191	#335	—	59	2	no	—
192	#334	—	59	2	no	—
193	#334	—	59	1	no	—
194	#334	—	59	1	no	—
195	#334	—	59	1	no	—
196	#334	—	59	1	no	—
197	#334	—	59	2	no	—
198	#334	—	59	2	no	—
199	C₅₉	#17824	1	1	yes	characteristic, complemented, Sylow
200	C₂×#41	C₂×Dic₅₉	1	1	yes	characteristic
201	C₂×M₁₆	—	118	2	no	—
202	C₂×M₁₆	—	118	2	no	—
203	C₂×M₁₆	C₂×Dic₅₉	1	2	yes	—
204	C₂²×C₈	—	118	2	no	—
205	C₂²×C₈	C₂×Dic₅₉	1	2	yes	—
206	C₄×Q₈	Dic₁₁₈	1	1	yes	characteristic
207	C₄×Q₈	D₂₃₆	1	1	yes	characteristic
208	C₄×D₄	Dic₁₁₈	1	1	yes	characteristic
209	C₄×D₄	Dic₁₁₈	1	2	yes	—
210	C₄×D₄	D₂₃₆	1	1	yes	characteristic
211	C₄×D₄	D₂₃₆	1	2	yes	—
212	#117	C₂×Dic₅₉	1	1	yes	characteristic
213	#117	C₂×Dic₅₉	1	2	yes	—
214	C₂×C₄²	C₂²×D₅₉	1	1	yes	characteristic
215	C₂×C₄²	C₂²×D₅₉	1	2	yes	—
216	C₄⋊C₈	—	59	2	no	—
217	C₄⋊C₈	—	59	2	no	—
218	C₄⋊C₈	—	59	2	no	—
219	C₄⋊C₈	—	59	2	no	—
220	C₄⋊C₈	—	118	2	no	—
221	C₄⋊C₈	—	118	2	no	—
222	C₄⋊C₈	—	118	2	no	—
223	C₄⋊C₈	C₂×Dic₅₉	1	4	yes	—
224	C₄⋊C₈	C₂×Dic₅₉	1	4	yes	—
225	C₂²⋊C₈	—	118	2	no	—
226	C₂²⋊C₈	—	118	2	no	—
227	C₂²⋊C₈	—	118	2	no	—
228	C₂²⋊C₈	C₂×Dic₅₉	1	4	yes	—
229	C₂²⋊C₈	C₂×Dic₅₉	1	4	yes	—
230	#97	—	118	2	no	—
231	#97	—	118	2	no	—
232	#97	C₂×Dic₅₉	1	2	yes	—
233	C₄×C₈	—	118	2	no	—
234	C₄×C₈	C₂×Dic₅₉	1	2	yes	—
235	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₄×C₈	C₂²×D₅₉	1	1	yes	characteristic
237	C₄×C₈	C₂²×D₅₉	1	2	yes	—
238	#41	—	4	2	no	—
239	C₂×Q₈	C₄⋊Dic₅₉	1	1	yes	characteristic
240	C₂×D₄	C₄⋊Dic₅₉	1	1	yes	characteristic
241	C₂×D₄	C₄⋊Dic₅₉	1	2	yes	—
242	C₂²×C₄	C₂×Dic₁₁₈	1	1	yes	characteristic
243	C₂²×C₄	C₂×Dic₁₁₈	1	2	yes	—
244	C₂²×C₄	C₂×D₂₃₆	1	2	yes	—
245	C₂²×C₄	C₂×D₂₃₆	1	1	yes	characteristic
246	C₂²×C₄	C₂²×Dic₅₉	1	2	yes	—
247	C₂²×C₄	C₂²×Dic₅₉	1	1	yes	characteristic
248	M₁₆	—	4	2	no	—
249	M₁₆	—	472	2	no	—
250	M₁₆	—	472	2	no	—
251	C₂×C₈	—	4	2	no	—
252	C₂×C₈	—	118	2	no	—
253	C₂×C₈	—	118	2	no	—
254	C₂×C₈	—	118	2	no	—
255	C₂×C₈	—	118	2	no	—
256	C₂×C₈	—	472	2	no	—
257	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
258	C₂×C₈	C₂²×Dic₅₉	1	2	yes	—
259	C₂×C₈	C₂²×Dic₅₉	1	4	yes	—
260	C₄⋊C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
261	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
262	C₄⋊C₄	C₄⋊Dic₅₉	1	1	yes	characteristic
263	C₄⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
264	C₂²⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
265	C₂²⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
266	C₂²⋊C₄	C₄⋊Dic₅₉	1	2	yes	—
267	C₄²	C₂×Dic₁₁₈	1	1	yes	characteristic
268	C₄²	C₂×Dic₁₁₈	1	2	yes	—
269	C₄²	C₂×D₂₃₆	1	2	yes	—
270	C₄²	C₂×D₂₃₆	1	1	yes	characteristic
271	C₄²	C₂²×Dic₅₉	1	1	yes	characteristic
272	C₄²	C₂²×Dic₅₉	1	2	yes	—
273	C₄²	C₂³×D₅₉	1	1	yes	characteristic
274	C₂³	C₂×C₄⋊Dic₅₉	1	2	yes	—
275	C₂³	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
276	Q₈	—	4	1	no	—
277	D₄	—	4	1	no	—
278	D₄	—	4	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
287	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
288	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
289	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
290	C₂×C₄	C₂×C₄⋊Dic₅₉	1	2	yes	—
291	C₂×C₄	C₂×C₄⋊Dic₅₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₅₉	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₂₃₆	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₁₁₈	1	1	yes	characteristic
295	C₈	—	2	2	no	—
296	C₈	—	2	2	no	—
297	C₈	—	2	4	no	—
298	C₈	—	236	2	no	—
299	C₈	—	236	2	no	—
300	C₈	—	236	2	no	—
301	C₈	—	236	2	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	1	no	—
308	C₂²	C₂²×C₄⋊Dic₅₉	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	#16838169	1	2	yes	—
317	C₄	#16838214	1	2	yes	—
318	C₂	—	4	1	no	—
319	C₂	—	4	2	no	—
320	C₂	#21432660	1	1	yes	characteristic
321	C₂	#21432691	1	1	yes	characteristic
322	C₂	#21432707	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂³×C₄
Frattini	C₄²	C₂³×D₅₉
Fitting	C₅₉×#2922	C₂
socle	C₂²×C₅₉	C₂²×C₄⋊C₄
prime radical	C₂×#41×C₅₉	C₂×C₄
cosocle	C₄²×C₅₉	C₂⁴
hypercenter	#2922	D₅₉
hypocenter	C₅₉	#17824
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×#41 ◁ #2922

Complements

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

N	H ≅ G/N	type
2. #21422885	C₂	semidirect
3. #21422881	C₂	semidirect
4. #21422876	C₂	semidirect
5. #21422867	C₂	semidirect
6. #21422863	C₂	semidirect
7. #21422859	C₂	semidirect
8. #21422852	C₂	semidirect
199. C₅₉	#17824	semidirect

Maximal subgroups

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

subgroup	index
#21433213	2
#21422885	2
#21422881	2
#21422876	2
#21422867	2
#21422863	2
#21422859	2
#21422852	2
C₅₉×#2922	2
#17824	59