Subgroups — C₅₉×#21710 #28433819

Order 15104.

1014 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉×#3150	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#444	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₉×#443	C₂	1	2	yes	maximal, complemented
4	C₅₉×#2932	C₂	1	2	yes	maximal, complemented
5	C₅₉×#1975	C₂	1	2	yes	maximal, complemented
6	C₅₉×#1974	C₂	1	2	yes	maximal, complemented
7	C₅₉×#1962	C₂	1	2	yes	maximal, complemented
8	C₅₉×#1849	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄×#97×C₅₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135×C₅₉	C₂²	1	2	yes	—
11	C₂×#135×C₅₉	C₂²	1	2	yes	—
12	C₂×#131×C₅₉	C₂²	1	4	yes	—
13	C₅₉×#532	C₂²	1	2	yes	—
14	C₅₉×#497	C₄	1	4	yes	complemented
15	C₅₉×#496	C₄	1	4	yes	complemented
16	C₅₉×#444	—	2	8	no	—
17	C₅₉×#443	—	2	8	no	—
18	C₅₉×#435	—	2	8	no	—
19	C₂×#108×C₅₉	C₂²	1	2	yes	—
20	C₂×#108×C₅₉	C₂²	1	2	yes	—
21	C₅₉×#423	C₂²	1	2	yes	—
22	C₂×#104×C₅₉	C₂²	1	4	yes	—
23	C₂×#104×C₅₉	C₂²	1	4	yes	—
24	C₅₉×#408	C₂²	1	4	yes	—
25	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
26	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
27	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
28	C₂×#97×C₅₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
31	C₄³×C₅₉	C₂²	1	1	yes	characteristic
32	C₂×#41×C₅₉	C₂³	1	4	yes	—
33	#135×C₅₉	—	2	4	no	—
34	#135×C₅₉	—	2	4	no	—
35	#135×C₅₉	—	2	4	no	—
36	#135×C₅₉	—	2	4	no	—
37	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
38	C₂×Q₁₆×C₅₉	C₂×C₄	1	2	yes	—
39	C₂×SD₁₆×C₅₉	C₂×C₄	1	4	yes	—
40	C₂×SD₁₆×C₅₉	C₂×C₄	1	4	yes	—
41	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
42	C₂×D₈×C₅₉	C₂×C₄	1	2	yes	—
43	#131×C₅₉	—	2	8	no	—
44	#131×C₅₉	—	2	8	no	—
45	C₂×M₁₆×C₅₉	—	2	4	no	—
46	C₂×M₁₆×C₅₉	C₂³	1	4	yes	—
47	C₂²×C₈×C₅₉	—	2	4	no	—
48	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
51	C₂²×C₈×C₅₉	C₂³	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₉	C₂×C₄	1	2	yes	—
53	#127×C₅₉	C₂×C₄	1	2	yes	—
54	#124×C₅₉	C₂×C₄	1	4	yes	—
55	#121×C₅₉	—	2	4	no	—
56	C₄×D₄×C₅₉	—	2	4	no	—
57	C₂×C₄²×C₅₉	—	2	2	no	—
58	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
59	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×C₄²×C₅₉	C₂³	1	1	yes	characteristic
61	#108×C₅₉	—	2	4	no	—
62	#108×C₅₉	—	2	4	no	—
63	C₄⋊C₈×C₅₉	—	2	4	no	—
64	#104×C₅₉	—	2	8	no	—
65	#104×C₅₉	—	2	8	no	—
66	C₂²⋊C₈×C₅₉	—	2	8	no	—
67	#97×C₅₉	—	2	2	no	—
68	#97×C₅₉	C₂×C₄	1	4	yes	—
69	#97×C₅₉	D₄	1	4	yes	—
70	#97×C₅₉	D₄	1	4	yes	complemented
71	C₄×C₈×C₅₉	—	2	2	no	—
72	C₄×C₈×C₅₉	D₄	1	4	yes	complemented
73	#41×C₅₉	—	2	4	no	—
74	#41×C₅₉	—	2	4	no	—
75	#41×C₅₉	—	2	4	no	—
76	#41×C₅₉	—	2	4	no	—
77	C₂×Q₈×C₅₉	C₂²×C₄	1	4	yes	—
78	C₂×D₄×C₅₉	—	2	4	no	—
79	C₂×D₄×C₅₉	—	2	4	no	—
80	C₂×D₄×C₅₉	C₂²×C₄	1	4	yes	—
81	C₂²×C₄×C₅₉	—	2	1	no	—
82	C₂²×C₄×C₅₉	—	2	1	no	—
83	C₂²×C₄×C₅₉	—	2	4	no	—
84	C₂²×C₄×C₅₉	#41	1	2	yes	—
85	C₂²×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
86	Q₁₆×C₅₉	—	2	4	no	—
87	Q₁₆×C₅₉	—	2	4	no	—
88	SD₁₆×C₅₉	—	2	8	no	—
89	SD₁₆×C₅₉	—	2	8	no	—
90	D₈×C₅₉	—	2	4	no	—
91	D₈×C₅₉	—	2	4	no	—
92	M₁₆×C₅₉	—	2	4	no	—
93	M₁₆×C₅₉	—	2	4	no	—
94	M₁₆×C₅₉	—	4	8	no	—
95	C₂×C₈×C₅₉	—	2	1	no	—
96	C₂×C₈×C₅₉	—	2	1	no	—
97	C₂×C₈×C₅₉	—	2	4	no	—
98	C₂×C₈×C₅₉	—	4	8	no	—
99	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₅₉	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
105	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₅₉	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₅₉	#41	1	2	yes	—
108	C₂×C₈×C₅₉	#41	1	2	yes	—
109	C₄⋊C₄×C₅₉	—	2	4	no	—
110	C₂²⋊C₄×C₅₉	—	2	8	no	—
111	C₄²×C₅₉	—	2	1	no	—
112	C₄²×C₅₉	—	2	1	no	—
113	C₄²×C₅₉	—	4	4	no	—
114	C₄²×C₅₉	C₂²×C₄	1	2	yes	—
115	C₄²×C₅₉	C₂×D₄	1	2	yes	—
116	C₄²×C₅₉	C₂×D₄	1	2	yes	—
117	C₄²×C₅₉	C₂×D₄	1	2	yes	—
118	C₂³×C₅₉	—	2	4	no	—
119	C₂³×C₅₉	C₂×#41	1	1	yes	characteristic
120	Q₈×C₅₉	—	2	4	no	—
121	Q₈×C₅₉	—	2	4	no	—
122	D₄×C₅₉	—	2	4	no	—
123	D₄×C₅₉	—	2	4	no	—
124	D₄×C₅₉	—	4	4	no	—
125	D₄×C₅₉	—	4	4	no	—
126	D₄×C₅₉	—	8	4	no	—
127	C₂×C₄×C₅₉	—	2	4	no	—
128	C₂×C₄×C₅₉	—	2	2	no	—
129	C₂×C₄×C₅₉	—	2	4	no	—
130	C₂×C₄×C₅₉	—	2	2	no	—
131	C₂×C₄×C₅₉	—	4	2	no	—
132	C₂×C₄×C₅₉	—	4	2	no	—
133	C₂×C₄×C₅₉	—	4	4	no	—
134	C₂×C₄×C₅₉	—	4	4	no	—
135	C₂×C₄×C₅₉	C₄×D₄	1	4	yes	—
136	C₂×C₄×C₅₉	C₂³×C₄	1	1	yes	characteristic
137	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₉	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
141	C₂×C₄×C₅₉	C₂×#41	1	1	yes	characteristic
142	C₈×C₅₉	—	2	2	no	—
143	C₈×C₅₉	—	4	4	no	—
144	C₈×C₅₉	C₄×D₄	1	4	yes	—
145	C₈×C₅₉	C₄×D₄	1	4	yes	—
146	C₈×C₅₉	#127	1	4	yes	—
147	#21710	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
148	C₂²×C₅₉	—	2	2	no	—
149	C₂²×C₅₉	—	4	4	no	—
150	C₂²×C₅₉	—	4	8	no	—
151	C₂²×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
152	C₂²×C₅₉	#532	1	1	yes	characteristic
153	C₂²×C₅₉	#532	1	1	yes	characteristic
154	C₄×C₅₉	—	2	4	no	—
155	C₄×C₅₉	—	2	4	no	—
156	C₄×C₅₉	—	4	2	no	—
157	C₄×C₅₉	—	4	4	no	—
158	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
159	C₄×C₅₉	C₂×C₄×D₄	1	1	yes	characteristic
160	C₄×C₅₉	C₂×#127	1	1	yes	characteristic
161	C₄×C₅₉	#532	1	1	yes	characteristic
162	#3150	C₂×C₅₉	1	1	yes	characteristic
163	C₂×#444	C₂×C₅₉	1	2	yes	complemented
164	C₂×#443	C₂×C₅₉	1	2	yes	complemented
165	#2932	C₂×C₅₉	1	2	yes	complemented
166	#1975	C₂×C₅₉	1	2	yes	complemented
167	#1974	C₂×C₅₉	1	2	yes	complemented
168	#1962	C₂×C₅₉	1	2	yes	complemented
169	#1849	C₂×C₅₉	1	1	yes	characteristic, complemented
170	C₄×#97	C₂×C₅₉	1	1	yes	characteristic, complemented
171	C₂×C₅₉	—	2	1	no	—
172	C₂×C₅₉	—	2	1	no	—
173	C₂×C₅₉	—	8	4	no	—
174	C₂×C₅₉	C₄×#127	1	1	yes	characteristic
175	C₂×C₅₉	#2956	1	2	yes	—
176	C₂×#135	C₂²×C₅₉	1	2	yes	—
177	C₂×#135	C₂²×C₅₉	1	2	yes	—
178	C₂×#131	C₂²×C₅₉	1	4	yes	—
179	#532	C₂²×C₅₉	1	2	yes	—
180	#497	C₄×C₅₉	1	4	yes	complemented
181	#496	C₄×C₅₉	1	4	yes	complemented
182	#444	—	2	8	no	—
183	#443	—	2	8	no	—
184	#435	—	2	8	no	—
185	C₂×#108	C₂²×C₅₉	1	2	yes	—
186	C₂×#108	C₂²×C₅₉	1	2	yes	—
187	#423	C₂²×C₅₉	1	2	yes	—
188	C₂×#104	C₂²×C₅₉	1	4	yes	—
189	C₂×#104	C₂²×C₅₉	1	4	yes	—
190	#408	C₂²×C₅₉	1	4	yes	—
191	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
192	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
193	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
194	C₂×#97	C₂²×C₅₉	1	1	yes	characteristic
195	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₅₉	1	1	yes	characteristic
197	C₄³	C₂²×C₅₉	1	1	yes	characteristic
198	C₅₉	#21710	1	1	yes	characteristic, direct product factor, Sylow
199	C₂×#41	C₂³×C₅₉	1	4	yes	—
200	#135	—	2	4	no	—
201	#135	—	2	4	no	—
202	#135	—	2	4	no	—
203	#135	—	2	4	no	—
204	C₂×Q₁₆	C₂×C₄×C₅₉	1	2	yes	—
205	C₂×Q₁₆	C₂×C₄×C₅₉	1	2	yes	—
206	C₂×SD₁₆	C₂×C₄×C₅₉	1	4	yes	—
207	C₂×SD₁₆	C₂×C₄×C₅₉	1	4	yes	—
208	C₂×D₈	C₂×C₄×C₅₉	1	2	yes	—
209	C₂×D₈	C₂×C₄×C₅₉	1	2	yes	—
210	#131	—	2	8	no	—
211	#131	—	2	8	no	—
212	C₂×M₁₆	—	2	4	no	—
213	C₂×M₁₆	C₂³×C₅₉	1	4	yes	—
214	C₂²×C₈	—	2	4	no	—
215	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
217	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₅₉	1	1	yes	characteristic
219	C₄⋊Q₈	C₂×C₄×C₅₉	1	2	yes	—
220	#127	C₂×C₄×C₅₉	1	2	yes	—
221	#124	C₂×C₄×C₅₉	1	4	yes	—
222	#121	—	2	4	no	—
223	C₄×D₄	—	2	4	no	—
224	C₂×C₄²	—	2	2	no	—
225	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
226	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₅₉	1	1	yes	characteristic
228	#108	—	2	4	no	—
229	#108	—	2	4	no	—
230	C₄⋊C₈	—	2	4	no	—
231	#104	—	2	8	no	—
232	#104	—	2	8	no	—
233	C₂²⋊C₈	—	2	8	no	—
234	#97	—	2	2	no	—
235	#97	C₂×C₄×C₅₉	1	4	yes	—
236	#97	D₄×C₅₉	1	4	yes	—
237	#97	D₄×C₅₉	1	4	yes	complemented
238	C₄×C₈	—	2	2	no	—
239	C₄×C₈	D₄×C₅₉	1	4	yes	complemented
240	#41	—	2	4	no	—
241	#41	—	2	4	no	—
242	#41	—	2	4	no	—
243	#41	—	2	4	no	—
244	C₂×Q₈	C₂²×C₄×C₅₉	1	4	yes	—
245	C₂×D₄	—	2	4	no	—
246	C₂×D₄	—	2	4	no	—
247	C₂×D₄	C₂²×C₄×C₅₉	1	4	yes	—
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	2	1	no	—
250	C₂²×C₄	—	2	4	no	—
251	C₂²×C₄	#41×C₅₉	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₅₉	1	1	yes	characteristic
253	Q₁₆	—	2	4	no	—
254	Q₁₆	—	2	4	no	—
255	SD₁₆	—	2	8	no	—
256	SD₁₆	—	2	8	no	—
257	D₈	—	2	4	no	—
258	D₈	—	2	4	no	—
259	M₁₆	—	2	4	no	—
260	M₁₆	—	2	4	no	—
261	M₁₆	—	4	8	no	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	4	no	—
265	C₂×C₈	—	4	8	no	—
266	C₂×C₈	C₂²×C₄×C₅₉	1	2	yes	—
267	C₂×C₈	C₂²×C₄×C₅₉	1	2	yes	—
268	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
269	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
270	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
271	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
272	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
273	C₂×C₈	C₂×D₄×C₅₉	1	2	yes	—
274	C₂×C₈	#41×C₅₉	1	2	yes	—
275	C₂×C₈	#41×C₅₉	1	2	yes	—
276	C₄⋊C₄	—	2	4	no	—
277	C₂²⋊C₄	—	2	8	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	2	1	no	—
280	C₄²	—	4	4	no	—
281	C₄²	C₂²×C₄×C₅₉	1	2	yes	—
282	C₄²	C₂×D₄×C₅₉	1	2	yes	—
283	C₄²	C₂×D₄×C₅₉	1	2	yes	—
284	C₄²	C₂×D₄×C₅₉	1	2	yes	—
285	C₂³	—	2	4	no	—
286	C₂³	C₂×#41×C₅₉	1	1	yes	characteristic
287	Q₈	—	2	4	no	—
288	Q₈	—	2	4	no	—
289	D₄	—	2	4	no	—
290	D₄	—	2	4	no	—
291	D₄	—	4	4	no	—
292	D₄	—	4	4	no	—
293	D₄	—	8	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	C₄×D₄×C₅₉	1	4	yes	—
303	C₂×C₄	C₂³×C₄×C₅₉	1	1	yes	characteristic
304	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₅₉	1	1	yes	characteristic
307	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₅₉	1	1	yes	characteristic
309	C₈	—	2	2	no	—
310	C₈	—	4	4	no	—
311	C₈	C₄×D₄×C₅₉	1	4	yes	—
312	C₈	C₄×D₄×C₅₉	1	4	yes	—
313	C₈	#127×C₅₉	1	4	yes	—
314	C₂²	—	2	2	no	—
315	C₂²	—	4	4	no	—
316	C₂²	—	4	8	no	—
317	C₂²	C₂×C₄×D₄×C₅₉	1	1	yes	characteristic
318	C₂²	C₅₉×#532	1	1	yes	characteristic
319	C₂²	C₅₉×#532	1	1	yes	characteristic
320	C₄	—	2	4	no	—
321	C₄	—	2	4	no	—
322	C₄	—	4	2	no	—
323	C₄	—	4	4	no	—
324	C₄	C₂×C₄×D₄×C₅₉	1	1	yes	characteristic
325	C₄	C₂×C₄×D₄×C₅₉	1	1	yes	characteristic
326	C₄	C₂×#127×C₅₉	1	1	yes	characteristic
327	C₄	C₅₉×#532	1	1	yes	characteristic
328	C₂	—	2	1	no	—
329	C₂	—	2	1	no	—
330	C₂	—	8	4	no	—
331	C₂	C₄×#127×C₅₉	1	1	yes	characteristic
332	C₂	C₅₉×#2956	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₅₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₅₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₅₉
socle	C₂²×C₅₉	C₂×C₄×D₄
prime radical	C₅₉×#3150	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₉ ◁ C₂×#97×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₅₉×#444	C₂	semidirect
3. C₂×C₅₉×#443	C₂	semidirect
4. C₅₉×#2932	C₂	semidirect
5. C₅₉×#1975	C₂	semidirect
6. C₅₉×#1974	C₂	semidirect
7. C₅₉×#1962	C₂	semidirect
8. C₅₉×#1849	C₂	semidirect
9. C₄×#97×C₅₉	C₂	semidirect
14. C₅₉×#497	C₄	semidirect
15. C₅₉×#496	C₄	semidirect
70. #97×C₅₉	D₄	semidirect
72. C₄×C₈×C₅₉	D₄	semidirect
147. #21710	C₅₉	direct (N × H)
163. C₂×#444	C₂×C₅₉	semidirect
164. C₂×#443	C₂×C₅₉	semidirect
165. #2932	C₂×C₅₉	semidirect
166. #1975	C₂×C₅₉	semidirect
167. #1974	C₂×C₅₉	semidirect
168. #1962	C₂×C₅₉	semidirect
169. #1849	C₂×C₅₉	semidirect
170. C₄×#97	C₂×C₅₉	semidirect
180. #497	C₄×C₅₉	semidirect
181. #496	C₄×C₅₉	semidirect
198. C₅₉	#21710	direct (N × H)
237. #97	D₄×C₅₉	semidirect
239. C₄×C₈	D₄×C₅₉	semidirect

Maximal subgroups

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

subgroup	index
C₅₉×#3150	2
C₂×C₅₉×#444	2
C₂×C₅₉×#443	2
C₅₉×#2932	2
C₅₉×#1975	2
C₅₉×#1974	2
C₅₉×#1962	2
C₅₉×#1849	2
C₄×#97×C₅₉	2
#21710	59

Subgroups — C59×#21710 #28433819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#21710 #28433819