Subgroups — C₅₉×#11377 #28423486

Order 15104.

814 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#107×C₅₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₉×#340	C₂	1	2	yes	maximal
3	C₂×C₅₉×#340	C₂	1	1	yes	maximal, characteristic
4	C₂×C₅₉×#336	C₂	1	1	yes	maximal, characteristic
5	C₅₉×C₄⋊#95	C₂	1	2	yes	maximal
6	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
7	C₂³×C₈×C₅₉	C₂²	1	1	yes	characteristic
8	C₂²×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
9	C₂²×C₄⋊C₄×C₅₉	C₂²	1	2	yes	—
10	C₂²×C₄²×C₅₉	C₂²	1	1	yes	characteristic
11	C₂×#107×C₅₉	—	2	2	no	—
12	C₂×#107×C₅₉	—	2	2	no	—
13	C₂×#107×C₅₉	C₄	1	4	yes	complemented
14	C₂×#95×C₅₉	—	2	4	no	—
15	C₅₉×#340	—	2	4	no	—
16	C₅₉×#340	—	2	2	no	—
17	C₅₉×#336	—	2	2	no	—
18	C₂³×C₄×C₅₉	—	2	2	no	—
19	C₂³×C₄×C₅₉	—	2	2	no	—
20	C₂³×C₄×C₅₉	D₄	1	2	yes	—
21	C₂³×C₄×C₅₉	C₂³	1	1	yes	characteristic
22	C₂²×C₈×C₅₉	—	2	2	no	—
23	C₂²×C₈×C₅₉	—	2	2	no	—
24	C₂²×C₈×C₅₉	—	2	1	no	—
25	C₂²×C₈×C₅₉	—	2	1	no	—
26	C₂²×C₈×C₅₉	—	2	2	no	—
27	C₂²×C₈×C₅₉	—	2	2	no	—
28	C₂²×C₈×C₅₉	C₂×C₄	1	2	yes	—
29	C₂²×C₈×C₅₉	D₄	1	2	yes	—
30	C₂²×C₈×C₅₉	D₄	1	2	yes	—
31	C₂²×C₈×C₅₉	D₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
37	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
40	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
41	C₂×C₄⋊C₄×C₅₉	—	2	2	no	—
42	C₂×C₄⋊C₄×C₅₉	C₂×C₄	1	4	yes	—
43	C₂×C₄⋊C₄×C₅₉	Q₈	1	4	yes	—
44	C₂×C₄²×C₅₉	—	2	1	no	—
45	C₂×C₄²×C₅₉	—	2	2	no	—
46	C₂×C₄²×C₅₉	—	2	2	no	—
47	C₂×C₄²×C₅₉	—	2	1	no	—
48	#107×C₅₉	—	2	8	no	—
49	#95×C₅₉	—	4	8	no	—
50	C₂⁴×C₅₉	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₉	—	2	2	no	—
52	C₂²×C₄×C₅₉	—	2	4	no	—
53	C₂²×C₄×C₅₉	—	2	2	no	—
54	C₂²×C₄×C₅₉	—	2	4	no	—
55	C₂²×C₄×C₅₉	—	2	4	no	—
56	C₂²×C₄×C₅₉	—	2	4	no	—
57	C₂²×C₄×C₅₉	—	4	2	no	—
58	C₂²×C₄×C₅₉	—	4	2	no	—
59	C₂²×C₄×C₅₉	—	4	2	no	—
60	C₂²×C₄×C₅₉	—	4	2	no	—
61	C₂²×C₄×C₅₉	—	4	4	no	—
62	C₂²×C₄×C₅₉	—	4	4	no	—
63	C₂²×C₄×C₅₉	—	4	4	no	—
64	C₂²×C₄×C₅₉	—	4	4	no	—
65	C₂²×C₄×C₅₉	Q₁₆	1	4	yes	—
66	C₂²×C₄×C₅₉	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₉	C₂×Q₈	1	2	yes	—
71	C₂²×C₄×C₅₉	#41	1	2	yes	—
72	C₂²×C₄×C₅₉	#41	1	2	yes	—
73	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
74	C₂²×C₄×C₅₉	#41	1	2	yes	—
75	C₂²×C₄×C₅₉	#41	1	1	yes	characteristic
76	C₂×C₈×C₅₉	—	2	4	no	—
77	C₂×C₈×C₅₉	—	2	2	no	—
78	C₂×C₈×C₅₉	—	2	2	no	—
79	C₂×C₈×C₅₉	—	2	4	no	—
80	C₂×C₈×C₅₉	—	2	2	no	—
81	C₂×C₈×C₅₉	—	2	4	no	—
82	C₂×C₈×C₅₉	—	4	1	no	—
83	C₂×C₈×C₅₉	—	4	1	no	—
84	C₂×C₈×C₅₉	—	4	2	no	—
85	C₂×C₈×C₅₉	C₂²⋊C₄	1	4	yes	—
86	C₄⋊C₄×C₅₉	—	2	4	no	—
87	C₄⋊C₄×C₅₉	—	2	4	no	—
88	C₄⋊C₄×C₅₉	—	2	4	no	—
89	C₄⋊C₄×C₅₉	—	2	4	no	—
90	C₄⋊C₄×C₅₉	—	4	2	no	—
91	C₄⋊C₄×C₅₉	—	4	2	no	—
92	C₄⋊C₄×C₅₉	—	4	2	no	—
93	C₄⋊C₄×C₅₉	—	4	2	no	—
94	C₄²×C₅₉	—	4	1	no	—
95	C₄²×C₅₉	—	4	1	no	—
96	C₄²×C₅₉	—	4	2	no	—
97	C₂³×C₅₉	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₅₉	#121	1	1	yes	characteristic
99	C₂³×C₅₉	C₂²⋊Q₈	1	2	yes	—
100	C₂³×C₅₉	#123	1	2	yes	—
101	C₂³×C₅₉	#127	1	1	yes	characteristic
102	C₂³×C₅₉	C₂×Q₁₆	1	2	yes	—
103	C₂³×C₅₉	#135	1	2	yes	—
104	C₂³×C₅₉	AGL(1,ℤ/8)	1	4	yes	—
105	C₂×C₄×C₅₉	—	2	4	no	—
106	C₂×C₄×C₅₉	—	2	8	no	—
107	C₂×C₄×C₅₉	—	4	2	no	—
108	C₂×C₄×C₅₉	—	4	2	no	—
109	C₂×C₄×C₅₉	—	4	4	no	—
110	C₂×C₄×C₅₉	—	4	4	no	—
111	C₂×C₄×C₅₉	—	4	4	no	—
112	C₂×C₄×C₅₉	—	4	8	no	—
113	C₂×C₄×C₅₉	—	4	8	no	—
114	C₂×C₄×C₅₉	—	8	4	no	—
115	C₂×C₄×C₅₉	—	8	4	no	—
116	C₂×C₄×C₅₉	—	8	2	no	—
117	C₂×C₄×C₅₉	—	8	2	no	—
118	C₂×C₄×C₅₉	—	8	2	no	—
119	C₂×C₄×C₅₉	—	8	2	no	—
120	C₂×C₄×C₅₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	#117	1	2	yes	—
122	C₂×C₄×C₅₉	C₄×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅₉	#121	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
126	C₂×C₄×C₅₉	C₂²⋊Q₈	1	2	yes	—
127	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
128	C₂×C₄×C₅₉	#123	1	1	yes	characteristic
129	C₂×C₄×C₅₉	#123	1	2	yes	—
130	C₂×C₄×C₅₉	#123	1	2	yes	—
131	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
132	C₂×C₄×C₅₉	#124	1	1	yes	characteristic
133	C₂×C₄×C₅₉	#125	1	2	yes	—
134	C₂×C₄×C₅₉	#125	1	2	yes	—
135	C₂×C₄×C₅₉	#126	1	2	yes	—
136	C₂×C₄×C₅₉	#126	1	2	yes	—
137	C₂×C₄×C₅₉	C₄⋊Q₈	1	1	yes	characteristic
138	C₈×C₅₉	—	2	4	no	—
139	C₈×C₅₉	—	4	2	no	—
140	#11377	C₅₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
141	C₂²×C₅₉	#387	1	2	yes	—
142	C₂²×C₅₉	#390	1	1	yes	characteristic
143	C₂²×C₅₉	#397	1	1	yes	characteristic
144	C₂²×C₅₉	#398	1	1	yes	characteristic
145	C₂²×C₅₉	#399	1	2	yes	—
146	C₂²×C₅₉	C₄×Q₁₆	1	2	yes	—
147	C₂²×C₅₉	#442	1	2	yes	—
148	C₂²×C₅₉	#467	1	2	yes	—
149	C₂²×C₅₉	#469	1	2	yes	—
150	C₂²×C₅₉	#477	1	4	yes	—
151	C₂²×C₅₉	#478	1	4	yes	—
152	C₂²×C₅₉	#482	1	4	yes	—
153	C₂²×C₅₉	#484	1	4	yes	—
154	C₂²×C₅₉	#494	1	1	yes	characteristic
155	C₂²×C₅₉	#495	1	1	yes	characteristic
156	C₂²×C₅₉	#496	1	2	yes	—
157	C₄×C₅₉	—	4	4	no	—
158	C₄×C₅₉	—	8	4	no	—
159	C₄×C₅₉	—	8	4	no	—
160	C₄×C₅₉	#381	1	1	yes	characteristic
161	C₄×C₅₉	#387	1	2	yes	—
162	C₄×C₅₉	#391	1	1	yes	characteristic
163	C₄×C₅₉	#397	1	1	yes	characteristic
164	C₄×C₅₉	#398	1	1	yes	characteristic
165	C₄×C₅₉	#400	1	2	yes	—
166	C₂²×#107	C₂×C₅₉	1	1	yes	characteristic
167	C₂×#340	C₂×C₅₉	1	2	yes	—
168	C₂×#340	C₂×C₅₉	1	1	yes	characteristic
169	C₂×#336	C₂×C₅₉	1	1	yes	characteristic
170	C₄⋊#95	C₂×C₅₉	1	2	yes	—
171	C₂×C₅₉	#1445	1	1	yes	characteristic
172	C₂×C₅₉	#1921	1	4	yes	—
173	C₂×C₅₉	#1969	1	1	yes	characteristic
174	C₂×C₅₉	#1972	1	1	yes	characteristic
175	C₂×C₅₉	#2076	1	2	yes	—
176	C₂×C₅₉	#2087	1	2	yes	—
177	C₂×C₅₉	#2092	1	2	yes	—
178	C₂×C₅₉	#2095	1	2	yes	—
179	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
180	C₂³×C₈	C₂²×C₅₉	1	1	yes	characteristic
181	C₂²×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
182	C₂²×C₄⋊C₄	C₂²×C₅₉	1	2	yes	—
183	C₂²×C₄²	C₂²×C₅₉	1	1	yes	characteristic
184	C₂×#107	—	2	2	no	—
185	C₂×#107	—	2	2	no	—
186	C₂×#107	C₄×C₅₉	1	4	yes	complemented
187	C₂×#95	—	2	4	no	—
188	#340	—	2	4	no	—
189	#340	—	2	2	no	—
190	#336	—	2	2	no	—
191	C₅₉	#11377	1	1	yes	characteristic, direct product factor, Sylow
192	C₂³×C₄	—	2	2	no	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	D₄×C₅₉	1	2	yes	—
195	C₂³×C₄	C₂³×C₅₉	1	1	yes	characteristic
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	2	2	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	2	1	no	—
200	C₂²×C₈	—	2	2	no	—
201	C₂²×C₈	—	2	2	no	—
202	C₂²×C₈	C₂×C₄×C₅₉	1	2	yes	—
203	C₂²×C₈	D₄×C₅₉	1	2	yes	—
204	C₂²×C₈	D₄×C₅₉	1	2	yes	—
205	C₂²×C₈	D₄×C₅₉	1	2	yes	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	—	2	2	no	—
211	C₂×C₄⋊C₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	2	2	no	—
213	C₂×C₄⋊C₄	—	2	2	no	—
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	C₂×C₄×C₅₉	1	4	yes	—
217	C₂×C₄⋊C₄	Q₈×C₅₉	1	4	yes	—
218	C₂×C₄²	—	2	1	no	—
219	C₂×C₄²	—	2	2	no	—
220	C₂×C₄²	—	2	2	no	—
221	C₂×C₄²	—	2	1	no	—
222	#107	—	2	8	no	—
223	#95	—	4	8	no	—
224	C₂⁴	C₂×D₄×C₅₉	1	1	yes	characteristic
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	4	no	—
229	C₂²×C₄	—	2	4	no	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	—	4	4	no	—
236	C₂²×C₄	—	4	4	no	—
237	C₂²×C₄	—	4	4	no	—
238	C₂²×C₄	—	4	4	no	—
239	C₂²×C₄	Q₁₆×C₅₉	1	4	yes	—
240	C₂²×C₄	C₂²×C₄×C₅₉	1	1	yes	characteristic
241	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₅₉	1	1	yes	characteristic
244	C₂²×C₄	C₂×Q₈×C₅₉	1	2	yes	—
245	C₂²×C₄	#41×C₅₉	1	2	yes	—
246	C₂²×C₄	#41×C₅₉	1	2	yes	—
247	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
248	C₂²×C₄	#41×C₅₉	1	2	yes	—
249	C₂²×C₄	#41×C₅₉	1	1	yes	characteristic
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	4	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	—	2	4	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	4	2	no	—
259	C₂×C₈	C₂²⋊C₄×C₅₉	1	4	yes	—
260	C₄⋊C₄	—	2	4	no	—
261	C₄⋊C₄	—	2	4	no	—
262	C₄⋊C₄	—	2	4	no	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	—	4	1	no	—
270	C₄²	—	4	2	no	—
271	C₂³	C₄×D₄×C₅₉	1	1	yes	characteristic
272	C₂³	#121×C₅₉	1	1	yes	characteristic
273	C₂³	C₂²⋊Q₈×C₅₉	1	2	yes	—
274	C₂³	#123×C₅₉	1	2	yes	—
275	C₂³	#127×C₅₉	1	1	yes	characteristic
276	C₂³	C₂×Q₁₆×C₅₉	1	2	yes	—
277	C₂³	#135×C₅₉	1	2	yes	—
278	C₂³	AGL(1,ℤ/8)×C₅₉	1	4	yes	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	8	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	4	4	no	—
286	C₂×C₄	—	4	8	no	—
287	C₂×C₄	—	4	8	no	—
288	C₂×C₄	—	8	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	8	2	no	—
291	C₂×C₄	—	8	2	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	—	8	2	no	—
294	C₂×C₄	C₂×C₂²⋊C₄×C₅₉	1	1	yes	characteristic
295	C₂×C₄	#117×C₅₉	1	2	yes	—
296	C₂×C₄	C₄×D₄×C₅₉	1	1	yes	characteristic
297	C₂×C₄	C₄×Q₈×C₅₉	1	2	yes	—
298	C₂×C₄	#121×C₅₉	1	1	yes	characteristic
299	C₂×C₄	C₂²⋊Q₈×C₅₉	1	2	yes	—
300	C₂×C₄	C₂²⋊Q₈×C₅₉	1	2	yes	—
301	C₂×C₄	#123×C₅₉	1	1	yes	characteristic
302	C₂×C₄	#123×C₅₉	1	1	yes	characteristic
303	C₂×C₄	#123×C₅₉	1	2	yes	—
304	C₂×C₄	#123×C₅₉	1	2	yes	—
305	C₂×C₄	#124×C₅₉	1	1	yes	characteristic
306	C₂×C₄	#124×C₅₉	1	1	yes	characteristic
307	C₂×C₄	#125×C₅₉	1	2	yes	—
308	C₂×C₄	#125×C₅₉	1	2	yes	—
309	C₂×C₄	#126×C₅₉	1	2	yes	—
310	C₂×C₄	#126×C₅₉	1	2	yes	—
311	C₂×C₄	C₄⋊Q₈×C₅₉	1	1	yes	characteristic
312	C₈	—	2	4	no	—
313	C₈	—	4	2	no	—
314	C₂²	C₅₉×#387	1	2	yes	—
315	C₂²	C₅₉×#390	1	1	yes	characteristic
316	C₂²	C₅₉×#397	1	1	yes	characteristic
317	C₂²	C₅₉×#398	1	1	yes	characteristic
318	C₂²	C₅₉×#399	1	2	yes	—
319	C₂²	C₄×Q₁₆×C₅₉	1	2	yes	—
320	C₂²	C₅₉×#442	1	2	yes	—
321	C₂²	C₅₉×#467	1	2	yes	—
322	C₂²	C₅₉×#469	1	2	yes	—
323	C₂²	C₅₉×#477	1	4	yes	—
324	C₂²	C₅₉×#478	1	4	yes	—
325	C₂²	C₅₉×#482	1	4	yes	—
326	C₂²	C₅₉×#484	1	4	yes	—
327	C₂²	C₅₉×#494	1	1	yes	characteristic
328	C₂²	C₅₉×#495	1	1	yes	characteristic
329	C₂²	C₅₉×#496	1	2	yes	—
330	C₄	—	4	4	no	—
331	C₄	—	8	4	no	—
332	C₄	—	8	4	no	—
333	C₄	C₅₉×#381	1	1	yes	characteristic
334	C₄	C₅₉×#387	1	2	yes	—
335	C₄	C₅₉×#391	1	1	yes	characteristic
336	C₄	C₅₉×#397	1	1	yes	characteristic
337	C₄	C₅₉×#398	1	1	yes	characteristic
338	C₄	C₅₉×#400	1	2	yes	—
339	C₂	C₅₉×#1445	1	1	yes	characteristic
340	C₂	C₅₉×#1921	1	4	yes	—
341	C₂	C₅₉×#1969	1	1	yes	characteristic
342	C₂	C₅₉×#1972	1	1	yes	characteristic
343	C₂	C₅₉×#2076	1	2	yes	—
344	C₂	C₅₉×#2087	1	2	yes	—
345	C₂	C₅₉×#2092	1	2	yes	—
346	C₂	C₅₉×#2095	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂⁴×C₅₉	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₅₉
Frattini, cosocle	C₂³×C₄	C₂³×C₅₉
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₅₉ ◁ C₂²×C₄²×C₅₉ ◁ G

Complements

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

N	H ≅ G/N	type
13. C₂×#107×C₅₉	C₄	semidirect
140. #11377	C₅₉	direct (N × H)
186. C₂×#107	C₄×C₅₉	semidirect
191. C₅₉	#11377	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#107×C₅₉	2
C₂×C₅₉×#340	2
C₂×C₅₉×#336	2
C₅₉×C₄⋊#95	2
#11377	59

Subgroups — C59×#11377 #28423486

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₉×#11377 #28423486