Subgroups — #21428876

Order 7552.

420 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₅₉⋊#401	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837344	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837327	C₂	1	1	yes	maximal, characteristic
4	#16837309	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837309	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837296	C₂	1	1	yes	maximal, characteristic
7	#16837290	C₂	1	1	yes	maximal, characteristic
8	#16837289	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837285	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837285	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837277	C₂	1	1	yes	maximal, characteristic
12	#16837263	C₂	1	1	yes	maximal, characteristic
13	C₂²⋊C₄×Dic₅₉	C₂	1	1	yes	maximal, characteristic
14	#16837247	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#393	C₂	1	1	yes	maximal, characteristic
16	C₂³×Dic₅₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
20	C₂×Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Dic₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₂²⋊Dic₅₉	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₅₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
29	C₅₉⋊#95	C₂²	1	1	yes	characteristic
30	C₅₉⋊#95	C₂²	1	1	yes	characteristic
31	C₅₉⋊#95	C₂²	1	1	yes	characteristic
32	C₅₉⋊#95	C₂²	1	1	yes	characteristic
33	C₅₉⋊#95	C₂²	1	1	yes	characteristic
34	C₅₉⋊#95	C₂²	1	1	yes	characteristic
35	C₅₉⋊#95	C₂²	1	1	yes	characteristic
36	C₅₉⋊#95	C₂²	1	1	yes	characteristic
37	C₅₉⋊#95	C₂²	1	1	yes	characteristic
38	C₅₉⋊#95	C₂²	1	1	yes	characteristic
39	C₅₉⋊#95	C₂²	1	1	yes	characteristic
40	C₅₉⋊#95	C₂²	1	1	yes	characteristic
41	C₅₉⋊#95	C₂²	1	1	yes	characteristic
42	C₅₉⋊#95	C₂²	1	1	yes	characteristic
43	C₅₉⋊#95	C₂²	1	1	yes	characteristic
44	C₂²×Q₈×C₅₉	C₂²	1	1	yes	characteristic
45	C₂×C₂²⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
46	C₂×C₂²⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
47	C₂×C₂²⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
48	#95×C₅₉	C₂²	1	1	yes	characteristic
49	#95×C₅₉	C₂²	1	1	yes	characteristic
50	#95×C₅₉	C₂²	1	1	yes	characteristic
51	C₂⁴×C₅₉	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₅₉	—	2	2	no	—
53	C₂×Q₈×C₅₉	—	2	2	no	—
54	C₂×Q₈×C₅₉	—	4	1	no	—
55	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×Dic₅₉	—	2	2	no	—
62	C₂²×Dic₅₉	—	2	2	no	—
63	C₂²×Dic₅₉	—	4	1	no	—
64	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
65	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
66	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
67	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
68	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
69	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
70	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
71	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
72	C₂²⋊C₄×C₅₉	—	2	2	no	—
73	C₂²⋊C₄×C₅₉	—	2	2	no	—
74	C₂²⋊C₄×C₅₉	—	4	1	no	—
75	C₂²⋊Dic₅₉	—	2	2	no	—
76	C₂²⋊Dic₅₉	—	2	2	no	—
77	C₂²⋊Dic₅₉	—	2	2	no	—
78	C₄⋊Dic₅₉	—	2	2	no	—
79	C₄⋊Dic₅₉	—	2	2	no	—
80	C₄⋊Dic₅₉	—	4	1	no	—
81	Dic₅₉⋊C₄	—	2	2	no	—
82	Dic₅₉⋊C₄	—	4	1	no	—
83	C₄×Dic₅₉	—	2	2	no	—
84	C₄×Dic₅₉	—	2	2	no	—
85	C₄×Dic₅₉	—	2	2	no	—
86	C₄×Dic₅₉	—	4	1	no	—
87	C₂³×C₅₉	—	2	1	no	—
88	C₂³×C₅₉	—	2	1	no	—
89	C₂³×C₅₉	—	2	1	no	—
90	C₂³×C₅₉	—	2	1	no	—
91	C₂³×C₅₉	—	2	1	no	—
92	C₂³×C₅₉	—	2	1	no	—
93	C₂³×C₅₉	#41	1	2	yes	—
94	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
95	Q₈×C₅₉	—	4	4	no	—
96	C₂×C₄×C₅₉	—	2	1	no	—
97	C₂×C₄×C₅₉	—	2	1	no	—
98	C₂×C₄×C₅₉	—	2	1	no	—
99	C₂×C₄×C₅₉	—	2	1	no	—
100	C₂×C₄×C₅₉	—	2	1	no	—
101	C₂×C₄×C₅₉	—	2	1	no	—
102	C₂×C₄×C₅₉	—	2	1	no	—
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	#41	1	2	yes	—
113	C₂×C₄×C₅₉	#41	1	2	yes	—
114	C₂×Dic₅₉	—	2	1	no	—
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	—	2	1	no	—
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	2	1	no	—
133	C₂×Dic₅₉	—	2	1	no	—
134	C₂×Dic₅₉	—	2	1	no	—
135	C₂×Dic₅₉	—	2	1	no	—
136	C₂×Dic₅₉	—	4	4	no	—
137	C₂×Dic₅₉	#41	1	2	yes	—
138	C₂×Dic₅₉	#41	1	2	yes	—
139	C₂²×C₅₉	—	2	2	no	—
140	C₂²×C₅₉	—	2	2	no	—
141	C₂²×C₅₉	—	2	2	no	—
142	C₂²×C₅₉	—	4	1	no	—
143	C₂²×C₅₉	—	4	1	no	—
144	C₂²×C₅₉	—	4	1	no	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
147	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
148	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
149	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
150	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
151	C₂²×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂²×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
153	C₄×C₅₉	—	2	2	no	—
154	C₄×C₅₉	—	2	2	no	—
155	C₄×C₅₉	—	4	1	no	—
156	C₄×C₅₉	—	4	1	no	—
157	C₄×C₅₉	—	4	1	no	—
158	C₄×C₅₉	—	4	1	no	—
159	Dic₅₉	—	2	2	no	—
160	Dic₅₉	—	2	2	no	—
161	Dic₅₉	—	4	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	—	4	1	no	—
167	#2527	—	59	1	no	maximal, Sylow
168	C₂×C₅₉	—	4	2	no	—
169	C₂×C₅₉	#529	1	1	yes	characteristic
170	C₂×C₅₉	#529	1	1	yes	characteristic
171	C₂×C₅₉	#542	1	1	yes	characteristic
172	C₂×C₅₉	#551	1	1	yes	characteristic
173	C₂×C₅₉	#551	1	1	yes	characteristic
174	C₂×C₅₉	#552	1	1	yes	characteristic
175	C₂×C₅₉	#556	1	1	yes	characteristic
176	#401	—	59	1	no	—
177	#400	—	59	1	no	—
178	#397	—	59	1	no	—
179	#393	D₅₉	1	1	yes	characteristic
180	#391	—	59	1	no	—
181	#391	—	59	1	no	—
182	#388	—	59	1	no	—
183	#388	—	59	1	no	—
184	#387	—	59	1	no	—
185	#387	—	59	1	no	—
186	#387	—	59	1	no	—
187	#385	—	59	1	no	—
188	#381	—	59	1	no	—
189	C₄×C₂²⋊C₄	—	59	1	no	—
190	#376	—	59	1	no	—
191	C₅₉	#2527	1	1	yes	characteristic, complemented, Sylow
192	C₂²×Q₈	C₂×D₅₉	1	1	yes	characteristic
193	C₂³×C₄	—	59	1	no	—
194	C₂×C₄⋊C₄	—	59	1	no	—
195	C₂×C₄⋊C₄	—	59	1	no	—
196	C₂×C₄⋊C₄	—	59	1	no	—
197	C₂×C₄⋊C₄	—	59	1	no	—
198	C₂×C₄⋊C₄	—	59	1	no	—
199	C₂×C₂²⋊C₄	—	59	1	no	—
200	C₂×C₂²⋊C₄	—	59	1	no	—
201	C₂×C₂²⋊C₄	—	59	1	no	—
202	C₂×C₂²⋊C₄	C₂×D₅₉	1	1	yes	characteristic
203	C₂×C₂²⋊C₄	C₂×D₅₉	1	1	yes	characteristic
204	C₂×C₂²⋊C₄	C₂×D₅₉	1	1	yes	characteristic
205	C₂×C₄²	—	59	1	no	—
206	C₂×C₄²	—	59	1	no	—
207	C₂×C₄²	—	59	1	no	—
208	C₂×C₄²	—	59	1	no	—
209	#95	—	59	1	no	—
210	#95	—	59	1	no	—
211	#95	—	59	1	no	—
212	#95	—	59	1	no	—
213	#95	—	59	1	no	—
214	#95	—	59	1	no	—
215	#95	—	59	1	no	—
216	#95	—	59	1	no	—
217	#95	—	59	1	no	—
218	#95	—	59	1	no	—
219	#95	—	59	1	no	—
220	#95	—	59	1	no	—
221	#95	—	59	1	no	—
222	#95	—	59	1	no	—
223	#95	—	59	1	no	—
224	#95	C₂×D₅₉	1	1	yes	characteristic
225	#95	C₂×D₅₉	1	1	yes	characteristic
226	#95	C₂×D₅₉	1	1	yes	characteristic
227	C₂⁴	C₂²×D₅₉	1	1	yes	characteristic
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	4	1	no	—
231	C₂²×C₄	—	59	1	no	—
232	C₂²×C₄	—	59	1	no	—
233	C₂²×C₄	—	59	1	no	—
234	C₂²×C₄	—	59	1	no	—
235	C₂²×C₄	—	59	1	no	—
236	C₂²×C₄	—	59	1	no	—
237	C₂²×C₄	—	59	1	no	—
238	C₂²×C₄	—	59	1	no	—
239	C₂²×C₄	—	118	2	no	—
240	C₂²×C₄	—	118	2	no	—
241	C₂²×C₄	—	236	1	no	—
242	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
243	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
244	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
245	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
246	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
247	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
248	C₄⋊C₄	—	118	2	no	—
249	C₄⋊C₄	—	118	2	no	—
250	C₄⋊C₄	—	118	2	no	—
251	C₄⋊C₄	—	236	1	no	—
252	C₄⋊C₄	—	236	1	no	—
253	C₂²⋊C₄	—	2	2	no	—
254	C₂²⋊C₄	—	2	2	no	—
255	C₂²⋊C₄	—	4	1	no	—
256	C₂²⋊C₄	—	118	2	no	—
257	C₂²⋊C₄	—	118	2	no	—
258	C₂²⋊C₄	—	118	2	no	—
259	C₄²	—	118	2	no	—
260	C₄²	—	118	2	no	—
261	C₄²	—	118	2	no	—
262	C₄²	—	236	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	—	2	1	no	—
269	C₂³	#11745689	1	2	yes	—
270	C₂³	C₂³×D₅₉	1	1	yes	characteristic
271	Q₈	—	4	4	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	59	2	no	—
289	C₂×C₄	—	59	2	no	—
290	C₂×C₄	—	118	1	no	—
291	C₂×C₄	—	118	1	no	—
292	C₂×C₄	—	118	1	no	—
293	C₂×C₄	—	118	1	no	—
294	C₂×C₄	—	118	1	no	—
295	C₂×C₄	—	118	1	no	—
296	C₂×C₄	—	118	1	no	—
297	C₂×C₄	—	118	1	no	—
298	C₂×C₄	—	118	1	no	—
299	C₂×C₄	—	118	1	no	—
300	C₂×C₄	—	118	1	no	—
301	C₂×C₄	—	118	1	no	—
302	C₂×C₄	—	118	1	no	—
303	C₂×C₄	—	118	1	no	—
304	C₂×C₄	—	118	1	no	—
305	C₂×C₄	—	118	1	no	—
306	C₂×C₄	—	118	1	no	—
307	C₂×C₄	—	118	1	no	—
308	C₂×C₄	—	118	1	no	—
309	C₂×C₄	—	118	1	no	—
310	C₂×C₄	—	118	1	no	—
311	C₂×C₄	—	118	1	no	—
312	C₂×C₄	—	236	4	no	—
313	C₂×C₄	#11745687	1	2	yes	—
314	C₂×C₄	#11745687	1	2	yes	—
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	#41×D₅₉	1	1	yes	characteristic
323	C₂²	#41×D₅₉	1	1	yes	characteristic
324	C₂²	C₂×#11745687	1	1	yes	characteristic
325	C₂²	C₂×#11745687	1	1	yes	characteristic
326	C₂²	C₂×#11745689	1	1	yes	characteristic
327	C₂²	#14273772	1	1	yes	characteristic
328	C₂²	#14273774	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	118	2	no	—
336	C₄	—	118	2	no	—
337	C₄	—	236	1	no	—
338	C₄	—	236	1	no	—
339	C₄	—	236	1	no	—
340	C₄	—	236	1	no	—
341	C₄	—	236	1	no	—
342	C₄	—	236	1	no	—
343	C₂	—	4	2	no	—
344	C₂	#16837931	1	1	yes	characteristic
345	C₂	#16837938	1	1	yes	characteristic
346	C₂	#16838016	1	1	yes	characteristic
347	C₂	#16838079	1	1	yes	characteristic
348	C₂	#16838083	1	1	yes	characteristic
349	C₂	#16838094	1	1	yes	characteristic
350	C₂	#16838121	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator, socle	C₂³×C₅₉	C₂⁴
Fitting	C₅₉×#393	C₂
hypercenter	#393	D₅₉
hypocenter	C₅₉	#2527
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ #393

Complements

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

N	H ≅ G/N	type
1. C₅₉⋊#401	C₂	semidirect
2. #16837344	C₂	semidirect
4. #16837309	C₂	semidirect
5. #16837309	C₂	semidirect
8. #16837289	C₂	semidirect
9. #16837285	C₂	semidirect
10. #16837285	C₂	semidirect
14. #16837247	C₂	semidirect
191. C₅₉	#2527	semidirect