Subgroups — #21432063

Order 7552.

408 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837344	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837341	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837338	C₂	1	1	yes	maximal, characteristic
4	#16837338	C₂	1	1	yes	maximal, characteristic
5	#16837336	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837334	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837301	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837300	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837300	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837299	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837295	C₂	1	1	yes	maximal, characteristic
12	#16837277	C₂	1	1	yes	maximal, characteristic
13	#16837277	C₂	1	1	yes	maximal, characteristic
14	#16837274	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#388	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×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
21	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
24	C₂×Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
25	C₂×Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
28	C₂×Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
29	C₂×Dic₅₉⋊C₄	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Dic₅₉	C₂²	1	1	yes	characteristic
31	C₂×C₂²⋊Dic₅₉	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊Dic₅₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×Dic₅₉	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	C₂×C₄⋊C₄×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	C₂×C₄²×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	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
54	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
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×Dic₅₉	—	4	1	no	—
59	C₂²×Dic₅₉	—	4	1	no	—
60	C₂²×Dic₅₉	—	4	1	no	—
61	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
62	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
63	C₂²×Dic₅₉	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₅₉	—	2	2	no	—
70	C₂²⋊C₄×C₅₉	—	2	2	no	—
71	C₂²⋊C₄×C₅₉	—	4	1	no	—
72	C₂²⋊C₄×C₅₉	Q₈	1	4	yes	—
73	C₄²×C₅₉	—	4	1	no	—
74	C₂²⋊Dic₅₉	—	2	2	no	—
75	C₂²⋊Dic₅₉	—	4	1	no	—
76	C₂²⋊Dic₅₉	—	4	1	no	—
77	C₄⋊Dic₅₉	—	2	2	no	—
78	C₄⋊Dic₅₉	—	2	2	no	—
79	C₄⋊Dic₅₉	—	4	1	no	—
80	C₄⋊Dic₅₉	—	4	1	no	—
81	C₄⋊Dic₅₉	—	4	1	no	—
82	C₄⋊Dic₅₉	—	4	1	no	—
83	C₄⋊Dic₅₉	—	4	1	no	—
84	Dic₅₉⋊C₄	—	2	2	no	—
85	Dic₅₉⋊C₄	—	4	1	no	—
86	Dic₅₉⋊C₄	—	4	1	no	—
87	Dic₅₉⋊C₄	—	4	1	no	—
88	Dic₅₉⋊C₄	—	4	1	no	—
89	Dic₅₉⋊C₄	—	4	1	no	—
90	C₄×Dic₅₉	—	4	1	no	—
91	C₂³×C₅₉	—	2	1	no	—
92	C₂³×C₅₉	—	2	1	no	—
93	C₂³×C₅₉	—	2	1	no	—
94	C₂³×C₅₉	—	2	1	no	—
95	C₂³×C₅₉	—	2	1	no	—
96	C₂³×C₅₉	—	2	1	no	—
97	C₂³×C₅₉	C₂×Q₈	1	2	yes	—
98	C₂³×C₅₉	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
115	C₂×C₄×C₅₉	C₂×Q₈	1	2	yes	—
116	C₂×C₄×C₅₉	#41	1	2	yes	—
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	—	2	1	no	—
137	C₂×Dic₅₉	—	2	1	no	—
138	C₂×Dic₅₉	—	2	1	no	—
139	C₂×Dic₅₉	—	2	1	no	—
140	C₂×Dic₅₉	—	4	4	no	—
141	C₂×Dic₅₉	#41	1	2	yes	—
142	C₂²×C₅₉	—	2	2	no	—
143	C₂²×C₅₉	—	2	2	no	—
144	C₂²×C₅₉	—	2	2	no	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	—	4	1	no	—
147	C₂²×C₅₉	—	4	1	no	—
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	C₂²×Q₈	1	1	yes	characteristic
150	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
151	C₂²×C₅₉	C₂×#41	1	1	yes	characteristic
152	C₂²×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
153	C₂²×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
154	C₂²×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂²×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
156	C₄×C₅₉	—	2	2	no	—
157	C₄×C₅₉	—	2	2	no	—
158	C₄×C₅₉	—	2	2	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	—	4	1	no	—
162	Dic₅₉	—	2	2	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	—	4	1	no	—
168	Dic₅₉	—	4	1	no	—
169	Dic₅₉	—	4	1	no	—
170	#2807	—	59	1	no	maximal, Sylow
171	C₂×C₅₉	—	4	2	no	—
172	C₂×C₅₉	#539	1	1	yes	characteristic
173	C₂×C₅₉	#539	1	1	yes	characteristic
174	C₂×C₅₉	#544	1	1	yes	characteristic
175	C₂×C₅₉	#554	1	1	yes	characteristic
176	C₂×C₅₉	#554	1	1	yes	characteristic
177	C₂×C₅₉	#559	1	1	yes	characteristic
178	C₂×C₅₉	#560	1	1	yes	characteristic
179	#400	—	59	1	no	—
180	#400	—	59	1	no	—
181	#399	—	59	1	no	—
182	#399	—	59	1	no	—
183	#398	—	59	1	no	—
184	#398	—	59	1	no	—
185	#389	—	59	1	no	—
186	#389	—	59	1	no	—
187	#389	—	59	1	no	—
188	#389	—	59	1	no	—
189	#388	—	59	1	no	—
190	#388	D₅₉	1	1	yes	characteristic
191	#385	—	59	1	no	—
192	#385	—	59	1	no	—
193	#385	—	59	1	no	—
194	C₅₉	#2807	1	1	yes	characteristic, complemented, Sylow
195	C₂³×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×C₄⋊C₄	—	59	1	no	—
200	C₂×C₄⋊C₄	—	59	1	no	—
201	C₂×C₄⋊C₄	—	59	1	no	—
202	C₂×C₄⋊C₄	—	59	1	no	—
203	C₂×C₄⋊C₄	—	59	1	no	—
204	C₂×C₄⋊C₄	—	59	1	no	—
205	C₂×C₄⋊C₄	—	59	1	no	—
206	C₂×C₄⋊C₄	—	59	1	no	—
207	C₂×C₄⋊C₄	—	59	1	no	—
208	C₂×C₄⋊C₄	—	59	1	no	—
209	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	—	59	1	no	—
211	C₂×C₂²⋊C₄	—	59	1	no	—
212	C₂×C₂²⋊C₄	—	59	1	no	—
213	C₂×C₂²⋊C₄	C₂×D₅₉	1	1	yes	characteristic
214	C₂×C₂²⋊C₄	C₂×D₅₉	1	1	yes	characteristic
215	C₂×C₂²⋊C₄	C₂×D₅₉	1	1	yes	characteristic
216	C₂×C₄²	—	59	1	no	—
217	C₂×C₄²	C₂×D₅₉	1	1	yes	characteristic
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	—	59	1	no	—
225	#95	—	59	1	no	—
226	#95	—	59	1	no	—
227	#95	—	59	1	no	—
228	#95	C₂×D₅₉	1	1	yes	characteristic
229	#95	C₂×D₅₉	1	1	yes	characteristic
230	C₂⁴	C₂²×D₅₉	1	1	yes	characteristic
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	—	236	1	no	—
240	C₂²×C₄	—	236	1	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	—	2	2	no	—
249	C₄⋊C₄	—	118	2	no	—
250	C₄⋊C₄	—	118	2	no	—
251	C₄⋊C₄	—	118	2	no	—
252	C₄⋊C₄	—	236	1	no	—
253	C₄⋊C₄	—	236	1	no	—
254	C₄⋊C₄	—	236	1	no	—
255	C₄⋊C₄	—	236	1	no	—
256	C₄⋊C₄	—	236	1	no	—
257	C₄⋊C₄	—	236	1	no	—
258	C₄⋊C₄	—	236	1	no	—
259	C₄⋊C₄	—	236	1	no	—
260	C₄⋊C₄	—	236	1	no	—
261	C₄⋊C₄	—	236	1	no	—
262	C₂²⋊C₄	—	2	2	no	—
263	C₂²⋊C₄	—	4	1	no	—
264	C₂²⋊C₄	—	118	2	no	—
265	C₂²⋊C₄	—	236	1	no	—
266	C₂²⋊C₄	—	236	1	no	—
267	C₂²⋊C₄	Dic₁₁₈	1	4	yes	—
268	C₄²	—	4	1	no	—
269	C₄²	—	236	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	2	1	no	—
272	C₂³	—	2	1	no	—
273	C₂³	—	2	1	no	—
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	C₂³	C₂×Dic₁₁₈	1	2	yes	—
277	C₂³	C₂³×D₅₉	1	1	yes	characteristic
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	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	59	2	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	—	118	1	no	—
313	C₂×C₄	—	118	1	no	—
314	C₂×C₄	—	118	1	no	—
315	C₂×C₄	—	118	1	no	—
316	C₂×C₄	—	118	1	no	—
317	C₂×C₄	—	236	4	no	—
318	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
319	C₂×C₄	C₂×Dic₁₁₈	1	2	yes	—
320	C₂×C₄	#11745687	1	2	yes	—
321	C₂²	—	2	2	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	2	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	C₂²×Dic₁₁₈	1	1	yes	characteristic
329	C₂²	#41×D₅₉	1	1	yes	characteristic
330	C₂²	C₂×#11745687	1	1	yes	characteristic
331	C₂²	#14273772	1	1	yes	characteristic
332	C₂²	#14273773	1	1	yes	characteristic
333	C₂²	#14273773	1	1	yes	characteristic
334	C₂²	#14273774	1	1	yes	characteristic
335	C₄	—	2	2	no	—
336	C₄	—	2	2	no	—
337	C₄	—	2	2	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	4	1	no	—
341	C₄	—	118	2	no	—
342	C₄	—	236	1	no	—
343	C₄	—	236	1	no	—
344	C₄	—	236	1	no	—
345	C₄	—	236	1	no	—
346	C₄	—	236	1	no	—
347	C₄	—	236	1	no	—
348	C₄	—	236	1	no	—
349	C₂	—	4	2	no	—
350	C₂	#16837993	1	1	yes	characteristic
351	C₂	#16837997	1	1	yes	characteristic
352	C₂	#16838030	1	1	yes	characteristic
353	C₂	#16838109	1	1	yes	characteristic
354	C₂	#16838111	1	1	yes	characteristic
355	C₂	#16838134	1	1	yes	characteristic
356	C₂	#16838142	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₅₉
commutator, socle	C₂³×C₅₉	C₂⁴
Fitting	C₅₉×#388	C₂
hypercenter	#388	D₅₉
hypocenter	C₅₉	#2807
LCS	G ▷ C₂³×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂³ ◁ #388

Complements

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

N	H ≅ G/N	type
1. #16837344	C₂	semidirect
2. #16837341	C₂	semidirect
5. #16837336	C₂	semidirect
6. #16837334	C₂	semidirect
7. #16837301	C₂	semidirect
8. #16837300	C₂	semidirect
9. #16837300	C₂	semidirect
10. #16837299	C₂	semidirect
194. C₅₉	#2807	semidirect