Subgroups — #21437152

Order 7552.

362 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	#16838134	C₂	1	1	yes	maximal, characteristic
2	#16838130	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837797	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837795	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837773	C₂	1	1	yes	maximal, characteristic
6	#16837627	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837610	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837608	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837605	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837601	C₂	1	1	yes	maximal, characteristic
11	#16837531	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837500	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837499	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837494	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#464	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745672	C₂²	1	1	yes	characteristic
17	C₂×#11745672	C₂²	1	1	yes	characteristic
18	C₂×#11745660	C₂²	1	1	yes	characteristic
19	C₂×#11745660	C₂²	1	1	yes	characteristic
20	C₂×#11745670	C₂²	1	1	yes	characteristic
21	C₂×#11745670	C₂²	1	1	yes	characteristic
22	C₂×#11745671	C₂²	1	1	yes	characteristic
23	#14273727	C₂²	1	1	yes	characteristic
24	#14273725	C₂²	1	1	yes	characteristic
25	#14273725	C₂²	1	1	yes	characteristic
26	#14273723	C₂²	1	1	yes	characteristic
27	#14273719	—	2	1	no	—
28	#14273717	—	2	1	no	—
29	#14273717	—	2	1	no	—
30	#14273716	C₂²	1	1	yes	characteristic
31	#14273713	—	2	1	no	—
32	#14273711	—	2	1	no	—
33	#14273709	—	2	1	no	—
34	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
35	C₄×Dic₁₁₈	—	2	1	no	—
36	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
37	#14273692	C₂²	1	1	yes	characteristic
38	#14273692	C₂²	1	1	yes	characteristic
39	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
40	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
41	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
42	#14273660	C₂²	1	1	yes	characteristic
43	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	#14273652	C₂²	1	1	yes	characteristic
46	#14273652	C₂²	1	1	yes	characteristic
47	#14273651	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	#14273638	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	#124×C₅₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂²×Dic₅₉	—	2	1	no	—
65	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
68	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
70	Q₁₆×C₅₉	—	4	1	no	—
71	SD₁₆×C₅₉	—	4	1	no	—
72	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	—	2	1	no	—
75	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₅₉	—	2	1	no	—
77	C₂²⋊C₄×C₅₉	—	2	1	no	—
78	C₄²×C₅₉	—	2	1	no	—
79	C₄²×C₅₉	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₅₉	—	2	1	no	—
81	#11745672	—	2	2	no	—
82	#11745672	—	4	1	no	—
83	#11745671	—	2	2	no	—
84	#11745670	—	2	2	no	—
85	#11745670	—	4	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	C₄⋊Dic₅₉	—	2	1	no	—
90	C₄⋊Dic₅₉	—	2	1	no	—
91	C₄⋊Dic₅₉	—	2	1	no	—
92	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
93	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	—	2	1	no	—
96	Dic₅₉⋊C₄	—	2	1	no	—
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
101	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
106	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	#11745660	—	4	1	no	—
110	#11745660	—	4	1	no	—
111	C₂³×C₅₉	—	2	1	no	—
112	C₂²×D₅₉	—	2	1	no	—
113	Q₈×C₅₉	—	2	1	no	—
114	Q₈×C₅₉	—	2	1	no	—
115	Q₈×C₅₉	—	2	1	no	—
116	Q₈×C₅₉	—	2	1	no	—
117	D₄×C₅₉	—	2	1	no	—
118	D₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₅₉	—	4	2	no	—
126	C₂²⋊D₅₉	—	8	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	8	1	no	—
131	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
133	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
135	D₂₃₆	—	2	1	no	—
136	D₂₃₆	—	2	1	no	—
137	C₄×D₅₉	—	8	1	no	—
138	Dic₁₁₈	—	2	2	no	—
139	Dic₁₁₈	—	2	1	no	—
140	Dic₁₁₈	—	2	1	no	—
141	Dic₁₁₈	—	2	1	no	—
142	Dic₁₁₈	—	2	1	no	—
143	C₈×C₅₉	—	2	1	no	—
144	C₈×C₅₉	—	2	1	no	—
145	C₅₉⋊C₈	—	2	1	no	—
146	C₅₉⋊C₈	#41	1	2	yes	—
147	C₂²×C₅₉	—	4	1	no	—
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	—	4	1	no	—
150	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₅₉	—	4	1	no	—
152	C₂×D₅₉	—	4	1	no	—
153	C₂×D₅₉	—	4	1	no	—
154	C₄×C₅₉	—	2	1	no	—
155	C₄×C₅₉	—	2	1	no	—
156	C₄×C₅₉	—	4	1	no	—
157	C₄×C₅₉	—	4	1	no	—
158	C₄×C₅₉	—	4	1	no	—
159	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
160	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	2	1	no	—
163	Dic₅₉	—	2	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	#137	1	2	yes	—
168	#3400	—	59	1	no	maximal, Sylow
169	C₂×C₅₉	—	8	1	no	—
170	C₂×C₅₉	#550	1	1	yes	characteristic
171	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
172	C₂×C₅₉	#577	1	1	yes	characteristic
173	D₅₉	—	8	1	no	—
174	#559	—	59	1	no	—
175	#557	—	59	1	no	—
176	#497	—	59	1	no	—
177	#497	—	59	1	no	—
178	#492	—	59	1	no	—
179	#464	—	59	1	no	—
180	#464	D₅₉	1	1	yes	characteristic, complemented
181	#462	—	59	1	no	—
182	#461	—	59	1	no	—
183	#461	—	59	1	no	—
184	#460	—	59	1	no	—
185	#446	—	59	1	no	—
186	#440	—	59	1	no	—
187	#440	—	59	1	no	—
188	C₄×Q₁₆	—	59	1	no	—
189	C₅₉	#3400	1	1	yes	characteristic, complemented, Sylow
190	C₂×Q₁₆	—	59	1	no	—
191	C₂×Q₁₆	—	59	1	no	—
192	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
193	C₂×SD₁₆	—	59	1	no	—
194	C₂×SD₁₆	—	59	1	no	—
195	C₂×SD₁₆	—	59	1	no	—
196	C₂×SD₁₆	—	59	1	no	—
197	C₂×SD₁₆	—	59	1	no	—
198	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
199	C₄⋊Q₈	—	59	1	no	—
200	C₄⋊Q₈	—	59	1	no	—
201	C₄⋊Q₈	—	59	1	no	—
202	#127	—	59	1	no	—
203	#125	—	118	1	no	—
204	#125	—	118	1	no	—
205	#125	—	118	1	no	—
206	#124	—	59	1	no	—
207	#124	—	118	1	no	—
208	#124	C₂×D₅₉	1	1	yes	characteristic
209	#123	—	118	1	no	—
210	#123	—	118	1	no	—
211	C₄×Q₈	—	59	1	no	—
212	C₄×Q₈	—	59	1	no	—
213	C₄×Q₈	—	59	1	no	—
214	C₄×Q₈	—	59	1	no	—
215	C₄×Q₈	—	118	1	no	—
216	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
217	C₄×D₄	—	59	1	no	—
218	C₄×D₄	—	59	1	no	—
219	C₄×D₄	—	118	1	no	—
220	#107	—	59	1	no	—
221	C₄⋊C₈	—	59	1	no	—
222	C₄⋊C₈	—	59	1	no	—
223	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
224	#103	—	59	1	no	—
225	#103	—	59	1	no	—
226	#103	—	59	1	no	—
227	#103	C₂×D₅₉	1	1	yes	characteristic
228	#102	—	59	1	no	—
229	#102	C₂×D₅₉	1	1	yes	characteristic
230	#97	—	59	1	no	—
231	#97	—	59	1	no	—
232	C₄×C₈	—	59	1	no	—
233	C₂×Q₈	—	59	1	no	—
234	C₂×Q₈	—	59	1	no	—
235	C₂×Q₈	—	59	1	no	—
236	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
237	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
238	C₂×D₄	—	59	1	no	—
239	C₂×D₄	—	118	1	no	—
240	C₂×D₄	—	118	1	no	—
241	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
242	C₂²×C₄	—	118	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	Q₁₆	—	4	1	no	—
245	Q₁₆	—	118	2	no	—
246	Q₁₆	—	236	1	no	—
247	SD₁₆	—	4	1	no	—
248	SD₁₆	—	118	2	no	—
249	SD₁₆	—	118	2	no	—
250	SD₁₆	—	236	1	no	—
251	SD₁₆	—	236	1	no	—
252	SD₁₆	—	236	1	no	—
253	C₂×C₈	—	59	1	no	—
254	C₂×C₈	—	59	1	no	—
255	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	59	1	no	—
259	C₄⋊C₄	—	59	1	no	—
260	C₄⋊C₄	—	59	1	no	—
261	C₄⋊C₄	—	59	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	—	118	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	—	118	1	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	—	118	1	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	—	118	1	no	—
270	C₄⋊C₄	—	118	1	no	—
271	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₂²⋊C₄	—	118	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₄²	—	2	1	no	—
279	C₄²	—	59	1	no	—
280	C₄²	—	59	1	no	—
281	C₄²	—	118	1	no	—
282	C₄²	—	118	1	no	—
283	C₄²	—	118	1	no	—
284	C₄²	C₂²×D₅₉	1	1	yes	characteristic
285	C₂³	—	2	1	no	—
286	C₂³	—	118	1	no	—
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	118	2	no	—
292	Q₈	—	118	1	no	—
293	Q₈	—	118	1	no	—
294	Q₈	—	118	1	no	—
295	Q₈	—	118	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	118	1	no	—
299	D₄	—	118	1	no	—
300	D₄	—	472	1	no	—
301	D₄	—	236	2	no	—
302	C₂×C₄	—	2	1	no	—
303	C₂×C₄	—	2	1	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	59	1	no	—
306	C₂×C₄	—	59	1	no	—
307	C₂×C₄	—	59	1	no	—
308	C₂×C₄	—	59	1	no	—
309	C₂×C₄	—	118	1	no	—
310	C₂×C₄	—	118	1	no	—
311	C₂×C₄	—	118	1	no	—
312	C₂×C₄	—	472	1	no	—
313	C₂×C₄	—	472	1	no	—
314	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
315	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	59	2	no	—
320	C₈	—	118	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	236	1	no	—
325	C₂²	—	236	1	no	—
326	C₂²	—	236	1	no	—
327	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	59	2	no	—
334	C₄	—	118	1	no	—
335	C₄	—	118	1	no	—
336	C₄	—	118	1	no	—
337	C₄	—	236	1	no	—
338	C₄	—	236	1	no	—
339	C₄	—	236	1	no	—
340	C₄	#41×D₅₉	1	1	yes	characteristic
341	C₄	#14273772	1	1	yes	characteristic
342	C₂	—	8	1	no	—
343	C₂	—	472	1	no	—
344	C₂	#16838070	1	1	yes	characteristic
345	C₂	#137×D₅₉	1	1	yes	characteristic
346	C₂	#16838209	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#464	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#464	D₅₉
hypocenter	C₅₉	#3400
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. #16838130	C₂	semidirect
3. #16837797	C₂	semidirect
4. #16837795	C₂	semidirect
6. #16837627	C₂	semidirect
7. #16837610	C₂	semidirect
8. #16837608	C₂	semidirect
9. #16837605	C₂	semidirect
11. #16837531	C₂	semidirect
12. #16837500	C₂	semidirect
13. #16837499	C₂	semidirect
14. #16837494	C₂	semidirect
15. C₅₉×#464	C₂	semidirect
180. #464	D₅₉	semidirect
189. C₅₉	#3400	semidirect