Subgroups — #21436142

Order 7552.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838086	C₂	1	1	yes	maximal, characteristic
2	#16838065	C₂	1	1	yes	maximal, characteristic
3	#16837769	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837740	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837734	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837712	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837642	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837610	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837557	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837556	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837506	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837471	C₂	1	1	yes	maximal, characteristic
13	#16837412	C₂	1	1	yes	maximal, characteristic, complemented
14	#103×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#470	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
18	C₂×#11745672	C₂²	1	1	yes	characteristic
19	C₂×#11745672	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
21	#14273726	—	2	1	no	—
22	#14273725	C₂²	1	1	yes	characteristic
23	#14273720	—	2	1	no	—
24	#14273718	C₂²	1	1	yes	characteristic
25	#14273717	—	2	1	no	—
26	#14273709	—	2	1	no	—
27	#14273707	C₂²	1	1	yes	characteristic
28	#14273705	—	2	1	no	—
29	#14273704	—	2	1	no	—
30	#14273703	C₂²	1	1	yes	characteristic
31	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
32	#14273692	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
34	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
35	#14273686	C₂²	1	1	yes	characteristic
36	#14273685	—	2	1	no	—
37	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273658	C₂²	1	1	yes	characteristic
40	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
41	#14273652	C₂²	1	1	yes	characteristic
42	#14273652	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273650	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	#14273637	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#103×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	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
63	C₂²×Dic₅₉	—	2	1	no	—
64	Q₈×D₅₉	—	2	2	no	—
65	Q₈×D₅₉	—	4	1	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₁₈	—	2	2	no	—
70	C₂×Dic₁₁₈	—	2	1	no	—
71	C₂×Dic₁₁₈	—	2	1	no	—
72	C₂×Dic₁₁₈	—	4	1	no	—
73	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
75	Q₁₆×C₅₉	—	4	1	no	—
76	M₁₆×C₅₉	—	4	1	no	—
77	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	—	2	1	no	—
80	C₄⋊C₄×C₅₉	—	2	1	no	—
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₅₉	—	2	1	no	—
84	C₂²⋊C₄×C₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
87	#11745672	—	2	2	no	—
88	#11745672	—	4	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	—	2	1	no	—
94	C₄⋊Dic₅₉	—	2	1	no	—
95	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	—	2	1	no	—
103	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	—	2	1	no	—
106	C₄×Dic₅₉	—	2	1	no	—
107	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
110	C₈×D₅₉	—	4	1	no	—
111	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
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	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×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	—	2	1	no	—
123	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₅₉	—	2	2	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	—	4	2	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	C₄×D₅₉	—	2	1	no	—
135	C₄×D₅₉	—	2	1	no	—
136	C₄×D₅₉	—	4	2	no	—
137	C₄×D₅₉	—	4	2	no	—
138	Dic₁₁₈	—	2	1	no	—
139	Dic₁₁₈	—	2	1	no	—
140	Dic₁₁₈	—	2	1	no	—
141	Dic₁₁₈	—	4	2	no	—
142	Dic₁₁₈	—	4	2	no	—
143	Dic₁₁₈	#41	1	2	yes	—
144	C₈×C₅₉	—	2	1	no	—
145	C₈×C₅₉	—	2	1	no	—
146	C₅₉⋊C₈	—	2	1	no	—
147	C₅₉⋊C₈	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₅₉	—	2	1	no	—
153	C₂×D₅₉	—	2	1	no	—
154	C₂×D₅₉	#137	1	2	yes	—
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	—	4	1	no	—
160	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
161	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
162	Dic₅₉	—	2	2	no	—
163	Dic₅₉	—	2	1	no	—
164	Dic₅₉	—	2	1	no	—
165	Dic₅₉	—	2	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	—	4	1	no	—
168	Dic₅₉	—	4	1	no	—
169	#3312	—	59	1	no	maximal, Sylow
170	C₂×C₅₉	—	4	1	no	—
171	C₂×C₅₉	#546	1	1	yes	characteristic
172	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
173	C₂×C₅₉	#578	1	1	yes	characteristic
174	D₅₉	—	2	2	no	—
175	#552	—	59	1	no	—
176	D₄×Q₈	—	59	1	no	—
177	#491	—	59	1	no	—
178	#485	—	59	1	no	—
179	#483	—	59	1	no	—
180	#479	—	59	1	no	—
181	#470	D₅₉	1	1	yes	characteristic, complemented
182	#467	—	59	1	no	—
183	#462	—	59	1	no	—
184	#451	—	59	1	no	—
185	#451	—	59	1	no	—
186	#441	—	59	1	no	—
187	#435	—	59	1	no	—
188	#419	—	59	1	no	—
189	C₂×#103	—	59	1	no	—
190	C₅₉	#3312	1	1	yes	characteristic, complemented, Sylow
191	C₂²×Q₈	—	59	1	no	—
192	C₂²×Q₈	—	59	1	no	—
193	C₂×Q₁₆	—	59	1	no	—
194	C₂×Q₁₆	—	59	1	no	—
195	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
196	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
197	C₂²×C₈	—	59	1	no	—
198	C₄⋊Q₈	—	59	1	no	—
199	C₄⋊Q₈	—	118	1	no	—
200	#126	—	118	1	no	—
201	#125	—	59	1	no	—
202	#125	—	118	1	no	—
203	#123	—	118	1	no	—
204	C₂²⋊Q₈	—	59	1	no	—
205	C₂²⋊Q₈	—	59	1	no	—
206	C₂²⋊Q₈	—	118	1	no	—
207	C₂²⋊Q₈	—	118	1	no	—
208	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
209	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
210	C₄×Q₈	—	59	1	no	—
211	C₄×Q₈	—	59	1	no	—
212	C₄×D₄	—	59	1	no	—
213	C₄×D₄	—	118	1	no	—
214	#117	—	59	1	no	—
215	#117	—	118	1	no	—
216	C₂×C₄⋊C₄	—	59	1	no	—
217	#107	—	59	1	no	—
218	#106	—	59	1	no	—
219	#106	C₂×D₅₉	1	1	yes	characteristic
220	C₄⋊C₈	—	59	1	no	—
221	#103	—	59	1	no	—
222	#103	—	59	1	no	—
223	#103	—	59	1	no	—
224	#103	—	59	1	no	—
225	#103	—	59	1	no	—
226	#103	—	59	1	no	—
227	#103	—	59	1	no	—
228	#103	—	59	1	no	—
229	#103	C₂×D₅₉	1	1	yes	characteristic
230	#103	C₂×D₅₉	1	1	yes	characteristic
231	C₂²⋊C₈	—	59	1	no	—
232	C₂²⋊C₈	—	59	1	no	—
233	#97	—	59	1	no	—
234	C₂×Q₈	—	59	1	no	—
235	C₂×Q₈	—	59	1	no	—
236	C₂×Q₈	—	118	1	no	—
237	C₂×Q₈	—	118	2	no	—
238	C₂×Q₈	—	118	1	no	—
239	C₂×Q₈	—	118	2	no	—
240	C₂×Q₈	—	236	1	no	—
241	C₂×Q₈	—	236	1	no	—
242	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
243	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
244	C₂×D₄	—	59	1	no	—
245	C₂²×C₄	—	59	1	no	—
246	C₂²×C₄	—	118	1	no	—
247	C₂²×C₄	—	118	1	no	—
248	C₂²×C₄	—	118	1	no	—
249	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
250	Q₁₆	—	4	1	no	—
251	Q₁₆	—	118	2	no	—
252	Q₁₆	—	236	1	no	—
253	M₁₆	—	4	1	no	—
254	C₂×C₈	—	59	1	no	—
255	C₂×C₈	—	59	1	no	—
256	C₂×C₈	—	236	1	no	—
257	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	59	1	no	—
262	C₄⋊C₄	—	59	1	no	—
263	C₄⋊C₄	—	59	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	—	118	1	no	—
272	C₄⋊C₄	—	118	1	no	—
273	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	59	1	no	—
278	C₂²⋊C₄	—	59	1	no	—
279	C₂²⋊C₄	—	118	1	no	—
280	C₂²⋊C₄	—	118	1	no	—
281	C₂²⋊C₄	—	118	1	no	—
282	C₄²	—	59	1	no	—
283	C₄²	—	118	1	no	—
284	C₄²	—	118	1	no	—
285	C₄²	—	118	1	no	—
286	C₂³	—	59	1	no	—
287	C₂³	D₄×D₅₉	1	1	yes	characteristic
288	Q₈	—	2	1	no	—
289	Q₈	—	2	1	no	—
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	59	2	no	—
293	Q₈	—	118	1	no	—
294	Q₈	—	118	1	no	—
295	Q₈	—	118	1	no	—
296	Q₈	—	236	2	no	—
297	Q₈	—	236	2	no	—
298	D₄	—	118	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	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	—	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	—	236	2	no	—
315	C₂×C₄	—	236	2	no	—
316	C₂×C₄	—	236	2	no	—
317	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
318	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	118	1	no	—
322	C₈	—	118	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	2	1	no	—
326	C₂²	—	59	2	no	—
327	C₂²	—	118	1	no	—
328	C₂²	—	118	1	no	—
329	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	118	1	no	—
336	C₄	—	118	1	no	—
337	C₄	—	118	1	no	—
338	C₄	—	118	2	no	—
339	C₄	—	236	1	no	—
340	C₄	—	236	1	no	—
341	C₄	—	236	1	no	—
342	C₄	#41×D₅₉	1	1	yes	characteristic
343	C₄	#14273772	1	1	yes	characteristic
344	C₂	—	4	1	no	—
345	C₂	—	118	2	no	—
346	C₂	#16838041	1	1	yes	characteristic
347	C₂	#137×D₅₉	1	1	yes	characteristic
348	C₂	#16838218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#470	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#470	D₅₉
hypocenter	C₅₉	#3312
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #470

Complements

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

N	H ≅ G/N	type
3. #16837769	C₂	semidirect
4. #16837740	C₂	semidirect
5. #16837734	C₂	semidirect
6. #16837712	C₂	semidirect
7. #16837642	C₂	semidirect
8. #16837610	C₂	semidirect
9. #16837557	C₂	semidirect
10. #16837556	C₂	semidirect
11. #16837506	C₂	semidirect
13. #16837412	C₂	semidirect
14. #103×D₅₉	C₂	semidirect
15. C₅₉×#470	C₂	semidirect
181. #470	D₅₉	semidirect
190. C₅₉	#3312	semidirect