Subgroups — #21434266

Order 7552.

386 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837931	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837736	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837719	C₂	1	1	yes	maximal, characteristic
5	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837709	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837621	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837556	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837539	C₂	1	1	yes	maximal, characteristic
10	#16837509	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837506	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837503	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837499	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837458	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#464	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745672	C₂²	1	1	yes	characteristic
18	C₂×#11745670	C₂²	1	1	yes	characteristic
19	#14273720	—	2	1	no	—
20	#14273719	C₂²	1	1	yes	characteristic
21	#14273709	—	2	1	no	—
22	#14273703	C₂²	1	1	yes	characteristic
23	#14273698	C₂²	1	1	yes	characteristic
24	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
26	C₄×Dic₁₁₈	—	2	1	no	—
27	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
28	#14273692	—	2	1	no	—
29	#14273692	C₂²	1	1	yes	characteristic
30	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
32	#14273683	—	2	1	no	—
33	C₄⋊C₄×D₅₉	—	2	1	no	—
34	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
35	C₄²×D₅₉	—	2	1	no	—
36	C₄²×D₅₉	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273659	C₂²	1	1	yes	characteristic
40	#14273658	C₂²	1	1	yes	characteristic
41	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
42	#14273652	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	#14273637	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	C₂²×Dic₅₉	—	2	1	no	—
64	Q₈×D₅₉	—	2	2	no	—
65	Q₈×D₅₉	—	2	2	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
69	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
70	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
71	C₂×Dic₁₁₈	—	2	1	no	—
72	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
73	Q₁₆×C₅₉	—	2	2	no	—
74	SD₁₆×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	—	2	1	no	—
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₅₉	—	2	1	no	—
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	—	2	1	no	—
82	C₄²×C₅₉	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	#11745672	—	2	2	no	—
86	#11745670	—	2	2	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	C₄⋊Dic₅₉	—	2	1	no	—
89	C₄⋊Dic₅₉	—	2	1	no	—
90	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
93	Dic₅₉⋊C₄	—	2	1	no	—
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	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	D₄	1	1	yes	characteristic
105	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
106	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂³×C₅₉	—	2	1	no	—
109	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₅₉	—	2	1	no	—
111	Q₈×C₅₉	—	2	1	no	—
112	Q₈×C₅₉	—	2	1	no	—
113	Q₈×C₅₉	#41	1	2	yes	—
114	D₄×C₅₉	—	2	1	no	—
115	D₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₅₉	—	4	2	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	—	4	2	no	—
128	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
131	C₄×D₅₉	—	2	4	no	—
132	C₄×D₅₉	—	2	2	no	—
133	C₄×D₅₉	—	2	1	no	—
134	C₄×D₅₉	—	2	1	no	—
135	C₄×D₅₉	—	4	1	no	—
136	C₄×D₅₉	—	4	2	no	—
137	Dic₁₁₈	—	2	4	no	—
138	Dic₁₁₈	—	2	1	no	—
139	Dic₁₁₈	#41	1	2	yes	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	—	2	1	no	—
144	C₂²×C₅₉	—	4	1	no	—
145	C₂²×C₅₉	—	4	1	no	—
146	C₂²×C₅₉	—	4	1	no	—
147	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₅₉	—	2	1	no	—
149	C₂×D₅₉	—	2	1	no	—
150	C₂×D₅₉	#137	1	2	yes	—
151	C₄×C₅₉	—	2	2	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	—	4	1	no	—
155	C₄×C₅₉	—	4	1	no	—
156	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
157	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
158	Dic₅₉	—	2	1	no	—
159	Dic₅₉	—	2	2	no	—
160	Dic₅₉	—	2	1	no	—
161	Dic₅₉	—	2	1	no	—
162	Dic₅₉	—	4	1	no	—
163	Dic₅₉	—	4	1	no	—
164	Dic₅₉	—	4	1	no	—
165	#3117	—	59	1	no	maximal, Sylow
166	C₂×C₅₉	—	8	1	no	—
167	C₂×C₅₉	#525	1	1	yes	characteristic
168	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
169	C₂×C₅₉	#575	1	1	yes	characteristic
170	D₅₉	—	2	2	no	—
171	#529	—	59	1	no	—
172	C₂×C₄×Q₈	—	59	1	no	—
173	#484	—	59	1	no	—
174	#481	—	59	1	no	—
175	#479	—	59	1	no	—
176	#479	—	59	1	no	—
177	#464	—	59	1	no	—
178	#464	D₅₉	1	1	yes	characteristic, complemented
179	#451	—	59	1	no	—
180	#448	—	59	1	no	—
181	#441	—	59	1	no	—
182	#441	—	59	1	no	—
183	#440	—	59	1	no	—
184	#440	—	59	1	no	—
185	#432	—	59	1	no	—
186	C₅₉	#3117	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₈	—	59	1	no	—
188	C₂×Q₁₆	—	59	1	no	—
189	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
190	C₂×SD₁₆	—	59	1	no	—
191	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
192	#126	—	118	1	no	—
193	#125	—	59	1	no	—
194	#124	C₂×D₅₉	1	1	yes	characteristic
195	#123	—	118	1	no	—
196	C₂²⋊Q₈	—	59	1	no	—
197	#121	—	59	1	no	—
198	C₄×Q₈	—	59	1	no	—
199	C₄×Q₈	—	59	1	no	—
200	C₄×Q₈	—	59	1	no	—
201	C₄×Q₈	—	59	1	no	—
202	C₄×Q₈	—	118	1	no	—
203	C₄×Q₈	—	118	1	no	—
204	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
205	C₄×D₄	—	59	1	no	—
206	C₄×D₄	—	118	1	no	—
207	#117	—	118	1	no	—
208	C₂×C₄⋊C₄	—	59	1	no	—
209	C₂×C₄⋊C₄	—	118	1	no	—
210	C₂×C₄²	—	59	1	no	—
211	C₂×C₄²	—	118	1	no	—
212	#107	—	59	1	no	—
213	#107	—	59	1	no	—
214	#106	—	59	1	no	—
215	#106	—	59	1	no	—
216	C₄⋊C₈	—	59	1	no	—
217	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
218	#103	—	59	1	no	—
219	#103	—	59	1	no	—
220	#103	—	59	1	no	—
221	#103	—	59	1	no	—
222	#103	—	59	1	no	—
223	#103	C₂×D₅₉	1	1	yes	characteristic
224	#102	—	59	1	no	—
225	#102	C₂×D₅₉	1	1	yes	characteristic
226	C₂²⋊C₈	—	59	1	no	—
227	C₂²⋊C₈	—	59	1	no	—
228	#97	—	59	1	no	—
229	#97	—	59	1	no	—
230	C₂×Q₈	—	59	1	no	—
231	C₂×Q₈	—	118	1	no	—
232	C₂×Q₈	—	118	2	no	—
233	C₂×Q₈	—	118	2	no	—
234	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₂×D₄	—	118	1	no	—
237	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
238	C₂²×C₄	—	59	1	no	—
239	C₂²×C₄	—	59	1	no	—
240	C₂²×C₄	—	59	1	no	—
241	C₂²×C₄	—	118	1	no	—
242	C₂²×C₄	—	118	1	no	—
243	C₂²×C₄	—	118	1	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	118	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	118	2	no	—
248	C₂×C₈	—	59	1	no	—
249	C₂×C₈	—	59	1	no	—
250	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	59	1	no	—
254	C₄⋊C₄	—	59	1	no	—
255	C₄⋊C₄	—	59	1	no	—
256	C₄⋊C₄	—	118	1	no	—
257	C₄⋊C₄	—	118	1	no	—
258	C₄⋊C₄	—	118	1	no	—
259	C₄⋊C₄	—	118	1	no	—
260	C₄⋊C₄	—	118	1	no	—
261	C₄⋊C₄	—	118	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	—	118	1	no	—
264	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	118	1	no	—
268	C₂²⋊C₄	—	118	1	no	—
269	C₂²⋊C₄	—	118	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	59	1	no	—
272	C₄²	—	59	1	no	—
273	C₄²	—	59	1	no	—
274	C₄²	—	118	1	no	—
275	C₄²	—	118	1	no	—
276	C₄²	—	118	1	no	—
277	C₄²	—	118	1	no	—
278	C₄²	C₂²×D₅₉	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	59	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	59	2	no	—
285	Q₈	—	118	4	no	—
286	Q₈	—	118	1	no	—
287	Q₈	#11745687	1	2	yes	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	236	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	59	1	no	—
295	C₂×C₄	—	59	1	no	—
296	C₂×C₄	—	59	1	no	—
297	C₂×C₄	—	118	2	no	—
298	C₂×C₄	—	118	4	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	—	236	1	no	—
306	C₂×C₄	—	236	2	no	—
307	C₂×C₄	—	236	2	no	—
308	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
309	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	118	1	no	—
314	C₈	—	118	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	59	2	no	—
319	C₂²	—	118	1	no	—
320	C₂²	—	118	1	no	—
321	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	118	1	no	—
328	C₄	—	118	2	no	—
329	C₄	—	118	1	no	—
330	C₄	—	118	1	no	—
331	C₄	—	236	1	no	—
332	C₄	—	236	1	no	—
333	C₄	—	236	1	no	—
334	C₄	#41×D₅₉	1	1	yes	characteristic
335	C₄	C₂×#11745687	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	118	2	no	—
338	C₂	#16837912	1	1	yes	characteristic
339	C₂	#137×D₅₉	1	1	yes	characteristic
340	C₂	#16838199	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	#3117
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. C₄×Q₈×D₅₉	C₂	semidirect
3. #16837736	C₂	semidirect
5. #16837710	C₂	semidirect
6. #16837709	C₂	semidirect
7. #16837621	C₂	semidirect
8. #16837556	C₂	semidirect
10. #16837509	C₂	semidirect
11. #16837506	C₂	semidirect
12. #16837503	C₂	semidirect
13. #16837499	C₂	semidirect
14. #16837458	C₂	semidirect
15. C₅₉×#464	C₂	semidirect
178. #464	D₅₉	semidirect
186. C₅₉	#3117	semidirect