Subgroups — #21436809

Order 7552.

372 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838121	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837761	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837760	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837751	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837698	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837680	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837615	C₂	1	1	yes	maximal, characteristic
10	#16837605	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837531	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837503	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837502	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₅₉	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#463	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₁₁₈	—	2	1	no	—
17	C₂×#11745670	C₂²	1	1	yes	characteristic
18	#14273727	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273725	C₂²	1	1	yes	characteristic
22	#14273720	—	2	1	no	—
23	#14273720	—	2	1	no	—
24	#14273714	C₂²	1	1	yes	characteristic
25	#14273706	—	2	1	no	—
26	#14273704	—	2	1	no	—
27	#14273704	—	2	1	no	—
28	#14273704	—	2	1	no	—
29	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
32	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
34	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
35	#14273683	—	2	1	no	—
36	#14273661	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	Dic₅₉⋊C₈	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	#14273651	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273650	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273648	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×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
54	#124×C₅₉	C₂²	1	1	yes	characteristic
55	C₄×D₄×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×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	—	2	1	no	—
63	C₂²×Dic₅₉	—	2	1	no	—
64	C₂²×Dic₅₉	—	2	1	no	—
65	C₂²×Dic₅₉	—	2	1	no	—
66	C₂×Dic₁₁₈	—	2	1	no	—
67	C₂×Dic₁₁₈	—	2	1	no	—
68	C₂×Dic₁₁₈	—	2	1	no	—
69	C₂×Dic₁₁₈	—	4	4	no	—
70	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
71	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
72	SD₁₆×C₅₉	—	2	2	no	—
73	D₈×C₅₉	—	2	2	no	—
74	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₅₉	—	2	1	no	—
78	C₂²⋊C₄×C₅₉	—	2	1	no	—
79	C₂²⋊C₄×C₅₉	—	2	1	no	—
80	C₄²×C₅₉	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	C₂²⋊Dic₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	#11745670	—	2	2	no	—
87	C₄⋊Dic₅₉	—	2	1	no	—
88	C₄⋊Dic₅₉	—	2	1	no	—
89	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
99	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
103	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₅₉	—	2	1	no	—
107	C₂³×C₅₉	—	2	1	no	—
108	Q₈×C₅₉	—	2	1	no	—
109	Q₈×C₅₉	—	2	1	no	—
110	D₄×C₅₉	—	2	1	no	—
111	D₄×C₅₉	—	2	1	no	—
112	D₄×C₅₉	—	2	1	no	—
113	D₄×C₅₉	#41	1	2	yes	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
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	—	4	2	no	—
125	C₂×Dic₅₉	—	4	2	no	—
126	C₂×Dic₅₉	—	4	2	no	—
127	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
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	Dic₁₁₈	—	2	2	no	—
132	Dic₁₁₈	—	2	1	no	—
133	Dic₁₁₈	—	2	1	no	—
134	Dic₁₁₈	—	4	2	no	—
135	Dic₁₁₈	—	4	2	no	—
136	Dic₁₁₈	—	4	2	no	—
137	C₈×C₅₉	—	2	1	no	—
138	C₈×C₅₉	—	2	1	no	—
139	C₅₉⋊C₈	—	2	1	no	—
140	C₅₉⋊C₈	—	2	1	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	—	4	1	no	—
147	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	4	1	no	—
151	C₄×C₅₉	—	4	1	no	—
152	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
153	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₅₉	—	2	1	no	—
155	Dic₅₉	—	2	1	no	—
156	Dic₅₉	—	2	1	no	—
157	Dic₅₉	—	4	1	no	—
158	Dic₅₉	—	4	1	no	—
159	Dic₅₉	—	4	1	no	—
160	Dic₅₉	—	4	1	no	—
161	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
162	#3367	—	59	1	no	maximal, Sylow
163	C₂×C₅₉	—	4	2	no	—
164	C₂×C₅₉	—	8	1	no	—
165	C₂×C₅₉	#548	1	1	yes	characteristic
166	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₅₉	#578	1	1	yes	characteristic
168	#556	—	59	1	no	—
169	D₄×Q₈	—	59	1	no	—
170	#489	—	59	1	no	—
171	#489	—	59	1	no	—
172	#487	—	59	1	no	—
173	#477	—	59	1	no	—
174	#477	—	59	1	no	—
175	#474	—	59	1	no	—
176	#463	—	59	1	no	—
177	#463	D₅₉	1	1	yes	characteristic
178	#461	—	59	1	no	—
179	#446	—	59	1	no	—
180	#440	—	59	1	no	—
181	#440	—	59	1	no	—
182	C₄×D₈	—	59	1	no	—
183	C₅₉	#3367	1	1	yes	characteristic, complemented, Sylow
184	C₂²×Q₈	—	118	1	no	—
185	C₂×SD₁₆	—	59	1	no	—
186	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
187	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
188	C₄⋊Q₈	—	59	1	no	—
189	C₄⋊Q₈	—	59	1	no	—
190	C₄⋊Q₈	—	59	1	no	—
191	C₄⋊Q₈	—	59	1	no	—
192	#126	—	118	1	no	—
193	#126	—	118	1	no	—
194	#124	—	59	1	no	—
195	#124	C₂×D₅₉	1	1	yes	characteristic
196	C₂²⋊Q₈	—	118	1	no	—
197	C₂²⋊Q₈	—	118	1	no	—
198	C₂²⋊Q₈	—	118	1	no	—
199	C₂²⋊Q₈	—	118	1	no	—
200	C₄×Q₈	—	59	1	no	—
201	C₄×Q₈	—	59	1	no	—
202	C₄×Q₈	—	59	1	no	—
203	C₄×D₄	—	59	1	no	—
204	C₄×D₄	—	59	1	no	—
205	C₄×D₄	—	59	1	no	—
206	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
207	#117	—	118	1	no	—
208	#107	—	59	1	no	—
209	#107	—	59	1	no	—
210	#107	—	59	1	no	—
211	C₄⋊C₈	—	59	1	no	—
212	C₄⋊C₈	—	59	1	no	—
213	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
214	#103	—	59	1	no	—
215	#103	—	59	1	no	—
216	#103	—	59	1	no	—
217	#103	—	59	1	no	—
218	#103	—	59	1	no	—
219	#103	C₂×D₅₉	1	1	yes	characteristic
220	#102	—	59	1	no	—
221	#102	—	59	1	no	—
222	#102	—	59	1	no	—
223	#102	C₂×D₅₉	1	1	yes	characteristic
224	#97	—	59	1	no	—
225	#97	—	59	1	no	—
226	C₄×C₈	—	59	1	no	—
227	C₂×Q₈	—	59	1	no	—
228	C₂×Q₈	—	59	1	no	—
229	C₂×Q₈	—	118	1	no	—
230	C₂×Q₈	—	118	1	no	—
231	C₂×Q₈	—	118	1	no	—
232	C₂×Q₈	—	236	4	no	—
233	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
235	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
236	C₂²×C₄	—	2	1	no	—
237	C₂²×C₄	—	118	1	no	—
238	C₂²×C₄	—	118	1	no	—
239	C₂²×C₄	—	118	1	no	—
240	SD₁₆	—	2	2	no	—
241	SD₁₆	—	118	2	no	—
242	D₈	—	2	2	no	—
243	C₂×C₈	—	59	1	no	—
244	C₂×C₈	—	59	1	no	—
245	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
247	C₄⋊C₄	—	59	1	no	—
248	C₄⋊C₄	—	59	1	no	—
249	C₄⋊C₄	—	59	1	no	—
250	C₄⋊C₄	—	59	1	no	—
251	C₄⋊C₄	—	59	1	no	—
252	C₄⋊C₄	—	59	1	no	—
253	C₄⋊C₄	—	118	1	no	—
254	C₄⋊C₄	—	118	1	no	—
255	C₄⋊C₄	—	118	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	C₂²×D₅₉	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	118	1	no	—
265	C₂²⋊C₄	—	118	1	no	—
266	C₂²⋊C₄	—	118	1	no	—
267	C₂²⋊C₄	—	118	1	no	—
268	C₂²⋊C₄	—	118	1	no	—
269	C₄²	—	59	1	no	—
270	C₄²	—	59	1	no	—
271	C₄²	—	118	1	no	—
272	C₄²	—	118	1	no	—
273	C₄²	C₂²×D₅₉	1	1	yes	characteristic
274	C₂³	—	2	1	no	—
275	C₂³	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	118	2	no	—
279	Q₈	—	118	1	no	—
280	Q₈	—	118	1	no	—
281	Q₈	—	236	2	no	—
282	Q₈	—	236	2	no	—
283	Q₈	—	236	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	2	1	no	—
287	D₄	#11745687	1	2	yes	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	59	1	no	—
292	C₂×C₄	—	59	1	no	—
293	C₂×C₄	—	59	1	no	—
294	C₂×C₄	—	59	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	—	236	2	no	—
300	C₂×C₄	—	236	2	no	—
301	C₂×C₄	—	236	2	no	—
302	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
303	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
304	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	118	1	no	—
308	C₈	—	118	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	4	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	59	2	no	—
321	C₄	—	118	1	no	—
322	C₄	—	118	1	no	—
323	C₄	—	118	1	no	—
324	C₄	—	236	1	no	—
325	C₄	—	236	1	no	—
326	C₄	—	236	1	no	—
327	C₄	—	236	1	no	—
328	C₄	C₂×#11745687	1	1	yes	characteristic
329	C₄	#14273774	1	1	yes	characteristic
330	C₂	—	4	2	no	—
331	C₂	—	8	1	no	—
332	C₂	#16838058	1	1	yes	characteristic
333	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
334	C₂	#16838216	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#463	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#463	D₅₉
hypocenter	C₅₉	#3367
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #463

Complements

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

N	H ≅ G/N	type
1. #16838121	C₂	semidirect
2. D₄×Dic₁₁₈	C₂	semidirect
3. #16837761	C₂	semidirect
4. #16837760	C₂	semidirect
5. #16837751	C₂	semidirect
6. #16837698	C₂	semidirect
7. #16837695	C₂	semidirect
8. #16837680	C₂	semidirect
10. #16837605	C₂	semidirect
11. #16837531	C₂	semidirect
12. #16837503	C₂	semidirect
13. #16837502	C₂	semidirect
183. C₅₉	#3367	semidirect