Subgroups — #21436795

Order 7552.

348 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838135	C₂	1	1	yes	maximal, characteristic
2	#16838129	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837759	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837755	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837747	C₂	1	1	yes	maximal, characteristic
6	#16837705	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837686	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837625	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837623	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837601	C₂	1	1	yes	maximal, characteristic
11	#16837526	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837496	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837489	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#11745670	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#478	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×#11745670	C₂²	1	1	yes	characteristic
19	#14273726	—	2	1	no	—
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273723	C₂²	1	1	yes	characteristic
22	#14273718	C₂²	1	1	yes	characteristic
23	#14273717	—	2	1	no	—
24	#14273717	—	2	1	no	—
25	#14273715	C₂²	1	1	yes	characteristic
26	#14273714	C₂²	1	1	yes	characteristic
27	#14273713	—	2	1	no	—
28	#14273710	—	2	1	no	—
29	#14273709	—	2	1	no	—
30	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
32	#14273692	—	2	1	no	—
33	#14273692	C₂²	1	1	yes	characteristic
34	#14273692	C₂²	1	1	yes	characteristic
35	#14273692	C₂²	1	1	yes	characteristic
36	#14273689	C₂²	1	1	yes	characteristic
37	C₄×C₂²⋊D₅₉	—	2	1	no	—
38	#14273658	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	#14273650	C₂²	1	1	yes	characteristic
43	#14273650	C₂²	1	1	yes	characteristic
44	#14273650	C₂²	1	1	yes	characteristic
45	#14273649	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273647	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	C₄×C₅₉⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	#125×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂×C₂²⋊D₅₉	—	2	1	no	—
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
67	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
68	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₅₉	—	2	1	no	—
72	C₄⋊C₄×C₅₉	—	2	1	no	—
73	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₅₉	—	2	1	no	—
79	C₂²⋊Dic₅₉	—	2	1	no	—
80	#11745670	—	2	2	no	—
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
86	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
87	Dic₅₉⋊C₄	—	2	1	no	—
88	Dic₅₉⋊C₄	—	2	1	no	—
89	Dic₅₉⋊C₄	—	2	1	no	—
90	Dic₅₉⋊C₄	—	2	1	no	—
91	Dic₅₉⋊C₄	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
94	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
95	C₄×Dic₅₉	—	2	1	no	—
96	C₄×Dic₅₉	—	2	1	no	—
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
100	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	Dic₂₃₆	—	2	2	no	—
104	#11745660	—	2	2	no	—
105	C₂³×C₅₉	—	2	1	no	—
106	C₂²×D₅₉	—	2	1	no	—
107	D₄×C₅₉	—	2	1	no	—
108	D₄×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	—	8	1	no	—
113	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₅₉	—	4	2	no	—
117	C₂²⋊D₅₉	—	8	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
125	D₂₃₆	—	2	1	no	—
126	D₂₃₆	—	2	1	no	—
127	C₄×D₅₉	—	8	1	no	—
128	Dic₁₁₈	—	2	1	no	—
129	Dic₁₁₈	—	2	2	no	—
130	Dic₁₁₈	—	2	1	no	—
131	Dic₁₁₈	—	2	1	no	—
132	Dic₁₁₈	—	4	2	no	—
133	Dic₁₁₈	#41	1	2	yes	—
134	C₈×C₅₉	—	2	1	no	—
135	C₈×C₅₉	—	2	1	no	—
136	C₅₉⋊C₈	—	2	1	no	—
137	C₅₉⋊C₈	—	2	1	no	—
138	C₂²×C₅₉	—	4	1	no	—
139	C₂²×C₅₉	—	4	1	no	—
140	C₂²×C₅₉	—	4	1	no	—
141	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
142	C₂×D₅₉	—	4	1	no	—
143	C₂×D₅₉	—	4	1	no	—
144	C₂×D₅₉	—	4	1	no	—
145	C₄×C₅₉	—	2	1	no	—
146	C₄×C₅₉	—	2	1	no	—
147	C₄×C₅₉	—	4	1	no	—
148	C₄×C₅₉	—	4	1	no	—
149	C₄×C₅₉	—	4	1	no	—
150	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
151	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
152	Dic₅₉	—	2	2	no	—
153	Dic₅₉	—	2	1	no	—
154	Dic₅₉	—	2	1	no	—
155	Dic₅₉	—	2	1	no	—
156	Dic₅₉	—	4	1	no	—
157	Dic₅₉	—	4	1	no	—
158	Dic₅₉	#135	1	2	yes	—
159	#3365	—	59	1	no	maximal, Sylow
160	C₂×C₅₉	—	8	1	no	—
161	C₂×C₅₉	#548	1	1	yes	characteristic
162	C₂×C₅₉	C₂×#135	1	1	yes	characteristic
163	C₂×C₅₉	#577	1	1	yes	characteristic
164	D₅₉	—	8	1	no	—
165	#559	—	59	1	no	—
166	#557	—	59	1	no	—
167	#488	—	59	1	no	—
168	#488	—	59	1	no	—
169	#486	—	59	1	no	—
170	#478	—	59	1	no	—
171	#478	D₅₉	1	1	yes	characteristic, complemented
172	#475	—	59	1	no	—
173	#464	—	59	1	no	—
174	#464	—	59	1	no	—
175	#460	—	59	1	no	—
176	C₈×Q₈	—	59	1	no	—
177	C₄×Q₁₆	—	59	1	no	—
178	C₄×SD₁₆	—	59	1	no	—
179	C₄×SD₁₆	—	59	1	no	—
180	C₅₉	#3365	1	1	yes	characteristic, complemented, Sylow
181	C₂×Q₁₆	—	59	1	no	—
182	C₂×SD₁₆	—	59	1	no	—
183	C₂×SD₁₆	—	59	1	no	—
184	C₄⋊Q₈	—	59	1	no	—
185	C₄⋊Q₈	—	118	1	no	—
186	#127	—	59	1	no	—
187	#125	—	59	1	no	—
188	#125	—	118	1	no	—
189	#125	—	118	1	no	—
190	#125	C₂×D₅₉	1	1	yes	characteristic
191	#124	—	59	1	no	—
192	#124	—	59	1	no	—
193	#124	—	118	1	no	—
194	#123	—	118	1	no	—
195	#123	—	118	1	no	—
196	C₄×Q₈	—	59	1	no	—
197	C₄×Q₈	—	59	1	no	—
198	C₄×Q₈	—	59	1	no	—
199	C₄×Q₈	—	59	1	no	—
200	C₄×Q₈	—	59	1	no	—
201	C₄×Q₈	—	118	1	no	—
202	C₄×D₄	—	59	1	no	—
203	C₄×D₄	—	118	1	no	—
204	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
205	#107	C₂×D₅₉	1	1	yes	characteristic
206	#106	—	59	1	no	—
207	#106	C₂×D₅₉	1	1	yes	characteristic
208	C₄⋊C₈	—	59	1	no	—
209	C₄⋊C₈	—	59	1	no	—
210	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
211	#103	—	59	1	no	—
212	#103	—	59	1	no	—
213	#103	—	59	1	no	—
214	#103	—	59	1	no	—
215	#102	—	59	1	no	—
216	#102	—	59	1	no	—
217	#102	—	59	1	no	—
218	#102	—	59	1	no	—
219	#102	C₂×D₅₉	1	1	yes	characteristic
220	#102	C₂×D₅₉	1	1	yes	characteristic
221	C₄×C₈	—	59	1	no	—
222	C₄×C₈	—	59	1	no	—
223	C₄×C₈	—	59	1	no	—
224	C₂×Q₈	—	59	1	no	—
225	C₂×Q₈	—	59	1	no	—
226	C₂×Q₈	—	59	1	no	—
227	C₂×Q₈	—	118	1	no	—
228	C₂×D₄	—	59	1	no	—
229	C₂×D₄	—	118	1	no	—
230	C₂×D₄	—	118	1	no	—
231	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	118	1	no	—
234	Q₁₆	—	118	2	no	—
235	SD₁₆	—	118	2	no	—
236	SD₁₆	—	118	2	no	—
237	C₂×C₈	—	59	1	no	—
238	C₂×C₈	—	59	1	no	—
239	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	2	1	no	—
243	C₄⋊C₄	—	59	1	no	—
244	C₄⋊C₄	—	59	1	no	—
245	C₄⋊C₄	—	59	1	no	—
246	C₄⋊C₄	—	59	1	no	—
247	C₄⋊C₄	—	118	1	no	—
248	C₄⋊C₄	—	118	1	no	—
249	C₄⋊C₄	—	118	1	no	—
250	C₄⋊C₄	—	118	1	no	—
251	C₄⋊C₄	—	118	1	no	—
252	C₄⋊C₄	—	118	1	no	—
253	C₄⋊C₄	—	118	1	no	—
254	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
257	C₂²⋊C₄	—	2	1	no	—
258	C₂²⋊C₄	—	118	1	no	—
259	C₂²⋊C₄	—	118	1	no	—
260	C₂²⋊C₄	—	118	1	no	—
261	C₂²⋊C₄	—	118	1	no	—
262	C₂²⋊C₄	—	118	1	no	—
263	C₄²	—	59	1	no	—
264	C₄²	—	59	1	no	—
265	C₄²	—	118	1	no	—
266	C₄²	—	118	1	no	—
267	C₄²	—	118	1	no	—
268	C₄²	—	118	1	no	—
269	C₄²	C₂²×D₅₉	1	1	yes	characteristic
270	C₂³	—	2	1	no	—
271	C₂³	—	118	1	no	—
272	Q₈	—	59	2	no	—
273	Q₈	—	118	2	no	—
274	Q₈	—	118	1	no	—
275	Q₈	—	118	1	no	—
276	Q₈	—	118	1	no	—
277	Q₈	—	236	2	no	—
278	D₄	—	2	1	no	—
279	D₄	—	2	1	no	—
280	D₄	—	118	1	no	—
281	D₄	—	118	1	no	—
282	D₄	—	472	1	no	—
283	D₄	—	236	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	8	1	no	—
288	C₂×C₄	—	59	1	no	—
289	C₂×C₄	—	59	1	no	—
290	C₂×C₄	—	59	1	no	—
291	C₂×C₄	—	59	1	no	—
292	C₂×C₄	—	118	1	no	—
293	C₂×C₄	—	118	1	no	—
294	C₂×C₄	—	118	1	no	—
295	C₂×C₄	—	472	1	no	—
296	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
297	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	2	1	no	—
301	C₈	—	118	1	no	—
302	C₈	—	118	1	no	—
303	C₂²	—	4	1	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	236	1	no	—
307	C₂²	—	236	1	no	—
308	C₂²	—	236	1	no	—
309	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	59	2	no	—
316	C₄	—	118	1	no	—
317	C₄	—	118	1	no	—
318	C₄	—	118	1	no	—
319	C₄	—	118	2	no	—
320	C₄	—	236	1	no	—
321	C₄	—	236	1	no	—
322	C₄	#41×D₅₉	1	1	yes	characteristic
323	C₄	#14273775	1	1	yes	characteristic
324	C₂	—	8	1	no	—
325	C₂	—	472	1	no	—
326	C₂	#16838062	1	1	yes	characteristic
327	C₂	#135×D₅₉	1	1	yes	characteristic
328	C₂	#16838212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#478	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#478	D₅₉
hypocenter	C₅₉	#3365
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
2. #16838129	C₂	semidirect
3. #16837759	C₂	semidirect
4. #16837755	C₂	semidirect
6. #16837705	C₂	semidirect
7. #16837686	C₂	semidirect
8. #16837625	C₂	semidirect
9. #16837623	C₂	semidirect
11. #16837526	C₂	semidirect
12. #16837496	C₂	semidirect
13. #16837489	C₂	semidirect
14. C₄×#11745670	C₂	semidirect
15. C₅₉×#478	C₂	semidirect
171. #478	D₅₉	semidirect
180. C₅₉	#3365	semidirect