Subgroups — #21436907

Order 7552.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 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	#16838132	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837763	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837747	C₂	1	1	yes	maximal, characteristic
5	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837680	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837679	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837627	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837621	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837601	C₂	1	1	yes	maximal, characteristic
11	#16837530	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837504	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×#489	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745672	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	#14273726	C₂²	1	1	yes	characteristic
22	#14273725	—	2	1	no	—
23	#14273725	—	2	1	no	—
24	#14273723	C₂²	1	1	yes	characteristic
25	#14273716	C₂²	1	1	yes	characteristic
26	#14273713	—	2	1	no	—
27	#14273711	—	2	1	no	—
28	#14273709	—	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	#14273692	—	2	1	no	—
33	#14273692	C₂²	1	1	yes	characteristic
34	#14273692	C₂²	1	1	yes	characteristic
35	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
36	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
37	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
38	#14273661	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273649	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	#14273647	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	#124×C₅₉	C₂²	1	1	yes	characteristic
54	#103×C₅₉	C₂²	1	1	yes	characteristic
55	#103×C₅₉	C₂²	1	1	yes	characteristic
56	#102×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	#97×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	—	2	1	no	—
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	—	2	1	no	—
68	C₂×Dic₁₁₈	—	2	1	no	—
69	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
70	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	—	2	1	no	—
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	C₂²⋊C₄×C₅₉	—	2	1	no	—
78	C₄²×C₅₉	C₂³	1	1	yes	characteristic
79	C₂²⋊Dic₅₉	—	2	1	no	—
80	#11745672	—	2	2	no	—
81	#11745670	—	2	2	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
87	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
94	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
95	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₅₉	—	2	1	no	—
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
101	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	#11745660	—	2	2	no	—
105	C₂³×C₅₉	—	2	1	no	—
106	C₂²×D₅₉	—	2	1	no	—
107	Q₈×C₅₉	—	2	1	no	—
108	Q₈×C₅₉	—	2	1	no	—
109	Q₈×C₅₉	—	8	1	no	—
110	D₄×C₅₉	—	2	1	no	—
111	D₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₅₉	—	4	2	no	—
119	C₂²⋊D₅₉	—	8	1	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	8	1	no	—
124	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
128	D₂₃₆	—	2	1	no	—
129	D₂₃₆	—	2	1	no	—
130	C₄×D₅₉	—	8	1	no	—
131	Dic₁₁₈	—	2	1	no	—
132	Dic₁₁₈	—	2	2	no	—
133	Dic₁₁₈	—	8	1	no	—
134	Dic₁₁₈	—	8	1	no	—
135	Dic₁₁₈	#41	1	2	yes	—
136	C₈×C₅₉	—	2	1	no	—
137	C₈×C₅₉	—	2	1	no	—
138	C₅₉⋊C₈	—	2	1	no	—
139	C₅₉⋊C₈	—	2	1	no	—
140	C₂²×C₅₉	—	4	1	no	—
141	C₂²×C₅₉	—	4	1	no	—
142	C₂²×C₅₉	—	4	1	no	—
143	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₅₉	—	4	1	no	—
145	C₂×D₅₉	—	4	1	no	—
146	C₂×D₅₉	—	4	1	no	—
147	C₄×C₅₉	—	2	1	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	4	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	2	no	—
155	Dic₅₉	—	2	1	no	—
156	Dic₅₉	—	2	1	no	—
157	Dic₅₉	—	2	1	no	—
158	Dic₅₉	—	4	1	no	—
159	Dic₅₉	—	4	1	no	—
160	Dic₅₉	#137	1	2	yes	—
161	#3375	—	59	1	no	maximal, Sylow
162	C₂×C₅₉	—	8	1	no	—
163	C₂×C₅₉	#548	1	1	yes	characteristic
164	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
165	C₂×C₅₉	#576	1	1	yes	characteristic
166	D₅₉	—	8	1	no	—
167	#559	—	59	1	no	—
168	Q₈²	—	59	1	no	—
169	#489	—	59	1	no	—
170	#489	D₅₉	1	1	yes	characteristic, complemented
171	#486	—	59	1	no	—
172	#477	—	59	1	no	—
173	#474	—	59	1	no	—
174	#474	—	59	1	no	—
175	#464	—	59	1	no	—
176	#464	—	59	1	no	—
177	#460	—	59	1	no	—
178	#446	—	59	1	no	—
179	#440	—	59	1	no	—
180	#440	—	59	1	no	—
181	C₄×Q₁₆	—	59	1	no	—
182	C₅₉	#3375	1	1	yes	characteristic, complemented, Sylow
183	C₂×Q₁₆	—	59	1	no	—
184	C₂×SD₁₆	—	59	1	no	—
185	C₂×SD₁₆	—	59	1	no	—
186	C₄⋊Q₈	—	59	1	no	—
187	C₄⋊Q₈	—	59	1	no	—
188	C₄⋊Q₈	—	118	1	no	—
189	C₄⋊Q₈	—	118	1	no	—
190	C₄⋊Q₈	—	118	1	no	—
191	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
192	#127	—	59	1	no	—
193	#124	—	59	1	no	—
194	#124	—	118	1	no	—
195	#124	C₂×D₅₉	1	1	yes	characteristic
196	#123	—	118	1	no	—
197	#123	—	118	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	—	59	1	no	—
203	C₄×Q₈	—	118	1	no	—
204	C₄×D₄	—	59	1	no	—
205	C₄×D₄	—	59	1	no	—
206	C₄×D₄	—	118	1	no	—
207	#107	—	59	1	no	—
208	#107	—	59	1	no	—
209	#107	—	59	1	no	—
210	C₄⋊C₈	—	59	1	no	—
211	C₄⋊C₈	—	59	1	no	—
212	C₄⋊C₈	—	59	1	no	—
213	#103	—	59	1	no	—
214	#103	—	59	1	no	—
215	#103	C₂×D₅₉	1	1	yes	characteristic
216	#103	C₂×D₅₉	1	1	yes	characteristic
217	#102	—	59	1	no	—
218	#102	—	59	1	no	—
219	#102	—	59	1	no	—
220	#102	—	59	1	no	—
221	#102	C₂×D₅₉	1	1	yes	characteristic
222	#102	C₂×D₅₉	1	1	yes	characteristic
223	#97	—	59	1	no	—
224	#97	C₂×D₅₉	1	1	yes	characteristic
225	C₄×C₈	—	59	1	no	—
226	C₂×Q₈	—	2	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	C₂²×D₅₉	1	1	yes	characteristic
232	C₂×D₄	—	59	1	no	—
233	C₂×D₄	—	118	1	no	—
234	C₂×D₄	—	118	1	no	—
235	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
236	C₂²×C₄	—	118	1	no	—
237	C₂²×C₄	—	118	1	no	—
238	Q₁₆	—	118	2	no	—
239	SD₁₆	—	118	2	no	—
240	SD₁₆	—	118	2	no	—
241	C₂×C₈	—	59	1	no	—
242	C₂×C₈	—	59	1	no	—
243	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
244	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	59	1	no	—
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	—	118	1	no	—
252	C₄⋊C₄	—	118	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	C₂²×D₅₉	1	1	yes	characteristic
258	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	118	1	no	—
262	C₂²⋊C₄	—	118	1	no	—
263	C₂²⋊C₄	—	118	1	no	—
264	C₂²⋊C₄	—	118	1	no	—
265	C₄²	—	59	1	no	—
266	C₄²	—	59	1	no	—
267	C₄²	—	118	1	no	—
268	C₄²	—	118	1	no	—
269	C₄²	—	118	1	no	—
270	C₄²	—	118	1	no	—
271	C₄²	C₂²×D₅₉	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	118	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	8	1	no	—
277	Q₈	—	59	2	no	—
278	Q₈	—	118	2	no	—
279	Q₈	—	118	1	no	—
280	Q₈	—	472	1	no	—
281	Q₈	—	472	1	no	—
282	D₄	—	2	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	118	1	no	—
285	D₄	—	118	1	no	—
286	D₄	—	472	1	no	—
287	D₄	—	236	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	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	—	472	1	no	—
299	C₂×C₄	—	472	1	no	—
300	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
301	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	118	1	no	—
306	C₈	—	118	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	236	1	no	—
311	C₂²	—	236	1	no	—
312	C₂²	—	236	1	no	—
313	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	59	2	no	—
320	C₄	—	118	1	no	—
321	C₄	—	118	1	no	—
322	C₄	—	118	1	no	—
323	C₄	—	118	2	no	—
324	C₄	—	236	1	no	—
325	C₄	—	236	1	no	—
326	C₄	#41×D₅₉	1	1	yes	characteristic
327	C₄	#14273774	1	1	yes	characteristic
328	C₂	—	8	1	no	—
329	C₂	—	472	1	no	—
330	C₂	#16838060	1	1	yes	characteristic
331	C₂	#137×D₅₉	1	1	yes	characteristic
332	C₂	#16838206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#489	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#489	D₅₉
hypocenter	C₅₉	#3375
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #489

Complements

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

N	H ≅ G/N	type
2. #16838132	C₂	semidirect
3. #16837763	C₂	semidirect
5. #16837695	C₂	semidirect
6. #16837680	C₂	semidirect
7. #16837679	C₂	semidirect
8. #16837627	C₂	semidirect
9. #16837621	C₂	semidirect
11. #16837530	C₂	semidirect
12. #16837504	C₂	semidirect
13. #16837499	C₂	semidirect
14. #16837494	C₂	semidirect
15. C₅₉×#489	C₂	semidirect
170. #489	D₅₉	semidirect
182. C₅₉	#3375	semidirect