Subgroups — #21437049

Order 7552.

348 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838113	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838111	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837810	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837808	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837803	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837705	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837704	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837703	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837700	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837681	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837531	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837512	C₂	1	1	yes	maximal, characteristic
13	#16837511	C₂	1	1	yes	maximal, characteristic
14	#16837482	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#474	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745669	C₂²	1	1	yes	characteristic
17	#14273726	C₂²	1	1	yes	characteristic
18	#14273719	C₂²	1	1	yes	characteristic
19	#14273718	C₂²	1	1	yes	characteristic
20	#14273718	C₂²	1	1	yes	characteristic
21	#14273718	C₂²	1	1	yes	characteristic
22	#14273707	—	2	1	no	—
23	#14273706	—	2	1	no	—
24	#14273704	—	2	1	no	—
25	#14273704	—	2	1	no	—
26	#14273703	—	2	1	no	—
27	#14273703	—	2	1	no	—
28	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
29	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
30	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
31	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
32	#14273689	C₂²	1	1	yes	characteristic
33	#14273689	C₂²	1	1	yes	characteristic
34	C₂×Dic₅₉⋊C₄	—	2	1	no	—
35	C₄⋊C₄×D₅₉	—	2	1	no	—
36	#14273661	C₂²	1	1	yes	characteristic
37	#14273660	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	#14273658	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	#14273649	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	#14273638	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	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#107×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×Q₈×C₅₉	—	2	1	no	—
60	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂²×Dic₅₉	—	2	1	no	—
63	C₂²×Dic₅₉	—	2	1	no	—
64	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×C₄×D₅₉	—	2	1	no	—
67	C₂×C₄×D₅₉	—	2	1	no	—
68	C₂×Dic₁₁₈	—	2	1	no	—
69	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
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	#11745669	Q₈	1	4	yes	complemented
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	—	2	1	no	—
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
91	Dic₅₉⋊C₄	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
98	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
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	C₂³×C₅₉	—	2	1	no	—
104	C₂²×D₅₉	—	2	1	no	—
105	Q₈×C₅₉	—	8	1	no	—
106	D₄×C₅₉	—	2	1	no	—
107	D₄×C₅₉	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	8	1	no	—
112	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₅₉	—	2	1	no	—
116	C₂×Dic₅₉	—	2	1	no	—
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	4	2	no	—
119	C₂×Dic₅₉	—	8	1	no	—
120	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
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	D₂₃₆	—	2	1	no	—
125	D₂₃₆	C₂×Q₈	1	2	yes	—
126	C₄×D₅₉	—	4	2	no	—
127	C₄×D₅₉	—	8	1	no	—
128	C₄×D₅₉	—	8	1	no	—
129	Dic₁₁₈	—	4	1	no	—
130	Dic₁₁₈	—	8	1	no	—
131	C₈×C₅₉	—	2	1	no	—
132	C₈×C₅₉	—	2	1	no	—
133	C₅₉⋊C₈	—	2	1	no	—
134	C₅₉⋊C₈	C₂×Q₈	1	2	yes	—
135	C₂²×C₅₉	—	2	2	no	—
136	C₂²×C₅₉	—	4	1	no	—
137	C₂²×C₅₉	—	4	1	no	—
138	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₅₉	—	2	2	no	—
140	C₂×D₅₉	—	4	1	no	—
141	C₂×D₅₉	—	4	1	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	—	4	1	no	—
146	C₄×C₅₉	—	4	1	no	—
147	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
148	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	2	1	no	—
151	Dic₅₉	—	2	1	no	—
152	Dic₅₉	—	2	1	no	—
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	—	4	1	no	—
155	Dic₅₉	—	4	1	no	—
156	#3389	—	59	1	no	maximal, Sylow
157	C₂×C₅₉	—	4	2	no	—
158	C₂×C₅₉	D₄×Q₈	1	1	yes	characteristic
159	C₂×C₅₉	#575	1	1	yes	characteristic
160	C₂×C₅₉	#576	1	1	yes	characteristic
161	D₅₉	—	4	2	no	—
162	#554	—	59	1	no	—
163	#554	—	59	1	no	—
164	#501	—	59	1	no	—
165	#501	—	59	1	no	—
166	#499	—	59	1	no	—
167	#478	—	59	1	no	—
168	#478	—	59	1	no	—
169	#478	—	59	1	no	—
170	#478	—	59	1	no	—
171	#474	—	59	1	no	—
172	#474	D₅₉	1	1	yes	characteristic, complemented
173	#446	—	59	1	no	—
174	#442	—	59	1	no	—
175	#442	—	59	1	no	—
176	C₄×D₈	—	59	1	no	—
177	C₅₉	#3389	1	1	yes	characteristic, complemented, Sylow
178	C₂×D₈	—	59	1	no	—
179	C₄⋊Q₈	—	59	1	no	—
180	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
181	#125	—	59	1	no	—
182	#125	—	59	1	no	—
183	#125	—	59	1	no	—
184	#125	—	59	1	no	—
185	C₂²⋊Q₈	—	118	1	no	—
186	C₂²⋊Q₈	—	118	1	no	—
187	C₂²⋊Q₈	—	118	1	no	—
188	C₂²⋊Q₈	—	118	1	no	—
189	C₂²⋊Q₈	—	118	1	no	—
190	C₂²⋊Q₈	—	118	1	no	—
191	C₄×Q₈	—	59	1	no	—
192	C₄×D₄	—	59	1	no	—
193	C₄×D₄	—	59	1	no	—
194	C₄×D₄	—	59	1	no	—
195	C₄×D₄	—	59	1	no	—
196	C₄×D₄	—	59	1	no	—
197	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
198	C₂×C₄⋊C₄	—	118	1	no	—
199	C₂×C₄⋊C₄	—	118	1	no	—
200	#107	—	59	1	no	—
201	#107	—	59	1	no	—
202	#107	—	59	1	no	—
203	#107	C₂×D₅₉	1	1	yes	characteristic
204	#107	C₂×D₅₉	1	1	yes	characteristic
205	#106	—	59	1	no	—
206	#106	—	59	1	no	—
207	#106	—	59	1	no	—
208	#106	—	59	1	no	—
209	C₄⋊C₈	—	59	1	no	—
210	C₄⋊C₈	—	59	1	no	—
211	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
212	#102	—	59	1	no	—
213	#102	—	59	1	no	—
214	#102	—	59	1	no	—
215	#102	—	59	1	no	—
216	#102	C₂×D₅₉	1	1	yes	characteristic
217	#102	C₂×D₅₉	1	1	yes	characteristic
218	#97	—	59	1	no	—
219	#97	—	59	1	no	—
220	C₄×C₈	—	59	1	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	59	1	no	—
223	C₂×Q₈	—	118	1	no	—
224	C₂×D₄	—	59	1	no	—
225	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	118	1	no	—
228	C₂²×C₄	—	118	1	no	—
229	C₂²×C₄	—	118	1	no	—
230	C₂²×C₄	—	118	1	no	—
231	C₂²×C₄	—	118	1	no	—
232	D₈	—	59	4	no	—
233	C₂×C₈	—	59	1	no	—
234	C₂×C₈	—	59	1	no	—
235	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	59	1	no	—
239	C₄⋊C₄	—	59	1	no	—
240	C₄⋊C₄	—	59	1	no	—
241	C₄⋊C₄	—	59	1	no	—
242	C₄⋊C₄	—	59	1	no	—
243	C₄⋊C₄	—	59	1	no	—
244	C₄⋊C₄	—	118	1	no	—
245	C₄⋊C₄	—	118	1	no	—
246	C₄⋊C₄	—	118	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	C₂²×D₅₉	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	118	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	C₄²	—	59	1	no	—
263	C₄²	—	59	1	no	—
264	C₄²	C₂²×D₅₉	1	1	yes	characteristic
265	C₂³	—	2	1	no	—
266	C₂³	—	118	1	no	—
267	Q₈	—	8	1	no	—
268	Q₈	—	472	1	no	—
269	Q₈	—	236	1	no	—
270	D₄	—	2	1	no	—
271	D₄	—	59	2	no	—
272	D₄	—	118	1	no	—
273	D₄	Q₈×D₅₉	1	2	yes	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	8	1	no	—
278	C₂×C₄	—	59	1	no	—
279	C₂×C₄	—	59	1	no	—
280	C₂×C₄	—	59	1	no	—
281	C₂×C₄	—	59	1	no	—
282	C₂×C₄	—	118	1	no	—
283	C₂×C₄	—	118	1	no	—
284	C₂×C₄	—	118	1	no	—
285	C₂×C₄	—	472	1	no	—
286	C₂×C₄	—	472	1	no	—
287	C₂×C₄	—	472	1	no	—
288	C₂×C₄	—	236	2	no	—
289	C₂×C₄	—	236	2	no	—
290	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
291	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	—	59	2	no	—
296	C₈	—	118	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	118	2	no	—
301	C₂²	—	236	1	no	—
302	C₂²	—	236	1	no	—
303	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	118	1	no	—
310	C₄	—	118	1	no	—
311	C₄	—	118	1	no	—
312	C₄	—	118	1	no	—
313	C₄	—	236	1	no	—
314	C₄	—	236	1	no	—
315	C₄	—	236	1	no	—
316	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
317	C₄	#14273774	1	1	yes	characteristic
318	C₂	—	4	2	no	—
319	C₂	—	236	2	no	—
320	C₂	#16838065	1	1	yes	characteristic
321	C₂	#16838201	1	1	yes	characteristic
322	C₂	#16838203	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#474	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#474	D₅₉
hypocenter	C₅₉	#3389
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. #16838113	C₂	semidirect
2. #16838111	C₂	semidirect
3. #16837810	C₂	semidirect
4. #16837808	C₂	semidirect
5. #16837803	C₂	semidirect
6. #16837705	C₂	semidirect
7. #16837704	C₂	semidirect
8. #16837703	C₂	semidirect
9. #16837700	C₂	semidirect
10. #16837681	C₂	semidirect
11. #16837531	C₂	semidirect
15. C₅₉×#474	C₂	semidirect
80. #11745669	Q₈	semidirect
172. #474	D₅₉	semidirect
177. C₅₉	#3389	semidirect