Subgroups — #21433931

Order 7552.

362 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838029	C₂	1	1	yes	maximal, characteristic
2	#16837878	C₂	1	1	yes	maximal, characteristic
3	#16837712	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837709	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837707	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837694	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837686	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837441	C₂	1	1	yes	maximal, characteristic
11	#16837437	C₂	1	1	yes	maximal, characteristic
12	#16837427	C₂	1	1	yes	maximal, characteristic
13	#16837409	C₂	1	1	yes	maximal, characteristic
14	#103×D₅₉	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#475	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745659	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273725	—	2	1	no	—
21	#14273719	C₂²	1	1	yes	characteristic
22	#14273718	C₂²	1	1	yes	characteristic
23	#14273717	—	2	1	no	—
24	#14273705	—	2	1	no	—
25	#14273703	—	2	1	no	—
26	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
27	C₄×Dic₁₁₈	—	2	1	no	—
28	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
29	#14273692	—	2	1	no	—
30	#14273692	C₂²	1	1	yes	characteristic
31	#14273685	—	2	1	no	—
32	#14273685	C₂²	1	1	yes	characteristic
33	#14273685	C₂²	1	1	yes	characteristic
34	#14273684	—	2	1	no	—
35	#14273684	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	#14273659	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
46	#14273652	C₂²	1	1	yes	characteristic
47	#14273652	C₂²	1	1	yes	characteristic
48	#14273651	C₂²	1	1	yes	characteristic
49	#14273651	C₂²	1	1	yes	characteristic
50	#14273650	C₂²	1	1	yes	characteristic
51	#14273650	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	#106×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#103×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	—	2	1	no	—
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	Q₈×D₅₉	—	2	2	no	—
62	Q₈×D₅₉	Q₈	1	4	yes	complemented
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₁₈	—	2	1	no	—
67	C₂×Dic₁₁₈	—	2	1	no	—
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	C₄²×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	C₂³	1	1	yes	characteristic
78	(C₂×D₅₉)⋊C₄	—	2	1	no	—
79	(C₂×D₅₉)⋊C₄	—	2	1	no	—
80	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
81	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
82	C₄⋊Dic₅₉	—	2	1	no	—
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₅₉	C₂³	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	—	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	—	2	1	no	—
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	#11745659	—	2	2	no	—
105	C₈×D₅₉	—	2	2	no	—
106	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₅₉	—	2	1	no	—
108	Q₈×C₅₉	—	8	1	no	—
109	Q₈×C₅₉	C₂×Q₈	1	2	yes	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₅₉	—	2	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	C₄×D₅₉	—	2	4	no	—
125	C₄×D₅₉	—	8	1	no	—
126	C₄×D₅₉	C₂×Q₈	1	2	yes	—
127	C₄×D₅₉	#41	1	2	yes	—
128	Dic₁₁₈	—	2	1	no	—
129	Dic₁₁₈	—	2	4	no	—
130	Dic₁₁₈	—	8	1	no	—
131	Dic₁₁₈	C₂×Q₈	1	2	yes	—
132	C₈×C₅₉	—	2	1	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₅₉⋊C₈	—	2	1	no	—
135	C₅₉⋊C₈	—	2	1	no	—
136	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
137	C₂×D₅₉	—	2	1	no	—
138	C₂×D₅₉	—	2	1	no	—
139	C₂×D₅₉	C₂²⋊Q₈	1	2	yes	—
140	C₄×C₅₉	—	2	2	no	—
141	C₄×C₅₉	—	2	1	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	4	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	—	4	1	no	—
146	C₄×C₅₉	C₂²×Q₈	1	1	yes	characteristic
147	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
148	Dic₅₉	—	2	2	no	—
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	2	1	no	—
151	Dic₅₉	—	4	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	C₂²⋊Q₈	1	2	yes	—
155	#3082	—	59	1	no	maximal, Sylow
156	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
157	C₂×C₅₉	#577	1	1	yes	characteristic
158	C₂×C₅₉	#578	1	1	yes	characteristic
159	D₅₉	—	2	2	no	—
160	#544	—	59	1	no	—
161	#519	—	59	1	no	—
162	#479	—	59	1	no	—
163	#479	—	59	1	no	—
164	#479	—	59	1	no	—
165	#479	—	59	1	no	—
166	#477	—	59	1	no	—
167	#477	—	59	1	no	—
168	#475	—	59	1	no	—
169	#475	D₅₉	1	1	yes	characteristic, complemented
170	#428	—	59	1	no	—
171	#427	—	59	1	no	—
172	#424	—	59	1	no	—
173	#419	—	59	1	no	—
174	C₂×#103	—	59	1	no	—
175	C₅₉	#3082	1	1	yes	characteristic, complemented, Sylow
176	C₂²×Q₈	—	59	1	no	—
177	C₂×M₁₆	—	59	1	no	—
178	C₂²×C₈	—	59	1	no	—
179	C₄⋊Q₈	—	59	1	no	—
180	C₄⋊Q₈	—	118	1	no	—
181	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
182	#125	—	59	1	no	—
183	#125	—	59	1	no	—
184	#125	—	118	1	no	—
185	C₂²⋊Q₈	—	118	1	no	—
186	C₂²⋊Q₈	—	118	1	no	—
187	C₄×Q₈	—	59	1	no	—
188	C₄×Q₈	—	59	1	no	—
189	C₄×Q₈	—	59	1	no	—
190	C₄×Q₈	—	118	1	no	—
191	C₄×Q₈	—	118	1	no	—
192	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
193	#117	—	59	1	no	—
194	#117	—	59	1	no	—
195	#117	—	59	1	no	—
196	#117	—	118	1	no	—
197	#117	—	118	1	no	—
198	C₂×C₄⋊C₄	—	59	1	no	—
199	#107	—	59	1	no	—
200	#107	—	59	1	no	—
201	#107	—	59	1	no	—
202	#107	—	59	1	no	—
203	#106	—	59	1	no	—
204	#106	—	59	1	no	—
205	#106	C₂×D₅₉	1	1	yes	characteristic
206	#106	C₂×D₅₉	1	1	yes	characteristic
207	C₄⋊C₈	—	59	1	no	—
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	#103	—	59	1	no	—
216	#103	—	59	1	no	—
217	#103	C₂×D₅₉	1	1	yes	characteristic
218	#103	C₂×D₅₉	1	1	yes	characteristic
219	C₂×Q₈	—	2	1	no	—
220	C₂×Q₈	—	59	4	no	—
221	C₂×Q₈	—	59	1	no	—
222	C₂×Q₈	—	118	1	no	—
223	C₂×Q₈	—	118	1	no	—
224	C₂×Q₈	—	118	2	no	—
225	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
226	C₂²×C₄	—	59	1	no	—
227	C₂²×C₄	—	118	1	no	—
228	C₂²×C₄	—	118	1	no	—
229	M₁₆	—	118	2	no	—
230	C₂×C₈	—	59	1	no	—
231	C₂×C₈	—	59	1	no	—
232	C₂×C₈	—	118	2	no	—
233	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
234	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₄⋊C₄	—	2	1	no	—
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	59	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	—	118	1	no	—
243	C₄⋊C₄	—	118	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	C₂²×D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
254	C₂²⋊C₄	—	59	1	no	—
255	C₂²⋊C₄	—	59	1	no	—
256	C₂²⋊C₄	—	118	1	no	—
257	C₂²⋊C₄	—	118	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	59	1	no	—
260	C₄²	—	118	1	no	—
261	C₄²	—	118	1	no	—
262	C₄²	—	118	1	no	—
263	C₄²	—	118	1	no	—
264	C₄²	—	118	1	no	—
265	C₄²	C₂²×D₅₉	1	1	yes	characteristic
266	C₂³	—	59	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	8	1	no	—
269	Q₈	—	59	2	no	—
270	Q₈	—	118	4	no	—
271	Q₈	—	118	1	no	—
272	Q₈	—	472	1	no	—
273	Q₈	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	—	2	1	no	—
278	C₂×C₄	—	59	1	no	—
279	C₂×C₄	—	59	2	no	—
280	C₂×C₄	—	59	2	no	—
281	C₂×C₄	—	59	1	no	—
282	C₂×C₄	—	59	1	no	—
283	C₂×C₄	—	118	4	no	—
284	C₂×C₄	—	118	1	no	—
285	C₂×C₄	—	118	1	no	—
286	C₂×C₄	—	118	1	no	—
287	C₂×C₄	—	118	1	no	—
288	C₂×C₄	—	472	1	no	—
289	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
290	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	118	1	no	—
295	C₈	—	118	1	no	—
296	C₂²	—	59	2	no	—
297	C₂²	—	118	1	no	—
298	C₂²	—	118	1	no	—
299	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
300	C₄	—	2	2	no	—
301	C₄	—	2	1	no	—
302	C₄	—	2	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	59	2	no	—
307	C₄	—	118	2	no	—
308	C₄	—	118	1	no	—
309	C₄	—	118	1	no	—
310	C₄	—	236	1	no	—
311	C₄	—	236	1	no	—
312	C₄	—	236	1	no	—
313	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
314	C₄	#41×D₅₉	1	1	yes	characteristic
315	C₂	—	118	2	no	—
316	C₂	C₂²⋊Q₈×D₅₉	1	1	yes	characteristic
317	C₂	#16838208	1	1	yes	characteristic
318	C₂	#16838218	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#475	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#475	D₅₉
hypocenter	C₅₉	#3082
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
3. #16837712	C₂	semidirect
4. #16837710	C₂	semidirect
5. #16837709	C₂	semidirect
6. #16837707	C₂	semidirect
7. #16837695	C₂	semidirect
8. #16837694	C₂	semidirect
9. #16837686	C₂	semidirect
15. C₅₉×#475	C₂	semidirect
62. Q₈×D₅₉	Q₈	semidirect
169. #475	D₅₉	semidirect
175. C₅₉	#3082	semidirect