Subgroups — #21435367

Order 7552.

308 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838032	C₂	1	1	yes	maximal, characteristic
2	#16838015	C₂	1	1	yes	maximal, characteristic
3	#16837812	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837772	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837753	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837746	C₂	1	1	yes	maximal, characteristic
7	#16837745	C₂	1	1	yes	maximal, characteristic
8	#16837739	C₂	1	1	yes	maximal, characteristic
9	#16837734	C₂	1	1	yes	maximal, characteristic
10	#16837712	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837708	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837696	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837686	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837461	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#499	C₂	1	1	yes	maximal, characteristic, complemented
16	#14273727	C₂²	1	1	yes	characteristic
17	#14273727	C₂²	1	1	yes	characteristic
18	#14273726	—	2	1	no	—
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273722	—	2	1	no	—
21	#14273722	—	2	1	no	—
22	#14273718	—	2	1	no	—
23	#14273718	—	2	1	no	—
24	#14273718	C₂²	1	1	yes	characteristic
25	#14273717	—	2	1	no	—
26	#14273707	—	2	1	no	—
27	#14273707	C₂²	1	1	yes	characteristic
28	#14273707	C₂²	1	1	yes	characteristic
29	#14273705	—	2	1	no	—
30	#14273692	C₂²	1	1	yes	characteristic
31	#14273692	C₂²	1	1	yes	characteristic
32	#14273685	C₂²	1	1	yes	characteristic
33	#14273684	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
35	#14273660	C₂²	1	1	yes	characteristic
36	#14273660	C₂²	1	1	yes	characteristic
37	#14273658	C₂²	1	1	yes	characteristic
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	#14273652	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273650	C₂²	1	1	yes	characteristic
46	#14273650	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273650	C₂²	1	1	yes	characteristic
49	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	#14273638	C₂²	1	1	yes	characteristic
52	#125×C₅₉	C₂²	1	1	yes	characteristic
53	#125×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#106×C₅₉	C₂²	1	1	yes	characteristic
57	#106×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×C₄×D₅₉	—	2	1	no	—
60	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₁₈	—	2	1	no	—
62	C₂×Dic₁₁₈	—	2	1	no	—
63	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
65	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
66	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
67	C₄⋊C₄×C₅₉	—	2	1	no	—
68	C₄⋊C₄×C₅₉	—	2	1	no	—
69	C₄⋊C₄×C₅₉	—	2	1	no	—
70	C₄⋊C₄×C₅₉	—	2	1	no	—
71	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
75	C₄²×C₅₉	C₂³	1	1	yes	characteristic
76	(C₂×D₅₉)⋊C₄	—	2	1	no	—
77	(C₂×D₅₉)⋊C₄	—	2	1	no	—
78	(C₂×D₅₉)⋊C₄	—	2	1	no	—
79	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
80	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
81	C₄⋊Dic₅₉	—	2	1	no	—
82	C₄⋊Dic₅₉	—	2	1	no	—
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	—	2	1	no	—
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	C₂³	1	1	yes	characteristic
100	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₅₉	—	2	1	no	—
104	C₂×C₄×C₅₉	—	2	1	no	—
105	C₂×C₄×C₅₉	—	2	1	no	—
106	C₂×C₄×C₅₉	—	2	1	no	—
107	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
110	C₂×Dic₅₉	—	2	1	no	—
111	C₂×Dic₅₉	—	2	1	no	—
112	C₂×Dic₅₉	—	2	1	no	—
113	C₂×Dic₅₉	—	2	1	no	—
114	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
117	C₄×D₅₉	—	2	1	no	—
118	C₄×D₅₉	—	2	1	no	—
119	C₄×D₅₉	—	8	1	no	—
120	Dic₁₁₈	—	2	1	no	—
121	Dic₁₁₈	—	2	1	no	—
122	Dic₁₁₈	—	2	1	no	—
123	Dic₁₁₈	—	2	1	no	—
124	Dic₁₁₈	—	8	1	no	—
125	Dic₁₁₈	—	8	1	no	—
126	C₈×C₅₉	—	2	1	no	—
127	C₈×C₅₉	—	2	1	no	—
128	C₅₉⋊C₈	—	2	1	no	—
129	C₅₉⋊C₈	—	2	1	no	—
130	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₅₉	—	2	1	no	—
132	C₂×D₅₉	—	2	1	no	—
133	C₂×D₅₉	—	2	1	no	—
134	C₄×C₅₉	—	2	1	no	—
135	C₄×C₅₉	—	2	1	no	—
136	C₄×C₅₉	—	4	1	no	—
137	C₄×C₅₉	—	4	1	no	—
138	C₄×C₅₉	—	4	1	no	—
139	C₄×C₅₉	—	4	1	no	—
140	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
141	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
142	Dic₅₉	—	2	1	no	—
143	Dic₅₉	—	2	1	no	—
144	Dic₅₉	—	2	1	no	—
145	Dic₅₉	—	4	1	no	—
146	Dic₅₉	—	4	1	no	—
147	Dic₅₉	—	4	1	no	—
148	Dic₅₉	—	4	1	no	—
149	#3242	—	59	1	no	maximal, Sylow
150	C₂×C₅₉	#536	1	1	yes	characteristic
151	C₂×C₅₉	#577	1	1	yes	characteristic
152	C₂×C₅₉	#578	1	1	yes	characteristic
153	D₅₉	—	4	1	no	—
154	#544	—	59	1	no	—
155	#541	—	59	1	no	—
156	#501	—	59	1	no	—
157	#499	D₅₉	1	1	yes	characteristic, complemented
158	#491	—	59	1	no	—
159	#487	—	59	1	no	—
160	#485	—	59	1	no	—
161	#485	—	59	1	no	—
162	#484	—	59	1	no	—
163	#483	—	59	1	no	—
164	#479	—	59	1	no	—
165	#479	—	59	1	no	—
166	#477	—	59	1	no	—
167	#475	—	59	1	no	—
168	#433	—	59	1	no	—
169	C₅₉	#3242	1	1	yes	characteristic, complemented, Sylow
170	C₄⋊Q₈	—	59	1	no	—
171	C₄⋊Q₈	—	59	1	no	—
172	C₄⋊Q₈	—	59	1	no	—
173	C₄⋊Q₈	—	118	1	no	—
174	#126	—	118	1	no	—
175	#126	—	118	1	no	—
176	#125	—	59	1	no	—
177	#125	—	118	1	no	—
178	#125	—	118	1	no	—
179	#125	—	118	1	no	—
180	#125	C₂×D₅₉	1	1	yes	characteristic
181	#125	C₂×D₅₉	1	1	yes	characteristic
182	C₂²⋊Q₈	—	59	1	no	—
183	C₂²⋊Q₈	—	59	1	no	—
184	C₂²⋊Q₈	—	118	1	no	—
185	C₂²⋊Q₈	—	118	1	no	—
186	C₄×Q₈	—	59	1	no	—
187	C₄×Q₈	—	59	1	no	—
188	#117	—	59	1	no	—
189	#117	—	59	1	no	—
190	C₂×C₄⋊C₄	—	59	1	no	—
191	#107	—	59	1	no	—
192	#107	—	59	1	no	—
193	#107	C₂×D₅₉	1	1	yes	characteristic
194	#107	C₂×D₅₉	1	1	yes	characteristic
195	#106	—	59	1	no	—
196	#106	—	59	1	no	—
197	#106	C₂×D₅₉	1	1	yes	characteristic
198	#106	C₂×D₅₉	1	1	yes	characteristic
199	C₄⋊C₈	—	59	1	no	—
200	C₄⋊C₈	—	59	1	no	—
201	#103	—	59	1	no	—
202	#103	—	59	1	no	—
203	#103	—	59	1	no	—
204	#103	—	59	1	no	—
205	#103	—	59	1	no	—
206	#103	—	59	1	no	—
207	#103	—	59	1	no	—
208	#103	—	59	1	no	—
209	C₂²⋊C₈	—	59	1	no	—
210	C₂²⋊C₈	—	59	1	no	—
211	#97	—	59	1	no	—
212	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
213	C₂×Q₈	—	59	1	no	—
214	C₂×Q₈	—	59	1	no	—
215	C₂×Q₈	—	118	1	no	—
216	C₂×Q₈	—	118	1	no	—
217	C₂²×C₄	—	59	1	no	—
218	C₂²×C₄	—	118	1	no	—
219	C₂×C₈	—	59	1	no	—
220	C₂×C₈	—	59	1	no	—
221	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
222	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	2	1	no	—
225	C₄⋊C₄	—	2	1	no	—
226	C₄⋊C₄	—	2	1	no	—
227	C₄⋊C₄	—	59	1	no	—
228	C₄⋊C₄	—	59	1	no	—
229	C₄⋊C₄	—	59	1	no	—
230	C₄⋊C₄	—	59	1	no	—
231	C₄⋊C₄	—	118	1	no	—
232	C₄⋊C₄	—	118	1	no	—
233	C₄⋊C₄	—	118	1	no	—
234	C₄⋊C₄	—	118	1	no	—
235	C₄⋊C₄	—	118	1	no	—
236	C₄⋊C₄	—	118	1	no	—
237	C₄⋊C₄	—	118	1	no	—
238	C₄⋊C₄	—	118	1	no	—
239	C₄⋊C₄	—	118	1	no	—
240	C₄⋊C₄	—	118	1	no	—
241	C₄⋊C₄	—	118	1	no	—
242	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
243	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
244	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
245	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
246	C₂²⋊C₄	—	59	1	no	—
247	C₂²⋊C₄	—	59	1	no	—
248	C₂²⋊C₄	—	118	1	no	—
249	C₂²⋊C₄	—	118	1	no	—
250	C₂²⋊C₄	—	118	1	no	—
251	C₄²	—	59	1	no	—
252	C₄²	—	118	1	no	—
253	C₄²	—	118	1	no	—
254	C₄²	—	118	1	no	—
255	C₄²	C₂²×D₅₉	1	1	yes	characteristic
256	C₂³	—	59	1	no	—
257	Q₈	—	118	1	no	—
258	Q₈	—	118	1	no	—
259	Q₈	—	118	1	no	—
260	Q₈	—	118	1	no	—
261	Q₈	—	472	1	no	—
262	Q₈	—	472	1	no	—
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	2	1	no	—
267	C₂×C₄	—	59	1	no	—
268	C₂×C₄	—	59	1	no	—
269	C₂×C₄	—	59	1	no	—
270	C₂×C₄	—	118	1	no	—
271	C₂×C₄	—	118	1	no	—
272	C₂×C₄	—	118	1	no	—
273	C₂×C₄	—	118	1	no	—
274	C₂×C₄	—	118	1	no	—
275	C₂×C₄	—	118	1	no	—
276	C₂×C₄	—	472	1	no	—
277	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
278	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
279	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
280	C₈	—	2	1	no	—
281	C₈	—	2	1	no	—
282	C₈	—	118	1	no	—
283	C₈	—	118	1	no	—
284	C₂²	—	118	1	no	—
285	C₂²	—	118	1	no	—
286	C₂²	—	118	1	no	—
287	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	4	1	no	—
291	C₄	—	4	1	no	—
292	C₄	—	4	1	no	—
293	C₄	—	4	1	no	—
294	C₄	—	118	1	no	—
295	C₄	—	118	1	no	—
296	C₄	—	118	1	no	—
297	C₄	—	236	1	no	—
298	C₄	—	236	1	no	—
299	C₄	—	236	1	no	—
300	C₄	—	236	1	no	—
301	C₄	#14273775	1	1	yes	characteristic
302	C₄	#14273775	1	1	yes	characteristic
303	C₂	—	236	1	no	—
304	C₂	#16837977	1	1	yes	characteristic
305	C₂	#16838209	1	1	yes	characteristic
306	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×#499	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#499	D₅₉
hypocenter	C₅₉	#3242
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

Complements

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

N	H ≅ G/N	type
3. #16837812	C₂	semidirect
4. #16837772	C₂	semidirect
5. #16837753	C₂	semidirect
10. #16837712	C₂	semidirect
11. #16837708	C₂	semidirect
12. #16837696	C₂	semidirect
13. #16837686	C₂	semidirect
15. C₅₉×#499	C₂	semidirect
157. #499	D₅₉	semidirect
169. C₅₉	#3242	semidirect