Subgroups — #21436925

Order 7552.

346 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	#16838125	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838112	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837771	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837762	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837757	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837703	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837688	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837618	C₂	1	1	yes	maximal, characteristic
10	#16837618	C₂	1	1	yes	maximal, characteristic
11	#16837608	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837515	C₂	1	1	yes	maximal, characteristic
13	#16837504	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#11745660	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#446	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745660	C₂²	1	1	yes	characteristic
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×D₄₇₂	C₂²	1	1	yes	characteristic
19	#14273727	C₂²	1	1	yes	characteristic
20	#14273727	C₂²	1	1	yes	characteristic
21	#14273722	—	2	1	no	—
22	#14273722	—	2	1	no	—
23	#14273719	C₂²	1	1	yes	characteristic
24	#14273719	C₂²	1	1	yes	characteristic
25	#14273715	C₂²	1	1	yes	characteristic
26	#14273715	C₂²	1	1	yes	characteristic
27	#14273705	—	2	1	no	—
28	#14273705	—	2	1	no	—
29	#14273703	—	2	1	no	—
30	#14273703	—	2	1	no	—
31	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
33	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
34	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
35	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
36	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
37	#14273685	—	2	1	no	—
38	C₄⋊C₄×D₅₉	—	2	1	no	—
39	#14273661	C₂²	1	1	yes	characteristic
40	#14273659	C₂²	1	1	yes	characteristic
41	#14273659	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	#14273652	C₂²	1	1	yes	characteristic
46	#14273652	C₂²	1	1	yes	characteristic
47	#14273652	C₂²	1	1	yes	characteristic
48	#14273649	C₂²	1	1	yes	characteristic
49	#14273649	C₂²	1	1	yes	characteristic
50	#14273649	C₂²	1	1	yes	characteristic
51	#14273649	C₂²	1	1	yes	characteristic
52	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
56	#97×C₅₉	C₂²	1	1	yes	characteristic
57	#97×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	D₄	1	1	yes	characteristic
60	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
61	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₅₉	—	2	1	no	—
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	—	2	1	no	—
65	C₂×C₄×D₅₉	—	2	1	no	—
66	C₂×Dic₁₁₈	—	2	1	no	—
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	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
76	C₄²×C₅₉	C₂³	1	1	yes	characteristic
77	C₄²×C₅₉	C₂³	1	1	yes	characteristic
78	C₄²×C₅₉	C₂³	1	1	yes	characteristic
79	(C₂×D₅₉)⋊C₄	—	2	1	no	—
80	(C₂×D₅₉)⋊C₄	—	2	1	no	—
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	(C₂×D₅₉)⋊C₄	—	2	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	C₄⋊Dic₅₉	—	2	1	no	—
88	C₄⋊Dic₅₉	—	2	1	no	—
89	C₄⋊Dic₅₉	—	2	1	no	—
90	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	—	2	1	no	—
96	Dic₅₉⋊C₄	—	2	1	no	—
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	D₄₇₂	—	2	2	no	—
103	#11745660	—	2	2	no	—
104	#11745660	—	2	2	no	—
105	C₂²×D₅₉	—	2	1	no	—
106	C₂²×D₅₉	—	2	1	no	—
107	Q₈×C₅₉	—	4	1	no	—
108	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
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	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	D₂₃₆	—	2	1	no	—
122	D₂₃₆	—	2	1	no	—
123	D₂₃₆	—	2	1	no	—
124	D₂₃₆	#41	1	2	yes	—
125	C₄×D₅₉	—	4	2	no	—
126	C₄×D₅₉	—	4	2	no	—
127	C₄×D₅₉	—	8	1	no	—
128	C₄×D₅₉	—	8	1	no	—
129	Dic₁₁₈	—	2	1	no	—
130	Dic₁₁₈	—	2	1	no	—
131	Dic₁₁₈	—	2	1	no	—
132	Dic₁₁₈	—	2	1	no	—
133	Dic₁₁₈	—	8	1	no	—
134	C₈×C₅₉	—	2	1	no	—
135	C₈×C₅₉	—	2	1	no	—
136	C₈×C₅₉	—	2	1	no	—
137	C₈×C₅₉	—	2	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	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	—	2	1	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
152	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	—	4	1	no	—
155	Dic₅₉	—	4	1	no	—
156	Dic₅₉	—	4	1	no	—
157	Dic₅₉	—	4	1	no	—
158	Dic₅₉	—	4	1	no	—
159	#3376	—	59	1	no	maximal, Sylow
160	C₂×C₅₉	#548	1	1	yes	characteristic
161	C₂×C₅₉	#575	1	1	yes	characteristic
162	C₂×C₅₉	#577	1	1	yes	characteristic
163	D₅₉	—	4	2	no	—
164	D₅₉	—	8	1	no	—
165	#556	—	59	1	no	—
166	#554	—	59	1	no	—
167	#491	—	59	1	no	—
168	#489	—	59	1	no	—
169	#488	—	59	1	no	—
170	#479	—	59	1	no	—
171	#478	—	59	1	no	—
172	#475	—	59	1	no	—
173	#463	—	59	1	no	—
174	#463	—	59	1	no	—
175	#461	—	59	1	no	—
176	#446	D₅₉	1	1	yes	characteristic, complemented
177	#442	—	59	1	no	—
178	#440	—	59	1	no	—
179	C₄×SD₁₆	—	59	1	no	—
180	C₅₉	#3376	1	1	yes	characteristic, complemented, Sylow
181	C₂×SD₁₆	—	59	1	no	—
182	C₂×SD₁₆	—	59	1	no	—
183	C₂×D₈	—	59	1	no	—
184	C₄⋊Q₈	—	59	1	no	—
185	C₄⋊Q₈	—	59	1	no	—
186	#126	—	118	1	no	—
187	#126	—	118	1	no	—
188	#125	—	59	1	no	—
189	#125	—	59	1	no	—
190	#124	—	59	1	no	—
191	#124	—	59	1	no	—
192	C₂²⋊Q₈	—	118	1	no	—
193	C₂²⋊Q₈	—	118	1	no	—
194	C₂²⋊Q₈	—	118	1	no	—
195	C₂²⋊Q₈	—	118	1	no	—
196	C₄×Q₈	—	59	1	no	—
197	C₄×Q₈	—	59	1	no	—
198	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
199	C₄×D₄	—	59	1	no	—
200	C₄×D₄	—	59	1	no	—
201	C₄×D₄	—	59	1	no	—
202	C₄×D₄	—	59	1	no	—
203	#117	—	118	1	no	—
204	C₂×C₄⋊C₄	—	118	1	no	—
205	#107	—	59	1	no	—
206	#106	—	59	1	no	—
207	#106	—	59	1	no	—
208	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
209	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
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	#102	—	59	1	no	—
218	#102	—	59	1	no	—
219	#102	—	59	1	no	—
220	#102	—	59	1	no	—
221	#97	C₂×D₅₉	1	1	yes	characteristic
222	#97	C₂×D₅₉	1	1	yes	characteristic
223	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
224	C₂×Q₈	—	59	1	no	—
225	C₂×Q₈	—	59	1	no	—
226	C₂×Q₈	—	118	1	no	—
227	C₂×Q₈	D₂₃₆	1	1	yes	characteristic
228	C₂×D₄	—	59	1	no	—
229	C₂×D₄	—	59	1	no	—
230	C₂²×C₄	—	118	1	no	—
231	C₂²×C₄	—	118	1	no	—
232	C₂²×C₄	—	118	1	no	—
233	C₂²×C₄	—	118	1	no	—
234	SD₁₆	—	118	2	no	—
235	SD₁₆	—	118	2	no	—
236	D₈	—	118	2	no	—
237	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
239	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
240	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
241	C₄⋊C₄	—	59	1	no	—
242	C₄⋊C₄	—	59	1	no	—
243	C₄⋊C₄	—	59	1	no	—
244	C₄⋊C₄	—	59	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	—	118	1	no	—
254	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
255	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
256	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
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	C₂²⋊C₄	—	118	1	no	—
263	C₂²⋊C₄	—	118	1	no	—
264	C₂²⋊C₄	—	118	1	no	—
265	C₄²	—	118	1	no	—
266	C₄²	—	118	1	no	—
267	C₄²	C₂²×D₅₉	1	1	yes	characteristic
268	C₄²	C₂²×D₅₉	1	1	yes	characteristic
269	C₄²	C₂²×D₅₉	1	1	yes	characteristic
270	C₂³	—	118	1	no	—
271	C₂³	—	118	1	no	—
272	Q₈	—	4	1	no	—
273	Q₈	—	118	1	no	—
274	Q₈	—	118	1	no	—
275	Q₈	—	118	1	no	—
276	Q₈	—	118	1	no	—
277	Q₈	—	472	1	no	—
278	D₄	—	59	2	no	—
279	D₄	—	118	1	no	—
280	D₄	—	118	1	no	—
281	D₄	—	118	1	no	—
282	C₂×C₄	—	118	1	no	—
283	C₂×C₄	—	118	1	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	—	472	1	no	—
290	C₂×C₄	—	236	2	no	—
291	C₂×C₄	—	236	2	no	—
292	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
293	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
294	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
295	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
296	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
297	C₂×C₄	C₂×D₂₃₆	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	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₂²	—	118	2	no	—
304	C₂²	—	236	1	no	—
305	C₂²	—	236	1	no	—
306	C₂²	—	236	1	no	—
307	C₂²	—	236	1	no	—
308	C₂²	—	236	1	no	—
309	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	236	1	no	—
317	C₄	—	236	1	no	—
318	C₄	—	236	1	no	—
319	C₄	—	236	1	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	—	472	1	no	—
325	C₂	—	236	2	no	—
326	C₂	#16838064	1	1	yes	characteristic
327	C₂	#16838202	1	1	yes	characteristic
328	C₂	#16838214	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#446	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#446	D₅₉
hypocenter	C₅₉	#3376
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #446

Complements

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

N	H ≅ G/N	type
1. #16838125	C₂	semidirect
2. #16838112	C₂	semidirect
3. #16837771	C₂	semidirect
4. #16837762	C₂	semidirect
5. #16837757	C₂	semidirect
6. #16837710	C₂	semidirect
7. #16837703	C₂	semidirect
8. #16837688	C₂	semidirect
11. #16837608	C₂	semidirect
13. #16837504	C₂	semidirect
14. C₄×#11745660	C₂	semidirect
15. C₅₉×#446	C₂	semidirect
176. #446	D₅₉	semidirect
180. C₅₉	#3376	semidirect