Subgroups — #21434327

Order 7552.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838023	C₂	1	1	yes	maximal, characteristic
2	#16837879	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837744	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837727	C₂	1	1	yes	maximal, characteristic
5	#16837698	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837688	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837625	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837625	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837560	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837543	C₂	1	1	yes	maximal, characteristic
11	#16837510	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837504	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×Dic₂₃₆	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#11745660	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#433	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂²×Dic₁₁₈	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
18	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
19	C₂×#11745660	C₂²	1	1	yes	characteristic
20	C₂×#11745660	C₂²	1	1	yes	characteristic
21	#14273727	C₂²	1	1	yes	characteristic
22	#14273722	—	2	1	no	—
23	#14273715	C₂²	1	1	yes	characteristic
24	#14273713	—	2	1	no	—
25	#14273709	—	2	1	no	—
26	#14273706	C₂²	1	1	yes	characteristic
27	#14273705	—	2	1	no	—
28	#14273701	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
34	#14273692	—	2	1	no	—
35	#14273692	—	2	1	no	—
36	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
37	#14273686	C₂²	1	1	yes	characteristic
38	#14273683	—	2	1	no	—
39	#14273683	—	2	1	no	—
40	#14273661	C₂²	1	1	yes	characteristic
41	#14273661	C₂²	1	1	yes	characteristic
42	#14273659	C₂²	1	1	yes	characteristic
43	#14273659	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	#14273652	C₂²	1	1	yes	characteristic
49	#14273652	C₂²	1	1	yes	characteristic
50	#14273649	C₂²	1	1	yes	characteristic
51	#14273649	C₂²	1	1	yes	characteristic
52	#117×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	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#97×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₅₉	—	2	1	no	—
61	C₂²×Dic₅₉	—	2	1	no	—
62	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×Dic₁₁₈	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	2	no	—
66	C₂×Dic₁₁₈	—	2	2	no	—
67	C₂×Dic₁₁₈	—	2	1	no	—
68	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
69	C₂×Dic₁₁₈	C₂³	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	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
76	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
77	C₂²⋊C₄×C₅₉	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₅₉	D₄	1	1	yes	characteristic
79	C₄²×C₅₉	C₂³	1	1	yes	characteristic
80	C₄²×C₅₉	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₅₉	—	2	1	no	—
82	C₂²⋊Dic₅₉	—	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	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	Dic₂₃₆	—	2	2	no	—
105	Dic₂₃₆	—	2	2	no	—
106	#11745660	—	2	2	no	—
107	#11745660	—	2	2	no	—
108	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	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	C₂×D₄	1	1	yes	characteristic
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	—	8	1	no	—
119	C₂×Dic₅₉	—	2	4	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	C₂×Dic₅₉	—	2	1	no	—
122	C₂×Dic₅₉	—	2	1	no	—
123	C₂×Dic₅₉	—	2	1	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	D₂₃₆	—	2	1	no	—
128	D₂₃₆	—	2	1	no	—
129	C₄×D₅₉	—	8	1	no	—
130	Dic₁₁₈	—	2	4	no	—
131	Dic₁₁₈	—	2	1	no	—
132	Dic₁₁₈	—	2	1	no	—
133	Dic₁₁₈	—	2	1	no	—
134	Dic₁₁₈	—	2	1	no	—
135	Dic₁₁₈	—	8	1	no	—
136	Dic₁₁₈	#41	1	2	yes	—
137	Dic₁₁₈	#41	1	2	yes	—
138	C₈×C₅₉	—	2	1	no	—
139	C₈×C₅₉	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₂²×C₅₉	—	2	1	no	—
143	C₂²×C₅₉	—	2	1	no	—
144	C₂²×C₅₉	—	2	1	no	—
145	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₅₉	—	4	1	no	—
147	C₂×D₅₉	—	4	1	no	—
148	C₂×D₅₉	—	4	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	2	1	no	—
151	C₄×C₅₉	—	2	1	no	—
152	C₄×C₅₉	—	2	1	no	—
153	C₄×C₅₉	—	2	1	no	—
154	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
156	Dic₅₉	—	2	2	no	—
157	Dic₅₉	—	2	2	no	—
158	Dic₅₉	—	4	1	no	—
159	Dic₅₉	—	4	1	no	—
160	Dic₅₉	—	4	1	no	—
161	Dic₅₉	—	4	1	no	—
162	Dic₅₉	—	4	1	no	—
163	#3121	—	59	1	no	maximal, Sylow
164	C₂×C₅₉	—	4	1	no	—
165	C₂×C₅₉	#525	1	1	yes	characteristic
166	C₂×C₅₉	#577	1	1	yes	characteristic
167	C₂×C₅₉	#578	1	1	yes	characteristic
168	D₅₉	—	8	1	no	—
169	#542	—	59	1	no	—
170	#519	—	59	1	no	—
171	#485	—	59	1	no	—
172	#482	—	59	1	no	—
173	#477	—	59	1	no	—
174	#475	—	59	1	no	—
175	#464	—	59	1	no	—
176	#464	—	59	1	no	—
177	#451	—	59	1	no	—
178	#448	—	59	1	no	—
179	#441	—	59	1	no	—
180	#440	—	59	1	no	—
181	C₄×Q₁₆	—	59	1	no	—
182	C₄×SD₁₆	—	59	1	no	—
183	#433	D₅₉	1	1	yes	characteristic, complemented
184	C₅₉	#3121	1	1	yes	characteristic, complemented, Sylow
185	C₂²×Q₈	—	59	1	no	—
186	C₂×Q₁₆	—	59	1	no	—
187	C₂×Q₁₆	—	59	1	no	—
188	C₂×SD₁₆	—	59	1	no	—
189	C₂×SD₁₆	—	59	1	no	—
190	C₄⋊Q₈	—	59	1	no	—
191	#126	—	118	1	no	—
192	#124	—	59	1	no	—
193	#124	—	118	1	no	—
194	#123	—	118	1	no	—
195	C₂²⋊Q₈	—	59	1	no	—
196	C₂²⋊Q₈	—	118	1	no	—
197	#121	—	59	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×Q₈	—	118	1	no	—
205	C₄×D₄	—	59	1	no	—
206	#117	—	59	1	no	—
207	#117	—	118	1	no	—
208	#117	—	118	1	no	—
209	#117	C₂×D₅₉	1	1	yes	characteristic
210	#107	—	59	1	no	—
211	#107	—	59	1	no	—
212	#106	—	59	1	no	—
213	#106	—	59	1	no	—
214	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
215	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
216	#103	—	59	1	no	—
217	#103	—	59	1	no	—
218	#103	—	59	1	no	—
219	#103	—	59	1	no	—
220	#103	—	59	1	no	—
221	#103	—	59	1	no	—
222	#102	—	59	1	no	—
223	#102	—	59	1	no	—
224	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
225	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
226	#97	C₂×D₅₉	1	1	yes	characteristic
227	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
228	C₂×Q₈	—	59	1	no	—
229	C₂×Q₈	—	59	1	no	—
230	C₂×Q₈	—	59	1	no	—
231	C₂×Q₈	—	118	2	no	—
232	C₂×Q₈	—	118	2	no	—
233	C₂×Q₈	—	118	1	no	—
234	C₂×Q₈	—	118	1	no	—
235	C₂×D₄	—	59	1	no	—
236	C₂×D₄	—	118	1	no	—
237	C₂²×C₄	—	118	1	no	—
238	C₂²×C₄	—	118	1	no	—
239	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
240	Q₁₆	—	118	2	no	—
241	Q₁₆	—	118	2	no	—
242	SD₁₆	—	118	2	no	—
243	SD₁₆	—	118	2	no	—
244	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
245	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
246	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
248	C₄⋊C₄	—	59	1	no	—
249	C₄⋊C₄	—	59	1	no	—
250	C₄⋊C₄	—	59	1	no	—
251	C₄⋊C₄	—	59	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	—	118	1	no	—
258	C₄⋊C₄	—	118	1	no	—
259	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
260	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
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	C₂²⋊C₄	—	118	1	no	—
266	C₂²⋊C₄	—	118	1	no	—
267	C₂²⋊C₄	D₂₃₆	1	1	yes	characteristic
268	C₂²⋊C₄	D₂₃₆	1	1	yes	characteristic
269	C₄²	—	118	1	no	—
270	C₄²	—	118	1	no	—
271	C₄²	—	118	1	no	—
272	C₄²	—	118	1	no	—
273	C₄²	—	118	1	no	—
274	C₄²	—	118	1	no	—
275	C₄²	C₂²×D₅₉	1	1	yes	characteristic
276	C₄²	C₂²×D₅₉	1	1	yes	characteristic
277	C₂³	—	118	1	no	—
278	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
279	Q₈	—	59	2	no	—
280	Q₈	—	59	2	no	—
281	Q₈	—	118	4	no	—
282	Q₈	—	118	1	no	—
283	Q₈	—	118	1	no	—
284	Q₈	—	118	1	no	—
285	Q₈	—	118	1	no	—
286	Q₈	—	472	1	no	—
287	D₄	—	118	1	no	—
288	D₄	—	118	1	no	—
289	D₄	—	472	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	118	4	no	—
293	C₂×C₄	—	118	1	no	—
294	C₂×C₄	—	118	1	no	—
295	C₂×C₄	—	118	1	no	—
296	C₂×C₄	—	118	1	no	—
297	C₂×C₄	—	118	1	no	—
298	C₂×C₄	—	118	1	no	—
299	C₂×C₄	—	118	1	no	—
300	C₂×C₄	—	472	1	no	—
301	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
302	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
303	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
304	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
305	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	236	1	no	—
315	C₂²	—	236	1	no	—
316	C₂²	—	236	1	no	—
317	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	118	2	no	—
324	C₄	—	118	2	no	—
325	C₄	—	236	1	no	—
326	C₄	—	236	1	no	—
327	C₄	—	236	1	no	—
328	C₄	—	236	1	no	—
329	C₄	—	236	1	no	—
330	C₄	#41×D₅₉	1	1	yes	characteristic
331	C₄	#41×D₅₉	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	472	1	no	—
334	C₂	#16837914	1	1	yes	characteristic
335	C₂	#16838214	1	1	yes	characteristic
336	C₂	#16838219	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₃₆
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#433	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#433	D₅₉
hypocenter	C₅₉	#3121
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #433

Complements

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

N	H ≅ G/N	type
2. #16837879	C₂	semidirect
3. #16837744	C₂	semidirect
5. #16837698	C₂	semidirect
6. #16837688	C₂	semidirect
7. #16837625	C₂	semidirect
8. #16837625	C₂	semidirect
9. #16837560	C₂	semidirect
11. #16837510	C₂	semidirect
12. #16837504	C₂	semidirect
13. C₄×Dic₂₃₆	C₂	semidirect
14. C₄×#11745660	C₂	semidirect
15. C₅₉×#433	C₂	semidirect
183. #433	D₅₉	semidirect
184. C₅₉	#3121	semidirect