Subgroups — #21434064

Order 7552.

386 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837975	C₂	1	1	yes	maximal, characteristic
2	#16837883	C₂	1	1	yes	maximal, characteristic
3	#16837745	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837744	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837742	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837740	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837739	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837732	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837731	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837443	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837439	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837409	C₂	1	1	yes	maximal, characteristic, complemented
13	#103×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837367	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#484	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745685	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
18	C₂×#11745659	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₅₉	C₂²	1	1	yes	characteristic
20	#14273726	—	2	1	no	—
21	#14273713	—	2	1	no	—
22	#14273712	—	2	1	no	—
23	#14273709	—	2	1	no	—
24	#14273707	C₂²	1	1	yes	characteristic
25	#14273706	C₂²	1	1	yes	characteristic
26	#14273703	C₂²	1	1	yes	characteristic
27	#14273692	—	2	1	no	—
28	#14273689	—	2	1	no	—
29	C₄×C₂²⋊D₅₉	—	2	1	no	—
30	C₄×C₂²⋊D₅₉	—	2	1	no	—
31	#14273686	C₂²	1	1	yes	characteristic
32	#14273685	C₂²	1	1	yes	characteristic
33	#14273685	C₂²	1	1	yes	characteristic
34	#14273685	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₅₉	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	#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	#14273652	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273650	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₅₉)⋊C₈	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	—	2	1	no	—
61	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
64	Q₈×D₅₉	—	2	2	no	—
65	Q₈×D₅₉	—	2	2	no	—
66	#11745685	—	2	2	no	—
67	#11745685	—	2	2	no	—
68	C₂×D₂₃₆	D₄	1	1	yes	characteristic
69	C₂×C₄×D₅₉	—	2	1	no	—
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
74	C₂×Dic₁₁₈	—	2	1	no	—
75	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
76	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₅₉	—	2	1	no	—
80	C₄⋊C₄×C₅₉	—	2	1	no	—
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
84	C₂²⋊C₄×C₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	C₂²⋊Dic₅₉	—	2	1	no	—
87	(C₂×D₅₉)⋊C₄	—	2	1	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	#11745659	—	2	2	no	—
107	C₈×D₅₉	—	2	2	no	—
108	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₅₉	—	2	1	no	—
112	Q₈×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	2	1	no	—
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	2	1	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	8	1	no	—
120	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₅₉	—	2	2	no	—
123	C₂²⋊D₅₉	—	2	2	no	—
124	C₂×Dic₅₉	—	2	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	D₂₃₆	—	2	2	no	—
131	C₄×D₅₉	—	2	1	no	—
132	C₄×D₅₉	—	2	1	no	—
133	C₄×D₅₉	—	4	2	no	—
134	C₄×D₅₉	—	8	1	no	—
135	C₄×D₅₉	—	8	1	no	—
136	C₄×D₅₉	#41	1	2	yes	—
137	C₄×D₅₉	#41	1	2	yes	—
138	Dic₁₁₈	—	2	2	no	—
139	Dic₁₁₈	—	2	1	no	—
140	Dic₁₁₈	—	2	1	no	—
141	Dic₁₁₈	—	4	2	no	—
142	C₈×C₅₉	—	2	1	no	—
143	C₈×C₅₉	—	2	1	no	—
144	C₅₉⋊C₈	—	2	1	no	—
145	C₅₉⋊C₈	—	2	1	no	—
146	C₂²×C₅₉	—	2	1	no	—
147	C₂²×C₅₉	—	2	1	no	—
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₅₉	—	2	1	no	—
151	C₂×D₅₉	—	2	1	no	—
152	C₂×D₅₉	—	2	1	no	—
153	C₂×D₅₉	—	2	1	no	—
154	C₂×D₅₉	—	2	1	no	—
155	C₂×D₅₉	#123	1	2	yes	—
156	C₄×C₅₉	—	2	1	no	—
157	C₄×C₅₉	—	4	1	no	—
158	C₄×C₅₉	—	4	1	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
162	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
163	Dic₅₉	—	2	1	no	—
164	Dic₅₉	—	4	1	no	—
165	Dic₅₉	—	4	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	—	4	1	no	—
168	Dic₅₉	#123	1	2	yes	—
169	#3097	—	59	1	no	maximal, Sylow
170	C₂×C₅₉	—	4	1	no	—
171	C₂×C₅₉	C₂×#123	1	1	yes	characteristic
172	C₂×C₅₉	#577	1	1	yes	characteristic
173	C₂×C₅₉	#578	1	1	yes	characteristic
174	D₅₉	—	2	2	no	—
175	D₅₉	—	4	1	no	—
176	#536	—	59	1	no	—
177	#520	—	59	1	no	—
178	#485	—	59	1	no	—
179	#485	—	59	1	no	—
180	#485	—	59	1	no	—
181	#485	—	59	1	no	—
182	#484	—	59	1	no	—
183	#484	D₅₉	1	1	yes	characteristic, complemented
184	#483	—	59	1	no	—
185	#483	—	59	1	no	—
186	#428	—	59	1	no	—
187	#427	—	59	1	no	—
188	#419	—	59	1	no	—
189	C₂×#103	—	59	1	no	—
190	#408	—	59	1	no	—
191	C₅₉	#3097	1	1	yes	characteristic, complemented, Sylow
192	C₂×#41	—	59	1	no	—
193	C₂²×Q₈	—	59	1	no	—
194	C₂×M₁₆	—	59	1	no	—
195	C₂²×C₈	—	59	1	no	—
196	C₄⋊Q₈	—	118	1	no	—
197	#124	—	118	1	no	—
198	#123	—	118	1	no	—
199	#123	—	118	1	no	—
200	C₂²⋊Q₈	—	59	1	no	—
201	C₂²⋊Q₈	—	59	1	no	—
202	C₂²⋊Q₈	—	59	1	no	—
203	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
204	C₄×Q₈	—	118	1	no	—
205	C₄×D₄	—	118	1	no	—
206	C₄×D₄	—	118	1	no	—
207	C₄×D₄	—	118	1	no	—
208	#117	—	59	1	no	—
209	#117	—	59	1	no	—
210	#117	—	59	1	no	—
211	#117	—	59	1	no	—
212	C₂×C₄⋊C₄	—	59	1	no	—
213	C₂×C₄⋊C₄	—	59	1	no	—
214	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
215	#107	—	59	1	no	—
216	#107	—	59	1	no	—
217	#107	C₂×D₅₉	1	1	yes	characteristic
218	#107	C₂×D₅₉	1	1	yes	characteristic
219	#106	—	59	1	no	—
220	#106	—	59	1	no	—
221	#106	—	59	1	no	—
222	#106	—	59	1	no	—
223	#103	—	59	1	no	—
224	#103	—	59	1	no	—
225	#103	—	59	1	no	—
226	#103	—	59	1	no	—
227	#103	—	59	1	no	—
228	#103	—	59	1	no	—
229	#103	C₂×D₅₉	1	1	yes	characteristic
230	#103	C₂×D₅₉	1	1	yes	characteristic
231	C₂²⋊C₈	—	59	1	no	—
232	C₂²⋊C₈	—	59	1	no	—
233	C₂²⋊C₈	—	59	1	no	—
234	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
235	#41	—	118	2	no	—
236	#41	—	118	2	no	—
237	C₂×Q₈	—	59	1	no	—
238	C₂×Q₈	—	59	1	no	—
239	C₂×Q₈	—	118	1	no	—
240	C₂×Q₈	—	118	2	no	—
241	C₂×Q₈	—	118	2	no	—
242	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
243	C₂×D₄	—	59	1	no	—
244	C₂×D₄	—	59	1	no	—
245	C₂×D₄	—	59	1	no	—
246	C₂²×C₄	—	2	1	no	—
247	C₂²×C₄	—	59	1	no	—
248	C₂²×C₄	—	59	1	no	—
249	C₂²×C₄	—	118	1	no	—
250	C₂²×C₄	—	118	1	no	—
251	C₂²×C₄	—	118	1	no	—
252	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
253	M₁₆	—	118	2	no	—
254	C₂×C₈	—	59	1	no	—
255	C₂×C₈	—	59	1	no	—
256	C₂×C₈	—	118	2	no	—
257	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	1	no	—
261	C₄⋊C₄	—	59	1	no	—
262	C₄⋊C₄	—	59	1	no	—
263	C₄⋊C₄	—	59	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	—	118	1	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	—	118	1	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	118	1	no	—
274	C₂²⋊C₄	—	118	1	no	—
275	C₂²⋊C₄	—	118	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₂²⋊C₄	—	118	1	no	—
279	C₂²⋊C₄	—	118	1	no	—
280	C₄²	—	118	1	no	—
281	C₄²	—	118	1	no	—
282	C₄²	—	118	1	no	—
283	C₄²	—	118	1	no	—
284	C₂³	—	59	1	no	—
285	C₂³	—	59	1	no	—
286	C₂³	D₄×D₅₉	1	1	yes	characteristic
287	Q₈	—	2	1	no	—
288	Q₈	—	2	1	no	—
289	Q₈	—	118	2	no	—
290	Q₈	—	118	1	no	—
291	Q₈	—	118	1	no	—
292	Q₈	—	236	2	no	—
293	D₄	—	118	2	no	—
294	D₄	—	118	2	no	—
295	D₄	—	118	2	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	8	1	no	—
303	C₂×C₄	—	59	1	no	—
304	C₂×C₄	—	59	2	no	—
305	C₂×C₄	—	59	2	no	—
306	C₂×C₄	—	59	1	no	—
307	C₂×C₄	—	118	1	no	—
308	C₂×C₄	—	118	1	no	—
309	C₂×C₄	—	118	1	no	—
310	C₂×C₄	—	118	1	no	—
311	C₂×C₄	—	118	1	no	—
312	C₂×C₄	—	118	1	no	—
313	C₂×C₄	—	472	1	no	—
314	C₂×C₄	—	472	1	no	—
315	C₂×C₄	—	236	2	no	—
316	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	118	1	no	—
321	C₈	—	118	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	59	2	no	—
326	C₂²	—	118	1	no	—
327	C₂²	—	118	1	no	—
328	C₂²	—	118	1	no	—
329	C₂²	—	118	1	no	—
330	C₂²	—	118	1	no	—
331	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
332	C₄	—	2	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	59	2	no	—
338	C₄	—	118	1	no	—
339	C₄	—	236	1	no	—
340	C₄	—	236	1	no	—
341	C₄	—	236	1	no	—
342	C₄	—	236	1	no	—
343	C₄	#41×D₅₉	1	1	yes	characteristic
344	C₄	#41×D₅₉	1	1	yes	characteristic
345	C₂	—	4	1	no	—
346	C₂	—	118	2	no	—
347	C₂	—	236	1	no	—
348	C₂	#123×D₅₉	1	1	yes	characteristic
349	C₂	#16838212	1	1	yes	characteristic
350	C₂	#16838215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#484	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#484	D₅₉
hypocenter	C₅₉	#3097
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
3. #16837745	C₂	semidirect
4. #16837744	C₂	semidirect
5. #16837742	C₂	semidirect
6. #16837740	C₂	semidirect
7. #16837739	C₂	semidirect
8. #16837732	C₂	semidirect
9. #16837731	C₂	semidirect
10. #16837443	C₂	semidirect
11. #16837439	C₂	semidirect
12. #16837409	C₂	semidirect
13. #103×D₅₉	C₂	semidirect
15. C₅₉×#484	C₂	semidirect
183. #484	D₅₉	semidirect
191. C₅₉	#3097	semidirect