Subgroups — #21434809

Order 7552.

402 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837954	C₂	1	1	yes	maximal, characteristic, complemented
2	#16837947	C₂	1	1	yes	maximal, characteristic
3	#16837806	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837773	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837751	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837747	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837650	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837648	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837646	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837505	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837503	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837501	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837499	C₂	1	1	yes	maximal, characteristic, complemented
14	M₁₆×Dic₅₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#468	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745687	—	2	1	no	—
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×#11745670	C₂²	1	1	yes	characteristic
19	C₂×#11745671	C₂²	1	1	yes	characteristic
20	C₂×#11745664	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273726	C₂²	1	1	yes	characteristic
23	#14273725	C₂²	1	1	yes	characteristic
24	#14273723	C₂²	1	1	yes	characteristic
25	#14273717	—	2	1	no	—
26	#14273713	—	2	1	no	—
27	#14273706	C₂²	1	1	yes	characteristic
28	#14273704	—	2	1	no	—
29	#14273701	C₂²	1	1	yes	characteristic
30	#14273699	—	2	1	no	—
31	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
32	#14273692	—	2	1	no	—
33	#14273692	C₂²	1	1	yes	characteristic
34	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
35	#14273689	C₂²	1	1	yes	characteristic
36	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
37	#14273683	—	2	1	no	—
38	C₂×C₄×Dic₅₉	C₂²	1	1	yes	characteristic
39	#14273661	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	#14273660	C₂²	1	1	yes	characteristic
42	#14273652	C₂²	1	1	yes	characteristic
43	#14273651	C₂²	1	1	yes	characteristic
44	#14273650	C₂²	1	1	yes	characteristic
45	#14273649	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273647	C₂²	1	1	yes	characteristic
48	#14273637	C₂²	1	1	yes	characteristic
49	#14273637	C₂²	1	1	yes	characteristic
50	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	#121×C₅₉	C₂²	1	1	yes	characteristic
56	#107×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#102×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₅₉	—	2	1	no	—
61	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	C₂×C₂²⋊D₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	2	1	no	—
67	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
68	C₂²×Dic₅₉	D₄	1	1	yes	characteristic
69	#11745687	—	4	4	no	—
70	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×Dic₁₁₈	—	2	1	no	—
73	C₂×Dic₁₁₈	—	2	1	no	—
74	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
75	SD₁₆×C₅₉	—	2	2	no	—
76	M₁₆×C₅₉	—	2	2	no	—
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	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₅₉	—	2	1	no	—
83	C₂²⋊C₄×C₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	#11745671	—	2	2	no	—
87	#11745670	—	2	2	no	—
88	(C₂×D₅₉)⋊C₄	—	2	1	no	—
89	(C₂×D₅₉)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₅₉	—	2	1	no	—
91	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	Dic₅₉⋊C₄	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	D₄	1	1	yes	characteristic
103	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
104	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
105	#11745664	—	2	2	no	—
106	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	#11745660	—	2	2	no	—
109	C₂³×C₅₉	—	2	1	no	—
110	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₅₉	—	2	1	no	—
112	Q₈×C₅₉	—	2	1	no	—
113	Q₈×C₅₉	—	2	1	no	—
114	D₄×C₅₉	—	2	1	no	—
115	D₄×C₅₉	—	2	1	no	—
116	D₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	—	2	1	no	—
118	C₂×C₄×C₅₉	—	2	1	no	—
119	C₂×C₄×C₅₉	—	2	1	no	—
120	C₂×C₄×C₅₉	—	2	1	no	—
121	C₂×C₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₅₉	—	4	2	no	—
125	C₂²⋊D₅₉	—	4	2	no	—
126	C₂²⋊D₅₉	—	4	2	no	—
127	C₂×Dic₅₉	—	2	2	no	—
128	C₂×Dic₅₉	—	2	2	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	4	2	no	—
133	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
137	D₂₃₆	—	2	1	no	—
138	D₂₃₆	—	2	1	no	—
139	C₄×D₅₉	—	4	2	no	—
140	Dic₁₁₈	—	2	1	no	—
141	Dic₁₁₈	—	2	1	no	—
142	Dic₁₁₈	—	4	2	no	—
143	Dic₁₁₈	—	4	2	no	—
144	C₈×C₅₉	—	2	1	no	—
145	C₈×C₅₉	#41	1	2	yes	complemented
146	C₅₉⋊C₈	—	2	1	no	—
147	C₅₉⋊C₈	#41	1	2	yes	complemented
148	C₂²×C₅₉	—	2	1	no	—
149	C₂²×C₅₉	—	2	1	no	—
150	C₂²×C₅₉	—	2	1	no	—
151	C₂²×C₅₉	—	4	1	no	—
152	C₂²×C₅₉	—	4	1	no	—
153	C₂²×C₅₉	—	4	1	no	—
154	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₅₉	—	4	1	no	—
156	C₂×D₅₉	—	4	1	no	—
157	C₂×D₅₉	—	4	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	4	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	—	4	1	no	—
162	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
164	Dic₅₉	—	2	1	no	—
165	Dic₅₉	—	2	1	no	—
166	Dic₅₉	—	4	1	no	—
167	Dic₅₉	—	4	1	no	—
168	Dic₅₉	—	4	1	no	—
169	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
170	Dic₅₉	#137	1	2	yes	—
171	#3182	—	59	1	no	maximal, Sylow
172	C₂×C₅₉	—	4	1	no	—
173	C₂×C₅₉	—	8	1	no	—
174	C₂×C₅₉	#532	1	1	yes	characteristic
175	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
176	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
177	D₅₉	—	8	1	no	—
178	#533	—	59	1	no	—
179	#532	—	59	1	no	—
180	#500	—	59	1	no	—
181	#492	—	59	1	no	—
182	#487	—	59	1	no	—
183	#486	—	59	1	no	—
184	#468	—	59	1	no	—
185	#468	—	59	1	no	—
186	#468	—	59	1	no	—
187	#468	D₅₉	1	1	yes	characteristic, complemented
188	#440	—	59	1	no	—
189	#440	—	59	1	no	—
190	#440	—	59	1	no	—
191	#440	—	59	1	no	—
192	C₄×M₁₆	—	59	1	no	—
193	C₅₉	#3182	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	118	1	no	—
195	C₂×SD₁₆	—	59	1	no	—
196	C₂×SD₁₆	—	59	1	no	—
197	C₂×SD₁₆	—	59	1	no	—
198	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
199	C₂×M₁₆	—	59	1	no	—
200	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
201	C₄⋊Q₈	—	59	1	no	—
202	C₄⋊Q₈	—	59	1	no	—
203	C₄⋊Q₈	—	59	1	no	—
204	#127	—	59	1	no	—
205	#125	—	118	1	no	—
206	#124	—	118	1	no	—
207	C₂²⋊Q₈	—	59	1	no	—
208	C₂²⋊Q₈	—	118	1	no	—
209	C₂²⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
210	#121	—	59	1	no	—
211	#121	—	118	1	no	—
212	#121	C₂×D₅₉	1	1	yes	characteristic
213	C₄×Q₈	—	59	1	no	—
214	C₄×Q₈	—	59	1	no	—
215	C₄×Q₈	—	118	1	no	—
216	C₄×D₄	—	59	1	no	—
217	C₄×D₄	—	59	1	no	—
218	C₄×D₄	—	118	1	no	—
219	#117	—	118	1	no	—
220	C₂×C₄²	—	59	1	no	—
221	#107	—	59	1	no	—
222	#107	—	59	1	no	—
223	#107	—	59	1	no	—
224	#107	C₂×D₅₉	1	1	yes	characteristic
225	#103	—	59	1	no	—
226	#103	—	59	1	no	—
227	#103	—	59	1	no	—
228	#103	C₂×D₅₉	1	1	yes	characteristic
229	#102	—	59	1	no	—
230	#102	—	59	1	no	—
231	#102	—	59	1	no	—
232	#102	C₂×D₅₉	1	1	yes	characteristic
233	#97	—	59	1	no	—
234	#97	—	59	1	no	—
235	C₄×C₈	—	59	1	no	—
236	C₄×C₈	—	59	1	no	—
237	#41	—	236	4	no	—
238	C₂×Q₈	—	59	1	no	—
239	C₂×Q₈	—	118	1	no	—
240	C₂×Q₈	—	118	1	no	—
241	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	59	1	no	—
244	C₂×D₄	—	118	1	no	—
245	C₂×D₄	—	118	1	no	—
246	C₂×D₄	—	118	1	no	—
247	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
248	C₂²×C₄	—	59	1	no	—
249	C₂²×C₄	—	59	1	no	—
250	C₂²×C₄	—	118	1	no	—
251	C₂²×C₄	—	118	1	no	—
252	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	118	2	no	—
255	SD₁₆	—	118	2	no	—
256	SD₁₆	—	118	2	no	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	118	2	no	—
259	C₂×C₈	—	59	1	no	—
260	C₂×C₈	—	59	1	no	—
261	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	59	1	no	—
265	C₄⋊C₄	—	59	1	no	—
266	C₄⋊C₄	—	118	1	no	—
267	C₄⋊C₄	—	118	1	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	—	118	1	no	—
270	C₄⋊C₄	—	118	1	no	—
271	C₄⋊C₄	—	118	1	no	—
272	C₄⋊C₄	—	118	1	no	—
273	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₂²⋊C₄	—	118	1	no	—
279	C₂²⋊C₄	—	118	1	no	—
280	C₂²⋊C₄	—	118	1	no	—
281	C₄²	—	59	1	no	—
282	C₄²	—	59	1	no	—
283	C₄²	—	59	1	no	—
284	C₄²	—	59	1	no	—
285	C₄²	—	118	1	no	—
286	C₄²	—	118	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	118	1	no	—
289	C₂³	D₄×D₅₉	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	118	1	no	—
293	Q₈	—	118	1	no	—
294	Q₈	—	236	2	no	—
295	Q₈	—	236	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	118	1	no	—
300	D₄	—	118	1	no	—
301	D₄	—	236	2	no	—
302	D₄	—	236	2	no	—
303	D₄	—	236	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	59	1	no	—
310	C₂×C₄	—	59	1	no	—
311	C₂×C₄	—	59	1	no	—
312	C₂×C₄	—	59	1	no	—
313	C₂×C₄	—	118	2	no	—
314	C₂×C₄	—	118	2	no	—
315	C₂×C₄	—	118	1	no	—
316	C₂×C₄	—	118	1	no	—
317	C₂×C₄	—	118	1	no	—
318	C₂×C₄	—	236	2	no	—
319	C₂×C₄	—	236	2	no	—
320	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	59	2	no	—
324	C₈	—	118	1	no	—
325	C₈	#11745687	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	236	1	no	—
333	C₂²	—	236	1	no	—
334	C₂²	—	236	1	no	—
335	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	59	2	no	—
341	C₄	—	59	2	no	—
342	C₄	—	118	1	no	—
343	C₄	—	118	1	no	—
344	C₄	—	236	1	no	—
345	C₄	—	236	1	no	—
346	C₄	—	236	1	no	—
347	C₄	#41×D₅₉	1	1	yes	characteristic
348	C₄	C₂×#11745687	1	1	yes	characteristic
349	C₂	—	4	1	no	—
350	C₂	—	8	1	no	—
351	C₂	—	472	1	no	—
352	C₂	#16837947	1	1	yes	characteristic
353	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
354	C₂	#137×D₅₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#468	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#468	D₅₉
hypocenter	C₅₉	#3182
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #16837954	C₂	semidirect
3. #16837806	C₂	semidirect
4. #16837773	C₂	semidirect
5. #16837751	C₂	semidirect
6. #16837747	C₂	semidirect
7. #16837650	C₂	semidirect
8. #16837648	C₂	semidirect
9. #16837646	C₂	semidirect
10. #16837505	C₂	semidirect
11. #16837503	C₂	semidirect
12. #16837501	C₂	semidirect
13. #16837499	C₂	semidirect
14. M₁₆×Dic₅₉	C₂	semidirect
15. C₅₉×#468	C₂	semidirect
145. C₈×C₅₉	#41	semidirect
147. C₅₉⋊C₈	#41	semidirect
187. #468	D₅₉	semidirect
193. C₅₉	#3182	semidirect
325. C₈	#11745687	semidirect