Subgroups — #21434216

Order 7552.

424 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837949	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₅₉	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837717	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837714	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837696	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837601	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837600	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837542	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837539	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837504	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837502	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837500	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837457	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#440	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745685	—	2	1	no	—
17	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
18	C₂×#11745660	C₂²	1	1	yes	characteristic
19	C₂×#11745670	C₂²	1	1	yes	characteristic
20	C₂×#11745671	C₂²	1	1	yes	characteristic
21	#14273726	C₂²	1	1	yes	characteristic
22	#14273723	C₂²	1	1	yes	characteristic
23	#14273713	—	2	1	no	—
24	#14273702	C₂²	1	1	yes	characteristic
25	#14273700	—	2	1	no	—
26	#14273698	C₂²	1	1	yes	characteristic
27	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₁₈	—	2	1	no	—
29	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
30	#14273692	—	2	1	no	—
31	#14273692	C₂²	1	1	yes	characteristic
32	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₅₉	—	2	1	no	—
34	C₄⋊C₄×D₅₉	—	2	1	no	—
35	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
36	C₄²×D₅₉	—	2	1	no	—
37	C₄²×D₅₉	C₂²	1	1	yes	characteristic
38	#14273661	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273660	C₂²	1	1	yes	characteristic
41	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273650	C₂²	1	1	yes	characteristic
46	#14273649	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273647	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	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	#97×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	Q₈×D₅₉	—	2	2	no	—
66	Q₈×D₅₉	—	2	2	no	—
67	#11745685	—	4	4	no	—
68	C₂×D₂₃₆	—	2	1	no	—
69	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₅₉	—	2	1	no	—
71	C₂×C₄×D₅₉	—	2	1	no	—
72	C₂×C₄×D₅₉	—	2	1	no	—
73	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
74	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
75	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
76	C₂×Dic₁₁₈	—	2	1	no	—
77	C₂×Dic₁₁₈	—	2	1	no	—
78	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
79	SD₁₆×C₅₉	—	2	2	no	—
80	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	—	2	1	no	—
83	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₅₉	—	2	1	no	—
86	C₄²×C₅₉	—	2	1	no	—
87	C₄²×C₅₉	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₅₉	—	2	1	no	—
89	#11745671	—	2	2	no	—
90	#11745670	—	2	2	no	—
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	(C₂×D₅₉)⋊C₄	—	2	1	no	—
93	C₄⋊Dic₅₉	—	2	1	no	—
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	D₄	1	1	yes	characteristic
104	C₄×Dic₅₉	D₄	1	1	yes	characteristic
105	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
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	C₂²×D₅₉	—	2	1	no	—
111	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₅₉	—	2	1	no	—
113	Q₈×C₅₉	#41	1	2	yes	complemented
114	D₄×C₅₉	—	2	1	no	—
115	D₄×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	—	4	2	no	—
120	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₅₉	—	4	2	no	—
124	C₂²⋊D₅₉	—	4	2	no	—
125	C₂²⋊D₅₉	—	4	2	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	—	2	1	no	—
129	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
132	D₂₃₆	—	2	1	no	—
133	D₂₃₆	—	2	1	no	—
134	D₂₃₆	—	4	2	no	—
135	C₄×D₅₉	—	2	4	no	—
136	C₄×D₅₉	—	2	2	no	—
137	C₄×D₅₉	—	2	1	no	—
138	C₄×D₅₉	—	2	1	no	—
139	C₄×D₅₉	—	2	2	no	—
140	C₄×D₅₉	—	4	2	no	—
141	C₄×D₅₉	—	4	2	no	—
142	Dic₁₁₈	—	2	4	no	—
143	Dic₁₁₈	—	2	1	no	—
144	Dic₁₁₈	—	4	2	no	—
145	Dic₁₁₈	#41	1	2	yes	complemented
146	C₈×C₅₉	—	2	1	no	—
147	C₈×C₅₉	—	2	1	no	—
148	C₅₉⋊C₈	—	2	1	no	—
149	C₅₉⋊C₈	—	2	1	no	—
150	C₂²×C₅₉	—	4	1	no	—
151	C₂²×C₅₉	—	4	1	no	—
152	C₂²×C₅₉	—	4	1	no	—
153	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₅₉	—	2	1	no	—
155	C₂×D₅₉	—	2	1	no	—
156	C₂×D₅₉	—	4	1	no	—
157	C₂×D₅₉	—	4	1	no	—
158	C₂×D₅₉	—	4	1	no	—
159	C₂×D₅₉	AGL(1,ℤ/8)	1	2	yes	—
160	C₄×C₅₉	—	2	2	no	—
161	C₄×C₅₉	—	2	1	no	—
162	C₄×C₅₉	—	2	1	no	—
163	C₄×C₅₉	—	4	1	no	—
164	C₄×C₅₉	—	4	1	no	—
165	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
166	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
167	Dic₅₉	—	2	1	no	—
168	Dic₅₉	—	2	2	no	—
169	Dic₅₉	—	2	1	no	—
170	Dic₅₉	—	4	1	no	—
171	Dic₅₉	—	4	1	no	—
172	Dic₅₉	#137	1	2	yes	—
173	#3112	—	59	1	no	maximal, Sylow
174	C₂×C₅₉	—	8	1	no	—
175	C₂×C₅₉	#525	1	1	yes	characteristic
176	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
178	D₅₉	—	2	2	no	—
179	D₅₉	—	8	1	no	—
180	#532	—	59	1	no	—
181	C₂×C₄×Q₈	—	59	1	no	—
182	#480	—	59	1	no	—
183	#480	—	59	1	no	—
184	#477	—	59	1	no	—
185	#477	—	59	1	no	—
186	#460	—	59	1	no	—
187	#460	—	59	1	no	—
188	#448	—	59	1	no	—
189	#448	—	59	1	no	—
190	#440	—	59	1	no	—
191	#440	—	59	1	no	—
192	#440	—	59	1	no	—
193	#440	D₅₉	1	1	yes	characteristic, complemented
194	#432	—	59	1	no	—
195	C₅₉	#3112	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	—	118	1	no	—
197	C₂²×Q₈	—	59	1	no	—
198	C₂×SD₁₆	—	59	1	no	—
199	C₂×SD₁₆	—	59	1	no	—
200	C₂×SD₁₆	—	59	1	no	—
201	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
202	C₄⋊Q₈	—	59	1	no	—
203	#127	—	59	1	no	—
204	#124	—	118	1	no	—
205	#121	—	59	1	no	—
206	#121	—	59	1	no	—
207	#121	—	118	1	no	—
208	C₄×Q₈	—	59	1	no	—
209	C₄×Q₈	—	59	1	no	—
210	C₄×Q₈	—	59	1	no	—
211	C₄×Q₈	—	118	1	no	—
212	C₄×Q₈	—	118	1	no	—
213	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	118	1	no	—
216	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	59	1	no	—
218	C₂×C₄⋊C₄	—	118	1	no	—
219	C₂×C₄²	—	59	1	no	—
220	C₂×C₄²	—	118	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	C₄⋊C₈	—	59	1	no	—
226	C₄⋊C₈	—	59	1	no	—
227	#103	—	59	1	no	—
228	#103	—	59	1	no	—
229	#103	—	59	1	no	—
230	#103	C₂×D₅₉	1	1	yes	characteristic
231	#102	—	59	1	no	—
232	#102	—	59	1	no	—
233	#102	—	59	1	no	—
234	#102	C₂×D₅₉	1	1	yes	characteristic
235	C₂²⋊C₈	—	59	1	no	—
236	C₂²⋊C₈	—	59	1	no	—
237	#97	—	59	1	no	—
238	#97	C₂×D₅₉	1	1	yes	characteristic
239	#41	—	236	4	no	—
240	C₂×Q₈	—	59	1	no	—
241	C₂×Q₈	—	118	1	no	—
242	C₂×Q₈	—	118	1	no	—
243	C₂×Q₈	—	118	2	no	—
244	C₂×Q₈	—	118	2	no	—
245	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
246	C₂×D₄	—	59	1	no	—
247	C₂×D₄	—	118	1	no	—
248	C₂×D₄	—	118	1	no	—
249	C₂×D₄	—	118	1	no	—
250	C₂×D₄	—	118	1	no	—
251	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	59	1	no	—
254	C₂²×C₄	—	59	1	no	—
255	C₂²×C₄	—	59	1	no	—
256	C₂²×C₄	—	118	1	no	—
257	C₂²×C₄	—	118	1	no	—
258	C₂²×C₄	—	118	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	118	2	no	—
261	SD₁₆	—	118	2	no	—
262	SD₁₆	—	118	2	no	—
263	C₂×C₈	—	59	1	no	—
264	C₂×C₈	—	59	1	no	—
265	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	59	1	no	—
269	C₄⋊C₄	—	59	1	no	—
270	C₄⋊C₄	—	118	1	no	—
271	C₄⋊C₄	—	118	1	no	—
272	C₄⋊C₄	—	118	1	no	—
273	C₄⋊C₄	—	118	1	no	—
274	C₄⋊C₄	—	118	1	no	—
275	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	118	1	no	—
279	C₂²⋊C₄	—	118	1	no	—
280	C₂²⋊C₄	—	118	1	no	—
281	C₄²	—	2	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	C₄²	—	118	1	no	—
288	C₄²	C₂²×D₅₉	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	59	1	no	—
291	C₂³	—	118	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	59	2	no	—
294	Q₈	—	118	4	no	—
295	Q₈	—	118	1	no	—
296	Q₈	—	236	2	no	—
297	Q₈	#11745685	1	2	yes	complemented
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	118	1	no	—
301	D₄	—	118	1	no	—
302	D₄	—	236	2	no	—
303	D₄	—	236	2	no	—
304	D₄	—	236	2	no	—
305	D₄	—	236	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	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	4	no	—
316	C₂×C₄	—	118	1	no	—
317	C₂×C₄	—	118	1	no	—
318	C₂×C₄	—	118	1	no	—
319	C₂×C₄	—	118	1	no	—
320	C₂×C₄	—	118	1	no	—
321	C₂×C₄	—	236	2	no	—
322	C₂×C₄	—	236	2	no	—
323	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
324	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	118	1	no	—
329	C₈	—	118	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	59	2	no	—
334	C₂²	—	118	1	no	—
335	C₂²	—	118	1	no	—
336	C₂²	—	236	1	no	—
337	C₂²	—	236	1	no	—
338	C₂²	—	236	1	no	—
339	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	2	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	4	1	no	—
345	C₄	—	59	2	no	—
346	C₄	—	118	1	no	—
347	C₄	—	118	2	no	—
348	C₄	—	118	1	no	—
349	C₄	—	236	1	no	—
350	C₄	—	236	1	no	—
351	C₄	#41×D₅₉	1	1	yes	characteristic
352	C₄	C₂×#11745685	1	1	yes	characteristic
353	C₂	—	8	1	no	—
354	C₂	—	118	2	no	—
355	C₂	—	472	1	no	—
356	C₂	#16837911	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
358	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×#440	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#440	D₅₉
hypocenter	C₅₉	#3112
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₅₉	C₂	semidirect
3. #16837717	C₂	semidirect
4. #16837714	C₂	semidirect
5. #16837696	C₂	semidirect
6. #16837695	C₂	semidirect
7. #16837601	C₂	semidirect
8. #16837600	C₂	semidirect
9. #16837542	C₂	semidirect
10. #16837539	C₂	semidirect
11. #16837504	C₂	semidirect
12. #16837502	C₂	semidirect
13. #16837500	C₂	semidirect
14. #16837457	C₂	semidirect
15. C₅₉×#440	C₂	semidirect
113. Q₈×C₅₉	#41	semidirect
145. Dic₁₁₈	#41	semidirect
193. #440	D₅₉	semidirect
195. C₅₉	#3112	semidirect
297. Q₈	#11745685	semidirect