Subgroups — #21436224

Order 7552.

392 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838092	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838073	C₂	1	1	yes	maximal, characteristic
3	#16837767	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837727	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837721	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837648	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837631	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837603	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837566	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837566	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837510	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837404	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837399	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#435	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745685	C₂²	1	1	yes	characteristic
17	C₂×#11745685	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	#14273724	C₂²	1	1	yes	characteristic
22	#14273723	—	2	1	no	—
23	#14273720	—	2	1	no	—
24	#14273719	C₂²	1	1	yes	characteristic
25	#14273717	—	2	1	no	—
26	#14273708	—	2	1	no	—
27	#14273706	C₂²	1	1	yes	characteristic
28	#14273706	C₂²	1	1	yes	characteristic
29	#14273701	C₂²	1	1	yes	characteristic
30	#14273701	C₂²	1	1	yes	characteristic
31	#14273700	—	2	1	no	—
32	#14273700	—	2	1	no	—
33	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
35	#14273689	—	2	1	no	—
36	#14273686	C₂²	1	1	yes	characteristic
37	#14273683	—	2	1	no	—
38	C₂×C₄⋊Dic₅₉	C₂²	1	1	yes	characteristic
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	#14273649	C₂²	1	1	yes	characteristic
47	#14273649	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	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
58	#97×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₅₉	D₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₉	—	2	1	no	—
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	#11745685	—	2	2	no	—
68	#11745685	—	2	2	no	—
69	#11745685	—	4	1	no	—
70	#11745685	—	4	1	no	—
71	C₂×D₂₃₆	—	2	1	no	—
72	C₂×D₂₃₆	—	2	1	no	—
73	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
74	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₅₉	—	2	1	no	—
76	C₂×C₄×D₅₉	—	2	1	no	—
77	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
78	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
79	M₁₆×C₅₉	—	4	1	no	—
80	C₂×C₈×C₅₉	—	4	1	no	—
81	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
82	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
84	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
85	C₄⋊C₄×C₅₉	D₄	1	1	yes	characteristic
86	C₂²⋊C₄×C₅₉	D₄	1	1	yes	characteristic
87	C₂²⋊C₄×C₅₉	D₄	1	1	yes	characteristic
88	C₄²×C₅₉	C₂³	1	1	yes	characteristic
89	C₂²⋊Dic₅₉	—	2	1	no	—
90	C₂²⋊Dic₅₉	—	2	1	no	—
91	C₂²⋊Dic₅₉	—	2	1	no	—
92	(C₂×D₅₉)⋊C₄	—	2	1	no	—
93	(C₂×D₅₉)⋊C₄	—	2	1	no	—
94	C₄⋊Dic₅₉	—	2	1	no	—
95	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	—	2	1	no	—
102	Dic₅₉⋊C₄	—	2	1	no	—
103	Dic₅₉⋊C₄	—	2	1	no	—
104	Dic₅₉⋊C₄	—	2	1	no	—
105	C₄×Dic₅₉	—	2	1	no	—
106	C₄×Dic₅₉	—	2	1	no	—
107	C₄×Dic₅₉	—	2	1	no	—
108	Dic₂₃₆	—	2	2	no	—
109	#11745660	—	4	1	no	—
110	#11745660	—	4	1	no	—
111	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
112	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
113	C₂²×D₅₉	—	2	1	no	—
114	C₂²×D₅₉	—	2	1	no	—
115	D₄×C₅₉	—	4	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	—	2	1	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	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
125	C₂²⋊D₅₉	—	4	2	no	—
126	C₂²⋊D₅₉	—	4	2	no	—
127	C₂²⋊D₅₉	—	8	1	no	—
128	C₂²⋊D₅₉	—	8	1	no	—
129	C₂×Dic₅₉	—	2	1	no	—
130	C₂×Dic₅₉	—	2	1	no	—
131	C₂×Dic₅₉	—	2	1	no	—
132	C₂×Dic₅₉	—	2	1	no	—
133	C₂×Dic₅₉	—	2	1	no	—
134	C₂×Dic₅₉	—	2	1	no	—
135	C₂×Dic₅₉	—	4	2	no	—
136	D₂₃₆	—	2	1	no	—
137	D₂₃₆	—	2	1	no	—
138	D₂₃₆	—	2	1	no	—
139	D₂₃₆	—	2	1	no	—
140	D₂₃₆	—	4	2	no	—
141	D₂₃₆	—	8	1	no	—
142	C₄×D₅₉	—	4	2	no	—
143	C₄×D₅₉	—	4	2	no	—
144	Dic₁₁₈	—	2	1	no	—
145	Dic₁₁₈	—	2	1	no	—
146	Dic₁₁₈	—	2	1	no	—
147	Dic₁₁₈	#41	1	2	yes	—
148	C₈×C₅₉	—	2	1	no	—
149	C₈×C₅₉	—	2	1	no	—
150	C₈×C₅₉	—	2	1	no	—
151	C₈×C₅₉	—	2	1	no	—
152	C₂²×C₅₉	—	2	1	no	—
153	C₂²×C₅₉	—	2	1	no	—
154	C₂²×C₅₉	—	2	1	no	—
155	C₂²×C₅₉	—	2	1	no	—
156	C₂²×C₅₉	—	2	1	no	—
157	C₂²×C₅₉	—	2	1	no	—
158	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
159	C₂×D₅₉	—	4	1	no	—
160	C₂×D₅₉	—	4	1	no	—
161	C₂×D₅₉	—	4	1	no	—
162	C₂×D₅₉	—	4	1	no	—
163	C₂×D₅₉	—	4	1	no	—
164	C₂×D₅₉	—	4	1	no	—
165	C₄×C₅₉	—	2	1	no	—
166	C₄×C₅₉	—	2	1	no	—
167	C₄×C₅₉	—	2	1	no	—
168	C₄×C₅₉	—	2	1	no	—
169	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
170	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
171	Dic₅₉	—	2	2	no	—
172	Dic₅₉	—	4	1	no	—
173	Dic₅₉	—	4	1	no	—
174	Dic₅₉	—	4	1	no	—
175	Dic₅₉	—	4	1	no	—
176	Dic₅₉	—	4	1	no	—
177	#3317	—	59	1	no	maximal, Sylow
178	C₂×C₅₉	—	4	1	no	—
179	C₂×C₅₉	—	4	1	no	—
180	C₂×C₅₉	#546	1	1	yes	characteristic
181	C₂×C₅₉	#575	1	1	yes	characteristic
182	C₂×C₅₉	#577	1	1	yes	characteristic
183	D₅₉	—	8	1	no	—
184	D₅₉	—	8	1	no	—
185	#552	—	59	1	no	—
186	#550	—	59	1	no	—
187	#490	—	59	1	no	—
188	#482	—	59	1	no	—
189	#481	—	59	1	no	—
190	#479	—	59	1	no	—
191	#468	—	59	1	no	—
192	#465	—	59	1	no	—
193	#460	—	59	1	no	—
194	#452	—	59	1	no	—
195	#452	—	59	1	no	—
196	#441	—	59	1	no	—
197	#435	D₅₉	1	1	yes	characteristic, complemented
198	#417	—	59	1	no	—
199	#416	—	59	1	no	—
200	C₅₉	#3317	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	59	1	no	—
202	C₂×#41	—	59	1	no	—
203	C₂×Q₁₆	—	59	1	no	—
204	C₂×SD₁₆	—	59	1	no	—
205	C₂×SD₁₆	—	59	1	no	—
206	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
207	C₂²×C₈	C₂×D₅₉	1	1	yes	characteristic
208	#127	—	59	1	no	—
209	#127	—	118	1	no	—
210	#126	—	118	1	no	—
211	#125	—	59	1	no	—
212	#125	—	118	1	no	—
213	#123	—	118	1	no	—
214	C₂²⋊Q₈	—	59	1	no	—
215	C₂²⋊Q₈	—	59	1	no	—
216	#121	—	59	1	no	—
217	#121	—	59	1	no	—
218	#121	—	118	1	no	—
219	#121	—	118	1	no	—
220	C₄×Q₈	—	59	1	no	—
221	C₄×Q₈	—	59	1	no	—
222	C₄×D₄	—	118	1	no	—
223	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
224	#117	—	59	1	no	—
225	#117	—	118	1	no	—
226	C₂×C₄⋊C₄	—	59	1	no	—
227	#107	—	59	1	no	—
228	#106	—	59	1	no	—
229	#106	—	59	1	no	—
230	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
231	#103	—	59	1	no	—
232	#103	—	59	1	no	—
233	#103	—	59	1	no	—
234	#103	—	59	1	no	—
235	#102	—	59	1	no	—
236	#102	—	59	1	no	—
237	#102	—	59	1	no	—
238	#102	—	59	1	no	—
239	#102	—	59	1	no	—
240	#102	—	59	1	no	—
241	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
242	C₂²⋊C₈	C₂×D₅₉	1	1	yes	characteristic
243	#97	C₂×D₅₉	1	1	yes	characteristic
244	#41	—	118	2	no	—
245	#41	—	118	2	no	—
246	#41	—	236	1	no	—
247	#41	—	236	1	no	—
248	C₂×Q₈	—	59	1	no	—
249	C₂×Q₈	—	59	1	no	—
250	C₂×D₄	—	59	1	no	—
251	C₂×D₄	—	59	1	no	—
252	C₂×D₄	—	118	1	no	—
253	C₂×D₄	—	118	1	no	—
254	C₂×D₄	—	118	1	no	—
255	C₂×D₄	—	118	1	no	—
256	C₂×D₄	—	118	1	no	—
257	C₂×D₄	—	118	1	no	—
258	C₂×D₄	D₂₃₆	1	1	yes	characteristic
259	C₂²×C₄	—	118	1	no	—
260	C₂²×C₄	—	118	1	no	—
261	C₂²×C₄	—	118	1	no	—
262	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
263	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
264	Q₁₆	—	118	2	no	—
265	SD₁₆	—	236	1	no	—
266	SD₁₆	—	236	1	no	—
267	M₁₆	—	4	1	no	—
268	C₂×C₈	—	4	1	no	—
269	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
270	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
273	C₄⋊C₄	—	59	1	no	—
274	C₄⋊C₄	—	59	1	no	—
275	C₄⋊C₄	—	59	1	no	—
276	C₄⋊C₄	—	59	1	no	—
277	C₄⋊C₄	—	118	1	no	—
278	C₄⋊C₄	—	118	1	no	—
279	C₄⋊C₄	—	118	1	no	—
280	C₄⋊C₄	—	118	1	no	—
281	C₄⋊C₄	—	118	1	no	—
282	C₄⋊C₄	—	118	1	no	—
283	C₄⋊C₄	—	118	1	no	—
284	C₄⋊C₄	D₂₃₆	1	1	yes	characteristic
285	C₂²⋊C₄	—	118	1	no	—
286	C₂²⋊C₄	—	118	1	no	—
287	C₂²⋊C₄	—	118	1	no	—
288	C₂²⋊C₄	—	118	1	no	—
289	C₂²⋊C₄	—	118	1	no	—
290	C₂²⋊C₄	D₂₃₆	1	1	yes	characteristic
291	C₂²⋊C₄	D₂₃₆	1	1	yes	characteristic
292	C₄²	—	118	1	no	—
293	C₄²	—	118	1	no	—
294	C₄²	—	118	1	no	—
295	C₄²	C₂²×D₅₉	1	1	yes	characteristic
296	C₂³	—	118	1	no	—
297	C₂³	—	118	1	no	—
298	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
299	C₂³	C₂×D₂₃₆	1	1	yes	characteristic
300	Q₈	—	59	2	no	—
301	Q₈	—	118	1	no	—
302	Q₈	—	118	1	no	—
303	Q₈	—	118	1	no	—
304	D₄	—	4	1	no	—
305	D₄	—	118	1	no	—
306	D₄	—	118	1	no	—
307	D₄	—	118	1	no	—
308	D₄	—	118	1	no	—
309	D₄	—	472	1	no	—
310	D₄	—	472	1	no	—
311	D₄	—	472	1	no	—
312	D₄	—	236	2	no	—
313	D₄	—	236	2	no	—
314	D₄	—	236	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	118	1	no	—
320	C₂×C₄	—	118	1	no	—
321	C₂×C₄	—	118	1	no	—
322	C₂×C₄	—	118	1	no	—
323	C₂×C₄	—	118	1	no	—
324	C₂×C₄	—	118	1	no	—
325	C₂×C₄	—	236	2	no	—
326	C₂×C₄	—	236	2	no	—
327	C₂×C₄	—	236	2	no	—
328	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
329	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
330	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
331	C₂×C₄	C₂×D₂₃₆	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	2	1	no	—
336	C₈	—	2	1	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	—	2	1	no	—
342	C₂²	—	2	1	no	—
343	C₂²	—	236	1	no	—
344	C₂²	—	236	1	no	—
345	C₂²	—	236	1	no	—
346	C₂²	—	236	1	no	—
347	C₂²	—	236	1	no	—
348	C₂²	—	236	1	no	—
349	C₂²	C₂²×D₂₃₆	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	2	1	no	—
353	C₄	—	2	1	no	—
354	C₄	—	118	2	no	—
355	C₄	—	236	1	no	—
356	C₄	—	236	1	no	—
357	C₄	—	236	1	no	—
358	C₄	—	236	1	no	—
359	C₄	—	236	1	no	—
360	C₄	#41×D₅₉	1	1	yes	characteristic
361	C₄	#14273772	1	1	yes	characteristic
362	C₂	—	4	1	no	—
363	C₂	—	4	1	no	—
364	C₂	—	472	1	no	—
365	C₂	—	472	1	no	—
366	C₂	#16838047	1	1	yes	characteristic
367	C₂	#16838202	1	1	yes	characteristic
368	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×#435	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#435	D₅₉
hypocenter	C₅₉	#3317
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ #435

Complements

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

N	H ≅ G/N	type
1. #16838092	C₂	semidirect
3. #16837767	C₂	semidirect
4. #16837727	C₂	semidirect
5. #16837721	C₂	semidirect
6. #16837710	C₂	semidirect
7. #16837648	C₂	semidirect
8. #16837631	C₂	semidirect
9. #16837603	C₂	semidirect
10. #16837566	C₂	semidirect
11. #16837566	C₂	semidirect
12. #16837510	C₂	semidirect
13. #16837404	C₂	semidirect
14. #16837399	C₂	semidirect
15. C₅₉×#435	C₂	semidirect
197. #435	D₅₉	semidirect
200. C₅₉	#3317	semidirect