Subgroups — #21436804

Order 7552.

396 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	#16838111	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838073	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837767	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837765	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837747	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837702	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837701	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837691	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837597	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837595	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837595	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837530	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837511	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837482	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#442	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745685	—	2	1	no	—
17	C₂×#11745669	C₂²	1	1	yes	characteristic
18	C₂×#11745669	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273724	C₂²	1	1	yes	characteristic
21	#14273723	C₂²	1	1	yes	characteristic
22	#14273723	C₂²	1	1	yes	characteristic
23	#14273719	C₂²	1	1	yes	characteristic
24	#14273718	C₂²	1	1	yes	characteristic
25	#14273706	—	2	1	no	—
26	#14273704	—	2	1	no	—
27	#14273704	—	2	1	no	—
28	#14273701	—	2	1	no	—
29	#14273700	—	2	1	no	—
30	#14273700	—	2	1	no	—
31	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
32	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
33	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
34	#14273689	C₂²	1	1	yes	characteristic
35	#14273689	C₂²	1	1	yes	characteristic
36	C₂×Dic₅₉⋊C₄	—	2	1	no	—
37	#14273660	C₂²	1	1	yes	characteristic
38	#14273658	C₂²	1	1	yes	characteristic
39	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
40	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
41	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
42	#14273649	C₂²	1	1	yes	characteristic
43	#14273649	C₂²	1	1	yes	characteristic
44	#14273649	C₂²	1	1	yes	characteristic
45	#14273649	C₂²	1	1	yes	characteristic
46	#14273648	C₂²	1	1	yes	characteristic
47	#14273648	C₂²	1	1	yes	characteristic
48	#14273647	C₂²	1	1	yes	characteristic
49	#14273647	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	#102×C₅₉	C₂²	1	1	yes	characteristic
57	#102×C₅₉	C₂²	1	1	yes	characteristic
58	#97×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₅₉	—	2	1	no	—
62	C₂²×C₄×C₅₉	—	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	C₂×C₂²⋊D₅₉	—	2	1	no	—
67	C₂²×Dic₅₉	—	2	1	no	—
68	C₂²×Dic₅₉	—	2	1	no	—
69	#11745685	—	4	4	no	—
70	C₂×D₂₃₆	—	2	1	no	—
71	C₂×D₂₃₆	—	2	1	no	—
72	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
73	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
74	C₂×C₄×D₅₉	—	2	1	no	—
75	C₂×C₄×D₅₉	—	2	1	no	—
76	C₂×Dic₁₁₈	—	2	1	no	—
77	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
78	D₈×C₅₉	—	2	2	no	—
79	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₅₉	—	2	1	no	—
84	C₂²⋊C₄×C₅₉	—	2	1	no	—
85	C₄²×C₅₉	C₂³	1	1	yes	characteristic
86	C₂²⋊Dic₅₉	—	2	1	no	—
87	C₂²⋊Dic₅₉	—	2	1	no	—
88	#11745669	—	2	2	no	—
89	#11745669	—	2	2	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	C₄⋊Dic₅₉	—	2	1	no	—
93	C₄⋊Dic₅₉	D₄	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	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
101	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
103	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₅₉	—	2	1	no	—
107	C₂³×C₅₉	—	2	1	no	—
108	C₂²×D₅₉	—	2	1	no	—
109	C₂²×D₅₉	—	2	1	no	—
110	D₄×C₅₉	—	2	1	no	—
111	D₄×C₅₉	—	2	1	no	—
112	D₄×C₅₉	—	2	1	no	—
113	D₄×C₅₉	#41	1	2	yes	complemented
114	C₂×C₄×C₅₉	—	2	1	no	—
115	C₂×C₄×C₅₉	—	2	1	no	—
116	C₂×C₄×C₅₉	—	4	2	no	—
117	C₂×C₄×C₅₉	—	8	1	no	—
118	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₅₉	—	4	2	no	—
122	C₂²⋊D₅₉	—	4	2	no	—
123	C₂²⋊D₅₉	—	8	1	no	—
124	C₂²⋊D₅₉	—	8	1	no	—
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	4	2	no	—
128	C₂×Dic₅₉	—	8	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	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
133	D₂₃₆	—	2	1	no	—
134	D₂₃₆	—	2	1	no	—
135	D₂₃₆	—	2	1	no	—
136	D₂₃₆	—	2	1	no	—
137	D₂₃₆	—	4	2	no	—
138	D₂₃₆	—	8	1	no	—
139	C₄×D₅₉	—	4	2	no	—
140	C₄×D₅₉	—	4	2	no	—
141	Dic₁₁₈	—	2	2	no	—
142	Dic₁₁₈	—	8	1	no	—
143	C₈×C₅₉	—	2	1	no	—
144	C₈×C₅₉	—	2	1	no	—
145	C₅₉⋊C₈	—	2	1	no	—
146	C₅₉⋊C₈	—	2	1	no	—
147	C₂²×C₅₉	—	2	2	no	—
148	C₂²×C₅₉	—	4	1	no	—
149	C₂²×C₅₉	—	4	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	—	4	1	no	—
155	C₂×D₅₉	—	4	1	no	—
156	C₂×D₅₉	—	4	1	no	—
157	C₂×D₅₉	—	4	1	no	—
158	C₂×D₅₉	—	4	1	no	—
159	C₂×D₅₉	—	4	1	no	—
160	C₄×C₅₉	—	2	1	no	—
161	C₄×C₅₉	—	2	1	no	—
162	C₄×C₅₉	—	4	1	no	—
163	C₄×C₅₉	—	4	1	no	—
164	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
165	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
166	Dic₅₉	—	2	1	no	—
167	Dic₅₉	—	2	1	no	—
168	Dic₅₉	—	2	1	no	—
169	Dic₅₉	—	4	1	no	—
170	Dic₅₉	—	4	1	no	—
171	Dic₅₉	AGL(1,ℤ/8)	1	2	yes	—
172	#3366	—	59	1	no	maximal, Sylow
173	C₂×C₅₉	—	4	2	no	—
174	C₂×C₅₉	—	8	1	no	—
175	C₂×C₅₉	#548	1	1	yes	characteristic
176	C₂×C₅₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
177	C₂×C₅₉	#576	1	1	yes	characteristic
178	D₅₉	—	8	1	no	—
179	D₅₉	—	8	1	no	—
180	#554	—	59	1	no	—
181	#550	—	59	1	no	—
182	#490	—	59	1	no	—
183	#490	—	59	1	no	—
184	#486	—	59	1	no	—
185	#478	—	59	1	no	—
186	#478	—	59	1	no	—
187	#476	—	59	1	no	—
188	#459	—	59	1	no	—
189	#459	—	59	1	no	—
190	#459	—	59	1	no	—
191	#446	—	59	1	no	—
192	#442	—	59	1	no	—
193	#442	D₅₉	1	1	yes	characteristic, complemented
194	C₄×D₈	—	59	1	no	—
195	C₅₉	#3366	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	—	118	1	no	—
197	C₂×D₈	—	59	1	no	—
198	C₂×D₈	—	59	1	no	—
199	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
200	C₄⋊Q₈	—	59	1	no	—
201	#127	—	59	1	no	—
202	#127	—	59	1	no	—
203	#127	—	59	1	no	—
204	#125	—	59	1	no	—
205	#125	—	59	1	no	—
206	C₂²⋊Q₈	—	118	1	no	—
207	C₂²⋊Q₈	—	118	1	no	—
208	C₂²⋊Q₈	—	118	1	no	—
209	#121	—	118	1	no	—
210	#121	—	118	1	no	—
211	#121	—	118	1	no	—
212	C₄×Q₈	—	59	1	no	—
213	C₄×D₄	—	59	1	no	—
214	C₄×D₄	—	59	1	no	—
215	C₄×D₄	—	59	1	no	—
216	C₄×D₄	—	59	1	no	—
217	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
218	C₄×D₄	C₂×D₅₉	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	118	1	no	—
220	#107	—	59	1	no	—
221	#106	—	59	1	no	—
222	#106	C₂×D₅₉	1	1	yes	characteristic
223	C₄⋊C₈	—	59	1	no	—
224	C₄⋊C₈	—	59	1	no	—
225	C₄⋊C₈	—	59	1	no	—
226	#102	—	59	1	no	—
227	#102	—	59	1	no	—
228	#102	—	59	1	no	—
229	#102	—	59	1	no	—
230	#102	—	59	1	no	—
231	#102	—	59	1	no	—
232	#102	—	59	1	no	—
233	#102	—	59	1	no	—
234	#102	C₂×D₅₉	1	1	yes	characteristic
235	#102	C₂×D₅₉	1	1	yes	characteristic
236	#97	—	59	1	no	—
237	#97	C₂×D₅₉	1	1	yes	characteristic
238	C₄×C₈	—	59	1	no	—
239	#41	—	236	4	no	—
240	C₂×Q₈	—	59	1	no	—
241	C₂×Q₈	—	118	1	no	—
242	C₂×D₄	—	59	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	—	118	1	no	—
248	C₂×D₄	—	118	1	no	—
249	C₂×D₄	—	118	1	no	—
250	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
251	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	2	1	no	—
254	C₂²×C₄	—	118	1	no	—
255	C₂²×C₄	—	118	1	no	—
256	C₂²×C₄	—	118	1	no	—
257	C₂²×C₄	—	118	1	no	—
258	D₈	—	2	2	no	—
259	D₈	—	118	2	no	—
260	D₈	—	118	2	no	—
261	C₂×C₈	—	59	1	no	—
262	C₂×C₈	—	59	1	no	—
263	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
265	C₄⋊C₄	—	59	1	no	—
266	C₄⋊C₄	—	59	1	no	—
267	C₄⋊C₄	—	59	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	—	2	1	no	—
279	C₂²⋊C₄	—	118	1	no	—
280	C₂²⋊C₄	—	118	1	no	—
281	C₂²⋊C₄	—	118	1	no	—
282	C₂²⋊C₄	—	118	1	no	—
283	C₄²	—	59	1	no	—
284	C₄²	—	59	1	no	—
285	C₄²	C₂²×D₅₉	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	118	1	no	—
289	C₂³	—	118	1	no	—
290	Q₈	—	118	2	no	—
291	Q₈	—	472	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	118	1	no	—
296	D₄	—	118	1	no	—
297	D₄	—	118	1	no	—
298	D₄	—	118	1	no	—
299	D₄	—	472	1	no	—
300	D₄	—	472	1	no	—
301	D₄	—	472	1	no	—
302	D₄	—	236	2	no	—
303	D₄	—	236	2	no	—
304	D₄	—	236	2	no	—
305	D₄	#11745685	1	2	yes	complemented
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	2	no	—
309	C₂×C₄	—	8	1	no	—
310	C₂×C₄	—	59	1	no	—
311	C₂×C₄	—	59	1	no	—
312	C₂×C₄	—	59	1	no	—
313	C₂×C₄	—	59	1	no	—
314	C₂×C₄	—	118	1	no	—
315	C₂×C₄	—	118	1	no	—
316	C₂×C₄	—	472	1	no	—
317	C₂×C₄	—	236	2	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	D₄×D₅₉	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	118	1	no	—
326	C₈	—	118	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	236	1	no	—
334	C₂²	—	236	1	no	—
335	C₂²	—	236	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	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	59	2	no	—
345	C₄	—	118	1	no	—
346	C₄	—	118	1	no	—
347	C₄	—	118	1	no	—
348	C₄	—	236	1	no	—
349	C₄	—	236	1	no	—
350	C₄	C₂×#11745685	1	1	yes	characteristic
351	C₄	#14273774	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	8	1	no	—
354	C₂	—	472	1	no	—
355	C₂	—	472	1	no	—
356	C₂	#16838061	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₅₉	1	1	yes	characteristic
358	C₂	#16838206	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#442	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#442	D₅₉
hypocenter	C₅₉	#3366
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #16838111	C₂	semidirect
2. #16838073	C₂	semidirect
3. #16837767	C₂	semidirect
4. #16837765	C₂	semidirect
5. #16837747	C₂	semidirect
6. #16837702	C₂	semidirect
7. #16837701	C₂	semidirect
8. #16837691	C₂	semidirect
9. #16837597	C₂	semidirect
10. #16837595	C₂	semidirect
11. #16837595	C₂	semidirect
12. #16837530	C₂	semidirect
13. #16837511	C₂	semidirect
14. #16837482	C₂	semidirect
15. C₅₉×#442	C₂	semidirect
113. D₄×C₅₉	#41	semidirect
193. #442	D₅₉	semidirect
195. C₅₉	#3366	semidirect
305. D₄	#11745685	semidirect