Subgroups — #21436707

Order 7552.

342 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838120	C₂	1	1	yes	maximal, characteristic
2	#16838107	C₂	1	1	yes	maximal, characteristic
3	#16837799	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837745	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837743	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837734	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837732	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837696	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837686	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837661	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837660	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837496	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837478	C₂	1	1	yes	maximal, characteristic
14	#16837443	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#428	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
17	C₂×#11745659	C₂²	1	1	yes	characteristic
18	#14273726	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273721	—	2	1	no	—
21	#14273720	—	2	1	no	—
22	#14273717	—	2	1	no	—
23	#14273713	—	2	1	no	—
24	#14273707	C₂²	1	1	yes	characteristic
25	#14273707	C₂²	1	1	yes	characteristic
26	#14273706	—	2	1	no	—
27	#14273706	C₂²	1	1	yes	characteristic
28	#14273706	C₂²	1	1	yes	characteristic
29	#14273705	—	2	1	no	—
30	C₄×Dic₁₁₈	—	2	1	no	—
31	#14273692	C₂²	1	1	yes	characteristic
32	#14273692	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₅₉	—	2	1	no	—
34	C₄×C₂²⋊D₅₉	C₂²	1	1	yes	characteristic
35	#14273685	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
37	#14273660	C₂²	1	1	yes	characteristic
38	#14273659	C₂²	1	1	yes	characteristic
39	#14273658	C₂²	1	1	yes	characteristic
40	#14273658	C₂²	1	1	yes	characteristic
41	#14273658	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273650	C₂²	1	1	yes	characteristic
46	#14273650	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	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	#117×C₅₉	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₅₉	C₂²	1	1	yes	characteristic
55	#107×C₅₉	C₂²	1	1	yes	characteristic
56	#107×C₅₉	C₂²	1	1	yes	characteristic
57	#106×C₅₉	C₂²	1	1	yes	characteristic
58	#106×C₅₉	C₂²	1	1	yes	characteristic
59	C₂²×C₄×C₅₉	—	2	1	no	—
60	C₂²×C₄×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₅₉	D₄	1	1	yes	characteristic
62	C₂×C₄×D₅₉	—	2	1	no	—
63	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
64	C₂×Dic₁₁₈	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
68	M₁₆×C₅₉	—	2	2	no	—
69	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
70	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
71	C₄⋊C₄×C₅₉	—	2	1	no	—
72	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
75	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
76	C₂²⋊C₄×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	—	2	1	no	—
78	C₂²⋊Dic₅₉	—	2	1	no	—
79	C₂²⋊Dic₅₉	—	2	1	no	—
80	C₂²⋊Dic₅₉	D₄	1	1	yes	characteristic
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	D₄	1	1	yes	characteristic
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
88	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
89	Dic₅₉⋊C₄	—	2	1	no	—
90	Dic₅₉⋊C₄	—	2	1	no	—
91	Dic₅₉⋊C₄	—	2	1	no	—
92	Dic₅₉⋊C₄	—	2	1	no	—
93	Dic₅₉⋊C₄	—	2	1	no	—
94	Dic₅₉⋊C₄	—	2	1	no	—
95	Dic₅₉⋊C₄	—	2	1	no	—
96	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
101	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	Dic₂₃₆	—	2	2	no	—
104	#11745659	—	2	2	no	—
105	C₂³×C₅₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₅₉	—	2	1	no	—
108	C₂×C₄×C₅₉	—	2	1	no	—
109	C₂×C₄×C₅₉	—	2	1	no	—
110	C₂×C₄×C₅₉	—	2	1	no	—
111	C₂×C₄×C₅₉	—	2	1	no	—
112	C₂×C₄×C₅₉	—	2	1	no	—
113	C₂×C₄×C₅₉	—	8	1	no	—
114	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
116	C₂²⋊D₅₉	—	2	2	no	—
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	—	2	1	no	—
121	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
124	C₄×D₅₉	—	2	1	no	—
125	C₄×D₅₉	—	2	1	no	—
126	C₄×D₅₉	—	8	1	no	—
127	Dic₁₁₈	—	2	1	no	—
128	Dic₁₁₈	—	2	1	no	—
129	Dic₁₁₈	—	2	1	no	—
130	Dic₁₁₈	—	2	1	no	—
131	Dic₁₁₈	—	4	2	no	—
132	Dic₁₁₈	—	8	1	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	#41	1	2	yes	—
135	C₅₉⋊C₈	—	2	1	no	—
136	C₅₉⋊C₈	—	2	1	no	—
137	C₂²×C₅₉	—	2	1	no	—
138	C₂²×C₅₉	—	2	1	no	—
139	C₂²×C₅₉	—	2	1	no	—
140	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
141	C₂×D₅₉	—	2	1	no	—
142	C₂×D₅₉	—	2	1	no	—
143	C₂×D₅₉	—	2	1	no	—
144	C₄×C₅₉	—	2	1	no	—
145	C₄×C₅₉	—	4	1	no	—
146	C₄×C₅₉	—	4	1	no	—
147	C₄×C₅₉	—	4	1	no	—
148	C₄×C₅₉	—	4	1	no	—
149	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
150	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₅₉	—	2	1	no	—
152	Dic₅₉	—	2	1	no	—
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	—	4	1	no	—
155	Dic₅₉	—	4	1	no	—
156	Dic₅₉	—	4	1	no	—
157	Dic₅₉	#137	1	2	yes	—
158	#3356	—	59	1	no	maximal, Sylow
159	C₂×C₅₉	—	4	1	no	—
160	C₂×C₅₉	#547	1	1	yes	characteristic
161	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
162	C₂×C₅₉	#577	1	1	yes	characteristic
163	D₅₉	—	4	1	no	—
164	#556	—	59	1	no	—
165	#554	—	59	1	no	—
166	#498	—	59	1	no	—
167	#485	—	59	1	no	—
168	#485	—	59	1	no	—
169	#483	—	59	1	no	—
170	#483	—	59	1	no	—
171	#477	—	59	1	no	—
172	#475	—	59	1	no	—
173	#470	—	59	1	no	—
174	#470	—	59	1	no	—
175	C₄×Q₁₆	—	59	1	no	—
176	#436	—	59	1	no	—
177	#428	—	59	1	no	—
178	#428	D₅₉	1	1	yes	characteristic, complemented
179	C₅₉	#3356	1	1	yes	characteristic, complemented, Sylow
180	C₂×Q₁₆	—	59	1	no	—
181	C₂×M₁₆	—	59	1	no	—
182	C₂×M₁₆	C₂×D₅₉	1	1	yes	characteristic
183	C₄⋊Q₈	—	59	1	no	—
184	C₄⋊Q₈	—	59	1	no	—
185	#126	—	118	1	no	—
186	#126	—	118	1	no	—
187	#125	—	118	1	no	—
188	#124	—	118	1	no	—
189	C₂²⋊Q₈	—	59	1	no	—
190	C₂²⋊Q₈	—	59	1	no	—
191	C₂²⋊Q₈	—	59	1	no	—
192	C₂²⋊Q₈	—	59	1	no	—
193	C₂²⋊Q₈	—	118	1	no	—
194	C₂²⋊Q₈	—	118	1	no	—
195	C₄×Q₈	—	59	1	no	—
196	C₄×Q₈	—	59	1	no	—
197	C₄×Q₈	—	118	1	no	—
198	C₄×D₄	—	59	1	no	—
199	C₄×D₄	—	118	1	no	—
200	#117	—	59	1	no	—
201	#117	C₂×D₅₉	1	1	yes	characteristic
202	C₂×C₄⋊C₄	—	59	1	no	—
203	C₂×C₄⋊C₄	C₂×D₅₉	1	1	yes	characteristic
204	#107	—	59	1	no	—
205	#107	C₂×D₅₉	1	1	yes	characteristic
206	#107	C₂×D₅₉	1	1	yes	characteristic
207	#106	—	59	1	no	—
208	#106	—	59	1	no	—
209	#106	—	59	1	no	—
210	#106	—	59	1	no	—
211	#106	C₂×D₅₉	1	1	yes	characteristic
212	#106	C₂×D₅₉	1	1	yes	characteristic
213	C₄⋊C₈	—	59	1	no	—
214	#103	—	59	1	no	—
215	#103	—	59	1	no	—
216	#103	—	59	1	no	—
217	#103	—	59	1	no	—
218	#103	—	59	1	no	—
219	#103	—	59	1	no	—
220	C₂²⋊C₈	—	59	1	no	—
221	C₂²⋊C₈	—	59	1	no	—
222	C₄×C₈	—	59	1	no	—
223	C₂×Q₈	—	59	1	no	—
224	C₂×Q₈	—	59	1	no	—
225	C₂×Q₈	—	118	1	no	—
226	C₂×Q₈	—	118	1	no	—
227	C₂×D₄	—	59	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	59	1	no	—
230	C₂²×C₄	—	118	1	no	—
231	C₂²×C₄	C₂²×D₅₉	1	1	yes	characteristic
232	Q₁₆	—	118	2	no	—
233	M₁₆	—	2	2	no	—
234	M₁₆	—	118	2	no	—
235	C₂×C₈	—	59	1	no	—
236	C₂×C₈	—	59	1	no	—
237	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	59	1	no	—
241	C₄⋊C₄	—	59	1	no	—
242	C₄⋊C₄	—	59	1	no	—
243	C₄⋊C₄	—	118	1	no	—
244	C₄⋊C₄	—	118	1	no	—
245	C₄⋊C₄	—	118	1	no	—
246	C₄⋊C₄	—	118	1	no	—
247	C₄⋊C₄	—	118	1	no	—
248	C₄⋊C₄	—	118	1	no	—
249	C₄⋊C₄	—	118	1	no	—
250	C₄⋊C₄	—	118	1	no	—
251	C₄⋊C₄	—	118	1	no	—
252	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
253	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
256	C₂²⋊C₄	—	2	1	no	—
257	C₂²⋊C₄	—	59	1	no	—
258	C₂²⋊C₄	—	59	1	no	—
259	C₂²⋊C₄	—	118	1	no	—
260	C₂²⋊C₄	—	118	1	no	—
261	C₂²⋊C₄	—	118	1	no	—
262	C₂²⋊C₄	—	118	1	no	—
263	C₂²⋊C₄	—	118	1	no	—
264	C₄²	—	2	1	no	—
265	C₄²	—	59	1	no	—
266	C₄²	—	118	1	no	—
267	C₄²	—	118	1	no	—
268	C₄²	—	118	1	no	—
269	C₂³	—	59	1	no	—
270	C₂³	D₄×D₅₉	1	1	yes	characteristic
271	Q₈	—	118	1	no	—
272	Q₈	—	118	1	no	—
273	Q₈	—	118	1	no	—
274	Q₈	—	118	1	no	—
275	Q₈	—	472	1	no	—
276	Q₈	—	236	2	no	—
277	D₄	—	118	2	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	—	59	1	no	—
286	C₂×C₄	—	59	1	no	—
287	C₂×C₄	—	59	1	no	—
288	C₂×C₄	—	118	1	no	—
289	C₂×C₄	—	118	1	no	—
290	C₂×C₄	—	118	1	no	—
291	C₂×C₄	—	118	1	no	—
292	C₂×C₄	—	118	1	no	—
293	C₂×C₄	—	118	1	no	—
294	C₂×C₄	—	472	1	no	—
295	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
297	C₈	—	2	1	no	—
298	C₈	—	118	1	no	—
299	C₈	—	118	1	no	—
300	C₈	#11745685	1	2	yes	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	118	1	no	—
305	C₂²	—	118	1	no	—
306	C₂²	—	118	1	no	—
307	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	59	2	no	—
314	C₄	—	118	1	no	—
315	C₄	—	118	1	no	—
316	C₄	—	236	1	no	—
317	C₄	—	236	1	no	—
318	C₄	—	236	1	no	—
319	C₄	—	236	1	no	—
320	C₄	C₂×#11745685	1	1	yes	characteristic
321	C₄	#14273775	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	236	1	no	—
324	C₂	#16838053	1	1	yes	characteristic
325	C₂	#137×D₅₉	1	1	yes	characteristic
326	C₂	#16838212	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#428	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#428	D₅₉
hypocenter	C₅₉	#3356
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
3. #16837799	C₂	semidirect
4. #16837745	C₂	semidirect
5. #16837743	C₂	semidirect
6. #16837734	C₂	semidirect
7. #16837732	C₂	semidirect
8. #16837696	C₂	semidirect
9. #16837686	C₂	semidirect
10. #16837661	C₂	semidirect
11. #16837660	C₂	semidirect
12. #16837496	C₂	semidirect
14. #16837443	C₂	semidirect
15. C₅₉×#428	C₂	semidirect
178. #428	D₅₉	semidirect
179. C₅₉	#3356	semidirect