Subgroups — #21437122

Order 7552.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838127	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838124	C₂	1	1	yes	maximal, characteristic
3	#16837795	C₂	1	1	yes	maximal, characteristic
4	#16837794	C₂	1	1	yes	maximal, characteristic
5	#16837780	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837626	C₂	1	1	yes	maximal, characteristic
7	#16837625	C₂	1	1	yes	maximal, characteristic
8	#16837611	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837610	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837606	C₂	1	1	yes	maximal, characteristic
11	#16837531	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837507	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837506	C₂	1	1	yes	maximal, characteristic, complemented
14	#16837487	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#462	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₂₃₆	C₂²	1	1	yes	characteristic
17	C₂×#11745672	C₂²	1	1	yes	characteristic
18	C₂×#11745672	C₂²	1	1	yes	characteristic
19	C₂×#11745672	C₂²	1	1	yes	characteristic
20	C₂×#11745660	C₂²	1	1	yes	characteristic
21	C₂×#11745671	C₂²	1	1	yes	characteristic
22	C₂×#11745671	C₂²	1	1	yes	characteristic
23	#14273725	C₂²	1	1	yes	characteristic
24	#14273725	C₂²	1	1	yes	characteristic
25	#14273725	C₂²	1	1	yes	characteristic
26	#14273722	—	2	1	no	—
27	#14273721	—	2	1	no	—
28	#14273718	—	2	1	no	—
29	#14273717	—	2	1	no	—
30	#14273717	—	2	1	no	—
31	#14273716	C₂²	1	1	yes	characteristic
32	#14273715	C₂²	1	1	yes	characteristic
33	#14273705	—	2	1	no	—
34	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
35	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
36	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
37	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
38	#14273692	—	2	1	no	—
39	#14273692	C₂²	1	1	yes	characteristic
40	#14273689	C₂²	1	1	yes	characteristic
41	#14273684	—	2	1	no	—
42	#14273658	C₂²	1	1	yes	characteristic
43	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	#14273652	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273650	C₂²	1	1	yes	characteristic
48	#14273649	C₂²	1	1	yes	characteristic
49	#14273638	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	C₈×Dic₅₉	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₄⋊Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#103×C₅₉	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
65	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
66	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
67	Q₁₆×C₅₉	—	4	1	no	—
68	Q₁₆×C₅₉	—	4	1	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	—	2	1	no	—
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
75	C₄²×C₅₉	—	2	1	no	—
76	C₄²×C₅₉	C₂³	1	1	yes	characteristic
77	#11745672	—	2	2	no	—
78	#11745672	—	2	2	no	—
79	#11745672	—	4	1	no	—
80	#11745671	—	2	2	no	—
81	#11745671	—	4	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	(C₂×D₅₉)⋊C₄	—	2	1	no	—
84	(C₂×D₅₉)⋊C₄	—	2	1	no	—
85	(C₂×D₅₉)⋊C₄	—	2	1	no	—
86	(C₂×D₅₉)⋊C₄	—	2	1	no	—
87	C₄⋊Dic₅₉	—	2	1	no	—
88	C₄⋊Dic₅₉	—	2	1	no	—
89	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
90	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
97	Dic₅₉⋊C₄	—	2	1	no	—
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
100	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
101	C₄×Dic₅₉	—	2	1	no	—
102	C₄×Dic₅₉	—	2	1	no	—
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
106	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
107	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	Dic₂₃₆	—	4	1	no	—
110	#11745660	—	4	1	no	—
111	C₂²×D₅₉	—	2	1	no	—
112	Q₈×C₅₉	—	2	1	no	—
113	Q₈×C₅₉	—	2	1	no	—
114	Q₈×C₅₉	—	2	1	no	—
115	Q₈×C₅₉	—	2	1	no	—
116	Q₈×C₅₉	—	2	1	no	—
117	Q₈×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	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₅₉	—	2	1	no	—
126	C₂×Dic₅₉	—	2	1	no	—
127	C₂×Dic₅₉	—	2	1	no	—
128	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
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	C₄×D₅₉	—	4	2	no	—
135	Dic₁₁₈	—	2	2	no	—
136	Dic₁₁₈	—	2	1	no	—
137	Dic₁₁₈	—	2	1	no	—
138	Dic₁₁₈	—	2	1	no	—
139	Dic₁₁₈	—	2	1	no	—
140	C₈×C₅₉	—	2	1	no	—
141	C₈×C₅₉	—	2	1	no	—
142	C₅₉⋊C₈	—	2	1	no	—
143	C₅₉⋊C₈	#41	1	2	yes	—
144	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₅₉	—	4	1	no	—
146	C₂×D₅₉	—	4	1	no	—
147	C₂×D₅₉	—	4	1	no	—
148	C₄×C₅₉	—	2	1	no	—
149	C₄×C₅₉	—	2	1	no	—
150	C₄×C₅₉	—	4	1	no	—
151	C₄×C₅₉	—	4	1	no	—
152	C₄×C₅₉	—	4	1	no	—
153	C₄×C₅₉	—	4	1	no	—
154	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
155	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
156	Dic₅₉	—	2	1	no	—
157	Dic₅₉	—	2	1	no	—
158	Dic₅₉	—	2	1	no	—
159	Dic₅₉	—	4	1	no	—
160	Dic₅₉	—	4	1	no	—
161	Dic₅₉	—	4	1	no	—
162	Dic₅₉	#137	1	2	yes	—
163	#3398	—	59	1	no	maximal, Sylow
164	C₂×C₅₉	#550	1	1	yes	characteristic
165	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
166	C₂×C₅₉	#578	1	1	yes	characteristic
167	D₅₉	—	8	1	no	—
168	#557	—	59	1	no	—
169	#556	—	59	1	no	—
170	#497	—	59	1	no	—
171	#497	—	59	1	no	—
172	#494	—	59	1	no	—
173	#464	—	59	1	no	—
174	#464	—	59	1	no	—
175	#462	—	59	1	no	—
176	#462	—	59	1	no	—
177	#462	D₅₉	1	1	yes	characteristic, complemented
178	#461	—	59	1	no	—
179	#446	—	59	1	no	—
180	#441	—	59	1	no	—
181	#441	—	59	1	no	—
182	C₄×SD₁₆	—	59	1	no	—
183	C₅₉	#3398	1	1	yes	characteristic, complemented, Sylow
184	C₂×Q₁₆	—	59	1	no	—
185	C₂×Q₁₆	—	59	1	no	—
186	C₂×Q₁₆	—	59	1	no	—
187	C₂×Q₁₆	—	59	1	no	—
188	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
189	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
190	C₂×SD₁₆	—	59	1	no	—
191	C₂×SD₁₆	—	59	1	no	—
192	C₂×SD₁₆	—	59	1	no	—
193	C₄⋊Q₈	—	59	1	no	—
194	C₄⋊Q₈	—	59	1	no	—
195	C₄⋊Q₈	—	59	1	no	—
196	C₄⋊Q₈	C₂×D₅₉	1	1	yes	characteristic
197	#126	—	118	1	no	—
198	#126	—	118	1	no	—
199	#125	—	118	1	no	—
200	#125	—	118	1	no	—
201	#125	—	118	1	no	—
202	#124	—	59	1	no	—
203	#124	—	59	1	no	—
204	C₂²⋊Q₈	—	118	1	no	—
205	C₄×Q₈	—	59	1	no	—
206	C₄×Q₈	—	59	1	no	—
207	C₄×Q₈	—	59	1	no	—
208	C₄×Q₈	—	59	1	no	—
209	C₄×Q₈	—	59	1	no	—
210	C₄×Q₈	—	118	1	no	—
211	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
212	C₄×D₄	—	59	1	no	—
213	#117	—	118	1	no	—
214	#106	—	59	1	no	—
215	C₄⋊C₈	—	59	1	no	—
216	C₄⋊C₈	—	59	1	no	—
217	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
218	#103	—	59	1	no	—
219	#103	—	59	1	no	—
220	#103	—	59	1	no	—
221	#103	C₂×D₅₉	1	1	yes	characteristic
222	#103	C₂×D₅₉	1	1	yes	characteristic
223	#102	—	59	1	no	—
224	#97	—	59	1	no	—
225	#97	—	59	1	no	—
226	C₄×C₈	—	59	1	no	—
227	C₂×Q₈	—	59	1	no	—
228	C₂×Q₈	—	59	1	no	—
229	C₂×Q₈	—	59	1	no	—
230	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
233	C₂×D₄	—	59	1	no	—
234	C₂²×C₄	—	118	1	no	—
235	Q₁₆	—	4	1	no	—
236	Q₁₆	—	4	1	no	—
237	Q₁₆	—	118	2	no	—
238	Q₁₆	—	118	2	no	—
239	Q₁₆	—	236	1	no	—
240	Q₁₆	—	236	1	no	—
241	SD₁₆	—	118	2	no	—
242	SD₁₆	—	236	1	no	—
243	SD₁₆	—	236	1	no	—
244	C₂×C₈	—	59	1	no	—
245	C₂×C₈	—	59	1	no	—
246	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	59	1	no	—
252	C₄⋊C₄	—	59	1	no	—
253	C₄⋊C₄	—	59	1	no	—
254	C₄⋊C₄	—	59	1	no	—
255	C₄⋊C₄	—	118	1	no	—
256	C₄⋊C₄	—	118	1	no	—
257	C₄⋊C₄	—	118	1	no	—
258	C₄⋊C₄	—	118	1	no	—
259	C₄⋊C₄	—	118	1	no	—
260	C₄⋊C₄	—	118	1	no	—
261	C₄⋊C₄	—	118	1	no	—
262	C₄⋊C₄	—	118	1	no	—
263	C₄⋊C₄	—	118	1	no	—
264	C₄⋊C₄	—	118	1	no	—
265	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
266	C₂²⋊C₄	—	118	1	no	—
267	C₂²⋊C₄	—	118	1	no	—
268	C₂²⋊C₄	—	118	1	no	—
269	C₂²⋊C₄	—	118	1	no	—
270	C₂²⋊C₄	—	118	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	59	1	no	—
273	C₄²	—	59	1	no	—
274	C₄²	—	118	1	no	—
275	C₄²	—	118	1	no	—
276	C₄²	—	118	1	no	—
277	C₄²	—	118	1	no	—
278	C₄²	C₂²×D₅₉	1	1	yes	characteristic
279	C₂³	—	118	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	118	2	no	—
287	Q₈	—	118	1	no	—
288	Q₈	—	118	1	no	—
289	Q₈	—	118	1	no	—
290	Q₈	—	118	1	no	—
291	D₄	—	118	1	no	—
292	D₄	—	118	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	59	1	no	—
298	C₂×C₄	—	59	1	no	—
299	C₂×C₄	—	59	1	no	—
300	C₂×C₄	—	59	1	no	—
301	C₂×C₄	—	118	1	no	—
302	C₂×C₄	—	118	1	no	—
303	C₂×C₄	—	118	1	no	—
304	C₂×C₄	—	236	2	no	—
305	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
306	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	59	2	no	—
311	C₈	—	118	1	no	—
312	C₂²	—	236	1	no	—
313	C₂²	—	236	1	no	—
314	C₂²	—	236	1	no	—
315	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	59	2	no	—
323	C₄	—	118	1	no	—
324	C₄	—	118	1	no	—
325	C₄	—	118	1	no	—
326	C₄	—	236	1	no	—
327	C₄	—	236	1	no	—
328	C₄	—	236	1	no	—
329	C₄	#41×D₅₉	1	1	yes	characteristic
330	C₄	#14273772	1	1	yes	characteristic
331	C₂	—	472	1	no	—
332	C₂	#16838070	1	1	yes	characteristic
333	C₂	#137×D₅₉	1	1	yes	characteristic
334	C₂	#16838215	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#462	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#462	D₅₉
hypocenter	C₅₉	#3398
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

Complements

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

N	H ≅ G/N	type
1. #16838127	C₂	semidirect
5. #16837780	C₂	semidirect
8. #16837611	C₂	semidirect
9. #16837610	C₂	semidirect
11. #16837531	C₂	semidirect
12. #16837507	C₂	semidirect
13. #16837506	C₂	semidirect
15. C₅₉×#462	C₂	semidirect
177. #462	D₅₉	semidirect
183. C₅₉	#3398	semidirect