Subgroups — #21437168

Order 7552.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16838134	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838121	C₂	1	1	yes	maximal, characteristic, complemented
3	#16837795	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837790	C₂	1	1	yes	maximal, characteristic
5	#16837622	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837621	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837619	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837614	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837605	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837600	C₂	1	1	yes	maximal, characteristic, complemented
11	#16837532	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837514	C₂	1	1	yes	maximal, characteristic, complemented
13	#16837509	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×#11745660	C₂	1	1	yes	maximal, characteristic, complemented
15	C₅₉×#495	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745672	C₂²	1	1	yes	characteristic
17	C₂×#11745660	C₂²	1	1	yes	characteristic
18	C₂×#11745670	C₂²	1	1	yes	characteristic
19	C₂×#11745671	C₂²	1	1	yes	characteristic
20	C₂×#11745669	C₂²	1	1	yes	characteristic
21	#14273725	C₂²	1	1	yes	characteristic
22	#14273723	C₂²	1	1	yes	characteristic
23	#14273720	—	2	1	no	—
24	#14273720	—	2	1	no	—
25	#14273716	C₂²	1	1	yes	characteristic
26	#14273714	C₂²	1	1	yes	characteristic
27	#14273713	—	2	1	no	—
28	#14273711	—	2	1	no	—
29	#14273709	—	2	1	no	—
30	#14273704	—	2	1	no	—
31	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
32	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
33	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
34	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
35	C₄×D₂₃₆	C₂²	1	1	yes	characteristic
36	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
37	D₄×Dic₅₉	C₂²	1	1	yes	characteristic
38	(C₂²⋊D₅₉)⋊C₄	—	2	1	no	—
39	#14273683	—	2	1	no	—
40	#14273659	C₂²	1	1	yes	characteristic
41	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
44	#14273652	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273651	C₂²	1	1	yes	characteristic
47	#14273649	C₂²	1	1	yes	characteristic
48	#14273648	C₂²	1	1	yes	characteristic
49	#14273648	C₂²	1	1	yes	characteristic
50	#14273637	C₂²	1	1	yes	characteristic
51	#14273637	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₅₉	C₂²	1	1	yes	characteristic
55	C₂×D₈×C₅₉	C₂²	1	1	yes	characteristic
56	#124×C₅₉	C₂²	1	1	yes	characteristic
57	#124×C₅₉	C₂²	1	1	yes	characteristic
58	C₄×C₈×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×D₄×C₅₉	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₅₉	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₅₉	—	2	1	no	—
64	C₂×C₂²⋊D₅₉	—	2	1	no	—
65	C₂²×Dic₅₉	—	2	1	no	—
66	C₂²×Dic₅₉	—	2	1	no	—
67	C₂×D₂₃₆	C₂³	1	1	yes	characteristic
68	C₂×C₄×D₅₉	—	2	1	no	—
69	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
70	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
71	Q₁₆×C₅₉	—	2	2	no	—
72	SD₁₆×C₅₉	—	4	1	no	—
73	SD₁₆×C₅₉	—	4	1	no	—
74	D₈×C₅₉	—	2	2	no	—
75	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
77	C₂²⋊C₄×C₅₉	—	2	1	no	—
78	C₂²⋊C₄×C₅₉	—	2	1	no	—
79	C₂²⋊C₄×C₅₉	—	2	1	no	—
80	C₂²⋊C₄×C₅₉	—	2	1	no	—
81	C₄²×C₅₉	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₅₉	—	2	1	no	—
83	C₂²⋊Dic₅₉	—	2	1	no	—
84	C₂²⋊Dic₅₉	—	2	1	no	—
85	C₂²⋊Dic₅₉	—	2	1	no	—
86	#11745672	—	4	1	no	—
87	#11745671	—	4	1	no	—
88	#11745670	—	4	1	no	—
89	#11745669	—	4	1	no	—
90	(C₂×D₅₉)⋊C₄	—	2	1	no	—
91	(C₂×D₅₉)⋊C₄	—	2	1	no	—
92	(C₂×D₅₉)⋊C₄	—	2	1	no	—
93	C₄⋊Dic₅₉	—	2	1	no	—
94	C₄⋊Dic₅₉	—	2	1	no	—
95	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
96	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
98	Dic₅₉⋊C₄	—	2	1	no	—
99	Dic₅₉⋊C₄	—	2	1	no	—
100	Dic₅₉⋊C₄	—	2	1	no	—
101	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
102	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
103	C₄×Dic₅₉	—	2	1	no	—
104	C₄×Dic₅₉	—	2	1	no	—
105	C₄×Dic₅₉	—	2	1	no	—
106	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
107	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
108	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
109	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
110	#11745660	—	2	2	no	—
111	C₂³×C₅₉	—	2	1	no	—
112	C₂³×C₅₉	—	2	1	no	—
113	C₂²×D₅₉	—	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	D₄×C₅₉	—	2	1	no	—
119	D₄×C₅₉	—	2	1	no	—
120	D₄×C₅₉	—	2	1	no	—
121	D₄×C₅₉	—	2	1	no	—
122	C₂×C₄×C₅₉	—	2	1	no	—
123	C₂×C₄×C₅₉	—	2	1	no	—
124	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
127	C₂²⋊D₅₉	—	4	2	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	—	4	2	no	—
133	C₂×Dic₅₉	—	8	1	no	—
134	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
138	D₂₃₆	—	2	1	no	—
139	D₂₃₆	—	2	1	no	—
140	C₄×D₅₉	—	8	1	no	—
141	Dic₁₁₈	—	2	1	no	—
142	Dic₁₁₈	—	2	1	no	—
143	Dic₁₁₈	—	4	1	no	—
144	C₈×C₅₉	—	2	1	no	—
145	C₈×C₅₉	#41	1	2	yes	—
146	C₅₉⋊C₈	—	2	1	no	—
147	C₅₉⋊C₈	—	2	1	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	—	4	1	no	—
154	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₅₉	—	4	1	no	—
156	C₂×D₅₉	—	4	1	no	—
157	C₂×D₅₉	—	4	1	no	—
158	C₄×C₅₉	—	2	1	no	—
159	C₄×C₅₉	—	2	1	no	—
160	C₄×C₅₉	—	4	1	no	—
161	C₄×C₅₉	—	4	1	no	—
162	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
163	C₄×C₅₉	H(5,𝔽₂)	1	1	yes	characteristic
164	Dic₅₉	—	2	1	no	—
165	Dic₅₉	—	2	1	no	—
166	Dic₅₉	—	2	1	no	—
167	Dic₅₉	—	2	1	no	—
168	Dic₅₉	—	4	1	no	—
169	Dic₅₉	—	4	1	no	—
170	Dic₅₉	—	4	1	no	—
171	#3401	—	59	1	no	maximal, Sylow
172	C₂×C₅₉	—	8	1	no	—
173	C₂×C₅₉	—	8	1	no	—
174	C₂×C₅₉	#550	1	1	yes	characteristic
175	C₂×C₅₉	#575	1	1	yes	characteristic
176	C₂×C₅₉	#577	1	1	yes	characteristic
177	D₅₉	—	8	1	no	—
178	#559	—	59	1	no	—
179	#556	—	59	1	no	—
180	#497	—	59	1	no	—
181	#496	—	59	1	no	—
182	#495	D₅₉	1	1	yes	characteristic, complemented
183	#464	—	59	1	no	—
184	#464	—	59	1	no	—
185	#463	—	59	1	no	—
186	#463	—	59	1	no	—
187	#461	—	59	1	no	—
188	#460	—	59	1	no	—
189	#446	—	59	1	no	—
190	#442	—	59	1	no	—
191	#441	—	59	1	no	—
192	C₄×SD₁₆	—	59	1	no	—
193	C₅₉	#3401	1	1	yes	characteristic, complemented, Sylow
194	C₂×Q₁₆	—	59	1	no	—
195	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
196	C₂×SD₁₆	—	59	1	no	—
197	C₂×SD₁₆	—	59	1	no	—
198	C₂×SD₁₆	—	59	1	no	—
199	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
200	C₂×SD₁₆	C₂×D₅₉	1	1	yes	characteristic
201	C₂×D₈	—	59	1	no	—
202	C₂×D₈	C₂×D₅₉	1	1	yes	characteristic
203	C₄⋊Q₈	—	59	1	no	—
204	#127	—	59	1	no	—
205	#126	—	118	1	no	—
206	#126	—	118	1	no	—
207	#124	—	59	1	no	—
208	#124	—	59	1	no	—
209	#124	—	118	1	no	—
210	#124	C₂×D₅₉	1	1	yes	characteristic
211	#124	C₂×D₅₉	1	1	yes	characteristic
212	#123	—	118	1	no	—
213	#123	—	118	1	no	—
214	C₂²⋊Q₈	—	118	1	no	—
215	C₄×Q₈	—	59	1	no	—
216	C₄×Q₈	—	59	1	no	—
217	C₄×Q₈	—	59	1	no	—
218	C₄×Q₈	—	59	1	no	—
219	C₄×D₄	—	59	1	no	—
220	C₄×D₄	—	59	1	no	—
221	C₄×D₄	—	59	1	no	—
222	C₄×D₄	—	118	1	no	—
223	#117	—	118	1	no	—
224	#106	—	59	1	no	—
225	C₄⋊C₈	—	59	1	no	—
226	C₄⋊C₈	—	59	1	no	—
227	C₄⋊C₈	—	59	1	no	—
228	#103	—	59	1	no	—
229	#103	—	59	1	no	—
230	#103	—	59	1	no	—
231	#102	—	59	1	no	—
232	#102	—	59	1	no	—
233	#102	—	59	1	no	—
234	#97	—	59	1	no	—
235	#97	—	59	1	no	—
236	C₄×C₈	C₂×D₅₉	1	1	yes	characteristic
237	C₂×Q₈	—	59	1	no	—
238	C₂×Q₈	—	59	1	no	—
239	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
240	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
241	C₂×D₄	—	59	1	no	—
242	C₂×D₄	—	118	1	no	—
243	C₂×D₄	—	118	1	no	—
244	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
245	C₂×D₄	C₂²×D₅₉	1	1	yes	characteristic
246	C₂²×C₄	—	118	1	no	—
247	C₂²×C₄	—	118	1	no	—
248	C₂²×C₄	—	118	1	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	236	1	no	—
251	SD₁₆	—	4	1	no	—
252	SD₁₆	—	4	1	no	—
253	SD₁₆	—	118	2	no	—
254	SD₁₆	—	236	1	no	—
255	SD₁₆	—	236	1	no	—
256	D₈	—	2	2	no	—
257	D₈	—	236	1	no	—
258	C₂×C₈	—	59	1	no	—
259	C₂×C₈	—	59	1	no	—
260	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
261	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
262	C₄⋊C₄	—	59	1	no	—
263	C₄⋊C₄	—	59	1	no	—
264	C₄⋊C₄	—	59	1	no	—
265	C₄⋊C₄	—	59	1	no	—
266	C₄⋊C₄	—	59	1	no	—
267	C₄⋊C₄	—	118	1	no	—
268	C₄⋊C₄	—	118	1	no	—
269	C₄⋊C₄	—	118	1	no	—
270	C₄⋊C₄	—	118	1	no	—
271	C₄⋊C₄	—	118	1	no	—
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	2	1	no	—
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	118	1	no	—
277	C₂²⋊C₄	—	118	1	no	—
278	C₂²⋊C₄	—	118	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	—	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	—	2	1	no	—
291	C₂³	—	118	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	2	1	no	—
294	Q₈	—	2	1	no	—
295	Q₈	—	2	1	no	—
296	Q₈	—	118	1	no	—
297	Q₈	—	118	1	no	—
298	Q₈	—	236	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	2	1	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	118	1	no	—
304	D₄	—	118	1	no	—
305	D₄	—	472	1	no	—
306	D₄	—	236	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	59	1	no	—
310	C₂×C₄	—	59	1	no	—
311	C₂×C₄	—	59	1	no	—
312	C₂×C₄	—	59	1	no	—
313	C₂×C₄	—	118	1	no	—
314	C₂×C₄	—	118	1	no	—
315	C₂×C₄	—	118	1	no	—
316	C₂×C₄	—	472	1	no	—
317	C₂×C₄	—	472	1	no	—
318	C₂×C₄	—	236	2	no	—
319	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
320	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	118	1	no	—
324	C₈	—	118	1	no	—
325	C₈	#11745687	1	2	yes	—
326	C₂²	—	4	1	no	—
327	C₂²	—	4	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	236	1	no	—
333	C₂²	—	236	1	no	—
334	C₂²	—	236	1	no	—
335	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	118	1	no	—
341	C₄	—	118	1	no	—
342	C₄	—	118	1	no	—
343	C₄	—	118	1	no	—
344	C₄	—	236	1	no	—
345	C₄	—	236	1	no	—
346	C₄	—	236	1	no	—
347	C₄	C₂×#11745687	1	1	yes	characteristic
348	C₄	#14273772	1	1	yes	characteristic
349	C₂	—	8	1	no	—
350	C₂	—	8	1	no	—
351	C₂	—	472	1	no	—
352	C₂	#16838072	1	1	yes	characteristic
353	C₂	#16838199	1	1	yes	characteristic
354	C₂	#16838209	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#495	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#495	D₅₉
hypocenter	C₅₉	#3401
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #495

Complements

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

N	H ≅ G/N	type
1. #16838134	C₂	semidirect
2. #16838121	C₂	semidirect
3. #16837795	C₂	semidirect
5. #16837622	C₂	semidirect
6. #16837621	C₂	semidirect
7. #16837619	C₂	semidirect
8. #16837614	C₂	semidirect
9. #16837605	C₂	semidirect
10. #16837600	C₂	semidirect
11. #16837532	C₂	semidirect
12. #16837514	C₂	semidirect
13. #16837509	C₂	semidirect
14. C₄×#11745660	C₂	semidirect
15. C₅₉×#495	C₂	semidirect
182. #495	D₅₉	semidirect
193. C₅₉	#3401	semidirect