Subgroups — #21436837

Order 7552.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₁₈	C₂	1	1	yes	maximal, characteristic, complemented
2	#16838124	C₂	1	1	yes	maximal, characteristic
3	#16837763	C₂	1	1	yes	maximal, characteristic
4	#16837762	C₂	1	1	yes	maximal, characteristic
5	#16837751	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837691	C₂	1	1	yes	maximal, characteristic
7	#16837687	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837686	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837626	C₂	1	1	yes	maximal, characteristic
10	#16837624	C₂	1	1	yes	maximal, characteristic
11	#16837610	C₂	1	1	yes	maximal, characteristic, complemented
12	#16837530	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×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11745672	C₂²	1	1	yes	characteristic
17	C₂×#11745671	C₂²	1	1	yes	characteristic
18	#14273727	—	2	1	no	—
19	#14273727	C₂²	1	1	yes	characteristic
20	#14273726	—	2	1	no	—
21	#14273726	—	2	1	no	—
22	#14273726	C₂²	1	1	yes	characteristic
23	#14273726	C₂²	1	1	yes	characteristic
24	#14273725	C₂²	1	1	yes	characteristic
25	#14273722	—	2	1	no	—
26	#14273721	—	2	1	no	—
27	#14273716	C₂²	1	1	yes	characteristic
28	#14273715	C₂²	1	1	yes	characteristic
29	#14273705	—	2	1	no	—
30	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
31	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₁₈	—	2	1	no	—
33	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
34	#14273692	C₂²	1	1	yes	characteristic
35	#14273689	C₂²	1	1	yes	characteristic
36	#14273684	—	2	1	no	—
37	#14273658	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	#14273652	C₂²	1	1	yes	characteristic
43	#14273652	C₂²	1	1	yes	characteristic
44	#14273651	C₂²	1	1	yes	characteristic
45	#14273651	C₂²	1	1	yes	characteristic
46	#14273650	C₂²	1	1	yes	characteristic
47	#14273649	C₂²	1	1	yes	characteristic
48	#14273649	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×Q₁₆×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
55	#106×C₅₉	C₂²	1	1	yes	characteristic
56	#103×C₅₉	C₂²	1	1	yes	characteristic
57	#103×C₅₉	C₂²	1	1	yes	characteristic
58	#97×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×D₂₃₆	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₅₉	—	2	1	no	—
63	C₂×Dic₁₁₈	—	2	1	no	—
64	C₂×Dic₁₁₈	—	2	1	no	—
65	C₂×Dic₁₁₈	—	2	1	no	—
66	C₂×Dic₁₁₈	D₄	1	1	yes	characteristic
67	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
68	Q₁₆×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	—	2	1	no	—
73	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
75	C₄²×C₅₉	—	2	1	no	—
76	C₄²×C₅₉	—	2	1	no	—
77	C₄²×C₅₉	C₂³	1	1	yes	characteristic
78	#11745672	—	2	2	no	—
79	#11745671	—	2	2	no	—
80	(C₂×D₅₉)⋊C₄	—	2	1	no	—
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	—	2	1	no	—
85	C₄⋊Dic₅₉	—	2	1	no	—
86	C₄⋊Dic₅₉	—	2	1	no	—
87	C₄⋊Dic₅₉	—	2	1	no	—
88	C₄⋊Dic₅₉	D₄	1	1	yes	characteristic
89	C₄⋊Dic₅₉	C₂³	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	D₄	1	1	yes	characteristic
97	Dic₅₉⋊C₄	D₄	1	1	yes	characteristic
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	—	2	1	no	—
100	C₄×Dic₅₉	—	2	1	no	—
101	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
102	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
105	C₂²×D₅₉	—	2	1	no	—
106	Q₈×C₅₉	—	2	1	no	—
107	Q₈×C₅₉	—	2	1	no	—
108	Q₈×C₅₉	—	2	1	no	—
109	Q₈×C₅₉	#41	1	2	yes	—
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	—	2	1	no	—
114	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₅₉	—	2	1	no	—
118	C₂×Dic₅₉	—	2	1	no	—
119	C₂×Dic₅₉	—	2	1	no	—
120	C₂×Dic₅₉	C₂×D₄	1	1	yes	characteristic
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	D₂₃₆	—	2	1	no	—
125	D₂₃₆	—	2	1	no	—
126	C₄×D₅₉	—	4	2	no	—
127	Dic₁₁₈	—	2	2	no	—
128	Dic₁₁₈	—	2	1	no	—
129	Dic₁₁₈	—	2	1	no	—
130	Dic₁₁₈	—	8	1	no	—
131	Dic₁₁₈	—	8	1	no	—
132	Dic₁₁₈	—	8	1	no	—
133	C₈×C₅₉	—	2	1	no	—
134	C₈×C₅₉	—	2	1	no	—
135	C₅₉⋊C₈	—	2	1	no	—
136	C₅₉⋊C₈	—	2	1	no	—
137	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
138	C₂×D₅₉	—	4	1	no	—
139	C₂×D₅₉	—	4	1	no	—
140	C₂×D₅₉	—	4	1	no	—
141	C₄×C₅₉	—	2	2	no	—
142	C₄×C₅₉	—	2	1	no	—
143	C₄×C₅₉	—	2	1	no	—
144	C₄×C₅₉	—	4	1	no	—
145	C₄×C₅₉	—	4	1	no	—
146	C₄×C₅₉	—	4	1	no	—
147	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
148	C₄×C₅₉	2₋¹⁺⁴	1	1	yes	characteristic
149	Dic₅₉	—	2	1	no	—
150	Dic₅₉	—	2	1	no	—
151	Dic₅₉	—	2	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	—	4	1	no	—
155	Dic₅₉	#137	1	2	yes	—
156	#3369	—	59	1	no	maximal, Sylow
157	C₂×C₅₉	#548	1	1	yes	characteristic
158	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
159	C₂×C₅₉	#577	1	1	yes	characteristic
160	D₅₉	—	8	1	no	—
161	Q₈²	—	59	1	no	—
162	#556	—	59	1	no	—
163	#489	—	59	1	no	—
164	#489	—	59	1	no	—
165	#487	—	59	1	no	—
166	#476	—	59	1	no	—
167	#475	—	59	1	no	—
168	#475	—	59	1	no	—
169	#464	—	59	1	no	—
170	#464	—	59	1	no	—
171	#462	—	59	1	no	—
172	#446	—	59	1	no	—
173	#441	—	59	1	no	—
174	#441	D₅₉	1	1	yes	characteristic, complemented
175	C₄×SD₁₆	—	59	1	no	—
176	C₅₉	#3369	1	1	yes	characteristic, complemented, Sylow
177	C₂×Q₁₆	—	59	1	no	—
178	C₂×Q₁₆	C₂×D₅₉	1	1	yes	characteristic
179	C₂×SD₁₆	—	59	1	no	—
180	C₄⋊Q₈	—	59	1	no	—
181	C₄⋊Q₈	—	59	1	no	—
182	C₄⋊Q₈	—	59	1	no	—
183	C₄⋊Q₈	—	59	1	no	—
184	C₄⋊Q₈	—	118	1	no	—
185	C₄⋊Q₈	—	118	1	no	—
186	C₄⋊Q₈	—	118	1	no	—
187	#126	—	118	1	no	—
188	#126	—	118	1	no	—
189	#124	—	59	1	no	—
190	#124	—	59	1	no	—
191	C₂²⋊Q₈	—	118	1	no	—
192	C₄×Q₈	—	59	1	no	—
193	C₄×Q₈	—	59	1	no	—
194	C₄×Q₈	—	59	1	no	—
195	C₄×Q₈	—	59	1	no	—
196	C₄×Q₈	—	118	1	no	—
197	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
198	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
199	C₄×D₄	—	59	1	no	—
200	#117	—	118	1	no	—
201	#106	—	59	1	no	—
202	#106	—	59	1	no	—
203	#106	C₂×D₅₉	1	1	yes	characteristic
204	C₄⋊C₈	—	59	1	no	—
205	C₄⋊C₈	—	59	1	no	—
206	C₄⋊C₈	—	59	1	no	—
207	#103	—	59	1	no	—
208	#103	—	59	1	no	—
209	#103	—	59	1	no	—
210	#103	—	59	1	no	—
211	#103	—	59	1	no	—
212	#103	C₂×D₅₉	1	1	yes	characteristic
213	#103	C₂×D₅₉	1	1	yes	characteristic
214	#102	—	59	1	no	—
215	#102	—	59	1	no	—
216	#102	—	59	1	no	—
217	#97	—	59	1	no	—
218	#97	C₂×D₅₉	1	1	yes	characteristic
219	C₄×C₈	—	59	1	no	—
220	C₂×Q₈	—	59	1	no	—
221	C₂×Q₈	—	59	1	no	—
222	C₂×Q₈	—	118	1	no	—
223	C₂×Q₈	—	118	1	no	—
224	C₂×Q₈	—	118	1	no	—
225	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
226	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
227	C₂×D₄	—	59	1	no	—
228	C₂²×C₄	—	118	1	no	—
229	Q₁₆	—	2	2	no	—
230	Q₁₆	—	118	2	no	—
231	SD₁₆	—	118	2	no	—
232	C₂×C₈	—	59	1	no	—
233	C₂×C₈	—	59	1	no	—
234	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₅₉	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	59	1	no	—
239	C₄⋊C₄	—	59	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	C₂²×D₅₉	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₅₉	1	1	yes	characteristic
253	C₂²⋊C₄	—	118	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	C₄²	—	2	1	no	—
259	C₄²	—	2	1	no	—
260	C₄²	—	59	1	no	—
261	C₄²	—	59	1	no	—
262	C₄²	—	118	1	no	—
263	C₄²	—	118	1	no	—
264	C₄²	—	118	1	no	—
265	C₄²	C₂²×D₅₉	1	1	yes	characteristic
266	C₂³	—	118	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	118	2	no	—
271	Q₈	—	118	1	no	—
272	Q₈	—	118	1	no	—
273	Q₈	—	472	1	no	—
274	Q₈	—	472	1	no	—
275	Q₈	—	472	1	no	—
276	Q₈	#11745685	1	2	yes	—
277	D₄	—	118	1	no	—
278	D₄	—	118	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	—	59	1	no	—
284	C₂×C₄	—	59	1	no	—
285	C₂×C₄	—	59	1	no	—
286	C₂×C₄	—	59	1	no	—
287	C₂×C₄	—	118	1	no	—
288	C₂×C₄	—	118	1	no	—
289	C₂×C₄	—	118	1	no	—
290	C₂×C₄	—	236	2	no	—
291	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
292	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	118	1	no	—
297	C₈	—	118	1	no	—
298	C₂²	—	236	1	no	—
299	C₂²	—	236	1	no	—
300	C₂²	—	236	1	no	—
301	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	59	2	no	—
309	C₄	—	118	1	no	—
310	C₄	—	118	1	no	—
311	C₄	—	118	1	no	—
312	C₄	—	236	1	no	—
313	C₄	—	236	1	no	—
314	C₄	—	236	1	no	—
315	C₄	C₂×#11745685	1	1	yes	characteristic
316	C₄	#14273774	1	1	yes	characteristic
317	C₂	—	472	1	no	—
318	C₂	#16838061	1	1	yes	characteristic
319	C₂	#137×D₅₉	1	1	yes	characteristic
320	C₂	#16838211	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#441	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#441	D₅₉
hypocenter	C₅₉	#3369
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₁₁₈	C₂	semidirect
5. #16837751	C₂	semidirect
7. #16837687	C₂	semidirect
8. #16837686	C₂	semidirect
11. #16837610	C₂	semidirect
12. #16837530	C₂	semidirect
13. #16837506	C₂	semidirect
15. C₅₉×#441	C₂	semidirect
174. #441	D₅₉	semidirect
176. C₅₉	#3369	semidirect