Subgroups — #21433899

Order 7552.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16837958	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₅₉	C₂	1	1	yes	maximal, characteristic
3	#16837710	C₂	1	1	yes	maximal, characteristic, complemented
4	#16837709	C₂	1	1	yes	maximal, characteristic, complemented
5	#16837706	C₂	1	1	yes	maximal, characteristic, complemented
6	#16837696	C₂	1	1	yes	maximal, characteristic, complemented
7	#16837695	C₂	1	1	yes	maximal, characteristic, complemented
8	#16837686	C₂	1	1	yes	maximal, characteristic, complemented
9	#16837685	C₂	1	1	yes	maximal, characteristic, complemented
10	#16837443	C₂	1	1	yes	maximal, characteristic
11	#16837441	C₂	1	1	yes	maximal, characteristic
12	#16837424	C₂	1	1	yes	maximal, characteristic
13	#16837409	C₂	1	1	yes	maximal, characteristic
14	#16837409	C₂	1	1	yes	maximal, characteristic
15	C₅₉×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₅₉	C₂²	1	1	yes	characteristic
17	C₂×#11745659	C₂²	1	1	yes	characteristic
18	C₂×#11745659	C₂²	1	1	yes	characteristic
19	#14273726	C₂²	1	1	yes	characteristic
20	#14273726	C₂²	1	1	yes	characteristic
21	#14273719	C₂²	1	1	yes	characteristic
22	#14273705	—	2	1	no	—
23	#14273705	—	2	1	no	—
24	Q₈×Dic₅₉	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₁₈	—	2	1	no	—
26	C₄×Dic₁₁₈	C₂²	1	1	yes	characteristic
27	#14273692	—	2	1	no	—
28	#14273692	—	2	1	no	—
29	#14273692	—	2	1	no	—
30	#14273692	C₂²	1	1	yes	characteristic
31	#14273685	C₂²	1	1	yes	characteristic
32	#14273685	C₂²	1	1	yes	characteristic
33	C₄⋊C₄×D₅₉	—	2	1	no	—
34	C₄⋊C₄×D₅₉	C₂²	1	1	yes	characteristic
35	C₄²×D₅₉	—	2	1	no	—
36	C₄²×D₅₉	C₂²	1	1	yes	characteristic
37	#14273661	C₂²	1	1	yes	characteristic
38	#14273660	C₂²	1	1	yes	characteristic
39	#14273660	C₂²	1	1	yes	characteristic
40	#14273659	C₂²	1	1	yes	characteristic
41	#14273658	C₂²	1	1	yes	characteristic
42	#14273658	C₂²	1	1	yes	characteristic
43	C₄⋊(C₅₉⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₅₉⋊C₈	C₂²	1	1	yes	characteristic
46	#14273652	C₂²	1	1	yes	characteristic
47	#14273652	C₂²	1	1	yes	characteristic
48	#14273651	C₂²	1	1	yes	characteristic
49	#14273651	C₂²	1	1	yes	characteristic
50	#14273650	C₂²	1	1	yes	characteristic
51	#14273650	C₂²	1	1	yes	characteristic
52	#125×C₅₉	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₅₉	C₂²	1	1	yes	characteristic
54	#107×C₅₉	C₂²	1	1	yes	characteristic
55	#106×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	Q₈×D₅₉	—	2	2	no	—
61	Q₈×D₅₉	Q₈	1	4	yes	complemented
62	C₂×C₄×D₅₉	—	2	1	no	—
63	C₂×C₄×D₅₉	—	2	1	no	—
64	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
65	C₂×C₄×D₅₉	D₄	1	1	yes	characteristic
66	C₂×C₄×D₅₉	C₂³	1	1	yes	characteristic
67	C₂×Dic₁₁₈	—	2	1	no	—
68	C₂×Dic₁₁₈	—	2	1	no	—
69	C₂×Dic₁₁₈	—	2	1	no	—
70	C₂×Dic₁₁₈	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₅₉	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₅₉	—	2	1	no	—
74	C₄⋊C₄×C₅₉	—	2	1	no	—
75	C₄⋊C₄×C₅₉	—	2	1	no	—
76	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₅₉	C₂³	1	1	yes	characteristic
79	C₄²×C₅₉	—	2	1	no	—
80	C₄²×C₅₉	C₂³	1	1	yes	characteristic
81	(C₂×D₅₉)⋊C₄	—	2	1	no	—
82	(C₂×D₅₉)⋊C₄	—	2	1	no	—
83	C₄⋊Dic₅₉	—	2	1	no	—
84	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₅₉	C₂³	1	1	yes	characteristic
87	Dic₅₉⋊C₄	—	2	1	no	—
88	Dic₅₉⋊C₄	—	2	1	no	—
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	C₄×Dic₅₉	—	2	1	no	—
95	C₄×Dic₅₉	—	2	1	no	—
96	C₄×Dic₅₉	—	2	1	no	—
97	C₄×Dic₅₉	—	2	1	no	—
98	C₄×Dic₅₉	—	2	1	no	—
99	C₄×Dic₅₉	D₄	1	1	yes	characteristic
100	C₄×Dic₅₉	D₄	1	1	yes	characteristic
101	C₄×Dic₅₉	C₂³	1	1	yes	characteristic
102	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₅₉⋊C₈	C₂³	1	1	yes	characteristic
104	#11745659	—	2	2	no	—
105	#11745659	—	2	2	no	—
106	C₂²×D₅₉	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₅₉	—	2	1	no	—
108	Q₈×C₅₉	C₂×Q₈	1	2	yes	—
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	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₉	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₅₉	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₅₉	—	2	1	no	—
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	C₄×D₅₉	—	2	4	no	—
124	C₄×D₅₉	—	2	2	no	—
125	C₄×D₅₉	—	4	1	no	—
126	C₄×D₅₉	—	4	2	no	—
127	C₄×D₅₉	C₂×Q₈	1	2	yes	—
128	C₄×D₅₉	#41	1	2	yes	—
129	Dic₁₁₈	—	2	1	no	—
130	Dic₁₁₈	—	2	4	no	—
131	Dic₁₁₈	—	4	2	no	—
132	Dic₁₁₈	—	4	2	no	—
133	Dic₁₁₈	C₂×Q₈	1	2	yes	—
134	C₈×C₅₉	—	2	1	no	—
135	C₈×C₅₉	—	2	1	no	—
136	C₅₉⋊C₈	—	2	1	no	—
137	C₅₉⋊C₈	—	2	1	no	—
138	C₂²×C₅₉	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₅₉	—	2	1	no	—
140	C₂×D₅₉	—	2	1	no	—
141	C₂×D₅₉	C₂²⋊Q₈	1	2	yes	—
142	C₄×C₅₉	—	2	2	no	—
143	C₄×C₅₉	—	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	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₅₉	C₂×#41	1	1	yes	characteristic
150	Dic₅₉	—	2	2	no	—
151	Dic₅₉	—	2	1	no	—
152	Dic₅₉	—	4	1	no	—
153	Dic₅₉	—	4	1	no	—
154	Dic₅₉	—	4	1	no	—
155	Dic₅₉	C₂²⋊Q₈	1	2	yes	—
156	Dic₅₉	#137	1	2	yes	—
157	#3078	—	59	1	no	maximal, Sylow
158	C₂×C₅₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₅₉	C₂×#137	1	1	yes	characteristic
160	C₂×C₅₉	#575	1	1	yes	characteristic
161	D₅₉	—	2	2	no	—
162	#533	—	59	1	no	—
163	C₂×C₄×Q₈	—	59	1	no	—
164	#479	—	59	1	no	—
165	#479	—	59	1	no	—
166	#479	—	59	1	no	—
167	#479	D₅₉	1	1	yes	characteristic, complemented
168	#477	—	59	1	no	—
169	#477	—	59	1	no	—
170	#475	—	59	1	no	—
171	#475	—	59	1	no	—
172	#428	—	59	1	no	—
173	#428	—	59	1	no	—
174	#423	—	59	1	no	—
175	#419	—	59	1	no	—
176	#419	—	59	1	no	—
177	C₅₉	#3078	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	59	1	no	—
179	C₂×M₁₆	—	59	1	no	—
180	C₂×M₁₆	—	59	1	no	—
181	C₄⋊Q₈	—	59	1	no	—
182	C₄⋊Q₈	—	59	1	no	—
183	#125	—	59	1	no	—
184	#125	C₂×D₅₉	1	1	yes	characteristic
185	C₂²⋊Q₈	—	118	1	no	—
186	C₂²⋊Q₈	—	118	1	no	—
187	C₄×Q₈	—	59	1	no	—
188	C₄×Q₈	—	59	1	no	—
189	C₄×Q₈	—	59	1	no	—
190	C₄×Q₈	—	118	1	no	—
191	C₄×Q₈	—	118	1	no	—
192	C₄×Q₈	—	118	1	no	—
193	C₄×Q₈	—	118	1	no	—
194	C₄×Q₈	C₂×D₅₉	1	1	yes	characteristic
195	#117	—	59	1	no	—
196	#117	—	59	1	no	—
197	C₂×C₄⋊C₄	—	59	1	no	—
198	C₂×C₄⋊C₄	—	118	1	no	—
199	C₂×C₄²	—	59	1	no	—
200	C₂×C₄²	—	118	1	no	—
201	#107	—	59	1	no	—
202	#107	—	59	1	no	—
203	#107	—	59	1	no	—
204	#107	C₂×D₅₉	1	1	yes	characteristic
205	#106	—	59	1	no	—
206	#106	—	59	1	no	—
207	#106	—	59	1	no	—
208	#106	C₂×D₅₉	1	1	yes	characteristic
209	C₄⋊C₈	—	59	1	no	—
210	C₄⋊C₈	—	59	1	no	—
211	C₄⋊C₈	—	59	1	no	—
212	C₄⋊C₈	C₂×D₅₉	1	1	yes	characteristic
213	#103	—	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	C₂×D₅₉	1	1	yes	characteristic
220	#103	C₂×D₅₉	1	1	yes	characteristic
221	C₂×Q₈	—	59	4	no	—
222	C₂×Q₈	—	59	1	no	—
223	C₂×Q₈	—	118	1	no	—
224	C₂×Q₈	—	118	1	no	—
225	C₂×Q₈	—	118	1	no	—
226	C₂×Q₈	—	118	2	no	—
227	C₂×Q₈	C₂²×D₅₉	1	1	yes	characteristic
228	C₂²×C₄	—	59	1	no	—
229	C₂²×C₄	—	59	1	no	—
230	C₂²×C₄	—	59	1	no	—
231	C₂²×C₄	—	118	1	no	—
232	C₂²×C₄	—	118	1	no	—
233	M₁₆	—	118	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	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	59	1	no	—
243	C₄⋊C₄	—	59	1	no	—
244	C₄⋊C₄	—	59	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	—	118	1	no	—
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	—	118	1	no	—
257	C₂²⋊C₄	—	118	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	59	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	—	118	1	no	—
266	C₄²	—	118	1	no	—
267	C₄²	C₂²×D₅₉	1	1	yes	characteristic
268	C₂³	—	59	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	59	2	no	—
271	Q₈	—	118	4	no	—
272	Q₈	—	118	1	no	—
273	Q₈	—	236	2	no	—
274	Q₈	—	236	2	no	—
275	Q₈	Q₈×D₅₉	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	59	1	no	—
281	C₂×C₄	—	59	1	no	—
282	C₂×C₄	—	59	2	no	—
283	C₂×C₄	—	59	2	no	—
284	C₂×C₄	—	59	1	no	—
285	C₂×C₄	—	118	2	no	—
286	C₂×C₄	—	118	4	no	—
287	C₂×C₄	—	118	1	no	—
288	C₂×C₄	—	118	1	no	—
289	C₂×C₄	—	118	1	no	—
290	C₂×C₄	—	118	1	no	—
291	C₂×C₄	—	236	1	no	—
292	C₂×C₄	—	236	2	no	—
293	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
294	C₂×C₄	D₄×D₅₉	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₅₉	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	118	1	no	—
299	C₈	—	118	1	no	—
300	C₂²	—	59	2	no	—
301	C₂²	—	118	1	no	—
302	C₂²	—	118	1	no	—
303	C₂²	C₂×D₄×D₅₉	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	59	2	no	—
311	C₄	—	59	2	no	—
312	C₄	—	118	2	no	—
313	C₄	—	118	1	no	—
314	C₄	—	236	1	no	—
315	C₄	—	236	1	no	—
316	C₄	—	236	1	no	—
317	C₄	C₂×Q₈×D₅₉	1	1	yes	characteristic
318	C₄	#41×D₅₉	1	1	yes	characteristic
319	C₂	—	118	2	no	—
320	C₂	C₂²⋊Q₈×D₅₉	1	1	yes	characteristic
321	C₂	#137×D₅₉	1	1	yes	characteristic
322	C₂	#16838199	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₉
commutator	C₂×C₄×C₅₉	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₉
Fitting	C₅₉×#479	C₂
socle	C₂²×C₅₉	C₂²×D₄
hypercenter	#479	D₅₉
hypocenter	C₅₉	#3078
LCS	G ▷ C₂×C₄×C₅₉ ▷ C₂×C₅₉ ▷ C₅₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #16837710	C₂	semidirect
4. #16837709	C₂	semidirect
5. #16837706	C₂	semidirect
6. #16837696	C₂	semidirect
7. #16837695	C₂	semidirect
8. #16837686	C₂	semidirect
9. #16837685	C₂	semidirect
15. C₅₉×#479	C₂	semidirect
61. Q₈×D₅₉	Q₈	semidirect
167. #479	D₅₉	semidirect
177. C₅₉	#3078	semidirect