Subgroups — #20255175

Order 6016.

454 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689940	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689930	C₂	1	2	yes	maximal, complemented
3	#15689928	C₂	1	1	yes	maximal, characteristic, complemented
4	#15689507	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689398	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689393	C₂	1	1	yes	maximal, characteristic, complemented
7	#15689387	C₂	1	2	yes	maximal, complemented
8	#15689302	C₂	1	2	yes	maximal, complemented
9	#15689297	C₂	1	2	yes	maximal, complemented
10	#15689241	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#443	C₂	1	1	yes	maximal, characteristic, complemented
12	#13094071	C₂²	1	2	yes	complemented
13	#13094070	C₂²	1	2	yes	—
14	#13094057	—	2	2	no	—
15	#13094057	C₂²	1	2	yes	complemented
16	#13094056	—	2	2	no	—
17	#13094054	C₂²	1	2	yes	—
18	#13094052	—	2	2	no	—
19	#13094052	C₂²	1	2	yes	—
20	#13094049	C₂²	1	2	yes	complemented
21	#13094048	—	2	2	no	—
22	#13094046	—	2	1	no	—
23	#13094046	—	2	1	no	—
24	#13094046	C₂²	1	1	yes	characteristic
25	#13094045	—	2	2	no	—
26	C₂×Dic₁₈₈	—	2	1	no	—
27	C₂×Dic₁₈₈	C₂²	1	1	yes	characteristic, complemented
28	C₂×#10625325	C₂²	1	2	yes	complemented
29	SD₁₆×D₄₇	—	2	2	no	—
30	C₂×D₃₇₆	—	2	1	no	—
31	C₂×D₃₇₆	C₂²	1	1	yes	characteristic
32	#13094030	C₂²	1	2	yes	complemented
33	#13094012	C₂²	1	2	yes	complemented
34	#13093960	C₂²	1	1	yes	characteristic, complemented
35	#13093952	C₂²	1	2	yes	complemented
36	#13093942	C₂²	1	2	yes	complemented
37	#13093940	C₂²	1	2	yes	complemented
38	#135×C₄₇	C₂²	1	1	yes	characteristic
39	#131×C₄₇	C₂²	1	1	yes	characteristic
40	#131×C₄₇	C₂²	1	1	yes	characteristic, complemented
41	#108×C₄₇	C₂²	1	1	yes	characteristic, complemented
42	#104×C₄₇	C₂²	1	1	yes	characteristic
43	#104×C₄₇	C₂²	1	1	yes	characteristic, complemented
44	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
45	#41×C₄₇	C₂³	1	1	yes	characteristic
46	#41×C₄₇	C₂³	1	1	yes	characteristic
47	#10625354	—	2	2	no	—
48	Q₈×D₄₇	—	2	2	no	—
49	#10625352	—	2	2	no	—
50	#10625352	—	4	2	no	—
51	D₄×D₄₇	—	2	2	no	—
52	D₄×D₄₇	—	4	2	no	—
53	#10625350	—	2	2	no	—
54	#10625350	—	2	2	no	—
55	#10625350	C₂³	1	2	yes	—
56	C₂×D₁₈₈	—	2	2	no	—
57	C₂×D₁₈₈	C₂³	1	2	yes	—
58	C₂×Dic₉₄	—	2	2	no	—
59	C₂×Dic₉₄	C₂³	1	2	yes	—
60	Q₁₆×C₄₇	D₄	1	1	yes	characteristic
61	SD₁₆×C₄₇	D₄	1	1	yes	characteristic
62	SD₁₆×C₄₇	D₄	1	1	yes	characteristic
63	D₈×C₄₇	D₄	1	1	yes	characteristic
64	M₁₆×C₄₇	—	2	1	no	—
65	M₁₆×C₄₇	—	2	1	no	—
66	M₁₆×C₄₇	C₂³	1	1	yes	characteristic
67	M₁₆×C₄₇	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₄₇	—	2	1	no	—
69	C₂×C₈×C₄₇	—	2	1	no	—
70	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
72	C₄²×C₄₇	C₂³	1	1	yes	characteristic
73	#10625337	—	2	2	no	—
74	#10625336	—	2	2	no	—
75	#10625335	—	2	2	no	—
76	#10625334	—	2	2	no	—
77	(C₂×D₄₇)⋊C₄	—	4	2	no	—
78	#10625329	C₂³	1	2	yes	—
79	Dic₁₈₈	—	2	2	no	—
80	Dic₁₈₈	—	2	1	no	—
81	Dic₁₈₈	—	2	1	no	—
82	Dic₁₈₈	D₄	1	1	yes	characteristic
83	D₃₇₆	—	2	2	no	—
84	D₃₇₆	—	2	1	no	—
85	D₃₇₆	—	2	1	no	—
86	D₃₇₆	D₄	1	1	yes	characteristic
87	#10625325	—	2	2	no	—
88	#10625325	—	2	2	no	—
89	#10625325	—	2	2	no	—
90	#10625325	—	2	2	no	—
91	#10625325	D₄	1	2	yes	—
92	#10625324	—	2	2	no	—
93	C₈×D₄₇	—	2	2	no	—
94	C₂²×D₄₇	—	2	2	no	—
95	C₂²×D₄₇	—	4	2	no	—
96	Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₄₇	C₂×D₄	1	1	yes	characteristic
98	D₄×C₄₇	—	2	1	no	—
99	D₄×C₄₇	—	2	1	no	—
100	D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
101	D₄×C₄₇	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₄₇	—	2	1	no	—
103	C₂×C₄×C₄₇	—	2	1	no	—
104	C₂×C₄×C₄₇	—	2	1	no	—
105	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
106	C₂²⋊D₄₇	—	2	2	no	—
107	C₂²⋊D₄₇	—	4	2	no	—
108	C₂²⋊D₄₇	—	4	2	no	—
109	C₂×Dic₄₇	—	2	2	no	—
110	C₂×Dic₄₇	—	4	2	no	—
111	D₁₈₈	—	2	2	no	—
112	D₁₈₈	—	2	2	no	—
113	D₁₈₈	—	4	2	no	—
114	D₁₈₈	C₂×D₄	1	2	yes	—
115	C₄×D₄₇	—	2	2	no	—
116	C₄×D₄₇	—	4	2	no	—
117	C₄×D₄₇	—	4	2	no	—
118	Dic₉₄	—	2	2	no	—
119	Dic₉₄	—	2	2	no	—
120	Dic₉₄	—	4	2	no	—
121	Dic₉₄	C₂×D₄	1	2	yes	—
122	C₈×C₄₇	—	2	1	no	—
123	C₈×C₄₇	—	2	1	no	—
124	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
125	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
126	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
127	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
128	C₄₇⋊C₈	—	2	2	no	—
129	C₂²×C₄₇	—	2	1	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	H(5,𝔽₂)	1	1	yes	characteristic
132	C₂×D₄₇	—	2	2	no	—
133	C₂×D₄₇	—	4	2	no	—
134	C₂×D₄₇	—	8	2	no	—
135	C₂×D₄₇	—	16	2	no	—
136	C₄×C₄₇	—	2	1	no	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	—	4	1	no	—
139	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
140	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
141	Dic₄₇	—	2	2	no	—
142	Dic₄₇	—	4	2	no	—
143	#3290	—	47	1	no	maximal, Sylow
144	C₂×C₄₇	—	2	1	no	—
145	C₂×C₄₇	—	4	1	no	—
146	C₂×C₄₇	—	4	1	no	—
147	C₂×C₄₇	D₄²	1	1	yes	characteristic
148	D₄₇	—	4	2	no	—
149	D₄₇	—	8	2	no	—
150	#578	—	47	1	no	—
151	#577	—	47	2	no	—
152	#576	—	47	1	no	—
153	#495	—	47	1	no	—
154	#473	—	47	1	no	—
155	#472	—	47	1	no	—
156	#471	—	47	2	no	—
157	#455	—	47	2	no	—
158	#454	—	47	2	no	—
159	#443	—	47	1	no	—
160	#443	D₄₇	1	1	yes	characteristic, complemented
161	C₄₇	#3290	1	1	yes	characteristic, complemented, Sylow
162	2₋¹⁺⁴	—	47	2	no	—
163	H(5,𝔽₂)	—	47	2	no	—
164	#137	—	47	2	no	—
165	#137	—	47	2	no	—
166	#137	—	94	2	no	—
167	#137	—	94	2	no	—
168	AGL(1,ℤ/8)	—	47	2	no	—
169	AGL(1,ℤ/8)	—	47	2	no	—
170	AGL(1,ℤ/8)	—	94	2	no	—
171	AGL(1,ℤ/8)	—	94	2	no	—
172	#135	—	47	1	no	—
173	#135	—	94	1	no	—
174	#135	—	94	1	no	—
175	#135	—	94	2	no	—
176	#135	C₂×D₄₇	1	1	yes	characteristic
177	C₂×Q₁₆	—	47	1	no	—
178	C₂×Q₁₆	—	94	1	no	—
179	C₂×SD₁₆	—	47	2	no	—
180	C₂×SD₁₆	—	94	2	no	—
181	C₂×D₈	—	47	1	no	—
182	C₂×D₈	—	94	1	no	—
183	#131	—	47	2	no	—
184	#131	C₂×D₄₇	1	1	yes	characteristic
185	#131	C₂×D₄₇	1	1	yes	characteristic, complemented
186	#124	—	47	2	no	—
187	#108	—	47	1	no	—
188	#108	C₂×D₄₇	1	1	yes	characteristic, complemented
189	#104	—	47	2	no	—
190	#104	C₂×D₄₇	1	1	yes	characteristic
191	#104	C₂×D₄₇	1	1	yes	characteristic, complemented
192	#101	—	47	2	no	—
193	#100	—	47	2	no	—
194	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic, complemented
195	#41	—	47	2	no	—
196	#41	—	94	2	no	—
197	#41	—	94	2	no	—
198	#41	—	94	2	no	—
199	#41	—	94	2	no	—
200	#41	—	188	2	no	—
201	#41	C₂²×D₄₇	1	1	yes	characteristic
202	#41	C₂²×D₄₇	1	1	yes	characteristic
203	C₂×Q₈	—	47	2	no	—
204	C₂×Q₈	—	94	2	no	—
205	C₂×Q₈	—	94	2	no	—
206	C₂×D₄	—	47	2	no	—
207	C₂×D₄	—	94	2	no	—
208	C₂×D₄	—	94	2	no	—
209	C₂×D₄	—	188	2	no	—
210	Q₁₆	—	47	1	no	—
211	Q₁₆	—	94	2	no	—
212	Q₁₆	—	94	2	no	—
213	Q₁₆	—	94	1	no	—
214	Q₁₆	—	94	1	no	—
215	Q₁₆	D₁₈₈	1	1	yes	characteristic
216	SD₁₆	—	47	2	no	—
217	SD₁₆	—	94	2	no	—
218	SD₁₆	—	94	2	no	—
219	SD₁₆	—	94	2	no	—
220	SD₁₆	—	94	2	no	—
221	SD₁₆	—	94	2	no	—
222	SD₁₆	—	94	2	no	—
223	SD₁₆	D₁₈₈	1	1	yes	characteristic
224	SD₁₆	D₁₈₈	1	1	yes	characteristic
225	D₈	—	47	1	no	—
226	D₈	—	94	2	no	—
227	D₈	—	94	2	no	—
228	D₈	—	94	1	no	—
229	D₈	—	94	1	no	—
230	D₈	D₁₈₈	1	1	yes	characteristic
231	M₁₆	—	2	1	no	—
232	M₁₆	—	2	1	no	—
233	M₁₆	—	47	2	no	—
234	M₁₆	—	94	2	no	—
235	M₁₆	C₂²×D₄₇	1	1	yes	characteristic
236	M₁₆	C₂²×D₄₇	1	1	yes	characteristic
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	94	2	no	—
240	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
241	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
242	C₂²⋊C₄	—	188	2	no	—
243	C₄²	C₂²×D₄₇	1	1	yes	characteristic
244	C₂³	—	94	2	no	—
245	C₂³	—	188	2	no	—
246	Q₈	—	47	2	no	—
247	Q₈	—	94	2	no	—
248	Q₈	—	94	2	no	—
249	Q₈	—	188	2	no	—
250	Q₈	C₂×D₁₈₈	1	1	yes	characteristic
251	Q₈	C₂×D₁₈₈	1	1	yes	characteristic
252	D₄	—	2	1	no	—
253	D₄	—	2	1	no	—
254	D₄	—	47	2	no	—
255	D₄	—	94	2	no	—
256	D₄	—	94	2	no	—
257	D₄	—	94	2	no	—
258	D₄	—	188	2	no	—
259	D₄	—	188	2	no	—
260	D₄	—	188	2	no	—
261	D₄	C₂×D₁₈₈	1	1	yes	characteristic
262	D₄	C₂×D₁₈₈	1	1	yes	characteristic
263	C₂×C₄	—	2	1	no	—
264	C₂×C₄	—	2	1	no	—
265	C₂×C₄	—	2	1	no	—
266	C₂×C₄	—	94	2	no	—
267	C₂×C₄	—	94	2	no	—
268	C₂×C₄	—	188	2	no	—
269	C₂×C₄	—	188	2	no	—
270	C₂×C₄	—	188	2	no	—
271	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
272	C₈	—	2	1	no	—
273	C₈	—	2	1	no	—
274	C₈	—	94	2	no	—
275	C₈	C₂×D₁₈₈	1	1	yes	characteristic
276	C₈	C₂×D₁₈₈	1	1	yes	characteristic
277	C₈	D₄×D₄₇	1	1	yes	characteristic
278	C₈	D₄×D₄₇	1	1	yes	characteristic
279	C₂²	—	2	1	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	94	2	no	—
282	C₂²	—	376	2	no	—
283	C₂²	—	188	2	no	—
284	C₂²	—	752	2	no	—
285	C₂²	#13094070	1	1	yes	characteristic
286	C₄	—	2	1	no	—
287	C₄	—	2	1	no	—
288	C₄	—	4	1	no	—
289	C₄	—	94	2	no	—
290	C₄	—	188	2	no	—
291	C₄	C₂×D₄×D₄₇	1	1	yes	characteristic
292	C₄	C₂²×D₁₈₈	1	1	yes	characteristic
293	C₂	—	2	1	no	—
294	C₂	—	4	1	no	—
295	C₂	—	4	1	no	—
296	C₂	—	376	2	no	—
297	C₂	—	188	2	no	—
298	C₂	D₄×D₁₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₄×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₄₇×#443	C₂
socle	C₂×C₄₇	D₄²
hypercenter	#443	D₄₇
hypocenter	C₄₇	#3290
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #443

Complements

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

N	H ≅ G/N	type
1. #15689940	C₂	semidirect
2. #15689930	C₂	semidirect
3. #15689928	C₂	semidirect
4. #15689507	C₂	semidirect
5. #15689398	C₂	semidirect
6. #15689393	C₂	semidirect
7. #15689387	C₂	semidirect
8. #15689302	C₂	semidirect
9. #15689297	C₂	semidirect
10. #15689241	C₂	semidirect
11. C₄₇×#443	C₂	semidirect
12. #13094071	C₂²	semidirect
15. #13094057	C₂²	semidirect
20. #13094049	C₂²	semidirect
27. C₂×Dic₁₈₈	C₂²	semidirect
28. C₂×#10625325	C₂²	semidirect
32. #13094030	C₂²	semidirect
33. #13094012	C₂²	semidirect
34. #13093960	C₂²	semidirect
35. #13093952	C₂²	semidirect
36. #13093942	C₂²	semidirect
37. #13093940	C₂²	semidirect
40. #131×C₄₇	C₂²	semidirect
41. #108×C₄₇	C₂²	semidirect
43. #104×C₄₇	C₂²	semidirect
44. C₄×C₈×C₄₇	C₂²	semidirect
160. #443	D₄₇	semidirect
161. C₄₇	#3290	semidirect
185. #131	C₂×D₄₇	semidirect
188. #108	C₂×D₄₇	semidirect
191. #104	C₂×D₄₇	semidirect
194. C₄×C₈	C₂×D₄₇	semidirect