Subgroups — #20257790

Order 6016.

1258 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689866	C₂	1	4	yes	maximal, complemented
2	#15689838	C₂	1	2	yes	maximal, complemented
3	#15689837	C₂	1	4	yes	maximal, complemented
4	#15689758	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄×C₂²⋊D₄₇	C₂	1	4	yes	maximal, complemented
6	#15689690	C₂	1	4	yes	maximal, complemented
7	#15689689	C₂	1	4	yes	maximal, complemented
8	#15689688	C₂	1	4	yes	maximal, complemented
9	#15689671	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689670	C₂	1	2	yes	maximal, complemented
11	D₄²×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	4	yes	complemented
13	C₂×#10625352	C₂²	1	2	yes	complemented
14	C₂²×C₂²⋊D₄₇	C₂²	1	4	yes	complemented
15	C₂×D₄×D₄₇	C₂²	1	4	yes	complemented
16	#13094024	C₂²	1	1	yes	characteristic, complemented
17	#13094023	C₂²	1	2	yes	complemented
18	#13094020	C₂²	1	2	yes	—
19	#13094020	C₂²	1	8	yes	complemented
20	#13094011	C₂²	1	8	yes	complemented
21	#13094011	C₂²	1	2	yes	complemented
22	#13094010	C₂²	1	8	yes	complemented
23	#13094010	C₂²	1	8	yes	complemented
24	#13094010	C₂²	1	4	yes	complemented
25	#13093999	C₂²	1	2	yes	complemented
26	#13093998	C₂²	1	4	yes	complemented
27	#13093997	C₂²	1	4	yes	complemented
28	#13093996	C₂²	1	8	yes	complemented
29	#13093995	C₂²	1	4	yes	complemented
30	#13093995	C₂²	1	4	yes	complemented
31	#13093995	C₂²	1	2	yes	complemented
32	#13093995	C₂²	1	4	yes	complemented
33	#13093994	C₂²	1	4	yes	complemented
34	#13093993	C₂²	1	8	yes	complemented
35	#13093993	C₂²	1	4	yes	complemented
36	#13093992	C₂²	1	8	yes	complemented
37	#13093992	C₂²	1	4	yes	complemented
38	C₄×D₁₈₈	C₂²	1	2	yes	complemented
39	D₄×Dic₄₇	C₂²	1	4	yes	complemented
40	C₄×C₂²⋊D₄₇	C₂²	1	4	yes	complemented
41	#13093983	C₂²	1	4	yes	complemented
42	#13093982	C₂²	1	4	yes	complemented
43	#13093980	C₂²	1	4	yes	complemented
44	C₄²×D₄₇	C₂²	1	1	yes	characteristic, complemented
45	C₂²×D₄×C₄₇	C₂²	1	4	yes	complemented
46	#127×C₄₇	C₂²	1	1	yes	characteristic
47	#121×C₄₇	C₂²	1	4	yes	complemented
48	#120×C₄₇	C₂²	1	4	yes	complemented
49	C₄×D₄×C₄₇	C₂²	1	2	yes	complemented
50	C₂⁴×C₄₇	C₂³	1	4	yes	—
51	C₂³×D₄₇	C₂³	1	4	yes	—
52	C₂×D₄×C₄₇	—	4	8	no	—
53	C₂×D₄×C₄₇	C₂³	1	2	yes	—
54	C₂×D₄×C₄₇	C₂³	1	4	yes	—
55	C₂×D₄×C₄₇	C₂³	1	2	yes	—
56	C₂×D₄×C₄₇	C₂³	1	8	yes	—
57	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₄₇	—	4	8	no	—
59	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
60	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
61	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
62	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
64	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
65	C₂²×Dic₄₇	C₂³	1	4	yes	—
66	#10625352	—	4	4	no	—
67	D₄×D₄₇	—	4	8	no	—
68	#10625350	—	4	8	no	—
69	C₂×D₁₈₈	C₂³	1	2	yes	—
70	C₂×D₁₈₈	C₂³	1	4	yes	—
71	C₂×C₄×D₄₇	C₂³	1	2	yes	—
72	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₄₇	C₂³	1	4	yes	—
74	C₂×Dic₉₄	C₂³	1	2	yes	—
75	C₂×Dic₉₄	C₂³	1	4	yes	—
76	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
79	C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
81	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
82	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
83	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
84	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
85	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
86	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
87	C₄⋊Dic₄₇	C₂³	1	4	yes	—
88	C₄⋊Dic₄₇	C₂³	1	2	yes	—
89	Dic₄₇⋊C₄	C₂³	1	4	yes	—
90	C₄×Dic₄₇	C₂³	1	2	yes	—
91	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
92	C₄×Dic₄₇	C₂³	1	8	yes	—
93	C₂³×C₄₇	—	4	4	no	—
94	C₂³×C₄₇	—	4	8	no	—
95	C₂³×C₄₇	C₂⁴	1	4	yes	—
96	C₂³×C₄₇	C₂⁴	1	4	yes	—
97	C₂²×D₄₇	—	4	8	no	—
98	C₂²×D₄₇	—	4	4	no	—
99	C₂²×D₄₇	C₂⁴	1	4	yes	—
100	C₂²×D₄₇	C₂⁴	1	1	yes	characteristic
101	D₄×C₄₇	—	2	8	no	—
102	D₄×C₄₇	—	2	4	no	—
103	D₄×C₄₇	—	4	2	no	—
104	D₄×C₄₇	—	4	8	no	—
105	C₂×C₄×C₄₇	—	2	8	no	—
106	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
107	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
109	C₂²⋊D₄₇	—	2	8	no	—
110	C₂²⋊D₄₇	—	4	4	no	—
111	C₂²⋊D₄₇	—	4	4	no	—
112	C₂²⋊D₄₇	—	4	8	no	—
113	C₂²⋊D₄₇	—	4	4	no	—
114	C₂²⋊D₄₇	—	4	8	no	—
115	C₂×Dic₄₇	—	2	8	no	—
116	C₂×Dic₄₇	C₂⁴	1	4	yes	—
117	C₂×Dic₄₇	C₂⁴	1	4	yes	—
118	C₂×Dic₄₇	C₂⁴	1	2	yes	—
119	C₂×Dic₄₇	C₂⁴	1	1	yes	characteristic
120	D₁₈₈	—	2	4	no	—
121	D₁₈₈	—	4	4	no	—
122	C₄×D₄₇	—	2	4	no	—
123	C₄×D₄₇	—	2	8	no	—
124	C₄×D₄₇	—	2	2	no	—
125	Dic₉₄	—	4	4	no	—
126	Dic₉₄	—	4	2	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	4	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	2	8	no	—
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	—	8	8	no	—
133	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
134	C₂×D₄₇	—	2	2	no	—
135	C₂×D₄₇	—	2	4	no	—
136	C₂×D₄₇	—	2	1	no	—
137	C₂×D₄₇	—	2	4	no	—
138	C₂×D₄₇	—	2	4	no	—
139	C₂×D₄₇	—	8	8	no	—
140	C₄×C₄₇	—	2	1	no	—
141	C₄×C₄₇	—	2	4	no	—
142	C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
143	Dic₄₇	—	2	2	no	—
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	2	4	no	—
146	Dic₄₇	H(5,𝔽₂)	1	2	yes	—
147	#3549	—	47	1	no	maximal, Sylow
148	C₂×C₄₇	—	4	4	no	—
149	C₂×C₄₇	—	4	4	no	—
150	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
151	C₂×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
152	D₄₇	—	4	1	no	—
153	D₄₇	—	4	4	no	—
154	H(3,𝔽₄)	—	47	4	no	—
155	#555	—	47	4	no	—
156	#555	—	47	2	no	—
157	D₄²	—	47	4	no	—
158	D₄²	—	47	1	no	—
159	D₄²	D₄₇	1	1	yes	characteristic, complemented
160	#535	—	47	4	no	—
161	#535	—	47	4	no	—
162	#535	—	47	4	no	—
163	#532	—	47	1	no	—
164	#532	—	47	2	no	—
165	C₄₇	#3549	1	1	yes	characteristic, complemented, Sylow
166	C₂×#41	—	47	2	no	—
167	C₂×#41	—	47	4	no	—
168	C₂²×D₄	—	47	4	no	—
169	C₂²×D₄	—	47	4	no	—
170	C₂²×D₄	C₂×D₄₇	1	4	yes	complemented
171	C₄⋊Q₈	—	47	2	no	—
172	C₄⋊Q₈	—	47	1	no	—
173	#127	—	47	2	no	—
174	#127	—	47	8	no	—
175	#127	C₂×D₄₇	1	1	yes	characteristic
176	#124	—	47	2	no	—
177	#124	—	47	8	no	—
178	#124	—	47	4	no	—
179	#124	—	47	8	no	—
180	#124	—	47	8	no	—
181	#121	—	47	4	no	—
182	#121	—	47	2	no	—
183	#121	—	47	4	no	—
184	#121	—	47	4	no	—
185	#121	—	47	8	no	—
186	#121	—	47	4	no	—
187	#121	—	47	2	no	—
188	#121	—	47	4	no	—
189	#121	C₂×D₄₇	1	4	yes	complemented
190	#120	—	47	4	no	—
191	#120	—	47	8	no	—
192	#120	—	47	4	no	—
193	#120	—	47	8	no	—
194	#120	—	47	4	no	—
195	#120	C₂×D₄₇	1	4	yes	complemented
196	C₄×D₄	—	47	4	no	—
197	C₄×D₄	—	47	4	no	—
198	C₄×D₄	—	47	2	no	—
199	C₄×D₄	C₂×D₄₇	1	2	yes	complemented
200	#117	—	47	4	no	—
201	#117	—	47	4	no	—
202	#117	—	47	4	no	—
203	C₂×C₄²	—	47	1	no	—
204	C₂⁴	—	47	4	no	—
205	C₂⁴	C₂²×D₄₇	1	4	yes	—
206	#41	—	188	8	no	—
207	#41	—	188	4	no	—
208	C₂×Q₈	—	47	4	no	—
209	C₂×Q₈	—	47	2	no	—
210	C₂×D₄	—	4	8	no	—
211	C₂×D₄	—	47	4	no	—
212	C₂×D₄	—	47	4	no	—
213	C₂×D₄	—	47	8	no	—
214	C₂×D₄	—	47	8	no	—
215	C₂×D₄	—	47	4	no	—
216	C₂×D₄	—	47	4	no	—
217	C₂×D₄	—	47	2	no	—
218	C₂×D₄	—	47	4	no	—
219	C₂×D₄	—	188	8	no	—
220	C₂×D₄	—	188	8	no	—
221	C₂×D₄	C₂²×D₄₇	1	4	yes	—
222	C₂×D₄	C₂²×D₄₇	1	2	yes	—
223	C₂×D₄	C₂²×D₄₇	1	2	yes	—
224	C₂×D₄	C₂²×D₄₇	1	8	yes	—
225	C₂²×C₄	—	47	4	no	—
226	C₂²×C₄	—	47	1	no	—
227	C₂²×C₄	—	47	2	no	—
228	C₂²×C₄	—	47	4	no	—
229	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
230	C₄⋊C₄	—	47	4	no	—
231	C₄⋊C₄	—	47	4	no	—
232	C₄⋊C₄	—	47	2	no	—
233	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
234	C₂²⋊C₄	—	47	4	no	—
235	C₂²⋊C₄	—	47	8	no	—
236	C₂²⋊C₄	—	47	8	no	—
237	C₂²⋊C₄	—	47	4	no	—
238	C₂²⋊C₄	—	47	8	no	—
239	C₂²⋊C₄	—	47	4	no	—
240	C₂²⋊C₄	—	47	4	no	—
241	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
242	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
243	C₄²	—	47	2	no	—
244	C₄²	—	47	1	no	—
245	C₄²	—	47	8	no	—
246	C₄²	C₂²×D₄₇	1	1	yes	characteristic
247	C₂³	—	4	8	no	—
248	C₂³	—	4	4	no	—
249	C₂³	—	47	4	no	—
250	C₂³	—	47	1	no	—
251	C₂³	—	188	4	no	—
252	C₂³	—	188	8	no	—
253	C₂³	C₂³×D₄₇	1	4	yes	—
254	C₂³	C₂³×D₄₇	1	4	yes	—
255	Q₈	—	188	2	no	—
256	Q₈	—	188	4	no	—
257	D₄	—	2	4	no	—
258	D₄	—	2	8	no	—
259	D₄	—	4	8	no	—
260	D₄	—	4	2	no	—
261	D₄	—	94	8	no	—
262	D₄	—	94	4	no	—
263	D₄	—	188	4	no	—
264	D₄	—	188	8	no	—
265	D₄	—	188	8	no	—
266	D₄	—	188	4	no	—
267	D₄	—	188	4	no	—
268	D₄	—	188	4	no	—
269	C₂×C₄	—	2	8	no	—
270	C₂×C₄	—	47	1	no	—
271	C₂×C₄	—	47	2	no	—
272	C₂×C₄	—	47	4	no	—
273	C₂×C₄	—	47	4	no	—
274	C₂×C₄	—	94	2	no	—
275	C₂×C₄	—	94	8	no	—
276	C₂×C₄	—	94	8	no	—
277	C₂×C₄	—	94	4	no	—
278	C₂×C₄	C₂³×D₄₇	1	4	yes	—
279	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₄₇	1	2	yes	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	8	no	—
286	C₂²	—	8	8	no	—
287	C₂²	—	94	1	no	—
288	C₂²	—	94	4	no	—
289	C₂²	—	94	2	no	—
290	C₂²	—	94	4	no	—
291	C₂²	—	94	4	no	—
292	C₂²	—	376	8	no	—
293	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
294	C₄	—	2	1	no	—
295	C₄	—	2	4	no	—
296	C₄	—	47	2	no	—
297	C₄	—	94	4	no	—
298	C₄	—	94	4	no	—
299	C₄	—	94	2	no	—
300	C₄	#13094069	1	4	yes	—
301	C₂	—	4	4	no	—
302	C₂	—	4	4	no	—
303	C₂	—	188	4	no	—
304	C₂	—	188	1	no	—
305	C₂	H(5,𝔽₂)×D₄₇	1	1	yes	characteristic
306	C₂	C₂×#13094069	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	D₄²×C₄₇	C₂
hypercenter	D₄²	D₄₇
hypocenter	C₄₇	#3549
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ D₄²

Complements

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

N	H ≅ G/N	type
1. #15689866	C₂	semidirect
2. #15689838	C₂	semidirect
3. #15689837	C₂	semidirect
4. #15689758	C₂	semidirect
5. D₄×C₂²⋊D₄₇	C₂	semidirect
6. #15689690	C₂	semidirect
7. #15689689	C₂	semidirect
8. #15689688	C₂	semidirect
9. #15689671	C₂	semidirect
10. #15689670	C₂	semidirect
11. D₄²×C₄₇	C₂	semidirect
12. C₂×#10625350	C₂²	semidirect
13. C₂×#10625352	C₂²	semidirect
14. C₂²×C₂²⋊D₄₇	C₂²	semidirect
15. C₂×D₄×D₄₇	C₂²	semidirect
16. #13094024	C₂²	semidirect
17. #13094023	C₂²	semidirect
19. #13094020	C₂²	semidirect
20. #13094011	C₂²	semidirect
21. #13094011	C₂²	semidirect
22. #13094010	C₂²	semidirect
23. #13094010	C₂²	semidirect
24. #13094010	C₂²	semidirect
25. #13093999	C₂²	semidirect
26. #13093998	C₂²	semidirect
27. #13093997	C₂²	semidirect
28. #13093996	C₂²	semidirect
29. #13093995	C₂²	semidirect
30. #13093995	C₂²	semidirect
31. #13093995	C₂²	semidirect
32. #13093995	C₂²	semidirect
33. #13093994	C₂²	semidirect
34. #13093993	C₂²	semidirect
35. #13093993	C₂²	semidirect
36. #13093992	C₂²	semidirect
37. #13093992	C₂²	semidirect
38. C₄×D₁₈₈	C₂²	semidirect
39. D₄×Dic₄₇	C₂²	semidirect
40. C₄×C₂²⋊D₄₇	C₂²	semidirect
41. #13093983	C₂²	semidirect
42. #13093982	C₂²	semidirect
43. #13093980	C₂²	semidirect
44. C₄²×D₄₇	C₂²	semidirect
45. C₂²×D₄×C₄₇	C₂²	semidirect
47. #121×C₄₇	C₂²	semidirect
48. #120×C₄₇	C₂²	semidirect
49. C₄×D₄×C₄₇	C₂²	semidirect
159. D₄²	D₄₇	semidirect
165. C₄₇	#3549	semidirect
170. C₂²×D₄	C₂×D₄₇	semidirect
189. #121	C₂×D₄₇	semidirect
195. #120	C₂×D₄₇	semidirect
199. C₄×D₄	C₂×D₄₇	semidirect