Subgroups — #20256878

Order 6016.

1648 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#41×D₄₇	C₂	1	2	yes	maximal, complemented
2	Q₈×D₁₈₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689780	C₂	1	2	yes	maximal, complemented
4	#15689778	C₂	1	3	yes	maximal, complemented
5	#15689770	C₂	1	6	yes	maximal, complemented
6	#15689758	C₂	1	3	yes	maximal, complemented
7	#15689671	C₂	1	3	yes	maximal, complemented
8	#15689633	C₂	1	6	yes	maximal, complemented
9	C₄×#10625354	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×D₄×D₄₇	C₂	1	3	yes	maximal, complemented
11	C₄₇×#550	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625354	C₂²	1	1	yes	characteristic
13	C₂×#10625350	C₂²	1	6	yes	complemented
14	C₂×#10625352	C₂²	1	6	yes	complemented
15	#41×D₄₇	—	2	16	no	—
16	C₂×Q₈×D₄₇	C₂²	1	2	yes	complemented
17	C₂×D₄×D₄₇	C₂²	1	3	yes	—
18	C₂×D₄×D₄₇	C₂²	1	6	yes	complemented
19	C₂²×C₄×D₄₇	C₂²	1	6	yes	complemented
20	#13094024	C₂²	1	3	yes	complemented
21	#13094011	C₂²	1	6	yes	complemented
22	#13094006	C₂²	1	12	yes	complemented
23	#13094004	C₂²	1	6	yes	complemented
24	#13094003	C₂²	1	6	yes	complemented
25	#13093995	C₂²	1	6	yes	complemented
26	#13093995	C₂²	1	6	yes	complemented
27	#13093995	C₂²	1	6	yes	complemented
28	#13093992	C₂²	1	12	yes	complemented
29	Q₈×Dic₄₇	C₂²	1	1	yes	characteristic, complemented
30	C₄×D₁₈₈	C₂²	1	3	yes	—
31	C₄×D₁₈₈	C₂²	1	3	yes	complemented
32	D₄×Dic₄₇	C₂²	1	3	yes	complemented
33	D₄×Dic₄₇	C₂²	1	6	yes	complemented
34	#13093986	C₂²	1	1	yes	characteristic
35	#13093986	C₂²	1	3	yes	complemented
36	C₄×C₂²⋊D₄₇	C₂²	1	6	yes	complemented
37	(C₂²⋊D₄₇)⋊C₄	C₂²	1	6	yes	complemented
38	#13093982	C₂²	1	3	yes	complemented
39	C₄⋊C₄×D₄₇	C₂²	1	3	yes	complemented
40	C₂²⋊C₄×D₄₇	C₂²	1	6	yes	complemented
41	C₄²×D₄₇	C₂²	1	3	yes	complemented
42	C₂×#41×C₄₇	C₂²	1	2	yes	complemented
43	#127×C₄₇	C₂²	1	3	yes	complemented
44	#121×C₄₇	C₂²	1	6	yes	complemented
45	C₄×Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
46	C₄×D₄×C₄₇	C₂²	1	3	yes	complemented
47	C₂³×D₄₇	C₂³	1	6	yes	—
48	#41×C₄₇	—	2	8	no	—
49	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₄₇	C₂³	1	3	yes	—
51	C₂×D₄×C₄₇	C₂³	1	6	yes	—
52	C₂×D₄×C₄₇	C₂³	1	6	yes	—
53	C₂²×C₄×C₄₇	C₂³	1	6	yes	—
54	C₂×C₂²⋊D₄₇	C₂³	1	6	yes	—
55	C₂×C₂²⋊D₄₇	C₂³	1	12	yes	—
56	C₂²×Dic₄₇	C₂³	1	6	yes	—
57	#10625354	D₄	1	8	yes	complemented
58	Q₈×D₄₇	—	2	8	no	—
59	#10625352	—	2	24	no	—
60	D₄×D₄₇	—	2	24	no	—
61	D₄×D₄₇	—	2	12	no	—
62	#10625350	—	2	24	no	—
63	C₂×D₁₈₈	C₂³	1	3	yes	—
64	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₄₇	—	2	24	no	—
66	C₂×C₄×D₄₇	C₂³	1	3	yes	—
67	C₂×C₄×D₄₇	C₂³	1	6	yes	—
68	C₂×C₄×D₄₇	C₂³	1	2	yes	—
69	C₂×C₄×D₄₇	C₂³	1	6	yes	—
70	C₂×C₄×D₄₇	C₂³	1	3	yes	—
71	C₂×C₄×D₄₇	C₂³	1	3	yes	—
72	C₂×Dic₉₄	C₂³	1	6	yes	—
73	C₄⋊C₄×C₄₇	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₄₇	C₂³	1	3	yes	—
75	C₂²⋊C₄×C₄₇	C₂³	1	6	yes	—
76	C₄²×C₄₇	C₂³	1	3	yes	—
77	C₂²⋊Dic₄₇	C₂³	1	6	yes	—
78	C₂²⋊Dic₄₇	C₂³	1	12	yes	—
79	(C₂×D₄₇)⋊C₄	C₂³	1	2	yes	—
80	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
81	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
82	C₄⋊Dic₄₇	C₂³	1	3	yes	—
83	C₄⋊Dic₄₇	C₂³	1	3	yes	—
84	C₄⋊Dic₄₇	C₂³	1	6	yes	—
85	Dic₄₇⋊C₄	C₂³	1	6	yes	—
86	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
87	C₄×Dic₄₇	C₂³	1	3	yes	—
88	C₄×Dic₄₇	C₂³	1	3	yes	—
89	C₂³×C₄₇	C₂⁴	1	6	yes	—
90	C₂²×D₄₇	—	2	12	no	—
91	C₂²×D₄₇	—	2	12	no	—
92	C₂²×D₄₇	—	4	6	no	—
93	C₂²×D₄₇	C₂⁴	1	3	yes	—
94	C₂²×D₄₇	C₂⁴	1	2	yes	—
95	Q₈×C₄₇	C₂×D₄	1	4	yes	complemented
96	D₄×C₄₇	—	2	12	no	—
97	D₄×C₄₇	—	2	6	no	—
98	D₄×C₄₇	—	4	6	no	—
99	C₂×C₄×C₄₇	—	2	12	no	—
100	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
101	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
102	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
103	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
104	C₂²⋊D₄₇	—	2	24	no	—
105	C₂²⋊D₄₇	—	2	12	no	—
106	C₂×Dic₄₇	—	2	12	no	—
107	C₂×Dic₄₇	C₂⁴	1	1	yes	characteristic
108	C₂×Dic₄₇	C₂⁴	1	3	yes	—
109	C₂×Dic₄₇	C₂⁴	1	6	yes	—
110	C₂×Dic₄₇	C₂⁴	1	1	yes	characteristic
111	D₁₈₈	C₂×D₄	1	12	yes	complemented
112	D₁₈₈	#41	1	4	yes	complemented
113	C₄×D₄₇	—	2	12	no	—
114	C₄×D₄₇	—	2	12	no	—
115	C₄×D₄₇	—	2	6	no	—
116	C₄×D₄₇	—	2	4	no	—
117	C₄×D₄₇	—	4	3	no	—
118	C₄×D₄₇	C₂×D₄	1	12	yes	complemented
119	Dic₉₄	—	2	12	no	—
120	C₂²×C₄₇	—	2	6	no	—
121	C₂²×C₄₇	—	2	6	no	—
122	C₂²×C₄₇	—	2	6	no	—
123	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
124	C₂×D₄₇	—	2	2	no	—
125	C₂×D₄₇	—	2	3	no	—
126	C₂×D₄₇	—	2	3	no	—
127	C₂×D₄₇	—	2	2	no	—
128	C₂×D₄₇	—	4	24	no	—
129	C₂×D₄₇	C₂²×D₄	1	6	yes	—
130	C₂×D₄₇	C₂×#41	1	4	yes	complemented
131	C₄×C₄₇	—	2	2	no	—
132	C₄×C₄₇	—	2	3	no	—
133	C₄×C₄₇	C₂²×D₄	1	6	yes	—
134	C₄×C₄₇	C₂×#41	1	2	yes	complemented
135	Dic₄₇	—	2	6	no	—
136	Dic₄₇	—	2	3	no	—
137	Dic₄₇	—	2	1	no	—
138	Dic₄₇	C₂²×D₄	1	2	yes	—
139	D₄×#41	—	47	1	no	maximal, Sylow
140	C₂×C₄₇	—	4	6	no	—
141	C₂×C₄₇	C₂³×D₄	1	1	yes	characteristic
142	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
143	C₂×C₄₇	#585	1	1	yes	characteristic
144	D₄₇	—	2	4	no	—
145	D₄₇	—	2	6	no	—
146	C₂²×#41	—	47	2	no	—
147	#550	D₄₇	1	1	yes	characteristic, complemented
148	D₄×Q₈	—	47	1	no	—
149	#548	—	47	2	no	—
150	#547	—	47	3	no	—
151	#546	—	47	6	no	—
152	D₄²	—	47	3	no	—
153	#532	—	47	3	no	—
154	#525	—	47	6	no	—
155	C₄×#41	—	47	1	no	—
156	C₂×C₄×D₄	—	47	3	no	—
157	C₄₇	D₄×#41	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	47	1	no	—
159	C₂×#41	—	47	6	no	—
160	C₂×#41	—	47	6	no	—
161	C₂×#41	—	94	16	no	—
162	C₂×#41	C₂×D₄₇	1	2	yes	complemented
163	C₂²×Q₈	—	47	2	no	—
164	C₂²×D₄	—	47	3	no	—
165	C₂²×D₄	—	47	6	no	—
166	C₂³×C₄	—	47	6	no	—
167	C₄⋊Q₈	—	47	3	no	—
168	#127	C₂×D₄₇	1	3	yes	complemented
169	#124	—	47	6	no	—
170	#123	—	47	12	no	—
171	C₂²⋊Q₈	—	47	6	no	—
172	C₂²⋊Q₈	—	47	6	no	—
173	#121	—	47	6	no	—
174	#121	—	47	6	no	—
175	#121	—	47	6	no	—
176	#121	C₂×D₄₇	1	6	yes	complemented
177	#120	—	47	12	no	—
178	C₄×Q₈	—	47	1	no	—
179	C₄×Q₈	C₂×D₄₇	1	1	yes	characteristic, complemented
180	C₄×D₄	—	47	3	no	—
181	C₄×D₄	—	47	6	no	—
182	C₄×D₄	—	47	6	no	—
183	C₄×D₄	—	47	1	no	—
184	C₄×D₄	—	47	3	no	—
185	C₄×D₄	—	47	6	no	—
186	C₄×D₄	—	47	3	no	—
187	C₄×D₄	—	47	3	no	—
188	C₄×D₄	C₂×D₄₇	1	3	yes	complemented
189	#117	—	47	3	no	—
190	C₂×C₄⋊C₄	—	47	3	no	—
191	C₂×C₂²⋊C₄	—	47	6	no	—
192	C₂×C₄²	—	47	3	no	—
193	C₂⁴	—	47	6	no	—
194	#41	—	2	8	no	—
195	#41	—	47	8	no	—
196	#41	—	94	24	no	—
197	#41	—	94	24	no	—
198	C₂×Q₈	—	47	6	no	—
199	C₂×Q₈	—	94	8	no	—
200	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
201	C₂×D₄	—	47	6	no	—
202	C₂×D₄	—	47	12	no	—
203	C₂×D₄	—	47	1	no	—
204	C₂×D₄	—	47	3	no	—
205	C₂×D₄	—	94	12	no	—
206	C₂×D₄	—	94	24	no	—
207	C₂×D₄	C₂²×D₄₇	1	3	yes	—
208	C₂×D₄	C₂²×D₄₇	1	6	yes	—
209	C₂×D₄	C₂²×D₄₇	1	6	yes	—
210	C₂²×C₄	—	47	3	no	—
211	C₂²×C₄	—	47	6	no	—
212	C₂²×C₄	—	47	3	no	—
213	C₂²×C₄	—	47	6	no	—
214	C₂²×C₄	—	47	2	no	—
215	C₂²×C₄	—	47	6	no	—
216	C₂²×C₄	—	47	3	no	—
217	C₂²×C₄	—	94	24	no	—
218	C₂²×C₄	C₂²×D₄₇	1	6	yes	—
219	C₄⋊C₄	—	47	6	no	—
220	C₄⋊C₄	—	47	3	no	—
221	C₄⋊C₄	—	47	3	no	—
222	C₄⋊C₄	—	47	6	no	—
223	C₄⋊C₄	C₂²×D₄₇	1	1	yes	characteristic
224	C₄⋊C₄	C₂²×D₄₇	1	3	yes	—
225	C₂²⋊C₄	—	47	6	no	—
226	C₂²⋊C₄	—	47	12	no	—
227	C₂²⋊C₄	—	47	2	no	—
228	C₂²⋊C₄	—	47	6	no	—
229	C₂²⋊C₄	—	47	6	no	—
230	C₂²⋊C₄	C₂²×D₄₇	1	6	yes	—
231	C₄²	—	47	1	no	—
232	C₄²	—	47	3	no	—
233	C₄²	—	47	3	no	—
234	C₄²	C₂²×D₄₇	1	3	yes	—
235	C₂³	—	47	2	no	—
236	C₂³	—	47	3	no	—
237	C₂³	—	94	12	no	—
238	C₂³	—	94	12	no	—
239	C₂³	—	188	6	no	—
240	C₂³	C₂³×D₄₇	1	6	yes	—
241	Q₈	—	94	12	no	—
242	Q₈	D₄×D₄₇	1	4	yes	complemented
243	D₄	—	2	12	no	—
244	D₄	—	2	6	no	—
245	D₄	—	4	6	no	—
246	D₄	—	47	12	no	—
247	D₄	—	47	4	no	—
248	D₄	—	94	12	no	—
249	D₄	—	94	24	no	—
250	C₂×C₄	—	2	12	no	—
251	C₂×C₄	—	47	1	no	—
252	C₂×C₄	—	47	12	no	—
253	C₂×C₄	—	47	1	no	—
254	C₂×C₄	—	47	3	no	—
255	C₂×C₄	—	47	6	no	—
256	C₂×C₄	—	94	4	no	—
257	C₂×C₄	—	94	12	no	—
258	C₂×C₄	—	94	12	no	—
259	C₂×C₄	—	94	12	no	—
260	C₂×C₄	—	94	6	no	—
261	C₂×C₄	—	188	3	no	—
262	C₂×C₄	C₂³×D₄₇	1	3	yes	—
263	C₂×C₄	C₂³×D₄₇	1	2	yes	—
264	C₂×C₄	C₂³×D₄₇	1	3	yes	—
265	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
266	C₂²	—	2	6	no	—
267	C₂²	—	2	6	no	—
268	C₂²	—	2	6	no	—
269	C₂²	—	47	6	no	—
270	C₂²	—	47	4	no	—
271	C₂²	—	94	2	no	—
272	C₂²	—	94	3	no	—
273	C₂²	—	94	3	no	—
274	C₂²	—	94	2	no	—
275	C₂²	—	188	24	no	—
276	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
277	C₄	—	2	2	no	—
278	C₄	—	2	3	no	—
279	C₄	—	47	2	no	—
280	C₄	—	94	6	no	—
281	C₄	—	94	3	no	—
282	C₄	—	94	1	no	—
283	C₄	C₂×D₄×D₄₇	1	6	yes	—
284	C₄	#41×D₄₇	1	2	yes	complemented
285	C₂	—	4	6	no	—
286	C₂	—	94	4	no	—
287	C₂	—	94	6	no	—
288	C₂	C₂²×D₄×D₄₇	1	1	yes	characteristic
289	C₂	C₂×#41×D₄₇	1	1	yes	characteristic
290	C₂	#15689958	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#550	C₂
hypercenter	#550	D₄₇
hypocenter	C₄₇	D₄×#41
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #550

Complements

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

N	H ≅ G/N	type
1. C₂×#41×D₄₇	C₂	semidirect
2. Q₈×D₁₈₈	C₂	semidirect
3. #15689780	C₂	semidirect
4. #15689778	C₂	semidirect
5. #15689770	C₂	semidirect
6. #15689758	C₂	semidirect
7. #15689671	C₂	semidirect
8. #15689633	C₂	semidirect
9. C₄×#10625354	C₂	semidirect
10. C₄×D₄×D₄₇	C₂	semidirect
11. C₄₇×#550	C₂	semidirect
13. C₂×#10625350	C₂²	semidirect
14. C₂×#10625352	C₂²	semidirect
16. C₂×Q₈×D₄₇	C₂²	semidirect
18. C₂×D₄×D₄₇	C₂²	semidirect
19. C₂²×C₄×D₄₇	C₂²	semidirect
20. #13094024	C₂²	semidirect
21. #13094011	C₂²	semidirect
22. #13094006	C₂²	semidirect
23. #13094004	C₂²	semidirect
24. #13094003	C₂²	semidirect
25. #13093995	C₂²	semidirect
26. #13093995	C₂²	semidirect
27. #13093995	C₂²	semidirect
28. #13093992	C₂²	semidirect
29. Q₈×Dic₄₇	C₂²	semidirect
31. C₄×D₁₈₈	C₂²	semidirect
32. D₄×Dic₄₇	C₂²	semidirect
33. D₄×Dic₄₇	C₂²	semidirect
35. #13093986	C₂²	semidirect
36. C₄×C₂²⋊D₄₇	C₂²	semidirect
37. (C₂²⋊D₄₇)⋊C₄	C₂²	semidirect
38. #13093982	C₂²	semidirect
39. C₄⋊C₄×D₄₇	C₂²	semidirect
40. C₂²⋊C₄×D₄₇	C₂²	semidirect
41. C₄²×D₄₇	C₂²	semidirect
42. C₂×#41×C₄₇	C₂²	semidirect
43. #127×C₄₇	C₂²	semidirect
44. #121×C₄₇	C₂²	semidirect
45. C₄×Q₈×C₄₇	C₂²	semidirect
46. C₄×D₄×C₄₇	C₂²	semidirect
57. #10625354	D₄	semidirect
95. Q₈×C₄₇	C₂×D₄	semidirect
111. D₁₈₈	C₂×D₄	semidirect
112. D₁₈₈	#41	semidirect
118. C₄×D₄₇	C₂×D₄	semidirect
130. C₂×D₄₇	C₂×#41	semidirect
134. C₄×C₄₇	C₂×#41	semidirect
147. #550	D₄₇	semidirect
157. C₄₇	D₄×#41	semidirect
162. C₂×#41	C₂×D₄₇	semidirect
168. #127	C₂×D₄₇	semidirect
176. #121	C₂×D₄₇	semidirect
179. C₄×Q₈	C₂×D₄₇	semidirect
188. C₄×D₄	C₂×D₄₇	semidirect
242. Q₈	D₄×D₄₇	semidirect
284. C₄	#41×D₄₇	semidirect