Subgroups — #562×D₄₇ #20257510

Order 6016.

1152 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	#15689871	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689870	C₂	1	4	yes	maximal, complemented
3	#15689869	C₂	1	4	yes	maximal, complemented
4	#15689868	C₂	1	4	yes	maximal, complemented
5	#15689867	C₂	1	2	yes	maximal, complemented
6	#125×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	#124×D₄₇	C₂	1	2	yes	maximal, complemented
8	#123×D₄₇	C₂	1	4	yes	maximal, complemented
9	C₂²⋊Q₈×D₄₇	C₂	1	4	yes	maximal, complemented
10	#121×D₄₇	C₂	1	4	yes	maximal, complemented
11	C₄₇×#562	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₄₇	C₂²	1	2	yes	—
13	C₂×D₄×D₄₇	C₂²	1	4	yes	—
14	C₂×D₄×D₄₇	C₂²	1	2	yes	—
15	C₂²×C₄×D₄₇	C₂²	1	4	yes	—
16	#13094016	C₂²	1	1	yes	characteristic, complemented
17	#13094015	C₂²	1	2	yes	complemented
18	#13094014	C₂²	1	4	yes	complemented
19	#13094013	C₂²	1	2	yes	complemented
20	#13094012	C₂²	1	2	yes	complemented
21	#13094011	C₂²	1	2	yes	complemented
22	#13094010	C₂²	1	4	yes	complemented
23	#13094010	C₂²	1	4	yes	complemented
24	#13094009	C₂²	1	4	yes	complemented
25	#13094009	C₂²	1	4	yes	complemented
26	#13094008	C₂²	1	4	yes	complemented
27	#13094007	C₂²	1	4	yes	complemented
28	#13094006	C₂²	1	4	yes	complemented
29	#13094006	C₂²	1	4	yes	complemented
30	#13094005	C₂²	1	4	yes	complemented
31	#13094004	C₂²	1	4	yes	complemented
32	#13094003	C₂²	1	4	yes	complemented
33	#13094002	C₂²	1	4	yes	complemented
34	#13094002	C₂²	1	4	yes	complemented
35	#13094001	C₂²	1	4	yes	complemented
36	#13094001	C₂²	1	4	yes	complemented
37	#13094000	C₂²	1	4	yes	complemented
38	#13093999	C₂²	1	4	yes	complemented
39	#13093998	C₂²	1	4	yes	complemented
40	#13093997	C₂²	1	4	yes	complemented
41	#13093997	C₂²	1	4	yes	complemented
42	#13093996	C₂²	1	4	yes	complemented
43	#13093996	C₂²	1	4	yes	complemented
44	#13093995	C₂²	1	4	yes	complemented
45	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
46	C₄⋊C₄×D₄₇	C₂²	1	4	yes	—
47	C₄⋊C₄×D₄₇	C₂²	1	2	yes	—
48	C₂²⋊C₄×D₄₇	C₂²	1	4	yes	—
49	C₂²⋊C₄×D₄₇	C₂²	1	4	yes	—
50	C₂²⋊C₄×D₄₇	C₂²	1	4	yes	—
51	C₄²×D₄₇	C₂²	1	1	yes	characteristic
52	#125×C₄₇	C₂²	1	1	yes	characteristic, complemented
53	#124×C₄₇	C₂²	1	2	yes	complemented
54	#123×C₄₇	C₂²	1	4	yes	complemented
55	C₂²⋊Q₈×C₄₇	C₂²	1	4	yes	complemented
56	#121×C₄₇	C₂²	1	4	yes	complemented
57	C₂³×D₄₇	C₂³	1	4	yes	—
58	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
59	C₂×D₄×C₄₇	C₂³	1	4	yes	—
60	C₂×D₄×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
62	C₂×C₂²⋊D₄₇	C₂³	1	4	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	Q₈×D₄₇	—	4	4	no	—
67	D₄×D₄₇	—	4	8	no	—
68	D₄×D₄₇	—	4	4	no	—
69	C₂×D₁₈₈	C₂³	1	2	yes	—
70	C₂×D₁₈₈	C₂³	1	4	yes	—
71	C₂×C₄×D₄₇	—	4	8	no	—
72	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₄₇	C₂³	1	2	yes	—
74	C₂×C₄×D₄₇	C₂³	1	4	yes	—
75	C₂×C₄×D₄₇	C₂³	1	4	yes	—
76	C₂×C₄×D₄₇	C₂³	1	4	yes	—
77	C₂×C₄×D₄₇	C₂³	1	4	yes	—
78	C₂×Dic₉₄	C₂³	1	2	yes	—
79	C₂×Dic₉₄	C₂³	1	4	yes	—
80	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
81	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
82	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
84	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
85	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
86	C₄²×C₄₇	C₂³	1	1	yes	characteristic
87	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
88	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
89	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
90	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
91	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
92	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
93	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
94	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
95	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
96	C₄⋊Dic₄₇	C₂³	1	4	yes	—
97	C₄⋊Dic₄₇	C₂³	1	4	yes	—
98	C₄⋊Dic₄₇	C₂³	1	2	yes	—
99	Dic₄₇⋊C₄	C₂³	1	4	yes	—
100	Dic₄₇⋊C₄	C₂³	1	4	yes	—
101	Dic₄₇⋊C₄	C₂³	1	4	yes	—
102	Dic₄₇⋊C₄	C₂³	1	4	yes	—
103	Dic₄₇⋊C₄	C₂³	1	4	yes	—
104	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
105	C₄×Dic₄₇	C₂³	1	2	yes	—
106	C₂³×C₄₇	C₂⁴	1	4	yes	—
107	C₂²×D₄₇	—	2	8	no	—
108	C₂²×D₄₇	—	2	8	no	—
109	C₂²×D₄₇	—	2	8	no	—
110	C₂²×D₄₇	C₂⁴	1	4	yes	—
111	C₂²×D₄₇	C₂⁴	1	1	yes	characteristic
112	Q₈×C₄₇	—	4	2	no	—
113	D₄×C₄₇	—	4	2	no	—
114	D₄×C₄₇	—	4	4	no	—
115	C₂×C₄×C₄₇	—	4	4	no	—
116	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
117	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
119	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
120	C₂²⋊D₄₇	—	4	4	no	—
121	C₂²⋊D₄₇	—	4	4	no	—
122	C₂²⋊D₄₇	—	4	4	no	—
123	C₂×Dic₄₇	—	4	4	no	—
124	C₂×Dic₄₇	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₄₇	C₂⁴	1	2	yes	—
126	C₂×Dic₄₇	C₂⁴	1	4	yes	—
127	C₂×Dic₄₇	C₂⁴	1	4	yes	—
128	D₁₈₈	—	4	2	no	—
129	D₁₈₈	—	4	4	no	—
130	C₄×D₄₇	—	2	8	no	—
131	C₄×D₄₇	—	2	8	no	—
132	C₄×D₄₇	—	2	4	no	—
133	C₄×D₄₇	—	2	2	no	—
134	C₄×D₄₇	—	4	4	no	—
135	C₄×D₄₇	—	4	4	no	—
136	Dic₉₄	—	4	2	no	—
137	Dic₉₄	—	4	4	no	—
138	C₂²×C₄₇	—	2	4	no	—
139	C₂²×C₄₇	—	2	4	no	—
140	C₂²×C₄₇	—	2	4	no	—
141	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
142	C₂×D₄₇	—	2	4	no	—
143	C₂×D₄₇	—	2	4	no	—
144	C₂×D₄₇	—	2	4	no	—
145	C₂×D₄₇	—	4	16	no	—
146	C₂×D₄₇	H(5,𝔽₂)	1	4	yes	—
147	C₂×D₄₇	2₋¹⁺⁴	1	2	yes	—
148	C₄×C₄₇	—	2	4	no	—
149	C₄×C₄₇	—	2	2	no	—
150	C₄×C₄₇	—	2	4	no	—
151	C₄×C₄₇	—	2	1	no	—
152	Dic₄₇	—	2	4	no	—
153	Dic₄₇	—	2	2	no	—
154	Dic₄₇	—	2	4	no	—
155	Dic₄₇	—	2	1	no	—
156	C₂×#562	—	47	1	no	maximal, Sylow
157	C₂×C₄₇	—	4	4	no	—
158	C₂×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
159	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
160	D₄₇	—	4	4	no	—
161	D₄₇	#562	1	4	yes	direct product factor
162	#562	—	47	1	no	—
163	#562	—	47	2	no	—
164	#562	—	47	4	no	—
165	#562	—	47	4	no	—
166	#562	—	47	4	no	—
167	#562	D₄₇	1	1	yes	characteristic, direct product factor
168	C₂×#125	—	47	1	no	—
169	C₂×#124	—	47	2	no	—
170	C₂×#123	—	47	4	no	—
171	C₂×C₂²⋊Q₈	—	47	4	no	—
172	C₂×#121	—	47	4	no	—
173	C₄₇	C₂×#562	1	1	yes	characteristic, complemented, Sylow
174	C₂²×Q₈	—	47	2	no	—
175	C₂²×D₄	—	47	4	no	—
176	C₂²×D₄	—	47	2	no	—
177	C₂³×C₄	—	47	4	no	—
178	#125	—	47	4	no	—
179	#125	—	47	2	no	—
180	#125	—	47	1	no	—
181	#125	C₂×D₄₇	1	1	yes	characteristic, complemented
182	#124	—	47	2	no	—
183	#124	—	47	2	no	—
184	#124	—	47	4	no	—
185	#124	—	47	4	no	—
186	#124	—	47	2	no	—
187	#124	C₂×D₄₇	1	2	yes	complemented
188	#123	—	47	4	no	—
189	#123	—	47	4	no	—
190	#123	—	47	4	no	—
191	#123	—	47	4	no	—
192	#123	—	47	4	no	—
193	#123	—	47	4	no	—
194	#123	—	47	4	no	—
195	#123	C₂×D₄₇	1	4	yes	complemented
196	C₂²⋊Q₈	—	47	4	no	—
197	C₂²⋊Q₈	—	47	4	no	—
198	C₂²⋊Q₈	—	47	4	no	—
199	C₂²⋊Q₈	—	47	4	no	—
200	C₂²⋊Q₈	—	47	4	no	—
201	C₂²⋊Q₈	—	47	4	no	—
202	C₂²⋊Q₈	—	47	4	no	—
203	C₂²⋊Q₈	C₂×D₄₇	1	4	yes	complemented
204	#121	—	47	4	no	—
205	#121	—	47	4	no	—
206	#121	—	47	4	no	—
207	#121	—	47	4	no	—
208	#121	—	47	4	no	—
209	#121	—	47	4	no	—
210	#121	—	47	4	no	—
211	#121	C₂×D₄₇	1	4	yes	complemented
212	C₂×C₄⋊C₄	—	47	4	no	—
213	C₂×C₄⋊C₄	—	47	4	no	—
214	C₂×C₄⋊C₄	—	47	2	no	—
215	C₂×C₂²⋊C₄	—	47	4	no	—
216	C₂×C₂²⋊C₄	—	47	4	no	—
217	C₂×C₂²⋊C₄	—	47	4	no	—
218	C₂×C₄²	—	47	1	no	—
219	C₂⁴	—	47	4	no	—
220	C₂×Q₈	—	47	4	no	—
221	C₂×Q₈	—	47	2	no	—
222	C₂×Q₈	—	188	4	no	—
223	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
224	C₂×D₄	—	47	4	no	—
225	C₂×D₄	—	47	4	no	—
226	C₂×D₄	—	47	4	no	—
227	C₂×D₄	—	47	4	no	—
228	C₂×D₄	—	47	2	no	—
229	C₂×D₄	—	188	8	no	—
230	C₂×D₄	—	188	4	no	—
231	C₂×D₄	C₂²×D₄₇	1	4	yes	—
232	C₂×D₄	C₂²×D₄₇	1	2	yes	—
233	C₂²×C₄	—	47	4	no	—
234	C₂²×C₄	—	47	1	no	—
235	C₂²×C₄	—	47	2	no	—
236	C₂²×C₄	—	47	4	no	—
237	C₂²×C₄	—	47	4	no	—
238	C₂²×C₄	—	47	4	no	—
239	C₂²×C₄	—	47	4	no	—
240	C₂²×C₄	—	188	8	no	—
241	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
242	C₄⋊C₄	—	47	4	no	—
243	C₄⋊C₄	—	47	4	no	—
244	C₄⋊C₄	—	47	4	no	—
245	C₄⋊C₄	—	47	4	no	—
246	C₄⋊C₄	—	47	4	no	—
247	C₄⋊C₄	—	47	4	no	—
248	C₄⋊C₄	—	47	4	no	—
249	C₄⋊C₄	—	47	2	no	—
250	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
251	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
252	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
253	C₂²⋊C₄	—	47	4	no	—
254	C₂²⋊C₄	—	47	4	no	—
255	C₂²⋊C₄	—	47	4	no	—
256	C₂²⋊C₄	—	47	4	no	—
257	C₂²⋊C₄	—	47	4	no	—
258	C₂²⋊C₄	—	47	4	no	—
259	C₂²⋊C₄	—	47	4	no	—
260	C₂²⋊C₄	—	47	4	no	—
261	C₂²⋊C₄	—	47	4	no	—
262	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
263	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
264	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
265	C₄²	—	47	2	no	—
266	C₄²	—	47	1	no	—
267	C₄²	C₂²×D₄₇	1	1	yes	characteristic
268	C₂³	—	47	1	no	—
269	C₂³	—	47	4	no	—
270	C₂³	—	94	8	no	—
271	C₂³	—	94	8	no	—
272	C₂³	—	94	8	no	—
273	C₂³	C₂³×D₄₇	1	4	yes	—
274	Q₈	—	4	2	no	—
275	Q₈	—	188	4	no	—
276	Q₈	—	188	2	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	2	no	—
279	D₄	—	188	4	no	—
280	D₄	—	188	2	no	—
281	D₄	—	188	4	no	—
282	D₄	—	188	4	no	—
283	D₄	—	188	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	47	1	no	—
286	C₂×C₄	—	47	2	no	—
287	C₂×C₄	—	47	4	no	—
288	C₂×C₄	—	47	4	no	—
289	C₂×C₄	—	94	2	no	—
290	C₂×C₄	—	94	8	no	—
291	C₂×C₄	—	94	8	no	—
292	C₂×C₄	—	94	4	no	—
293	C₂×C₄	—	188	4	no	—
294	C₂×C₄	—	188	4	no	—
295	C₂×C₄	—	188	4	no	—
296	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₄₇	1	2	yes	—
298	C₂×C₄	C₂³×D₄₇	1	4	yes	—
299	C₂×C₄	C₂³×D₄₇	1	4	yes	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	47	4	no	—
304	C₂²	—	47	2	no	—
305	C₂²	—	94	4	no	—
306	C₂²	—	94	4	no	—
307	C₂²	—	94	4	no	—
308	C₂²	—	188	16	no	—
309	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	94	4	no	—
315	C₄	—	94	4	no	—
316	C₄	—	94	2	no	—
317	C₄	—	94	1	no	—
318	C₂	—	4	4	no	—
319	C₂	—	47	4	no	—
320	C₂	—	188	4	no	—
321	C₂	H(5,𝔽₂)×D₄₇	1	2	yes	—
322	C₂	2₋¹⁺⁴×D₄₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#562	C₂
hypercenter	#562	D₄₇
hypocenter	C₄₇	C₂×#562
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #562

Complements

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

N	H ≅ G/N	type
1. #15689871	C₂	semidirect
2. #15689870	C₂	semidirect
3. #15689869	C₂	semidirect
4. #15689868	C₂	semidirect
5. #15689867	C₂	semidirect
6. #125×D₄₇	C₂	semidirect
7. #124×D₄₇	C₂	semidirect
8. #123×D₄₇	C₂	semidirect
9. C₂²⋊Q₈×D₄₇	C₂	semidirect
10. #121×D₄₇	C₂	semidirect
11. C₄₇×#562	C₂	semidirect
16. #13094016	C₂²	semidirect
17. #13094015	C₂²	semidirect
18. #13094014	C₂²	semidirect
19. #13094013	C₂²	semidirect
20. #13094012	C₂²	semidirect
21. #13094011	C₂²	semidirect
22. #13094010	C₂²	semidirect
23. #13094010	C₂²	semidirect
24. #13094009	C₂²	semidirect
25. #13094009	C₂²	semidirect
26. #13094008	C₂²	semidirect
27. #13094007	C₂²	semidirect
28. #13094006	C₂²	semidirect
29. #13094006	C₂²	semidirect
30. #13094005	C₂²	semidirect
31. #13094004	C₂²	semidirect
32. #13094003	C₂²	semidirect
33. #13094002	C₂²	semidirect
34. #13094002	C₂²	semidirect
35. #13094001	C₂²	semidirect
36. #13094001	C₂²	semidirect
37. #13094000	C₂²	semidirect
38. #13093999	C₂²	semidirect
39. #13093998	C₂²	semidirect
40. #13093997	C₂²	semidirect
41. #13093997	C₂²	semidirect
42. #13093996	C₂²	semidirect
43. #13093996	C₂²	semidirect
44. #13093995	C₂²	semidirect
52. #125×C₄₇	C₂²	semidirect
53. #124×C₄₇	C₂²	semidirect
54. #123×C₄₇	C₂²	semidirect
55. C₂²⋊Q₈×C₄₇	C₂²	semidirect
56. #121×C₄₇	C₂²	semidirect
161. D₄₇	#562	direct (N × H)
167. #562	D₄₇	direct (N × H)
173. C₄₇	C₂×#562	semidirect
181. #125	C₂×D₄₇	semidirect
187. #124	C₂×D₄₇	semidirect
195. #123	C₂×D₄₇	semidirect
203. C₂²⋊Q₈	C₂×D₄₇	semidirect
211. #121	C₂×D₄₇	semidirect

Subgroups — #562×D47 #20257510

Conjugacy classes of subgroups

Complements

Subgroups — #562×D₄₇ #20257510