Subgroups — #20257719

Order 6016.

1260 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689862	C₂	1	4	yes	maximal, complemented
2	#15689859	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689794	C₂	1	2	yes	maximal, complemented
4	#15689770	C₂	1	4	yes	maximal, complemented
5	D₄×C₂²⋊D₄₇	C₂	1	4	yes	maximal, complemented
6	#15689726	C₂	1	4	yes	maximal, complemented
7	#15689690	C₂	1	4	yes	maximal, complemented
8	#15689683	C₂	1	4	yes	maximal, complemented
9	#127×D₄₇	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689655	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	C₂²×C₂²⋊D₄₇	C₂²	1	4	yes	complemented
14	C₂×D₄×D₄₇	C₂²	1	4	yes	complemented
15	C₂×D₄×D₄₇	C₂²	1	2	yes	complemented
16	#13094021	C₂²	1	1	yes	characteristic, complemented
17	#13094020	C₂²	1	4	yes	complemented
18	#13094020	C₂²	1	2	yes	complemented
19	#13094020	C₂²	1	4	yes	complemented
20	#13094017	C₂²	1	4	yes	complemented
21	#13094015	C₂²	1	2	yes	complemented
22	#13094011	C₂²	1	4	yes	complemented
23	#13094011	C₂²	1	2	yes	complemented
24	#13094008	C₂²	1	4	yes	complemented
25	#13094006	C₂²	1	8	yes	complemented
26	#13094006	C₂²	1	4	yes	complemented
27	#13094005	C₂²	1	4	yes	complemented
28	#13094004	C₂²	1	2	yes	complemented
29	#13093998	C₂²	1	4	yes	complemented
30	#13093997	C₂²	1	8	yes	complemented
31	#13093996	C₂²	1	8	yes	complemented
32	#13093995	C₂²	1	2	yes	—
33	#13093995	C₂²	1	8	yes	complemented
34	#13093995	C₂²	1	8	yes	complemented
35	#13093993	C₂²	1	8	yes	complemented
36	#13093993	C₂²	1	4	yes	complemented
37	#13093992	C₂²	1	8	yes	complemented
38	C₄×Dic₉₄	C₂²	1	2	yes	complemented
39	D₄×Dic₄₇	C₂²	1	4	yes	complemented
40	#13093986	C₂²	1	4	yes	complemented
41	C₄×C₂²⋊D₄₇	C₂²	1	4	yes	complemented
42	#13093983	C₂²	1	4	yes	complemented
43	C₂²⋊C₄×D₄₇	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	8	yes	—
61	C₂×C₂²⋊D₄₇	C₂³	1	4	yes	—
62	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
64	C₂²×Dic₄₇	C₂³	1	4	yes	—
65	D₄×D₄₇	—	4	8	no	—
66	D₄×D₄₇	—	4	4	no	—
67	#10625350	—	4	8	no	—
68	C₂×D₁₈₈	C₂³	1	4	yes	—
69	C₂×D₁₈₈	C₂³	1	2	yes	—
70	C₂×C₄×D₄₇	C₂³	1	2	yes	—
71	C₂×C₄×D₄₇	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₄₇	C₂³	1	8	yes	—
73	C₂×Dic₉₄	C₂³	1	2	yes	—
74	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
75	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
76	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
77	C₄²×C₄₇	C₂³	1	1	yes	characteristic
78	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
79	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
80	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
81	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
82	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
83	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
84	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
85	C₄⋊Dic₄₇	C₂³	1	4	yes	—
86	C₄⋊Dic₄₇	C₂³	1	4	yes	—
87	C₄⋊Dic₄₇	C₂³	1	2	yes	—
88	Dic₄₇⋊C₄	C₂³	1	4	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	4	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	—	2	8	no	—
98	C₂²×D₄₇	—	4	8	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	1	yes	characteristic
107	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
108	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
109	C₂²⋊D₄₇	—	2	8	no	—
110	C₂²⋊D₄₇	—	4	8	no	—
111	C₂²⋊D₄₇	—	4	8	no	—
112	C₂²⋊D₄₇	—	4	4	no	—
113	C₂²⋊D₄₇	—	4	8	no	—
114	C₂×Dic₄₇	—	4	4	no	—
115	C₂×Dic₄₇	C₂⁴	1	2	yes	—
116	C₂×Dic₄₇	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₄₇	C₂⁴	1	4	yes	—
118	C₂×Dic₄₇	C₂⁴	1	4	yes	—
119	D₁₈₈	—	2	8	no	—
120	D₁₈₈	—	4	2	no	—
121	C₄×D₄₇	—	2	2	no	—
122	C₄×D₄₇	—	2	4	no	—
123	C₄×D₄₇	—	4	8	no	—
124	Dic₉₄	—	2	4	no	—
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	8	no	—
129	C₂²×C₄₇	—	2	4	no	—
130	C₂²×C₄₇	—	8	8	no	—
131	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
132	C₂×D₄₇	—	2	2	no	—
133	C₂×D₄₇	—	2	8	no	—
134	C₂×D₄₇	—	2	4	no	—
135	C₂×D₄₇	—	8	8	no	—
136	C₂×D₄₇	H(5,𝔽₂)	1	2	yes	—
137	C₄×C₄₇	—	2	1	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	H(5,𝔽₂)	1	4	yes	—
140	Dic₄₇	—	2	4	no	—
141	Dic₄₇	—	2	1	no	—
142	Dic₄₇	—	2	2	no	—
143	Dic₄₇	—	2	4	no	—
144	#3543	—	47	1	no	maximal, Sylow
145	C₂×C₄₇	—	4	4	no	—
146	C₂×C₄₇	—	4	4	no	—
147	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
149	D₄₇	—	2	2	no	—
150	D₄₇	—	4	4	no	—
151	#560	—	47	4	no	—
152	#559	—	47	1	no	—
153	#550	—	47	2	no	—
154	#546	—	47	4	no	—
155	D₄²	—	47	4	no	—
156	D₄²	D₄₇	1	1	yes	characteristic, complemented
157	#540	—	47	4	no	—
158	#535	—	47	4	no	—
159	#534	—	47	4	no	—
160	C₂×#127	—	47	1	no	—
161	#529	—	47	2	no	—
162	C₄₇	#3543	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	47	4	no	—
164	C₂²×D₄	—	47	4	no	—
165	C₂²×D₄	—	47	2	no	—
166	C₂²×D₄	—	47	4	no	—
167	C₂²×D₄	C₂×D₄₇	1	4	yes	complemented
168	#127	—	47	4	no	—
169	#127	—	47	2	no	—
170	#127	—	47	4	no	—
171	#127	—	47	1	no	—
172	#127	C₂×D₄₇	1	1	yes	characteristic
173	#126	—	47	4	no	—
174	#125	—	47	2	no	—
175	#124	—	47	2	no	—
176	#124	—	47	4	no	—
177	#123	—	47	4	no	—
178	#123	—	47	4	no	—
179	#123	—	47	4	no	—
180	#123	—	47	8	no	—
181	C₂²⋊Q₈	—	47	2	no	—
182	#121	—	47	2	no	—
183	#121	—	47	8	no	—
184	#121	—	47	8	no	—
185	#121	—	47	8	no	—
186	#121	—	47	4	no	—
187	#121	—	47	8	no	—
188	#121	C₂×D₄₇	1	4	yes	complemented
189	#120	—	47	4	no	—
190	#120	—	47	8	no	—
191	#120	—	47	8	no	—
192	#120	C₂×D₄₇	1	4	yes	complemented
193	C₄×Q₈	—	47	2	no	—
194	C₄×D₄	—	47	4	no	—
195	C₄×D₄	—	47	4	no	—
196	C₄×D₄	—	47	4	no	—
197	C₄×D₄	C₂×D₄₇	1	2	yes	complemented
198	#117	—	47	4	no	—
199	C₂×C₂²⋊C₄	—	47	4	no	—
200	C₂×C₄²	—	47	1	no	—
201	C₂⁴	—	47	4	no	—
202	C₂⁴	C₂²×D₄₇	1	4	yes	—
203	#41	—	188	8	no	—
204	C₂×Q₈	—	47	2	no	—
205	C₂×D₄	—	4	8	no	—
206	C₂×D₄	—	47	4	no	—
207	C₂×D₄	—	47	8	no	—
208	C₂×D₄	—	47	8	no	—
209	C₂×D₄	—	47	4	no	—
210	C₂×D₄	—	47	8	no	—
211	C₂×D₄	—	47	4	no	—
212	C₂×D₄	—	47	2	no	—
213	C₂×D₄	—	188	8	no	—
214	C₂×D₄	—	188	4	no	—
215	C₂×D₄	—	188	8	no	—
216	C₂×D₄	C₂²×D₄₇	1	4	yes	—
217	C₂×D₄	C₂²×D₄₇	1	2	yes	—
218	C₂×D₄	C₂²×D₄₇	1	8	yes	—
219	C₂×D₄	C₂²×D₄₇	1	2	yes	—
220	C₂²×C₄	—	47	4	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	47	1	no	—
223	C₂²×C₄	—	47	8	no	—
224	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
225	C₄⋊C₄	—	47	4	no	—
226	C₄⋊C₄	—	47	4	no	—
227	C₄⋊C₄	—	47	4	no	—
228	C₄⋊C₄	—	47	4	no	—
229	C₄⋊C₄	—	47	2	no	—
230	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
231	C₂²⋊C₄	—	47	4	no	—
232	C₂²⋊C₄	—	47	4	no	—
233	C₂²⋊C₄	—	47	4	no	—
234	C₂²⋊C₄	—	47	4	no	—
235	C₂²⋊C₄	—	47	8	no	—
236	C₂²⋊C₄	—	47	4	no	—
237	C₂²⋊C₄	—	47	8	no	—
238	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
239	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
240	C₄²	—	47	2	no	—
241	C₄²	—	47	1	no	—
242	C₄²	—	47	4	no	—
243	C₄²	C₂²×D₄₇	1	1	yes	characteristic
244	C₂³	—	4	8	no	—
245	C₂³	—	4	4	no	—
246	C₂³	—	47	1	no	—
247	C₂³	—	47	4	no	—
248	C₂³	—	94	8	no	—
249	C₂³	—	188	8	no	—
250	C₂³	C₂³×D₄₇	1	4	yes	—
251	C₂³	C₂³×D₄₇	1	4	yes	—
252	Q₈	—	94	4	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	2	no	—
257	D₄	—	94	8	no	—
258	D₄	—	94	8	no	—
259	D₄	—	188	2	no	—
260	D₄	—	188	4	no	—
261	D₄	—	188	8	no	—
262	D₄	—	188	8	no	—
263	D₄	—	188	8	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	47	1	no	—
266	C₂×C₄	—	47	2	no	—
267	C₂×C₄	—	47	4	no	—
268	C₂×C₄	—	47	4	no	—
269	C₂×C₄	—	94	2	no	—
270	C₂×C₄	—	94	4	no	—
271	C₂×C₄	—	188	8	no	—
272	C₂×C₄	—	188	4	no	—
273	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
274	C₂×C₄	C₂³×D₄₇	1	4	yes	—
275	C₂×C₄	C₂³×D₄₇	1	2	yes	—
276	C₂²	—	2	4	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	8	no	—
281	C₂²	—	8	8	no	—
282	C₂²	—	47	2	no	—
283	C₂²	—	94	2	no	—
284	C₂²	—	94	4	no	—
285	C₂²	—	94	8	no	—
286	C₂²	—	376	8	no	—
287	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	4	no	—
290	C₄	—	94	1	no	—
291	C₄	—	94	4	no	—
292	C₄	—	94	2	no	—
293	C₄	—	94	4	no	—
294	C₄	#13094069	1	4	yes	—
295	C₂	—	4	4	no	—
296	C₂	—	4	4	no	—
297	C₂	—	94	2	no	—
298	C₂	—	188	4	no	—
299	C₂	H(5,𝔽₂)×D₄₇	1	1	yes	characteristic
300	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	#3543
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. #15689862	C₂	semidirect
2. #15689859	C₂	semidirect
3. #15689794	C₂	semidirect
4. #15689770	C₂	semidirect
5. D₄×C₂²⋊D₄₇	C₂	semidirect
6. #15689726	C₂	semidirect
7. #15689690	C₂	semidirect
8. #15689683	C₂	semidirect
9. #127×D₄₇	C₂	semidirect
10. #15689655	C₂	semidirect
11. D₄²×C₄₇	C₂	semidirect
12. C₂×#10625350	C₂²	semidirect
13. C₂²×C₂²⋊D₄₇	C₂²	semidirect
14. C₂×D₄×D₄₇	C₂²	semidirect
15. C₂×D₄×D₄₇	C₂²	semidirect
16. #13094021	C₂²	semidirect
17. #13094020	C₂²	semidirect
18. #13094020	C₂²	semidirect
19. #13094020	C₂²	semidirect
20. #13094017	C₂²	semidirect
21. #13094015	C₂²	semidirect
22. #13094011	C₂²	semidirect
23. #13094011	C₂²	semidirect
24. #13094008	C₂²	semidirect
25. #13094006	C₂²	semidirect
26. #13094006	C₂²	semidirect
27. #13094005	C₂²	semidirect
28. #13094004	C₂²	semidirect
29. #13093998	C₂²	semidirect
30. #13093997	C₂²	semidirect
31. #13093996	C₂²	semidirect
33. #13093995	C₂²	semidirect
34. #13093995	C₂²	semidirect
35. #13093993	C₂²	semidirect
36. #13093993	C₂²	semidirect
37. #13093992	C₂²	semidirect
38. C₄×Dic₉₄	C₂²	semidirect
39. D₄×Dic₄₇	C₂²	semidirect
40. #13093986	C₂²	semidirect
41. C₄×C₂²⋊D₄₇	C₂²	semidirect
42. #13093983	C₂²	semidirect
43. C₂²⋊C₄×D₄₇	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
156. D₄²	D₄₇	semidirect
162. C₄₇	#3543	semidirect
167. C₂²×D₄	C₂×D₄₇	semidirect
188. #121	C₂×D₄₇	semidirect
192. #120	C₂×D₄₇	semidirect
197. C₄×D₄	C₂×D₄₇	semidirect