Subgroups — #20256810

Order 6016.

1140 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689754	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689738	C₂	1	4	yes	maximal, complemented
3	#15689724	C₂	1	4	yes	maximal, complemented
4	#15689711	C₂	1	4	yes	maximal, complemented
5	#15689704	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689690	C₂	1	2	yes	maximal, complemented
7	#15689670	C₂	1	4	yes	maximal, complemented
8	#15689655	C₂	1	4	yes	maximal, complemented
9	#15689634	C₂	1	4	yes	maximal, complemented
10	C₄×#10625350	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10625350	C₂²	1	2	yes	—
13	C₂×#10625350	C₂²	1	4	yes	complemented
14	#13094023	C₂²	1	4	yes	complemented
15	#13094020	C₂²	1	4	yes	—
16	#13094017	C₂²	1	8	yes	complemented
17	#13094015	C₂²	1	4	yes	complemented
18	#13094011	C₂²	1	4	yes	—
19	#13094010	C₂²	1	8	yes	complemented
20	#13094007	C₂²	1	8	yes	complemented
21	#13094006	C₂²	1	8	yes	complemented
22	#13094004	C₂²	1	4	yes	complemented
23	#13094003	C₂²	1	4	yes	—
24	#13094003	C₂²	1	4	yes	complemented
25	#13093999	C₂²	1	4	yes	complemented
26	#13093998	C₂²	1	4	yes	—
27	#13093998	C₂²	1	4	yes	complemented
28	#13093997	C₂²	1	8	yes	complemented
29	#13093995	C₂²	1	4	yes	—
30	#13093995	C₂²	1	4	yes	—
31	#13093993	C₂²	1	8	yes	—
32	C₄×Dic₉₄	C₂²	1	2	yes	complemented
33	C₄×Dic₉₄	C₂²	1	2	yes	complemented
34	C₄×D₁₈₈	C₂²	1	2	yes	complemented
35	C₄×D₁₈₈	C₂²	1	2	yes	complemented
36	C₄×C₂²⋊D₄₇	C₂²	1	4	yes	—
37	C₄×C₂²⋊D₄₇	C₂²	1	4	yes	—
38	C₄×C₂²⋊D₄₇	C₂²	1	8	yes	complemented
39	#13093983	C₂²	1	2	yes	—
40	#13093983	C₂²	1	4	yes	complemented
41	#13093981	C₂²	1	4	yes	complemented
42	C₄²×D₄₇	C₂²	1	4	yes	complemented
43	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
44	C₂³×C₄×C₄₇	C₂²	1	2	yes	—
45	C₄×D₄×C₄₇	C₂²	1	4	yes	—
46	C₄×D₄×C₄₇	C₂²	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
49	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
50	C₂⁴×C₄₇	C₂³	1	2	yes	—
51	C₂×D₄×C₄₇	—	2	4	no	—
52	C₂×D₄×C₄₇	C₂³	1	4	yes	—
53	C₂²×C₄×C₄₇	—	4	4	no	—
54	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
59	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
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	#10625350	—	2	8	no	—
64	#10625350	—	4	8	no	—
65	C₂×D₁₈₈	C₂³	1	2	yes	—
66	C₂×D₁₈₈	C₂³	1	4	yes	—
67	C₂×C₄×D₄₇	C₂³	1	4	yes	—
68	C₂×C₄×D₄₇	C₂³	1	4	yes	—
69	C₂×C₄×D₄₇	C₂³	1	4	yes	—
70	C₂×Dic₉₄	C₂³	1	2	yes	—
71	C₂×Dic₉₄	C₂³	1	4	yes	—
72	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
74	C₄²×C₄₇	C₂³	1	2	yes	—
75	C₄²×C₄₇	C₂³	1	2	yes	—
76	C₂²⋊Dic₄₇	C₂³	1	4	yes	—
77	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
78	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
79	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
80	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
81	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
82	C₄⋊Dic₄₇	C₂³	1	2	yes	—
83	C₄⋊Dic₄₇	C₂³	1	2	yes	—
84	C₄⋊Dic₄₇	C₂³	1	4	yes	—
85	C₄⋊Dic₄₇	C₂³	1	4	yes	—
86	Dic₄₇⋊C₄	C₂³	1	4	yes	—
87	Dic₄₇⋊C₄	C₂³	1	4	yes	—
88	Dic₄₇⋊C₄	C₂³	1	8	yes	—
89	C₄×Dic₄₇	C₂³	1	4	yes	—
90	C₄×Dic₄₇	C₂³	1	4	yes	—
91	C₄×Dic₄₇	C₂³	1	4	yes	—
92	C₂³×C₄₇	—	2	4	no	—
93	C₂³×C₄₇	—	4	2	no	—
94	C₂³×C₄₇	—	4	2	no	—
95	C₂³×C₄₇	C₂⁴	1	4	yes	—
96	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
97	C₂²×D₄₇	C₂⁴	1	4	yes	—
98	D₄×C₄₇	—	2	8	no	—
99	C₂×C₄×C₄₇	—	2	8	no	—
100	C₂×C₄×C₄₇	—	2	2	no	—
101	C₂×C₄×C₄₇	—	4	2	no	—
102	C₂×C₄×C₄₇	—	4	4	no	—
103	C₂×C₄×C₄₇	—	4	1	no	—
104	C₂×C₄×C₄₇	#41	1	4	yes	complemented
105	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
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	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₄₇	—	2	8	no	—
111	C₂²⋊D₄₇	—	4	8	no	—
112	C₂²⋊D₄₇	—	4	8	no	—
113	C₂×Dic₄₇	C₂⁴	1	4	yes	—
114	C₂×Dic₄₇	C₂⁴	1	4	yes	—
115	C₂×Dic₄₇	C₂⁴	1	4	yes	—
116	D₁₈₈	—	2	4	no	—
117	D₁₈₈	—	4	4	no	—
118	C₄×D₄₇	—	2	8	no	—
119	C₄×D₄₇	—	2	8	no	—
120	C₄×D₄₇	—	4	4	no	—
121	Dic₉₄	—	2	4	no	—
122	Dic₉₄	—	4	4	no	—
123	C₂²×C₄₇	—	2	1	no	—
124	C₂²×C₄₇	—	2	1	no	—
125	C₂²×C₄₇	—	2	4	no	—
126	C₂²×C₄₇	—	2	4	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	4	8	no	—
129	C₂²×C₄₇	C₂×#41	1	2	yes	—
130	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
131	C₂×D₄₇	—	2	4	no	—
132	C₂×D₄₇	—	2	4	no	—
133	C₂×D₄₇	—	2	4	no	—
134	C₄×C₄₇	—	2	4	no	—
135	C₄×C₄₇	—	2	1	no	—
136	C₄×C₄₇	—	2	2	no	—
137	C₄×C₄₇	C₂×#41	1	4	yes	—
138	C₄×C₄₇	H(5,𝔽₂)	1	2	yes	—
139	Dic₄₇	—	2	4	no	—
140	Dic₄₇	—	2	4	no	—
141	Dic₄₇	—	2	4	no	—
142	#3460	—	47	1	no	maximal, Sylow
143	C₂×C₄₇	—	2	2	no	—
144	C₂×C₄₇	—	4	4	no	—
145	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
146	C₂×C₄₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄₇	#585	1	1	yes	characteristic
148	D₄₇	—	4	4	no	—
149	#544	—	47	1	no	—
150	#542	—	47	4	no	—
151	#540	—	47	4	no	—
152	#538	—	47	4	no	—
153	#537	—	47	1	no	—
154	#535	—	47	2	no	—
155	#532	—	47	4	no	—
156	#529	—	47	4	no	—
157	#525	—	47	4	no	—
158	C₄×#41	—	47	2	no	—
159	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic, complemented
160	C₄₇	#3460	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	47	2	no	—
162	C₂×#41	—	47	4	no	—
163	C₂²×D₄	C₂×D₄₇	1	1	yes	characteristic
164	C₂³×C₄	C₂×D₄₇	1	2	yes	—
165	C₄⋊Q₈	—	47	4	no	—
166	#127	—	47	4	no	—
167	#126	—	47	8	no	—
168	#125	—	47	4	no	—
169	#124	—	47	4	no	—
170	#124	—	47	8	no	—
171	#123	—	47	8	no	—
172	#123	—	47	8	no	—
173	C₂²⋊Q₈	—	47	4	no	—
174	C₂²⋊Q₈	—	47	4	no	—
175	C₂²⋊Q₈	—	47	4	no	—
176	#121	—	47	4	no	—
177	#121	—	47	4	no	—
178	#121	—	47	4	no	—
179	#121	—	47	4	no	—
180	#121	—	47	4	no	—
181	#121	—	47	8	no	—
182	#120	—	47	8	no	—
183	C₄×Q₈	—	47	2	no	—
184	C₄×Q₈	—	47	2	no	—
185	C₄×D₄	—	47	4	no	—
186	C₄×D₄	—	47	4	no	—
187	C₄×D₄	—	47	8	no	—
188	C₄×D₄	—	47	2	no	—
189	C₄×D₄	—	47	2	no	—
190	C₄×D₄	C₂×D₄₇	1	4	yes	—
191	C₄×D₄	C₂×D₄₇	1	4	yes	—
192	#117	—	47	2	no	—
193	#117	—	47	4	no	—
194	#117	—	47	4	no	—
195	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
197	C₂×C₄²	—	47	4	no	—
198	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
199	C₂⁴	C₂²×D₄₇	1	2	yes	—
200	#41	—	94	8	no	—
201	#41	—	188	8	no	—
202	C₂×Q₈	—	47	2	no	—
203	C₂×Q₈	—	47	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	47	4	no	—
206	C₂×D₄	—	47	8	no	—
207	C₂×D₄	—	47	8	no	—
208	C₂×D₄	—	47	2	no	—
209	C₂×D₄	—	47	4	no	—
210	C₂×D₄	C₂²×D₄₇	1	4	yes	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	47	4	no	—
213	C₂²×C₄	—	47	4	no	—
214	C₂²×C₄	—	47	4	no	—
215	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
217	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
219	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
220	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
221	C₄⋊C₄	—	47	4	no	—
222	C₄⋊C₄	—	47	4	no	—
223	C₄⋊C₄	—	47	8	no	—
224	C₄⋊C₄	—	47	2	no	—
225	C₄⋊C₄	—	47	2	no	—
226	C₄⋊C₄	—	47	4	no	—
227	C₄⋊C₄	—	47	4	no	—
228	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
229	C₂²⋊C₄	—	47	4	no	—
230	C₂²⋊C₄	—	47	8	no	—
231	C₂²⋊C₄	—	47	8	no	—
232	C₂²⋊C₄	—	47	4	no	—
233	C₂²⋊C₄	—	47	4	no	—
234	C₂²⋊C₄	—	47	8	no	—
235	C₂²⋊C₄	C₂²×D₄₇	1	8	yes	—
236	C₄²	—	47	4	no	—
237	C₄²	—	47	4	no	—
238	C₄²	—	47	4	no	—
239	C₄²	C₂²×D₄₇	1	2	yes	—
240	C₄²	C₂²×D₄₇	1	2	yes	—
241	C₂³	—	2	4	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	47	4	no	—
245	C₂³	C₂³×D₄₇	1	4	yes	—
246	C₂³	C₂³×D₄₇	1	1	yes	characteristic
247	Q₈	—	94	4	no	—
248	Q₈	—	188	4	no	—
249	D₄	—	2	8	no	—
250	D₄	—	94	8	no	—
251	D₄	—	94	4	no	—
252	D₄	—	188	4	no	—
253	D₄	—	188	8	no	—
254	D₄	—	188	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	47	4	no	—
261	C₂×C₄	—	47	4	no	—
262	C₂×C₄	—	47	4	no	—
263	C₂×C₄	—	94	8	no	—
264	C₂×C₄	—	94	8	no	—
265	C₂×C₄	—	188	4	no	—
266	C₂×C₄	#10625350	1	4	yes	complemented
267	C₂×C₄	C₂³×D₄₇	1	2	yes	—
268	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
269	C₂×C₄	C₂³×D₄₇	1	4	yes	—
270	C₂×C₄	C₂³×D₄₇	1	2	yes	—
271	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	4	8	no	—
278	C₂²	—	94	4	no	—
279	C₂²	—	94	4	no	—
280	C₂²	—	94	4	no	—
281	C₂²	C₂×#10625350	1	2	yes	—
282	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	94	4	no	—
287	C₄	—	94	4	no	—
288	C₄	—	94	4	no	—
289	C₄	C₂×#10625350	1	4	yes	—
290	C₄	#13094069	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	4	4	no	—
293	C₂	—	188	4	no	—
294	C₂	C₂²×#10625350	1	1	yes	characteristic
295	C₂	C₂×#13094069	1	1	yes	characteristic
296	C₂	#15689959	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Frattini	C₂²	C₂⁴×D₄₇
Fitting	C₂×C₄×D₄×C₄₇	C₂
hypercenter	C₂×C₄×D₄	D₄₇
hypocenter	C₄₇	#3460
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #15689754	C₂	semidirect
2. #15689738	C₂	semidirect
3. #15689724	C₂	semidirect
4. #15689711	C₂	semidirect
5. #15689704	C₂	semidirect
6. #15689690	C₂	semidirect
7. #15689670	C₂	semidirect
8. #15689655	C₂	semidirect
9. #15689634	C₂	semidirect
10. C₄×#10625350	C₂	semidirect
11. C₂×C₄×D₄×C₄₇	C₂	semidirect
13. C₂×#10625350	C₂²	semidirect
14. #13094023	C₂²	semidirect
16. #13094017	C₂²	semidirect
17. #13094015	C₂²	semidirect
19. #13094010	C₂²	semidirect
20. #13094007	C₂²	semidirect
21. #13094006	C₂²	semidirect
22. #13094004	C₂²	semidirect
24. #13094003	C₂²	semidirect
25. #13093999	C₂²	semidirect
27. #13093998	C₂²	semidirect
28. #13093997	C₂²	semidirect
32. C₄×Dic₉₄	C₂²	semidirect
33. C₄×Dic₉₄	C₂²	semidirect
34. C₄×D₁₈₈	C₂²	semidirect
35. C₄×D₁₈₈	C₂²	semidirect
38. C₄×C₂²⋊D₄₇	C₂²	semidirect
40. #13093983	C₂²	semidirect
41. #13093981	C₂²	semidirect
42. C₄²×D₄₇	C₂²	semidirect
104. C₂×C₄×C₄₇	#41	semidirect
159. C₂×C₄×D₄	D₄₇	semidirect
160. C₄₇	#3460	semidirect
266. C₂×C₄	#10625350	semidirect