Subgroups — #20257284

Order 6016.

1040 conjugacy classes of subgroups.

The subgroup lattice has 304 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#15689856	C₂	1	4	yes	maximal, complemented
2	#15689845	C₂	1	4	yes	maximal, complemented
3	#15689802	C₂	1	8	yes	maximal, complemented
4	#15689744	C₂	1	4	yes	maximal, complemented
5	#15689694	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689687	C₂	1	1	yes	maximal, characteristic
7	#15689680	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689657	C₂	1	4	yes	maximal, complemented
9	C₂×#13094012	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689600	C₂	1	1	yes	maximal, characteristic, complemented
11	#15689597	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
14	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
15	#13094024	C₂²	1	2	yes	complemented
16	#13094022	C₂²	1	2	yes	complemented
17	#13094020	C₂²	1	2	yes	—
18	#13094019	C₂²	1	8	yes	complemented
19	#13094018	C₂²	1	8	yes	complemented
20	#13094016	C₂²	1	2	yes	complemented
21	#13094013	C₂²	1	2	yes	—
22	#13094012	C₂²	1	2	yes	—
23	#13094012	C₂²	1	4	yes	—
24	#13094012	C₂²	1	2	yes	—
25	#13094011	C₂²	1	2	yes	complemented
26	#13094010	C₂²	1	8	yes	complemented
27	#13094007	C₂²	1	8	yes	complemented
28	#13094006	C₂²	1	8	yes	complemented
29	#13094005	C₂²	1	4	yes	—
30	#13094003	C₂²	1	4	yes	—
31	#13094003	C₂²	1	4	yes	complemented
32	#13094002	C₂²	1	8	yes	complemented
33	#13093998	C₂²	1	4	yes	—
34	#13093992	C₂²	1	4	yes	—
35	C₄×Dic₉₄	C₂²	1	4	yes	complemented
36	C₄×Dic₉₄	C₂²	1	4	yes	complemented
37	#13093989	C₂²	1	4	yes	complemented
38	C₄×D₁₈₈	C₂²	1	4	yes	—
39	#13093986	C₂²	1	4	yes	—
40	C₄×C₂²⋊D₄₇	C₂²	1	8	yes	complemented
41	#13093983	C₂²	1	1	yes	characteristic
42	#13093983	C₂²	1	1	yes	characteristic
43	#13093981	C₂²	1	8	yes	complemented
44	#13093980	C₂²	1	4	yes	—
45	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
46	C₂²⋊C₄×D₄₇	C₂²	1	4	yes	—
47	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
48	C₄×Q₈×C₄₇	C₂²	1	2	yes	complemented
49	C₄×D₄×C₄₇	C₂²	1	4	yes	—
50	C₄×D₄×C₄₇	C₂²	1	2	yes	complemented
51	#117×C₄₇	C₂²	1	1	yes	characteristic
52	#117×C₄₇	C₂²	1	2	yes	complemented
53	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
54	C₂×C₄²×C₄₇	C₂²	1	2	yes	complemented
55	C₂³×D₄₇	C₂³	1	2	yes	—
56	#41×C₄₇	—	2	4	no	—
57	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
58	C₂×D₄×C₄₇	C₂³	1	2	yes	—
59	C₂×D₄×C₄₇	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
61	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
62	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
63	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
65	C₂²×Dic₄₇	C₂³	1	2	yes	—
66	C₂×D₁₈₈	—	2	4	no	—
67	C₂×D₁₈₈	C₂³	1	2	yes	—
68	C₂×D₁₈₈	C₂³	1	2	yes	—
69	C₂×C₄×D₄₇	C₂³	1	8	yes	—
70	C₂×Dic₉₄	—	2	4	no	—
71	C₂×Dic₉₄	C₂³	1	2	yes	—
72	C₂×Dic₉₄	C₂³	1	2	yes	—
73	C₂×Dic₉₄	C₂³	1	4	yes	—
74	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
75	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
76	C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₄₇	C₂³	1	4	yes	—
78	C₄²×C₄₇	C₂³	1	1	yes	characteristic
79	C₄²×C₄₇	C₂³	1	1	yes	characteristic
80	C₄²×C₄₇	C₂³	1	2	yes	—
81	C₄²×C₄₇	C₂³	1	2	yes	—
82	C₄²×C₄₇	C₂³	1	4	yes	—
83	C₂²⋊Dic₄₇	C₂³	1	2	yes	—
84	C₂²⋊Dic₄₇	C₂³	1	2	yes	—
85	C₂²⋊Dic₄₇	C₂³	1	8	yes	—
86	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
87	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
88	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
89	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
90	C₄⋊Dic₄₇	C₂³	1	2	yes	—
91	C₄⋊Dic₄₇	C₂³	1	2	yes	—
92	C₄⋊Dic₄₇	C₂³	1	4	yes	—
93	Dic₄₇⋊C₄	C₂³	1	8	yes	—
94	Dic₄₇⋊C₄	C₂³	1	4	yes	—
95	Dic₄₇⋊C₄	C₂³	1	8	yes	—
96	Dic₄₇⋊C₄	C₂³	1	4	yes	—
97	C₄×Dic₄₇	C₂³	1	8	yes	—
98	C₄×Dic₄₇	C₂³	1	2	yes	—
99	C₄×Dic₄₇	C₂³	1	2	yes	—
100	C₂³×C₄₇	C₂⁴	1	2	yes	—
101	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
102	C₂²×D₄₇	—	2	4	no	—
103	C₂²×D₄₇	—	4	4	no	—
104	C₂²×D₄₇	C₂⁴	1	4	yes	—
105	Q₈×C₄₇	—	2	2	no	—
106	D₄×C₄₇	—	2	4	no	—
107	D₄×C₄₇	—	2	2	no	—
108	C₂×C₄×C₄₇	—	2	4	no	—
109	C₂×C₄×C₄₇	—	4	2	no	—
110	C₂×C₄×C₄₇	—	4	4	no	—
111	C₂×C₄×C₄₇	#41	1	4	yes	—
112	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
113	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
114	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
115	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
116	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₄₇	—	4	8	no	—
119	C₂×Dic₄₇	—	2	4	no	—
120	C₂×Dic₄₇	C₂⁴	1	4	yes	—
121	C₂×Dic₄₇	C₂⁴	1	4	yes	—
122	C₂×Dic₄₇	C₂⁴	1	4	yes	—
123	D₁₈₈	—	2	4	no	—
124	D₁₈₈	—	2	4	no	—
125	C₄×D₄₇	—	4	8	no	—
126	Dic₉₄	—	2	4	no	—
127	Dic₉₄	—	2	4	no	—
128	Dic₉₄	—	4	4	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	—	2	2	no	—
131	C₂²×C₄₇	—	2	4	no	—
132	C₂²×C₄₇	C₂×#41	1	2	yes	—
133	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
134	C₂×D₄₇	—	2	4	no	—
135	C₂×D₄₇	—	2	8	no	—
136	C₂×D₄₇	—	8	4	no	—
137	C₄×C₄₇	—	2	2	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	—	2	4	no	—
140	C₄×C₄₇	—	2	2	no	—
141	C₄×C₄₇	C₂×#41	1	2	yes	—
142	C₄×C₄₇	C₂×#41	1	2	yes	—
143	Dic₄₇	—	2	4	no	—
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	2	4	no	—
146	#3510	—	47	1	no	maximal, Sylow
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	—	4	2	no	—
149	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
150	C₂×C₄₇	#585	1	2	yes	—
151	D₄₇	—	4	4	no	—
152	#559	—	47	4	no	—
153	#556	—	47	4	no	—
154	#551	—	47	8	no	—
155	#542	—	47	4	no	—
156	#536	—	47	1	no	—
157	#535	—	47	1	no	—
158	#533	—	47	1	no	—
159	#529	—	47	4	no	—
160	C₂×#124	—	47	1	no	—
161	#519	—	47	1	no	—
162	#518	—	47	1	no	—
163	C₄×#41	D₄₇	1	1	yes	characteristic, complemented
164	C₄₇	#3510	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
166	C₂²×Q₈	—	47	1	no	—
167	C₂²×D₄	—	47	1	no	—
168	C₄⋊Q₈	—	47	2	no	—
169	C₄⋊Q₈	—	47	2	no	—
170	#127	—	47	2	no	—
171	#126	—	47	8	no	—
172	#126	—	47	8	no	—
173	#125	—	47	2	no	—
174	#124	—	47	2	no	—
175	#124	—	47	2	no	—
176	#124	—	47	8	no	—
177	#124	—	47	2	no	—
178	#124	—	47	2	no	—
179	#124	—	47	4	no	—
180	#123	—	47	4	no	—
181	#123	—	47	8	no	—
182	#123	—	47	8	no	—
183	C₂²⋊Q₈	—	47	4	no	—
184	C₂²⋊Q₈	—	47	4	no	—
185	C₂²⋊Q₈	—	47	8	no	—
186	#121	—	47	4	no	—
187	#120	—	47	4	no	—
188	C₄×Q₈	—	47	4	no	—
189	C₄×Q₈	—	47	4	no	—
190	C₄×Q₈	—	47	4	no	—
191	C₄×Q₈	C₂×D₄₇	1	2	yes	complemented
192	C₄×D₄	—	47	8	no	—
193	C₄×D₄	—	47	4	no	—
194	C₄×D₄	—	47	4	no	—
195	C₄×D₄	C₂×D₄₇	1	4	yes	—
196	C₄×D₄	C₂×D₄₇	1	2	yes	complemented
197	#117	—	47	1	no	—
198	#117	—	47	1	no	—
199	#117	—	47	4	no	—
200	#117	—	47	8	no	—
201	#117	C₂×D₄₇	1	1	yes	characteristic
202	#117	C₂×D₄₇	1	2	yes	complemented
203	C₂×C₂²⋊C₄	—	47	4	no	—
204	C₂×C₂²⋊C₄	—	47	4	no	—
205	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
206	C₂×C₄²	C₂×D₄₇	1	2	yes	complemented
207	C₂⁴	—	47	2	no	—
208	#41	—	2	4	no	—
209	C₂×Q₈	—	47	2	no	—
210	C₂×Q₈	—	47	2	no	—
211	C₂×Q₈	—	47	4	no	—
212	C₂×Q₈	—	94	4	no	—
213	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
214	C₂×D₄	—	47	8	no	—
215	C₂×D₄	—	47	2	no	—
216	C₂×D₄	—	47	2	no	—
217	C₂×D₄	—	94	4	no	—
218	C₂×D₄	C₂²×D₄₇	1	2	yes	—
219	C₂×D₄	C₂²×D₄₇	1	1	yes	characteristic
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	47	8	no	—
222	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
223	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
224	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
225	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
226	C₄⋊C₄	—	47	4	no	—
227	C₄⋊C₄	—	47	8	no	—
228	C₄⋊C₄	—	47	8	no	—
229	C₄⋊C₄	—	47	4	no	—
230	C₄⋊C₄	—	47	2	no	—
231	C₄⋊C₄	—	47	2	no	—
232	C₄⋊C₄	—	47	4	no	—
233	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
234	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
235	C₂²⋊C₄	—	47	2	no	—
236	C₂²⋊C₄	—	47	2	no	—
237	C₂²⋊C₄	—	47	8	no	—
238	C₂²⋊C₄	—	47	8	no	—
239	C₂²⋊C₄	—	47	8	no	—
240	C₂²⋊C₄	—	47	8	no	—
241	C₂²⋊C₄	—	47	8	no	—
242	C₂²⋊C₄	C₂²×D₄₇	1	2	yes	—
243	C₂²⋊C₄	C₂²×D₄₇	1	4	yes	—
244	C₄²	—	47	8	no	—
245	C₄²	—	47	2	no	—
246	C₄²	—	47	2	no	—
247	C₄²	C₂²×D₄₇	1	1	yes	characteristic
248	C₄²	C₂²×D₄₇	1	1	yes	characteristic
249	C₄²	C₂²×D₄₇	1	2	yes	—
250	C₄²	C₂²×D₄₇	1	4	yes	—
251	C₄²	C₂²×D₄₇	1	2	yes	—
252	C₂³	—	47	4	no	—
253	C₂³	—	94	4	no	—
254	C₂³	—	188	4	no	—
255	C₂³	C₂³×D₄₇	1	2	yes	—
256	C₂³	C₂³×D₄₇	1	1	yes	characteristic
257	Q₈	—	2	2	no	—
258	Q₈	—	94	4	no	—
259	Q₈	—	94	4	no	—
260	Q₈	—	188	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	2	no	—
263	D₄	—	94	4	no	—
264	D₄	—	94	4	no	—
265	D₄	—	188	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	—	4	2	no	—
269	C₂×C₄	—	47	4	no	—
270	C₂×C₄	—	47	4	no	—
271	C₂×C₄	—	47	4	no	—
272	C₂×C₄	—	94	4	no	—
273	C₂×C₄	—	188	8	no	—
274	C₂×C₄	#10625350	1	4	yes	—
275	C₂×C₄	C₂³×D₄₇	1	4	yes	—
276	C₂×C₄	C₂³×D₄₇	1	2	yes	—
277	C₂×C₄	C₂³×D₄₇	1	2	yes	—
278	C₂×C₄	C₂³×D₄₇	1	2	yes	—
279	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	2	no	—
283	C₂²	—	2	4	no	—
284	C₂²	—	94	4	no	—
285	C₂²	—	94	8	no	—
286	C₂²	—	376	4	no	—
287	C₂²	C₂×#10625350	1	2	yes	—
288	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	4	no	—
292	C₄	—	2	2	no	—
293	C₄	—	94	4	no	—
294	C₄	—	94	4	no	—
295	C₄	—	94	4	no	—
296	C₄	C₂×#10625350	1	2	yes	—
297	C₄	C₂×#10625350	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	2	no	—
300	C₂	—	188	4	no	—
301	C₂	C₂²×#10625350	1	1	yes	characteristic
302	C₂	#15689959	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄×#41×C₄₇	C₂
hypercenter	C₄×#41	D₄₇
hypocenter	C₄₇	#3510
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄×#41

Complements

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

N	H ≅ G/N	type
1. #15689856	C₂	semidirect
2. #15689845	C₂	semidirect
3. #15689802	C₂	semidirect
4. #15689744	C₂	semidirect
5. #15689694	C₂	semidirect
7. #15689680	C₂	semidirect
8. #15689657	C₂	semidirect
9. C₂×#13094012	C₂	semidirect
10. #15689600	C₂	semidirect
11. #15689597	C₂	semidirect
12. C₄×#41×C₄₇	C₂	semidirect
15. #13094024	C₂²	semidirect
16. #13094022	C₂²	semidirect
18. #13094019	C₂²	semidirect
19. #13094018	C₂²	semidirect
20. #13094016	C₂²	semidirect
25. #13094011	C₂²	semidirect
26. #13094010	C₂²	semidirect
27. #13094007	C₂²	semidirect
28. #13094006	C₂²	semidirect
31. #13094003	C₂²	semidirect
32. #13094002	C₂²	semidirect
35. C₄×Dic₉₄	C₂²	semidirect
36. C₄×Dic₉₄	C₂²	semidirect
37. #13093989	C₂²	semidirect
40. C₄×C₂²⋊D₄₇	C₂²	semidirect
43. #13093981	C₂²	semidirect
48. C₄×Q₈×C₄₇	C₂²	semidirect
50. C₄×D₄×C₄₇	C₂²	semidirect
52. #117×C₄₇	C₂²	semidirect
54. C₂×C₄²×C₄₇	C₂²	semidirect
163. C₄×#41	D₄₇	semidirect
164. C₄₇	#3510	semidirect
191. C₄×Q₈	C₂×D₄₇	semidirect
196. C₄×D₄	C₂×D₄₇	semidirect
202. #117	C₂×D₄₇	semidirect
206. C₂×C₄²	C₂×D₄₇	semidirect