Subgroups — #20257285

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	#15689858	C₂	1	4	yes	maximal, complemented
2	#15689845	C₂	1	4	yes	maximal, complemented
3	#15689797	C₂	1	8	yes	maximal, complemented
4	#15689745	C₂	1	4	yes	maximal, complemented
5	#15689697	C₂	1	1	yes	maximal, characteristic, complemented
6	#15689692	C₂	1	1	yes	maximal, characteristic
7	#15689676	C₂	1	1	yes	maximal, characteristic, complemented
8	#15689660	C₂	1	4	yes	maximal, complemented
9	C₂×#13094013	C₂	1	1	yes	maximal, characteristic, complemented
10	#15689597	C₂	1	1	yes	maximal, characteristic, complemented
11	#15689591	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₇×#519	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#10625350	C₂²	1	1	yes	characteristic
14	C₂²×D₁₈₈	C₂²	1	1	yes	characteristic
15	#13094023	C₂²	1	2	yes	complemented
16	#13094022	C₂²	1	2	yes	complemented
17	#13094021	C₂²	1	2	yes	—
18	#13094019	C₂²	1	8	yes	complemented
19	#13094018	C₂²	1	8	yes	complemented
20	#13094015	C₂²	1	2	yes	complemented
21	#13094013	C₂²	1	2	yes	—
22	#13094013	C₂²	1	4	yes	—
23	#13094013	C₂²	1	2	yes	—
24	#13094013	C₂²	1	2	yes	complemented
25	#13094012	C₂²	1	2	yes	—
26	#13094012	C₂²	1	8	yes	complemented
27	#13094009	C₂²	1	8	yes	complemented
28	#13094009	C₂²	1	8	yes	complemented
29	#13094007	C₂²	1	4	yes	—
30	#13094002	C₂²	1	8	yes	complemented
31	#13094001	C₂²	1	4	yes	—
32	#13094000	C₂²	1	4	yes	complemented
33	#13093999	C₂²	1	4	yes	—
34	#13093994	C₂²	1	4	yes	—
35	C₄×Dic₉₄	C₂²	1	4	yes	complemented
36	#13093989	C₂²	1	2	yes	complemented
37	C₄×D₁₈₈	C₂²	1	4	yes	—
38	C₄×D₁₈₈	C₂²	1	8	yes	complemented
39	#13093986	C₂²	1	4	yes	—
40	#13093986	C₂²	1	2	yes	complemented
41	(C₂²⋊D₄₇)⋊C₄	C₂²	1	4	yes	—
42	#13093983	C₂²	1	1	yes	characteristic
43	#13093981	C₂²	1	2	yes	complemented
44	#13093981	C₂²	1	8	yes	complemented
45	C₂×(C₂×D₄₇)⋊C₄	C₂²	1	4	yes	—
46	C₂²⋊C₄×D₄₇	C₂²	1	4	yes	—
47	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
48	C₄²×D₄₇	C₂²	1	2	yes	complemented
49	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
50	C₄×Q₈×C₄₇	C₂²	1	4	yes	complemented
51	C₄×Q₈×C₄₇	C₂²	1	4	yes	complemented
52	#117×C₄₇	C₂²	1	4	yes	—
53	#117×C₄₇	C₂²	1	1	yes	characteristic
54	#117×C₄₇	C₂²	1	1	yes	characteristic
55	C₂³×D₄₇	C₂³	1	2	yes	—
56	C₂×Q₈×C₄₇	—	2	4	no	—
57	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
58	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
60	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₄₇	C₂³	1	2	yes	—
62	C₂²×Dic₄₇	C₂³	1	2	yes	—
63	#10625350	—	2	4	no	—
64	C₂×D₁₈₈	—	2	4	no	—
65	C₂×D₁₈₈	C₂³	1	2	yes	—
66	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
67	C₂×D₁₈₈	C₂³	1	2	yes	—
68	C₂×D₁₈₈	C₂³	1	8	yes	—
69	C₂×C₄×D₄₇	C₂³	1	2	yes	—
70	C₂×C₄×D₄₇	C₂³	1	4	yes	—
71	C₂×C₄×D₄₇	C₂³	1	8	yes	—
72	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
73	C₂×Dic₉₄	C₂³	1	4	yes	—
74	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
75	C₄⋊C₄×C₄₇	C₂³	1	2	yes	—
76	C₄⋊C₄×C₄₇	C₂³	1	8	yes	—
77	C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
78	C₂²⋊C₄×C₄₇	C₂³	1	2	yes	—
79	C₄²×C₄₇	C₂³	1	2	yes	—
80	C₄²×C₄₇	C₂³	1	8	yes	—
81	C₄²×C₄₇	C₂³	1	2	yes	—
82	C₂²⋊Dic₄₇	C₂³	1	2	yes	—
83	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
84	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
85	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
86	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
87	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
88	(C₂×D₄₇)⋊C₄	C₂³	1	8	yes	—
89	C₄⋊Dic₄₇	C₂³	1	2	yes	—
90	C₄⋊Dic₄₇	C₂³	1	4	yes	—
91	Dic₄₇⋊C₄	C₂³	1	4	yes	—
92	Dic₄₇⋊C₄	C₂³	1	4	yes	—
93	Dic₄₇⋊C₄	C₂³	1	8	yes	—
94	Dic₄₇⋊C₄	C₂³	1	4	yes	—
95	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
96	C₄×Dic₄₇	C₂³	1	4	yes	—
97	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
98	C₄×Dic₄₇	C₂³	1	2	yes	—
99	C₄×Dic₄₇	C₂³	1	2	yes	—
100	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
101	C₂²×D₄₇	—	2	4	no	—
102	C₂²×D₄₇	—	4	4	no	—
103	C₂²×D₄₇	C₂⁴	1	4	yes	—
104	C₂²×D₄₇	C₂⁴	1	2	yes	—
105	Q₈×C₄₇	—	2	4	no	—
106	Q₈×C₄₇	—	2	4	no	—
107	C₂×C₄×C₄₇	—	2	4	no	—
108	C₂×C₄×C₄₇	#41	1	4	yes	—
109	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
110	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
111	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
112	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₄₇	—	2	4	no	—
115	C₂×Dic₄₇	—	4	2	no	—
116	C₂×Dic₄₇	C₂⁴	1	2	yes	—
117	C₂×Dic₄₇	C₂⁴	1	2	yes	—
118	C₂×Dic₄₇	C₂⁴	1	2	yes	—
119	C₂×Dic₄₇	C₂⁴	1	4	yes	—
120	D₁₈₈	—	2	4	no	—
121	D₁₈₈	—	2	4	no	—
122	D₁₈₈	—	2	2	no	—
123	D₁₈₈	—	4	8	no	—
124	C₄×D₄₇	—	2	4	no	—
125	C₄×D₄₇	—	4	8	no	—
126	C₄×D₄₇	—	4	4	no	—
127	Dic₉₄	—	2	2	no	—
128	Dic₉₄	—	4	4	no	—
129	C₂²×C₄₇	—	2	2	no	—
130	C₂²×C₄₇	C₂×#41	1	2	yes	—
131	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
132	C₂×D₄₇	—	2	2	no	—
133	C₂×D₄₇	—	2	4	no	—
134	C₂×D₄₇	—	2	4	no	—
135	C₂×D₄₇	—	2	8	no	—
136	C₂×D₄₇	—	8	4	no	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	—	2	4	no	—
139	C₄×C₄₇	—	2	4	no	—
140	C₄×C₄₇	C₂×#41	1	2	yes	—
141	C₄×C₄₇	C₂×#41	1	2	yes	—
142	Dic₄₇	—	2	2	no	—
143	Dic₄₇	—	2	2	no	—
144	Dic₄₇	—	2	4	no	—
145	Dic₄₇	—	2	2	no	—
146	#3510	—	47	1	no	maximal, Sylow
147	C₂×C₄₇	—	2	2	no	—
148	C₂×C₄₇	C₂²×#41	1	1	yes	characteristic
149	C₂×C₄₇	#585	1	2	yes	—
150	D₄₇	—	4	2	no	—
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	D₄₇	1	1	yes	characteristic, complemented
162	#518	—	47	1	no	—
163	C₄×#41	—	47	1	no	—
164	C₄₇	#3510	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	—	47	1	no	—
166	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
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	4	no	—
177	#124	—	47	2	no	—
178	#124	—	47	2	no	—
179	#124	—	47	8	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	8	no	—
185	C₂²⋊Q₈	—	47	4	no	—
186	#121	—	47	4	no	—
187	#120	—	47	4	no	—
188	C₄×Q₈	—	47	2	no	—
189	C₄×Q₈	—	47	4	no	—
190	C₄×Q₈	C₂×D₄₇	1	4	yes	complemented
191	C₄×Q₈	C₂×D₄₇	1	4	yes	complemented
192	C₄×D₄	—	47	2	no	—
193	C₄×D₄	—	47	4	no	—
194	C₄×D₄	—	47	4	no	—
195	C₄×D₄	—	47	4	no	—
196	C₄×D₄	—	47	8	no	—
197	#117	—	47	1	no	—
198	#117	—	47	2	no	—
199	#117	—	47	8	no	—
200	#117	C₂×D₄₇	1	1	yes	characteristic
201	#117	C₂×D₄₇	1	1	yes	characteristic
202	#117	C₂×D₄₇	1	4	yes	—
203	C₂×C₂²⋊C₄	—	47	4	no	—
204	C₂×C₂²⋊C₄	—	47	4	no	—
205	C₂×C₄²	—	47	1	no	—
206	C₂×C₄²	—	47	2	no	—
207	C₂⁴	—	47	2	no	—
208	#41	—	94	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	47	1	no	—
211	C₂×Q₈	—	47	4	no	—
212	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
213	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
214	C₂×D₄	—	47	2	no	—
215	C₂×D₄	—	47	2	no	—
216	C₂×D₄	—	47	2	no	—
217	C₂×D₄	—	47	1	no	—
218	C₂×D₄	—	47	8	no	—
219	C₂×D₄	—	94	4	no	—
220	C₂²×C₄	—	47	2	no	—
221	C₂²×C₄	—	47	2	no	—
222	C₂²×C₄	—	47	4	no	—
223	C₂²×C₄	—	47	8	no	—
224	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
225	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
226	C₄⋊C₄	—	47	4	no	—
227	C₄⋊C₄	—	47	4	no	—
228	C₄⋊C₄	—	47	8	no	—
229	C₄⋊C₄	—	47	4	no	—
230	C₄⋊C₄	—	47	2	no	—
231	C₄⋊C₄	—	47	4	no	—
232	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
233	C₄⋊C₄	C₂²×D₄₇	1	2	yes	—
234	C₄⋊C₄	C₂²×D₄₇	1	8	yes	—
235	C₂²⋊C₄	—	47	2	no	—
236	C₂²⋊C₄	—	47	8	no	—
237	C₂²⋊C₄	—	47	8	no	—
238	C₂²⋊C₄	—	47	8	no	—
239	C₂²⋊C₄	—	47	4	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	2	yes	—
244	C₄²	—	47	1	no	—
245	C₄²	—	47	4	no	—
246	C₄²	—	47	1	no	—
247	C₄²	—	47	2	no	—
248	C₄²	—	47	2	no	—
249	C₄²	C₂²×D₄₇	1	2	yes	—
250	C₄²	C₂²×D₄₇	1	8	yes	—
251	C₄²	C₂²×D₄₇	1	2	yes	—
252	C₂³	—	47	4	no	—
253	C₂³	—	47	2	no	—
254	C₂³	—	94	4	no	—
255	C₂³	—	188	4	no	—
256	C₂³	C₂³×D₄₇	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	94	2	no	—
260	Q₈	—	188	4	no	—
261	D₄	—	94	4	no	—
262	D₄	—	94	4	no	—
263	D₄	—	94	4	no	—
264	D₄	—	94	2	no	—
265	D₄	—	188	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	47	2	no	—
268	C₂×C₄	—	47	2	no	—
269	C₂×C₄	—	47	2	no	—
270	C₂×C₄	—	47	4	no	—
271	C₂×C₄	—	94	4	no	—
272	C₂×C₄	—	188	4	no	—
273	C₂×C₄	—	188	8	no	—
274	C₂×C₄	—	188	2	no	—
275	C₂×C₄	#10625350	1	4	yes	—
276	C₂×C₄	C₂³×D₄₇	1	4	yes	—
277	C₂×C₄	C₂³×D₄₇	1	4	yes	—
278	C₂×C₄	C₂³×D₄₇	1	4	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	—	94	2	no	—
283	C₂²	—	94	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	4	no	—
290	C₄	—	2	4	no	—
291	C₄	—	2	4	no	—
292	C₄	—	94	2	no	—
293	C₄	—	94	2	no	—
294	C₄	—	94	4	no	—
295	C₄	—	94	2	no	—
296	C₄	C₂×#10625350	1	2	yes	—
297	C₄	C₂×#10625350	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	188	4	no	—
300	C₂	—	188	2	no	—
301	C₂	C₂²×#10625350	1	1	yes	characteristic
302	C₂	#15689960	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#519	C₂
hypercenter	#519	D₄₇
hypocenter	C₄₇	#3510
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #519

Complements

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

N	H ≅ G/N	type
1. #15689858	C₂	semidirect
2. #15689845	C₂	semidirect
3. #15689797	C₂	semidirect
4. #15689745	C₂	semidirect
5. #15689697	C₂	semidirect
7. #15689676	C₂	semidirect
8. #15689660	C₂	semidirect
9. C₂×#13094013	C₂	semidirect
10. #15689597	C₂	semidirect
11. #15689591	C₂	semidirect
12. C₄₇×#519	C₂	semidirect
15. #13094023	C₂²	semidirect
16. #13094022	C₂²	semidirect
18. #13094019	C₂²	semidirect
19. #13094018	C₂²	semidirect
20. #13094015	C₂²	semidirect
24. #13094013	C₂²	semidirect
26. #13094012	C₂²	semidirect
27. #13094009	C₂²	semidirect
28. #13094009	C₂²	semidirect
30. #13094002	C₂²	semidirect
32. #13094000	C₂²	semidirect
35. C₄×Dic₉₄	C₂²	semidirect
36. #13093989	C₂²	semidirect
38. C₄×D₁₈₈	C₂²	semidirect
40. #13093986	C₂²	semidirect
43. #13093981	C₂²	semidirect
44. #13093981	C₂²	semidirect
48. C₄²×D₄₇	C₂²	semidirect
50. C₄×Q₈×C₄₇	C₂²	semidirect
51. C₄×Q₈×C₄₇	C₂²	semidirect
161. #519	D₄₇	semidirect
164. C₄₇	#3510	semidirect
190. C₄×Q₈	C₂×D₄₇	semidirect
191. C₄×Q₈	C₂×D₄₇	semidirect