Subgroups — #20257068

Order 6016.

1522 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13094071	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₂²⋊D₄₇	C₂	1	2	yes	maximal, complemented
3	#15689780	C₂	1	2	yes	maximal, complemented
4	#15689772	C₂	1	6	yes	maximal, complemented
5	#15689767	C₂	1	6	yes	maximal, complemented
6	#15689701	C₂	1	3	yes	maximal, complemented
7	#15689694	C₂	1	3	yes	maximal, complemented
8	#15689634	C₂	1	3	yes	maximal, complemented
9	C₂×#13094003	C₂	1	3	yes	maximal, complemented
10	#15689606	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×#41×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
12	#13094071	—	2	8	no	—
13	C₂×#10625350	C₂²	1	3	yes	complemented
14	C₂×#10625352	C₂²	1	6	yes	complemented
15	C₂²×Dic₉₄	C₂²	1	3	yes	complemented
16	C₂×Q₈×D₄₇	C₂²	1	2	yes	complemented
17	#13094022	C₂²	1	6	yes	complemented
18	#13094011	C₂²	1	6	yes	complemented
19	#13094007	C₂²	1	6	yes	complemented
20	#13094005	C₂²	1	12	yes	complemented
21	#13094003	C₂²	1	6	yes	—
22	#13094003	C₂²	1	3	yes	—
23	#13094003	C₂²	1	6	yes	complemented
24	#13094003	C₂²	1	6	yes	complemented
25	#13094003	C₂²	1	3	yes	complemented
26	#13094000	C₂²	1	6	yes	complemented
27	#13093998	C₂²	1	1	yes	characteristic
28	#13093998	C₂²	1	3	yes	complemented
29	#13093996	C₂²	1	12	yes	complemented
30	#13093995	C₂²	1	6	yes	complemented
31	#13093993	C₂²	1	6	yes	—
32	Q₈×Dic₄₇	C₂²	1	2	yes	complemented
33	D₄×Dic₄₇	C₂²	1	6	yes	complemented
34	C₄×C₂²⋊D₄₇	C₂²	1	6	yes	—
35	C₄×C₂²⋊D₄₇	C₂²	1	6	yes	complemented
36	#13093983	C₂²	1	3	yes	complemented
37	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	complemented
38	C₂×Dic₄₇⋊C₄	C₂²	1	6	yes	complemented
39	C₂×C₂²⋊Dic₄₇	C₂²	1	6	yes	—
40	C₂×#41×C₄₇	—	2	8	no	—
41	C₂×#41×C₄₇	C₂²	1	3	yes	—
42	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
43	C₂×#41×C₄₇	C₂²	1	3	yes	—
44	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
46	C₂²×D₄×C₄₇	C₂²	1	3	yes	—
47	C₂³×C₄×C₄₇	C₂²	1	3	yes	—
48	C₂⁴×C₄₇	C₂³	1	3	yes	—
49	#41×C₄₇	—	2	4	no	—
50	#41×C₄₇	—	2	12	no	—
51	#41×C₄₇	—	2	12	no	—
52	#41×C₄₇	D₄	1	8	yes	complemented
53	C₂×Q₈×C₄₇	—	2	4	no	—
54	C₂×Q₈×C₄₇	C₂³	1	3	yes	—
55	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₄₇	—	2	12	no	—
57	C₂×D₄×C₄₇	C₂³	1	3	yes	—
58	C₂×D₄×C₄₇	C₂³	1	6	yes	—
59	C₂×D₄×C₄₇	C₂³	1	3	yes	—
60	C₂²×C₄×C₄₇	—	2	12	no	—
61	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
63	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
64	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
65	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
66	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
67	C₂×C₂²⋊D₄₇	C₂³	1	2	yes	—
68	C₂×C₂²⋊D₄₇	C₂³	1	6	yes	—
69	C₂×C₂²⋊D₄₇	C₂³	1	6	yes	—
70	C₂²×Dic₄₇	C₂³	1	6	yes	—
71	C₂²×Dic₄₇	C₂³	1	6	yes	—
72	Q₈×D₄₇	—	2	8	no	—
73	#10625352	—	2	24	no	—
74	#10625350	—	2	12	no	—
75	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₄₇	C₂³	1	6	yes	—
77	C₂×Dic₉₄	—	2	12	no	—
78	C₂×Dic₉₄	C₂³	1	3	yes	—
79	C₂×Dic₉₄	C₂³	1	6	yes	—
80	C₂²⋊Dic₄₇	C₂³	1	6	yes	—
81	C₂²⋊Dic₄₇	C₂³	1	12	yes	—
82	C₂²⋊Dic₄₇	C₂³	1	6	yes	—
83	C₂²⋊Dic₄₇	C₂³	1	6	yes	—
84	(C₂×D₄₇)⋊C₄	C₂³	1	6	yes	—
85	(C₂×D₄₇)⋊C₄	C₂³	1	2	yes	—
86	C₄⋊Dic₄₇	C₂³	1	3	yes	—
87	C₄⋊Dic₄₇	C₂³	1	6	yes	—
88	C₄⋊Dic₄₇	C₂³	1	1	yes	characteristic
89	Dic₄₇⋊C₄	C₂³	1	6	yes	—
90	Dic₄₇⋊C₄	C₂³	1	12	yes	—
91	Dic₄₇⋊C₄	C₂³	1	6	yes	—
92	C₄×Dic₄₇	C₂³	1	6	yes	—
93	C₂³×C₄₇	—	2	6	no	—
94	C₂³×C₄₇	—	2	6	no	—
95	C₂³×C₄₇	—	4	3	no	—
96	C₂³×C₄₇	C₂⁴	1	3	yes	—
97	C₂³×C₄₇	C₂⁴	1	3	yes	—
98	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
99	C₂²×D₄₇	C₂⁴	1	2	yes	—
100	Q₈×C₄₇	—	2	6	no	—
101	Q₈×C₄₇	C₂×D₄	1	4	yes	—
102	D₄×C₄₇	—	2	12	no	—
103	D₄×C₄₇	—	2	6	no	—
104	D₄×C₄₇	C₂×D₄	1	12	yes	—
105	C₂×C₄×C₄₇	—	2	6	no	—
106	C₂×C₄×C₄₇	—	2	2	no	—
107	C₂×C₄×C₄₇	—	2	6	no	—
108	C₂×C₄×C₄₇	—	2	6	no	—
109	C₂×C₄×C₄₇	—	2	6	no	—
110	C₂×C₄×C₄₇	C₂×D₄	1	12	yes	—
111	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
113	C₂×C₄×C₄₇	C₂⁴	1	3	yes	—
114	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
115	C₂²⋊D₄₇	—	2	12	no	—
116	C₂²⋊D₄₇	—	2	4	no	—
117	C₂²⋊D₄₇	—	4	6	no	—
118	C₂×Dic₄₇	—	2	12	no	—
119	C₂×Dic₄₇	—	4	6	no	—
120	C₂×Dic₄₇	C₂⁴	1	2	yes	—
121	C₂×Dic₄₇	C₂⁴	1	6	yes	—
122	C₂×Dic₄₇	C₂⁴	1	6	yes	—
123	D₁₈₈	—	2	2	no	—
124	C₄×D₄₇	—	2	12	no	—
125	Dic₉₄	—	2	6	no	—
126	Dic₉₄	—	2	12	no	—
127	C₂²×C₄₇	—	2	1	no	—
128	C₂²×C₄₇	—	2	3	no	—
129	C₂²×C₄₇	—	2	3	no	—
130	C₂²×C₄₇	—	2	1	no	—
131	C₂²×C₄₇	—	2	3	no	—
132	C₂²×C₄₇	—	2	3	no	—
133	C₂²×C₄₇	—	2	3	no	—
134	C₂²×C₄₇	—	4	12	no	—
135	C₂²×C₄₇	C₂²×D₄	1	6	yes	—
136	C₂²×C₄₇	2₋¹⁺⁴	1	2	yes	complemented
137	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
138	C₂×D₄₇	—	2	2	no	—
139	C₂×D₄₇	—	2	2	no	—
140	C₂×D₄₇	—	2	2	no	—
141	C₄×C₄₇	—	2	1	no	—
142	C₄×C₄₇	—	2	3	no	—
143	C₄×C₄₇	C₂²×D₄	1	2	yes	—
144	C₄×C₄₇	C₂²×D₄	1	6	yes	—
145	Dic₄₇	—	2	6	no	—
146	Dic₄₇	—	2	2	no	—
147	Dic₄₇	—	2	6	no	—
148	#3489	—	47	1	no	maximal, Sylow
149	C₂×C₄₇	—	2	2	no	—
150	C₂×C₄₇	—	2	6	no	—
151	C₂×C₄₇	—	4	3	no	—
152	C₂×C₄₇	C₂³×D₄	1	1	yes	characteristic
153	C₂×C₄₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
154	C₂×C₄₇	#585	1	1	yes	characteristic
155	D₄₇	—	4	2	no	—
156	C₂×2₋¹⁺⁴	—	47	1	no	—
157	C₂²×#41	D₄₇	1	1	yes	characteristic, complemented
158	D₄×Q₈	—	47	2	no	—
159	#548	—	47	2	no	—
160	#547	—	47	6	no	—
161	#546	—	47	6	no	—
162	#537	—	47	3	no	—
163	#536	—	47	3	no	—
164	#525	—	47	3	no	—
165	C₂×C₂²⋊Q₈	—	47	3	no	—
166	#520	—	47	1	no	—
167	C₄₇	#3489	1	1	yes	characteristic, complemented, Sylow
168	2₋¹⁺⁴	—	94	8	no	—
169	C₂×#41	—	2	8	no	—
170	C₂×#41	—	47	6	no	—
171	C₂×#41	—	47	3	no	—
172	C₂×#41	C₂×D₄₇	1	3	yes	—
173	C₂×#41	C₂×D₄₇	1	3	yes	—
174	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
175	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
176	C₂²×Q₈	—	47	2	no	—
177	C₂²×Q₈	—	47	3	no	—
178	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
179	C₂²×D₄	C₂×D₄₇	1	3	yes	—
180	C₂³×C₄	C₂×D₄₇	1	3	yes	—
181	C₄⋊Q₈	—	47	6	no	—
182	#124	—	47	6	no	—
183	#123	—	47	12	no	—
184	#123	—	47	6	no	—
185	C₂²⋊Q₈	—	47	6	no	—
186	C₂²⋊Q₈	—	47	3	no	—
187	C₂²⋊Q₈	—	47	6	no	—
188	C₂²⋊Q₈	—	47	6	no	—
189	C₂²⋊Q₈	—	47	6	no	—
190	C₂²⋊Q₈	—	47	3	no	—
191	#121	—	47	12	no	—
192	#121	—	47	6	no	—
193	#121	—	47	1	no	—
194	#121	—	47	3	no	—
195	#120	—	47	6	no	—
196	C₄×Q₈	—	47	2	no	—
197	C₄×D₄	—	47	6	no	—
198	C₄×D₄	—	47	6	no	—
199	C₄×D₄	—	47	6	no	—
200	#117	—	47	3	no	—
201	C₂×C₄⋊C₄	—	47	3	no	—
202	C₂×C₄⋊C₄	—	47	6	no	—
203	C₂×C₂²⋊C₄	—	47	6	no	—
204	C₂⁴	C₂²×D₄₇	1	3	yes	—
205	#41	—	2	4	no	—
206	#41	—	2	12	no	—
207	#41	—	2	12	no	—
208	#41	—	94	24	no	—
209	#41	—	94	12	no	—
210	#41	C₂²⋊D₄₇	1	8	yes	complemented
211	C₂×Q₈	—	2	4	no	—
212	C₂×Q₈	—	47	3	no	—
213	C₂×Q₈	—	47	6	no	—
214	C₂×Q₈	—	94	8	no	—
215	C₂×Q₈	—	94	12	no	—
216	C₂×Q₈	C₂²×D₄₇	1	3	yes	—
217	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
218	C₂×D₄	—	2	12	no	—
219	C₂×D₄	—	47	2	no	—
220	C₂×D₄	—	47	6	no	—
221	C₂×D₄	—	47	6	no	—
222	C₂×D₄	—	47	1	no	—
223	C₂×D₄	C₂²×D₄₇	1	6	yes	—
224	C₂×D₄	C₂²×D₄₇	1	3	yes	—
225	C₂×D₄	C₂²×D₄₇	1	3	yes	—
226	C₂²×C₄	—	2	12	no	—
227	C₂²×C₄	—	47	6	no	—
228	C₂²×C₄	—	47	6	no	—
229	C₂²×C₄	—	47	6	no	—
230	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
231	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
232	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
233	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
234	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
235	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
236	C₄⋊C₄	—	47	6	no	—
237	C₄⋊C₄	—	47	12	no	—
238	C₄⋊C₄	—	47	6	no	—
239	C₄⋊C₄	—	47	3	no	—
240	C₄⋊C₄	—	47	6	no	—
241	C₄⋊C₄	—	47	1	no	—
242	C₂²⋊C₄	—	47	6	no	—
243	C₂²⋊C₄	—	47	12	no	—
244	C₂²⋊C₄	—	47	6	no	—
245	C₂²⋊C₄	—	47	6	no	—
246	C₂²⋊C₄	—	47	2	no	—
247	C₂²⋊C₄	—	47	6	no	—
248	C₄²	—	47	6	no	—
249	C₂³	—	2	6	no	—
250	C₂³	—	2	6	no	—
251	C₂³	—	4	3	no	—
252	C₂³	—	47	2	no	—
253	C₂³	C₂³×D₄₇	1	3	yes	—
254	C₂³	C₂³×D₄₇	1	3	yes	—
255	C₂³	C₂³×D₄₇	1	1	yes	characteristic
256	Q₈	—	2	6	no	—
257	Q₈	—	94	6	no	—
258	Q₈	—	94	12	no	—
259	Q₈	C₂×C₂²⋊D₄₇	1	4	yes	—
260	D₄	—	2	12	no	—
261	D₄	—	2	6	no	—
262	D₄	—	94	12	no	—
263	D₄	—	94	4	no	—
264	D₄	—	94	2	no	—
265	D₄	—	188	6	no	—
266	D₄	C₂×C₂²⋊D₄₇	1	12	yes	—
267	C₂×C₄	—	2	6	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	6	no	—
270	C₂×C₄	—	2	6	no	—
271	C₂×C₄	—	2	6	no	—
272	C₂×C₄	—	47	2	no	—
273	C₂×C₄	—	47	6	no	—
274	C₂×C₄	—	47	6	no	—
275	C₂×C₄	—	94	12	no	—
276	C₂×C₄	—	94	12	no	—
277	C₂×C₄	—	188	6	no	—
278	C₂×C₄	C₂×C₂²⋊D₄₇	1	12	yes	—
279	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₄₇	1	3	yes	—
281	C₂×C₄	C₂³×D₄₇	1	3	yes	—
282	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
283	C₂²	—	2	1	no	—
284	C₂²	—	2	3	no	—
285	C₂²	—	2	3	no	—
286	C₂²	—	2	1	no	—
287	C₂²	—	2	3	no	—
288	C₂²	—	2	3	no	—
289	C₂²	—	2	3	no	—
290	C₂²	—	4	12	no	—
291	C₂²	—	94	2	no	—
292	C₂²	—	94	2	no	—
293	C₂²	—	94	2	no	—
294	C₂²	C₂²×C₂²⋊D₄₇	1	6	yes	—
295	C₂²	#13094071	1	2	yes	complemented
296	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
297	C₄	—	2	1	no	—
298	C₄	—	2	3	no	—
299	C₄	—	94	6	no	—
300	C₄	—	94	2	no	—
301	C₄	—	94	6	no	—
302	C₄	C₂²×C₂²⋊D₄₇	1	2	yes	—
303	C₄	C₂²×C₂²⋊D₄₇	1	6	yes	—
304	C₂	—	2	2	no	—
305	C₂	—	2	6	no	—
306	C₂	—	4	3	no	—
307	C₂	—	188	2	no	—
308	C₂	C₂³×C₂²⋊D₄₇	1	1	yes	characteristic
309	C₂	C₂×#13094071	1	1	yes	characteristic
310	C₂	#15689959	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₂²×#41×C₄₇	C₂
hypercenter	C₂²×#41	D₄₇
hypocenter	C₄₇	#3489
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#13094071	C₂	semidirect
2. Q₈×C₂²⋊D₄₇	C₂	semidirect
3. #15689780	C₂	semidirect
4. #15689772	C₂	semidirect
5. #15689767	C₂	semidirect
6. #15689701	C₂	semidirect
7. #15689694	C₂	semidirect
8. #15689634	C₂	semidirect
9. C₂×#13094003	C₂	semidirect
10. #15689606	C₂	semidirect
11. C₂²×#41×C₄₇	C₂	semidirect
13. C₂×#10625350	C₂²	semidirect
14. C₂×#10625352	C₂²	semidirect
15. C₂²×Dic₉₄	C₂²	semidirect
16. C₂×Q₈×D₄₇	C₂²	semidirect
17. #13094022	C₂²	semidirect
18. #13094011	C₂²	semidirect
19. #13094007	C₂²	semidirect
20. #13094005	C₂²	semidirect
23. #13094003	C₂²	semidirect
24. #13094003	C₂²	semidirect
25. #13094003	C₂²	semidirect
26. #13094000	C₂²	semidirect
28. #13093998	C₂²	semidirect
29. #13093996	C₂²	semidirect
30. #13093995	C₂²	semidirect
32. Q₈×Dic₄₇	C₂²	semidirect
33. D₄×Dic₄₇	C₂²	semidirect
35. C₄×C₂²⋊D₄₇	C₂²	semidirect
36. #13093983	C₂²	semidirect
37. C₂×C₄⋊Dic₄₇	C₂²	semidirect
38. C₂×Dic₄₇⋊C₄	C₂²	semidirect
52. #41×C₄₇	D₄	semidirect
136. C₂²×C₄₇	2₋¹⁺⁴	semidirect
157. C₂²×#41	D₄₇	semidirect
167. C₄₇	#3489	semidirect
210. #41	C₂²⋊D₄₇	semidirect
295. C₂²	#13094071	semidirect