Subgroups — #20257161

Order 6016.

1748 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	C₂×#13094070	C₂	1	2	yes	maximal, complemented
2	#15689794	C₂	1	4	yes	maximal, complemented
3	D₄×D₁₈₈	C₂	1	4	yes	maximal, complemented
4	#15689757	C₂	1	8	yes	maximal, complemented
5	#15689691	C₂	1	4	yes	maximal, complemented
6	#15689687	C₂	1	2	yes	maximal, complemented
7	#15689674	C₂	1	2	yes	maximal, complemented
8	C₂×#13094021	C₂	1	1	yes	maximal, characteristic, complemented
9	#127×D₄₇	C₂	1	2	yes	maximal, complemented
10	C₄×#10625350	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₇×#532	C₂	1	1	yes	maximal, characteristic, complemented
12	#13094070	—	2	16	no	—
13	C₂×#10625354	C₂²	1	4	yes	complemented
14	C₂×#10625350	C₂²	1	1	yes	characteristic
15	C₂×#10625350	C₂²	1	4	yes	complemented
16	C₂²×D₁₈₈	C₂²	1	4	yes	complemented
17	C₂²×D₁₈₈	C₂²	1	2	yes	complemented
18	C₂²×D₁₈₈	C₂²	1	4	yes	complemented
19	C₂×D₄×D₄₇	C₂²	1	8	yes	complemented
20	C₂×D₄×D₄₇	C₂²	1	4	yes	complemented
21	C₂×D₄×D₄₇	C₂²	1	8	yes	complemented
22	#13094021	C₂²	1	4	yes	complemented
23	#13094021	C₂²	1	2	yes	complemented
24	#13094021	C₂²	1	2	yes	complemented
25	#13094020	C₂²	1	8	yes	complemented
26	#13094020	C₂²	1	4	yes	complemented
27	#13094013	C₂²	1	4	yes	complemented
28	#13093999	C₂²	1	8	yes	complemented
29	#13093998	C₂²	1	8	yes	complemented
30	#13093997	C₂²	1	16	yes	complemented
31	#13093994	C₂²	1	16	yes	complemented
32	#13093992	C₂²	1	8	yes	complemented
33	C₄×Dic₉₄	C₂²	1	2	yes	complemented
34	C₄×D₁₈₈	C₂²	1	2	yes	complemented
35	#13093986	C₂²	1	8	yes	complemented
36	C₄×C₂²⋊D₄₇	C₂²	1	4	yes	complemented
37	#13093983	C₂²	1	1	yes	characteristic, complemented
38	#13093981	C₂²	1	2	yes	complemented
39	C₄²×D₄₇	C₂²	1	2	yes	complemented
40	C₂×#41×C₄₇	C₂²	1	2	yes	complemented
41	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
42	#127×C₄₇	C₂²	1	1	yes	characteristic, complemented
43	#124×C₄₇	C₂²	1	2	yes	complemented
44	#121×C₄₇	C₂²	1	4	yes	complemented
45	C₄×D₄×C₄₇	C₂²	1	4	yes	complemented
46	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic, complemented
47	C₂³×D₄₇	C₂³	1	4	yes	—
48	C₂³×D₄₇	C₂³	1	8	yes	—
49	#41×C₄₇	—	2	8	no	—
50	C₂×Q₈×C₄₇	C₂³	1	2	yes	—
51	C₂×D₄×C₄₇	C₂³	1	4	yes	—
52	C₂×D₄×C₄₇	C₂³	1	4	yes	—
53	C₂×D₄×C₄₇	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₇	C₂³	1	2	yes	—
55	C₂²×C₄×C₄₇	C₂³	1	4	yes	—
56	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₄₇	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₄₇	C₂³	1	16	yes	—
59	C₂×C₂²⋊D₄₇	C₂³	1	2	yes	—
60	#10625354	—	2	16	no	—
61	D₄×D₄₇	—	2	32	no	—
62	D₄×D₄₇	—	2	16	no	—
63	D₄×D₄₇	—	4	16	no	—
64	#10625350	—	2	16	no	—
65	#10625350	D₄	1	8	yes	complemented
66	C₂×D₁₈₈	—	2	8	no	—
67	C₂×D₁₈₈	—	2	16	no	—
68	C₂×D₁₈₈	—	4	8	no	—
69	C₂×D₁₈₈	C₂³	1	8	yes	—
70	C₂×D₁₈₈	C₂³	1	8	yes	—
71	C₂×D₁₈₈	C₂³	1	4	yes	—
72	C₂×D₁₈₈	C₂³	1	4	yes	—
73	C₂×D₁₈₈	C₂³	1	8	yes	—
74	C₂×D₁₈₈	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₄₇	C₂³	1	4	yes	—
76	C₂×C₄×D₄₇	C₂³	1	8	yes	—
77	C₂×C₄×D₄₇	C₂³	1	2	yes	—
78	C₂×Dic₉₄	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₄₇	C₂³	1	4	yes	—
80	C₂²⋊C₄×C₄₇	C₂³	1	8	yes	—
81	C₄²×C₄₇	C₂³	1	2	yes	—
82	C₄²×C₄₇	C₂³	1	1	yes	characteristic
83	C₄²×C₄₇	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₄₇	C₂³	1	2	yes	—
85	(C₂×D₄₇)⋊C₄	C₂³	1	4	yes	—
86	(C₂×D₄₇)⋊C₄	C₂³	1	16	yes	—
87	C₄⋊Dic₄₇	C₂³	1	2	yes	—
88	Dic₄₇⋊C₄	C₂³	1	4	yes	—
89	C₄×Dic₄₇	C₂³	1	2	yes	—
90	C₄×Dic₄₇	C₂³	1	4	yes	—
91	C₂³×C₄₇	C₂⁴	1	4	yes	—
92	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
93	C₂²×D₄₇	—	2	8	no	—
94	C₂²×D₄₇	—	2	16	no	—
95	C₂²×D₄₇	—	4	8	no	—
96	C₂²×D₄₇	—	4	16	no	—
97	C₂²×D₄₇	C₂⁴	1	8	yes	—
98	C₂²×D₄₇	C₂⁴	1	2	yes	—
99	Q₈×C₄₇	—	2	4	no	—
100	D₄×C₄₇	—	2	8	no	—
101	D₄×C₄₇	—	2	4	no	—
102	D₄×C₄₇	—	2	8	no	—
103	C₂×C₄×C₄₇	—	2	8	no	—
104	C₂×C₄×C₄₇	—	2	4	no	—
105	C₂×C₄×C₄₇	C₂×D₄	1	4	yes	complemented
106	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
107	C₂×C₄×C₄₇	C₂⁴	1	4	yes	—
108	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
109	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
110	C₂×C₄×C₄₇	C₂⁴	1	2	yes	—
111	C₂²⋊D₄₇	—	2	16	no	—
112	C₂²⋊D₄₇	—	4	16	no	—
113	C₂²⋊D₄₇	C₂×D₄	1	8	yes	complemented
114	C₂×Dic₄₇	C₂⁴	1	4	yes	—
115	C₂×Dic₄₇	C₂⁴	1	2	yes	—
116	D₁₈₈	—	2	16	no	—
117	D₁₈₈	—	2	8	no	—
118	D₁₈₈	—	2	8	no	—
119	D₁₈₈	—	2	16	no	—
120	D₁₈₈	—	4	8	no	—
121	D₁₈₈	C₂×D₄	1	4	yes	complemented
122	C₄×D₄₇	—	2	8	no	—
123	C₄×D₄₇	—	2	16	no	—
124	C₄×D₄₇	C₂×D₄	1	8	yes	complemented
125	Dic₉₄	C₂×D₄	1	4	yes	complemented
126	C₂²×C₄₇	—	2	2	no	—
127	C₂²×C₄₇	—	2	4	no	—
128	C₂²×C₄₇	—	2	8	no	—
129	C₂²×C₄₇	C₂²×D₄	1	2	yes	—
130	C₂²×C₄₇	C₂⁵	1	1	yes	characteristic
131	C₂×D₄₇	—	2	4	no	—
132	C₂×D₄₇	—	2	8	no	—
133	C₂×D₄₇	—	2	16	no	—
134	C₂×D₄₇	—	8	16	no	—
135	C₂×D₄₇	—	8	8	no	—
136	C₂×D₄₇	C₂²×D₄	1	4	yes	—
137	C₄×C₄₇	—	2	4	no	—
138	C₄×C₄₇	—	2	2	no	—
139	C₄×C₄₇	C₂²×D₄	1	2	yes	—
140	C₄×C₄₇	C₂²×D₄	1	2	yes	—
141	C₄×C₄₇	H(5,𝔽₂)	1	4	yes	complemented
142	Dic₄₇	—	2	4	no	—
143	Dic₄₇	C₂²×D₄	1	4	yes	—
144	#3498	—	47	1	no	maximal, Sylow
145	C₂×C₄₇	—	2	2	no	—
146	C₂×C₄₇	—	4	4	no	—
147	C₂×C₄₇	C₂³×D₄	1	1	yes	characteristic
148	C₂×C₄₇	C₂×H(5,𝔽₂)	1	2	yes	—
149	D₄₇	—	2	4	no	—
150	D₄₇	—	4	8	no	—
151	C₂×H(5,𝔽₂)	—	47	2	no	—
152	#550	—	47	4	no	—
153	D₄²	—	47	8	no	—
154	D₄²	—	47	4	no	—
155	#535	—	47	2	no	—
156	#535	—	47	4	no	—
157	#532	—	47	2	no	—
158	#532	D₄₇	1	1	yes	characteristic, complemented
159	C₂×#127	—	47	2	no	—
160	C₂×#127	—	47	1	no	—
161	C₄×#41	—	47	1	no	—
162	C₄₇	#3498	1	1	yes	characteristic, complemented, Sylow
163	H(5,𝔽₂)	—	94	16	no	—
164	C₂×#41	—	47	4	no	—
165	C₂×#41	—	47	1	no	—
166	C₂×#41	—	47	4	no	—
167	C₂×#41	C₂×D₄₇	1	2	yes	complemented
168	C₂²×D₄	—	47	4	no	—
169	C₂²×D₄	—	47	8	no	—
170	C₂²×D₄	—	47	8	no	—
171	C₂²×D₄	—	47	2	no	—
172	C₂²×D₄	—	47	4	no	—
173	C₂²×D₄	—	47	4	no	—
174	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic, complemented
175	#127	—	47	4	no	—
176	#127	—	47	8	no	—
177	#127	—	47	2	no	—
178	#127	—	47	2	no	—
179	#127	—	47	4	no	—
180	#127	C₂×D₄₇	1	1	yes	characteristic, complemented
181	#124	—	47	4	no	—
182	#124	C₂×D₄₇	1	2	yes	complemented
183	#121	—	47	8	no	—
184	#121	—	47	8	no	—
185	#121	—	47	16	no	—
186	#121	C₂×D₄₇	1	4	yes	complemented
187	#120	—	47	8	no	—
188	#120	—	47	16	no	—
189	C₄×Q₈	—	47	2	no	—
190	C₄×D₄	—	47	4	no	—
191	C₄×D₄	—	47	8	no	—
192	C₄×D₄	—	47	2	no	—
193	C₄×D₄	C₂×D₄₇	1	4	yes	complemented
194	#117	—	47	1	no	—
195	#117	—	47	2	no	—
196	C₂×C₄²	—	47	2	no	—
197	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic, complemented
198	C₂⁴	—	47	8	no	—
199	C₂⁴	—	47	4	no	—
200	#41	—	2	8	no	—
201	#41	—	47	8	no	—
202	#41	—	94	16	no	—
203	#41	—	94	16	no	—
204	C₂×Q₈	—	47	1	no	—
205	C₂×Q₈	C₂²×D₄₇	1	2	yes	—
206	C₂×D₄	—	47	2	no	—
207	C₂×D₄	—	47	8	no	—
208	C₂×D₄	—	47	16	no	—
209	C₂×D₄	—	47	1	no	—
210	C₂×D₄	—	47	8	no	—
211	C₂×D₄	—	47	8	no	—
212	C₂×D₄	—	47	4	no	—
213	C₂×D₄	—	47	4	no	—
214	C₂×D₄	—	47	8	no	—
215	C₂×D₄	—	94	16	no	—
216	C₂×D₄	—	94	16	no	—
217	C₂×D₄	—	94	32	no	—
218	C₂×D₄	—	94	8	no	—
219	C₂×D₄	—	188	8	no	—
220	C₂×D₄	—	188	16	no	—
221	C₂×D₄	C₂²×D₄₇	1	4	yes	—
222	C₂×D₄	C₂²×D₄₇	1	4	yes	—
223	C₂×D₄	C₂²×D₄₇	1	2	yes	—
224	C₂²×C₄	—	47	2	no	—
225	C₂²×C₄	—	47	4	no	—
226	C₂²×C₄	—	47	8	no	—
227	C₂²×C₄	C₂²×D₄₇	1	2	yes	—
228	C₂²×C₄	C₂²×D₄₇	1	4	yes	—
229	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
230	C₄⋊C₄	—	47	4	no	—
231	C₄⋊C₄	—	47	2	no	—
232	C₄⋊C₄	C₂²×D₄₇	1	4	yes	—
233	C₂²⋊C₄	—	47	2	no	—
234	C₂²⋊C₄	—	47	4	no	—
235	C₂²⋊C₄	—	47	16	no	—
236	C₂²⋊C₄	C₂²×D₄₇	1	8	yes	—
237	C₄²	—	47	2	no	—
238	C₄²	—	47	4	no	—
239	C₄²	C₂²×D₄₇	1	1	yes	characteristic
240	C₄²	C₂²×D₄₇	1	1	yes	characteristic
241	C₄²	C₂²×D₄₇	1	2	yes	—
242	C₂³	—	47	2	no	—
243	C₂³	—	47	8	no	—
244	C₂³	—	94	16	no	—
245	C₂³	—	94	8	no	—
246	C₂³	—	188	8	no	—
247	C₂³	—	188	16	no	—
248	C₂³	C₂³×D₄₇	1	4	yes	—
249	C₂³	C₂³×D₄₇	1	1	yes	characteristic
250	Q₈	—	2	4	no	—
251	Q₈	—	47	4	no	—
252	D₄	—	2	8	no	—
253	D₄	—	2	4	no	—
254	D₄	—	2	8	no	—
255	D₄	—	47	8	no	—
256	D₄	—	47	4	no	—
257	D₄	—	94	16	no	—
258	D₄	—	94	16	no	—
259	D₄	—	94	8	no	—
260	D₄	—	94	8	no	—
261	D₄	—	94	16	no	—
262	D₄	—	188	8	no	—
263	D₄	—	188	16	no	—
264	C₂×C₄	—	2	8	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	47	2	no	—
267	C₂×C₄	—	47	8	no	—
268	C₂×C₄	—	47	4	no	—
269	C₂×C₄	—	94	16	no	—
270	C₂×C₄	—	94	8	no	—
271	C₂×C₄	D₄×D₄₇	1	4	yes	complemented
272	C₂×C₄	C₂³×D₄₇	1	2	yes	—
273	C₂×C₄	C₂³×D₄₇	1	4	yes	—
274	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
275	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
276	C₂×C₄	C₂³×D₄₇	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	47	4	no	—
281	C₂²	—	94	4	no	—
282	C₂²	—	94	8	no	—
283	C₂²	—	94	16	no	—
284	C₂²	—	376	8	no	—
285	C₂²	—	376	16	no	—
286	C₂²	C₂×D₄×D₄₇	1	2	yes	—
287	C₂²	C₂⁴×D₄₇	1	1	yes	characteristic
288	C₄	—	2	4	no	—
289	C₄	—	2	2	no	—
290	C₄	—	47	4	no	—
291	C₄	—	94	4	no	—
292	C₄	C₂×D₄×D₄₇	1	2	yes	—
293	C₄	C₂×D₄×D₄₇	1	2	yes	—
294	C₄	#13094070	1	4	yes	complemented
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	94	4	no	—
298	C₂	—	188	8	no	—
299	C₂	C₂²×D₄×D₄₇	1	1	yes	characteristic
300	C₂	C₂×#13094070	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₇
commutator, socle	C₂²×C₄₇	C₂⁵
Fitting	C₄₇×#532	C₂
hypercenter	#532	D₄₇
hypocenter	C₄₇	#3498
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. C₂×#13094070	C₂	semidirect
2. #15689794	C₂	semidirect
3. D₄×D₁₈₈	C₂	semidirect
4. #15689757	C₂	semidirect
5. #15689691	C₂	semidirect
6. #15689687	C₂	semidirect
7. #15689674	C₂	semidirect
8. C₂×#13094021	C₂	semidirect
9. #127×D₄₇	C₂	semidirect
10. C₄×#10625350	C₂	semidirect
11. C₄₇×#532	C₂	semidirect
13. C₂×#10625354	C₂²	semidirect
15. C₂×#10625350	C₂²	semidirect
16. C₂²×D₁₈₈	C₂²	semidirect
17. C₂²×D₁₈₈	C₂²	semidirect
18. C₂²×D₁₈₈	C₂²	semidirect
19. C₂×D₄×D₄₇	C₂²	semidirect
20. C₂×D₄×D₄₇	C₂²	semidirect
21. C₂×D₄×D₄₇	C₂²	semidirect
22. #13094021	C₂²	semidirect
23. #13094021	C₂²	semidirect
24. #13094021	C₂²	semidirect
25. #13094020	C₂²	semidirect
26. #13094020	C₂²	semidirect
27. #13094013	C₂²	semidirect
28. #13093999	C₂²	semidirect
29. #13093998	C₂²	semidirect
30. #13093997	C₂²	semidirect
31. #13093994	C₂²	semidirect
32. #13093992	C₂²	semidirect
33. C₄×Dic₉₄	C₂²	semidirect
34. C₄×D₁₈₈	C₂²	semidirect
35. #13093986	C₂²	semidirect
36. C₄×C₂²⋊D₄₇	C₂²	semidirect
37. #13093983	C₂²	semidirect
38. #13093981	C₂²	semidirect
39. C₄²×D₄₇	C₂²	semidirect
40. C₂×#41×C₄₇	C₂²	semidirect
41. C₄⋊Q₈×C₄₇	C₂²	semidirect
42. #127×C₄₇	C₂²	semidirect
43. #124×C₄₇	C₂²	semidirect
44. #121×C₄₇	C₂²	semidirect
45. C₄×D₄×C₄₇	C₂²	semidirect
46. C₂×C₄²×C₄₇	C₂²	semidirect
65. #10625350	D₄	semidirect
105. C₂×C₄×C₄₇	C₂×D₄	semidirect
113. C₂²⋊D₄₇	C₂×D₄	semidirect
121. D₁₈₈	C₂×D₄	semidirect
124. C₄×D₄₇	C₂×D₄	semidirect
125. Dic₉₄	C₂×D₄	semidirect
141. C₄×C₄₇	H(5,𝔽₂)	semidirect
158. #532	D₄₇	semidirect
162. C₄₇	#3498	semidirect
167. C₂×#41	C₂×D₄₇	semidirect
174. C₄⋊Q₈	C₂×D₄₇	semidirect
180. #127	C₂×D₄₇	semidirect
182. #124	C₂×D₄₇	semidirect
186. #121	C₂×D₄₇	semidirect
193. C₄×D₄	C₂×D₄₇	semidirect
197. C₂×C₄²	C₂×D₄₇	semidirect
271. C₂×C₄	D₄×D₄₇	semidirect
294. C₄	#13094070	semidirect