Subgroups — #19148996

Order 5504.

1258 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583641	C₂	1	4	yes	maximal, complemented
2	#14583615	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583612	C₂	1	4	yes	maximal, complemented
4	D₄×D₁₇₂	C₂	1	2	yes	maximal, complemented
5	#14583533	C₂	1	4	yes	maximal, complemented
6	#14583468	C₂	1	4	yes	maximal, complemented
7	#14583467	C₂	1	4	yes	maximal, complemented
8	#14583465	C₂	1	4	yes	maximal, complemented
9	#14583450	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583446	C₂	1	2	yes	maximal, complemented
11	C₄₃×#555	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621902	C₂²	1	2	yes	complemented
13	C₂×#10621898	C₂²	1	4	yes	complemented
14	C₂²×D₁₇₂	C₂²	1	4	yes	complemented
15	C₂×D₄×D₄₃	C₂²	1	4	yes	complemented
16	C₂×D₄×D₄₃	C₂²	1	4	yes	complemented
17	#13063324	C₂²	1	2	yes	complemented
18	#13063322	C₂²	1	1	yes	characteristic
19	#13063322	C₂²	1	4	yes	complemented
20	#13063321	C₂²	1	2	yes	—
21	#13063321	C₂²	1	4	yes	complemented
22	#13063314	C₂²	1	4	yes	complemented
23	#13063314	C₂²	1	2	yes	complemented
24	#13063313	C₂²	1	8	yes	complemented
25	#13063311	C₂²	1	8	yes	complemented
26	#13063311	C₂²	1	4	yes	complemented
27	#13063300	C₂²	1	4	yes	complemented
28	#13063300	C₂²	1	4	yes	complemented
29	#13063300	C₂²	1	2	yes	complemented
30	#13063300	C₂²	1	4	yes	complemented
31	#13063299	C₂²	1	4	yes	complemented
32	#13063298	C₂²	1	8	yes	complemented
33	#13063298	C₂²	1	4	yes	complemented
34	#13063296	C₂²	1	2	yes	complemented
35	#13063295	C₂²	1	8	yes	complemented
36	#13063295	C₂²	1	8	yes	complemented
37	#13063295	C₂²	1	8	yes	complemented
38	#13063293	C₂²	1	8	yes	complemented
39	C₄×D₁₇₂	C₂²	1	2	yes	complemented
40	#13063287	C₂²	1	4	yes	complemented
41	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	complemented
42	#13063283	C₂²	1	4	yes	complemented
43	#13063282	C₂²	1	4	yes	complemented
44	C₄²×D₄₃	C₂²	1	1	yes	characteristic, complemented
45	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
46	#124×C₄₃	C₂²	1	4	yes	complemented
47	#121×C₄₃	C₂²	1	4	yes	complemented
48	C₄×D₄×C₄₃	C₂²	1	2	yes	complemented
49	#117×C₄₃	C₂²	1	4	yes	complemented
50	C₂³×D₄₃	C₂³	1	8	yes	—
51	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
52	C₂×D₄×C₄₃	C₂³	1	2	yes	—
53	C₂×D₄×C₄₃	C₂³	1	4	yes	—
54	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
55	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
56	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
57	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
58	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
59	#10621902	—	4	4	no	—
60	D₄×D₄₃	—	4	8	no	—
61	D₄×D₄₃	—	4	8	no	—
62	#10621898	—	4	8	no	—
63	C₂×D₁₇₂	—	4	8	no	—
64	C₂×D₁₇₂	C₂³	1	4	yes	—
65	C₂×D₁₇₂	C₂³	1	2	yes	—
66	C₂×D₁₇₂	C₂³	1	2	yes	—
67	C₂×D₁₇₂	C₂³	1	8	yes	—
68	C₂×D₁₇₂	C₂³	1	4	yes	—
69	C₂×D₁₇₂	C₂³	1	4	yes	—
70	C₂×D₁₇₂	C₂³	1	4	yes	—
71	C₂×C₄×D₄₃	C₂³	1	2	yes	—
72	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₄×D₄₃	C₂³	1	4	yes	—
74	C₂×C₄×D₄₃	C₂³	1	4	yes	—
75	C₂×Dic₈₆	C₂³	1	4	yes	—
76	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
77	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
79	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
80	C₄²×C₄₃	C₂³	1	4	yes	—
81	C₄²×C₄₃	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
83	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
84	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
85	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
86	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
87	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
88	C₄⋊Dic₄₃	C₂³	1	2	yes	—
89	Dic₄₃⋊C₄	C₂³	1	4	yes	—
90	C₄×Dic₄₃	C₂³	1	2	yes	—
91	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
92	C₄×Dic₄₃	C₂³	1	4	yes	—
93	C₂³×C₄₃	C₂⁴	1	4	yes	—
94	C₂²×D₄₃	—	4	8	no	—
95	C₂²×D₄₃	—	4	8	no	—
96	C₂²×D₄₃	—	4	8	no	—
97	C₂²×D₄₃	C₂⁴	1	4	yes	—
98	C₂²×D₄₃	C₂⁴	1	4	yes	—
99	C₂²×D₄₃	C₂⁴	1	1	yes	characteristic
100	Q₈×C₄₃	—	4	2	no	—
101	D₄×C₄₃	—	2	4	no	—
102	D₄×C₄₃	—	4	4	no	—
103	C₂×C₄×C₄₃	—	2	8	no	—
104	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
106	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
107	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
108	C₂²⋊D₄₃	—	2	8	no	—
109	C₂²⋊D₄₃	—	4	4	no	—
110	C₂²⋊D₄₃	—	4	4	no	—
111	C₂²⋊D₄₃	—	4	8	no	—
112	C₂×Dic₄₃	C₂⁴	1	4	yes	—
113	C₂×Dic₄₃	C₂⁴	1	2	yes	—
114	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
115	D₁₇₂	—	2	4	no	—
116	D₁₇₂	—	2	8	no	—
117	D₁₇₂	—	4	8	no	—
118	D₁₇₂	—	4	4	no	—
119	D₁₇₂	—	4	4	no	—
120	D₁₇₂	—	4	4	no	—
121	D₁₇₂	—	4	2	no	—
122	C₄×D₄₃	—	2	4	no	—
123	C₄×D₄₃	—	2	8	no	—
124	C₄×D₄₃	—	2	8	no	—
125	C₄×D₄₃	—	2	2	no	—
126	Dic₈₆	—	4	4	no	—
127	C₂²×C₄₃	—	2	4	no	—
128	C₂²×C₄₃	—	2	4	no	—
129	C₂²×C₄₃	—	2	4	no	—
130	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
131	C₂×D₄₃	—	2	4	no	—
132	C₂×D₄₃	—	2	4	no	—
133	C₂×D₄₃	—	2	4	no	—
134	C₂×D₄₃	—	2	8	no	—
135	C₂×D₄₃	—	2	4	no	—
136	C₂×D₄₃	—	2	1	no	—
137	C₂×D₄₃	—	2	2	no	—
138	C₂×D₄₃	—	8	16	no	—
139	C₄×C₄₃	—	2	4	no	—
140	C₄×C₄₃	—	2	4	no	—
141	C₄×C₄₃	—	2	1	no	—
142	C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
143	Dic₄₃	—	2	2	no	—
144	Dic₄₃	—	2	4	no	—
145	Dic₄₃	H(5,𝔽₂)	1	2	yes	—
146	#3549	—	43	1	no	maximal, Sylow
147	C₂×C₄₃	—	4	4	no	—
148	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
150	D₄₃	—	4	1	no	—
151	D₄₃	—	4	4	no	—
152	D₄₃	—	4	4	no	—
153	H(3,𝔽₄)	—	43	4	no	—
154	#555	—	43	1	no	—
155	#555	—	43	4	no	—
156	#555	D₄₃	1	1	yes	characteristic, complemented
157	D₄²	—	43	4	no	—
158	D₄²	—	43	2	no	—
159	#535	—	43	4	no	—
160	#535	—	43	4	no	—
161	#535	—	43	4	no	—
162	#532	—	43	1	no	—
163	#532	—	43	2	no	—
164	C₄₃	#3549	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	—	43	2	no	—
166	C₂×#41	—	43	4	no	—
167	C₂²×D₄	—	43	4	no	—
168	C₂²×D₄	—	43	4	no	—
169	C₂²×D₄	—	43	4	no	—
170	C₄⋊Q₈	—	43	2	no	—
171	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
172	#127	—	43	2	no	—
173	#127	—	43	4	no	—
174	#127	—	43	1	no	—
175	#127	—	43	4	no	—
176	#124	—	43	2	no	—
177	#124	—	43	4	no	—
178	#124	—	43	4	no	—
179	#124	—	43	8	no	—
180	#124	—	43	8	no	—
181	#124	C₂×D₄₃	1	4	yes	complemented
182	#121	—	43	2	no	—
183	#121	—	43	4	no	—
184	#121	—	43	4	no	—
185	#121	—	43	2	no	—
186	#121	—	43	4	no	—
187	#121	—	43	4	no	—
188	#121	—	43	4	no	—
189	#121	—	43	8	no	—
190	#121	C₂×D₄₃	1	4	yes	complemented
191	#120	—	43	8	no	—
192	#120	—	43	8	no	—
193	#120	—	43	8	no	—
194	#120	—	43	8	no	—
195	C₄×D₄	—	43	4	no	—
196	C₄×D₄	—	43	4	no	—
197	C₄×D₄	—	43	2	no	—
198	C₄×D₄	C₂×D₄₃	1	2	yes	complemented
199	#117	—	43	4	no	—
200	#117	—	43	4	no	—
201	#117	C₂×D₄₃	1	4	yes	complemented
202	C₂×C₄²	—	43	1	no	—
203	C₂⁴	—	43	8	no	—
204	#41	—	172	8	no	—
205	#41	—	172	4	no	—
206	C₂×Q₈	—	43	4	no	—
207	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
208	C₂×D₄	—	43	4	no	—
209	C₂×D₄	—	43	4	no	—
210	C₂×D₄	—	43	8	no	—
211	C₂×D₄	—	43	4	no	—
212	C₂×D₄	—	43	4	no	—
213	C₂×D₄	—	43	2	no	—
214	C₂×D₄	—	43	2	no	—
215	C₂×D₄	—	43	4	no	—
216	C₂×D₄	—	43	8	no	—
217	C₂×D₄	—	43	4	no	—
218	C₂×D₄	—	43	4	no	—
219	C₂×D₄	—	172	8	no	—
220	C₂×D₄	—	172	8	no	—
221	C₂×D₄	—	172	8	no	—
222	C₂×D₄	C₂²×D₄₃	1	2	yes	—
223	C₂×D₄	C₂²×D₄₃	1	4	yes	—
224	C₂²×C₄	—	43	1	no	—
225	C₂²×C₄	—	43	2	no	—
226	C₂²×C₄	—	43	4	no	—
227	C₂²×C₄	—	43	4	no	—
228	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
229	C₄⋊C₄	—	43	4	no	—
230	C₄⋊C₄	—	43	2	no	—
231	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
232	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
233	C₂²⋊C₄	—	43	4	no	—
234	C₂²⋊C₄	—	43	8	no	—
235	C₂²⋊C₄	—	43	8	no	—
236	C₂²⋊C₄	—	43	8	no	—
237	C₂²⋊C₄	—	43	4	no	—
238	C₂²⋊C₄	—	43	4	no	—
239	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
240	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
241	C₄²	—	43	2	no	—
242	C₄²	—	43	1	no	—
243	C₄²	—	43	4	no	—
244	C₄²	C₂²×D₄₃	1	1	yes	characteristic
245	C₄²	C₂²×D₄₃	1	4	yes	—
246	C₂³	—	43	4	no	—
247	C₂³	—	43	4	no	—
248	C₂³	—	43	1	no	—
249	C₂³	—	172	8	no	—
250	C₂³	—	172	8	no	—
251	C₂³	—	172	8	no	—
252	C₂³	C₂³×D₄₃	1	4	yes	—
253	Q₈	—	4	2	no	—
254	Q₈	—	172	4	no	—
255	D₄	—	2	4	no	—
256	D₄	—	4	4	no	—
257	D₄	—	86	8	no	—
258	D₄	—	86	4	no	—
259	D₄	—	86	8	no	—
260	D₄	—	172	4	no	—
261	D₄	—	172	8	no	—
262	D₄	—	172	4	no	—
263	D₄	—	172	2	no	—
264	D₄	—	172	4	no	—
265	D₄	—	172	8	no	—
266	D₄	—	172	4	no	—
267	D₄	—	172	4	no	—
268	C₂×C₄	—	2	8	no	—
269	C₂×C₄	—	43	1	no	—
270	C₂×C₄	—	43	2	no	—
271	C₂×C₄	—	43	4	no	—
272	C₂×C₄	—	86	8	no	—
273	C₂×C₄	—	86	2	no	—
274	C₂×C₄	—	86	8	no	—
275	C₂×C₄	—	86	4	no	—
276	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
277	C₂×C₄	C₂³×D₄₃	1	4	yes	—
278	C₂×C₄	C₂³×D₄₃	1	4	yes	—
279	C₂×C₄	C₂³×D₄₃	1	2	yes	—
280	C₂²	—	2	4	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	86	1	no	—
284	C₂²	—	86	4	no	—
285	C₂²	—	86	4	no	—
286	C₂²	—	86	2	no	—
287	C₂²	—	86	4	no	—
288	C₂²	—	86	4	no	—
289	C₂²	—	86	8	no	—
290	C₂²	—	344	16	no	—
291	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
292	C₄	—	2	4	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	4	no	—
295	C₄	—	43	2	no	—
296	C₄	—	86	4	no	—
297	C₄	—	86	2	no	—
298	C₄	#13063371	1	4	yes	—
299	C₂	—	4	4	no	—
300	C₂	—	172	4	no	—
301	C₂	—	172	4	no	—
302	C₂	—	172	1	no	—
303	C₂	H(5,𝔽₂)×D₄₃	1	1	yes	characteristic
304	C₂	C₂×#13063371	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#555	C₂
hypercenter	#555	D₄₃
hypocenter	C₄₃	#3549
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #555

Complements

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

N	H ≅ G/N	type
1. #14583641	C₂	semidirect
2. #14583615	C₂	semidirect
3. #14583612	C₂	semidirect
4. D₄×D₁₇₂	C₂	semidirect
5. #14583533	C₂	semidirect
6. #14583468	C₂	semidirect
7. #14583467	C₂	semidirect
8. #14583465	C₂	semidirect
9. #14583450	C₂	semidirect
10. #14583446	C₂	semidirect
11. C₄₃×#555	C₂	semidirect
12. C₂×#10621902	C₂²	semidirect
13. C₂×#10621898	C₂²	semidirect
14. C₂²×D₁₇₂	C₂²	semidirect
15. C₂×D₄×D₄₃	C₂²	semidirect
16. C₂×D₄×D₄₃	C₂²	semidirect
17. #13063324	C₂²	semidirect
19. #13063322	C₂²	semidirect
21. #13063321	C₂²	semidirect
22. #13063314	C₂²	semidirect
23. #13063314	C₂²	semidirect
24. #13063313	C₂²	semidirect
25. #13063311	C₂²	semidirect
26. #13063311	C₂²	semidirect
27. #13063300	C₂²	semidirect
28. #13063300	C₂²	semidirect
29. #13063300	C₂²	semidirect
30. #13063300	C₂²	semidirect
31. #13063299	C₂²	semidirect
32. #13063298	C₂²	semidirect
33. #13063298	C₂²	semidirect
34. #13063296	C₂²	semidirect
35. #13063295	C₂²	semidirect
36. #13063295	C₂²	semidirect
37. #13063295	C₂²	semidirect
38. #13063293	C₂²	semidirect
39. C₄×D₁₇₂	C₂²	semidirect
40. #13063287	C₂²	semidirect
41. C₄×C₂²⋊D₄₃	C₂²	semidirect
42. #13063283	C₂²	semidirect
43. #13063282	C₂²	semidirect
44. C₄²×D₄₃	C₂²	semidirect
45. C₄⋊Q₈×C₄₃	C₂²	semidirect
46. #124×C₄₃	C₂²	semidirect
47. #121×C₄₃	C₂²	semidirect
48. C₄×D₄×C₄₃	C₂²	semidirect
49. #117×C₄₃	C₂²	semidirect
156. #555	D₄₃	semidirect
164. C₄₃	#3549	semidirect
171. C₄⋊Q₈	C₂×D₄₃	semidirect
181. #124	C₂×D₄₃	semidirect
190. #121	C₂×D₄₃	semidirect
198. C₄×D₄	C₂×D₄₃	semidirect
201. #117	C₂×D₄₃	semidirect