Subgroups — #19148489

Order 5504.

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	#14583631	C₂	1	4	yes	maximal, complemented
2	#14583618	C₂	1	4	yes	maximal, complemented
3	#14583575	C₂	1	8	yes	maximal, complemented
4	#14583518	C₂	1	4	yes	maximal, complemented
5	#14583473	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583466	C₂	1	1	yes	maximal, characteristic
7	#14583456	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583428	C₂	1	4	yes	maximal, complemented
9	#14583377	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583374	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#10621898	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#124×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#10621898	C₂²	1	1	yes	characteristic
14	#13063325	C₂²	1	2	yes	complemented
15	#13063323	C₂²	1	2	yes	complemented
16	#13063321	C₂²	1	2	yes	—
17	#13063318	C₂²	1	8	yes	complemented
18	#13063318	C₂²	1	8	yes	complemented
19	#13063317	C₂²	1	2	yes	complemented
20	#13063314	C₂²	1	2	yes	complemented
21	#13063312	C₂²	1	2	yes	—
22	#13063311	C₂²	1	8	yes	complemented
23	#13063309	C₂²	1	8	yes	complemented
24	#13063308	C₂²	1	4	yes	—
25	#13063307	C₂²	1	8	yes	complemented
26	#13063304	C₂²	1	4	yes	—
27	#13063302	C₂²	1	8	yes	complemented
28	#13063301	C₂²	1	4	yes	complemented
29	#13063296	C₂²	1	4	yes	—
30	#13063294	C₂²	1	4	yes	—
31	Q₈×Dic₄₃	C₂²	1	4	yes	complemented
32	Q₈×Dic₄₃	C₂²	1	4	yes	complemented
33	C₄×Dic₈₆	C₂²	1	2	yes	complemented
34	C₄×Dic₈₆	C₂²	1	4	yes	complemented
35	C₄×D₁₇₂	C₂²	1	2	yes	complemented
36	D₄×Dic₄₃	C₂²	1	4	yes	—
37	D₄×Dic₄₃	C₂²	1	4	yes	—
38	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	—
39	(C₂²⋊D₄₃)⋊C₄	C₂²	1	8	yes	complemented
40	#13063284	C₂²	1	4	yes	—
41	#13063284	C₂²	1	1	yes	characteristic
42	#13063284	C₂²	1	1	yes	characteristic
43	#13063284	C₂²	1	1	yes	characteristic
44	#13063282	C₂²	1	2	yes	complemented
45	#13063281	C₂²	1	8	yes	complemented
46	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	—
47	C₄²×D₄₃	C₂²	1	2	yes	complemented
48	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic
49	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
50	#124×C₄₃	C₂²	1	4	yes	—
51	#124×C₄₃	C₂²	1	2	yes	—
52	#124×C₄₃	C₂²	1	2	yes	—
53	C₂×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
54	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
55	C₂⁴×C₄₃	C₂³	1	2	yes	—
56	C₂×Q₈×C₄₃	—	2	4	no	—
57	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
58	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
59	C₂×D₄×C₄₃	—	2	4	no	—
60	C₂×D₄×C₄₃	C₂³	1	2	yes	—
61	C₂×D₄×C₄₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₄₃	C₂³	1	2	yes	—
66	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
67	C₂²×Dic₄₃	C₂³	1	8	yes	—
68	#10621898	—	2	4	no	—
69	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₄₃	C₂³	1	2	yes	—
71	C₂×C₄×D₄₃	C₂³	1	4	yes	—
72	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
73	C₂×Dic₈₆	C₂³	1	4	yes	—
74	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
75	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
76	C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	C₄²×C₄₃	C₂³	1	1	yes	characteristic
78	C₄²×C₄₃	C₂³	1	2	yes	—
79	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
80	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
81	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
82	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
83	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
84	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
85	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
86	C₄⋊Dic₄₃	C₂³	1	2	yes	—
87	C₄⋊Dic₄₃	C₂³	1	2	yes	—
88	C₄⋊Dic₄₃	C₂³	1	2	yes	—
89	C₄⋊Dic₄₃	C₂³	1	8	yes	—
90	C₄⋊Dic₄₃	C₂³	1	4	yes	—
91	Dic₄₃⋊C₄	C₂³	1	4	yes	—
92	Dic₄₃⋊C₄	C₂³	1	8	yes	—
93	Dic₄₃⋊C₄	C₂³	1	4	yes	—
94	Dic₄₃⋊C₄	C₂³	1	4	yes	—
95	C₄×Dic₄₃	C₂³	1	4	yes	—
96	C₄×Dic₄₃	C₂³	1	2	yes	—
97	C₄×Dic₄₃	C₂³	1	8	yes	—
98	C₄×Dic₄₃	C₂³	1	2	yes	—
99	C₄×Dic₄₃	C₂³	1	2	yes	—
100	C₂³×C₄₃	—	2	4	no	—
101	C₂³×C₄₃	—	4	4	no	—
102	C₂³×C₄₃	C₂⁴	1	4	yes	—
103	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₂²×D₄₃	C₂⁴	1	2	yes	—
105	Q₈×C₄₃	—	2	4	no	—
106	Q₈×C₄₃	—	2	4	no	—
107	D₄×C₄₃	—	2	4	no	—
108	D₄×C₄₃	—	2	4	no	—
109	C₂×C₄×C₄₃	—	2	4	no	—
110	C₂×C₄×C₄₃	—	4	2	no	—
111	C₂×C₄×C₄₃	#41	1	4	yes	—
112	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
113	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
114	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
115	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₄₃	—	2	4	no	—
118	C₂²⋊D₄₃	—	4	8	no	—
119	C₂×Dic₄₃	—	4	8	no	—
120	C₂×Dic₄₃	C₂⁴	1	4	yes	—
121	C₂×Dic₄₃	C₂⁴	1	4	yes	—
122	C₂×Dic₄₃	C₂⁴	1	2	yes	—
123	C₂×Dic₄₃	C₂⁴	1	4	yes	—
124	D₁₇₂	—	2	2	no	—
125	C₄×D₄₃	—	2	4	no	—
126	C₄×D₄₃	—	4	4	no	—
127	Dic₈₆	—	2	2	no	—
128	Dic₈₆	—	4	4	no	—
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	2	4	no	—
131	C₂²×C₄₃	—	2	8	no	—
132	C₂²×C₄₃	—	8	4	no	—
133	C₂²×C₄₃	C₂×#41	1	2	yes	—
134	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
135	C₂×D₄₃	—	2	2	no	—
136	C₂×D₄₃	—	2	4	no	—
137	C₄×C₄₃	—	2	4	no	—
138	C₄×C₄₃	—	2	2	no	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	C₂×#41	1	2	yes	—
141	C₄×C₄₃	C₂×#41	1	2	yes	—
142	Dic₄₃	—	2	2	no	—
143	Dic₄₃	—	2	4	no	—
144	Dic₄₃	—	2	4	no	—
145	Dic₄₃	—	2	4	no	—
146	#3510	—	43	1	no	maximal, Sylow
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	—	4	4	no	—
149	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
150	C₂×C₄₃	#585	1	2	yes	—
151	D₄₃	—	4	2	no	—
152	#559	—	43	4	no	—
153	#556	—	43	4	no	—
154	#551	—	43	8	no	—
155	#542	—	43	4	no	—
156	#536	—	43	1	no	—
157	#535	—	43	1	no	—
158	#533	—	43	1	no	—
159	#529	—	43	4	no	—
160	C₂×#124	D₄₃	1	1	yes	characteristic, complemented
161	#519	—	43	1	no	—
162	#518	—	43	1	no	—
163	C₄×#41	—	43	1	no	—
164	C₄₃	#3510	1	1	yes	characteristic, complemented, Sylow
165	C₂×#41	—	43	1	no	—
166	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic
167	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
168	C₄⋊Q₈	—	43	2	no	—
169	C₄⋊Q₈	—	43	2	no	—
170	#127	—	43	2	no	—
171	#126	—	43	8	no	—
172	#126	—	43	8	no	—
173	#125	—	43	2	no	—
174	#124	—	43	2	no	—
175	#124	—	43	2	no	—
176	#124	—	43	8	no	—
177	#124	C₂×D₄₃	1	2	yes	—
178	#124	C₂×D₄₃	1	2	yes	—
179	#124	C₂×D₄₃	1	4	yes	—
180	#123	—	43	4	no	—
181	#123	—	43	8	no	—
182	#123	—	43	8	no	—
183	C₂²⋊Q₈	—	43	4	no	—
184	C₂²⋊Q₈	—	43	4	no	—
185	C₂²⋊Q₈	—	43	8	no	—
186	#121	—	43	4	no	—
187	#120	—	43	4	no	—
188	C₄×Q₈	—	43	4	no	—
189	C₄×Q₈	—	43	4	no	—
190	C₄×Q₈	—	43	2	no	—
191	C₄×Q₈	—	43	4	no	—
192	C₄×D₄	—	43	4	no	—
193	C₄×D₄	—	43	4	no	—
194	C₄×D₄	—	43	4	no	—
195	C₄×D₄	—	43	8	no	—
196	C₄×D₄	—	43	2	no	—
197	#117	—	43	1	no	—
198	#117	—	43	1	no	—
199	#117	—	43	1	no	—
200	#117	—	43	4	no	—
201	#117	—	43	8	no	—
202	#117	—	43	2	no	—
203	C₂×C₂²⋊C₄	—	43	4	no	—
204	C₂×C₂²⋊C₄	C₂×D₄₃	1	4	yes	—
205	C₂×C₄²	—	43	2	no	—
206	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
207	C₂⁴	C₂²×D₄₃	1	2	yes	—
208	#41	—	86	4	no	—
209	C₂×Q₈	—	2	4	no	—
210	C₂×Q₈	—	43	1	no	—
211	C₂×Q₈	—	43	4	no	—
212	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
213	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	43	2	no	—
216	C₂×D₄	—	43	8	no	—
217	C₂×D₄	—	43	1	no	—
218	C₂×D₄	C₂²×D₄₃	1	2	yes	—
219	C₂×D₄	C₂²×D₄₃	1	2	yes	—
220	C₂²×C₄	—	43	8	no	—
221	C₂²×C₄	—	43	2	no	—
222	C₂²×C₄	—	43	4	no	—
223	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
224	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
225	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
226	C₄⋊C₄	—	43	4	no	—
227	C₄⋊C₄	—	43	2	no	—
228	C₄⋊C₄	—	43	2	no	—
229	C₄⋊C₄	—	43	4	no	—
230	C₄⋊C₄	—	43	2	no	—
231	C₄⋊C₄	—	43	8	no	—
232	C₄⋊C₄	—	43	8	no	—
233	C₄⋊C₄	—	43	4	no	—
234	C₄⋊C₄	—	43	4	no	—
235	C₂²⋊C₄	—	43	2	no	—
236	C₂²⋊C₄	—	43	2	no	—
237	C₂²⋊C₄	—	43	8	no	—
238	C₂²⋊C₄	—	43	2	no	—
239	C₂²⋊C₄	—	43	8	no	—
240	C₂²⋊C₄	—	43	4	no	—
241	C₂²⋊C₄	—	43	8	no	—
242	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
243	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
244	C₄²	—	43	2	no	—
245	C₄²	—	43	4	no	—
246	C₄²	—	43	8	no	—
247	C₄²	—	43	2	no	—
248	C₄²	—	43	2	no	—
249	C₄²	C₂²×D₄₃	1	1	yes	characteristic
250	C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	C₄²	C₂²×D₄₃	1	2	yes	—
252	C₂³	—	2	4	no	—
253	C₂³	—	4	4	no	—
254	C₂³	—	43	2	no	—
255	C₂³	C₂³×D₄₃	1	4	yes	—
256	C₂³	C₂³×D₄₃	1	1	yes	characteristic
257	Q₈	—	2	4	no	—
258	Q₈	—	2	4	no	—
259	Q₈	—	86	2	no	—
260	Q₈	—	172	4	no	—
261	D₄	—	2	4	no	—
262	D₄	—	2	4	no	—
263	D₄	—	86	4	no	—
264	D₄	—	86	2	no	—
265	D₄	—	172	8	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	4	2	no	—
268	C₂×C₄	—	43	4	no	—
269	C₂×C₄	—	43	4	no	—
270	C₂×C₄	—	43	2	no	—
271	C₂×C₄	—	43	4	no	—
272	C₂×C₄	—	86	4	no	—
273	C₂×C₄	—	172	4	no	—
274	C₂×C₄	—	172	8	no	—
275	C₂×C₄	#10621900	1	4	yes	—
276	C₂×C₄	C₂³×D₄₃	1	2	yes	—
277	C₂×C₄	C₂³×D₄₃	1	4	yes	—
278	C₂×C₄	C₂³×D₄₃	1	2	yes	—
279	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
280	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
281	C₂²	—	2	2	no	—
282	C₂²	—	2	4	no	—
283	C₂²	—	2	8	no	—
284	C₂²	—	8	4	no	—
285	C₂²	—	86	2	no	—
286	C₂²	—	86	4	no	—
287	C₂²	C₂×#10621900	1	2	yes	—
288	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
289	C₄	—	2	4	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	86	2	no	—
293	C₄	—	86	4	no	—
294	C₄	—	86	4	no	—
295	C₄	—	86	4	no	—
296	C₄	C₂×#10621900	1	2	yes	—
297	C₄	C₂×#10621900	1	2	yes	—
298	C₂	—	2	2	no	—
299	C₂	—	4	4	no	—
300	C₂	—	172	2	no	—
301	C₂	C₂²×#10621900	1	1	yes	characteristic
302	C₂	#14583735	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₂×#124×C₄₃	C₂
hypercenter	C₂×#124	D₄₃
hypocenter	C₄₃	#3510
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#124

Complements

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

N	H ≅ G/N	type
1. #14583631	C₂	semidirect
2. #14583618	C₂	semidirect
3. #14583575	C₂	semidirect
4. #14583518	C₂	semidirect
5. #14583473	C₂	semidirect
7. #14583456	C₂	semidirect
8. #14583428	C₂	semidirect
9. #14583377	C₂	semidirect
10. #14583374	C₂	semidirect
11. C₄×#10621898	C₂	semidirect
12. C₂×#124×C₄₃	C₂	semidirect
14. #13063325	C₂²	semidirect
15. #13063323	C₂²	semidirect
17. #13063318	C₂²	semidirect
18. #13063318	C₂²	semidirect
19. #13063317	C₂²	semidirect
20. #13063314	C₂²	semidirect
22. #13063311	C₂²	semidirect
23. #13063309	C₂²	semidirect
25. #13063307	C₂²	semidirect
27. #13063302	C₂²	semidirect
28. #13063301	C₂²	semidirect
31. Q₈×Dic₄₃	C₂²	semidirect
32. Q₈×Dic₄₃	C₂²	semidirect
33. C₄×Dic₈₆	C₂²	semidirect
34. C₄×Dic₈₆	C₂²	semidirect
35. C₄×D₁₇₂	C₂²	semidirect
39. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
44. #13063282	C₂²	semidirect
45. #13063281	C₂²	semidirect
47. C₄²×D₄₃	C₂²	semidirect
160. C₂×#124	D₄₃	semidirect
164. C₄₃	#3510	semidirect