Subgroups — #19148466

Order 5504.

1232 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583632	C₂	1	4	yes	maximal, complemented
2	#14583570	C₂	1	4	yes	maximal, complemented
3	#14583539	C₂	1	8	yes	maximal, complemented
4	#14583500	C₂	1	4	yes	maximal, complemented
5	#14583468	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583466	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583452	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583446	C₂	1	4	yes	maximal, complemented
9	C₂×#13063321	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583373	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583367	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄₃×#518	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#10621898	C₂²	1	1	yes	characteristic
14	C₂×#10621898	C₂²	1	8	yes	complemented
15	C₂²×D₁₇₂	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊D₄₃	C₂²	1	4	yes	—
17	#13063324	C₂²	1	2	yes	complemented
18	#13063322	C₂²	1	2	yes	complemented
19	#13063321	C₂²	1	4	yes	—
20	#13063321	C₂²	1	2	yes	—
21	#13063321	C₂²	1	2	yes	—
22	#13063321	C₂²	1	2	yes	complemented
23	#13063316	C₂²	1	2	yes	complemented
24	#13063313	C₂²	1	2	yes	complemented
25	#13063312	C₂²	1	2	yes	complemented
26	#13063311	C₂²	1	8	yes	complemented
27	#13063308	C₂²	1	8	yes	complemented
28	#13063307	C₂²	1	8	yes	complemented
29	#13063302	C₂²	1	4	yes	—
30	#13063300	C₂²	1	4	yes	complemented
31	#13063299	C₂²	1	8	yes	complemented
32	#13063298	C₂²	1	8	yes	complemented
33	#13063298	C₂²	1	8	yes	complemented
34	#13063296	C₂²	1	4	yes	complemented
35	#13063295	C₂²	1	4	yes	—
36	#13063294	C₂²	1	4	yes	—
37	C₄×Dic₈₆	C₂²	1	4	yes	complemented
38	#13063290	C₂²	1	2	yes	complemented
39	C₄×D₁₇₂	C₂²	1	4	yes	complemented
40	#13063287	C₂²	1	2	yes	complemented
41	#13063287	C₂²	1	4	yes	complemented
42	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
43	(C₂²⋊D₄₃)⋊C₄	C₂²	1	4	yes	—
44	#13063284	C₂²	1	1	yes	characteristic
45	#13063282	C₂²	1	2	yes	complemented
46	C₂²⋊C₄×D₄₃	C₂²	1	4	yes	—
47	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
48	C₄²×D₄₃	C₂²	1	2	yes	complemented
49	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
50	C₄×D₄×C₄₃	C₂²	1	4	yes	complemented
51	C₄×D₄×C₄₃	C₂²	1	4	yes	complemented
52	#117×C₄₃	C₂²	1	1	yes	characteristic
53	#117×C₄₃	C₂²	1	1	yes	characteristic
54	C₂×C₂²⋊C₄×C₄₃	C₂²	1	4	yes	—
55	C₂⁴×C₄₃	C₂³	1	2	yes	—
56	C₂³×D₄₃	C₂³	1	2	yes	—
57	C₂×D₄×C₄₃	—	2	4	no	—
58	C₂×D₄×C₄₃	C₂³	1	2	yes	—
59	C₂×D₄×C₄₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂³	1	8	yes	—
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	4	8	no	—
63	C₂×C₂²⋊D₄₃	C₂³	1	2	yes	—
64	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
65	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
66	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
67	C₂²×Dic₄₃	C₂³	1	2	yes	—
68	#10621898	—	2	4	no	—
69	#10621898	—	4	16	no	—
70	C₂×D₁₇₂	—	2	4	no	—
71	C₂×D₁₇₂	C₂³	1	2	yes	—
72	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
73	C₂×D₁₇₂	C₂³	1	2	yes	—
74	C₂×D₁₇₂	C₂³	1	8	yes	—
75	C₂×C₄×D₄₃	C₂³	1	2	yes	—
76	C₂×C₄×D₄₃	C₂³	1	4	yes	—
77	C₂×C₄×D₄₃	C₂³	1	8	yes	—
78	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
79	C₂×Dic₈₆	C₂³	1	4	yes	—
80	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
81	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
82	C₂²⋊C₄×C₄₃	C₂³	1	2	yes	—
83	C₂²⋊C₄×C₄₃	C₂³	1	2	yes	—
84	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
85	C₄²×C₄₃	C₂³	1	2	yes	—
86	C₄²×C₄₃	C₂³	1	2	yes	—
87	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
88	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
89	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
90	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
91	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
92	C₄⋊Dic₄₃	C₂³	1	2	yes	—
93	C₄⋊Dic₄₃	C₂³	1	4	yes	—
94	Dic₄₃⋊C₄	C₂³	1	4	yes	—
95	Dic₄₃⋊C₄	C₂³	1	4	yes	—
96	Dic₄₃⋊C₄	C₂³	1	4	yes	—
97	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
98	C₄×Dic₄₃	C₂³	1	4	yes	—
99	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₄₃	C₂³	1	2	yes	—
101	C₄×Dic₄₃	C₂³	1	2	yes	—
102	C₂³×C₄₃	—	2	4	no	—
103	C₂³×C₄₃	—	4	4	no	—
104	C₂³×C₄₃	C₂⁴	1	4	yes	—
105	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
106	C₂²×D₄₃	—	2	4	no	—
107	C₂²×D₄₃	—	4	4	no	—
108	C₂²×D₄₃	C₂⁴	1	4	yes	—
109	C₂²×D₄₃	C₂⁴	1	2	yes	—
110	D₄×C₄₃	—	2	4	no	—
111	D₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	4	8	no	—
113	C₂×C₄×C₄₃	#41	1	4	yes	complemented
114	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
116	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₄₃	—	2	4	no	—
119	C₂²⋊D₄₃	—	2	16	no	—
120	C₂²⋊D₄₃	—	4	8	no	—
121	C₂²⋊D₄₃	—	4	8	no	—
122	C₂×Dic₄₃	—	2	4	no	—
123	C₂×Dic₄₃	C₂⁴	1	2	yes	—
124	C₂×Dic₄₃	C₂⁴	1	2	yes	—
125	C₂×Dic₄₃	C₂⁴	1	4	yes	—
126	C₂×Dic₄₃	C₂⁴	1	2	yes	—
127	D₁₇₂	—	2	4	no	—
128	D₁₇₂	—	2	4	no	—
129	D₁₇₂	—	2	2	no	—
130	D₁₇₂	—	4	8	no	—
131	C₄×D₄₃	—	2	4	no	—
132	C₄×D₄₃	—	2	8	no	—
133	C₄×D₄₃	—	4	8	no	—
134	Dic₈₆	—	2	2	no	—
135	Dic₈₆	—	2	8	no	—
136	C₂²×C₄₃	—	2	2	no	—
137	C₂²×C₄₃	—	2	4	no	—
138	C₂²×C₄₃	—	2	8	no	—
139	C₂²×C₄₃	—	8	4	no	—
140	C₂²×C₄₃	C₂×#41	1	2	yes	—
141	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
142	C₂×D₄₃	—	2	4	no	—
143	C₂×D₄₃	—	2	2	no	—
144	C₂×D₄₃	—	2	8	no	—
145	C₂×D₄₃	—	2	4	no	—
146	C₂×D₄₃	—	8	4	no	—
147	C₄×C₄₃	—	2	4	no	—
148	C₄×C₄₃	—	2	4	no	—
149	C₄×C₄₃	C₂×#41	1	2	yes	—
150	C₄×C₄₃	C₂×#41	1	2	yes	—
151	Dic₄₃	—	2	2	no	—
152	Dic₄₃	—	2	4	no	—
153	Dic₄₃	—	2	2	no	—
154	Dic₄₃	H(5,𝔽₂)	1	4	yes	—
155	#3508	—	43	1	no	maximal, Sylow
156	C₂×C₄₃	—	2	2	no	—
157	C₂×C₄₃	—	4	4	no	—
158	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
159	C₂×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
160	D₄₃	—	4	2	no	—
161	D₄₃	—	4	4	no	—
162	#559	—	43	4	no	—
163	#550	—	43	4	no	—
164	#546	—	43	8	no	—
165	#540	—	43	4	no	—
166	#535	—	43	1	no	—
167	#535	—	43	1	no	—
168	#533	—	43	1	no	—
169	#532	—	43	4	no	—
170	C₂×#127	—	43	1	no	—
171	#518	—	43	1	no	—
172	#518	D₄₃	1	1	yes	characteristic, complemented
173	C₄×#41	—	43	1	no	—
174	C₄₃	#3508	1	1	yes	characteristic, complemented, Sylow
175	C₂×#41	—	43	1	no	—
176	C₂×#41	—	43	8	no	—
177	C₂²×D₄	—	43	4	no	—
178	C₂²×D₄	—	43	1	no	—
179	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
180	C₄⋊Q₈	—	43	2	no	—
181	#127	—	43	2	no	—
182	#127	—	43	2	no	—
183	#127	—	43	4	no	—
184	#127	—	43	2	no	—
185	#127	—	43	2	no	—
186	#125	—	43	2	no	—
187	#124	—	43	2	no	—
188	#124	—	43	8	no	—
189	#124	—	43	2	no	—
190	#123	—	43	8	no	—
191	#123	—	43	8	no	—
192	C₂²⋊Q₈	—	43	4	no	—
193	#121	—	43	4	no	—
194	#121	—	43	8	no	—
195	#121	—	43	4	no	—
196	#121	—	43	8	no	—
197	#121	—	43	8	no	—
198	#120	—	43	4	no	—
199	#120	—	43	4	no	—
200	C₄×Q₈	—	43	2	no	—
201	C₄×Q₈	—	43	4	no	—
202	C₄×D₄	—	43	8	no	—
203	C₄×D₄	—	43	2	no	—
204	C₄×D₄	—	43	4	no	—
205	C₄×D₄	—	43	4	no	—
206	C₄×D₄	—	43	4	no	—
207	C₄×D₄	C₂×D₄₃	1	4	yes	complemented
208	C₄×D₄	C₂×D₄₃	1	4	yes	complemented
209	#117	—	43	1	no	—
210	#117	—	43	2	no	—
211	#117	C₂×D₄₃	1	1	yes	characteristic
212	#117	C₂×D₄₃	1	1	yes	characteristic
213	C₂×C₂²⋊C₄	—	43	4	no	—
214	C₂×C₂²⋊C₄	C₂×D₄₃	1	4	yes	—
215	C₂×C₄²	—	43	1	no	—
216	C₂×C₄²	—	43	2	no	—
217	C₂⁴	—	43	2	no	—
218	C₂⁴	C₂²×D₄₃	1	2	yes	—
219	#41	—	86	4	no	—
220	#41	—	172	16	no	—
221	C₂×Q₈	—	43	1	no	—
222	C₂×Q₈	—	43	4	no	—
223	C₂×D₄	—	2	4	no	—
224	C₂×D₄	—	43	2	no	—
225	C₂×D₄	—	43	8	no	—
226	C₂×D₄	—	43	8	no	—
227	C₂×D₄	—	43	8	no	—
228	C₂×D₄	—	43	2	no	—
229	C₂×D₄	—	43	2	no	—
230	C₂×D₄	—	43	1	no	—
231	C₂×D₄	—	43	8	no	—
232	C₂×D₄	—	86	4	no	—
233	C₂×D₄	—	172	8	no	—
234	C₂×D₄	C₂²×D₄₃	1	2	yes	—
235	C₂×D₄	C₂²×D₄₃	1	2	yes	—
236	C₂²×C₄	—	43	2	no	—
237	C₂²×C₄	—	43	2	no	—
238	C₂²×C₄	—	43	4	no	—
239	C₂²×C₄	—	43	8	no	—
240	C₂²×C₄	C₂²×D₄₃	1	8	yes	—
241	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
242	C₄⋊C₄	—	43	2	no	—
243	C₄⋊C₄	—	43	4	no	—
244	C₄⋊C₄	—	43	4	no	—
245	C₄⋊C₄	—	43	4	no	—
246	C₄⋊C₄	—	43	4	no	—
247	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
248	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
249	C₂²⋊C₄	—	43	2	no	—
250	C₂²⋊C₄	—	43	8	no	—
251	C₂²⋊C₄	—	43	8	no	—
252	C₂²⋊C₄	—	43	4	no	—
253	C₂²⋊C₄	—	43	8	no	—
254	C₂²⋊C₄	C₂²×D₄₃	1	2	yes	—
255	C₂²⋊C₄	C₂²×D₄₃	1	2	yes	—
256	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
257	C₄²	—	43	1	no	—
258	C₄²	—	43	1	no	—
259	C₄²	—	43	4	no	—
260	C₄²	—	43	2	no	—
261	C₄²	—	43	2	no	—
262	C₄²	C₂²×D₄₃	1	2	yes	—
263	C₄²	C₂²×D₄₃	1	2	yes	—
264	C₂³	—	2	4	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	43	4	no	—
267	C₂³	—	43	2	no	—
268	C₂³	—	86	4	no	—
269	C₂³	—	172	4	no	—
270	C₂³	C₂³×D₄₃	1	4	yes	—
271	C₂³	C₂³×D₄₃	1	1	yes	characteristic
272	Q₈	—	86	8	no	—
273	Q₈	—	86	2	no	—
274	D₄	—	2	4	no	—
275	D₄	—	2	4	no	—
276	D₄	—	86	16	no	—
277	D₄	—	86	4	no	—
278	D₄	—	86	4	no	—
279	D₄	—	86	4	no	—
280	D₄	—	86	2	no	—
281	D₄	—	172	8	no	—
282	D₄	—	172	8	no	—
283	D₄	—	172	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	43	2	no	—
286	C₂×C₄	—	43	2	no	—
287	C₂×C₄	—	43	2	no	—
288	C₂×C₄	—	43	4	no	—
289	C₂×C₄	—	86	8	no	—
290	C₂×C₄	—	86	4	no	—
291	C₂×C₄	—	86	4	no	—
292	C₂×C₄	—	172	8	no	—
293	C₂×C₄	#10621898	1	4	yes	complemented
294	C₂×C₄	C₂³×D₄₃	1	4	yes	—
295	C₂×C₄	C₂³×D₄₃	1	4	yes	—
296	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	2	8	no	—
301	C₂²	—	8	4	no	—
302	C₂²	—	86	2	no	—
303	C₂²	—	86	4	no	—
304	C₂²	—	86	4	no	—
305	C₂²	—	86	8	no	—
306	C₂²	—	344	4	no	—
307	C₂²	C₂×#10621898	1	2	yes	—
308	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
309	C₄	—	2	4	no	—
310	C₄	—	2	4	no	—
311	C₄	—	43	4	no	—
312	C₄	—	86	2	no	—
313	C₄	—	86	2	no	—
314	C₄	—	86	4	no	—
315	C₄	C₂×#10621898	1	2	yes	—
316	C₄	C₂×#10621898	1	2	yes	—
317	C₂	—	2	2	no	—
318	C₂	—	4	4	no	—
319	C₂	—	172	4	no	—
320	C₂	—	172	2	no	—
321	C₂	C₂²×#10621898	1	1	yes	characteristic
322	C₂	H(5,𝔽₂)×D₄₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#518	C₂
hypercenter	#518	D₄₃
hypocenter	C₄₃	#3508
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #518

Complements

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

N	H ≅ G/N	type
1. #14583632	C₂	semidirect
2. #14583570	C₂	semidirect
3. #14583539	C₂	semidirect
4. #14583500	C₂	semidirect
5. #14583468	C₂	semidirect
6. #14583466	C₂	semidirect
7. #14583452	C₂	semidirect
8. #14583446	C₂	semidirect
9. C₂×#13063321	C₂	semidirect
10. #14583373	C₂	semidirect
11. #14583367	C₂	semidirect
12. C₄₃×#518	C₂	semidirect
14. C₂×#10621898	C₂²	semidirect
17. #13063324	C₂²	semidirect
18. #13063322	C₂²	semidirect
22. #13063321	C₂²	semidirect
23. #13063316	C₂²	semidirect
24. #13063313	C₂²	semidirect
25. #13063312	C₂²	semidirect
26. #13063311	C₂²	semidirect
27. #13063308	C₂²	semidirect
28. #13063307	C₂²	semidirect
30. #13063300	C₂²	semidirect
31. #13063299	C₂²	semidirect
32. #13063298	C₂²	semidirect
33. #13063298	C₂²	semidirect
34. #13063296	C₂²	semidirect
37. C₄×Dic₈₆	C₂²	semidirect
38. #13063290	C₂²	semidirect
39. C₄×D₁₇₂	C₂²	semidirect
40. #13063287	C₂²	semidirect
41. #13063287	C₂²	semidirect
42. C₄×C₂²⋊D₄₃	C₂²	semidirect
45. #13063282	C₂²	semidirect
48. C₄²×D₄₃	C₂²	semidirect
50. C₄×D₄×C₄₃	C₂²	semidirect
51. C₄×D₄×C₄₃	C₂²	semidirect
113. C₂×C₄×C₄₃	#41	semidirect
172. #518	D₄₃	semidirect
174. C₄₃	#3508	semidirect
207. C₄×D₄	C₂×D₄₃	semidirect
208. C₄×D₄	C₂×D₄₃	semidirect
293. C₂×C₄	#10621898	semidirect