Subgroups — #19148017

Order 5504.

1140 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583530	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583514	C₂	1	4	yes	maximal, complemented
3	#14583500	C₂	1	4	yes	maximal, complemented
4	#14583487	C₂	1	4	yes	maximal, complemented
5	#14583480	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583466	C₂	1	2	yes	maximal, complemented
7	#14583446	C₂	1	4	yes	maximal, complemented
8	#14583431	C₂	1	4	yes	maximal, complemented
9	#14583410	C₂	1	4	yes	maximal, complemented
10	C₄×#10621898	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	2	yes	—
13	C₂×#10621898	C₂²	1	4	yes	complemented
14	#13063324	C₂²	1	4	yes	complemented
15	#13063321	C₂²	1	4	yes	—
16	#13063318	C₂²	1	8	yes	complemented
17	#13063316	C₂²	1	4	yes	complemented
18	#13063312	C₂²	1	4	yes	—
19	#13063311	C₂²	1	8	yes	complemented
20	#13063308	C₂²	1	8	yes	complemented
21	#13063307	C₂²	1	8	yes	complemented
22	#13063305	C₂²	1	4	yes	complemented
23	#13063304	C₂²	1	4	yes	—
24	#13063304	C₂²	1	4	yes	complemented
25	#13063300	C₂²	1	4	yes	complemented
26	#13063299	C₂²	1	4	yes	—
27	#13063299	C₂²	1	4	yes	complemented
28	#13063298	C₂²	1	8	yes	complemented
29	#13063296	C₂²	1	4	yes	—
30	#13063296	C₂²	1	4	yes	—
31	#13063294	C₂²	1	8	yes	—
32	C₄×Dic₈₆	C₂²	1	2	yes	complemented
33	C₄×Dic₈₆	C₂²	1	2	yes	complemented
34	C₄×D₁₇₂	C₂²	1	2	yes	complemented
35	C₄×D₁₇₂	C₂²	1	2	yes	complemented
36	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	—
37	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	—
38	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
39	#13063284	C₂²	1	2	yes	—
40	#13063284	C₂²	1	4	yes	complemented
41	#13063282	C₂²	1	4	yes	complemented
42	C₄²×D₄₃	C₂²	1	4	yes	complemented
43	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
44	C₂³×C₄×C₄₃	C₂²	1	2	yes	—
45	C₄×D₄×C₄₃	C₂²	1	4	yes	—
46	C₄×D₄×C₄₃	C₂²	1	4	yes	—
47	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
49	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic
50	C₂⁴×C₄₃	C₂³	1	2	yes	—
51	C₂×D₄×C₄₃	—	2	4	no	—
52	C₂×D₄×C₄₃	C₂³	1	4	yes	—
53	C₂²×C₄×C₄₃	—	4	4	no	—
54	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
56	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
61	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
62	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
63	#10621898	—	2	8	no	—
64	#10621898	—	4	8	no	—
65	C₂×D₁₇₂	C₂³	1	2	yes	—
66	C₂×D₁₇₂	C₂³	1	4	yes	—
67	C₂×C₄×D₄₃	C₂³	1	4	yes	—
68	C₂×C₄×D₄₃	C₂³	1	4	yes	—
69	C₂×C₄×D₄₃	C₂³	1	4	yes	—
70	C₂×Dic₈₆	C₂³	1	2	yes	—
71	C₂×Dic₈₆	C₂³	1	4	yes	—
72	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
73	C₂²⋊C₄×C₄₃	C₂³	1	8	yes	—
74	C₄²×C₄₃	C₂³	1	2	yes	—
75	C₄²×C₄₃	C₂³	1	2	yes	—
76	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
77	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
78	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
79	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
80	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
81	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
82	C₄⋊Dic₄₃	C₂³	1	2	yes	—
83	C₄⋊Dic₄₃	C₂³	1	2	yes	—
84	C₄⋊Dic₄₃	C₂³	1	4	yes	—
85	C₄⋊Dic₄₃	C₂³	1	4	yes	—
86	Dic₄₃⋊C₄	C₂³	1	4	yes	—
87	Dic₄₃⋊C₄	C₂³	1	4	yes	—
88	Dic₄₃⋊C₄	C₂³	1	8	yes	—
89	C₄×Dic₄₃	C₂³	1	4	yes	—
90	C₄×Dic₄₃	C₂³	1	4	yes	—
91	C₄×Dic₄₃	C₂³	1	4	yes	—
92	C₂³×C₄₃	—	2	4	no	—
93	C₂³×C₄₃	—	4	2	no	—
94	C₂³×C₄₃	—	4	2	no	—
95	C₂³×C₄₃	C₂⁴	1	4	yes	—
96	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂²×D₄₃	C₂⁴	1	4	yes	—
98	D₄×C₄₃	—	2	8	no	—
99	C₂×C₄×C₄₃	—	2	8	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	—	4	4	no	—
103	C₂×C₄×C₄₃	—	4	1	no	—
104	C₂×C₄×C₄₃	#41	1	4	yes	complemented
105	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
106	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
108	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
109	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
110	C₂²⋊D₄₃	—	2	8	no	—
111	C₂²⋊D₄₃	—	4	8	no	—
112	C₂²⋊D₄₃	—	4	8	no	—
113	C₂×Dic₄₃	C₂⁴	1	4	yes	—
114	C₂×Dic₄₃	C₂⁴	1	4	yes	—
115	C₂×Dic₄₃	C₂⁴	1	4	yes	—
116	D₁₇₂	—	2	4	no	—
117	D₁₇₂	—	4	4	no	—
118	C₄×D₄₃	—	2	8	no	—
119	C₄×D₄₃	—	2	8	no	—
120	C₄×D₄₃	—	4	4	no	—
121	Dic₈₆	—	2	4	no	—
122	Dic₈₆	—	4	4	no	—
123	C₂²×C₄₃	—	2	1	no	—
124	C₂²×C₄₃	—	2	1	no	—
125	C₂²×C₄₃	—	2	4	no	—
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	4	no	—
128	C₂²×C₄₃	—	4	8	no	—
129	C₂²×C₄₃	C₂×#41	1	2	yes	—
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	C₄×C₄₃	—	2	4	no	—
135	C₄×C₄₃	—	2	1	no	—
136	C₄×C₄₃	—	2	2	no	—
137	C₄×C₄₃	C₂×#41	1	4	yes	—
138	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
139	Dic₄₃	—	2	4	no	—
140	Dic₄₃	—	2	4	no	—
141	Dic₄₃	—	2	4	no	—
142	#3460	—	43	1	no	maximal, Sylow
143	C₂×C₄₃	—	2	2	no	—
144	C₂×C₄₃	—	4	4	no	—
145	C₂×C₄₃	C₂²×#41	1	1	yes	characteristic
146	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄₃	#585	1	1	yes	characteristic
148	D₄₃	—	4	4	no	—
149	#544	—	43	1	no	—
150	#542	—	43	4	no	—
151	#540	—	43	4	no	—
152	#538	—	43	4	no	—
153	#537	—	43	1	no	—
154	#535	—	43	2	no	—
155	#532	—	43	4	no	—
156	#529	—	43	4	no	—
157	#525	—	43	4	no	—
158	C₄×#41	—	43	2	no	—
159	C₂×C₄×D₄	D₄₃	1	1	yes	characteristic, complemented
160	C₄₃	#3460	1	1	yes	characteristic, complemented, Sylow
161	C₂×#41	—	43	2	no	—
162	C₂×#41	—	43	4	no	—
163	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
164	C₂³×C₄	C₂×D₄₃	1	2	yes	—
165	C₄⋊Q₈	—	43	4	no	—
166	#127	—	43	4	no	—
167	#126	—	43	8	no	—
168	#125	—	43	4	no	—
169	#124	—	43	4	no	—
170	#124	—	43	8	no	—
171	#123	—	43	8	no	—
172	#123	—	43	8	no	—
173	C₂²⋊Q₈	—	43	4	no	—
174	C₂²⋊Q₈	—	43	4	no	—
175	C₂²⋊Q₈	—	43	4	no	—
176	#121	—	43	4	no	—
177	#121	—	43	4	no	—
178	#121	—	43	4	no	—
179	#121	—	43	4	no	—
180	#121	—	43	4	no	—
181	#121	—	43	8	no	—
182	#120	—	43	8	no	—
183	C₄×Q₈	—	43	2	no	—
184	C₄×Q₈	—	43	2	no	—
185	C₄×D₄	—	43	4	no	—
186	C₄×D₄	—	43	4	no	—
187	C₄×D₄	—	43	8	no	—
188	C₄×D₄	—	43	2	no	—
189	C₄×D₄	—	43	2	no	—
190	C₄×D₄	C₂×D₄₃	1	4	yes	—
191	C₄×D₄	C₂×D₄₃	1	4	yes	—
192	#117	—	43	2	no	—
193	#117	—	43	4	no	—
194	#117	—	43	4	no	—
195	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
196	C₂×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
197	C₂×C₄²	—	43	4	no	—
198	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic
199	C₂⁴	C₂²×D₄₃	1	2	yes	—
200	#41	—	86	8	no	—
201	#41	—	172	8	no	—
202	C₂×Q₈	—	43	2	no	—
203	C₂×Q₈	—	43	4	no	—
204	C₂×D₄	—	2	4	no	—
205	C₂×D₄	—	43	4	no	—
206	C₂×D₄	—	43	8	no	—
207	C₂×D₄	—	43	8	no	—
208	C₂×D₄	—	43	2	no	—
209	C₂×D₄	—	43	4	no	—
210	C₂×D₄	C₂²×D₄₃	1	4	yes	—
211	C₂²×C₄	—	4	4	no	—
212	C₂²×C₄	—	43	4	no	—
213	C₂²×C₄	—	43	4	no	—
214	C₂²×C₄	—	43	4	no	—
215	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
217	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
218	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
219	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
220	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
221	C₄⋊C₄	—	43	2	no	—
222	C₄⋊C₄	—	43	2	no	—
223	C₄⋊C₄	—	43	4	no	—
224	C₄⋊C₄	—	43	4	no	—
225	C₄⋊C₄	—	43	4	no	—
226	C₄⋊C₄	—	43	4	no	—
227	C₄⋊C₄	—	43	8	no	—
228	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
229	C₂²⋊C₄	—	43	4	no	—
230	C₂²⋊C₄	—	43	8	no	—
231	C₂²⋊C₄	—	43	8	no	—
232	C₂²⋊C₄	—	43	4	no	—
233	C₂²⋊C₄	—	43	4	no	—
234	C₂²⋊C₄	—	43	8	no	—
235	C₂²⋊C₄	C₂²×D₄₃	1	8	yes	—
236	C₄²	—	43	4	no	—
237	C₄²	—	43	4	no	—
238	C₄²	—	43	4	no	—
239	C₄²	C₂²×D₄₃	1	2	yes	—
240	C₄²	C₂²×D₄₃	1	2	yes	—
241	C₂³	—	2	4	no	—
242	C₂³	—	4	2	no	—
243	C₂³	—	4	2	no	—
244	C₂³	—	43	4	no	—
245	C₂³	C₂³×D₄₃	1	4	yes	—
246	C₂³	C₂³×D₄₃	1	1	yes	characteristic
247	Q₈	—	86	4	no	—
248	Q₈	—	172	4	no	—
249	D₄	—	2	8	no	—
250	D₄	—	86	8	no	—
251	D₄	—	86	4	no	—
252	D₄	—	172	4	no	—
253	D₄	—	172	8	no	—
254	D₄	—	172	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	8	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	4	1	no	—
260	C₂×C₄	—	43	4	no	—
261	C₂×C₄	—	43	4	no	—
262	C₂×C₄	—	43	4	no	—
263	C₂×C₄	—	86	8	no	—
264	C₂×C₄	—	86	8	no	—
265	C₂×C₄	—	172	4	no	—
266	C₂×C₄	#10621898	1	4	yes	complemented
267	C₂×C₄	C₂³×D₄₃	1	2	yes	—
268	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
269	C₂×C₄	C₂³×D₄₃	1	4	yes	—
270	C₂×C₄	C₂³×D₄₃	1	2	yes	—
271	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	4	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	4	8	no	—
278	C₂²	—	86	4	no	—
279	C₂²	—	86	4	no	—
280	C₂²	—	86	4	no	—
281	C₂²	C₂×#10621898	1	2	yes	—
282	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
283	C₄	—	2	1	no	—
284	C₄	—	2	4	no	—
285	C₄	—	2	2	no	—
286	C₄	—	86	4	no	—
287	C₄	—	86	4	no	—
288	C₄	—	86	4	no	—
289	C₄	C₂×#10621898	1	4	yes	—
290	C₄	#13063370	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	4	4	no	—
293	C₂	—	172	4	no	—
294	C₂	C₂²×#10621898	1	1	yes	characteristic
295	C₂	C₂×#13063370	1	1	yes	characteristic
296	C₂	#14583735	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Frattini	C₂²	C₂⁴×D₄₃
Fitting	C₂×C₄×D₄×C₄₃	C₂
hypercenter	C₂×C₄×D₄	D₄₃
hypocenter	C₄₃	#3460
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×D₄

Complements

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

N	H ≅ G/N	type
1. #14583530	C₂	semidirect
2. #14583514	C₂	semidirect
3. #14583500	C₂	semidirect
4. #14583487	C₂	semidirect
5. #14583480	C₂	semidirect
6. #14583466	C₂	semidirect
7. #14583446	C₂	semidirect
8. #14583431	C₂	semidirect
9. #14583410	C₂	semidirect
10. C₄×#10621898	C₂	semidirect
11. C₂×C₄×D₄×C₄₃	C₂	semidirect
13. C₂×#10621898	C₂²	semidirect
14. #13063324	C₂²	semidirect
16. #13063318	C₂²	semidirect
17. #13063316	C₂²	semidirect
19. #13063311	C₂²	semidirect
20. #13063308	C₂²	semidirect
21. #13063307	C₂²	semidirect
22. #13063305	C₂²	semidirect
24. #13063304	C₂²	semidirect
25. #13063300	C₂²	semidirect
27. #13063299	C₂²	semidirect
28. #13063298	C₂²	semidirect
32. C₄×Dic₈₆	C₂²	semidirect
33. C₄×Dic₈₆	C₂²	semidirect
34. C₄×D₁₇₂	C₂²	semidirect
35. C₄×D₁₇₂	C₂²	semidirect
38. C₄×C₂²⋊D₄₃	C₂²	semidirect
40. #13063284	C₂²	semidirect
41. #13063282	C₂²	semidirect
42. C₄²×D₄₃	C₂²	semidirect
104. C₂×C₄×C₄₃	#41	semidirect
159. C₂×C₄×D₄	D₄₃	semidirect
160. C₄₃	#3460	semidirect
266. C₂×C₄	#10621898	semidirect