Subgroups — #19148834

Order 5504.

1270 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	#14583639	C₂	1	4	yes	maximal, complemented
2	#14583595	C₂	1	4	yes	maximal, complemented
3	D₄×D₁₇₂	C₂	1	4	yes	maximal, complemented
4	#14583531	C₂	1	4	yes	maximal, complemented
5	#14583504	C₂	1	2	yes	maximal, complemented
6	#14583489	C₂	1	4	yes	maximal, complemented
7	#14583463	C₂	1	2	yes	maximal, complemented
8	#14583457	C₂	1	4	yes	maximal, complemented
9	C₂×#13063299	C₂	1	2	yes	maximal, complemented
10	C₄₃×#525	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²×D₁₇₂	C₂²	1	2	yes	complemented
12	C₂²×D₁₇₂	C₂²	1	2	yes	complemented
13	C₂²×C₂²⋊D₄₃	C₂²	1	4	yes	complemented
14	C₂×D₄×D₄₃	C₂²	1	8	yes	complemented
15	#13063322	C₂²	1	2	yes	complemented
16	#13063321	C₂²	1	4	yes	complemented
17	#13063320	C₂²	1	8	yes	complemented
18	#13063317	C₂²	1	2	yes	complemented
19	#13063314	C₂²	1	4	yes	complemented
20	#13063310	C₂²	1	8	yes	complemented
21	#13063307	C₂²	1	8	yes	complemented
22	#13063300	C₂²	1	8	yes	complemented
23	#13063299	C₂²	1	4	yes	complemented
24	#13063299	C₂²	1	4	yes	complemented
25	#13063299	C₂²	1	4	yes	complemented
26	#13063299	C₂²	1	4	yes	complemented
27	#13063299	C₂²	1	4	yes	complemented
28	#13063298	C₂²	1	8	yes	complemented
29	#13063297	C₂²	1	8	yes	complemented
30	#13063295	C₂²	1	8	yes	complemented
31	#13063295	C₂²	1	8	yes	complemented
32	#13063293	C₂²	1	4	yes	complemented
33	#13063293	C₂²	1	4	yes	complemented
34	C₄×D₁₇₂	C₂²	1	4	yes	complemented
35	(C₂²⋊D₄₃)⋊C₄	C₂²	1	8	yes	complemented
36	#13063284	C₂²	1	2	yes	complemented
37	C₂×C₄⋊Dic₄₃	C₂²	1	2	yes	complemented
38	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	4	yes	complemented
39	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic, complemented
40	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic, complemented
41	#123×C₄₃	C₂²	1	2	yes	complemented
42	C₂²⋊Q₈×C₄₃	C₂²	1	2	yes	complemented
43	#121×C₄₃	C₂²	1	2	yes	complemented
44	#120×C₄₃	C₂²	1	2	yes	complemented
45	C₄×D₄×C₄₃	C₂²	1	4	yes	complemented
46	#117×C₄₃	C₂²	1	1	yes	characteristic, complemented
47	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
48	C₂³×D₄₃	C₂³	1	4	yes	—
49	C₂³×D₄₃	C₂³	1	4	yes	—
50	#41×C₄₃	—	4	2	no	—
51	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
52	C₂×D₄×C₄₃	C₂³	1	4	yes	—
53	C₂×D₄×C₄₃	C₂³	1	2	yes	—
54	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	—	4	2	no	—
56	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₄₃	—	4	8	no	—
62	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
63	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
64	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
65	C₂²×Dic₄₃	C₂³	1	4	yes	—
66	D₄×D₄₃	—	4	16	no	—
67	C₂×D₁₇₂	—	4	4	no	—
68	C₂×D₁₇₂	—	4	4	no	—
69	C₂×D₁₇₂	C₂³	1	4	yes	—
70	C₂×D₁₇₂	C₂³	1	8	yes	—
71	C₂×D₁₇₂	C₂³	1	4	yes	—
72	C₂×D₁₇₂	C₂³	1	4	yes	—
73	C₂×C₄×D₄₃	C₂³	1	8	yes	—
74	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
75	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
76	C₂²⋊C₄×C₄₃	C₂³	1	2	yes	—
77	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
78	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
79	C₄²×C₄₃	C₂³	1	2	yes	—
80	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
81	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
82	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
83	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
84	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
85	C₄⋊Dic₄₃	C₂³	1	4	yes	—
86	C₄⋊Dic₄₃	C₂³	1	4	yes	—
87	Dic₄₃⋊C₄	C₂³	1	8	yes	—
88	C₄×Dic₄₃	C₂³	1	4	yes	—
89	C₂³×C₄₃	—	2	4	no	—
90	C₂³×C₄₃	—	4	1	no	—
91	C₂³×C₄₃	C₂⁴	1	2	yes	—
92	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₂³×C₄₃	C₂⁴	1	2	yes	—
94	C₂²×D₄₃	—	2	8	no	—
95	C₂²×D₄₃	—	4	4	no	—
96	C₂²×D₄₃	—	4	4	no	—
97	C₂²×D₄₃	—	4	8	no	—
98	C₂²×D₄₃	—	4	4	no	—
99	C₂²×D₄₃	C₂⁴	1	8	yes	—
100	Q₈×C₄₃	—	4	1	no	—
101	D₄×C₄₃	—	2	8	no	—
102	D₄×C₄₃	—	4	2	no	—
103	D₄×C₄₃	—	4	1	no	—
104	C₂×C₄×C₄₃	—	2	4	no	—
105	C₂×C₄×C₄₃	—	2	4	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	4	1	no	—
108	C₂×C₄×C₄₃	—	4	2	no	—
109	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
111	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
114	C₂²⋊D₄₃	—	2	16	no	—
115	C₂²⋊D₄₃	—	4	8	no	—
116	C₂²⋊D₄₃	—	4	8	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	C₂⁴	1	8	yes	—
119	D₁₇₂	—	2	8	no	—
120	D₁₇₂	—	4	4	no	—
121	D₁₇₂	—	4	4	no	—
122	D₁₇₂	—	4	8	no	—
123	C₄×D₄₃	—	4	8	no	—
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	2	no	—
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	2	1	no	—
130	C₂²×C₄₃	—	4	4	no	—
131	C₂²×C₄₃	H(5,𝔽₂)	1	4	yes	—
132	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
133	C₂×D₄₃	—	2	8	no	—
134	C₂×D₄₃	—	2	8	no	—
135	C₂×D₄₃	—	2	8	no	—
136	C₂×D₄₃	—	8	8	no	—
137	C₂×D₄₃	—	8	8	no	—
138	C₄×C₄₃	—	2	4	no	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	—	2	2	no	—
141	C₄×C₄₃	—	2	1	no	—
142	C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
143	Dic₄₃	—	2	8	no	—
144	#3537	—	43	1	no	maximal, Sylow
145	C₂×C₄₃	—	2	4	no	—
146	C₂×C₄₃	—	4	2	no	—
147	C₂×C₄₃	—	4	1	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	8	no	—
151	#560	—	43	4	no	—
152	#553	—	43	4	no	—
153	D₄²	—	43	4	no	—
154	D₄²	—	43	4	no	—
155	#540	—	43	2	no	—
156	#538	—	43	4	no	—
157	#535	—	43	2	no	—
158	#534	—	43	4	no	—
159	#525	D₄₃	1	1	yes	characteristic, complemented
160	C₂×#121	—	43	2	no	—
161	C₄₃	#3537	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	C₂×D₄₃	1	1	yes	characteristic, complemented
163	C₂²×D₄	—	43	4	no	—
164	C₂²×D₄	—	43	8	no	—
165	C₂²×D₄	—	43	2	no	—
166	C₂²×D₄	—	43	2	no	—
167	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic, complemented
168	#127	—	43	4	no	—
169	#127	—	43	2	no	—
170	#126	—	43	8	no	—
171	#125	—	43	2	no	—
172	#124	—	43	4	no	—
173	#123	—	43	8	no	—
174	#123	—	43	8	no	—
175	#123	C₂×D₄₃	1	2	yes	complemented
176	C₂²⋊Q₈	C₂×D₄₃	1	2	yes	complemented
177	#121	—	43	8	no	—
178	#121	—	43	4	no	—
179	#121	—	43	4	no	—
180	#121	—	43	4	no	—
181	#121	—	43	4	no	—
182	#121	—	43	4	no	—
183	#121	—	43	8	no	—
184	#121	—	43	8	no	—
185	#121	C₂×D₄₃	1	2	yes	complemented
186	#120	—	43	4	no	—
187	#120	—	43	4	no	—
188	#120	—	43	8	no	—
189	#120	—	43	8	no	—
190	#120	C₂×D₄₃	1	2	yes	complemented
191	C₄×D₄	—	43	8	no	—
192	C₄×D₄	—	43	4	no	—
193	C₄×D₄	C₂×D₄₃	1	4	yes	complemented
194	#117	—	43	2	no	—
195	#117	C₂×D₄₃	1	1	yes	characteristic, complemented
196	C₂×C₄⋊C₄	—	43	2	no	—
197	C₂×C₂²⋊C₄	—	43	4	no	—
198	C₂⁴	—	43	4	no	—
199	C₂⁴	—	43	4	no	—
200	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
201	#41	—	4	2	no	—
202	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
203	C₂×D₄	—	43	8	no	—
204	C₂×D₄	—	43	8	no	—
205	C₂×D₄	—	43	8	no	—
206	C₂×D₄	—	43	4	no	—
207	C₂×D₄	—	43	4	no	—
208	C₂×D₄	—	43	8	no	—
209	C₂×D₄	—	43	4	no	—
210	C₂×D₄	—	172	4	no	—
211	C₂×D₄	—	172	4	no	—
212	C₂×D₄	—	172	16	no	—
213	C₂×D₄	—	172	8	no	—
214	C₂×D₄	C₂²×D₄₃	1	4	yes	—
215	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
216	C₂×D₄	C₂²×D₄₃	1	2	yes	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	43	4	no	—
219	C₂²×C₄	—	43	8	no	—
220	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
221	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
222	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
224	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
225	C₄⋊C₄	—	43	4	no	—
226	C₄⋊C₄	—	43	4	no	—
227	C₄⋊C₄	—	43	8	no	—
228	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
229	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
230	C₂²⋊C₄	—	43	4	no	—
231	C₂²⋊C₄	—	43	8	no	—
232	C₂²⋊C₄	—	43	8	no	—
233	C₂²⋊C₄	—	43	8	no	—
234	C₂²⋊C₄	—	43	8	no	—
235	C₂²⋊C₄	C₂²×D₄₃	1	2	yes	—
236	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
237	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
238	C₄²	—	43	4	no	—
239	C₄²	C₂²×D₄₃	1	2	yes	—
240	C₂³	—	2	4	no	—
241	C₂³	—	4	1	no	—
242	C₂³	—	43	8	no	—
243	C₂³	—	86	8	no	—
244	C₂³	—	172	4	no	—
245	C₂³	—	172	4	no	—
246	C₂³	—	172	4	no	—
247	C₂³	—	172	8	no	—
248	C₂³	C₂³×D₄₃	1	2	yes	—
249	C₂³	C₂³×D₄₃	1	1	yes	characteristic
250	C₂³	C₂³×D₄₃	1	2	yes	—
251	Q₈	—	4	1	no	—
252	D₄	—	2	8	no	—
253	D₄	—	4	1	no	—
254	D₄	—	4	2	no	—
255	D₄	—	86	16	no	—
256	D₄	—	86	8	no	—
257	D₄	—	172	4	no	—
258	D₄	—	172	4	no	—
259	D₄	—	172	8	no	—
260	D₄	—	172	8	no	—
261	D₄	—	172	8	no	—
262	C₂×C₄	—	2	4	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	2	no	—
266	C₂×C₄	—	4	1	no	—
267	C₂×C₄	—	43	8	no	—
268	C₂×C₄	—	172	8	no	—
269	C₂×C₄	—	172	4	no	—
270	C₂×C₄	C₂³×D₄₃	1	2	yes	—
271	C₂×C₄	C₂³×D₄₃	1	4	yes	—
272	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
273	C₂×C₄	C₂³×D₄₃	1	2	yes	—
274	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
275	C₂²	—	2	2	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	4	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	4	4	no	—
282	C₂²	—	86	8	no	—
283	C₂²	—	86	8	no	—
284	C₂²	—	86	8	no	—
285	C₂²	—	344	8	no	—
286	C₂²	—	344	8	no	—
287	C₂²	#13063371	1	4	yes	—
288	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	4	no	—
293	C₄	—	86	8	no	—
294	C₄	#13063370	1	2	yes	—
295	C₂	—	2	4	no	—
296	C₂	—	4	1	no	—
297	C₂	—	4	2	no	—
298	C₂	—	172	8	no	—
299	C₂	C₂×#13063370	1	1	yes	characteristic
300	C₂	C₂×#13063371	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#525	C₂
hypercenter	#525	D₄₃
hypocenter	C₄₃	#3537
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #525

Complements

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

N	H ≅ G/N	type
1. #14583639	C₂	semidirect
2. #14583595	C₂	semidirect
3. D₄×D₁₇₂	C₂	semidirect
4. #14583531	C₂	semidirect
5. #14583504	C₂	semidirect
6. #14583489	C₂	semidirect
7. #14583463	C₂	semidirect
8. #14583457	C₂	semidirect
9. C₂×#13063299	C₂	semidirect
10. C₄₃×#525	C₂	semidirect
11. C₂²×D₁₇₂	C₂²	semidirect
12. C₂²×D₁₇₂	C₂²	semidirect
13. C₂²×C₂²⋊D₄₃	C₂²	semidirect
14. C₂×D₄×D₄₃	C₂²	semidirect
15. #13063322	C₂²	semidirect
16. #13063321	C₂²	semidirect
17. #13063320	C₂²	semidirect
18. #13063317	C₂²	semidirect
19. #13063314	C₂²	semidirect
20. #13063310	C₂²	semidirect
21. #13063307	C₂²	semidirect
22. #13063300	C₂²	semidirect
23. #13063299	C₂²	semidirect
24. #13063299	C₂²	semidirect
25. #13063299	C₂²	semidirect
26. #13063299	C₂²	semidirect
27. #13063299	C₂²	semidirect
28. #13063298	C₂²	semidirect
29. #13063297	C₂²	semidirect
30. #13063295	C₂²	semidirect
31. #13063295	C₂²	semidirect
32. #13063293	C₂²	semidirect
33. #13063293	C₂²	semidirect
34. C₄×D₁₇₂	C₂²	semidirect
35. (C₂²⋊D₄₃)⋊C₄	C₂²	semidirect
36. #13063284	C₂²	semidirect
37. C₂×C₄⋊Dic₄₃	C₂²	semidirect
38. C₂×(C₂×D₄₃)⋊C₄	C₂²	semidirect
39. C₂×#41×C₄₃	C₂²	semidirect
40. C₂³×C₄×C₄₃	C₂²	semidirect
41. #123×C₄₃	C₂²	semidirect
42. C₂²⋊Q₈×C₄₃	C₂²	semidirect
43. #121×C₄₃	C₂²	semidirect
44. #120×C₄₃	C₂²	semidirect
45. C₄×D₄×C₄₃	C₂²	semidirect
46. #117×C₄₃	C₂²	semidirect
159. #525	D₄₃	semidirect
161. C₄₃	#3537	semidirect
162. C₂×#41	C₂×D₄₃	semidirect
167. C₂³×C₄	C₂×D₄₃	semidirect
175. #123	C₂×D₄₃	semidirect
176. C₂²⋊Q₈	C₂×D₄₃	semidirect
185. #121	C₂×D₄₃	semidirect
190. #120	C₂×D₄₃	semidirect
193. C₄×D₄	C₂×D₄₃	semidirect
195. #117	C₂×D₄₃	semidirect