Subgroups — #19148928

Order 5504.

1260 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	#14583636	C₂	1	4	yes	maximal, complemented
2	#14583635	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583569	C₂	1	2	yes	maximal, complemented
4	#14583543	C₂	1	4	yes	maximal, complemented
5	#14583534	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×C₂²⋊D₄₃	C₂	1	4	yes	maximal, complemented
7	#14583503	C₂	1	4	yes	maximal, complemented
8	#14583462	C₂	1	4	yes	maximal, complemented
9	#14583458	C₂	1	4	yes	maximal, complemented
10	#14583433	C₂	1	2	yes	maximal, complemented
11	C₂×#127×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621900	C₂²	1	4	yes	complemented
13	C₂²×C₂²⋊D₄₃	C₂²	1	4	yes	complemented
14	C₂×D₄×D₄₃	C₂²	1	4	yes	complemented
15	#13063321	C₂²	1	4	yes	complemented
16	#13063319	C₂²	1	4	yes	complemented
17	#13063317	C₂²	1	2	yes	complemented
18	#13063313	C₂²	1	2	yes	complemented
19	#13063312	C₂²	1	4	yes	complemented
20	#13063308	C₂²	1	4	yes	complemented
21	#13063306	C₂²	1	4	yes	complemented
22	#13063306	C₂²	1	8	yes	complemented
23	#13063306	C₂²	1	4	yes	complemented
24	#13063304	C₂²	1	2	yes	complemented
25	#13063299	C₂²	1	2	yes	—
26	#13063297	C₂²	1	8	yes	complemented
27	#13063297	C₂²	1	8	yes	complemented
28	#13063297	C₂²	1	8	yes	complemented
29	#13063297	C₂²	1	8	yes	complemented
30	#13063296	C₂²	1	4	yes	complemented
31	#13063296	C₂²	1	4	yes	complemented
32	#13063294	C₂²	1	8	yes	complemented
33	#13063293	C₂²	1	4	yes	complemented
34	#13063293	C₂²	1	8	yes	complemented
35	#13063293	C₂²	1	4	yes	complemented
36	C₄×Dic₈₆	C₂²	1	2	yes	complemented
37	C₄×D₁₇₂	C₂²	1	2	yes	complemented
38	D₄×Dic₄₃	C₂²	1	4	yes	complemented
39	D₄×Dic₄₃	C₂²	1	4	yes	complemented
40	C₄×C₂²⋊D₄₃	C₂²	1	4	yes	complemented
41	#13063282	C₂²	1	4	yes	complemented
42	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	complemented
43	C₂²×D₄×C₄₃	C₂²	1	2	yes	complemented
44	C₂²×D₄×C₄₃	C₂²	1	4	yes	complemented
45	#127×C₄₃	C₂²	1	1	yes	characteristic
46	#127×C₄₃	C₂²	1	1	yes	characteristic, complemented
47	#127×C₄₃	C₂²	1	2	yes	complemented
48	#127×C₄₃	C₂²	1	4	yes	complemented
49	C₂×C₄²×C₄₃	C₂²	1	1	yes	characteristic, complemented
50	C₂⁴×C₄₃	C₂³	1	4	yes	—
51	C₂³×D₄₃	C₂³	1	4	yes	—
52	C₂×D₄×C₄₃	—	4	4	no	—
53	C₂×D₄×C₄₃	—	4	8	no	—
54	C₂×D₄×C₄₃	C₂³	1	4	yes	—
55	C₂×D₄×C₄₃	C₂³	1	4	yes	—
56	C₂×D₄×C₄₃	C₂³	1	2	yes	—
57	C₂×D₄×C₄₃	C₂³	1	8	yes	—
58	C₂×D₄×C₄₃	C₂³	1	4	yes	—
59	C₂×D₄×C₄₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	4	8	no	—
63	C₂×C₂²⋊D₄₃	C₂³	1	4	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	8	yes	—
68	C₂²×Dic₄₃	C₂³	1	4	yes	—
69	#10621900	—	4	8	no	—
70	D₄×D₄₃	—	4	8	no	—
71	C₂×D₁₇₂	C₂³	1	2	yes	—
72	C₂×C₄×D₄₃	C₂³	1	4	yes	—
73	C₂×Dic₈₆	C₂³	1	2	yes	—
74	C₄²×C₄₃	C₂³	1	1	yes	characteristic
75	C₄²×C₄₃	C₂³	1	2	yes	—
76	C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
78	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
79	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
80	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
81	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
82	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
83	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
84	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
85	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
86	C₄⋊Dic₄₃	C₂³	1	2	yes	—
87	C₄⋊Dic₄₃	C₂³	1	2	yes	—
88	Dic₄₃⋊C₄	C₂³	1	4	yes	—
89	Dic₄₃⋊C₄	C₂³	1	4	yes	—
90	Dic₄₃⋊C₄	C₂³	1	4	yes	—
91	Dic₄₃⋊C₄	C₂³	1	4	yes	—
92	C₄×Dic₄₃	C₂³	1	4	yes	—
93	C₂³×C₄₃	—	2	8	no	—
94	C₂³×C₄₃	—	4	8	no	—
95	C₂³×C₄₃	C₂⁴	1	4	yes	—
96	C₂³×C₄₃	C₂⁴	1	4	yes	—
97	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₂²×D₄₃	—	4	8	no	—
99	C₂²×D₄₃	—	4	4	no	—
100	C₂²×D₄₃	C₂⁴	1	4	yes	—
101	D₄×C₄₃	—	2	8	no	—
102	D₄×C₄₃	—	2	8	no	—
103	D₄×C₄₃	—	4	2	no	—
104	D₄×C₄₃	—	4	4	no	—
105	D₄×C₄₃	—	4	8	no	—
106	D₄×C₄₃	—	4	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	4	no	—
109	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
113	C₂²⋊D₄₃	—	2	8	no	—
114	C₂²⋊D₄₃	—	4	8	no	—
115	C₂²⋊D₄₃	—	4	8	no	—
116	C₂²⋊D₄₃	—	4	8	no	—
117	C₂×Dic₄₃	—	4	8	no	—
118	C₂×Dic₄₃	—	4	4	no	—
119	C₂×Dic₄₃	C₂⁴	1	4	yes	—
120	C₂×Dic₄₃	C₂⁴	1	4	yes	—
121	C₂×Dic₄₃	C₂⁴	1	4	yes	—
122	D₁₇₂	—	2	4	no	—
123	C₄×D₄₃	—	2	8	no	—
124	Dic₈₆	—	2	4	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	8	no	—
128	C₂²×C₄₃	—	2	4	no	—
129	C₂²×C₄₃	—	2	8	no	—
130	C₂²×C₄₃	—	8	8	no	—
131	C₂²×C₄₃	H(5,𝔽₂)	1	2	yes	—
132	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
133	C₂×D₄₃	—	2	4	no	—
134	C₂×D₄₃	—	2	4	no	—
135	C₂×D₄₃	—	2	4	no	—
136	C₂×D₄₃	—	8	8	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	2	1	no	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	H(5,𝔽₂)	1	4	yes	—
141	Dic₄₃	—	2	4	no	—
142	Dic₄₃	—	2	4	no	—
143	Dic₄₃	—	2	4	no	—
144	#3543	—	43	1	no	maximal, Sylow
145	C₂×C₄₃	—	2	2	no	—
146	C₂×C₄₃	—	4	4	no	—
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	4	no	—
151	#560	—	43	4	no	—
152	#559	—	43	1	no	—
153	#550	—	43	2	no	—
154	#546	—	43	4	no	—
155	D₄²	—	43	4	no	—
156	D₄²	—	43	1	no	—
157	#540	—	43	4	no	—
158	#535	—	43	4	no	—
159	#534	—	43	4	no	—
160	C₂×#127	D₄₃	1	1	yes	characteristic, complemented
161	#529	—	43	2	no	—
162	C₄₃	#3543	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	43	4	no	—
164	C₂²×D₄	—	43	4	no	—
165	C₂²×D₄	—	43	4	no	—
166	C₂²×D₄	C₂×D₄₃	1	2	yes	complemented
167	C₂²×D₄	C₂×D₄₃	1	4	yes	complemented
168	#127	—	43	4	no	—
169	#127	C₂×D₄₃	1	1	yes	characteristic
170	#127	C₂×D₄₃	1	4	yes	complemented
171	#127	C₂×D₄₃	1	1	yes	characteristic, complemented
172	#127	C₂×D₄₃	1	2	yes	complemented
173	#126	—	43	4	no	—
174	#125	—	43	2	no	—
175	#124	—	43	4	no	—
176	#124	—	43	2	no	—
177	#123	—	43	4	no	—
178	#123	—	43	4	no	—
179	#123	—	43	8	no	—
180	#123	—	43	4	no	—
181	C₂²⋊Q₈	—	43	2	no	—
182	#121	—	43	8	no	—
183	#121	—	43	8	no	—
184	#121	—	43	8	no	—
185	#121	—	43	4	no	—
186	#121	—	43	4	no	—
187	#121	—	43	8	no	—
188	#121	—	43	2	no	—
189	#120	—	43	8	no	—
190	#120	—	43	4	no	—
191	#120	—	43	4	no	—
192	#120	—	43	8	no	—
193	C₄×Q₈	—	43	2	no	—
194	C₄×D₄	—	43	4	no	—
195	C₄×D₄	—	43	4	no	—
196	C₄×D₄	—	43	4	no	—
197	C₄×D₄	—	43	2	no	—
198	#117	—	43	4	no	—
199	C₂×C₂²⋊C₄	—	43	4	no	—
200	C₂×C₄²	C₂×D₄₃	1	1	yes	characteristic, complemented
201	C₂⁴	—	43	4	no	—
202	C₂⁴	C₂²×D₄₃	1	4	yes	—
203	#41	—	172	8	no	—
204	C₂×Q₈	—	43	2	no	—
205	C₂×D₄	—	4	8	no	—
206	C₂×D₄	—	4	4	no	—
207	C₂×D₄	—	43	4	no	—
208	C₂×D₄	—	43	8	no	—
209	C₂×D₄	—	43	8	no	—
210	C₂×D₄	—	43	8	no	—
211	C₂×D₄	—	43	2	no	—
212	C₂×D₄	—	172	8	no	—
213	C₂×D₄	—	172	8	no	—
214	C₂×D₄	C₂²×D₄₃	1	4	yes	—
215	C₂×D₄	C₂²×D₄₃	1	4	yes	—
216	C₂×D₄	C₂²×D₄₃	1	2	yes	—
217	C₂×D₄	C₂²×D₄₃	1	4	yes	—
218	C₂×D₄	C₂²×D₄₃	1	8	yes	—
219	C₂×D₄	C₂²×D₄₃	1	2	yes	—
220	C₂²×C₄	—	43	8	no	—
221	C₂²×C₄	—	43	4	no	—
222	C₂²×C₄	—	43	4	no	—
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	2	no	—
228	C₄⋊C₄	—	43	2	no	—
229	C₄⋊C₄	—	43	4	no	—
230	C₄⋊C₄	—	43	4	no	—
231	C₂²⋊C₄	—	43	4	no	—
232	C₂²⋊C₄	—	43	4	no	—
233	C₂²⋊C₄	—	43	4	no	—
234	C₂²⋊C₄	—	43	8	no	—
235	C₂²⋊C₄	—	43	8	no	—
236	C₂²⋊C₄	—	43	4	no	—
237	C₂²⋊C₄	—	43	4	no	—
238	C₂²⋊C₄	—	43	4	no	—
239	C₂²⋊C₄	—	43	4	no	—
240	C₄²	—	43	4	no	—
241	C₄²	C₂²×D₄₃	1	2	yes	—
242	C₄²	C₂²×D₄₃	1	1	yes	characteristic
243	C₄²	C₂²×D₄₃	1	1	yes	characteristic
244	C₂³	—	2	8	no	—
245	C₂³	—	4	8	no	—
246	C₂³	—	43	4	no	—
247	C₂³	—	172	4	no	—
248	C₂³	—	172	8	no	—
249	C₂³	C₂³×D₄₃	1	4	yes	—
250	C₂³	C₂³×D₄₃	1	4	yes	—
251	C₂³	C₂³×D₄₃	1	1	yes	characteristic
252	Q₈	—	86	4	no	—
253	D₄	—	2	8	no	—
254	D₄	—	2	8	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	4	no	—
258	D₄	—	4	2	no	—
259	D₄	—	86	8	no	—
260	D₄	—	86	4	no	—
261	D₄	—	172	8	no	—
262	D₄	—	172	8	no	—
263	D₄	—	172	8	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	43	4	no	—
267	C₂×C₄	—	43	4	no	—
268	C₂×C₄	—	43	4	no	—
269	C₂×C₄	—	86	8	no	—
270	C₂×C₄	—	172	4	no	—
271	C₂×C₄	—	172	8	no	—
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	C₂×C₄	C₂³×D₄₃	1	2	yes	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	8	no	—
278	C₂²	—	2	4	no	—
279	C₂²	—	2	8	no	—
280	C₂²	—	2	4	no	—
281	C₂²	—	8	8	no	—
282	C₂²	—	86	4	no	—
283	C₂²	—	86	4	no	—
284	C₂²	—	86	4	no	—
285	C₂²	—	344	8	no	—
286	C₂²	#13063370	1	2	yes	—
287	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	1	no	—
290	C₄	—	2	2	no	—
291	C₄	—	86	4	no	—
292	C₄	—	86	4	no	—
293	C₄	—	86	4	no	—
294	C₄	#13063370	1	4	yes	—
295	C₂	—	2	2	no	—
296	C₂	—	4	4	no	—
297	C₂	—	4	4	no	—
298	C₂	—	172	4	no	—
299	C₂	C₂×#13063370	1	1	yes	characteristic
300	C₂	C₂×#13063370	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₂×#127×C₄₃	C₂
hypercenter	C₂×#127	D₄₃
hypocenter	C₄₃	#3543
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#127

Complements

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

N	H ≅ G/N	type
1. #14583636	C₂	semidirect
2. #14583635	C₂	semidirect
3. #14583569	C₂	semidirect
4. #14583543	C₂	semidirect
5. #14583534	C₂	semidirect
6. D₄×C₂²⋊D₄₃	C₂	semidirect
7. #14583503	C₂	semidirect
8. #14583462	C₂	semidirect
9. #14583458	C₂	semidirect
10. #14583433	C₂	semidirect
11. C₂×#127×C₄₃	C₂	semidirect
12. C₂×#10621900	C₂²	semidirect
13. C₂²×C₂²⋊D₄₃	C₂²	semidirect
14. C₂×D₄×D₄₃	C₂²	semidirect
15. #13063321	C₂²	semidirect
16. #13063319	C₂²	semidirect
17. #13063317	C₂²	semidirect
18. #13063313	C₂²	semidirect
19. #13063312	C₂²	semidirect
20. #13063308	C₂²	semidirect
21. #13063306	C₂²	semidirect
22. #13063306	C₂²	semidirect
23. #13063306	C₂²	semidirect
24. #13063304	C₂²	semidirect
26. #13063297	C₂²	semidirect
27. #13063297	C₂²	semidirect
28. #13063297	C₂²	semidirect
29. #13063297	C₂²	semidirect
30. #13063296	C₂²	semidirect
31. #13063296	C₂²	semidirect
32. #13063294	C₂²	semidirect
33. #13063293	C₂²	semidirect
34. #13063293	C₂²	semidirect
35. #13063293	C₂²	semidirect
36. C₄×Dic₈₆	C₂²	semidirect
37. C₄×D₁₇₂	C₂²	semidirect
38. D₄×Dic₄₃	C₂²	semidirect
39. D₄×Dic₄₃	C₂²	semidirect
40. C₄×C₂²⋊D₄₃	C₂²	semidirect
41. #13063282	C₂²	semidirect
42. C₂×C₂²⋊Dic₄₃	C₂²	semidirect
43. C₂²×D₄×C₄₃	C₂²	semidirect
44. C₂²×D₄×C₄₃	C₂²	semidirect
46. #127×C₄₃	C₂²	semidirect
47. #127×C₄₃	C₂²	semidirect
48. #127×C₄₃	C₂²	semidirect
49. C₂×C₄²×C₄₃	C₂²	semidirect
160. C₂×#127	D₄₃	semidirect
162. C₄₃	#3543	semidirect
166. C₂²×D₄	C₂×D₄₃	semidirect
167. C₂²×D₄	C₂×D₄₃	semidirect
170. #127	C₂×D₄₃	semidirect
171. #127	C₂×D₄₃	semidirect
172. #127	C₂×D₄₃	semidirect
200. C₂×C₄²	C₂×D₄₃	semidirect