Subgroups — #19148239

Order 5504.

1560 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063370	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583548	C₂	1	8	yes	maximal, complemented
3	#14583537	C₂	1	8	yes	maximal, complemented
4	#14583473	C₂	1	2	yes	maximal, complemented
5	#14583458	C₂	1	4	yes	maximal, complemented
6	#14583410	C₂	1	2	yes	maximal, complemented
7	C₂×#13063308	C₂	1	4	yes	maximal, complemented
8	#14583374	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#41×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#13063370	—	2	8	no	—
11	C₂×#10621898	C₂²	1	2	yes	—
12	C₂×#10621900	C₂²	1	4	yes	complemented
13	C₂²×C₂²⋊D₄₃	C₂²	1	4	yes	—
14	C₂×D₄×D₄₃	C₂²	1	4	yes	complemented
15	#13063323	C₂²	1	4	yes	complemented
16	#13063314	C₂²	1	4	yes	complemented
17	#13063308	C₂²	1	4	yes	complemented
18	#13063308	C₂²	1	8	yes	complemented
19	#13063308	C₂²	1	16	yes	complemented
20	#13063306	C₂²	1	8	yes	complemented
21	#13063304	C₂²	1	4	yes	complemented
22	#13063301	C₂²	1	8	yes	complemented
23	#13063299	C₂²	1	4	yes	complemented
24	#13063297	C₂²	1	16	yes	complemented
25	#13063294	C₂²	1	4	yes	complemented
26	#13063293	C₂²	1	8	yes	complemented
27	D₄×Dic₄₃	C₂²	1	8	yes	complemented
28	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
29	#13063284	C₂²	1	2	yes	complemented
30	C₂×Dic₄₃⋊C₄	C₂²	1	8	yes	complemented
31	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	complemented
32	C₂×(C₂×D₄₃)⋊C₄	C₂²	1	8	yes	complemented
33	C₂×#41×C₄₃	—	2	8	no	—
34	C₂×#41×C₄₃	C₂²	1	8	yes	complemented
35	C₂²×Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
36	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
37	C₂²×D₄×C₄₃	C₂²	1	2	yes	complemented
38	C₂³×C₄×C₄₃	C₂²	1	2	yes	complemented
39	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic, complemented
40	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
41	C₂⁴×C₄₃	C₂³	1	2	yes	—
42	C₂³×D₄₃	C₂³	1	4	yes	—
43	#41×C₄₃	—	2	32	no	—
44	C₂×Q₈×C₄₃	—	2	4	no	—
45	C₂×Q₈×C₄₃	C₂³	1	4	yes	—
46	C₂×D₄×C₄₃	—	2	8	no	—
47	C₂×D₄×C₄₃	D₄	1	8	yes	complemented
48	C₂×D₄×C₄₃	C₂³	1	8	yes	—
49	C₂×D₄×C₄₃	C₂³	1	4	yes	—
50	C₂²×C₄×C₄₃	—	2	4	no	—
51	C₂²×C₄×C₄₃	—	2	8	no	—
52	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂³	1	8	yes	—
55	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
56	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂×C₂²⋊D₄₃	—	2	16	no	—
58	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
59	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
60	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
61	C₂²×Dic₄₃	C₂³	1	4	yes	—
62	C₂²×Dic₄₃	C₂³	1	8	yes	—
63	#10621900	—	2	16	no	—
64	D₄×D₄₃	—	2	16	no	—
65	#10621898	—	2	8	no	—
66	C₂×D₁₇₂	C₂³	1	2	yes	—
67	C₂×C₄×D₄₃	C₂³	1	4	yes	—
68	C₂×Dic₈₆	C₂³	1	2	yes	—
69	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
70	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
71	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
72	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	—
73	(C₂×D₄₃)⋊C₄	C₂³	1	16	yes	—
74	C₄⋊Dic₄₃	C₂³	1	4	yes	—
75	Dic₄₃⋊C₄	C₂³	1	8	yes	—
76	Dic₄₃⋊C₄	C₂³	1	16	yes	—
77	C₄×Dic₄₃	C₂³	1	4	yes	—
78	C₂³×C₄₃	—	2	2	no	—
79	C₂³×C₄₃	—	4	4	no	—
80	C₂³×C₄₃	—	4	2	no	—
81	C₂³×C₄₃	C₂×D₄	1	8	yes	—
82	C₂³×C₄₃	C₂⁴	1	2	yes	—
83	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
84	C₂³×C₄₃	C₂⁴	1	4	yes	—
85	C₂²×D₄₃	—	2	8	no	—
86	C₂²×D₄₃	—	4	8	no	—
87	C₂²×D₄₃	C₂⁴	1	4	yes	—
88	Q₈×C₄₃	—	2	8	no	—
89	D₄×C₄₃	—	2	16	no	—
90	D₄×C₄₃	C₂×D₄	1	16	yes	—
91	C₂×C₄×C₄₃	—	2	2	no	—
92	C₂×C₄×C₄₃	—	2	4	no	—
93	C₂×C₄×C₄₃	—	2	16	no	—
94	C₂×C₄×C₄₃	—	2	8	no	—
95	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	—
96	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
97	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
98	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
99	C₂²⋊D₄₃	—	2	8	no	—
100	C₂²⋊D₄₃	—	2	16	no	—
101	C₂²⋊D₄₃	—	4	8	no	—
102	C₂×Dic₄₃	—	2	8	no	—
103	C₂×Dic₄₃	—	4	8	no	—
104	C₂×Dic₄₃	C₂⁴	1	8	yes	—
105	C₂×Dic₄₃	C₂⁴	1	4	yes	—
106	D₁₇₂	—	2	4	no	—
107	C₄×D₄₃	—	2	8	no	—
108	Dic₈₆	—	2	4	no	—
109	C₂²×C₄₃	—	2	8	no	—
110	C₂²×C₄₃	—	2	2	no	—
111	C₂²×C₄₃	—	2	2	no	—
112	C₂²×C₄₃	—	2	4	no	—
113	C₂²×C₄₃	—	4	8	no	—
114	C₂²×C₄₃	—	4	4	no	—
115	C₂²×C₄₃	C₂²×D₄	1	8	yes	—
116	C₂²×C₄₃	C₂²×D₄	1	2	yes	—
117	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
118	C₂×D₄₃	—	2	4	no	—
119	C₂×D₄₃	—	2	8	no	—
120	C₂×D₄₃	—	8	8	no	—
121	C₄×C₄₃	—	2	2	no	—
122	C₄×C₄₃	—	2	4	no	—
123	C₄×C₄₃	C₂²×D₄	1	4	yes	—
124	Dic₄₃	—	2	4	no	—
125	Dic₄₃	—	2	8	no	—
126	#3485	—	43	1	no	maximal, Sylow
127	C₂×C₄₃	—	2	8	no	—
128	C₂×C₄₃	—	2	2	no	—
129	C₂×C₄₃	—	4	2	no	—
130	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
131	C₂×C₄₃	#585	1	2	yes	—
132	D₄₃	—	4	4	no	—
133	C₂×H(5,𝔽₂)	—	43	1	no	—
134	C₂²×#41	D₄₃	1	1	yes	characteristic, complemented
135	#547	—	43	8	no	—
136	#546	—	43	8	no	—
137	#536	—	43	2	no	—
138	#534	—	43	4	no	—
139	#525	—	43	2	no	—
140	C₂×#123	—	43	4	no	—
141	#518	—	43	1	no	—
142	C₄₃	#3485	1	1	yes	characteristic, complemented, Sylow
143	H(5,𝔽₂)	—	86	8	no	—
144	C₂×#41	—	2	8	no	—
145	C₂×#41	—	43	4	no	—
146	C₂×#41	—	43	2	no	—
147	C₂×#41	C₂×D₄₃	1	8	yes	complemented
148	C₂²×Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
149	C₂²×D₄	—	43	4	no	—
150	C₂²×D₄	—	43	4	no	—
151	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
152	C₂²×D₄	C₂×D₄₃	1	2	yes	complemented
153	C₂³×C₄	C₂×D₄₃	1	2	yes	complemented
154	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic, complemented
155	C₄⋊Q₈	—	43	4	no	—
156	#124	—	43	4	no	—
157	#123	—	43	8	no	—
158	#123	—	43	4	no	—
159	#123	—	43	8	no	—
160	#123	—	43	16	no	—
161	C₂²⋊Q₈	—	43	8	no	—
162	C₂²⋊Q₈	—	43	4	no	—
163	#121	—	43	16	no	—
164	#121	—	43	4	no	—
165	#120	—	43	4	no	—
166	#120	—	43	8	no	—
167	C₄×D₄	—	43	8	no	—
168	C₄×D₄	—	43	8	no	—
169	#117	—	43	2	no	—
170	C₂×C₄⋊C₄	—	43	8	no	—
171	C₂×C₂²⋊C₄	—	43	4	no	—
172	C₂×C₂²⋊C₄	—	43	8	no	—
173	C₂⁴	—	43	4	no	—
174	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
175	C₂⁴	C₂²×D₄₃	1	2	yes	—
176	#41	—	2	32	no	—
177	#41	—	86	16	no	—
178	#41	—	86	8	no	—
179	C₂×Q₈	—	2	4	no	—
180	C₂×Q₈	—	43	2	no	—
181	C₂×Q₈	C₂²×D₄₃	1	4	yes	—
182	C₂×D₄	—	2	8	no	—
183	C₂×D₄	—	43	4	no	—
184	C₂×D₄	—	43	8	no	—
185	C₂×D₄	—	43	8	no	—
186	C₂×D₄	—	43	2	no	—
187	C₂×D₄	—	86	16	no	—
188	C₂×D₄	—	86	16	no	—
189	C₂×D₄	C₂²⋊D₄₃	1	8	yes	complemented
190	C₂×D₄	C₂²×D₄₃	1	8	yes	—
191	C₂×D₄	C₂²×D₄₃	1	4	yes	—
192	C₂²×C₄	—	2	8	no	—
193	C₂²×C₄	—	2	4	no	—
194	C₂²×C₄	—	43	4	no	—
195	C₂²×C₄	—	43	8	no	—
196	C₂²×C₄	—	43	4	no	—
197	C₂²×C₄	C₂²×D₄₃	1	8	yes	—
198	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
199	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
200	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
201	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
202	C₄⋊C₄	—	43	8	no	—
203	C₄⋊C₄	—	43	16	no	—
204	C₄⋊C₄	—	43	4	no	—
205	C₂²⋊C₄	—	43	4	no	—
206	C₂²⋊C₄	—	43	8	no	—
207	C₂²⋊C₄	—	43	8	no	—
208	C₂²⋊C₄	—	43	8	no	—
209	C₂²⋊C₄	—	43	16	no	—
210	C₄²	—	43	4	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	4	4	no	—
213	C₂³	—	4	2	no	—
214	C₂³	—	43	4	no	—
215	C₂³	—	86	8	no	—
216	C₂³	—	172	8	no	—
217	C₂³	C₂×C₂²⋊D₄₃	1	8	yes	—
218	C₂³	C₂³×D₄₃	1	2	yes	—
219	C₂³	C₂³×D₄₃	1	1	yes	characteristic
220	C₂³	C₂³×D₄₃	1	4	yes	—
221	Q₈	—	2	8	no	—
222	Q₈	—	86	4	no	—
223	D₄	—	2	16	no	—
224	D₄	—	86	8	no	—
225	D₄	—	86	16	no	—
226	D₄	—	86	4	no	—
227	D₄	—	172	8	no	—
228	D₄	C₂×C₂²⋊D₄₃	1	16	yes	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	16	no	—
232	C₂×C₄	—	2	8	no	—
233	C₂×C₄	—	43	8	no	—
234	C₂×C₄	—	43	4	no	—
235	C₂×C₄	—	86	8	no	—
236	C₂×C₄	—	86	8	no	—
237	C₂×C₄	—	172	8	no	—
238	C₂×C₄	C₂×C₂²⋊D₄₃	1	4	yes	—
239	C₂×C₄	C₂³×D₄₃	1	2	yes	—
240	C₂×C₄	C₂³×D₄₃	1	4	yes	—
241	C₂×C₄	C₂³×D₄₃	1	2	yes	—
242	C₂²	—	2	8	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	8	no	—
247	C₂²	—	4	4	no	—
248	C₂²	—	86	4	no	—
249	C₂²	—	86	8	no	—
250	C₂²	—	344	8	no	—
251	C₂²	C₂²×C₂²⋊D₄₃	1	2	yes	—
252	C₂²	C₂²×C₂²⋊D₄₃	1	8	yes	—
253	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
254	C₄	—	2	2	no	—
255	C₄	—	2	4	no	—
256	C₄	—	86	4	no	—
257	C₄	—	86	8	no	—
258	C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
259	C₂	—	2	2	no	—
260	C₂	—	2	8	no	—
261	C₂	—	4	2	no	—
262	C₂	—	172	4	no	—
263	C₂	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂	#14583735	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₂²×#41×C₄₃	C₂
hypercenter	C₂²×#41	D₄₃
hypocenter	C₄₃	#3485
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×#41

Complements

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

N	H ≅ G/N	type
1. C₂×#13063370	C₂	semidirect
2. #14583548	C₂	semidirect
3. #14583537	C₂	semidirect
4. #14583473	C₂	semidirect
5. #14583458	C₂	semidirect
6. #14583410	C₂	semidirect
7. C₂×#13063308	C₂	semidirect
8. #14583374	C₂	semidirect
9. C₂²×#41×C₄₃	C₂	semidirect
12. C₂×#10621900	C₂²	semidirect
14. C₂×D₄×D₄₃	C₂²	semidirect
15. #13063323	C₂²	semidirect
16. #13063314	C₂²	semidirect
17. #13063308	C₂²	semidirect
18. #13063308	C₂²	semidirect
19. #13063308	C₂²	semidirect
20. #13063306	C₂²	semidirect
21. #13063304	C₂²	semidirect
22. #13063301	C₂²	semidirect
23. #13063299	C₂²	semidirect
24. #13063297	C₂²	semidirect
25. #13063294	C₂²	semidirect
26. #13063293	C₂²	semidirect
27. D₄×Dic₄₃	C₂²	semidirect
28. C₄×C₂²⋊D₄₃	C₂²	semidirect
29. #13063284	C₂²	semidirect
30. C₂×Dic₄₃⋊C₄	C₂²	semidirect
31. C₂×C₂²⋊Dic₄₃	C₂²	semidirect
32. C₂×(C₂×D₄₃)⋊C₄	C₂²	semidirect
34. C₂×#41×C₄₃	C₂²	semidirect
35. C₂²×Q₈×C₄₃	C₂²	semidirect
37. C₂²×D₄×C₄₃	C₂²	semidirect
38. C₂³×C₄×C₄₃	C₂²	semidirect
39. C₂³×C₄×C₄₃	C₂²	semidirect
47. C₂×D₄×C₄₃	D₄	semidirect
134. C₂²×#41	D₄₃	semidirect
142. C₄₃	#3485	semidirect
147. C₂×#41	C₂×D₄₃	semidirect
148. C₂²×Q₈	C₂×D₄₃	semidirect
152. C₂²×D₄	C₂×D₄₃	semidirect
153. C₂³×C₄	C₂×D₄₃	semidirect
154. C₂³×C₄	C₂×D₄₃	semidirect
189. C₂×D₄	C₂²⋊D₄₃	semidirect