Subgroups — #19149366

Order 5504.

1424 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583735	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583714	C₂	1	4	yes	maximal, complemented
3	#14583710	C₂	1	4	yes	maximal, complemented
4	#14583707	C₂	1	4	yes	maximal, complemented
5	#14583702	C₂	1	4	yes	maximal, complemented
6	#14583697	C₂	1	6	yes	maximal, complemented
7	C₂×#13063345	C₂	1	6	yes	maximal, complemented
8	C₂×#13063332	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#585	C₂	1	1	yes	maximal, characteristic, complemented
10	#13063373	—	2	1	no	—
11	#13063372	C₂²	1	4	yes	complemented
12	#13063371	C₂²	1	4	yes	complemented
13	#13063370	—	2	3	no	—
14	C₂×#10621898	C₂²	1	6	yes	complemented
15	#41×D₄₃	—	2	4	no	—
16	#13063356	C₂²	1	12	yes	complemented
17	#13063355	C₂²	1	12	yes	complemented
18	#13063351	C₂²	1	12	yes	complemented
19	#13063350	C₂²	1	12	yes	complemented
20	#13063345	C₂²	1	4	yes	complemented
21	#13063345	C₂²	1	12	yes	complemented
22	#13063345	C₂²	1	12	yes	complemented
23	#13063345	C₂²	1	12	yes	complemented
24	C₂×#10621885	C₂²	1	6	yes	complemented
25	C₂×#10621883	C₂²	1	6	yes	complemented
26	C₂×#10621884	C₂²	1	6	yes	complemented
27	C₂×#10621882	C₂²	1	6	yes	complemented
28	#13063332	C₂²	1	4	yes	complemented
29	#13063332	C₂²	1	4	yes	complemented
30	C₂×#10621877	C₂²	1	3	yes	complemented
31	C₂²×C₄₃⋊C₈	C₂²	1	3	yes	complemented
32	2₋¹⁺⁴×C₄₃	—	2	1	no	—
33	2₋¹⁺⁴×C₄₃	C₂²	1	4	yes	complemented
34	H(5,𝔽₂)×C₄₃	—	2	3	no	—
35	H(5,𝔽₂)×C₄₃	C₂²	1	4	yes	complemented
36	C₂×#41×C₄₃	—	2	4	no	—
37	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
38	C₂×#41×C₄₃	C₂²	1	6	yes	complemented
39	#41×C₄₃	—	2	4	no	—
40	#41×C₄₃	—	2	12	no	—
41	#41×C₄₃	—	2	6	no	—
42	#41×C₄₃	—	2	6	no	—
43	#41×C₄₃	D₄	1	4	yes	—
44	#41×C₄₃	C₂³	1	4	yes	—
45	#41×C₄₃	C₂³	1	12	yes	—
46	#41×C₄₃	C₂³	1	4	yes	—
47	C₂×Q₈×C₄₃	—	2	4	no	—
48	C₂×Q₈×C₄₃	D₄	1	1	yes	characteristic
49	C₂×Q₈×C₄₃	C₂³	1	6	yes	—
50	C₂×D₄×C₄₃	—	2	12	no	—
51	C₂×D₄×C₄₃	—	2	6	no	—
52	C₂×D₄×C₄₃	D₄	1	3	yes	—
53	C₂×D₄×C₄₃	C₂³	1	6	yes	—
54	C₂²×C₄×C₄₃	—	2	6	no	—
55	C₂²×C₄×C₄₃	C₂³	1	3	yes	—
56	C₂×C₂²⋊D₄₃	—	2	6	no	—
57	#10621902	—	2	4	no	—
58	Q₈×D₄₃	—	2	4	no	—
59	#10621900	—	2	12	no	—
60	D₄×D₄₃	—	2	12	no	—
61	#10621898	—	2	6	no	—
62	#10621898	—	2	6	no	—
63	#10621898	C₂³	1	4	yes	—
64	#10621898	C₂³	1	12	yes	—
65	C₂×D₁₇₂	C₂³	1	6	yes	—
66	C₂×C₄×D₄₃	—	2	6	no	—
67	C₂×Dic₈₆	C₂³	1	6	yes	—
68	#10621885	C₂³	1	4	yes	—
69	#10621885	C₂³	1	12	yes	—
70	#10621884	C₂³	1	4	yes	—
71	#10621884	C₂³	1	12	yes	—
72	#10621883	C₂³	1	4	yes	—
73	#10621883	C₂³	1	12	yes	—
74	#10621882	C₂³	1	4	yes	—
75	#10621882	C₂³	1	12	yes	—
76	#10621877	C₂³	1	12	yes	—
77	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₂×C₄₃⋊C₈	C₂³	1	3	yes	—
79	C₂×C₄₃⋊C₈	C₂³	1	6	yes	—
80	C₂×C₄₃⋊C₈	C₂³	1	6	yes	—
81	C₂³×C₄₃	—	2	6	no	—
82	C₂³×C₄₃	C₂×D₄	1	3	yes	—
83	C₂²×D₄₃	—	2	6	no	—
84	Q₈×C₄₃	—	2	6	no	—
85	Q₈×C₄₃	C₂×D₄	1	4	yes	—
86	Q₈×C₄₃	C₂⁴	1	4	yes	—
87	D₄×C₄₃	—	2	12	no	—
88	D₄×C₄₃	—	2	4	no	—
89	D₄×C₄₃	—	2	6	no	—
90	D₄×C₄₃	C₂×D₄	1	12	yes	—
91	D₄×C₄₃	C₂⁴	1	4	yes	—
92	C₂×C₄×C₄₃	—	2	6	no	—
93	C₂×C₄×C₄₃	—	2	4	no	—
94	C₂×C₄×C₄₃	—	2	12	no	—
95	C₂×C₄×C₄₃	C₂×D₄	1	6	yes	—
96	C₂×C₄×C₄₃	C₂×D₄	1	3	yes	—
97	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂⁴	1	6	yes	—
99	C₂²⋊D₄₃	—	2	4	no	—
100	C₂²⋊D₄₃	—	2	12	no	—
101	C₂×Dic₄₃	—	2	6	no	—
102	D₁₇₂	—	2	6	no	—
103	D₁₇₂	C₂⁴	1	4	yes	—
104	C₄×D₄₃	—	2	4	no	—
105	C₄×D₄₃	—	2	12	no	—
106	Dic₈₆	—	2	6	no	—
107	Dic₈₆	C₂⁴	1	4	yes	—
108	C₄₃⋊C₈	C₂⁴	1	1	yes	characteristic
109	C₄₃⋊C₈	C₂⁴	1	1	yes	characteristic
110	C₄₃⋊C₈	C₂⁴	1	6	yes	—
111	C₂²×C₄₃	—	2	4	no	—
112	C₂²×C₄₃	—	4	3	no	—
113	C₂²×C₄₃	—	8	6	no	—
114	C₂²×C₄₃	C₂²×D₄	1	6	yes	—
115	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₄₃	—	2	4	no	—
117	C₂×D₄₃	—	8	6	no	—
118	C₄×C₄₃	—	2	4	no	—
119	C₄×C₄₃	C₂²×D₄	1	6	yes	—
120	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
121	C₄×C₄₃	C₂⁵	1	1	yes	characteristic
122	Dic₄₃	—	2	4	no	—
123	#3584	—	43	1	no	maximal, Sylow
124	C₂×C₄₃	—	2	6	no	—
125	C₂×C₄₃	—	2	1	no	—
126	C₂×C₄₃	—	4	4	no	—
127	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
128	D₄₃	—	4	4	no	—
129	#585	—	43	1	no	—
130	#585	D₄₃	1	1	yes	characteristic, complemented
131	#578	—	43	4	no	—
132	#577	—	43	4	no	—
133	#577	—	43	4	no	—
134	#576	—	43	4	no	—
135	#575	—	43	6	no	—
136	C₂×#135	—	43	6	no	—
137	C₂×#131	—	43	1	no	—
138	C₄₃	#3584	1	1	yes	characteristic, complemented, Sylow
139	2₋¹⁺⁴	—	2	1	no	—
140	2₋¹⁺⁴	—	43	4	no	—
141	2₋¹⁺⁴	—	86	1	no	—
142	2₋¹⁺⁴	C₂×D₄₃	1	4	yes	complemented
143	H(5,𝔽₂)	—	2	3	no	—
144	H(5,𝔽₂)	—	43	4	no	—
145	H(5,𝔽₂)	—	86	3	no	—
146	H(5,𝔽₂)	C₂×D₄₃	1	4	yes	complemented
147	C₂×#41	—	2	4	no	—
148	C₂×#41	—	43	6	no	—
149	C₂×#41	—	86	4	no	—
150	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
151	C₂×#41	C₂×D₄₃	1	6	yes	complemented
152	#137	—	43	12	no	—
153	#137	—	43	12	no	—
154	AGL(1,ℤ/8)	—	43	12	no	—
155	AGL(1,ℤ/8)	—	43	12	no	—
156	#135	—	43	4	no	—
157	#135	—	43	12	no	—
158	#135	—	43	12	no	—
159	#135	—	43	12	no	—
160	C₂×Q₁₆	—	43	6	no	—
161	C₂×SD₁₆	—	43	6	no	—
162	C₂×SD₁₆	—	43	6	no	—
163	C₂×D₈	—	43	6	no	—
164	#131	—	43	4	no	—
165	#131	—	43	4	no	—
166	C₂×M₁₆	—	43	3	no	—
167	C₂²×C₈	—	43	3	no	—
168	#41	—	2	4	no	—
169	#41	—	2	12	no	—
170	#41	—	2	6	no	—
171	#41	—	2	6	no	—
172	#41	—	43	4	no	—
173	#41	—	43	12	no	—
174	#41	—	86	4	no	—
175	#41	—	86	12	no	—
176	#41	—	86	6	no	—
177	#41	—	86	6	no	—
178	#41	C₂²⋊D₄₃	1	4	yes	—
179	#41	C₂²×D₄₃	1	4	yes	—
180	#41	C₂²×D₄₃	1	12	yes	—
181	#41	C₂²×D₄₃	1	4	yes	—
182	C₂×Q₈	—	2	4	no	—
183	C₂×Q₈	—	43	6	no	—
184	C₂×Q₈	—	86	4	no	—
185	C₂×Q₈	C₂²⋊D₄₃	1	1	yes	characteristic
186	C₂×Q₈	C₂²×D₄₃	1	6	yes	—
187	C₂×D₄	—	2	6	no	—
188	C₂×D₄	—	2	12	no	—
189	C₂×D₄	—	43	6	no	—
190	C₂×D₄	—	86	6	no	—
191	C₂×D₄	—	86	12	no	—
192	C₂×D₄	C₂²⋊D₄₃	1	3	yes	—
193	C₂×D₄	C₂²×D₄₃	1	6	yes	—
194	C₂²×C₄	—	2	6	no	—
195	C₂²×C₄	—	86	6	no	—
196	C₂²×C₄	C₂²×D₄₃	1	3	yes	—
197	Q₁₆	—	43	4	no	—
198	Q₁₆	—	43	12	no	—
199	SD₁₆	—	43	4	no	—
200	SD₁₆	—	43	12	no	—
201	SD₁₆	—	43	4	no	—
202	SD₁₆	—	43	12	no	—
203	D₈	—	43	4	no	—
204	D₈	—	43	12	no	—
205	M₁₆	—	43	12	no	—
206	C₂×C₈	—	43	1	no	—
207	C₂×C₈	—	43	3	no	—
208	C₂×C₈	—	43	6	no	—
209	C₂×C₈	—	43	6	no	—
210	C₂³	—	2	6	no	—
211	C₂³	—	86	6	no	—
212	C₂³	C₂×C₂²⋊D₄₃	1	3	yes	—
213	Q₈	—	2	6	no	—
214	Q₈	—	43	4	no	—
215	Q₈	—	86	6	no	—
216	Q₈	C₂×C₂²⋊D₄₃	1	4	yes	—
217	Q₈	C₂³×D₄₃	1	4	yes	—
218	D₄	—	2	12	no	—
219	D₄	—	2	4	no	—
220	D₄	—	2	6	no	—
221	D₄	—	43	4	no	—
222	D₄	—	86	12	no	—
223	D₄	—	86	4	no	—
224	D₄	—	86	6	no	—
225	D₄	C₂×C₂²⋊D₄₃	1	12	yes	—
226	D₄	C₂³×D₄₃	1	4	yes	—
227	C₂×C₄	—	2	6	no	—
228	C₂×C₄	—	2	12	no	—
229	C₂×C₄	—	2	4	no	—
230	C₂×C₄	—	86	6	no	—
231	C₂×C₄	—	86	12	no	—
232	C₂×C₄	—	86	4	no	—
233	C₂×C₄	C₂×C₂²⋊D₄₃	1	3	yes	—
234	C₂×C₄	C₂×C₂²⋊D₄₃	1	6	yes	—
235	C₂×C₄	C₂³×D₄₃	1	6	yes	—
236	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
237	C₈	—	43	1	no	—
238	C₈	—	43	1	no	—
239	C₈	—	43	6	no	—
240	C₂²	—	2	4	no	—
241	C₂²	—	4	3	no	—
242	C₂²	—	8	6	no	—
243	C₂²	—	86	4	no	—
244	C₂²	—	344	6	no	—
245	C₂²	C₂²×C₂²⋊D₄₃	1	6	yes	—
246	C₂²	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	86	4	no	—
249	C₄	C₂²×C₂²⋊D₄₃	1	6	yes	—
250	C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
251	C₄	C₂⁴×D₄₃	1	1	yes	characteristic
252	C₂	—	2	1	no	—
253	C₂	—	2	6	no	—
254	C₂	—	4	4	no	—
255	C₂	—	172	4	no	—
256	C₂	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂²×C₂²⋊D₄₃
commutator	C₄×C₄₃	C₂⁵
Frattini	C₄	C₂⁴×D₄₃
Fitting	C₄₃×#585	C₂
socle	C₂×C₄₃	C₂³×D₄
hypercenter	#585	D₄₃
hypocenter	C₄₃	#3584
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂×#41 ◁ #585

Complements

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

N	H ≅ G/N	type
1. #14583735	C₂	semidirect
2. #14583714	C₂	semidirect
3. #14583710	C₂	semidirect
4. #14583707	C₂	semidirect
5. #14583702	C₂	semidirect
6. #14583697	C₂	semidirect
7. C₂×#13063345	C₂	semidirect
8. C₂×#13063332	C₂	semidirect
9. C₄₃×#585	C₂	semidirect
11. #13063372	C₂²	semidirect
12. #13063371	C₂²	semidirect
14. C₂×#10621898	C₂²	semidirect
16. #13063356	C₂²	semidirect
17. #13063355	C₂²	semidirect
18. #13063351	C₂²	semidirect
19. #13063350	C₂²	semidirect
20. #13063345	C₂²	semidirect
21. #13063345	C₂²	semidirect
22. #13063345	C₂²	semidirect
23. #13063345	C₂²	semidirect
24. C₂×#10621885	C₂²	semidirect
25. C₂×#10621883	C₂²	semidirect
26. C₂×#10621884	C₂²	semidirect
27. C₂×#10621882	C₂²	semidirect
28. #13063332	C₂²	semidirect
29. #13063332	C₂²	semidirect
30. C₂×#10621877	C₂²	semidirect
31. C₂²×C₄₃⋊C₈	C₂²	semidirect
33. 2₋¹⁺⁴×C₄₃	C₂²	semidirect
35. H(5,𝔽₂)×C₄₃	C₂²	semidirect
38. C₂×#41×C₄₃	C₂²	semidirect
130. #585	D₄₃	semidirect
138. C₄₃	#3584	semidirect
142. 2₋¹⁺⁴	C₂×D₄₃	semidirect
146. H(5,𝔽₂)	C₂×D₄₃	semidirect
151. C₂×#41	C₂×D₄₃	semidirect