Subgroups — #19148230

Order 5504.

1900 conjugacy classes of subgroups.

The subgroup lattice has 290 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	#14583543	C₂	1	8	yes	maximal, complemented
3	D₄×C₂²⋊D₄₃	C₂	1	8	yes	maximal, complemented
4	#14583466	C₂	1	2	yes	maximal, complemented
5	#14583458	C₂	1	4	yes	maximal, complemented
6	#14583410	C₂	1	2	yes	maximal, complemented
7	C₂×#13063294	C₂	1	4	yes	maximal, complemented
8	#14583374	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂³×D₄×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#13063370	—	2	8	no	—
11	C₂×#10621898	C₂²	1	2	yes	complemented
12	C₂×#10621900	C₂²	1	4	yes	complemented
13	C₂²×C₂²⋊D₄₃	C₂²	1	4	yes	complemented
14	C₂²×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
15	C₂×D₄×D₄₃	C₂²	1	4	yes	complemented
16	#13063321	C₂²	1	4	yes	complemented
17	#13063312	C₂²	1	4	yes	complemented
18	#13063308	C₂²	1	4	yes	complemented
19	#13063306	C₂²	1	8	yes	complemented
20	#13063304	C₂²	1	4	yes	complemented
21	#13063299	C₂²	1	4	yes	complemented
22	#13063297	C₂²	1	16	yes	complemented
23	#13063296	C₂²	1	8	yes	complemented
24	#13063294	C₂²	1	4	yes	complemented
25	#13063294	C₂²	1	8	yes	complemented
26	#13063294	C₂²	1	16	yes	complemented
27	#13063293	C₂²	1	8	yes	complemented
28	D₄×Dic₄₃	C₂²	1	8	yes	complemented
29	C₄×C₂²⋊D₄₃	C₂²	1	8	yes	complemented
30	#13063284	C₂²	1	2	yes	complemented
31	C₂×C₂²⋊Dic₄₃	C₂²	1	4	yes	complemented
32	C₂×C₂²⋊Dic₄₃	C₂²	1	8	yes	complemented
33	C₂⁵×C₄₃	C₂²	1	2	yes	complemented
34	C₂²×D₄×C₄₃	—	2	8	no	—
35	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic
36	C₂²×D₄×C₄₃	C₂²	1	2	yes	complemented
37	C₂²×D₄×C₄₃	C₂²	1	1	yes	characteristic, complemented
38	C₂²×D₄×C₄₃	C₂²	1	8	yes	complemented
39	C₂³×C₄×C₄₃	C₂²	1	1	yes	characteristic, complemented
40	C₂⁴×C₄₃	—	2	8	no	—
41	C₂⁴×C₄₃	—	4	8	no	—
42	C₂⁴×C₄₃	C₂³	1	1	yes	characteristic
43	C₂⁴×C₄₃	C₂³	1	8	yes	—
44	C₂⁴×C₄₃	C₂³	1	2	yes	—
45	C₂⁴×C₄₃	C₂³	1	2	yes	—
46	C₂³×D₄₃	C₂³	1	4	yes	complemented
47	C₂×D₄×C₄₃	—	2	32	no	—
48	C₂×D₄×C₄₃	—	2	4	no	—
49	C₂×D₄×C₄₃	—	2	8	no	—
50	C₂×D₄×C₄₃	D₄	1	8	yes	complemented
51	C₂×D₄×C₄₃	C₂³	1	8	yes	—
52	C₂×D₄×C₄₃	C₂³	1	4	yes	—
53	C₂×D₄×C₄₃	C₂³	1	4	yes	—
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	C₂³	1	4	yes	—
56	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₂²⋊D₄₃	—	2	16	no	—
59	C₂×C₂²⋊D₄₃	—	4	16	no	—
60	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	—
61	C₂×C₂²⋊D₄₃	C₂³	1	4	yes	—
62	C₂×C₂²⋊D₄₃	C₂³	1	8	yes	complemented
63	C₂×C₂²⋊D₄₃	C₂³	1	16	yes	complemented
64	C₂²×Dic₄₃	C₂³	1	4	yes	complemented
65	C₂²×Dic₄₃	C₂³	1	8	yes	complemented
66	#10621900	—	2	16	no	—
67	D₄×D₄₃	—	2	16	no	—
68	#10621898	—	2	8	no	—
69	C₂×D₁₇₂	C₂³	1	2	yes	complemented
70	C₂×C₄×D₄₃	C₂³	1	4	yes	complemented
71	C₂×Dic₈₆	C₂³	1	2	yes	complemented
72	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
73	C₂²⋊Dic₄₃	C₂³	1	8	yes	—
74	C₂²⋊Dic₄₃	C₂³	1	8	yes	complemented
75	C₂²⋊Dic₄₃	C₂³	1	16	yes	complemented
76	(C₂×D₄₃)⋊C₄	C₂³	1	8	yes	complemented
77	C₄⋊Dic₄₃	C₂³	1	4	yes	complemented
78	Dic₄₃⋊C₄	C₂³	1	8	yes	complemented
79	C₄×Dic₄₃	C₂³	1	4	yes	complemented
80	C₂³×C₄₃	—	2	2	no	—
81	C₂³×C₄₃	—	2	4	no	—
82	C₂³×C₄₃	—	2	16	no	—
83	C₂³×C₄₃	—	2	4	no	—
84	C₂³×C₄₃	—	2	16	no	—
85	C₂³×C₄₃	—	4	4	no	—
86	C₂³×C₄₃	—	4	8	no	—
87	C₂³×C₄₃	—	4	4	no	—
88	C₂³×C₄₃	—	4	32	no	—
89	C₂³×C₄₃	C₂×D₄	1	8	yes	complemented
90	C₂³×C₄₃	C₂⁴	1	4	yes	—
91	C₂³×C₄₃	C₂⁴	1	1	yes	characteristic
92	C₂³×C₄₃	C₂⁴	1	4	yes	—
93	C₂³×C₄₃	C₂⁴	1	2	yes	—
94	C₂²×D₄₃	—	2	8	no	—
95	C₂²×D₄₃	—	4	8	no	—
96	C₂²×D₄₃	C₂⁴	1	4	yes	—
97	D₄×C₄₃	—	2	16	no	—
98	D₄×C₄₃	—	2	8	no	—
99	D₄×C₄₃	C₂×D₄	1	16	yes	complemented
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	8	no	—
102	C₂×C₄×C₄₃	C₂×D₄	1	4	yes	complemented
103	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
104	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
105	C₂²⋊D₄₃	—	2	8	no	—
106	C₂²⋊D₄₃	—	2	16	no	—
107	C₂²⋊D₄₃	—	2	16	no	—
108	C₂²⋊D₄₃	—	4	16	no	—
109	C₂×Dic₄₃	—	2	8	no	—
110	C₂×Dic₄₃	—	4	8	no	—
111	C₂×Dic₄₃	C₂⁴	1	8	yes	—
112	C₂×Dic₄₃	C₂⁴	1	4	yes	—
113	D₁₇₂	—	2	4	no	—
114	C₄×D₄₃	—	2	8	no	—
115	Dic₈₆	—	2	4	no	—
116	C₂²×C₄₃	—	2	8	no	—
117	C₂²×C₄₃	—	2	2	no	—
118	C₂²×C₄₃	—	2	8	no	—
119	C₂²×C₄₃	—	2	2	no	—
120	C₂²×C₄₃	—	2	2	no	—
121	C₂²×C₄₃	—	2	4	no	—
122	C₂²×C₄₃	—	2	8	no	—
123	C₂²×C₄₃	—	4	8	no	—
124	C₂²×C₄₃	—	4	32	no	—
125	C₂²×C₄₃	—	4	8	no	—
126	C₂²×C₄₃	C₂²×D₄	1	8	yes	—
127	C₂²×C₄₃	C₂²×D₄	1	2	yes	—
128	C₂²×C₄₃	H(5,𝔽₂)	1	4	yes	complemented
129	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
130	C₂×D₄₃	—	2	4	no	—
131	C₂×D₄₃	—	2	8	no	—
132	C₂×D₄₃	—	8	8	no	—
133	C₄×C₄₃	—	2	2	no	—
134	C₄×C₄₃	C₂²×D₄	1	4	yes	—
135	Dic₄₃	—	2	4	no	—
136	Dic₄₃	—	2	8	no	—
137	#3484	—	43	1	no	maximal, Sylow
138	C₂×C₄₃	—	2	8	no	—
139	C₂×C₄₃	—	2	2	no	—
140	C₂×C₄₃	—	2	4	no	—
141	C₂×C₄₃	—	4	4	no	—
142	C₂×C₄₃	C₂³×D₄	1	1	yes	characteristic
143	C₂×C₄₃	C₂×H(5,𝔽₂)	1	2	yes	—
144	D₄₃	—	4	4	no	—
145	C₂×H(5,𝔽₂)	—	43	1	no	—
146	C₂³×D₄	D₄₃	1	1	yes	characteristic, complemented
147	#546	—	43	8	no	—
148	D₄²	—	43	8	no	—
149	#535	—	43	2	no	—
150	#534	—	43	4	no	—
151	#525	—	43	2	no	—
152	C₂×#120	—	43	4	no	—
153	#518	—	43	1	no	—
154	C₄₃	#3484	1	1	yes	characteristic, complemented, Sylow
155	C₂⁵	C₂×D₄₃	1	2	yes	complemented
156	H(5,𝔽₂)	—	86	8	no	—
157	C₂×#41	—	43	4	no	—
158	C₂×#41	—	43	2	no	—
159	C₂²×D₄	—	2	8	no	—
160	C₂²×D₄	—	43	4	no	—
161	C₂²×D₄	—	43	8	no	—
162	C₂²×D₄	—	43	4	no	—
163	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic
164	C₂²×D₄	C₂×D₄₃	1	2	yes	complemented
165	C₂²×D₄	C₂×D₄₃	1	1	yes	characteristic, complemented
166	C₂²×D₄	C₂×D₄₃	1	8	yes	complemented
167	C₂³×C₄	C₂×D₄₃	1	1	yes	characteristic, complemented
168	#127	—	43	4	no	—
169	#124	—	43	4	no	—
170	#123	—	43	8	no	—
171	#123	—	43	4	no	—
172	C₂²⋊Q₈	—	43	4	no	—
173	#121	—	43	16	no	—
174	#121	—	43	8	no	—
175	#121	—	43	4	no	—
176	#120	—	43	4	no	—
177	#120	—	43	8	no	—
178	#120	—	43	16	no	—
179	#120	—	43	8	no	—
180	C₄×D₄	—	43	8	no	—
181	C₄×D₄	—	43	8	no	—
182	#117	—	43	2	no	—
183	C₂×C₂²⋊C₄	—	43	4	no	—
184	C₂×C₂²⋊C₄	—	43	8	no	—
185	C₂⁴	—	2	8	no	—
186	C₂⁴	—	4	8	no	—
187	C₂⁴	—	43	4	no	—
188	C₂⁴	C₂²×D₄₃	1	1	yes	characteristic
189	C₂⁴	C₂²×D₄₃	1	8	yes	—
190	C₂⁴	C₂²×D₄₃	1	2	yes	—
191	C₂⁴	C₂²×D₄₃	1	2	yes	—
192	#41	—	86	16	no	—
193	#41	—	86	8	no	—
194	C₂×Q₈	—	43	2	no	—
195	C₂×D₄	—	2	8	no	—
196	C₂×D₄	—	2	32	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	43	8	no	—
199	C₂×D₄	—	43	4	no	—
200	C₂×D₄	—	43	8	no	—
201	C₂×D₄	—	43	16	no	—
202	C₂×D₄	—	43	2	no	—
203	C₂×D₄	—	86	16	no	—
204	C₂×D₄	—	86	16	no	—
205	C₂×D₄	—	172	16	no	—
206	C₂×D₄	C₂²⋊D₄₃	1	8	yes	complemented
207	C₂×D₄	C₂²×D₄₃	1	8	yes	—
208	C₂×D₄	C₂²×D₄₃	1	4	yes	—
209	C₂×D₄	C₂²×D₄₃	1	4	yes	—
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	43	4	no	—
212	C₂²×C₄	—	43	8	no	—
213	C₂²×C₄	—	43	4	no	—
214	C₂²×C₄	C₂²×D₄₃	1	4	yes	—
215	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
217	C₄⋊C₄	—	43	4	no	—
218	C₄⋊C₄	—	43	8	no	—
219	C₂²⋊C₄	—	43	4	no	—
220	C₂²⋊C₄	—	43	8	no	—
221	C₂²⋊C₄	—	43	8	no	—
222	C₂²⋊C₄	—	43	16	no	—
223	C₂²⋊C₄	—	43	8	no	—
224	C₄²	—	43	4	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	4	no	—
227	C₂³	—	2	16	no	—
228	C₂³	—	2	4	no	—
229	C₂³	—	2	16	no	—
230	C₂³	—	4	4	no	—
231	C₂³	—	4	8	no	—
232	C₂³	—	4	4	no	—
233	C₂³	—	4	32	no	—
234	C₂³	—	43	4	no	—
235	C₂³	—	86	8	no	—
236	C₂³	—	172	8	no	—
237	C₂³	C₂×C₂²⋊D₄₃	1	8	yes	complemented
238	C₂³	C₂³×D₄₃	1	4	yes	—
239	C₂³	C₂³×D₄₃	1	1	yes	characteristic
240	C₂³	C₂³×D₄₃	1	4	yes	—
241	C₂³	C₂³×D₄₃	1	2	yes	—
242	Q₈	—	86	4	no	—
243	D₄	—	2	16	no	—
244	D₄	—	2	8	no	—
245	D₄	—	86	8	no	—
246	D₄	—	86	16	no	—
247	D₄	—	86	16	no	—
248	D₄	—	86	4	no	—
249	D₄	—	172	16	no	—
250	D₄	C₂×C₂²⋊D₄₃	1	16	yes	complemented
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	43	8	no	—
254	C₂×C₄	—	43	4	no	—
255	C₂×C₄	—	86	8	no	—
256	C₂×C₄	—	86	8	no	—
257	C₂×C₄	—	172	8	no	—
258	C₂×C₄	C₂×C₂²⋊D₄₃	1	4	yes	complemented
259	C₂×C₄	C₂³×D₄₃	1	2	yes	—
260	C₂×C₄	C₂³×D₄₃	1	2	yes	—
261	C₂²	—	2	8	no	—
262	C₂²	—	2	2	no	—
263	C₂²	—	2	8	no	—
264	C₂²	—	2	2	no	—
265	C₂²	—	2	2	no	—
266	C₂²	—	2	4	no	—
267	C₂²	—	2	8	no	—
268	C₂²	—	4	8	no	—
269	C₂²	—	4	32	no	—
270	C₂²	—	4	8	no	—
271	C₂²	—	86	4	no	—
272	C₂²	—	86	8	no	—
273	C₂²	—	344	8	no	—
274	C₂²	C₂²×C₂²⋊D₄₃	1	2	yes	—
275	C₂²	C₂²×C₂²⋊D₄₃	1	8	yes	—
276	C₂²	#13063370	1	4	yes	complemented
277	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
278	C₄	—	2	2	no	—
279	C₄	—	86	4	no	—
280	C₄	—	86	8	no	—
281	C₄	C₂²×C₂²⋊D₄₃	1	4	yes	—
282	C₂	—	2	2	no	—
283	C₂	—	2	4	no	—
284	C₂	—	2	8	no	—
285	C₂	—	4	4	no	—
286	C₂	—	172	4	no	—
287	C₂	C₂³×C₂²⋊D₄₃	1	1	yes	characteristic
288	C₂	C₂×#13063370	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₂³×D₄×C₄₃	C₂
hypercenter	C₂³×D₄	D₄₃
hypocenter	C₄₃	#3484
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂³×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#13063370	C₂	semidirect
2. #14583543	C₂	semidirect
3. D₄×C₂²⋊D₄₃	C₂	semidirect
4. #14583466	C₂	semidirect
5. #14583458	C₂	semidirect
6. #14583410	C₂	semidirect
7. C₂×#13063294	C₂	semidirect
8. #14583374	C₂	semidirect
9. C₂³×D₄×C₄₃	C₂	semidirect
11. C₂×#10621898	C₂²	semidirect
12. C₂×#10621900	C₂²	semidirect
13. C₂²×C₂²⋊D₄₃	C₂²	semidirect
14. C₂²×C₂²⋊D₄₃	C₂²	semidirect
15. C₂×D₄×D₄₃	C₂²	semidirect
16. #13063321	C₂²	semidirect
17. #13063312	C₂²	semidirect
18. #13063308	C₂²	semidirect
19. #13063306	C₂²	semidirect
20. #13063304	C₂²	semidirect
21. #13063299	C₂²	semidirect
22. #13063297	C₂²	semidirect
23. #13063296	C₂²	semidirect
24. #13063294	C₂²	semidirect
25. #13063294	C₂²	semidirect
26. #13063294	C₂²	semidirect
27. #13063293	C₂²	semidirect
28. D₄×Dic₄₃	C₂²	semidirect
29. C₄×C₂²⋊D₄₃	C₂²	semidirect
30. #13063284	C₂²	semidirect
31. C₂×C₂²⋊Dic₄₃	C₂²	semidirect
32. C₂×C₂²⋊Dic₄₃	C₂²	semidirect
33. C₂⁵×C₄₃	C₂²	semidirect
36. C₂²×D₄×C₄₃	C₂²	semidirect
37. C₂²×D₄×C₄₃	C₂²	semidirect
38. C₂²×D₄×C₄₃	C₂²	semidirect
39. C₂³×C₄×C₄₃	C₂²	semidirect
46. C₂³×D₄₃	C₂³	semidirect
50. C₂×D₄×C₄₃	D₄	semidirect
62. C₂×C₂²⋊D₄₃	C₂³	semidirect
63. C₂×C₂²⋊D₄₃	C₂³	semidirect
64. C₂²×Dic₄₃	C₂³	semidirect
65. C₂²×Dic₄₃	C₂³	semidirect
69. C₂×D₁₇₂	C₂³	semidirect
70. C₂×C₄×D₄₃	C₂³	semidirect
71. C₂×Dic₈₆	C₂³	semidirect
74. C₂²⋊Dic₄₃	C₂³	semidirect
75. C₂²⋊Dic₄₃	C₂³	semidirect
76. (C₂×D₄₃)⋊C₄	C₂³	semidirect
77. C₄⋊Dic₄₃	C₂³	semidirect
78. Dic₄₃⋊C₄	C₂³	semidirect
79. C₄×Dic₄₃	C₂³	semidirect
89. C₂³×C₄₃	C₂×D₄	semidirect
99. D₄×C₄₃	C₂×D₄	semidirect
102. C₂×C₄×C₄₃	C₂×D₄	semidirect
128. C₂²×C₄₃	H(5,𝔽₂)	semidirect
146. C₂³×D₄	D₄₃	semidirect
154. C₄₃	#3484	semidirect
155. C₂⁵	C₂×D₄₃	semidirect
164. C₂²×D₄	C₂×D₄₃	semidirect
165. C₂²×D₄	C₂×D₄₃	semidirect
166. C₂²×D₄	C₂×D₄₃	semidirect
167. C₂³×C₄	C₂×D₄₃	semidirect
206. C₂×D₄	C₂²⋊D₄₃	semidirect
237. C₂³	C₂×C₂²⋊D₄₃	semidirect
250. D₄	C₂×C₂²⋊D₄₃	semidirect
258. C₂×C₄	C₂×C₂²⋊D₄₃	semidirect
276. C₂²	#13063370	semidirect