Subgroups — #19143413

Order 5504.

484 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063358	C₂	1	1	yes	maximal, characteristic
2	#14583367	C₂	1	1	yes	maximal, characteristic
3	C₂×#13063290	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583005	C₂	1	2	yes	maximal, complemented
5	#14582998	C₂	1	2	yes	maximal, complemented
6	#14582993	C₂	1	2	yes	maximal, complemented
7	#14582986	C₂	1	2	yes	maximal, complemented
8	#14582907	C₂	1	1	yes	maximal, characteristic, complemented
9	#14582898	C₂	1	1	yes	maximal, characteristic
10	M₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#428	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	1	yes	characteristic
13	C₂²×Dic₈₆	C₂²	1	1	yes	characteristic
14	#13063358	C₄	1	8	yes	complemented
15	C₂×Dic₁₇₂	C₂²	1	2	yes	—
16	C₂×#10621873	C₂²	1	2	yes	—
17	C₂×#10621877	C₂²	1	1	yes	characteristic
18	#13063290	—	2	1	no	—
19	#13063290	—	2	1	no	—
20	#13063290	C₂²	1	2	yes	—
21	#13063290	C₂²	1	2	yes	—
22	#13063290	C₂²	1	1	yes	characteristic
23	#13063290	C₂²	1	1	yes	characteristic
24	#13063287	C₂²	1	1	yes	characteristic
25	#13063287	C₂²	1	1	yes	characteristic
26	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
27	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
28	#13063282	—	2	1	no	—
29	C₂×Dic₄₃⋊C₄	—	2	1	no	—
30	C₂×C₄×Dic₄₃	—	2	1	no	—
31	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
32	C₄²×D₄₃	—	2	1	no	—
33	#13063248	C₂²	1	2	yes	—
34	#13063248	C₂²	1	2	yes	—
35	#13063248	C₂²	1	2	yes	—
36	#13063245	C₂²	1	2	yes	—
37	#13063235	C₂²	1	2	yes	—
38	C₈×Dic₄₃	C₂²	1	2	yes	—
39	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
40	#117×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
42	#107×C₄₃	C₂²	1	2	yes	—
43	#106×C₄₃	C₂²	1	2	yes	—
44	C₂²×C₄×C₄₃	—	2	1	no	—
45	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₄₃	—	2	1	no	—
47	C₂²×Dic₄₃	—	2	1	no	—
48	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
49	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
50	#10621898	—	2	2	no	—
51	#10621898	C₂×C₄	1	4	yes	—
52	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
53	C₂×C₄×D₄₃	—	2	1	no	—
54	C₂×C₄×D₄₃	—	2	1	no	—
55	C₂×C₄×D₄₃	—	2	1	no	—
56	C₂×Dic₈₆	—	2	1	no	—
57	C₂×Dic₈₆	—	2	2	no	—
58	C₂×Dic₈₆	C₂×C₄	1	4	yes	complemented
59	C₂×Dic₈₆	C₂³	1	2	yes	—
60	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
61	M₁₆×C₄₃	C₂×C₄	1	4	yes	complemented
62	C₂×C₈×C₄₃	C₂³	1	2	yes	—
63	C₄⋊C₄×C₄₃	—	2	1	no	—
64	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
65	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
66	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
67	C₂²⋊C₄×C₄₃	—	2	1	no	—
68	C₄²×C₄₃	—	2	1	no	—
69	(C₂×D₄₃)⋊C₄	—	2	1	no	—
70	(C₂×D₄₃)⋊C₄	—	2	1	no	—
71	Dic₄₃⋊C₄	—	2	2	no	—
72	Dic₄₃⋊C₄	—	2	2	no	—
73	Dic₄₃⋊C₄	—	2	1	no	—
74	Dic₄₃⋊C₄	—	2	1	no	—
75	C₄×Dic₄₃	—	2	1	no	—
76	C₄×Dic₄₃	—	2	1	no	—
77	C₄×Dic₄₃	—	2	2	no	—
78	C₄×Dic₄₃	—	2	2	no	—
79	C₄×Dic₄₃	D₄	1	1	yes	characteristic
80	C₄×Dic₄₃	D₄	1	1	yes	characteristic
81	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
82	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
83	#10621877	—	2	2	no	—
84	C₂×C₄₃⋊C₈	C₂³	1	2	yes	—
85	Dic₁₇₂	C₂×C₄	1	8	yes	complemented
86	#10621873	C₂×C₄	1	8	yes	complemented
87	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×D₄₃	—	2	1	no	—
89	C₂×C₄×C₄₃	—	2	2	no	—
90	C₂×C₄×C₄₃	—	2	1	no	—
91	C₂×C₄×C₄₃	—	2	1	no	—
92	C₂×C₄×C₄₃	—	4	2	no	—
93	C₂×C₄×C₄₃	C₂²×C₄	1	2	yes	—
94	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₄×C₄₃	#41	1	2	yes	—
96	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
97	C₂²⋊D₄₃	—	2	4	no	—
98	C₂×Dic₄₃	—	2	4	no	—
99	C₂×Dic₄₃	—	2	2	no	—
100	C₂×Dic₄₃	—	2	1	no	—
101	C₂×Dic₄₃	—	2	2	no	—
102	C₂×Dic₄₃	—	4	1	no	—
103	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
105	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
107	D₁₇₂	—	2	1	no	—
108	D₁₇₂	C₂²×C₄	1	2	yes	—
109	C₄×D₄₃	—	2	4	no	—
110	C₄×D₄₃	—	4	2	no	—
111	C₄×D₄₃	—	4	2	no	—
112	Dic₈₆	—	2	4	no	—
113	Dic₈₆	—	2	2	no	—
114	Dic₈₆	—	2	1	no	—
115	Dic₈₆	C₂²×C₄	1	4	yes	—
116	Dic₈₆	C₂²×C₄	1	2	yes	—
117	C₈×C₄₃	C₂²×C₄	1	4	yes	—
118	C₄₃⋊C₈	—	2	2	no	—
119	C₂²×C₄₃	—	2	1	no	—
120	C₂²×C₄₃	—	2	1	no	—
121	C₂²×C₄₃	C₄×D₄	1	2	yes	—
122	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×D₄₃	—	2	2	no	—
124	C₂×D₄₃	—	4	1	no	—
125	C₂×D₄₃	—	4	1	no	—
126	C₄×C₄₃	—	4	1	no	—
127	C₄×C₄₃	—	4	1	no	—
128	C₄×C₄₃	—	4	2	no	—
129	C₄×C₄₃	C₄×D₄	1	2	yes	—
130	C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
131	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
132	Dic₄₃	—	2	4	no	—
133	Dic₄₃	—	2	2	no	—
134	Dic₄₃	—	2	1	no	—
135	Dic₄₃	—	2	1	no	—
136	Dic₄₃	—	2	1	no	—
137	Dic₄₃	#137	1	2	yes	complemented
138	C₄×#137	—	43	1	no	maximal, Sylow
139	C₂×C₄₃	—	2	2	no	—
140	C₂×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
141	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
142	C₂×C₄₃	#575	1	1	yes	characteristic
143	D₄₃	—	4	2	no	—
144	C₂×#137	—	43	1	no	—
145	C₄×#41	—	43	1	no	—
146	C₂×C₄×Q₈	—	43	1	no	—
147	#441	—	43	2	no	—
148	#440	—	43	2	no	—
149	C₄×Q₁₆	—	43	2	no	—
150	C₄×SD₁₆	—	43	2	no	—
151	#428	D₄₃	1	1	yes	characteristic, complemented
152	#419	—	43	1	no	—
153	#417	—	43	1	no	—
154	C₄×M₁₆	—	43	1	no	—
155	C₄₃	C₄×#137	1	1	yes	characteristic, complemented, Sylow
156	C₂×#41	—	43	1	no	—
157	C₂²×Q₈	—	43	1	no	—
158	#137	—	43	8	no	—
159	C₂×Q₁₆	—	43	2	no	—
160	C₂×SD₁₆	—	43	2	no	—
161	C₂×M₁₆	—	43	1	no	—
162	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
163	C₄×Q₈	—	43	2	no	—
164	C₄×Q₈	—	43	2	no	—
165	C₄×Q₈	—	43	1	no	—
166	C₄×Q₈	—	43	1	no	—
167	C₄×Q₈	—	86	1	no	—
168	C₄×Q₈	—	86	1	no	—
169	C₄×D₄	—	43	1	no	—
170	C₄×D₄	—	43	1	no	—
171	C₄×D₄	—	86	1	no	—
172	C₄×D₄	—	86	1	no	—
173	#117	—	86	1	no	—
174	#117	C₂×D₄₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	86	1	no	—
176	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
177	C₂×C₄²	—	43	1	no	—
178	C₂×C₄²	—	86	1	no	—
179	C₂×C₄²	—	86	1	no	—
180	#107	C₂×D₄₃	1	2	yes	—
181	#106	C₂×D₄₃	1	2	yes	—
182	#103	—	43	2	no	—
183	#103	—	43	2	no	—
184	#103	—	43	2	no	—
185	#102	—	43	2	no	—
186	#97	—	43	2	no	—
187	C₄×C₈	—	43	2	no	—
188	#41	—	43	4	no	—
189	#41	—	86	2	no	—
190	C₂×Q₈	—	43	4	no	—
191	C₂×Q₈	—	43	2	no	—
192	C₂×Q₈	—	43	1	no	—
193	C₂×Q₈	—	86	1	no	—
194	C₂×Q₈	—	86	2	no	—
195	C₂×D₄	—	43	1	no	—
196	C₂×D₄	—	86	1	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	43	1	no	—
199	C₂²×C₄	—	43	1	no	—
200	C₂²×C₄	—	86	1	no	—
201	C₂²×C₄	—	86	1	no	—
202	C₂²×C₄	—	86	1	no	—
203	C₂²×C₄	—	86	1	no	—
204	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
205	Q₁₆	—	43	8	no	—
206	SD₁₆	—	43	8	no	—
207	M₁₆	—	86	2	no	—
208	M₁₆	C₄×D₄₃	1	4	yes	complemented
209	C₂×C₈	—	43	2	no	—
210	C₂×C₈	C₂²×D₄₃	1	2	yes	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	86	2	no	—
213	C₄⋊C₄	—	86	2	no	—
214	C₄⋊C₄	—	86	1	no	—
215	C₄⋊C₄	—	86	1	no	—
216	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
217	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
218	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
219	C₂²⋊C₄	—	2	1	no	—
220	C₂²⋊C₄	—	86	1	no	—
221	C₂²⋊C₄	—	86	1	no	—
222	C₄²	—	2	1	no	—
223	C₄²	—	43	1	no	—
224	C₄²	—	43	1	no	—
225	C₄²	—	43	1	no	—
226	C₄²	—	43	1	no	—
227	C₄²	—	86	1	no	—
228	C₄²	—	86	1	no	—
229	C₄²	—	86	2	no	—
230	C₄²	—	86	2	no	—
231	C₂³	—	86	1	no	—
232	C₂³	D₄×D₄₃	1	1	yes	characteristic
233	Q₈	—	43	4	no	—
234	Q₈	—	43	2	no	—
235	Q₈	—	86	4	no	—
236	Q₈	—	86	2	no	—
237	Q₈	—	86	1	no	—
238	D₄	—	43	2	no	—
239	D₄	—	86	4	no	—
240	D₄	—	86	1	no	—
241	C₂×C₄	—	2	1	no	—
242	C₂×C₄	—	2	1	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	43	1	no	—
246	C₂×C₄	—	43	1	no	—
247	C₂×C₄	—	43	1	no	—
248	C₂×C₄	—	43	1	no	—
249	C₂×C₄	—	86	2	no	—
250	C₂×C₄	—	86	4	no	—
251	C₂×C₄	—	86	1	no	—
252	C₂×C₄	—	86	2	no	—
253	C₂×C₄	—	86	4	no	—
254	C₂×C₄	—	172	2	no	—
255	C₂×C₄	—	172	2	no	—
256	C₂×C₄	—	172	1	no	—
257	C₂×C₄	C₂×C₄×D₄₃	1	2	yes	—
258	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
259	C₂×C₄	#10621902	1	2	yes	—
260	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
261	C₈	—	86	2	no	—
262	C₈	C₂×C₄×D₄₃	1	4	yes	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	86	2	no	—
266	C₂²	—	172	1	no	—
267	C₂²	—	172	1	no	—
268	C₂²	#13063287	1	2	yes	—
269	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
270	C₄	—	4	1	no	—
271	C₄	—	4	2	no	—
272	C₄	—	4	1	no	—
273	C₄	—	43	2	no	—
274	C₄	—	86	2	no	—
275	C₄	—	86	4	no	—
276	C₄	—	86	1	no	—
277	C₄	—	86	1	no	—
278	C₄	—	86	1	no	—
279	C₄	#13063287	1	2	yes	—
280	C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
281	C₄	C₂×#10621902	1	1	yes	characteristic
282	C₂	—	2	2	no	—
283	C₂	—	172	2	no	—
284	C₂	C₂×#13063287	1	1	yes	characteristic
285	C₂	#137×D₄₃	1	1	yes	characteristic
286	C₂	#14583695	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₄×C₄₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#428	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#428	D₄₃
hypocenter	C₄₃	C₄×#137
LCS	G ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
3. C₂×#13063290	C₂	semidirect
4. #14583005	C₂	semidirect
5. #14582998	C₂	semidirect
6. #14582993	C₂	semidirect
7. #14582986	C₂	semidirect
8. #14582907	C₂	semidirect
10. M₁₆×Dic₄₃	C₂	semidirect
11. C₄₃×#428	C₂	semidirect
14. #13063358	C₄	semidirect
58. C₂×Dic₈₆	C₂×C₄	semidirect
61. M₁₆×C₄₃	C₂×C₄	semidirect
85. Dic₁₇₂	C₂×C₄	semidirect
86. #10621873	C₂×C₄	semidirect
137. Dic₄₃	#137	semidirect
151. #428	D₄₃	semidirect
155. C₄₃	C₄×#137	semidirect
208. M₁₆	C₄×D₄₃	semidirect