Subgroups — #563×D₄₃ #19148723

Order 5504.

1024 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583653	C₂	1	2	yes	maximal, complemented
2	#14583652	C₂	1	2	yes	maximal, complemented
3	#14583651	C₂	1	4	yes	maximal, complemented
4	#14583650	C₂	1	4	yes	maximal, complemented
5	#14583649	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583648	C₂	1	2	yes	maximal, complemented
7	C₄⋊Q₈×D₄₃	C₂	1	2	yes	maximal, complemented
8	#126×D₄₃	C₂	1	4	yes	maximal, complemented
9	#125×D₄₃	C₂	1	2	yes	maximal, complemented
10	#124×D₄₃	C₂	1	1	yes	maximal, characteristic
11	#123×D₄₃	C₂	1	2	yes	maximal
12	C₂²⋊Q₈×D₄₃	C₂	1	4	yes	maximal, complemented
13	C₄₃×#563	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₄₃	C₂²	1	2	yes	—
15	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
16	C₂×D₄×D₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄×D₄₃	C₂²	1	2	yes	—
18	#13063325	C₂²	1	2	yes	complemented
19	#13063324	C₂²	1	4	yes	complemented
20	#13063324	C₂²	1	4	yes	complemented
21	#13063323	C₂²	1	2	yes	complemented
22	#13063323	C₂²	1	2	yes	complemented
23	#13063320	C₂²	1	4	yes	complemented
24	#13063320	C₂²	1	4	yes	complemented
25	#13063320	C₂²	1	4	yes	complemented
26	#13063319	C₂²	1	4	yes	complemented
27	#13063318	C₂²	1	4	yes	complemented
28	#13063318	C₂²	1	4	yes	complemented
29	#13063318	C₂²	1	4	yes	complemented
30	#13063317	C₂²	1	2	yes	complemented
31	#13063316	C₂²	1	2	yes	complemented
32	#13063316	C₂²	1	2	yes	complemented
33	#13063315	C₂²	1	4	yes	complemented
34	#13063315	C₂²	1	4	yes	complemented
35	#13063314	C₂²	1	1	yes	characteristic
36	#13063313	C₂²	1	1	yes	characteristic
37	#13063312	C₂²	1	1	yes	characteristic
38	#13063311	C₂²	1	4	yes	—
39	#13063310	C₂²	1	4	yes	—
40	#13063309	C₂²	1	2	yes	—
41	#13063308	C₂²	1	2	yes	—
42	#13063307	C₂²	1	4	yes	—
43	#13063306	C₂²	1	2	yes	—
44	#13063305	C₂²	1	4	yes	complemented
45	#13063304	C₂²	1	4	yes	complemented
46	#13063303	C₂²	1	4	yes	complemented
47	#13063303	C₂²	1	4	yes	complemented
48	#13063302	C₂²	1	4	yes	complemented
49	#13063302	C₂²	1	4	yes	complemented
50	#13063301	C₂²	1	4	yes	complemented
51	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
52	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
53	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
54	C₄⋊C₄×D₄₃	C₂²	1	4	yes	—
55	C₄⋊C₄×D₄₃	C₂²	1	2	yes	—
56	C₂²⋊C₄×D₄₃	C₂²	1	4	yes	—
57	C₂²⋊C₄×D₄₃	C₂²	1	4	yes	—
58	C₂²⋊C₄×D₄₃	C₂²	1	2	yes	—
59	C₄²×D₄₃	C₂²	1	1	yes	characteristic
60	C₄²×D₄₃	C₂²	1	2	yes	—
61	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	complemented
62	#126×C₄₃	C₂²	1	4	yes	complemented
63	#125×C₄₃	C₂²	1	2	yes	complemented
64	#124×C₄₃	C₂²	1	1	yes	characteristic
65	#123×C₄₃	C₂²	1	2	yes	—
66	C₂²⋊Q₈×C₄₃	C₂²	1	4	yes	complemented
67	C₂³×D₄₃	C₂³	1	2	yes	—
68	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
69	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂³	1	2	yes	—
72	C₂×C₂²⋊D₄₃	C₂³	1	2	yes	—
73	C₂²×Dic₄₃	C₂³	1	2	yes	—
74	Q₈×D₄₃	—	4	4	no	—
75	Q₈×D₄₃	—	4	2	no	—
76	D₄×D₄₃	—	4	2	no	—
77	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₄₃	—	4	4	no	—
79	C₂×C₄×D₄₃	C₂³	1	2	yes	—
80	C₂×C₄×D₄₃	C₂³	1	4	yes	—
81	C₂×C₄×D₄₃	C₂³	1	2	yes	—
82	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
83	C₂×C₄×D₄₃	C₂³	1	4	yes	—
84	C₂×C₄×D₄₃	C₂³	1	4	yes	—
85	C₂×Dic₈₆	C₂³	1	2	yes	—
86	C₂×Dic₈₆	C₂³	1	4	yes	—
87	C₂×Dic₈₆	C₂³	1	2	yes	—
88	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
90	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
91	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
92	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
93	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
94	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
95	C₂²⋊C₄×C₄₃	C₂³	1	4	yes	—
96	C₂²⋊C₄×C₄₃	C₂³	1	2	yes	—
97	C₄²×C₄₃	C₂³	1	1	yes	characteristic
98	C₄²×C₄₃	C₂³	1	2	yes	—
99	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
100	C₂²⋊Dic₄₃	C₂³	1	4	yes	—
101	C₂²⋊Dic₄₃	C₂³	1	2	yes	—
102	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
103	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
104	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
105	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
106	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
107	C₄⋊Dic₄₃	C₂³	1	4	yes	—
108	C₄⋊Dic₄₃	C₂³	1	2	yes	—
109	C₄⋊Dic₄₃	C₂³	1	4	yes	—
110	C₄⋊Dic₄₃	C₂³	1	4	yes	—
111	C₄⋊Dic₄₃	C₂³	1	2	yes	—
112	Dic₄₃⋊C₄	C₂³	1	4	yes	—
113	Dic₄₃⋊C₄	C₂³	1	4	yes	—
114	Dic₄₃⋊C₄	C₂³	1	4	yes	—
115	Dic₄₃⋊C₄	C₂³	1	4	yes	—
116	Dic₄₃⋊C₄	C₂³	1	4	yes	—
117	Dic₄₃⋊C₄	C₂³	1	4	yes	—
118	Dic₄₃⋊C₄	C₂³	1	4	yes	—
119	Dic₄₃⋊C₄	C₂³	1	4	yes	—
120	C₄×Dic₄₃	C₂³	1	2	yes	—
121	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
122	C₄×Dic₄₃	C₂³	1	2	yes	—
123	C₄×Dic₄₃	C₂³	1	4	yes	—
124	C₂³×C₄₃	C₂⁴	1	2	yes	—
125	C₂²×D₄₃	—	2	4	no	—
126	C₂²×D₄₃	—	2	8	no	—
127	C₂²×D₄₃	C₂⁴	1	2	yes	—
128	C₂²×D₄₃	C₂⁴	1	1	yes	characteristic
129	Q₈×C₄₃	—	4	1	no	—
130	Q₈×C₄₃	—	4	2	no	—
131	D₄×C₄₃	—	4	1	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
135	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
136	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
137	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
138	C₂²⋊D₄₃	—	4	2	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	C₂⁴	1	2	yes	—
141	C₂×Dic₄₃	C₂⁴	1	4	yes	—
142	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
143	C₂×Dic₄₃	C₂⁴	1	2	yes	—
144	C₂×Dic₄₃	C₂⁴	1	4	yes	—
145	D₁₇₂	—	4	1	no	—
146	C₄×D₄₃	—	2	8	no	—
147	C₄×D₄₃	—	2	2	no	—
148	C₄×D₄₃	—	2	4	no	—
149	C₄×D₄₃	—	2	4	no	—
150	C₄×D₄₃	—	2	8	no	—
151	C₄×D₄₃	—	4	4	no	—
152	Dic₈₆	—	4	1	no	—
153	Dic₈₆	—	4	4	no	—
154	Dic₈₆	—	4	2	no	—
155	Dic₈₆	—	4	2	no	—
156	C₂²×C₄₃	—	2	2	no	—
157	C₂²×C₄₃	—	2	4	no	—
158	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
159	C₂×D₄₃	—	2	2	no	—
160	C₂×D₄₃	—	2	4	no	—
161	C₂×D₄₃	—	4	8	no	—
162	C₂×D₄₃	H(5,𝔽₂)	1	2	yes	—
163	C₂×D₄₃	2₋¹⁺⁴	1	4	yes	—
164	C₄×C₄₃	—	2	2	no	—
165	C₄×C₄₃	—	2	4	no	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	2	2	no	—
168	C₄×C₄₃	—	2	4	no	—
169	Dic₄₃	—	2	2	no	—
170	Dic₄₃	—	2	4	no	—
171	Dic₄₃	—	2	1	no	—
172	Dic₄₃	—	2	2	no	—
173	Dic₄₃	—	2	4	no	—
174	C₂×#563	—	43	1	no	maximal, Sylow
175	C₂×C₄₃	—	4	2	no	—
176	C₂×C₄₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
177	C₂×C₄₃	C₂×2₋¹⁺⁴	1	2	yes	—
178	D₄₃	—	4	2	no	—
179	D₄₃	#563	1	4	yes	direct product factor
180	#563	—	43	2	no	—
181	#563	—	43	4	no	—
182	#563	—	43	2	no	—
183	#563	—	43	1	no	—
184	#563	—	43	2	no	—
185	#563	—	43	4	no	—
186	#563	D₄₃	1	1	yes	characteristic, direct product factor
187	C₂×C₄⋊Q₈	—	43	2	no	—
188	C₂×#126	—	43	4	no	—
189	C₂×#125	—	43	2	no	—
190	C₂×#124	—	43	1	no	—
191	C₂×#123	—	43	2	no	—
192	C₂×C₂²⋊Q₈	—	43	4	no	—
193	C₄₃	C₂×#563	1	1	yes	characteristic, complemented, Sylow
194	C₂²×Q₈	—	43	2	no	—
195	C₂²×Q₈	—	43	1	no	—
196	C₂²×D₄	—	43	1	no	—
197	C₂³×C₄	—	43	2	no	—
198	C₄⋊Q₈	—	43	2	no	—
199	C₄⋊Q₈	—	43	2	no	—
200	C₄⋊Q₈	—	43	4	no	—
201	C₄⋊Q₈	—	43	4	no	—
202	C₄⋊Q₈	—	43	2	no	—
203	C₄⋊Q₈	C₂×D₄₃	1	2	yes	complemented
204	#126	—	43	4	no	—
205	#126	—	43	4	no	—
206	#126	—	43	4	no	—
207	#126	—	43	4	no	—
208	#126	—	43	4	no	—
209	#126	—	43	4	no	—
210	#126	—	43	4	no	—
211	#126	C₂×D₄₃	1	4	yes	complemented
212	#125	—	43	4	no	—
213	#125	—	43	4	no	—
214	#125	—	43	2	no	—
215	#125	—	43	2	no	—
216	#125	—	43	2	no	—
217	#125	C₂×D₄₃	1	2	yes	complemented
218	#124	—	43	1	no	—
219	#124	—	43	1	no	—
220	#124	—	43	4	no	—
221	#124	—	43	1	no	—
222	#124	C₂×D₄₃	1	1	yes	characteristic
223	#123	—	43	2	no	—
224	#123	—	43	2	no	—
225	#123	—	43	4	no	—
226	#123	—	43	4	no	—
227	#123	—	43	2	no	—
228	#123	C₂×D₄₃	1	2	yes	—
229	C₂²⋊Q₈	—	43	4	no	—
230	C₂²⋊Q₈	—	43	4	no	—
231	C₂²⋊Q₈	—	43	4	no	—
232	C₂²⋊Q₈	—	43	4	no	—
233	C₂²⋊Q₈	—	43	4	no	—
234	C₂²⋊Q₈	—	43	4	no	—
235	C₂²⋊Q₈	—	43	4	no	—
236	C₂²⋊Q₈	C₂×D₄₃	1	4	yes	complemented
237	C₂×C₄⋊C₄	—	43	2	no	—
238	C₂×C₄⋊C₄	—	43	4	no	—
239	C₂×C₄⋊C₄	—	43	4	no	—
240	C₂×C₄⋊C₄	—	43	4	no	—
241	C₂×C₄⋊C₄	—	43	2	no	—
242	C₂×C₂²⋊C₄	—	43	4	no	—
243	C₂×C₂²⋊C₄	—	43	4	no	—
244	C₂×C₂²⋊C₄	—	43	2	no	—
245	C₂×C₄²	—	43	1	no	—
246	C₂×C₄²	—	43	2	no	—
247	C₂⁴	—	43	2	no	—
248	C₂×Q₈	—	43	4	no	—
249	C₂×Q₈	—	43	2	no	—
250	C₂×Q₈	—	43	1	no	—
251	C₂×Q₈	—	43	2	no	—
252	C₂×Q₈	—	172	4	no	—
253	C₂×Q₈	—	172	2	no	—
254	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
255	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₂×D₄	—	43	2	no	—
257	C₂×D₄	—	43	1	no	—
258	C₂×D₄	—	172	2	no	—
259	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
260	C₂²×C₄	—	43	2	no	—
261	C₂²×C₄	—	43	2	no	—
262	C₂²×C₄	—	43	4	no	—
263	C₂²×C₄	—	43	2	no	—
264	C₂²×C₄	—	43	1	no	—
265	C₂²×C₄	—	43	4	no	—
266	C₂²×C₄	—	43	4	no	—
267	C₂²×C₄	—	172	4	no	—
268	C₂²×C₄	C₂²×D₄₃	1	2	yes	—
269	C₄⋊C₄	—	43	4	no	—
270	C₄⋊C₄	—	43	2	no	—
271	C₄⋊C₄	—	43	4	no	—
272	C₄⋊C₄	—	43	4	no	—
273	C₄⋊C₄	—	43	4	no	—
274	C₄⋊C₄	—	43	4	no	—
275	C₄⋊C₄	—	43	4	no	—
276	C₄⋊C₄	—	43	4	no	—
277	C₄⋊C₄	—	43	4	no	—
278	C₄⋊C₄	—	43	4	no	—
279	C₄⋊C₄	—	43	4	no	—
280	C₄⋊C₄	—	43	4	no	—
281	C₄⋊C₄	—	43	2	no	—
282	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
283	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
284	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
285	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
286	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
287	C₂²⋊C₄	—	43	4	no	—
288	C₂²⋊C₄	—	43	4	no	—
289	C₂²⋊C₄	—	43	2	no	—
290	C₂²⋊C₄	—	43	4	no	—
291	C₂²⋊C₄	—	43	4	no	—
292	C₂²⋊C₄	—	43	4	no	—
293	C₂²⋊C₄	—	43	4	no	—
294	C₂²⋊C₄	—	43	4	no	—
295	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
296	C₂²⋊C₄	C₂²×D₄₃	1	4	yes	—
297	C₂²⋊C₄	C₂²×D₄₃	1	2	yes	—
298	C₄²	—	43	1	no	—
299	C₄²	—	43	2	no	—
300	C₄²	—	43	2	no	—
301	C₄²	—	43	4	no	—
302	C₄²	C₂²×D₄₃	1	1	yes	characteristic
303	C₄²	C₂²×D₄₃	1	2	yes	—
304	C₂³	—	43	1	no	—
305	C₂³	—	43	2	no	—
306	C₂³	—	86	4	no	—
307	C₂³	—	86	8	no	—
308	C₂³	C₂³×D₄₃	1	2	yes	—
309	Q₈	—	4	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	172	4	no	—
312	Q₈	—	172	1	no	—
313	Q₈	—	172	2	no	—
314	Q₈	—	172	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	172	1	no	—
317	D₄	—	172	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	43	2	no	—
320	C₂×C₄	—	43	4	no	—
321	C₂×C₄	—	43	2	no	—
322	C₂×C₄	—	43	1	no	—
323	C₂×C₄	—	43	4	no	—
324	C₂×C₄	—	86	8	no	—
325	C₂×C₄	—	86	2	no	—
326	C₂×C₄	—	86	4	no	—
327	C₂×C₄	—	86	4	no	—
328	C₂×C₄	—	86	8	no	—
329	C₂×C₄	—	172	4	no	—
330	C₂×C₄	—	172	2	no	—
331	C₂×C₄	C₂³×D₄₃	1	2	yes	—
332	C₂×C₄	C₂³×D₄₃	1	4	yes	—
333	C₂×C₄	C₂³×D₄₃	1	2	yes	—
334	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
335	C₂×C₄	C₂³×D₄₃	1	4	yes	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	4	no	—
338	C₂²	—	43	2	no	—
339	C₂²	—	43	4	no	—
340	C₂²	—	86	2	no	—
341	C₂²	—	86	4	no	—
342	C₂²	—	172	8	no	—
343	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
344	C₄	—	2	4	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	2	no	—
347	C₄	—	2	2	no	—
348	C₄	—	2	4	no	—
349	C₄	—	86	4	no	—
350	C₄	—	86	1	no	—
351	C₄	—	86	2	no	—
352	C₄	—	86	2	no	—
353	C₄	—	86	4	no	—
354	C₂	—	4	2	no	—
355	C₂	—	43	4	no	—
356	C₂	—	172	2	no	—
357	C₂	H(5,𝔽₂)×D₄₃	1	1	yes	characteristic
358	C₂	2₋¹⁺⁴×D₄₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#563	C₂
hypercenter	#563	D₄₃
hypocenter	C₄₃	C₂×#563
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #563

Complements

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

N	H ≅ G/N	type
1. #14583653	C₂	semidirect
2. #14583652	C₂	semidirect
3. #14583651	C₂	semidirect
4. #14583650	C₂	semidirect
5. #14583649	C₂	semidirect
6. #14583648	C₂	semidirect
7. C₄⋊Q₈×D₄₃	C₂	semidirect
8. #126×D₄₃	C₂	semidirect
9. #125×D₄₃	C₂	semidirect
12. C₂²⋊Q₈×D₄₃	C₂	semidirect
13. C₄₃×#563	C₂	semidirect
18. #13063325	C₂²	semidirect
19. #13063324	C₂²	semidirect
20. #13063324	C₂²	semidirect
21. #13063323	C₂²	semidirect
22. #13063323	C₂²	semidirect
23. #13063320	C₂²	semidirect
24. #13063320	C₂²	semidirect
25. #13063320	C₂²	semidirect
26. #13063319	C₂²	semidirect
27. #13063318	C₂²	semidirect
28. #13063318	C₂²	semidirect
29. #13063318	C₂²	semidirect
30. #13063317	C₂²	semidirect
31. #13063316	C₂²	semidirect
32. #13063316	C₂²	semidirect
33. #13063315	C₂²	semidirect
34. #13063315	C₂²	semidirect
44. #13063305	C₂²	semidirect
45. #13063304	C₂²	semidirect
46. #13063303	C₂²	semidirect
47. #13063303	C₂²	semidirect
48. #13063302	C₂²	semidirect
49. #13063302	C₂²	semidirect
50. #13063301	C₂²	semidirect
61. C₄⋊Q₈×C₄₃	C₂²	semidirect
62. #126×C₄₃	C₂²	semidirect
63. #125×C₄₃	C₂²	semidirect
66. C₂²⋊Q₈×C₄₃	C₂²	semidirect
179. D₄₃	#563	direct (N × H)
186. #563	D₄₃	direct (N × H)
193. C₄₃	C₂×#563	semidirect
203. C₄⋊Q₈	C₂×D₄₃	semidirect
211. #126	C₂×D₄₃	semidirect
217. #125	C₂×D₄₃	semidirect
236. C₂²⋊Q₈	C₂×D₄₃	semidirect

Subgroups — #563×D43 #19148723

Conjugacy classes of subgroups

Complements

Subgroups — #563×D₄₃ #19148723