Subgroups — #19144739

Order 5504.

430 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583444	C₂	1	1	yes	maximal, characteristic, complemented
2	#41×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583224	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583220	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583177	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583097	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583092	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583044	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583011	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583008	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583002	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583000	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582953	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#449	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	—	2	1	no	—
17	C₂×#10621884	C₂²	1	1	yes	characteristic
18	C₂×#10621882	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063321	C₂²	1	1	yes	characteristic
21	#13063311	—	2	1	no	—
22	#13063299	C₂²	1	1	yes	characteristic
23	#13063297	—	2	1	no	—
24	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
25	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
26	D₄×Dic₄₃	—	2	1	no	—
27	D₄×Dic₄₃	—	2	1	no	—
28	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
29	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
30	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
31	#13063287	C₂²	1	1	yes	characteristic
32	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
33	#13063284	—	2	1	no	—
34	#13063284	C₂²	1	1	yes	characteristic
35	C₂×C₄×Dic₄₃	—	2	1	no	—
36	C₂×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
37	#13063259	C₂²	1	1	yes	characteristic
38	#13063258	C₂²	1	1	yes	characteristic
39	#13063257	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	#13063249	C₂²	1	1	yes	characteristic
44	#13063247	C₂²	1	1	yes	characteristic
45	#13063247	C₂²	1	1	yes	characteristic
46	#13063246	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	#13063235	C₂²	1	1	yes	characteristic
52	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
54	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
55	#121×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
59	#41×C₄₃	—	2	2	no	—
60	#41×C₄₃	—	2	2	no	—
61	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₄₃	—	2	1	no	—
63	C₂×D₄×C₄₃	—	2	1	no	—
64	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	—	2	1	no	—
67	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×C₂²⋊D₄₃	—	2	1	no	—
69	C₂×C₂²⋊D₄₃	—	2	1	no	—
70	C₂×C₂²⋊D₄₃	—	2	1	no	—
71	C₂²×Dic₄₃	—	2	1	no	—
72	C₂²×Dic₄₃	—	2	1	no	—
73	C₂²×Dic₄₃	—	2	1	no	—
74	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
75	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
76	#10621900	—	4	4	no	—
77	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₄₃	—	2	1	no	—
79	C₂×Dic₈₆	—	2	1	no	—
80	SD₁₆×C₄₃	—	2	2	no	—
81	D₈×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₄₃	—	2	1	no	—
86	C₂²⋊Dic₄₃	—	2	1	no	—
87	C₂²⋊Dic₄₃	—	2	1	no	—
88	C₂²⋊Dic₄₃	—	2	1	no	—
89	C₂²⋊Dic₄₃	—	2	1	no	—
90	#10621884	—	2	2	no	—
91	#10621882	—	2	2	no	—
92	(C₂×D₄₃)⋊C₄	—	2	1	no	—
93	(C₂×D₄₃)⋊C₄	—	2	1	no	—
94	C₄⋊Dic₄₃	—	2	1	no	—
95	C₄⋊Dic₄₃	—	2	1	no	—
96	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
97	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
98	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
99	Dic₄₃⋊C₄	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	D₄	1	1	yes	characteristic
104	C₄×Dic₄₃	D₄	1	1	yes	characteristic
105	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
106	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
108	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
109	C₂³×C₄₃	—	2	1	no	—
110	C₂³×C₄₃	—	2	1	no	—
111	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂²×D₄₃	—	2	1	no	—
113	Q₈×C₄₃	—	2	1	no	—
114	Q₈×C₄₃	#41	1	2	yes	complemented
115	D₄×C₄₃	—	2	4	no	—
116	D₄×C₄₃	—	2	1	no	—
117	D₄×C₄₃	—	2	1	no	—
118	D₄×C₄₃	—	2	1	no	—
119	D₄×C₄₃	—	4	2	no	—
120	D₄×C₄₃	#41	1	2	yes	complemented
121	C₂×C₄×C₄₃	—	2	4	no	—
122	C₂×C₄×C₄₃	—	2	1	no	—
123	C₂×C₄×C₄₃	—	2	1	no	—
124	C₂×C₄×C₄₃	—	2	1	no	—
125	C₂×C₄×C₄₃	—	2	1	no	—
126	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₄₃	—	4	2	no	—
129	C₂²⋊D₄₃	—	4	2	no	—
130	C₂²⋊D₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	2	2	no	—
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	—	2	1	no	—
135	C₂×Dic₄₃	—	4	1	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
140	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
143	D₁₇₂	—	2	1	no	—
144	D₁₇₂	—	2	1	no	—
145	C₄×D₄₃	—	4	2	no	—
146	Dic₈₆	—	4	2	no	—
147	C₈×C₄₃	—	2	1	no	—
148	C₈×C₄₃	—	2	1	no	—
149	C₄₃⋊C₈	—	2	1	no	—
150	C₄₃⋊C₈	—	2	1	no	—
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	2	1	no	—
153	C₂²×C₄₃	—	2	1	no	—
154	C₂²×C₄₃	—	2	1	no	—
155	C₂²×C₄₃	—	4	1	no	—
156	C₂²×C₄₃	—	4	1	no	—
157	C₂²×C₄₃	—	4	1	no	—
158	C₂²×C₄₃	—	4	1	no	—
159	C₂²×C₄₃	—	4	1	no	—
160	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×D₄₃	—	4	1	no	—
162	C₂×D₄₃	—	4	1	no	—
163	C₂×D₄₃	—	4	1	no	—
164	C₄×C₄₃	—	2	1	no	—
165	C₄×C₄₃	—	2	2	no	—
166	C₄×C₄₃	—	4	1	no	—
167	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
168	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
169	Dic₄₃	—	2	1	no	—
170	Dic₄₃	—	2	1	no	—
171	Dic₄₃	—	2	1	no	—
172	Dic₄₃	—	4	1	no	—
173	Dic₄₃	—	4	1	no	—
174	Dic₄₃	—	4	1	no	—
175	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
176	#3115	—	43	1	no	maximal, Sylow
177	C₂×C₄₃	—	4	1	no	—
178	C₂×C₄₃	—	4	2	no	—
179	C₂×C₄₃	—	8	1	no	—
180	C₂×C₄₃	#525	1	1	yes	characteristic
181	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
182	C₂×C₄₃	#575	1	1	yes	characteristic
183	D₄₃	—	8	1	no	—
184	#532	—	43	1	no	—
185	C₄×#41	—	43	1	no	—
186	#482	—	43	1	no	—
187	#482	—	43	1	no	—
188	#475	—	43	1	no	—
189	#474	—	43	1	no	—
190	#460	—	43	1	no	—
191	#459	—	43	1	no	—
192	#449	—	43	1	no	—
193	#449	D₄₃	1	1	yes	characteristic, complemented
194	#442	—	43	1	no	—
195	#442	—	43	1	no	—
196	#440	—	43	1	no	—
197	#440	—	43	1	no	—
198	#432	—	43	1	no	—
199	C₄₃	#3115	1	1	yes	characteristic, complemented, Sylow
200	C₂×#41	—	86	1	no	—
201	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
202	C₂×SD₁₆	—	43	1	no	—
203	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
204	C₂×D₈	—	43	1	no	—
205	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	43	1	no	—
207	#127	—	43	1	no	—
208	#124	—	86	1	no	—
209	#121	—	43	1	no	—
210	#121	—	86	1	no	—
211	#121	C₂×D₄₃	1	1	yes	characteristic
212	C₄×Q₈	—	43	1	no	—
213	C₄×Q₈	—	43	1	no	—
214	C₄×D₄	—	43	1	no	—
215	C₄×D₄	—	43	1	no	—
216	C₄×D₄	—	43	1	no	—
217	C₄×D₄	—	43	1	no	—
218	C₄×D₄	—	86	1	no	—
219	C₄×D₄	—	86	1	no	—
220	C₄×D₄	—	86	1	no	—
221	#117	—	43	1	no	—
222	#117	—	86	1	no	—
223	C₂×C₄²	—	43	1	no	—
224	C₂×C₄²	—	86	1	no	—
225	#107	—	43	1	no	—
226	#107	—	43	1	no	—
227	#106	—	43	1	no	—
228	#106	—	43	1	no	—
229	C₄⋊C₈	—	43	1	no	—
230	C₄⋊C₈	—	43	1	no	—
231	#103	—	43	1	no	—
232	#103	C₂×D₄₃	1	1	yes	characteristic
233	#102	—	43	1	no	—
234	#102	—	43	1	no	—
235	#102	—	43	1	no	—
236	#102	—	43	1	no	—
237	#102	—	43	1	no	—
238	#102	C₂×D₄₃	1	1	yes	characteristic
239	C₂²⋊C₈	—	43	1	no	—
240	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
241	#97	—	43	1	no	—
242	#97	—	43	1	no	—
243	#41	—	2	2	no	—
244	#41	—	2	2	no	—
245	#41	—	172	4	no	—
246	C₂×Q₈	—	86	1	no	—
247	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	2	1	no	—
250	C₂×D₄	—	43	1	no	—
251	C₂×D₄	—	86	1	no	—
252	C₂×D₄	—	86	1	no	—
253	C₂×D₄	—	86	1	no	—
254	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
255	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
256	C₂²×C₄	—	2	1	no	—
257	C₂²×C₄	—	43	1	no	—
258	C₂²×C₄	—	43	1	no	—
259	C₂²×C₄	—	86	1	no	—
260	C₂²×C₄	—	86	1	no	—
261	C₂²×C₄	—	86	1	no	—
262	C₂²×C₄	—	86	1	no	—
263	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
264	SD₁₆	—	2	2	no	—
265	SD₁₆	—	86	2	no	—
266	D₈	—	2	2	no	—
267	D₈	—	86	2	no	—
268	C₂×C₈	—	43	1	no	—
269	C₂×C₈	—	43	1	no	—
270	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
272	C₄⋊C₄	—	43	1	no	—
273	C₄⋊C₄	—	43	1	no	—
274	C₄⋊C₄	—	43	1	no	—
275	C₄⋊C₄	—	86	1	no	—
276	C₄⋊C₄	—	86	1	no	—
277	C₄⋊C₄	—	86	1	no	—
278	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	86	1	no	—
281	C₂²⋊C₄	—	86	1	no	—
282	C₂²⋊C₄	—	86	1	no	—
283	C₂²⋊C₄	—	86	1	no	—
284	C₂²⋊C₄	—	86	1	no	—
285	C₂²⋊C₄	—	86	1	no	—
286	C₄²	—	43	1	no	—
287	C₄²	—	43	1	no	—
288	C₄²	—	43	1	no	—
289	C₄²	—	43	1	no	—
290	C₄²	—	86	1	no	—
291	C₄²	—	86	1	no	—
292	C₄²	—	86	1	no	—
293	C₂³	—	2	1	no	—
294	C₂³	—	2	1	no	—
295	C₂³	—	86	1	no	—
296	C₂³	D₄×D₄₃	1	1	yes	characteristic
297	Q₈	—	2	1	no	—
298	Q₈	—	172	2	no	—
299	Q₈	#10621900	1	2	yes	complemented
300	D₄	—	2	4	no	—
301	D₄	—	2	1	no	—
302	D₄	—	2	1	no	—
303	D₄	—	2	1	no	—
304	D₄	—	4	2	no	—
305	D₄	—	86	1	no	—
306	D₄	—	86	1	no	—
307	D₄	—	172	2	no	—
308	D₄	—	172	2	no	—
309	D₄	—	172	2	no	—
310	D₄	#10621900	1	2	yes	complemented
311	C₂×C₄	—	2	4	no	—
312	C₂×C₄	—	2	1	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	43	1	no	—
317	C₂×C₄	—	43	1	no	—
318	C₂×C₄	—	43	1	no	—
319	C₂×C₄	—	43	1	no	—
320	C₂×C₄	—	86	2	no	—
321	C₂×C₄	—	86	1	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	172	2	no	—
325	C₂×C₄	—	172	2	no	—
326	C₂×C₄	—	172	1	no	—
327	C₂×C₄	—	172	2	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
330	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
331	C₈	—	2	1	no	—
332	C₈	—	2	1	no	—
333	C₈	—	86	1	no	—
334	C₈	—	86	1	no	—
335	C₂²	—	2	1	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	1	no	—
338	C₂²	—	2	1	no	—
339	C₂²	—	4	1	no	—
340	C₂²	—	4	1	no	—
341	C₂²	—	4	1	no	—
342	C₂²	—	4	1	no	—
343	C₂²	—	4	1	no	—
344	C₂²	—	172	1	no	—
345	C₂²	—	172	1	no	—
346	C₂²	—	172	1	no	—
347	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
348	C₄	—	2	1	no	—
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	43	2	no	—
352	C₄	—	86	1	no	—
353	C₄	—	86	1	no	—
354	C₄	—	86	1	no	—
355	C₄	—	172	1	no	—
356	C₄	—	172	1	no	—
357	C₄	—	172	1	no	—
358	C₄	C₂×#10621900	1	1	yes	characteristic
359	C₄	C₂×#10621900	1	1	yes	characteristic
360	C₂	—	4	1	no	—
361	C₂	—	4	2	no	—
362	C₂	—	8	1	no	—
363	C₂	—	344	1	no	—
364	C₂	#14583405	1	1	yes	characteristic
365	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
366	C₂	#14583696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#449	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#449	D₄₃
hypocenter	C₄₃	#3115
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #449

Complements

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

N	H ≅ G/N	type
1. #14583444	C₂	semidirect
2. #41×Dic₄₃	C₂	semidirect
3. #14583224	C₂	semidirect
4. #14583220	C₂	semidirect
5. #14583182	C₂	semidirect
6. #14583177	C₂	semidirect
7. #14583097	C₂	semidirect
8. #14583092	C₂	semidirect
9. #14583044	C₂	semidirect
10. #14583011	C₂	semidirect
11. #14583008	C₂	semidirect
12. #14583002	C₂	semidirect
13. #14583000	C₂	semidirect
14. #14582953	C₂	semidirect
15. C₄₃×#449	C₂	semidirect
114. Q₈×C₄₃	#41	semidirect
120. D₄×C₄₃	#41	semidirect
193. #449	D₄₃	semidirect
199. C₄₃	#3115	semidirect
299. Q₈	#10621900	semidirect
310. D₄	#10621900	semidirect