Subgroups — #19147297

Order 5504.

396 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	#14583608	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583569	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583263	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583262	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583244	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583200	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583197	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583187	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583092	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583092	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583028	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583011	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583009	C₂	1	1	yes	maximal, characteristic, complemented
14	D₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	—	2	1	no	—
17	C₂×#10621882	C₂²	1	1	yes	characteristic
18	#13063324	C₂²	1	1	yes	characteristic
19	#13063321	C₂²	1	1	yes	characteristic
20	#13063321	C₂²	1	1	yes	characteristic
21	#13063317	C₂²	1	1	yes	characteristic
22	#13063316	C₂²	1	1	yes	characteristic
23	#13063304	—	2	1	no	—
24	#13063302	—	2	1	no	—
25	#13063302	—	2	1	no	—
26	#13063297	—	2	1	no	—
27	#13063297	—	2	1	no	—
28	#13063296	—	2	1	no	—
29	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
30	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
31	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
32	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	C₂×Dic₄₃⋊C₄	—	2	1	no	—
36	#13063259	C₂²	1	1	yes	characteristic
37	#13063257	C₂²	1	1	yes	characteristic
38	#13063256	C₂²	1	1	yes	characteristic
39	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
41	#13063247	C₂²	1	1	yes	characteristic
42	#13063247	C₂²	1	1	yes	characteristic
43	#13063246	C₂²	1	1	yes	characteristic
44	#13063246	C₂²	1	1	yes	characteristic
45	#13063246	C₂²	1	1	yes	characteristic
46	#13063246	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063245	C₂²	1	1	yes	characteristic
49	#13063236	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
54	#127×C₄₃	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₄₃	—	2	1	no	—
60	C₂×D₄×C₄₃	—	2	1	no	—
61	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	—	2	1	no	—
65	C₂×C₂²⋊D₄₃	—	2	1	no	—
66	C₂×C₂²⋊D₄₃	—	2	1	no	—
67	C₂×C₂²⋊D₄₃	—	2	1	no	—
68	C₂×C₂²⋊D₄₃	—	2	1	no	—
69	C₂²×Dic₄₃	—	2	1	no	—
70	C₂²×Dic₄₃	—	2	1	no	—
71	C₂²×Dic₄₃	—	2	1	no	—
72	C₂²×Dic₄₃	—	2	1	no	—
73	#10621900	—	4	4	no	—
74	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
75	C₂×C₄×D₄₃	—	2	1	no	—
76	C₂×Dic₈₆	—	2	1	no	—
77	C₂×Dic₈₆	D₄	1	1	yes	characteristic
78	D₈×C₄₃	—	2	2	no	—
79	D₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
83	C₂²⋊C₄×C₄₃	—	2	1	no	—
84	C₄²×C₄₃	C₂³	1	1	yes	characteristic
85	C₂²⋊Dic₄₃	—	2	1	no	—
86	C₂²⋊Dic₄₃	—	2	1	no	—
87	C₂²⋊Dic₄₃	—	2	1	no	—
88	C₂²⋊Dic₄₃	—	2	1	no	—
89	#10621882	—	2	2	no	—
90	(C₂×D₄₃)⋊C₄	—	2	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
93	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
94	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
96	Dic₄₃⋊C₄	—	2	1	no	—
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	—	2	1	no	—
99	Dic₄₃⋊C₄	—	2	1	no	—
100	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
101	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
102	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
103	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	—	2	1	no	—
108	C₂³×C₄₃	—	2	1	no	—
109	C₂²×D₄₃	—	2	1	no	—
110	D₄×C₄₃	—	2	1	no	—
111	D₄×C₄₃	—	2	1	no	—
112	D₄×C₄₃	—	2	1	no	—
113	D₄×C₄₃	—	2	1	no	—
114	D₄×C₄₃	—	2	1	no	—
115	D₄×C₄₃	—	4	2	no	—
116	D₄×C₄₃	—	8	1	no	—
117	D₄×C₄₃	#41	1	2	yes	complemented
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	8	1	no	—
120	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₄₃	—	4	2	no	—
124	C₂²⋊D₄₃	—	4	2	no	—
125	C₂²⋊D₄₃	—	8	1	no	—
126	C₂²⋊D₄₃	—	8	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	2	1	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	C₂×Dic₄₃	—	4	2	no	—
133	C₂×Dic₄₃	—	8	1	no	—
134	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
137	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
138	D₁₇₂	—	2	1	no	—
139	D₁₇₂	—	2	1	no	—
140	C₄×D₄₃	—	4	2	no	—
141	Dic₈₆	—	2	2	no	—
142	Dic₈₆	—	8	1	no	—
143	C₈×C₄₃	—	2	1	no	—
144	C₈×C₄₃	—	2	1	no	—
145	C₄₃⋊C₈	—	2	1	no	—
146	C₄₃⋊C₈	—	2	1	no	—
147	C₂²×C₄₃	—	2	2	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	—	4	1	no	—
152	C₂²×C₄₃	—	4	1	no	—
153	C₂²×C₄₃	—	4	1	no	—
154	C₂²×C₄₃	—	4	1	no	—
155	C₂²×C₄₃	—	4	1	no	—
156	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
157	C₂×D₄₃	—	4	1	no	—
158	C₂×D₄₃	—	4	1	no	—
159	C₂×D₄₃	—	4	1	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	—	4	1	no	—
163	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
164	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
165	Dic₄₃	—	2	1	no	—
166	Dic₄₃	—	2	1	no	—
167	Dic₄₃	—	2	1	no	—
168	Dic₄₃	—	4	1	no	—
169	Dic₄₃	—	4	1	no	—
170	Dic₄₃	—	4	1	no	—
171	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
172	#3366	—	43	1	no	maximal, Sylow
173	C₂×C₄₃	—	4	2	no	—
174	C₂×C₄₃	—	8	1	no	—
175	C₂×C₄₃	—	8	1	no	—
176	C₂×C₄₃	#548	1	1	yes	characteristic
177	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
178	C₂×C₄₃	#576	1	1	yes	characteristic
179	D₄₃	—	8	1	no	—
180	#554	—	43	1	no	—
181	#550	—	43	1	no	—
182	#490	—	43	1	no	—
183	#490	—	43	1	no	—
184	#486	—	43	1	no	—
185	#478	—	43	1	no	—
186	#478	—	43	1	no	—
187	#476	—	43	1	no	—
188	#459	—	43	1	no	—
189	#459	—	43	1	no	—
190	#459	D₄₃	1	1	yes	characteristic, complemented
191	#446	—	43	1	no	—
192	#442	—	43	1	no	—
193	#442	—	43	1	no	—
194	C₄×D₈	—	43	1	no	—
195	C₄₃	#3366	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	—	86	1	no	—
197	C₂×D₈	—	43	1	no	—
198	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	43	1	no	—
201	#127	—	43	1	no	—
202	#127	—	43	1	no	—
203	#127	C₂×D₄₃	1	1	yes	characteristic
204	#125	—	43	1	no	—
205	#125	—	43	1	no	—
206	C₂²⋊Q₈	—	86	1	no	—
207	C₂²⋊Q₈	—	86	1	no	—
208	C₂²⋊Q₈	—	86	1	no	—
209	#121	—	86	1	no	—
210	#121	—	86	1	no	—
211	#121	—	86	1	no	—
212	C₄×Q₈	—	43	1	no	—
213	C₄×D₄	—	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	C₂×D₄₃	1	1	yes	characteristic
219	C₂×C₄⋊C₄	—	86	1	no	—
220	#107	—	43	1	no	—
221	#106	—	43	1	no	—
222	#106	—	43	1	no	—
223	C₄⋊C₈	—	43	1	no	—
224	C₄⋊C₈	—	43	1	no	—
225	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
226	#102	—	43	1	no	—
227	#102	—	43	1	no	—
228	#102	—	43	1	no	—
229	#102	—	43	1	no	—
230	#102	—	43	1	no	—
231	#102	—	43	1	no	—
232	#102	—	43	1	no	—
233	#102	—	43	1	no	—
234	#102	C₂×D₄₃	1	1	yes	characteristic
235	#102	C₂×D₄₃	1	1	yes	characteristic
236	#97	—	43	1	no	—
237	#97	—	43	1	no	—
238	C₄×C₈	—	43	1	no	—
239	#41	—	172	4	no	—
240	C₂×Q₈	—	43	1	no	—
241	C₂×Q₈	—	86	1	no	—
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	2	1	no	—
244	C₂×D₄	—	43	1	no	—
245	C₂×D₄	—	86	1	no	—
246	C₂×D₄	—	86	1	no	—
247	C₂×D₄	—	86	1	no	—
248	C₂×D₄	—	86	1	no	—
249	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
250	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
251	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	86	1	no	—
254	C₂²×C₄	—	86	1	no	—
255	C₂²×C₄	—	86	1	no	—
256	C₂²×C₄	—	86	1	no	—
257	C₂²×C₄	—	86	1	no	—
258	D₈	—	2	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	86	2	no	—
261	C₂×C₈	—	43	1	no	—
262	C₂×C₈	—	43	1	no	—
263	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
265	C₄⋊C₄	—	43	1	no	—
266	C₄⋊C₄	—	43	1	no	—
267	C₄⋊C₄	—	43	1	no	—
268	C₄⋊C₄	—	43	1	no	—
269	C₄⋊C₄	—	43	1	no	—
270	C₄⋊C₄	—	43	1	no	—
271	C₄⋊C₄	—	86	1	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₄⋊C₄	—	86	1	no	—
274	C₄⋊C₄	—	86	1	no	—
275	C₄⋊C₄	—	86	1	no	—
276	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	86	1	no	—
279	C₂²⋊C₄	—	86	1	no	—
280	C₂²⋊C₄	—	86	1	no	—
281	C₂²⋊C₄	—	86	1	no	—
282	C₂²⋊C₄	—	86	1	no	—
283	C₄²	—	43	1	no	—
284	C₄²	—	43	1	no	—
285	C₄²	C₂²×D₄₃	1	1	yes	characteristic
286	C₂³	—	2	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	—	86	1	no	—
290	Q₈	—	86	2	no	—
291	Q₈	—	344	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	2	1	no	—
294	D₄	—	2	1	no	—
295	D₄	—	2	1	no	—
296	D₄	—	2	1	no	—
297	D₄	—	4	2	no	—
298	D₄	—	8	1	no	—
299	D₄	—	86	1	no	—
300	D₄	—	86	1	no	—
301	D₄	—	344	1	no	—
302	D₄	—	344	1	no	—
303	D₄	—	172	2	no	—
304	D₄	—	172	2	no	—
305	D₄	#10621900	1	2	yes	complemented
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	8	1	no	—
308	C₂×C₄	—	43	1	no	—
309	C₂×C₄	—	43	1	no	—
310	C₂×C₄	—	43	1	no	—
311	C₂×C₄	—	43	1	no	—
312	C₂×C₄	—	86	1	no	—
313	C₂×C₄	—	86	1	no	—
314	C₂×C₄	—	86	1	no	—
315	C₂×C₄	—	344	1	no	—
316	C₂×C₄	—	172	2	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	—	172	2	no	—
320	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
321	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
322	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
323	C₈	—	2	1	no	—
324	C₈	—	2	1	no	—
325	C₈	—	86	1	no	—
326	C₈	—	86	1	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	4	1	no	—
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	172	1	no	—
337	C₂²	—	172	1	no	—
338	C₂²	—	172	1	no	—
339	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
340	C₄	—	2	1	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	43	2	no	—
344	C₄	—	86	1	no	—
345	C₄	—	86	1	no	—
346	C₄	—	86	1	no	—
347	C₄	—	172	1	no	—
348	C₄	—	172	1	no	—
349	C₄	—	172	1	no	—
350	C₄	C₂×#10621900	1	1	yes	characteristic
351	C₄	#13063372	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	8	1	no	—
354	C₂	—	8	1	no	—
355	C₂	—	344	1	no	—
356	C₂	#14583555	1	1	yes	characteristic
357	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
358	C₂	#14583700	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#459	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#459	D₄₃
hypocenter	C₄₃	#3366
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #459

Complements

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

N	H ≅ G/N	type
1. #14583608	C₂	semidirect
2. #14583569	C₂	semidirect
3. #14583263	C₂	semidirect
4. #14583262	C₂	semidirect
5. #14583244	C₂	semidirect
6. #14583200	C₂	semidirect
7. #14583197	C₂	semidirect
8. #14583187	C₂	semidirect
9. #14583092	C₂	semidirect
10. #14583092	C₂	semidirect
11. #14583028	C₂	semidirect
12. #14583011	C₂	semidirect
13. #14583009	C₂	semidirect
14. D₈×Dic₄₃	C₂	semidirect
15. C₄₃×#459	C₂	semidirect
117. D₄×C₄₃	#41	semidirect
190. #459	D₄₃	semidirect
195. C₄₃	#3366	semidirect
305. D₄	#10621900	semidirect