Subgroups — #19147068

Order 5504.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583593	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583549	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583307	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583218	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583216	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583214	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583201	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583188	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583150	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583134	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583008	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582973	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582942	C₂	1	1	yes	maximal, characteristic, complemented
14	#106×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#482	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	—	2	1	no	—
17	C₂×#10621882	C₂²	1	1	yes	characteristic
18	C₂×#10621877	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063319	—	2	1	no	—
22	#13063316	C₂²	1	1	yes	characteristic
23	#13063307	—	2	1	no	—
24	#13063307	—	2	1	no	—
25	#13063306	—	2	1	no	—
26	#13063302	—	2	1	no	—
27	#13063300	C₂²	1	1	yes	characteristic
28	#13063299	C₂²	1	1	yes	characteristic
29	#13063298	—	2	1	no	—
30	#13063296	C₂²	1	1	yes	characteristic
31	C₄×D₁₇₂	—	2	1	no	—
32	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	#13063287	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₄₃	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
37	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
38	#13063258	C₂²	1	1	yes	characteristic
39	#13063258	C₂²	1	1	yes	characteristic
40	#13063257	C₂²	1	1	yes	characteristic
41	#13063257	C₂²	1	1	yes	characteristic
42	#13063256	C₂²	1	1	yes	characteristic
43	#13063256	C₂²	1	1	yes	characteristic
44	#13063256	C₂²	1	1	yes	characteristic
45	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
46	#13063247	C₂²	1	1	yes	characteristic
47	#13063247	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	#13063245	C₂²	1	1	yes	characteristic
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
51	#13063235	C₂²	1	1	yes	characteristic
52	#121×C₄₃	C₂²	1	1	yes	characteristic
53	#117×C₄₃	C₂²	1	1	yes	characteristic
54	#107×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	#102×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₄₃	—	2	1	no	—
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂×C₂²⋊D₄₃	D₄	1	1	yes	characteristic
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	1	no	—
67	#10621900	—	4	4	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
74	C₂×Dic₈₆	—	2	1	no	—
75	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
80	C₂²⋊C₄×C₄₃	—	2	1	no	—
81	C₂²⋊C₄×C₄₃	—	2	1	no	—
82	C₄²×C₄₃	—	2	1	no	—
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
86	#10621882	—	2	2	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	(C₂×D₄₃)⋊C₄	—	2	1	no	—
90	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄⋊Dic₄₃	—	2	1	no	—
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	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
102	#10621877	—	2	2	no	—
103	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
104	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
105	C₈×D₄₃	—	2	2	no	—
106	C₂³×C₄₃	—	2	1	no	—
107	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₄₃	—	2	1	no	—
109	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
110	D₄×C₄₃	—	2	1	no	—
111	D₄×C₄₃	—	2	1	no	—
112	D₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₄₃	—	2	2	no	—
121	C₂²⋊D₄₃	—	4	2	no	—
122	C₂²⋊D₄₃	—	8	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	4	2	no	—
128	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	D₁₇₂	—	2	1	no	—
132	D₁₇₂	—	2	1	no	—
133	D₁₇₂	—	4	2	no	—
134	C₄×D₄₃	—	2	1	no	—
135	C₄×D₄₃	—	2	1	no	—
136	C₄×D₄₃	—	4	2	no	—
137	C₄×D₄₃	—	8	1	no	—
138	C₄×D₄₃	—	8	1	no	—
139	Dic₈₆	—	4	2	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₄₃⋊C₈	—	2	1	no	—
143	C₄₃⋊C₈	#41	1	2	yes	complemented
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₂×D₄₃	—	2	1	no	—
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	4	1	no	—
154	C₂×D₄₃	—	4	1	no	—
155	C₂×D₄₃	—	4	1	no	—
156	C₂×D₄₃	AGL(1,ℤ/8)	1	2	yes	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	4	1	no	—
159	C₄×C₄₃	—	4	1	no	—
160	C₄×C₄₃	—	4	1	no	—
161	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
162	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
163	Dic₄₃	—	2	1	no	—
164	Dic₄₃	—	2	1	no	—
165	Dic₄₃	—	2	1	no	—
166	Dic₄₃	—	4	1	no	—
167	Dic₄₃	—	4	1	no	—
168	Dic₄₃	—	4	1	no	—
169	#3345	—	43	1	no	maximal, Sylow
170	C₂×C₄₃	—	4	1	no	—
171	C₂×C₄₃	—	8	1	no	—
172	C₂×C₄₃	#547	1	1	yes	characteristic
173	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
174	C₂×C₄₃	#577	1	1	yes	characteristic
175	D₄₃	—	2	2	no	—
176	D₄₃	—	8	1	no	—
177	#553	—	43	1	no	—
178	#547	—	43	1	no	—
179	#501	—	43	1	no	—
180	#482	D₄₃	1	1	yes	characteristic, complemented
181	#481	—	43	1	no	—
182	#481	—	43	1	no	—
183	#480	—	43	1	no	—
184	#478	—	43	1	no	—
185	#476	—	43	1	no	—
186	#469	—	43	1	no	—
187	#466	—	43	1	no	—
188	#442	—	43	1	no	—
189	#435	—	43	1	no	—
190	#428	—	43	1	no	—
191	C₂×#106	—	43	1	no	—
192	C₄₃	#3345	1	1	yes	characteristic, complemented, Sylow
193	C₂×#41	—	86	1	no	—
194	C₂×D₈	—	43	1	no	—
195	C₂×M₁₆	—	43	1	no	—
196	C₂²×C₈	—	43	1	no	—
197	C₄⋊Q₈	—	43	1	no	—
198	#126	—	86	1	no	—
199	#125	—	43	1	no	—
200	#123	—	86	1	no	—
201	#123	—	86	1	no	—
202	#123	—	86	1	no	—
203	C₂²⋊Q₈	—	86	1	no	—
204	#121	—	43	1	no	—
205	#121	—	43	1	no	—
206	#121	—	43	1	no	—
207	#121	—	86	1	no	—
208	#121	C₂×D₄₃	1	1	yes	characteristic
209	C₄×D₄	—	43	1	no	—
210	C₄×D₄	—	43	1	no	—
211	C₄×D₄	—	43	1	no	—
212	C₄×D₄	—	86	1	no	—
213	#117	C₂×D₄₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	—	43	1	no	—
215	C₂×C₄⋊C₄	—	43	1	no	—
216	C₂×C₄⋊C₄	—	43	1	no	—
217	#107	—	43	1	no	—
218	#107	—	43	1	no	—
219	#107	C₂×D₄₃	1	1	yes	characteristic
220	#106	—	43	1	no	—
221	#106	—	43	1	no	—
222	#106	—	43	1	no	—
223	#106	—	43	1	no	—
224	#106	—	43	1	no	—
225	#106	C₂×D₄₃	1	1	yes	characteristic
226	C₄⋊C₈	—	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	C₂×D₄₃	1	1	yes	characteristic
232	#102	C₂×D₄₃	1	1	yes	characteristic
233	C₂²⋊C₈	—	43	1	no	—
234	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
235	#97	—	43	1	no	—
236	#41	—	172	4	no	—
237	C₂×Q₈	—	86	1	no	—
238	C₂×D₄	—	2	1	no	—
239	C₂×D₄	—	43	1	no	—
240	C₂×D₄	—	43	1	no	—
241	C₂×D₄	—	86	1	no	—
242	C₂×D₄	—	86	1	no	—
243	C₂×D₄	—	86	1	no	—
244	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
245	C₂²×C₄	—	43	1	no	—
246	C₂²×C₄	—	86	1	no	—
247	C₂²×C₄	—	86	1	no	—
248	C₂²×C₄	—	86	1	no	—
249	C₂²×C₄	—	86	1	no	—
250	C₂²×C₄	—	86	1	no	—
251	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
252	D₈	—	86	2	no	—
253	M₁₆	—	86	2	no	—
254	C₂×C₈	—	43	1	no	—
255	C₂×C₈	—	43	1	no	—
256	C₂×C₈	—	86	2	no	—
257	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
258	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
259	C₄⋊C₄	—	43	1	no	—
260	C₄⋊C₄	—	43	1	no	—
261	C₄⋊C₄	—	43	1	no	—
262	C₄⋊C₄	—	43	1	no	—
263	C₄⋊C₄	—	86	1	no	—
264	C₄⋊C₄	—	86	1	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	—	86	1	no	—
267	C₄⋊C₄	—	86	1	no	—
268	C₄⋊C₄	—	86	1	no	—
269	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
270	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
271	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
272	C₂²⋊C₄	—	2	1	no	—
273	C₂²⋊C₄	—	2	1	no	—
274	C₂²⋊C₄	—	43	1	no	—
275	C₂²⋊C₄	—	43	1	no	—
276	C₂²⋊C₄	—	86	1	no	—
277	C₂²⋊C₄	—	86	1	no	—
278	C₂²⋊C₄	—	86	1	no	—
279	C₂²⋊C₄	—	86	1	no	—
280	C₂²⋊C₄	—	86	1	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	43	1	no	—
283	C₂³	—	2	1	no	—
284	C₂³	—	43	1	no	—
285	C₂³	—	86	1	no	—
286	C₂³	D₄×D₄₃	1	1	yes	characteristic
287	Q₈	—	172	2	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	4	2	no	—
291	D₄	—	86	2	no	—
292	D₄	—	86	1	no	—
293	D₄	—	86	1	no	—
294	D₄	—	344	1	no	—
295	D₄	—	172	2	no	—
296	D₄	—	172	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	2	1	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	43	1	no	—
304	C₂×C₄	—	43	1	no	—
305	C₂×C₄	—	86	1	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	86	1	no	—
308	C₂×C₄	—	86	1	no	—
309	C₂×C₄	—	86	1	no	—
310	C₂×C₄	—	344	1	no	—
311	C₂×C₄	—	344	1	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	—	172	2	no	—
314	C₂×C₄	—	172	2	no	—
315	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
316	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
317	C₈	—	2	1	no	—
318	C₈	—	2	1	no	—
319	C₈	—	43	2	no	—
320	C₈	—	86	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	2	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	4	1	no	—
327	C₂²	—	43	2	no	—
328	C₂²	—	86	1	no	—
329	C₂²	—	86	1	no	—
330	C₂²	—	172	1	no	—
331	C₂²	—	172	1	no	—
332	C₂²	—	172	1	no	—
333	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
334	C₄	—	2	1	no	—
335	C₄	—	4	1	no	—
336	C₄	—	4	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	86	1	no	—
339	C₄	—	86	1	no	—
340	C₄	—	86	1	no	—
341	C₄	—	172	1	no	—
342	C₄	—	172	1	no	—
343	C₄	—	172	1	no	—
344	C₄	#41×D₄₃	1	1	yes	characteristic
345	C₄	#13063372	1	1	yes	characteristic
346	C₂	—	4	1	no	—
347	C₂	—	8	1	no	—
348	C₂	—	86	2	no	—
349	C₂	—	344	1	no	—
350	C₂	#14583547	1	1	yes	characteristic
351	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
352	C₂	#14583706	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#482	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#482	D₄₃
hypocenter	C₄₃	#3345
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #482

Complements

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

N	H ≅ G/N	type
1. #14583593	C₂	semidirect
2. #14583549	C₂	semidirect
3. #14583307	C₂	semidirect
4. #14583218	C₂	semidirect
5. #14583216	C₂	semidirect
6. #14583214	C₂	semidirect
7. #14583201	C₂	semidirect
8. #14583188	C₂	semidirect
9. #14583150	C₂	semidirect
10. #14583134	C₂	semidirect
11. #14583008	C₂	semidirect
12. #14582973	C₂	semidirect
13. #14582942	C₂	semidirect
14. #106×D₄₃	C₂	semidirect
15. C₄₃×#482	C₂	semidirect
143. C₄₃⋊C₈	#41	semidirect
180. #482	D₄₃	semidirect
192. C₄₃	#3345	semidirect