Subgroups — #19147627

Order 5504.

350 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583627	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583621	C₂	1	1	yes	maximal, characteristic
3	#14583293	C₂	1	1	yes	maximal, characteristic
4	#14583292	C₂	1	1	yes	maximal, characteristic
5	#14583277	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583124	C₂	1	1	yes	maximal, characteristic
7	#14583121	C₂	1	1	yes	maximal, characteristic
8	#14583109	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583103	C₂	1	1	yes	maximal, characteristic
12	#14583027	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583005	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582986	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#441	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
18	C₂×#10621885	C₂²	1	1	yes	characteristic
19	C₂×#10621885	C₂²	1	1	yes	characteristic
20	C₂×#10621885	C₂²	1	1	yes	characteristic
21	C₂×#10621873	C₂²	1	1	yes	characteristic
22	C₂×#10621873	C₂²	1	1	yes	characteristic
23	C₂×#10621884	C₂²	1	1	yes	characteristic
24	#13063325	C₂²	1	1	yes	characteristic
25	#13063323	C₂²	1	1	yes	characteristic
26	#13063323	C₂²	1	1	yes	characteristic
27	#13063323	C₂²	1	1	yes	characteristic
28	#13063320	—	2	1	no	—
29	#13063319	—	2	1	no	—
30	#13063317	—	2	1	no	—
31	#13063315	—	2	1	no	—
32	#13063315	—	2	1	no	—
33	#13063314	C₂²	1	1	yes	characteristic
34	#13063313	C₂²	1	1	yes	characteristic
35	#13063303	—	2	1	no	—
36	C₄×Dic₈₆	—	2	1	no	—
37	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
38	#13063290	C₂²	1	1	yes	characteristic
39	#13063290	C₂²	1	1	yes	characteristic
40	#13063290	C₂²	1	1	yes	characteristic
41	#13063287	C₂²	1	1	yes	characteristic
42	#13063282	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
46	#13063248	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063248	C₂²	1	1	yes	characteristic
49	#13063245	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#97×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
62	C₂×C₄×D₄₃	—	2	1	no	—
63	C₂×Dic₈₆	D₄	1	1	yes	characteristic
64	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
65	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
66	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
67	Q₁₆×C₄₃	—	2	2	no	—
68	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	C₄⋊C₄×C₄₃	—	2	1	no	—
71	C₄⋊C₄×C₄₃	—	2	1	no	—
72	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
74	C₄²×C₄₃	—	2	1	no	—
75	C₄²×C₄₃	—	2	1	no	—
76	C₄²×C₄₃	C₂³	1	1	yes	characteristic
77	#10621885	—	4	1	no	—
78	#10621885	—	4	1	no	—
79	#10621885	—	4	1	no	—
80	#10621884	—	4	1	no	—
81	(C₂×D₄₃)⋊C₄	—	2	1	no	—
82	(C₂×D₄₃)⋊C₄	—	2	1	no	—
83	(C₂×D₄₃)⋊C₄	—	2	1	no	—
84	(C₂×D₄₃)⋊C₄	—	2	1	no	—
85	(C₂×D₄₃)⋊C₄	—	2	1	no	—
86	C₄⋊Dic₄₃	—	2	1	no	—
87	C₄⋊Dic₄₃	—	2	1	no	—
88	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
89	Dic₄₃⋊C₄	—	2	1	no	—
90	Dic₄₃⋊C₄	—	2	1	no	—
91	Dic₄₃⋊C₄	—	2	1	no	—
92	Dic₄₃⋊C₄	—	2	1	no	—
93	Dic₄₃⋊C₄	—	2	1	no	—
94	Dic₄₃⋊C₄	—	2	1	no	—
95	Dic₄₃⋊C₄	—	2	1	no	—
96	Dic₄₃⋊C₄	—	2	1	no	—
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
99	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
104	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	Dic₁₇₂	—	2	2	no	—
108	Dic₁₇₂	—	4	1	no	—
109	#10621873	—	2	2	no	—
110	#10621873	—	4	1	no	—
111	C₂²×D₄₃	—	2	1	no	—
112	Q₈×C₄₃	—	2	1	no	—
113	Q₈×C₄₃	—	2	1	no	—
114	Q₈×C₄₃	—	2	1	no	—
115	Q₈×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	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	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	D₁₇₂	—	2	1	no	—
131	D₁₇₂	—	2	1	no	—
132	C₄×D₄₃	—	4	2	no	—
133	Dic₈₆	—	2	2	no	—
134	Dic₈₆	—	2	1	no	—
135	Dic₈₆	—	2	1	no	—
136	Dic₈₆	—	2	1	no	—
137	Dic₈₆	—	2	1	no	—
138	Dic₈₆	—	2	1	no	—
139	Dic₈₆	—	2	1	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	#41	1	2	yes	—
142	C₄₃⋊C₈	—	2	1	no	—
143	C₄₃⋊C₈	—	2	1	no	—
144	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₄₃	—	4	1	no	—
146	C₂×D₄₃	—	4	1	no	—
147	C₂×D₄₃	—	4	1	no	—
148	C₄×C₄₃	—	2	1	no	—
149	C₄×C₄₃	—	2	1	no	—
150	C₄×C₄₃	—	4	1	no	—
151	C₄×C₄₃	—	4	1	no	—
152	C₄×C₄₃	—	4	1	no	—
153	C₄×C₄₃	—	4	1	no	—
154	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
155	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
156	Dic₄₃	—	2	1	no	—
157	Dic₄₃	—	2	1	no	—
158	Dic₄₃	—	2	1	no	—
159	Dic₄₃	—	4	1	no	—
160	Dic₄₃	—	4	1	no	—
161	Dic₄₃	—	4	1	no	—
162	Dic₄₃	#137	1	2	yes	—
163	#3398	—	43	1	no	maximal, Sylow
164	C₂×C₄₃	#550	1	1	yes	characteristic
165	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
166	C₂×C₄₃	#578	1	1	yes	characteristic
167	D₄₃	—	8	1	no	—
168	#557	—	43	1	no	—
169	#556	—	43	1	no	—
170	#497	—	43	1	no	—
171	#497	—	43	1	no	—
172	#494	—	43	1	no	—
173	#464	—	43	1	no	—
174	#464	—	43	1	no	—
175	#462	—	43	1	no	—
176	#462	—	43	1	no	—
177	#462	—	43	1	no	—
178	#461	—	43	1	no	—
179	#446	—	43	1	no	—
180	#441	—	43	1	no	—
181	#441	D₄₃	1	1	yes	characteristic, complemented
182	C₄×SD₁₆	—	43	1	no	—
183	C₄₃	#3398	1	1	yes	characteristic, complemented, Sylow
184	C₂×Q₁₆	—	43	1	no	—
185	C₂×Q₁₆	—	43	1	no	—
186	C₂×Q₁₆	—	43	1	no	—
187	C₂×Q₁₆	—	43	1	no	—
188	C₂×Q₁₆	—	43	1	no	—
189	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
190	C₂×SD₁₆	—	43	1	no	—
191	C₂×SD₁₆	—	43	1	no	—
192	C₂×SD₁₆	—	43	1	no	—
193	C₄⋊Q₈	—	43	1	no	—
194	C₄⋊Q₈	—	43	1	no	—
195	C₄⋊Q₈	—	43	1	no	—
196	C₄⋊Q₈	—	43	1	no	—
197	#126	—	86	1	no	—
198	#126	—	86	1	no	—
199	#125	—	86	1	no	—
200	#125	—	86	1	no	—
201	#125	—	86	1	no	—
202	#124	—	43	1	no	—
203	#124	—	43	1	no	—
204	C₂²⋊Q₈	—	86	1	no	—
205	C₄×Q₈	—	43	1	no	—
206	C₄×Q₈	—	43	1	no	—
207	C₄×Q₈	—	43	1	no	—
208	C₄×Q₈	—	43	1	no	—
209	C₄×Q₈	—	86	1	no	—
210	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
211	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
212	C₄×D₄	—	43	1	no	—
213	#117	—	86	1	no	—
214	#106	C₂×D₄₃	1	1	yes	characteristic
215	C₄⋊C₈	—	43	1	no	—
216	C₄⋊C₈	—	43	1	no	—
217	C₄⋊C₈	—	43	1	no	—
218	#103	—	43	1	no	—
219	#103	—	43	1	no	—
220	#103	—	43	1	no	—
221	#103	C₂×D₄₃	1	1	yes	characteristic
222	#103	C₂×D₄₃	1	1	yes	characteristic
223	#102	—	43	1	no	—
224	#97	—	43	1	no	—
225	#97	C₂×D₄₃	1	1	yes	characteristic
226	C₄×C₈	—	43	1	no	—
227	C₂×Q₈	—	43	1	no	—
228	C₂×Q₈	—	43	1	no	—
229	C₂×Q₈	—	43	1	no	—
230	C₂×Q₈	—	43	1	no	—
231	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
232	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₂×D₄	—	43	1	no	—
234	C₂²×C₄	—	86	1	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	86	2	no	—
237	Q₁₆	—	172	1	no	—
238	Q₁₆	—	172	1	no	—
239	Q₁₆	—	172	1	no	—
240	Q₁₆	—	172	1	no	—
241	SD₁₆	—	86	2	no	—
242	SD₁₆	—	172	1	no	—
243	SD₁₆	—	172	1	no	—
244	C₂×C₈	—	43	1	no	—
245	C₂×C₈	—	43	1	no	—
246	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	2	1	no	—
250	C₄⋊C₄	—	43	1	no	—
251	C₄⋊C₄	—	43	1	no	—
252	C₄⋊C₄	—	43	1	no	—
253	C₄⋊C₄	—	86	1	no	—
254	C₄⋊C₄	—	86	1	no	—
255	C₄⋊C₄	—	86	1	no	—
256	C₄⋊C₄	—	86	1	no	—
257	C₄⋊C₄	—	86	1	no	—
258	C₄⋊C₄	—	86	1	no	—
259	C₄⋊C₄	—	86	1	no	—
260	C₄⋊C₄	—	86	1	no	—
261	C₄⋊C₄	—	86	1	no	—
262	C₄⋊C₄	—	86	1	no	—
263	C₄⋊C₄	—	86	1	no	—
264	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
266	C₂²⋊C₄	—	86	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₄²	—	2	1	no	—
272	C₄²	—	2	1	no	—
273	C₄²	—	43	1	no	—
274	C₄²	—	43	1	no	—
275	C₄²	—	86	1	no	—
276	C₄²	—	86	1	no	—
277	C₄²	—	86	1	no	—
278	C₄²	C₂²×D₄₃	1	1	yes	characteristic
279	C₂³	—	86	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	86	2	no	—
285	Q₈	—	86	1	no	—
286	Q₈	—	86	1	no	—
287	Q₈	—	86	1	no	—
288	Q₈	—	86	1	no	—
289	Q₈	—	86	1	no	—
290	Q₈	—	86	1	no	—
291	D₄	—	86	1	no	—
292	D₄	—	86	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	43	1	no	—
298	C₂×C₄	—	43	1	no	—
299	C₂×C₄	—	43	1	no	—
300	C₂×C₄	—	43	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	86	1	no	—
304	C₂×C₄	—	172	2	no	—
305	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
306	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	86	1	no	—
310	C₈	—	86	1	no	—
311	C₈	#10621898	1	2	yes	—
312	C₂²	—	172	1	no	—
313	C₂²	—	172	1	no	—
314	C₂²	—	172	1	no	—
315	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	43	2	no	—
323	C₄	—	86	1	no	—
324	C₄	—	86	1	no	—
325	C₄	—	86	1	no	—
326	C₄	—	172	1	no	—
327	C₄	—	172	1	no	—
328	C₄	—	172	1	no	—
329	C₄	C₂×#10621898	1	1	yes	characteristic
330	C₄	#13063371	1	1	yes	characteristic
331	C₂	—	344	1	no	—
332	C₂	#14583570	1	1	yes	characteristic
333	C₂	#137×D₄₃	1	1	yes	characteristic
334	C₂	#14583715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#441	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#441	D₄₃
hypocenter	C₄₃	#3398
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
1. #14583627	C₂	semidirect
5. #14583277	C₂	semidirect
8. #14583109	C₂	semidirect
9. #14583108	C₂	semidirect
10. #14583108	C₂	semidirect
12. #14583027	C₂	semidirect
13. #14583005	C₂	semidirect
15. C₄₃×#441	C₂	semidirect
181. #441	D₄₃	semidirect
183. C₄₃	#3398	semidirect