Subgroups — #19144454

Order 5504.

376 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583526	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583381	C₂	1	1	yes	maximal, characteristic
3	#14583208	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583206	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583200	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583198	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583188	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583186	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582938	C₂	1	1	yes	maximal, characteristic, complemented
11	#106×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582924	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582899	C₂	1	1	yes	maximal, characteristic
14	#14582894	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621902	C₂²	1	1	yes	characteristic
17	C₂×#10621872	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063323	—	2	1	no	—
21	#13063317	C₂²	1	1	yes	characteristic
22	#13063316	C₂²	1	1	yes	characteristic
23	#13063315	—	2	1	no	—
24	#13063303	—	2	1	no	—
25	#13063301	—	2	1	no	—
26	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
27	C₄×D₁₇₂	—	2	1	no	—
28	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
29	#13063287	—	2	1	no	—
30	#13063287	C₂²	1	1	yes	characteristic
31	#13063283	C₂²	1	1	yes	characteristic
32	#13063282	—	2	1	no	—
33	#13063282	C₂²	1	1	yes	characteristic
34	C₄⋊C₄×D₄₃	—	2	1	no	—
35	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₄₃	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	#13063256	C₂²	1	1	yes	characteristic
42	#13063256	C₂²	1	1	yes	characteristic
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	#13063249	C₂²	1	1	yes	characteristic
47	#13063249	C₂²	1	1	yes	characteristic
48	#13063247	C₂²	1	1	yes	characteristic
49	#13063247	C₂²	1	1	yes	characteristic
50	#13063245	C₂²	1	1	yes	characteristic
51	#13063245	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
54	#106×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#103×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	#10621902	—	2	2	no	—
62	#10621902	Q₈	1	4	yes	complemented
63	C₂×D₁₇₂	—	2	1	no	—
64	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
65	C₂×C₄×D₄₃	—	2	1	no	—
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	—	2	1	no	—
70	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
71	C₂×Dic₈₆	—	2	1	no	—
72	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₄⋊C₄×C₄₃	—	2	1	no	—
75	C₄⋊C₄×C₄₃	—	2	1	no	—
76	C₄⋊C₄×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	C₄²×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
85	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
86	C₄⋊Dic₄₃	—	2	1	no	—
87	C₄⋊Dic₄₃	—	2	1	no	—
88	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
89	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
90	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
98	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
104	#10621872	—	2	2	no	—
105	C₈×D₄₃	—	2	2	no	—
106	C₂²×D₄₃	—	2	1	no	—
107	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
108	Q₈×C₄₃	—	2	1	no	—
109	Q₈×C₄₃	—	8	1	no	—
110	Q₈×C₄₃	C₂×Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	1	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	D₁₇₂	—	2	4	no	—
125	D₁₇₂	—	2	1	no	—
126	D₁₇₂	C₂×Q₈	1	2	yes	—
127	C₄×D₄₃	—	2	4	no	—
128	C₄×D₄₃	—	8	1	no	—
129	C₄×D₄₃	—	8	1	no	—
130	C₄×D₄₃	—	8	1	no	—
131	C₄×D₄₃	—	8	1	no	—
132	C₄×D₄₃	C₂×Q₈	1	2	yes	—
133	C₄×D₄₃	#41	1	2	yes	—
134	Dic₈₆	—	8	1	no	—
135	C₈×C₄₃	—	2	1	no	—
136	C₈×C₄₃	—	2	1	no	—
137	C₄₃⋊C₈	—	2	1	no	—
138	C₄₃⋊C₈	—	2	1	no	—
139	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₄₃	—	2	1	no	—
141	C₂×D₄₃	—	2	1	no	—
142	C₂×D₄₃	—	2	2	no	—
143	C₂×D₄₃	—	4	1	no	—
144	C₂×D₄₃	—	4	1	no	—
145	C₂×D₄₃	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₄₃	—	2	2	no	—
147	C₄×C₄₃	—	2	1	no	—
148	C₄×C₄₃	—	2	1	no	—
149	C₄×C₄₃	—	4	1	no	—
150	C₄×C₄₃	—	4	1	no	—
151	C₄×C₄₃	—	4	1	no	—
152	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
153	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
154	Dic₄₃	—	2	1	no	—
155	Dic₄₃	—	2	1	no	—
156	Dic₄₃	—	4	1	no	—
157	Dic₄₃	—	4	1	no	—
158	Dic₄₃	—	4	1	no	—
159	Dic₄₃	C₂²⋊Q₈	1	2	yes	—
160	#3084	—	43	1	no	maximal, Sylow
161	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₄₃	#577	1	1	yes	characteristic
163	C₂×C₄₃	#577	1	1	yes	characteristic
164	D₄₃	—	2	2	no	—
165	D₄₃	—	4	2	no	—
166	#544	—	43	1	no	—
167	#520	—	43	1	no	—
168	#479	—	43	1	no	—
169	#479	—	43	1	no	—
170	#478	—	43	1	no	—
171	#478	—	43	1	no	—
172	#476	—	43	1	no	—
173	#476	—	43	1	no	—
174	#475	—	43	1	no	—
175	#475	D₄₃	1	1	yes	characteristic, complemented
176	#428	—	43	1	no	—
177	C₂×#106	—	43	1	no	—
178	#424	—	43	1	no	—
179	#417	—	43	1	no	—
180	#416	—	43	1	no	—
181	C₄₃	#3084	1	1	yes	characteristic, complemented, Sylow
182	C₂×#41	—	43	1	no	—
183	C₂×M₁₆	—	43	1	no	—
184	C₂²×C₈	—	43	1	no	—
185	C₄⋊Q₈	—	43	1	no	—
186	C₄⋊Q₈	—	86	1	no	—
187	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
188	#125	—	43	1	no	—
189	#125	—	43	1	no	—
190	#125	—	86	1	no	—
191	C₂²⋊Q₈	—	86	1	no	—
192	C₂²⋊Q₈	—	86	1	no	—
193	C₄×Q₈	—	43	1	no	—
194	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
195	C₄×D₄	—	43	1	no	—
196	C₄×D₄	—	43	1	no	—
197	C₄×D₄	—	86	1	no	—
198	C₄×D₄	—	86	1	no	—
199	#117	—	43	1	no	—
200	#117	—	43	1	no	—
201	#117	—	86	1	no	—
202	C₂×C₄⋊C₄	—	43	1	no	—
203	C₂×C₄⋊C₄	—	43	1	no	—
204	C₂×C₄⋊C₄	—	86	1	no	—
205	#107	—	43	1	no	—
206	#107	—	43	1	no	—
207	#106	—	43	1	no	—
208	#106	—	43	1	no	—
209	#106	—	43	1	no	—
210	#106	—	43	1	no	—
211	#106	C₂×D₄₃	1	1	yes	characteristic
212	#106	C₂×D₄₃	1	1	yes	characteristic
213	C₄⋊C₈	—	43	1	no	—
214	C₄⋊C₈	—	43	1	no	—
215	C₄⋊C₈	—	43	1	no	—
216	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
217	#103	—	43	1	no	—
218	#103	—	43	1	no	—
219	#103	C₂×D₄₃	1	1	yes	characteristic
220	#103	C₂×D₄₃	1	1	yes	characteristic
221	#102	—	43	1	no	—
222	#102	—	43	1	no	—
223	#102	—	43	1	no	—
224	#102	—	43	1	no	—
225	#41	—	43	4	no	—
226	#41	—	86	2	no	—
227	C₂×Q₈	—	2	1	no	—
228	C₂×Q₈	—	86	1	no	—
229	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
230	C₂×D₄	—	43	1	no	—
231	C₂×D₄	—	86	1	no	—
232	C₂²×C₄	—	43	1	no	—
233	C₂²×C₄	—	86	1	no	—
234	C₂²×C₄	—	86	1	no	—
235	C₂²×C₄	—	86	1	no	—
236	C₂²×C₄	—	86	1	no	—
237	C₂²×C₄	—	86	1	no	—
238	M₁₆	—	86	2	no	—
239	C₂×C₈	—	43	1	no	—
240	C₂×C₈	—	43	1	no	—
241	C₂×C₈	—	86	2	no	—
242	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	43	1	no	—
247	C₄⋊C₄	—	43	1	no	—
248	C₄⋊C₄	—	43	1	no	—
249	C₄⋊C₄	—	43	1	no	—
250	C₄⋊C₄	—	43	1	no	—
251	C₄⋊C₄	—	86	1	no	—
252	C₄⋊C₄	—	86	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	C₂²×D₄₃	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
261	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
262	C₂²⋊C₄	—	43	1	no	—
263	C₂²⋊C₄	—	43	1	no	—
264	C₂²⋊C₄	—	86	1	no	—
265	C₂²⋊C₄	—	86	1	no	—
266	C₂²⋊C₄	—	86	1	no	—
267	C₄²	—	2	1	no	—
268	C₄²	—	43	1	no	—
269	C₄²	—	86	1	no	—
270	C₄²	—	86	1	no	—
271	C₄²	C₂²×D₄₃	1	1	yes	characteristic
272	C₂³	—	43	1	no	—
273	C₂³	—	86	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	8	1	no	—
276	Q₈	—	344	1	no	—
277	Q₈	Q₈×D₄₃	1	2	yes	—
278	D₄	—	43	2	no	—
279	D₄	—	86	4	no	—
280	D₄	—	86	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	43	1	no	—
286	C₂×C₄	—	43	2	no	—
287	C₂×C₄	—	43	2	no	—
288	C₂×C₄	—	43	1	no	—
289	C₂×C₄	—	43	1	no	—
290	C₂×C₄	—	86	4	no	—
291	C₂×C₄	—	86	1	no	—
292	C₂×C₄	—	86	1	no	—
293	C₂×C₄	—	86	1	no	—
294	C₂×C₄	—	344	1	no	—
295	C₂×C₄	—	344	1	no	—
296	C₂×C₄	—	344	1	no	—
297	C₂×C₄	—	344	1	no	—
298	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
299	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	86	1	no	—
304	C₈	—	86	1	no	—
305	C₂²	—	43	2	no	—
306	C₂²	—	86	1	no	—
307	C₂²	—	86	1	no	—
308	C₂²	—	86	2	no	—
309	C₂²	—	172	1	no	—
310	C₂²	—	172	1	no	—
311	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	43	2	no	—
319	C₄	—	86	1	no	—
320	C₄	—	86	1	no	—
321	C₄	—	172	1	no	—
322	C₄	—	172	1	no	—
323	C₄	—	172	1	no	—
324	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
325	C₄	#41×D₄₃	1	1	yes	characteristic
326	C₂	—	86	2	no	—
327	C₂	—	172	2	no	—
328	C₂	C₂²⋊Q₈×D₄₃	1	1	yes	characteristic
329	C₂	#14583705	1	1	yes	characteristic
330	C₂	#14583708	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#475	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#475	D₄₃
hypocenter	C₄₃	#3084
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #475

Complements

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

N	H ≅ G/N	type
1. #14583526	C₂	semidirect
3. #14583208	C₂	semidirect
4. #14583206	C₂	semidirect
5. #14583200	C₂	semidirect
6. #14583198	C₂	semidirect
7. #14583188	C₂	semidirect
8. #14583186	C₂	semidirect
9. #14583182	C₂	semidirect
10. #14582938	C₂	semidirect
11. #106×D₄₃	C₂	semidirect
12. #14582924	C₂	semidirect
15. C₄₃×#475	C₂	semidirect
62. #10621902	Q₈	semidirect
175. #475	D₄₃	semidirect
181. C₄₃	#3084	semidirect