Subgroups — #19144959

Order 5504.

372 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583529	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583469	C₂	1	1	yes	maximal, characteristic
3	#14583269	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583266	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583258	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583257	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583252	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583250	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582906	C₂	1	1	yes	maximal, characteristic, complemented
11	#14582897	C₂	1	1	yes	maximal, characteristic
12	#14582892	C₂	1	1	yes	maximal, characteristic
13	#103×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582853	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#488	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
18	C₂×#10621872	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
20	#13063325	C₂²	1	1	yes	characteristic
21	#13063325	C₂²	1	1	yes	characteristic
22	#13063324	—	2	1	no	—
23	#13063324	C₂²	1	1	yes	characteristic
24	#13063323	C₂²	1	1	yes	characteristic
25	#13063316	C₂²	1	1	yes	characteristic
26	#13063315	—	2	1	no	—
27	#13063312	C₂²	1	1	yes	characteristic
28	#13063307	—	2	1	no	—
29	#13063307	—	2	1	no	—
30	#13063305	—	2	1	no	—
31	#13063303	—	2	1	no	—
32	#13063302	—	2	1	no	—
33	#13063302	—	2	1	no	—
34	#13063283	C₂²	1	1	yes	characteristic
35	#13063282	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
37	#13063250	C₂²	1	1	yes	characteristic
38	#13063250	C₂²	1	1	yes	characteristic
39	#13063250	C₂²	1	1	yes	characteristic
40	#13063250	C₂²	1	1	yes	characteristic
41	#13063249	C₂²	1	1	yes	characteristic
42	#13063249	C₂²	1	1	yes	characteristic
43	#13063248	C₂²	1	1	yes	characteristic
44	#13063248	C₂²	1	1	yes	characteristic
45	#13063248	C₂²	1	1	yes	characteristic
46	#13063248	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063246	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	#125×C₄₃	C₂²	1	1	yes	characteristic
53	#124×C₄₃	C₂²	1	1	yes	characteristic
54	#103×C₄₃	C₂²	1	1	yes	characteristic
55	#103×C₄₃	C₂²	1	1	yes	characteristic
56	#102×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×C₂²⋊D₄₃	—	2	1	no	—
62	C₂²×Dic₄₃	—	2	1	no	—
63	Q₈×D₄₃	—	2	2	no	—
64	Q₈×D₄₃	—	2	2	no	—
65	#10621900	—	2	2	no	—
66	#10621900	—	2	2	no	—
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
70	C₂×Dic₈₆	—	2	1	no	—
71	C₂×Dic₈₆	—	2	1	no	—
72	C₂×Dic₈₆	—	2	1	no	—
73	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
74	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₄₃	—	2	1	no	—
78	C₄⋊C₄×C₄₃	—	2	1	no	—
79	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
80	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
81	C₂²⋊C₄×C₄₃	—	2	1	no	—
82	C₂²⋊C₄×C₄₃	—	2	1	no	—
83	C₄²×C₄₃	C₂³	1	1	yes	characteristic
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	C₂²⋊Dic₄₃	—	2	1	no	—
86	(C₂×D₄₃)⋊C₄	—	2	1	no	—
87	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
89	C₄⋊Dic₄₃	—	2	1	no	—
90	C₄⋊Dic₄₃	—	2	1	no	—
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	Dic₄₃⋊C₄	—	2	1	no	—
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	—	2	1	no	—
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	#10621872	—	2	2	no	—
107	C₈×D₄₃	—	2	2	no	—
108	C₂³×C₄₃	—	2	1	no	—
109	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₄₃	—	2	1	no	—
111	Q₈×C₄₃	—	2	1	no	—
112	D₄×C₄₃	—	2	1	no	—
113	D₄×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	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
120	C₂²⋊D₄₃	—	4	2	no	—
121	C₂×Dic₄₃	—	2	1	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	4	2	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	C₄×D₄₃	—	4	2	no	—
130	C₄×D₄₃	—	8	1	no	—
131	C₄×D₄₃	#41	1	2	yes	—
132	C₄×D₄₃	#41	1	2	yes	—
133	Dic₈₆	—	2	1	no	—
134	Dic₈₆	—	2	1	no	—
135	Dic₈₆	—	2	1	no	—
136	Dic₈₆	—	2	1	no	—
137	Dic₈₆	—	4	2	no	—
138	Dic₈₆	—	8	1	no	—
139	Dic₈₆	—	8	1	no	—
140	C₈×C₄₃	—	2	1	no	—
141	C₈×C₄₃	—	2	1	no	—
142	C₄₃⋊C₈	—	2	1	no	—
143	C₄₃⋊C₈	—	2	1	no	—
144	C₂²×C₄₃	—	4	1	no	—
145	C₂²×C₄₃	—	4	1	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₄₃	—	2	1	no	—
149	C₂×D₄₃	—	2	1	no	—
150	C₂×D₄₃	#124	1	2	yes	—
151	C₄×C₄₃	—	2	1	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	4	1	no	—
154	C₄×C₄₃	—	4	1	no	—
155	C₄×C₄₃	—	4	1	no	—
156	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
157	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
158	Dic₄₃	—	2	1	no	—
159	Dic₄₃	—	2	1	no	—
160	Dic₄₃	—	4	1	no	—
161	Dic₄₃	—	4	1	no	—
162	Dic₄₃	—	4	1	no	—
163	Dic₄₃	—	4	1	no	—
164	Dic₄₃	#124	1	2	yes	—
165	#3137	—	43	1	no	maximal, Sylow
166	C₂×C₄₃	—	8	1	no	—
167	C₂×C₄₃	C₂×#124	1	1	yes	characteristic
168	C₂×C₄₃	#577	1	1	yes	characteristic
169	C₂×C₄₃	#578	1	1	yes	characteristic
170	D₄₃	—	2	2	no	—
171	#544	—	43	1	no	—
172	#536	—	43	1	no	—
173	#491	—	43	1	no	—
174	#491	—	43	1	no	—
175	#489	—	43	1	no	—
176	#489	—	43	1	no	—
177	#488	—	43	1	no	—
178	#488	D₄₃	1	1	yes	characteristic, complemented
179	#487	—	43	1	no	—
180	#487	—	43	1	no	—
181	#419	—	43	1	no	—
182	#417	—	43	1	no	—
183	#416	—	43	1	no	—
184	C₂×#103	—	43	1	no	—
185	#405	—	43	1	no	—
186	C₄₃	#3137	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	43	1	no	—
188	C₂²×Q₈	—	43	1	no	—
189	C₂×M₁₆	—	43	1	no	—
190	C₂²×C₈	—	43	1	no	—
191	C₄⋊Q₈	—	43	1	no	—
192	C₄⋊Q₈	—	43	1	no	—
193	C₄⋊Q₈	—	43	1	no	—
194	C₄⋊Q₈	—	43	1	no	—
195	C₄⋊Q₈	—	86	1	no	—
196	#125	—	43	1	no	—
197	#125	—	86	1	no	—
198	#125	C₂×D₄₃	1	1	yes	characteristic
199	#124	—	43	1	no	—
200	#124	C₂×D₄₃	1	1	yes	characteristic
201	#123	—	86	1	no	—
202	#123	—	86	1	no	—
203	C₂²⋊Q₈	—	86	1	no	—
204	C₂²⋊Q₈	—	86	1	no	—
205	C₂²⋊Q₈	—	86	1	no	—
206	C₂²⋊Q₈	—	86	1	no	—
207	#117	—	43	1	no	—
208	#117	—	43	1	no	—
209	C₂×C₄⋊C₄	—	43	1	no	—
210	#103	—	43	1	no	—
211	#103	—	43	1	no	—
212	#103	—	43	1	no	—
213	#103	—	43	1	no	—
214	#103	—	43	1	no	—
215	#103	—	43	1	no	—
216	#103	—	43	1	no	—
217	#103	—	43	1	no	—
218	#103	—	43	1	no	—
219	#103	—	43	1	no	—
220	#103	C₂×D₄₃	1	1	yes	characteristic
221	#103	C₂×D₄₃	1	1	yes	characteristic
222	#102	—	43	1	no	—
223	#102	—	43	1	no	—
224	#102	C₂×D₄₃	1	1	yes	characteristic
225	#102	C₂×D₄₃	1	1	yes	characteristic
226	#97	—	43	1	no	—
227	#97	—	43	1	no	—
228	C₄×C₈	—	43	1	no	—
229	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
230	#41	—	86	2	no	—
231	#41	—	86	2	no	—
232	C₂×Q₈	—	43	1	no	—
233	C₂×Q₈	—	43	1	no	—
234	C₂×Q₈	—	86	1	no	—
235	C₂×Q₈	—	86	1	no	—
236	C₂×Q₈	—	86	2	no	—
237	C₂×Q₈	—	86	2	no	—
238	C₂×Q₈	—	86	1	no	—
239	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
240	C₂×D₄	—	86	1	no	—
241	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	—	43	1	no	—
243	C₂²×C₄	—	86	1	no	—
244	C₂²×C₄	—	86	1	no	—
245	C₂²×C₄	—	86	1	no	—
246	M₁₆	—	86	2	no	—
247	C₂×C₈	—	43	1	no	—
248	C₂×C₈	—	43	1	no	—
249	C₂×C₈	—	86	2	no	—
250	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
251	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	43	1	no	—
255	C₄⋊C₄	—	43	1	no	—
256	C₄⋊C₄	—	43	1	no	—
257	C₄⋊C₄	—	43	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	—	86	1	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	—	86	1	no	—
267	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	2	1	no	—
271	C₂²⋊C₄	—	43	1	no	—
272	C₂²⋊C₄	—	43	1	no	—
273	C₂²⋊C₄	—	86	1	no	—
274	C₂²⋊C₄	—	86	1	no	—
275	C₂²⋊C₄	—	86	1	no	—
276	C₄²	—	43	1	no	—
277	C₄²	—	86	1	no	—
278	C₄²	C₂²×D₄₃	1	1	yes	characteristic
279	C₂³	—	2	1	no	—
280	C₂³	—	43	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	86	1	no	—
284	Q₈	—	86	1	no	—
285	Q₈	—	86	1	no	—
286	Q₈	—	86	1	no	—
287	Q₈	—	344	1	no	—
288	Q₈	—	344	1	no	—
289	Q₈	—	172	2	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	172	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	43	1	no	—
297	C₂×C₄	—	43	2	no	—
298	C₂×C₄	—	43	2	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	—	86	1	no	—
305	C₂×C₄	—	344	1	no	—
306	C₂×C₄	—	172	2	no	—
307	C₂×C₄	—	172	2	no	—
308	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
309	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
310	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	86	1	no	—
314	C₈	—	86	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	43	2	no	—
319	C₂²	—	86	1	no	—
320	C₂²	—	86	1	no	—
321	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	43	2	no	—
328	C₄	—	86	1	no	—
329	C₄	—	86	1	no	—
330	C₄	—	172	1	no	—
331	C₄	—	172	1	no	—
332	C₄	—	172	1	no	—
333	C₄	—	172	1	no	—
334	C₄	#41×D₄₃	1	1	yes	characteristic
335	C₄	#41×D₄₃	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	86	2	no	—
338	C₂	#124×D₄₃	1	1	yes	characteristic
339	C₂	#14583706	1	1	yes	characteristic
340	C₂	#14583713	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#488	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#488	D₄₃
hypocenter	C₄₃	#3137
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #14583529	C₂	semidirect
3. #14583269	C₂	semidirect
4. #14583266	C₂	semidirect
5. #14583258	C₂	semidirect
6. #14583257	C₂	semidirect
7. #14583252	C₂	semidirect
8. #14583250	C₂	semidirect
9. #14583248	C₂	semidirect
10. #14582906	C₂	semidirect
13. #103×D₄₃	C₂	semidirect
14. #14582853	C₂	semidirect
15. C₄₃×#488	C₂	semidirect
178. #488	D₄₃	semidirect
186. C₄₃	#3137	semidirect