Subgroups — #19144804

Order 5504.

372 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583509	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583380	C₂	1	1	yes	maximal, characteristic
3	#14583229	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583205	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583202	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583103	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583063	C₂	1	1	yes	maximal, characteristic
9	#14583059	C₂	1	1	yes	maximal, characteristic
10	#14583003	C₂	1	1	yes	maximal, characteristic, complemented
11	#14582998	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582993	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582984	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582957	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#484	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
18	C₂×#10621885	C₂²	1	1	yes	characteristic
19	C₂×#10621873	C₂²	1	1	yes	characteristic
20	C₂×#10621884	C₂²	1	1	yes	characteristic
21	#13063323	C₂²	1	1	yes	characteristic
22	#13063318	—	2	1	no	—
23	#13063318	—	2	1	no	—
24	#13063316	—	2	1	no	—
25	#13063316	C₂²	1	1	yes	characteristic
26	#13063315	—	2	1	no	—
27	#13063304	C₂²	1	1	yes	characteristic
28	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
29	#13063290	C₂²	1	1	yes	characteristic
30	#13063290	C₂²	1	1	yes	characteristic
31	#13063290	C₂²	1	1	yes	characteristic
32	#13063287	C₂²	1	1	yes	characteristic
33	#13063287	C₂²	1	1	yes	characteristic
34	C₄×C₂²⋊D₄₃	—	2	1	no	—
35	C₄×C₂²⋊D₄₃	—	2	1	no	—
36	#13063284	C₂²	1	1	yes	characteristic
37	#13063283	—	2	1	no	—
38	C₄⋊C₄×D₄₃	—	2	1	no	—
39	#13063256	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	#13063250	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	#13063247	C₂²	1	1	yes	characteristic
48	#13063245	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
54	#107×C₄₃	C₂²	1	1	yes	characteristic
55	#107×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	C₂²⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	#10621898	—	2	2	no	—
64	#10621898	—	2	2	no	—
65	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	—	2	1	no	—
69	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
70	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
71	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
72	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₄⋊C₄×C₄₃	—	2	1	no	—
75	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
76	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
78	C₂²⋊C₄×C₄₃	—	2	1	no	—
79	C₂²⋊Dic₄₃	—	2	1	no	—
80	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
81	C₂²⋊Dic₄₃	D₄	1	1	yes	characteristic
82	#10621885	—	2	2	no	—
83	#10621884	—	2	2	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	C₄⋊Dic₄₃	D₄	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	—	2	1	no	—
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
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	Dic₁₇₂	—	2	2	no	—
107	#10621873	—	2	2	no	—
108	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂²×D₄₃	—	2	1	no	—
110	Q₈×C₄₃	—	2	1	no	—
111	Q₈×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	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	8	1	no	—
119	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	2	1	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
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	D₁₇₂	—	2	1	no	—
130	D₁₇₂	#41	1	2	yes	—
131	C₄×D₄₃	—	2	4	no	—
132	C₄×D₄₃	—	8	1	no	—
133	C₄×D₄₃	—	8	1	no	—
134	Dic₈₆	—	2	1	no	—
135	Dic₈₆	—	2	1	no	—
136	Dic₈₆	—	2	1	no	—
137	Dic₈₆	#41	1	2	yes	—
138	C₈×C₄₃	—	2	1	no	—
139	C₈×C₄₃	—	2	1	no	—
140	C₄₃⋊C₈	—	2	1	no	—
141	C₄₃⋊C₈	—	2	1	no	—
142	C₂²×C₄₃	—	2	1	no	—
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₄₃	—	2	2	no	—
147	C₂×D₄₃	—	4	1	no	—
148	C₂×D₄₃	—	4	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	C₂×#41	1	1	yes	characteristic
156	Dic₄₃	—	2	2	no	—
157	Dic₄₃	—	2	1	no	—
158	Dic₄₃	—	2	1	no	—
159	Dic₄₃	—	2	1	no	—
160	Dic₄₃	—	2	1	no	—
161	Dic₄₃	—	4	1	no	—
162	Dic₄₃	—	4	1	no	—
163	#3120	—	43	1	no	maximal, Sylow
164	C₂×C₄₃	—	4	1	no	—
165	C₂×C₄₃	#525	1	1	yes	characteristic
166	C₂×C₄₃	#577	1	1	yes	characteristic
167	C₂×C₄₃	#578	1	1	yes	characteristic
168	D₄₃	—	4	2	no	—
169	#541	—	43	1	no	—
170	#520	—	43	1	no	—
171	#484	D₄₃	1	1	yes	characteristic, complemented
172	#483	—	43	1	no	—
173	#479	—	43	1	no	—
174	#478	—	43	1	no	—
175	#462	—	43	1	no	—
176	#461	—	43	1	no	—
177	#452	—	43	1	no	—
178	#452	—	43	1	no	—
179	#441	—	43	1	no	—
180	#440	—	43	1	no	—
181	C₄×Q₁₆	—	43	1	no	—
182	C₄×SD₁₆	—	43	1	no	—
183	#433	—	43	1	no	—
184	C₄₃	#3120	1	1	yes	characteristic, complemented, Sylow
185	C₂×#41	—	43	1	no	—
186	C₂×Q₁₆	—	43	1	no	—
187	C₂×Q₁₆	—	43	1	no	—
188	C₂×SD₁₆	—	43	1	no	—
189	C₂×SD₁₆	—	43	1	no	—
190	C₄⋊Q₈	—	43	1	no	—
191	#126	—	86	1	no	—
192	#126	—	86	1	no	—
193	#125	—	43	1	no	—
194	#125	—	86	1	no	—
195	#125	—	86	1	no	—
196	C₂²⋊Q₈	—	43	1	no	—
197	C₂²⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
198	C₄×Q₈	—	43	1	no	—
199	C₄×Q₈	—	43	1	no	—
200	C₄×Q₈	—	43	1	no	—
201	C₄×Q₈	—	43	1	no	—
202	C₄×D₄	—	43	1	no	—
203	C₄×D₄	—	43	1	no	—
204	C₄×D₄	—	86	1	no	—
205	C₄×D₄	—	86	1	no	—
206	#117	—	43	1	no	—
207	#117	—	86	1	no	—
208	C₂×C₄⋊C₄	—	86	1	no	—
209	C₂×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
210	#107	C₂×D₄₃	1	1	yes	characteristic
211	#107	C₂×D₄₃	1	1	yes	characteristic
212	#106	—	43	1	no	—
213	#106	—	43	1	no	—
214	C₄⋊C₈	—	43	1	no	—
215	C₄⋊C₈	—	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	C₂²⋊C₈	—	43	1	no	—
225	C₂²⋊C₈	C₂×D₄₃	1	1	yes	characteristic
226	#97	—	43	1	no	—
227	C₄×C₈	—	43	1	no	—
228	#41	—	86	2	no	—
229	#41	—	86	2	no	—
230	C₂×Q₈	—	43	1	no	—
231	C₂×Q₈	—	43	1	no	—
232	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₂×D₄	—	43	1	no	—
234	C₂×D₄	—	86	1	no	—
235	C₂²×C₄	—	2	1	no	—
236	C₂²×C₄	—	86	1	no	—
237	C₂²×C₄	—	86	1	no	—
238	C₂²×C₄	—	86	1	no	—
239	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
240	Q₁₆	—	86	2	no	—
241	Q₁₆	—	86	2	no	—
242	SD₁₆	—	86	2	no	—
243	SD₁₆	—	86	2	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	C₂²×D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	43	1	no	—
267	C₂²⋊C₄	—	43	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₄²	—	43	1	no	—
272	C₄²	—	43	1	no	—
273	C₄²	—	86	1	no	—
274	C₄²	—	86	1	no	—
275	C₄²	—	86	1	no	—
276	C₄²	—	86	1	no	—
277	C₂³	—	86	1	no	—
278	C₂³	D₄×D₄₃	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	43	2	no	—
282	Q₈	—	86	1	no	—
283	Q₈	—	86	1	no	—
284	Q₈	—	86	1	no	—
285	D₄	—	43	2	no	—
286	D₄	—	86	4	no	—
287	D₄	—	86	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	8	1	no	—
295	C₂×C₄	—	43	1	no	—
296	C₂×C₄	—	43	1	no	—
297	C₂×C₄	—	43	1	no	—
298	C₂×C₄	—	43	1	no	—
299	C₂×C₄	—	86	4	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	344	1	no	—
304	C₂×C₄	—	344	1	no	—
305	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	86	1	no	—
310	C₈	—	86	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	86	2	no	—
315	C₂²	—	172	1	no	—
316	C₂²	—	172	1	no	—
317	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	86	2	no	—
324	C₄	—	86	1	no	—
325	C₄	—	86	1	no	—
326	C₄	—	86	1	no	—
327	C₄	—	86	1	no	—
328	C₄	—	172	1	no	—
329	C₄	—	172	1	no	—
330	C₄	#41×D₄₃	1	1	yes	characteristic
331	C₄	#41×D₄₃	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	172	2	no	—
334	C₂	#14583407	1	1	yes	characteristic
335	C₂	#14583709	1	1	yes	characteristic
336	C₂	#14583712	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#484	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#484	D₄₃
hypocenter	C₄₃	#3120
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #484

Complements

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

N	H ≅ G/N	type
1. #14583509	C₂	semidirect
3. #14583229	C₂	semidirect
4. #14583205	C₂	semidirect
5. #14583202	C₂	semidirect
6. #14583108	C₂	semidirect
7. #14583103	C₂	semidirect
10. #14583003	C₂	semidirect
11. #14582998	C₂	semidirect
12. #14582993	C₂	semidirect
13. #14582984	C₂	semidirect
14. #14582957	C₂	semidirect
15. C₄₃×#484	C₂	semidirect
171. #484	D₄₃	semidirect
184. C₄₃	#3120	semidirect