Subgroups — #19144674

Order 5504.

390 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	#14583428	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583228	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583211	C₂	1	1	yes	maximal, characteristic
5	#14583207	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583206	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583118	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583053	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583036	C₂	1	1	yes	maximal, characteristic
10	Q₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#14582991	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582990	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582951	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#464	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621885	C₂²	1	1	yes	characteristic
18	C₂×#10621883	C₂²	1	1	yes	characteristic
19	#13063318	—	2	1	no	—
20	#13063317	C₂²	1	1	yes	characteristic
21	#13063307	—	2	1	no	—
22	#13063301	C₂²	1	1	yes	characteristic
23	#13063296	C₂²	1	1	yes	characteristic
24	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
25	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
26	C₄×Dic₈₆	—	2	1	no	—
27	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
28	#13063290	—	2	1	no	—
29	#13063290	C₂²	1	1	yes	characteristic
30	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
31	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
32	#13063281	—	2	1	no	—
33	C₄⋊C₄×D₄₃	—	2	1	no	—
34	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
35	C₄²×D₄₃	—	2	1	no	—
36	C₄²×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	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
42	#13063250	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063249	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063249	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
54	#124×C₄₃	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#102×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×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	C₂²×Dic₄₃	—	2	1	no	—
64	Q₈×D₄₃	—	2	2	no	—
65	Q₈×D₄₃	—	2	2	no	—
66	C₂×C₄×D₄₃	—	2	1	no	—
67	C₂×C₄×D₄₃	—	2	1	no	—
68	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
69	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
71	C₂×Dic₈₆	—	2	1	no	—
72	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
73	Q₁₆×C₄₃	—	2	2	no	—
74	SD₁₆×C₄₃	—	2	2	no	—
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	C₂³	1	1	yes	characteristic
79	C₂²⋊C₄×C₄₃	—	2	1	no	—
80	C₂²⋊C₄×C₄₃	—	2	1	no	—
81	C₄²×C₄₃	—	2	1	no	—
82	C₄²×C₄₃	C₂³	1	1	yes	characteristic
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	#10621885	—	2	2	no	—
86	#10621883	—	2	2	no	—
87	(C₂×D₄₃)⋊C₄	—	2	1	no	—
88	C₄⋊Dic₄₃	—	2	1	no	—
89	C₄⋊Dic₄₃	—	2	1	no	—
90	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
91	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	D₄	1	1	yes	characteristic
104	C₄×Dic₄₃	D₄	1	1	yes	characteristic
105	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
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	Q₈×C₄₃	—	2	1	no	—
113	Q₈×C₄₃	#41	1	2	yes	—
114	D₄×C₄₃	—	2	1	no	—
115	D₄×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	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	2	1	no	—
124	C₂×Dic₄₃	—	2	1	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	2	1	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	C₄×D₄₃	—	2	4	no	—
132	C₄×D₄₃	—	2	2	no	—
133	C₄×D₄₃	—	2	1	no	—
134	C₄×D₄₃	—	2	1	no	—
135	C₄×D₄₃	—	2	2	no	—
136	C₄×D₄₃	—	4	2	no	—
137	Dic₈₆	—	2	4	no	—
138	Dic₈₆	—	2	1	no	—
139	Dic₈₆	#41	1	2	yes	—
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	#135	1	2	yes	—
151	C₄×C₄₃	—	2	2	no	—
152	C₄×C₄₃	—	2	1	no	—
153	C₄×C₄₃	—	2	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	2	no	—
160	Dic₄₃	—	2	1	no	—
161	Dic₄₃	—	4	1	no	—
162	Dic₄₃	—	4	1	no	—
163	Dic₄₃	—	4	1	no	—
164	Dic₄₃	#137	1	2	yes	—
165	#3108	—	43	1	no	maximal, Sylow
166	C₂×C₄₃	—	8	1	no	—
167	C₂×C₄₃	#525	1	1	yes	characteristic
168	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
169	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
170	D₄₃	—	2	2	no	—
171	#529	—	43	1	no	—
172	C₂×C₄×Q₈	—	43	1	no	—
173	#483	—	43	1	no	—
174	#480	—	43	1	no	—
175	#479	—	43	1	no	—
176	#479	—	43	1	no	—
177	#464	—	43	1	no	—
178	#464	D₄₃	1	1	yes	characteristic, complemented
179	#451	—	43	1	no	—
180	#448	—	43	1	no	—
181	C₄×Q₁₆	—	43	1	no	—
182	C₄×Q₁₆	—	43	1	no	—
183	C₄×SD₁₆	—	43	1	no	—
184	C₄×SD₁₆	—	43	1	no	—
185	#431	—	43	1	no	—
186	C₄₃	#3108	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₈	—	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	C₂×D₄₃	1	1	yes	characteristic
192	#126	—	86	1	no	—
193	#125	—	43	1	no	—
194	#124	C₂×D₄₃	1	1	yes	characteristic
195	#123	—	86	1	no	—
196	C₂²⋊Q₈	—	43	1	no	—
197	#121	—	43	1	no	—
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×Q₈	—	86	1	no	—
203	C₄×Q₈	—	86	1	no	—
204	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
205	C₄×D₄	—	43	1	no	—
206	C₄×D₄	—	86	1	no	—
207	#117	—	86	1	no	—
208	C₂×C₄⋊C₄	—	43	1	no	—
209	C₂×C₄⋊C₄	—	86	1	no	—
210	C₂×C₄²	—	43	1	no	—
211	C₂×C₄²	—	86	1	no	—
212	#107	—	43	1	no	—
213	#107	—	43	1	no	—
214	#106	—	43	1	no	—
215	#106	—	43	1	no	—
216	C₄⋊C₈	—	43	1	no	—
217	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
218	#103	—	43	1	no	—
219	#103	—	43	1	no	—
220	#103	—	43	1	no	—
221	#103	—	43	1	no	—
222	#103	—	43	1	no	—
223	#103	C₂×D₄₃	1	1	yes	characteristic
224	#102	—	43	1	no	—
225	#102	C₂×D₄₃	1	1	yes	characteristic
226	C₂²⋊C₈	—	43	1	no	—
227	C₂²⋊C₈	—	43	1	no	—
228	C₄×C₈	—	43	1	no	—
229	C₄×C₈	—	43	1	no	—
230	C₂×Q₈	—	43	1	no	—
231	C₂×Q₈	—	86	1	no	—
232	C₂×Q₈	—	86	2	no	—
233	C₂×Q₈	—	86	2	no	—
234	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
236	C₂×D₄	—	86	1	no	—
237	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	—	43	1	no	—
239	C₂²×C₄	—	43	1	no	—
240	C₂²×C₄	—	43	1	no	—
241	C₂²×C₄	—	86	1	no	—
242	C₂²×C₄	—	86	1	no	—
243	C₂²×C₄	—	86	1	no	—
244	Q₁₆	—	2	2	no	—
245	Q₁₆	—	86	2	no	—
246	SD₁₆	—	2	2	no	—
247	SD₁₆	—	86	2	no	—
248	C₂×C₈	—	43	1	no	—
249	C₂×C₈	—	43	1	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	—	43	1	no	—
254	C₄⋊C₄	—	43	1	no	—
255	C₄⋊C₄	—	43	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	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	—	86	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	43	1	no	—
272	C₄²	—	43	1	no	—
273	C₄²	—	43	1	no	—
274	C₄²	—	86	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	—	2	1	no	—
280	C₂³	—	43	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	2	1	no	—
284	Q₈	—	43	2	no	—
285	Q₈	—	86	4	no	—
286	Q₈	—	86	1	no	—
287	Q₈	#10621900	1	2	yes	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	172	2	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	43	1	no	—
295	C₂×C₄	—	43	1	no	—
296	C₂×C₄	—	43	1	no	—
297	C₂×C₄	—	86	2	no	—
298	C₂×C₄	—	86	2	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	—	86	1	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	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	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	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	2	no	—
330	C₄	—	86	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	C₂×#10621900	1	1	yes	characteristic
336	C₂	—	8	1	no	—
337	C₂	—	86	2	no	—
338	C₂	#14583409	1	1	yes	characteristic
339	C₂	#135×D₄₃	1	1	yes	characteristic
340	C₂	#137×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#464	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#464	D₄₃
hypocenter	C₄₃	#3108
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₄₃	C₂	semidirect
3. #14583228	C₂	semidirect
5. #14583207	C₂	semidirect
6. #14583206	C₂	semidirect
7. #14583118	C₂	semidirect
8. #14583053	C₂	semidirect
10. Q₁₆×Dic₄₃	C₂	semidirect
11. #14582991	C₂	semidirect
12. #14582990	C₂	semidirect
13. SD₁₆×Dic₄₃	C₂	semidirect
14. #14582951	C₂	semidirect
15. C₄₃×#464	C₂	semidirect
178. #464	D₄₃	semidirect
186. C₄₃	#3108	semidirect