Subgroups — #19147465

Order 5504.

338 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583626	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583608	C₂	1	1	yes	maximal, characteristic
3	#14583300	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583298	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583296	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583209	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583207	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583201	C₂	1	1	yes	maximal, characteristic
9	#14583193	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583177	C₂	1	1	yes	maximal, characteristic
11	#14583023	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582990	C₂	1	1	yes	maximal, characteristic
13	#14582990	C₂	1	1	yes	maximal, characteristic
14	C₄×#10621883	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#478	C₂	1	1	yes	maximal, characteristic
16	C₂×#10621883	C₂²	1	1	yes	characteristic
17	#13063324	—	2	1	no	—
18	#13063324	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063317	C₂²	1	1	yes	characteristic
21	#13063316	C₂²	1	1	yes	characteristic
22	#13063316	C₂²	1	1	yes	characteristic
23	#13063315	—	2	1	no	—
24	#13063315	—	2	1	no	—
25	#13063304	—	2	1	no	—
26	#13063302	—	2	1	no	—
27	#13063302	—	2	1	no	—
28	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
29	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
30	#13063290	—	2	1	no	—
31	#13063290	C₂²	1	1	yes	characteristic
32	#13063290	C₂²	1	1	yes	characteristic
33	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	C₂×Dic₄₃⋊C₄	—	2	1	no	—
36	#13063259	C₂²	1	1	yes	characteristic
37	#13063258	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	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
45	#13063250	C₂²	1	1	yes	characteristic
46	#13063250	C₂²	1	1	yes	characteristic
47	#13063248	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	#125×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
54	#107×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×Dic₄₃	—	2	1	no	—
62	C₂²×Dic₄₃	—	2	1	no	—
63	C₂×Dic₈₆	—	2	1	no	—
64	C₂×Dic₈₆	—	2	1	no	—
65	C₂×Dic₈₆	D₄	1	1	yes	characteristic
66	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
67	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₄₃	—	2	1	no	—
70	C₄⋊C₄×C₄₃	—	2	1	no	—
71	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
72	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
73	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
74	C₂²⋊C₄×C₄₃	—	2	1	no	—
75	C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	C₂²⋊Dic₄₃	—	2	1	no	—
77	C₂²⋊Dic₄₃	—	2	1	no	—
78	#10621883	Q₈	1	4	yes	complemented
79	C₄⋊Dic₄₃	—	2	1	no	—
80	C₄⋊Dic₄₃	—	2	1	no	—
81	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
82	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
83	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
85	Dic₄₃⋊C₄	—	2	1	no	—
86	Dic₄₃⋊C₄	—	2	1	no	—
87	Dic₄₃⋊C₄	—	2	1	no	—
88	Dic₄₃⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
95	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
96	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
103	C₂³×C₄₃	—	2	1	no	—
104	D₄×C₄₃	—	2	1	no	—
105	D₄×C₄₃	C₂×Q₈	1	2	yes	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	8	1	no	—
110	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
113	C₂×Dic₄₃	—	2	1	no	—
114	C₂×Dic₄₃	—	2	1	no	—
115	C₂×Dic₄₃	—	2	1	no	—
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	4	2	no	—
118	C₂×Dic₄₃	—	8	1	no	—
119	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
120	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
123	Dic₈₆	—	2	1	no	—
124	Dic₈₆	—	2	2	no	—
125	Dic₈₆	—	4	2	no	—
126	Dic₈₆	—	8	1	no	—
127	Dic₈₆	C₂×Q₈	1	2	yes	—
128	C₈×C₄₃	—	2	1	no	—
129	C₈×C₄₃	—	2	1	no	—
130	C₄₃⋊C₈	—	2	1	no	—
131	C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
132	C₂²×C₄₃	—	2	2	no	—
133	C₂²×C₄₃	—	4	1	no	—
134	C₂²×C₄₃	—	4	1	no	—
135	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₄₃	—	2	1	no	—
137	C₄×C₄₃	—	2	1	no	—
138	C₄×C₄₃	—	4	1	no	—
139	C₄×C₄₃	—	4	1	no	—
140	C₄×C₄₃	—	4	1	no	—
141	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
142	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
143	Dic₄₃	—	2	2	no	—
144	Dic₄₃	—	2	1	no	—
145	Dic₄₃	—	2	1	no	—
146	Dic₄₃	—	2	1	no	—
147	Dic₄₃	—	4	1	no	—
148	Dic₄₃	—	4	1	no	—
149	Dic₄₃	—	4	1	no	—
150	Dic₄₃	#135	1	2	yes	—
151	#3381	—	43	1	no	maximal, Sylow
152	C₂×C₄₃	—	4	2	no	—
153	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
154	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
155	C₂×C₄₃	#577	1	1	yes	characteristic
156	#557	—	43	1	no	—
157	#554	—	43	1	no	—
158	#499	—	43	1	no	—
159	#499	—	43	1	no	—
160	#498	—	43	1	no	—
161	#479	—	43	1	no	—
162	#479	—	43	1	no	—
163	#478	—	43	1	no	—
164	#478	D₄₃	1	1	yes	characteristic
165	#477	—	43	1	no	—
166	#474	—	43	1	no	—
167	C₈×Q₈	—	43	1	no	—
168	C₄×SD₁₆	—	43	1	no	—
169	C₄×SD₁₆	—	43	1	no	—
170	C₄×SD₁₆	—	43	1	no	—
171	C₄₃	#3381	1	1	yes	characteristic, complemented, Sylow
172	C₂×SD₁₆	—	43	1	no	—
173	C₄⋊Q₈	—	43	1	no	—
174	C₄⋊Q₈	—	43	1	no	—
175	C₄⋊Q₈	—	86	1	no	—
176	#125	—	43	1	no	—
177	#125	—	43	1	no	—
178	#125	—	43	1	no	—
179	#125	—	86	1	no	—
180	#125	—	86	1	no	—
181	#125	C₂×D₄₃	1	1	yes	characteristic
182	C₂²⋊Q₈	—	86	1	no	—
183	C₂²⋊Q₈	—	86	1	no	—
184	C₂²⋊Q₈	—	86	1	no	—
185	C₄×Q₈	—	43	1	no	—
186	C₄×Q₈	—	43	1	no	—
187	C₄×Q₈	—	43	1	no	—
188	C₄×Q₈	—	43	1	no	—
189	C₄×Q₈	—	86	1	no	—
190	C₄×D₄	—	43	1	no	—
191	C₄×D₄	—	43	1	no	—
192	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	86	1	no	—
194	#107	—	43	1	no	—
195	#107	—	43	1	no	—
196	#107	—	43	1	no	—
197	#107	C₂×D₄₃	1	1	yes	characteristic
198	#106	—	43	1	no	—
199	#106	—	43	1	no	—
200	#106	—	43	1	no	—
201	#106	—	43	1	no	—
202	#106	C₂×D₄₃	1	1	yes	characteristic
203	C₄⋊C₈	—	43	1	no	—
204	C₄⋊C₈	—	43	1	no	—
205	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
206	#103	—	43	1	no	—
207	#103	—	43	1	no	—
208	#103	—	43	1	no	—
209	#102	—	43	1	no	—
210	#102	C₂×D₄₃	1	1	yes	characteristic
211	#102	C₂×D₄₃	1	1	yes	characteristic
212	C₄×C₈	—	43	1	no	—
213	C₄×C₈	—	43	1	no	—
214	C₄×C₈	—	43	1	no	—
215	C₂×Q₈	—	43	1	no	—
216	C₂×Q₈	—	43	1	no	—
217	C₂×Q₈	—	86	1	no	—
218	C₂×Q₈	—	86	1	no	—
219	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	86	1	no	—
222	C₂²×C₄	—	86	1	no	—
223	SD₁₆	—	43	4	no	—
224	C₂×C₈	—	43	1	no	—
225	C₂×C₈	—	43	1	no	—
226	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
227	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	43	1	no	—
231	C₄⋊C₄	—	43	1	no	—
232	C₄⋊C₄	—	43	1	no	—
233	C₄⋊C₄	—	43	1	no	—
234	C₄⋊C₄	—	43	1	no	—
235	C₄⋊C₄	—	43	1	no	—
236	C₄⋊C₄	—	86	1	no	—
237	C₄⋊C₄	—	86	1	no	—
238	C₄⋊C₄	—	86	1	no	—
239	C₄⋊C₄	—	86	1	no	—
240	C₄⋊C₄	—	86	1	no	—
241	C₄⋊C₄	—	86	1	no	—
242	C₄⋊C₄	—	86	1	no	—
243	C₄⋊C₄	—	86	1	no	—
244	C₄⋊C₄	—	86	1	no	—
245	C₄⋊C₄	—	86	1	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	86	1	no	—
252	C₂²⋊C₄	—	86	1	no	—
253	C₄²	—	43	1	no	—
254	C₄²	—	43	1	no	—
255	C₄²	—	86	1	no	—
256	C₄²	—	86	1	no	—
257	C₄²	—	86	1	no	—
258	C₄²	C₂²×D₄₃	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	43	2	no	—
261	Q₈	—	86	2	no	—
262	Q₈	—	86	1	no	—
263	Q₈	—	344	1	no	—
264	Q₈	—	172	2	no	—
265	D₄	—	2	1	no	—
266	D₄	Q₈×D₄₃	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	8	1	no	—
271	C₂×C₄	—	43	1	no	—
272	C₂×C₄	—	43	1	no	—
273	C₂×C₄	—	43	1	no	—
274	C₂×C₄	—	43	1	no	—
275	C₂×C₄	—	86	1	no	—
276	C₂×C₄	—	86	1	no	—
277	C₂×C₄	—	86	1	no	—
278	C₂×C₄	—	86	1	no	—
279	C₂×C₄	—	344	1	no	—
280	C₂×C₄	—	172	2	no	—
281	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
282	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	—	43	2	no	—
287	C₈	—	86	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	4	1	no	—
295	C₄	—	4	1	no	—
296	C₄	—	4	1	no	—
297	C₄	—	43	2	no	—
298	C₄	—	86	2	no	—
299	C₄	—	86	1	no	—
300	C₄	—	86	1	no	—
301	C₄	—	86	1	no	—
302	C₄	—	172	1	no	—
303	C₄	—	172	1	no	—
304	C₄	—	172	1	no	—
305	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
306	C₄	#13063372	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	#14583562	1	1	yes	characteristic
309	C₂	#135×D₄₃	1	1	yes	characteristic
310	C₂	#14583709	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#478	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#478	D₄₃
hypocenter	C₄₃	#3381
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #14583626	C₂	semidirect
3. #14583300	C₂	semidirect
4. #14583298	C₂	semidirect
5. #14583296	C₂	semidirect
6. #14583209	C₂	semidirect
7. #14583207	C₂	semidirect
9. #14583193	C₂	semidirect
11. #14583023	C₂	semidirect
78. #10621883	Q₈	semidirect
171. C₄₃	#3381	semidirect