Subgroups — #19147495

Order 5504.

374 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583608	C₂	1	1	yes	maximal, characteristic, complemented
2	D₄×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583308	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583306	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583296	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583199	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583197	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583189	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583187	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583187	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583177	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583027	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583011	C₂	1	1	yes	maximal, characteristic
14	D₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#442	C₂	1	1	yes	maximal, characteristic
16	C₂²×Dic₈₆	—	2	1	no	—
17	#13063325	C₂²	1	1	yes	characteristic
18	#13063324	C₂²	1	1	yes	characteristic
19	#13063324	C₂²	1	1	yes	characteristic
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063317	C₂²	1	1	yes	characteristic
22	#13063316	C₂²	1	1	yes	characteristic
23	#13063304	—	2	1	no	—
24	#13063304	—	2	1	no	—
25	#13063302	—	2	1	no	—
26	#13063302	—	2	1	no	—
27	#13063302	—	2	1	no	—
28	#13063302	—	2	1	no	—
29	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
30	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
31	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
32	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
33	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
34	C₂×Dic₄₃⋊C₄	—	2	1	no	—
35	#13063259	C₂²	1	1	yes	characteristic
36	#13063259	C₂²	1	1	yes	characteristic
37	#13063258	C₂²	1	1	yes	characteristic
38	#13063257	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	#13063246	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	#13063246	C₂²	1	1	yes	characteristic
50	#13063235	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×D₈×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	#102×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	#97×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂²×C₄×C₄₃	—	2	1	no	—
63	C₂²×Dic₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	1	no	—
67	C₂×Dic₈₆	—	2	1	no	—
68	C₂×Dic₈₆	—	2	1	no	—
69	C₂×Dic₈₆	—	2	1	no	—
70	C₂×Dic₈₆	—	2	1	no	—
71	C₂×Dic₈₆	—	4	4	no	—
72	C₂×Dic₈₆	D₄	1	1	yes	characteristic
73	D₈×C₄₃	Q₈	1	4	yes	complemented
74	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
75	C₂×C₈×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⋊C₄×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	C₂²⋊Dic₄₃	—	2	1	no	—
82	C₂²⋊Dic₄₃	—	2	1	no	—
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	C₄⋊Dic₄₃	—	2	1	no	—
86	C₄⋊Dic₄₃	—	2	1	no	—
87	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
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	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
99	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
100	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
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	C₂³×C₄₃	—	2	1	no	—
105	C₂³×C₄₃	—	2	1	no	—
106	D₄×C₄₃	—	2	1	no	—
107	D₄×C₄₃	—	2	1	no	—
108	D₄×C₄₃	C₂×Q₈	1	2	yes	—
109	D₄×C₄₃	C₂×Q₈	1	2	yes	complemented
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	4	2	no	—
113	C₂×C₄×C₄₃	—	8	1	no	—
114	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	2	1	no	—
121	C₂×Dic₄₃	—	4	2	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	8	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	Dic₈₆	—	2	2	no	—
130	Dic₈₆	—	4	2	no	—
131	Dic₈₆	—	4	2	no	—
132	Dic₈₆	—	4	2	no	—
133	Dic₈₆	—	8	1	no	—
134	C₈×C₄₃	—	2	1	no	—
135	C₈×C₄₃	C₂×Q₈	1	2	yes	complemented
136	C₄₃⋊C₈	—	2	1	no	—
137	C₄₃⋊C₈	—	2	1	no	—
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	4	1	no	—
141	C₂²×C₄₃	—	4	1	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₄×C₄₃	—	2	1	no	—
146	C₄×C₄₃	—	2	1	no	—
147	C₄×C₄₃	—	4	1	no	—
148	C₄×C₄₃	—	4	1	no	—
149	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
150	C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
151	Dic₄₃	—	2	1	no	—
152	Dic₄₃	—	2	1	no	—
153	Dic₄₃	—	2	1	no	—
154	Dic₄₃	—	4	1	no	—
155	Dic₄₃	—	4	1	no	—
156	Dic₄₃	—	4	1	no	—
157	Dic₄₃	—	4	1	no	—
158	Dic₄₃	AGL(1,ℤ/8)	1	2	yes	—
159	#3384	—	43	1	no	maximal, Sylow
160	C₂×C₄₃	—	4	2	no	—
161	C₂×C₄₃	—	4	2	no	—
162	C₂×C₄₃	D₄×Q₈	1	1	yes	characteristic
163	C₂×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
164	C₂×C₄₃	#577	1	1	yes	characteristic
165	#554	—	43	1	no	—
166	D₄×Q₈	—	43	1	no	—
167	#501	—	43	1	no	—
168	#501	—	43	1	no	—
169	#498	—	43	1	no	—
170	#478	—	43	1	no	—
171	#478	—	43	1	no	—
172	#476	—	43	1	no	—
173	#476	—	43	1	no	—
174	#476	—	43	1	no	—
175	#474	—	43	1	no	—
176	#446	—	43	1	no	—
177	#442	—	43	1	no	—
178	#442	D₄₃	1	1	yes	characteristic
179	C₄×D₈	—	43	1	no	—
180	C₄₃	#3384	1	1	yes	characteristic, complemented, Sylow
181	C₂²×Q₈	—	86	1	no	—
182	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	43	1	no	—
184	C₄⋊Q₈	—	43	1	no	—
185	C₄⋊Q₈	—	43	1	no	—
186	C₄⋊Q₈	—	43	1	no	—
187	#125	—	43	1	no	—
188	#125	—	43	1	no	—
189	C₂²⋊Q₈	—	86	1	no	—
190	C₂²⋊Q₈	—	86	1	no	—
191	C₂²⋊Q₈	—	86	1	no	—
192	C₂²⋊Q₈	—	86	1	no	—
193	C₂²⋊Q₈	—	86	1	no	—
194	C₂²⋊Q₈	—	86	1	no	—
195	C₄×Q₈	—	43	1	no	—
196	C₄×D₄	—	43	1	no	—
197	C₄×D₄	—	43	1	no	—
198	C₄×D₄	—	43	1	no	—
199	C₄×D₄	—	43	1	no	—
200	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
201	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
202	C₂×C₄⋊C₄	—	86	1	no	—
203	#107	—	43	1	no	—
204	#107	—	43	1	no	—
205	#107	—	43	1	no	—
206	#106	—	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	C₄⋊C₈	—	43	1	no	—
213	C₄⋊C₈	—	43	1	no	—
214	C₄⋊C₈	—	43	1	no	—
215	#102	—	43	1	no	—
216	#102	—	43	1	no	—
217	#102	—	43	1	no	—
218	#102	—	43	1	no	—
219	#102	C₂×D₄₃	1	1	yes	characteristic
220	#102	C₂×D₄₃	1	1	yes	characteristic
221	#97	—	43	1	no	—
222	#97	C₂×D₄₃	1	1	yes	characteristic
223	C₄×C₈	—	43	1	no	—
224	C₂×Q₈	—	43	1	no	—
225	C₂×Q₈	—	86	1	no	—
226	C₂×Q₈	—	86	1	no	—
227	C₂×Q₈	—	86	1	no	—
228	C₂×Q₈	—	86	1	no	—
229	C₂×Q₈	—	172	4	no	—
230	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	2	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	D₈	Dic₈₆	1	4	yes	complemented
239	C₂×C₈	—	43	1	no	—
240	C₂×C₈	—	43	1	no	—
241	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
242	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
243	C₄⋊C₄	—	43	1	no	—
244	C₄⋊C₄	—	43	1	no	—
245	C₄⋊C₄	—	43	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	—	86	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	C₂²×D₄₃	1	1	yes	characteristic
259	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	86	1	no	—
263	C₂²⋊C₄	—	86	1	no	—
264	C₂²⋊C₄	—	86	1	no	—
265	C₂²⋊C₄	—	86	1	no	—
266	C₄²	—	43	1	no	—
267	C₄²	—	43	1	no	—
268	C₄²	C₂²×D₄₃	1	1	yes	characteristic
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	Q₈	—	86	2	no	—
272	Q₈	—	344	1	no	—
273	Q₈	—	172	2	no	—
274	Q₈	—	172	2	no	—
275	Q₈	—	172	2	no	—
276	D₄	—	2	1	no	—
277	D₄	—	2	1	no	—
278	D₄	C₂×Dic₈₆	1	2	yes	—
279	D₄	C₂×Dic₈₆	1	2	yes	complemented
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	4	2	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	43	1	no	—
285	C₂×C₄	—	43	1	no	—
286	C₂×C₄	—	43	1	no	—
287	C₂×C₄	—	43	1	no	—
288	C₂×C₄	—	86	1	no	—
289	C₂×C₄	—	86	1	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	86	1	no	—
292	C₂×C₄	—	344	1	no	—
293	C₂×C₄	—	172	2	no	—
294	C₂×C₄	—	172	2	no	—
295	C₂×C₄	—	172	2	no	—
296	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
297	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
298	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
299	C₈	—	2	1	no	—
300	C₈	—	86	1	no	—
301	C₈	—	86	1	no	—
302	C₈	C₂×Dic₈₆	1	2	yes	complemented
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	43	2	no	—
315	C₄	—	86	1	no	—
316	C₄	—	86	1	no	—
317	C₄	—	86	1	no	—
318	C₄	—	172	1	no	—
319	C₄	—	172	1	no	—
320	C₄	—	172	1	no	—
321	C₄	—	172	1	no	—
322	C₄	C₂²×Dic₈₆	1	1	yes	characteristic
323	C₄	#13063372	1	1	yes	characteristic
324	C₂	—	4	2	no	—
325	C₂	—	4	2	no	—
326	C₂	D₄×Dic₈₆	1	1	yes	characteristic
327	C₂	AGL(1,ℤ/8)×D₄₃	1	1	yes	characteristic
328	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×#442	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#442	D₄₃
hypocenter	C₄₃	#3384
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #442

Complements

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

N	H ≅ G/N	type
1. #14583608	C₂	semidirect
2. D₄×Dic₈₆	C₂	semidirect
3. #14583308	C₂	semidirect
4. #14583306	C₂	semidirect
5. #14583296	C₂	semidirect
6. #14583199	C₂	semidirect
7. #14583197	C₂	semidirect
8. #14583189	C₂	semidirect
9. #14583187	C₂	semidirect
10. #14583187	C₂	semidirect
11. #14583177	C₂	semidirect
12. #14583027	C₂	semidirect
73. D₈×C₄₃	Q₈	semidirect
109. D₄×C₄₃	C₂×Q₈	semidirect
135. C₈×C₄₃	C₂×Q₈	semidirect
180. C₄₃	#3384	semidirect
238. D₈	Dic₈₆	semidirect
279. D₄	C₂×Dic₈₆	semidirect
302. C₈	C₂×Dic₈₆	semidirect