Subgroups — #19144398

Order 5504.

376 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583455	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic
3	#14583207	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583206	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583203	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583193	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583192	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583183	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582940	C₂	1	1	yes	maximal, characteristic
11	#14582938	C₂	1	1	yes	maximal, characteristic
12	#14582921	C₂	1	1	yes	maximal, characteristic
13	#14582906	C₂	1	1	yes	maximal, characteristic
14	#14582906	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#479	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
17	C₂×#10621872	C₂²	1	1	yes	characteristic
18	C₂×#10621872	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	#13063303	—	2	1	no	—
23	#13063303	—	2	1	no	—
24	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
25	C₄×Dic₈₆	—	2	1	no	—
26	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
27	#13063290	—	2	1	no	—
28	#13063290	—	2	1	no	—
29	#13063290	—	2	1	no	—
30	#13063290	C₂²	1	1	yes	characteristic
31	#13063283	C₂²	1	1	yes	characteristic
32	#13063283	C₂²	1	1	yes	characteristic
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	#13063258	C₂²	1	1	yes	characteristic
40	#13063257	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	#13063250	C₂²	1	1	yes	characteristic
47	#13063250	C₂²	1	1	yes	characteristic
48	#13063249	C₂²	1	1	yes	characteristic
49	#13063249	C₂²	1	1	yes	characteristic
50	#13063248	C₂²	1	1	yes	characteristic
51	#13063248	C₂²	1	1	yes	characteristic
52	#125×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×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	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#103×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	Q₈×D₄₃	—	2	2	no	—
61	Q₈×D₄₃	Q₈	1	4	yes	complemented
62	C₂×C₄×D₄₃	—	2	1	no	—
63	C₂×C₄×D₄₃	—	2	1	no	—
64	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
65	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
66	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
67	C₂×Dic₈₆	—	2	1	no	—
68	C₂×Dic₈₆	—	2	1	no	—
69	C₂×Dic₈₆	—	2	1	no	—
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	—	2	1	no	—
76	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
79	C₄²×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	C₂³	1	1	yes	characteristic
81	(C₂×D₄₃)⋊C₄	—	2	1	no	—
82	(C₂×D₄₃)⋊C₄	—	2	1	no	—
83	C₄⋊Dic₄₃	—	2	1	no	—
84	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
85	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
86	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	C₄×Dic₄₃	—	2	1	no	—
95	C₄×Dic₄₃	—	2	1	no	—
96	C₄×Dic₄₃	—	2	1	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	D₄	1	1	yes	characteristic
100	C₄×Dic₄₃	D₄	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	#10621872	—	2	2	no	—
105	#10621872	—	2	2	no	—
106	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
107	Q₈×C₄₃	—	2	1	no	—
108	Q₈×C₄₃	C₂×Q₈	1	2	yes	—
109	C₂×C₄×C₄₃	—	2	1	no	—
110	C₂×C₄×C₄₃	—	2	1	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
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	C₄×D₄₃	—	2	4	no	—
124	C₄×D₄₃	—	2	2	no	—
125	C₄×D₄₃	—	4	1	no	—
126	C₄×D₄₃	—	4	2	no	—
127	C₄×D₄₃	C₂×Q₈	1	2	yes	—
128	C₄×D₄₃	#41	1	2	yes	—
129	Dic₈₆	—	2	1	no	—
130	Dic₈₆	—	2	4	no	—
131	Dic₈₆	—	4	2	no	—
132	Dic₈₆	—	4	2	no	—
133	Dic₈₆	C₂×Q₈	1	2	yes	—
134	C₈×C₄₃	—	2	1	no	—
135	C₈×C₄₃	—	2	1	no	—
136	C₄₃⋊C₈	—	2	1	no	—
137	C₄₃⋊C₈	—	2	1	no	—
138	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×D₄₃	—	2	1	no	—
140	C₂×D₄₃	—	2	1	no	—
141	C₂×D₄₃	C₂²⋊Q₈	1	2	yes	—
142	C₄×C₄₃	—	2	2	no	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	—	4	1	no	—
146	C₄×C₄₃	—	4	1	no	—
147	C₄×C₄₃	—	4	1	no	—
148	C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
149	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
150	Dic₄₃	—	2	2	no	—
151	Dic₄₃	—	2	1	no	—
152	Dic₄₃	—	4	1	no	—
153	Dic₄₃	—	4	1	no	—
154	Dic₄₃	—	4	1	no	—
155	Dic₄₃	C₂²⋊Q₈	1	2	yes	—
156	Dic₄₃	#137	1	2	yes	—
157	#3078	—	43	1	no	maximal, Sylow
158	C₂×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
159	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
160	C₂×C₄₃	#575	1	1	yes	characteristic
161	D₄₃	—	2	2	no	—
162	#533	—	43	1	no	—
163	C₂×C₄×Q₈	—	43	1	no	—
164	#479	—	43	1	no	—
165	#479	—	43	1	no	—
166	#479	—	43	1	no	—
167	#479	D₄₃	1	1	yes	characteristic, complemented
168	#477	—	43	1	no	—
169	#477	—	43	1	no	—
170	#475	—	43	1	no	—
171	#475	—	43	1	no	—
172	#428	—	43	1	no	—
173	#428	—	43	1	no	—
174	#423	—	43	1	no	—
175	#419	—	43	1	no	—
176	#419	—	43	1	no	—
177	C₄₃	#3078	1	1	yes	characteristic, complemented, Sylow
178	C₂²×Q₈	—	43	1	no	—
179	C₂×M₁₆	—	43	1	no	—
180	C₂×M₁₆	—	43	1	no	—
181	C₄⋊Q₈	—	43	1	no	—
182	C₄⋊Q₈	—	43	1	no	—
183	#125	—	43	1	no	—
184	#125	C₂×D₄₃	1	1	yes	characteristic
185	C₂²⋊Q₈	—	86	1	no	—
186	C₂²⋊Q₈	—	86	1	no	—
187	C₄×Q₈	—	43	1	no	—
188	C₄×Q₈	—	43	1	no	—
189	C₄×Q₈	—	43	1	no	—
190	C₄×Q₈	—	86	1	no	—
191	C₄×Q₈	—	86	1	no	—
192	C₄×Q₈	—	86	1	no	—
193	C₄×Q₈	—	86	1	no	—
194	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
195	#117	—	43	1	no	—
196	#117	—	43	1	no	—
197	C₂×C₄⋊C₄	—	43	1	no	—
198	C₂×C₄⋊C₄	—	86	1	no	—
199	C₂×C₄²	—	43	1	no	—
200	C₂×C₄²	—	86	1	no	—
201	#107	—	43	1	no	—
202	#107	—	43	1	no	—
203	#107	—	43	1	no	—
204	#107	C₂×D₄₃	1	1	yes	characteristic
205	#106	—	43	1	no	—
206	#106	—	43	1	no	—
207	#106	—	43	1	no	—
208	#106	C₂×D₄₃	1	1	yes	characteristic
209	C₄⋊C₈	—	43	1	no	—
210	C₄⋊C₈	—	43	1	no	—
211	C₄⋊C₈	—	43	1	no	—
212	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
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	C₂×D₄₃	1	1	yes	characteristic
220	#103	C₂×D₄₃	1	1	yes	characteristic
221	C₂×Q₈	—	43	4	no	—
222	C₂×Q₈	—	43	1	no	—
223	C₂×Q₈	—	86	1	no	—
224	C₂×Q₈	—	86	1	no	—
225	C₂×Q₈	—	86	1	no	—
226	C₂×Q₈	—	86	2	no	—
227	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
228	C₂²×C₄	—	43	1	no	—
229	C₂²×C₄	—	43	1	no	—
230	C₂²×C₄	—	43	1	no	—
231	C₂²×C₄	—	86	1	no	—
232	C₂²×C₄	—	86	1	no	—
233	M₁₆	—	86	2	no	—
234	M₁₆	—	86	2	no	—
235	C₂×C₈	—	43	1	no	—
236	C₂×C₈	—	43	1	no	—
237	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	2	1	no	—
242	C₄⋊C₄	—	43	1	no	—
243	C₄⋊C₄	—	43	1	no	—
244	C₄⋊C₄	—	43	1	no	—
245	C₄⋊C₄	—	86	1	no	—
246	C₄⋊C₄	—	86	1	no	—
247	C₄⋊C₄	—	86	1	no	—
248	C₄⋊C₄	—	86	1	no	—
249	C₄⋊C₄	—	86	1	no	—
250	C₄⋊C₄	—	86	1	no	—
251	C₄⋊C₄	—	86	1	no	—
252	C₄⋊C₄	—	86	1	no	—
253	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
254	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
255	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
256	C₂²⋊C₄	—	86	1	no	—
257	C₂²⋊C₄	—	86	1	no	—
258	C₄²	—	2	1	no	—
259	C₄²	—	43	1	no	—
260	C₄²	—	43	1	no	—
261	C₄²	—	43	1	no	—
262	C₄²	—	86	1	no	—
263	C₄²	—	86	1	no	—
264	C₄²	—	86	1	no	—
265	C₄²	—	86	1	no	—
266	C₄²	—	86	1	no	—
267	C₄²	C₂²×D₄₃	1	1	yes	characteristic
268	C₂³	—	43	1	no	—
269	Q₈	—	2	1	no	—
270	Q₈	—	43	2	no	—
271	Q₈	—	86	4	no	—
272	Q₈	—	86	1	no	—
273	Q₈	—	172	2	no	—
274	Q₈	—	172	2	no	—
275	Q₈	Q₈×D₄₃	1	2	yes	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	43	1	no	—
281	C₂×C₄	—	43	1	no	—
282	C₂×C₄	—	43	2	no	—
283	C₂×C₄	—	43	2	no	—
284	C₂×C₄	—	43	1	no	—
285	C₂×C₄	—	86	2	no	—
286	C₂×C₄	—	86	4	no	—
287	C₂×C₄	—	86	1	no	—
288	C₂×C₄	—	86	1	no	—
289	C₂×C₄	—	86	1	no	—
290	C₂×C₄	—	86	1	no	—
291	C₂×C₄	—	172	1	no	—
292	C₂×C₄	—	172	2	no	—
293	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
294	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	—	86	1	no	—
299	C₈	—	86	1	no	—
300	C₂²	—	43	2	no	—
301	C₂²	—	86	1	no	—
302	C₂²	—	86	1	no	—
303	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	1	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	1	no	—
308	C₄	—	4	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	43	2	no	—
311	C₄	—	43	2	no	—
312	C₄	—	86	2	no	—
313	C₄	—	86	1	no	—
314	C₄	—	172	1	no	—
315	C₄	—	172	1	no	—
316	C₄	—	172	1	no	—
317	C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
318	C₄	#41×D₄₃	1	1	yes	characteristic
319	C₂	—	86	2	no	—
320	C₂	C₂²⋊Q₈×D₄₃	1	1	yes	characteristic
321	C₂	#137×D₄₃	1	1	yes	characteristic
322	C₂	#14583696	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#479	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#479	D₄₃
hypocenter	C₄₃	#3078
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #479

Complements

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

N	H ≅ G/N	type
3. #14583207	C₂	semidirect
4. #14583206	C₂	semidirect
5. #14583203	C₂	semidirect
6. #14583193	C₂	semidirect
7. #14583192	C₂	semidirect
8. #14583183	C₂	semidirect
9. #14583182	C₂	semidirect
15. C₄₃×#479	C₂	semidirect
61. Q₈×D₄₃	Q₈	semidirect
167. #479	D₄₃	semidirect
177. C₄₃	#3078	semidirect