Subgroups — #19144314

Order 5504.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583699	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583697	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063345	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13063333	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583173	C₂	1	2	yes	maximal, complemented
6	#14583168	C₂	1	2	yes	maximal, complemented
7	#14583165	C₂	1	2	yes	maximal, complemented
8	#14583161	C₂	1	2	yes	maximal, complemented
9	C₂×#13063260	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582881	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#413	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#10621898	C₂²	1	1	yes	characteristic
13	C₂×#10621898	C₂²	1	1	yes	characteristic
14	#13063358	—	2	1	no	—
15	#13063358	C₂²	1	2	yes	—
16	#13063356	C₂²	1	2	yes	—
17	#13063355	—	2	1	no	—
18	#13063353	—	2	1	no	—
19	#13063353	C₂²	1	2	yes	—
20	#13063351	C₂²	1	2	yes	—
21	#13063350	—	2	1	no	—
22	#13063347	—	2	2	no	—
23	#13063345	—	2	2	no	—
24	#13063345	—	2	1	no	—
25	#13063345	—	2	1	no	—
26	#13063345	—	2	1	no	—
27	#13063345	—	2	1	no	—
28	C₂×#10621885	C₂²	1	1	yes	characteristic, complemented
29	C₂×#10621883	C₂²	1	1	yes	characteristic, complemented
30	C₂×#10621884	C₂²	1	1	yes	characteristic, complemented
31	C₂×#10621882	C₂²	1	1	yes	characteristic, complemented
32	#13063333	—	2	1	no	—
33	#13063333	—	2	1	no	—
34	#13063333	C₂²	1	2	yes	—
35	#13063333	C₂²	1	2	yes	—
36	C₂×#10621877	C₂²	1	1	yes	characteristic
37	C₂×#10621872	—	2	1	no	—
38	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
39	C₂×C₈×D₄₃	—	2	1	no	—
40	#13063260	C₂²	1	2	yes	complemented
41	#13063260	C₂²	1	2	yes	complemented
42	#13063243	C₂²	1	2	yes	complemented
43	#13063241	C₂²	1	2	yes	complemented
44	C₂×#41×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
46	C₂×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
47	#101×C₄₃	C₂²	1	2	yes	complemented
48	#100×C₄₃	C₂²	1	2	yes	complemented
49	#41×C₄₃	—	2	1	no	—
50	#41×C₄₃	—	2	1	no	—
51	#41×C₄₃	—	2	1	no	—
52	#41×C₄₃	—	2	1	no	—
53	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
54	C₂×D₄×C₄₃	—	2	1	no	—
55	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	—	2	1	no	—
57	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
58	C₂×C₂²⋊D₄₃	—	2	1	no	—
59	C₂×C₂²⋊D₄₃	—	2	1	no	—
60	#10621898	—	2	1	no	—
61	#10621898	—	2	1	no	—
62	#10621898	—	2	1	no	—
63	#10621898	—	2	1	no	—
64	#10621898	—	2	1	no	—
65	#10621898	—	2	1	no	—
66	#10621898	D₄	1	2	yes	—
67	#10621898	C₂³	1	2	yes	—
68	C₂×D₁₇₂	D₄	1	1	yes	characteristic
69	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×Dic₈₆	D₄	1	1	yes	characteristic
73	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
74	M₁₆×C₄₃	—	2	1	no	—
75	M₁₆×C₄₃	—	2	1	no	—
76	M₁₆×C₄₃	C₂³	1	2	yes	—
77	M₁₆×C₄₃	C₂³	1	2	yes	—
78	C₂×C₈×C₄₃	—	2	1	no	—
79	C₂×C₈×C₄₃	—	2	1	no	—
80	#10621885	—	2	2	no	—
81	#10621885	—	2	1	no	—
82	#10621885	—	2	1	no	—
83	#10621884	—	2	2	no	—
84	#10621884	—	2	1	no	—
85	#10621884	—	2	1	no	—
86	#10621883	—	2	2	no	—
87	#10621883	—	2	1	no	—
88	#10621883	—	2	1	no	—
89	#10621882	—	2	2	no	—
90	#10621882	—	2	1	no	—
91	#10621882	—	2	1	no	—
92	#10621877	—	2	1	no	—
93	#10621877	C₂³	1	2	yes	—
94	C₂×C₄₃⋊C₈	—	2	1	no	—
95	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
96	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
97	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
98	C₂×C₄₃⋊C₈	D₄	1	1	yes	characteristic
99	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
100	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
101	Dic₁₇₂	—	2	2	no	—
102	D₃₄₄	—	2	2	no	—
103	#10621873	—	2	2	no	—
104	#10621873	—	2	2	no	—
105	#10621872	—	2	2	no	—
106	#10621872	—	2	2	no	—
107	C₈×D₄₃	—	2	2	no	—
108	C₈×D₄₃	—	2	2	no	—
109	C₂³×C₄₃	—	2	1	no	—
110	C₂³×C₄₃	#41	1	1	yes	characteristic
111	C₂²×D₄₃	—	2	1	no	—
112	C₂²×D₄₃	—	2	1	no	—
113	Q₈×C₄₃	—	2	1	no	—
114	Q₈×C₄₃	—	2	1	no	—
115	D₄×C₄₃	—	2	1	no	—
116	D₄×C₄₃	—	2	1	no	—
117	D₄×C₄₃	—	4	1	no	—
118	D₄×C₄₃	—	4	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	4	1	no	—
121	C₂×C₄×C₄₃	—	4	1	no	—
122	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#41	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
128	C₂²⋊D₄₃	—	2	2	no	—
129	C₂²⋊D₄₃	—	2	2	no	—
130	C₂²⋊D₄₃	—	4	1	no	—
131	C₂²⋊D₄₃	—	4	1	no	—
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	D₁₇₂	—	2	1	no	—
135	D₁₇₂	—	2	1	no	—
136	D₁₇₂	—	2	1	no	—
137	D₁₇₂	C₂×D₄	1	2	yes	—
138	C₄×D₄₃	—	2	2	no	—
139	C₄×D₄₃	—	2	2	no	—
140	C₄×D₄₃	—	4	1	no	—
141	C₄×D₄₃	—	4	1	no	—
142	Dic₈₆	—	2	1	no	—
143	Dic₈₆	—	2	1	no	—
144	Dic₈₆	—	2	1	no	—
145	Dic₈₆	C₂×D₄	1	2	yes	—
146	C₈×C₄₃	—	2	2	no	—
147	C₈×C₄₃	—	2	2	no	—
148	C₄₃⋊C₈	—	2	2	no	—
149	C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
150	C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
151	C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
152	C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄₃	—	4	1	no	—
154	C₂²×C₄₃	—	4	1	no	—
155	C₂²×C₄₃	—	8	1	no	—
156	C₂²×C₄₃	#121	1	1	yes	characteristic
157	C₂²×C₄₃	#121	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×#41	1	1	yes	characteristic
159	C₂×D₄₃	—	2	2	no	—
160	C₂×D₄₃	—	4	2	no	—
161	C₂×D₄₃	—	4	1	no	—
162	C₂×D₄₃	—	8	1	no	—
163	C₄×C₄₃	—	4	1	no	—
164	C₄×C₄₃	#121	1	1	yes	characteristic
165	C₄×C₄₃	#121	1	1	yes	characteristic
166	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
167	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
168	Dic₄₃	—	2	2	no	—
169	Dic₄₃	—	4	1	no	—
170	#3067	—	43	1	no	maximal, Sylow
171	C₂×C₄₃	—	2	1	no	—
172	C₂×C₄₃	—	2	1	no	—
173	C₂×C₄₃	—	2	1	no	—
174	C₂×C₄₃	—	8	1	no	—
175	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
176	D₄₃	—	4	2	no	—
177	D₄₃	—	8	1	no	—
178	#575	—	43	1	no	—
179	#575	—	43	1	no	—
180	C₂×#135	—	43	1	no	—
181	C₂×#131	—	43	1	no	—
182	#473	—	43	2	no	—
183	#472	—	43	2	no	—
184	#471	—	43	2	no	—
185	#471	—	43	2	no	—
186	C₂×#108	—	43	1	no	—
187	#413	—	43	1	no	—
188	#413	D₄₃	1	1	yes	characteristic, complemented
189	C₄₃	#3067	1	1	yes	characteristic, complemented, Sylow
190	C₂×#41	—	43	1	no	—
191	C₂×#41	—	43	1	no	—
192	C₂×#41	C₂×D₄₃	1	1	yes	characteristic
193	#137	—	43	2	no	—
194	#137	—	43	2	no	—
195	#137	—	86	1	no	—
196	#137	—	86	1	no	—
197	AGL(1,ℤ/8)	—	43	2	no	—
198	AGL(1,ℤ/8)	—	43	2	no	—
199	AGL(1,ℤ/8)	—	86	1	no	—
200	AGL(1,ℤ/8)	—	86	1	no	—
201	#135	—	86	2	no	—
202	#135	—	86	1	no	—
203	#135	—	86	1	no	—
204	#135	—	86	1	no	—
205	#135	—	86	1	no	—
206	#135	—	86	2	no	—
207	C₂×Q₁₆	—	43	1	no	—
208	C₂×SD₁₆	—	43	1	no	—
209	C₂×SD₁₆	—	43	1	no	—
210	C₂×D₈	—	43	1	no	—
211	#131	—	43	2	no	—
212	#131	—	43	2	no	—
213	#131	—	86	1	no	—
214	#131	—	86	1	no	—
215	C₂×M₁₆	—	43	1	no	—
216	C₂×M₁₆	—	86	1	no	—
217	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
218	C₂×M₁₆	C₂×D₄₃	1	1	yes	characteristic
219	C₂²×C₈	—	43	1	no	—
220	C₂²×C₈	—	86	1	no	—
221	#108	—	43	2	no	—
222	#108	—	43	2	no	—
223	#101	—	43	2	no	—
224	#101	C₂×D₄₃	1	2	yes	complemented
225	#100	—	43	2	no	—
226	#100	C₂×D₄₃	1	2	yes	complemented
227	#41	—	2	1	no	—
228	#41	—	2	1	no	—
229	#41	—	2	1	no	—
230	#41	—	2	1	no	—
231	#41	—	43	2	no	—
232	#41	—	43	2	no	—
233	#41	—	86	1	no	—
234	#41	—	86	1	no	—
235	#41	—	86	1	no	—
236	#41	—	86	1	no	—
237	#41	—	86	1	no	—
238	#41	—	86	1	no	—
239	C₂×Q₈	—	43	1	no	—
240	C₂×Q₈	—	43	1	no	—
241	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂×D₄	—	43	1	no	—
244	C₂×D₄	—	43	1	no	—
245	C₂×D₄	—	86	1	no	—
246	C₂×D₄	—	86	1	no	—
247	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
248	C₂²×C₄	—	2	1	no	—
249	C₂²×C₄	—	86	1	no	—
250	C₂²×C₄	—	86	1	no	—
251	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
252	Q₁₆	—	86	2	no	—
253	Q₁₆	—	86	1	no	—
254	Q₁₆	—	86	1	no	—
255	Q₁₆	—	86	2	no	—
256	SD₁₆	—	86	2	no	—
257	SD₁₆	—	86	2	no	—
258	SD₁₆	—	86	1	no	—
259	SD₁₆	—	86	1	no	—
260	SD₁₆	—	86	1	no	—
261	SD₁₆	—	86	1	no	—
262	SD₁₆	—	86	2	no	—
263	SD₁₆	—	86	2	no	—
264	D₈	—	86	2	no	—
265	D₈	—	86	1	no	—
266	D₈	—	86	1	no	—
267	D₈	—	86	2	no	—
268	M₁₆	—	2	1	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	43	2	no	—
271	M₁₆	—	86	2	no	—
272	M₁₆	—	86	2	no	—
273	M₁₆	—	86	1	no	—
274	M₁₆	C₂²×D₄₃	1	2	yes	—
275	M₁₆	C₂²×D₄₃	1	2	yes	—
276	C₂×C₈	—	2	1	no	—
277	C₂×C₈	—	2	1	no	—
278	C₂×C₈	—	43	1	no	—
279	C₂×C₈	—	43	1	no	—
280	C₂×C₈	—	43	1	no	—
281	C₂×C₈	—	43	1	no	—
282	C₂×C₈	—	43	1	no	—
283	C₂×C₈	—	43	1	no	—
284	C₂×C₈	—	86	2	no	—
285	C₂×C₈	—	86	2	no	—
286	C₂×C₈	—	86	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	86	1	no	—
289	C₂³	—	86	1	no	—
290	C₂³	#10621898	1	1	yes	characteristic
291	Q₈	—	2	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	43	2	no	—
294	Q₈	—	86	1	no	—
295	Q₈	—	86	1	no	—
296	Q₈	—	86	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	2	1	no	—
299	D₄	—	4	1	no	—
300	D₄	—	4	1	no	—
301	D₄	—	43	2	no	—
302	D₄	—	86	2	no	—
303	D₄	—	86	2	no	—
304	D₄	—	86	1	no	—
305	D₄	—	86	1	no	—
306	D₄	—	86	1	no	—
307	D₄	—	172	1	no	—
308	D₄	—	172	1	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	1	no	—
312	C₂×C₄	—	86	1	no	—
313	C₂×C₄	—	86	2	no	—
314	C₂×C₄	—	86	2	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	172	1	no	—
317	C₂×C₄	—	172	1	no	—
318	C₂×C₄	#10621898	1	1	yes	characteristic
319	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
320	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
321	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
322	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
323	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
324	C₈	—	2	2	no	—
325	C₈	—	2	2	no	—
326	C₈	—	43	1	no	—
327	C₈	—	43	1	no	—
328	C₈	—	43	1	no	—
329	C₈	—	43	1	no	—
330	C₈	—	86	2	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	8	1	no	—
334	C₂²	—	86	2	no	—
335	C₂²	—	344	1	no	—
336	C₂²	—	172	2	no	—
337	C₂²	—	172	1	no	—
338	C₂²	#13063298	1	1	yes	characteristic
339	C₂²	#13063298	1	1	yes	characteristic
340	C₂²	C₂×#10621898	1	1	yes	characteristic
341	C₄	—	4	1	no	—
342	C₄	—	86	2	no	—
343	C₄	—	172	1	no	—
344	C₄	#13063298	1	1	yes	characteristic
345	C₄	#13063298	1	1	yes	characteristic
346	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
347	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
348	C₂	—	2	1	no	—
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	8	1	no	—
352	C₂	—	344	1	no	—
353	C₂	—	172	2	no	—
354	C₂	C₂×#13063298	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#13063298
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#413	C₂
socle	C₂×C₄₃	C₂×#121
hypercenter	#413	D₄₃
hypocenter	C₄₃	#3067
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ #413

Complements

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

N	H ≅ G/N	type
1. #14583699	C₂	semidirect
2. #14583697	C₂	semidirect
3. C₂×#13063345	C₂	semidirect
4. C₂×#13063333	C₂	semidirect
5. #14583173	C₂	semidirect
6. #14583168	C₂	semidirect
7. #14583165	C₂	semidirect
8. #14583161	C₂	semidirect
9. C₂×#13063260	C₂	semidirect
10. #14582881	C₂	semidirect
11. C₄₃×#413	C₂	semidirect
28. C₂×#10621885	C₂²	semidirect
29. C₂×#10621883	C₂²	semidirect
30. C₂×#10621884	C₂²	semidirect
31. C₂×#10621882	C₂²	semidirect
40. #13063260	C₂²	semidirect
41. #13063260	C₂²	semidirect
42. #13063243	C₂²	semidirect
43. #13063241	C₂²	semidirect
47. #101×C₄₃	C₂²	semidirect
48. #100×C₄₃	C₂²	semidirect
188. #413	D₄₃	semidirect
189. C₄₃	#3067	semidirect
224. #101	C₂×D₄₃	semidirect
226. #100	C₂×D₄₃	semidirect