Subgroups — #19144609

Order 5504.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583431	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#10621900	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583242	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583221	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583205	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583197	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583120	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583112	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583062	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583041	C₂	1	1	yes	maximal, characteristic
11	C₄×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#10621883	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#10621883	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582950	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄×D₈×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
18	C₂×#10621883	C₂²	1	1	yes	characteristic
19	C₂×#10621883	C₂²	1	1	yes	characteristic
20	#13063318	—	2	1	no	—
21	#13063316	C₂²	1	1	yes	characteristic
22	#13063312	C₂²	1	1	yes	characteristic
23	#13063307	—	2	1	no	—
24	#13063305	C₂²	1	1	yes	characteristic
25	#13063296	C₂²	1	1	yes	characteristic
26	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
27	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
28	#13063290	C₂²	1	1	yes	characteristic
29	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
30	C₄×C₂²⋊D₄₃	—	2	1	no	—
31	C₄×C₂²⋊D₄₃	—	2	1	no	—
32	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
33	#13063284	—	2	1	no	—
34	#13063283	C₂²	1	1	yes	characteristic
35	#13063281	—	2	1	no	—
36	C₂×C₄×Dic₄₃	—	2	1	no	—
37	C₄²×D₄₃	C₂²	1	1	yes	characteristic
38	#13063259	C₂²	1	1	yes	characteristic
39	#13063256	C₂²	1	1	yes	characteristic
40	#13063256	C₂²	1	1	yes	characteristic
41	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
43	#13063250	C₂²	1	1	yes	characteristic
44	#13063250	C₂²	1	1	yes	characteristic
45	#13063248	C₂²	1	1	yes	characteristic
46	#13063248	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063246	C₂²	1	1	yes	characteristic
49	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₄₃⋊C₈	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	#107×C₄₃	C₂²	1	1	yes	characteristic
56	#102×C₄₃	C₂²	1	1	yes	characteristic
57	#102×C₄₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×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	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	1	no	—
67	#10621900	—	2	2	no	—
68	#10621900	—	2	2	no	—
69	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
70	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
71	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
72	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
73	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
74	D₈×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	C₂³	1	1	yes	characteristic
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	C₂³	1	1	yes	characteristic
82	C₂²⋊Dic₄₃	—	2	1	no	—
83	C₂²⋊Dic₄₃	—	2	1	no	—
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	C₂²⋊Dic₄₃	—	2	1	no	—
86	#10621883	—	2	2	no	—
87	#10621883	—	2	2	no	—
88	(C₂×D₄₃)⋊C₄	—	2	1	no	—
89	(C₂×D₄₃)⋊C₄	—	2	1	no	—
90	C₄⋊Dic₄₃	—	2	1	no	—
91	C₄⋊Dic₄₃	—	2	1	no	—
92	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
93	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	D₄	1	1	yes	characteristic
103	C₄×Dic₄₃	D₄	1	1	yes	characteristic
104	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	Dic₁₇₂	—	2	2	no	—
108	C₂³×C₄₃	—	2	1	no	—
109	C₂³×C₄₃	—	2	1	no	—
110	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
111	D₄×C₄₃	—	2	1	no	—
112	D₄×C₄₃	—	2	1	no	—
113	D₄×C₄₃	—	2	1	no	—
114	D₄×C₄₃	#41	1	2	yes	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	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	—	2	4	no	—
123	C₂²⋊D₄₃	—	4	2	no	—
124	C₂×Dic₄₃	—	2	4	no	—
125	C₂×Dic₄₃	—	2	1	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	C₄×D₄₃	—	2	2	no	—
134	C₄×D₄₃	—	2	1	no	—
135	C₄×D₄₃	—	2	1	no	—
136	C₄×D₄₃	—	4	1	no	—
137	Dic₈₆	—	2	1	no	—
138	Dic₈₆	—	2	1	no	—
139	Dic₈₆	—	2	1	no	—
140	Dic₈₆	#41	1	2	yes	—
141	C₈×C₄₃	—	2	1	no	—
142	C₈×C₄₃	—	2	1	no	—
143	C₄₃⋊C₈	—	2	1	no	—
144	C₄₃⋊C₈	—	2	1	no	—
145	C₂²×C₄₃	—	2	2	no	—
146	C₂²×C₄₃	—	4	1	no	—
147	C₂²×C₄₃	—	4	1	no	—
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
152	C₂×D₄₃	—	2	1	no	—
153	C₂×D₄₃	—	2	1	no	—
154	C₂×D₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	—	4	1	no	—
157	C₄×C₄₃	—	4	1	no	—
158	C₄×C₄₃	#135	1	2	yes	—
159	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
160	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
161	Dic₄₃	—	2	2	no	—
162	Dic₄₃	—	2	1	no	—
163	Dic₄₃	—	2	1	no	—
164	Dic₄₃	—	2	1	no	—
165	Dic₄₃	—	4	1	no	—
166	Dic₄₃	—	4	1	no	—
167	Dic₄₃	—	4	1	no	—
168	#3102	—	43	1	no	maximal, Sylow
169	C₂×C₄₃	—	4	2	no	—
170	C₂×C₄₃	—	8	1	no	—
171	C₂×C₄₃	#525	1	1	yes	characteristic
172	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
173	C₂×C₄₃	#575	1	1	yes	characteristic
174	D₄₃	—	4	1	no	—
175	#529	—	43	1	no	—
176	C₄×#41	—	43	1	no	—
177	#485	—	43	1	no	—
178	#482	—	43	1	no	—
179	#479	—	43	1	no	—
180	#478	—	43	1	no	—
181	#464	—	43	1	no	—
182	#463	—	43	1	no	—
183	#452	—	43	1	no	—
184	#449	—	43	1	no	—
185	C₄×Q₁₆	—	43	1	no	—
186	C₄×SD₁₆	—	43	1	no	—
187	C₄×SD₁₆	—	43	1	no	—
188	C₄×D₈	D₄₃	1	1	yes	characteristic, complemented
189	#431	—	43	1	no	—
190	C₄₃	#3102	1	1	yes	characteristic, complemented, Sylow
191	C₂×#41	—	43	1	no	—
192	C₂×Q₁₆	—	43	1	no	—
193	C₂×SD₁₆	—	43	1	no	—
194	C₂×SD₁₆	—	43	1	no	—
195	C₂×D₈	C₂×D₄₃	1	1	yes	characteristic
196	#126	—	86	1	no	—
197	#125	—	43	1	no	—
198	#124	—	43	1	no	—
199	#123	—	86	1	no	—
200	C₂²⋊Q₈	—	43	1	no	—
201	#121	—	43	1	no	—
202	C₄×Q₈	—	43	1	no	—
203	C₄×Q₈	—	43	1	no	—
204	C₄×Q₈	—	43	1	no	—
205	C₄×D₄	—	43	1	no	—
206	C₄×D₄	—	86	1	no	—
207	C₄×D₄	—	86	1	no	—
208	C₄×D₄	—	86	1	no	—
209	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
210	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
211	#117	—	43	1	no	—
212	#117	—	86	1	no	—
213	#117	—	86	1	no	—
214	C₂×C₄²	—	43	1	no	—
215	C₂×C₄²	—	86	1	no	—
216	#107	—	43	1	no	—
217	#107	C₂×D₄₃	1	1	yes	characteristic
218	#106	—	43	1	no	—
219	#106	—	43	1	no	—
220	C₄⋊C₈	—	43	1	no	—
221	C₄⋊C₈	—	43	1	no	—
222	#103	—	43	1	no	—
223	#103	—	43	1	no	—
224	#103	—	43	1	no	—
225	#103	—	43	1	no	—
226	#102	—	43	1	no	—
227	#102	—	43	1	no	—
228	#102	C₂×D₄₃	1	1	yes	characteristic
229	#102	C₂×D₄₃	1	1	yes	characteristic
230	C₂²⋊C₈	—	43	1	no	—
231	C₂²⋊C₈	—	43	1	no	—
232	C₄×C₈	—	43	1	no	—
233	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
234	#41	—	86	2	no	—
235	#41	—	86	2	no	—
236	C₂×Q₈	—	43	1	no	—
237	C₂×Q₈	—	43	1	no	—
238	C₂×D₄	—	86	1	no	—
239	C₂×D₄	—	86	1	no	—
240	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
241	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	—	2	1	no	—
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	43	1	no	—
245	C₂²×C₄	—	43	1	no	—
246	C₂²×C₄	—	43	1	no	—
247	C₂²×C₄	—	86	1	no	—
248	C₂²×C₄	—	86	1	no	—
249	Q₁₆	—	86	2	no	—
250	SD₁₆	—	86	2	no	—
251	SD₁₆	—	86	2	no	—
252	D₈	—	2	2	no	—
253	C₂×C₈	—	43	1	no	—
254	C₂×C₈	—	43	1	no	—
255	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
257	C₄⋊C₄	—	43	1	no	—
258	C₄⋊C₄	—	43	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	—	86	1	no	—
265	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
266	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
267	C₂²⋊C₄	—	2	1	no	—
268	C₂²⋊C₄	—	2	1	no	—
269	C₂²⋊C₄	—	86	1	no	—
270	C₂²⋊C₄	—	86	1	no	—
271	C₂²⋊C₄	—	86	1	no	—
272	C₂²⋊C₄	—	86	1	no	—
273	C₂²⋊C₄	—	86	1	no	—
274	C₂²⋊C₄	—	86	1	no	—
275	C₄²	—	43	1	no	—
276	C₄²	—	43	1	no	—
277	C₄²	—	43	1	no	—
278	C₄²	—	86	1	no	—
279	C₄²	—	86	1	no	—
280	C₄²	—	86	1	no	—
281	C₄²	—	86	1	no	—
282	C₄²	C₂²×D₄₃	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	2	1	no	—
285	C₂³	—	43	1	no	—
286	Q₈	—	43	2	no	—
287	Q₈	—	86	1	no	—
288	Q₈	—	86	1	no	—
289	Q₈	—	86	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	2	1	no	—
293	D₄	—	86	4	no	—
294	D₄	—	172	2	no	—
295	D₄	#10621898	1	2	yes	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	43	1	no	—
301	C₂×C₄	—	43	1	no	—
302	C₂×C₄	—	43	1	no	—
303	C₂×C₄	—	86	4	no	—
304	C₂×C₄	—	86	1	no	—
305	C₂×C₄	—	86	2	no	—
306	C₂×C₄	—	86	1	no	—
307	C₂×C₄	—	86	1	no	—
308	C₂×C₄	—	86	1	no	—
309	C₂×C₄	—	86	1	no	—
310	C₂×C₄	—	86	1	no	—
311	C₂×C₄	—	172	1	no	—
312	C₂×C₄	—	172	2	no	—
313	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
314	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	86	1	no	—
319	C₈	—	86	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	4	1	no	—
324	C₂²	—	4	1	no	—
325	C₂²	—	4	1	no	—
326	C₂²	—	86	1	no	—
327	C₂²	—	86	1	no	—
328	C₂²	—	86	1	no	—
329	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	4	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	86	1	no	—
334	C₄	—	86	2	no	—
335	C₄	—	86	1	no	—
336	C₄	—	86	1	no	—
337	C₄	—	172	1	no	—
338	C₄	—	172	1	no	—
339	C₄	—	172	1	no	—
340	C₄	#13063344	1	2	yes	—
341	C₄	#41×D₄₃	1	1	yes	characteristic
342	C₄	C₂×#10621898	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	8	1	no	—
345	C₂	—	172	1	no	—
346	C₂	#14583408	1	1	yes	characteristic
347	C₂	C₂×#13063344	1	1	yes	characteristic
348	C₂	#14583696	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄×D₈×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×D₈	D₄₃
hypocenter	C₄₃	#3102
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₈

Complements

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

N	H ≅ G/N	type
1. #14583431	C₂	semidirect
2. C₄×#10621900	C₂	semidirect
3. #14583242	C₂	semidirect
4. #14583221	C₂	semidirect
5. #14583205	C₂	semidirect
6. #14583197	C₂	semidirect
7. #14583120	C₂	semidirect
8. #14583112	C₂	semidirect
9. #14583062	C₂	semidirect
11. C₄×Dic₁₇₂	C₂	semidirect
12. C₄×#10621883	C₂	semidirect
13. C₄×#10621883	C₂	semidirect
14. #14582950	C₂	semidirect
15. C₄×D₈×C₄₃	C₂	semidirect
188. C₄×D₈	D₄₃	semidirect
190. C₄₃	#3102	semidirect