Subgroups — #19145149

Order 5504.

412 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	#14583451	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583444	C₂	1	1	yes	maximal, characteristic
3	#14583296	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583270	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583248	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583244	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583130	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583128	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583126	C₂	1	1	yes	maximal, characteristic, complemented
10	#14582990	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582986	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582984	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#465	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	—	2	1	no	—
17	C₂×#10621873	C₂²	1	1	yes	characteristic
18	C₂×#10621883	C₂²	1	1	yes	characteristic
19	C₂×#10621884	C₂²	1	1	yes	characteristic
20	C₂²×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
21	#13063324	C₂²	1	1	yes	characteristic
22	#13063324	C₂²	1	1	yes	characteristic
23	#13063323	C₂²	1	1	yes	characteristic
24	#13063321	C₂²	1	1	yes	characteristic
25	#13063315	—	2	1	no	—
26	#13063311	—	2	1	no	—
27	#13063304	C₂²	1	1	yes	characteristic
28	#13063302	—	2	1	no	—
29	#13063299	C₂²	1	1	yes	characteristic
30	#13063297	—	2	1	no	—
31	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
32	#13063290	—	2	1	no	—
33	#13063290	C₂²	1	1	yes	characteristic
34	D₄×Dic₄₃	C₂²	1	1	yes	characteristic
35	#13063287	C₂²	1	1	yes	characteristic
36	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
37	#13063281	—	2	1	no	—
38	C₂×C₄×Dic₄₃	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	#13063250	C₂²	1	1	yes	characteristic
43	#13063249	C₂²	1	1	yes	characteristic
44	#13063248	C₂²	1	1	yes	characteristic
45	#13063247	C₂²	1	1	yes	characteristic
46	#13063246	C₂²	1	1	yes	characteristic
47	#13063245	C₂²	1	1	yes	characteristic
48	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
49	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
50	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
55	#121×C₄₃	C₂²	1	1	yes	characteristic
56	#106×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#102×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	—	2	1	no	—
61	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	C₂×C₂²⋊D₄₃	—	2	1	no	—
66	C₂²×Dic₄₃	—	2	1	no	—
67	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
68	C₂²×Dic₄₃	D₄	1	1	yes	characteristic
69	#10621900	—	4	4	no	—
70	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×Dic₈₆	—	2	1	no	—
73	C₂×Dic₈₆	—	2	1	no	—
74	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
75	SD₁₆×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₄⋊C₄×C₄₃	—	2	1	no	—
80	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
82	C₂²⋊C₄×C₄₃	—	2	1	no	—
83	C₂²⋊C₄×C₄₃	—	2	1	no	—
84	C₂²⋊Dic₄₃	—	2	1	no	—
85	C₂²⋊Dic₄₃	—	2	1	no	—
86	#10621884	—	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	C₂³	1	1	yes	characteristic
92	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
99	C₄×Dic₄₃	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	D₄	1	1	yes	characteristic
102	C₄×Dic₄₃	D₄	1	1	yes	characteristic
103	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
104	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
108	#10621873	—	2	2	no	—
109	C₂³×C₄₃	—	2	1	no	—
110	C₂³×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂²×D₄₃	—	2	1	no	—
112	Q₈×C₄₃	—	2	1	no	—
113	Q₈×C₄₃	—	2	1	no	—
114	D₄×C₄₃	—	2	1	no	—
115	D₄×C₄₃	—	2	1	no	—
116	D₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	1	no	—
122	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₄₃	—	4	2	no	—
125	C₂²⋊D₄₃	—	4	2	no	—
126	C₂²⋊D₄₃	—	4	2	no	—
127	C₂×Dic₄₃	—	2	2	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	—	2	1	no	—
130	C₂×Dic₄₃	—	2	1	no	—
131	C₂×Dic₄₃	—	4	2	no	—
132	C₂×Dic₄₃	SD₁₆	1	4	yes	complemented
133	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
134	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
135	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
136	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
137	D₁₇₂	—	2	1	no	—
138	D₁₇₂	—	2	1	no	—
139	C₄×D₄₃	—	4	2	no	—
140	Dic₈₆	—	2	1	no	—
141	Dic₈₆	—	2	1	no	—
142	Dic₈₆	—	4	2	no	—
143	Dic₈₆	—	4	2	no	—
144	C₈×C₄₃	—	2	1	no	—
145	C₈×C₄₃	#41	1	2	yes	complemented
146	C₄₃⋊C₈	—	2	1	no	—
147	C₄₃⋊C₈	#41	1	2	yes	complemented
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	—	4	1	no	—
152	C₂²×C₄₃	—	4	1	no	—
153	C₂²×C₄₃	C₂×SD₁₆	1	2	yes	—
154	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×D₄₃	—	4	1	no	—
156	C₂×D₄₃	—	4	1	no	—
157	C₂×D₄₃	—	4	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	4	1	no	—
160	C₄×C₄₃	—	4	1	no	—
161	C₄×C₄₃	—	4	1	no	—
162	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
163	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
164	Dic₄₃	—	2	1	no	—
165	Dic₄₃	—	4	1	no	—
166	Dic₄₃	—	4	1	no	—
167	Dic₄₃	—	4	1	no	—
168	Dic₄₃	C₂×SD₁₆	1	2	yes	—
169	Dic₄₃	C₂×SD₁₆	1	2	yes	—
170	Dic₄₃	#135	1	2	yes	—
171	#3167	—	43	1	no	maximal, Sylow
172	C₂×C₄₃	—	2	2	no	—
173	C₂×C₄₃	—	8	1	no	—
174	C₂×C₄₃	#532	1	1	yes	characteristic
175	C₂×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
176	C₂×C₄₃	C₂×#135	1	1	yes	characteristic
177	D₄₃	—	8	1	no	—
178	#533	—	43	1	no	—
179	#532	—	43	1	no	—
180	#498	—	43	1	no	—
181	#492	—	43	1	no	—
182	#487	—	43	1	no	—
183	#486	—	43	1	no	—
184	#465	—	43	1	no	—
185	#465	—	43	1	no	—
186	#465	—	43	1	no	—
187	#465	D₄₃	1	1	yes	characteristic, complemented
188	C₄×SD₁₆	—	43	1	no	—
189	C₄×SD₁₆	—	43	1	no	—
190	C₄×SD₁₆	—	43	1	no	—
191	C₄×SD₁₆	—	43	1	no	—
192	C₂×C₄×C₈	—	43	1	no	—
193	C₄₃	#3167	1	1	yes	characteristic, complemented, Sylow
194	C₂×#41	—	86	1	no	—
195	C₂×SD₁₆	—	43	1	no	—
196	C₂×SD₁₆	—	43	1	no	—
197	C₂×SD₁₆	—	43	1	no	—
198	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
199	C₂²×C₈	—	43	1	no	—
200	C₂²×C₈	C₂×D₄₃	1	1	yes	characteristic
201	C₄⋊Q₈	—	43	1	no	—
202	C₄⋊Q₈	—	43	1	no	—
203	C₄⋊Q₈	—	43	1	no	—
204	#127	—	43	1	no	—
205	#125	—	86	1	no	—
206	#124	—	86	1	no	—
207	C₂²⋊Q₈	—	43	1	no	—
208	C₂²⋊Q₈	—	86	1	no	—
209	C₂²⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
210	#121	—	43	1	no	—
211	#121	—	86	1	no	—
212	#121	C₂×D₄₃	1	1	yes	characteristic
213	C₄×Q₈	—	43	1	no	—
214	C₄×Q₈	—	43	1	no	—
215	C₄×Q₈	—	86	1	no	—
216	C₄×D₄	—	43	1	no	—
217	C₄×D₄	—	43	1	no	—
218	C₄×D₄	—	86	1	no	—
219	#117	—	86	1	no	—
220	C₂×C₄²	—	43	1	no	—
221	#106	—	43	1	no	—
222	#106	—	43	1	no	—
223	#106	—	43	1	no	—
224	#106	C₂×D₄₃	1	1	yes	characteristic
225	#103	—	43	1	no	—
226	#103	—	43	1	no	—
227	#103	—	43	1	no	—
228	#103	C₂×D₄₃	1	1	yes	characteristic
229	#102	—	43	1	no	—
230	#102	—	43	1	no	—
231	#102	—	43	1	no	—
232	#102	C₂×D₄₃	1	1	yes	characteristic
233	C₄×C₈	—	43	1	no	—
234	C₄×C₈	—	43	1	no	—
235	C₄×C₈	—	43	1	no	—
236	C₄×C₈	—	43	1	no	—
237	#41	—	172	4	no	—
238	C₂×Q₈	—	43	1	no	—
239	C₂×Q₈	—	86	1	no	—
240	C₂×Q₈	—	86	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	—	86	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	—	43	1	no	—
249	C₂²×C₄	—	43	1	no	—
250	C₂²×C₄	—	86	1	no	—
251	C₂²×C₄	—	86	1	no	—
252	C₂²×C₄	C₂²×D₄₃	1	1	yes	characteristic
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	86	2	no	—
255	SD₁₆	—	86	2	no	—
256	SD₁₆	—	86	2	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	43	1	no	—
259	C₂×C₈	—	43	1	no	—
260	C₂×C₈	—	86	2	no	—
261	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
263	C₄⋊C₄	—	2	1	no	—
264	C₄⋊C₄	—	43	1	no	—
265	C₄⋊C₄	—	43	1	no	—
266	C₄⋊C₄	—	86	1	no	—
267	C₄⋊C₄	—	86	1	no	—
268	C₄⋊C₄	—	86	1	no	—
269	C₄⋊C₄	—	86	1	no	—
270	C₄⋊C₄	—	86	1	no	—
271	C₄⋊C₄	—	86	1	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
274	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
275	C₂²⋊C₄	—	2	1	no	—
276	C₂²⋊C₄	—	2	1	no	—
277	C₂²⋊C₄	—	86	1	no	—
278	C₂²⋊C₄	—	86	1	no	—
279	C₂²⋊C₄	—	86	1	no	—
280	C₂²⋊C₄	—	86	1	no	—
281	C₄²	—	43	1	no	—
282	C₄²	—	43	1	no	—
283	C₄²	—	43	1	no	—
284	C₄²	—	43	1	no	—
285	C₄²	—	86	1	no	—
286	C₄²	—	86	1	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	86	1	no	—
289	C₂³	D₄×D₄₃	1	1	yes	characteristic
290	Q₈	—	2	1	no	—
291	Q₈	—	2	1	no	—
292	Q₈	—	86	1	no	—
293	Q₈	—	86	1	no	—
294	Q₈	—	172	2	no	—
295	Q₈	—	172	2	no	—
296	D₄	—	2	1	no	—
297	D₄	—	2	1	no	—
298	D₄	—	4	2	no	—
299	D₄	—	86	1	no	—
300	D₄	—	86	1	no	—
301	D₄	—	172	2	no	—
302	D₄	—	172	2	no	—
303	D₄	—	172	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	1	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	43	4	no	—
310	C₂×C₄	—	43	1	no	—
311	C₂×C₄	—	43	1	no	—
312	C₂×C₄	—	43	1	no	—
313	C₂×C₄	—	43	1	no	—
314	C₂×C₄	—	86	2	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	86	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	—	172	2	no	—
320	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
321	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
322	C₈	—	2	1	no	—
323	C₈	—	43	2	no	—
324	C₈	—	86	1	no	—
325	C₈	#10621900	1	2	yes	complemented
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	4	1	no	—
329	C₂²	—	4	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	172	1	no	—
332	C₂²	—	172	1	no	—
333	C₂²	—	172	1	no	—
334	C₂²	SD₁₆×D₄₃	1	2	yes	—
335	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	4	1	no	—
338	C₄	—	4	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	43	2	no	—
341	C₄	—	43	2	no	—
342	C₄	—	43	2	no	—
343	C₄	—	86	1	no	—
344	C₄	—	172	1	no	—
345	C₄	—	172	1	no	—
346	C₄	—	172	1	no	—
347	C₄	#41×D₄₃	1	1	yes	characteristic
348	C₄	C₂×#10621900	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	8	1	no	—
351	C₂	—	344	1	no	—
352	C₂	#14583444	1	1	yes	characteristic
353	C₂	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
354	C₂	#135×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#465	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#465	D₄₃
hypocenter	C₄₃	#3167
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #14583451	C₂	semidirect
3. #14583296	C₂	semidirect
4. #14583270	C₂	semidirect
5. #14583248	C₂	semidirect
6. #14583244	C₂	semidirect
7. #14583130	C₂	semidirect
8. #14583128	C₂	semidirect
9. #14583126	C₂	semidirect
10. #14582990	C₂	semidirect
11. SD₁₆×Dic₄₃	C₂	semidirect
12. #14582986	C₂	semidirect
13. #14582984	C₂	semidirect
14. C₂×C₈×Dic₄₃	C₂	semidirect
15. C₄₃×#465	C₂	semidirect
132. C₂×Dic₄₃	SD₁₆	semidirect
145. C₈×C₄₃	#41	semidirect
147. C₄₃⋊C₈	#41	semidirect
187. #465	D₄₃	semidirect
193. C₄₃	#3167	semidirect
325. C₈	#10621900	semidirect