Subgroups — #19146570

Order 5504.

394 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583602	C₂	1	1	yes	maximal, characteristic, complemented
2	#14583552	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583238	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583228	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583196	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583148	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583131	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583101	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583047	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583047	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583012	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582972	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582899	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582894	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621902	C₂²	1	1	yes	characteristic
17	C₂×#10621902	C₂²	1	1	yes	characteristic
18	C₂×#10621884	C₂²	1	1	yes	characteristic
19	C₂×#10621884	C₂²	1	1	yes	characteristic
20	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
21	C₂×#10621872	C₂²	1	1	yes	characteristic
22	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
23	#13063320	—	2	1	no	—
24	#13063310	—	2	1	no	—
25	#13063310	—	2	1	no	—
26	#13063310	—	2	1	no	—
27	#13063301	C₂²	1	1	yes	characteristic
28	#13063301	C₂²	1	1	yes	characteristic
29	#13063300	—	2	1	no	—
30	#13063300	—	2	1	no	—
31	#13063300	C₂²	1	1	yes	characteristic
32	#13063300	C₂²	1	1	yes	characteristic
33	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
34	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
35	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
36	#13063287	—	2	1	no	—
37	#13063283	C₂²	1	1	yes	characteristic
38	C₄⋊C₄×D₄₃	—	2	1	no	—
39	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
40	#13063258	C₂²	1	1	yes	characteristic
41	#13063257	C₂²	1	1	yes	characteristic
42	#13063256	C₂²	1	1	yes	characteristic
43	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
44	#13063249	C₂²	1	1	yes	characteristic
45	#13063249	C₂²	1	1	yes	characteristic
46	#13063247	C₂²	1	1	yes	characteristic
47	#13063247	C₂²	1	1	yes	characteristic
48	#13063245	C₂²	1	1	yes	characteristic
49	#13063245	C₂²	1	1	yes	characteristic
50	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
51	(C₂×D₄₃)⋊C₈	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
53	#125×C₄₃	C₂²	1	1	yes	characteristic
54	#103×C₄₃	C₂²	1	1	yes	characteristic
55	#103×C₄₃	C₂²	1	1	yes	characteristic
56	#103×C₄₃	C₂²	1	1	yes	characteristic
57	#103×C₄₃	C₂²	1	1	yes	characteristic
58	#97×C₄₃	C₂²	1	1	yes	characteristic
59	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	#10621902	—	2	2	no	—
62	#10621902	—	2	2	no	—
63	#10621902	—	4	1	no	—
64	#10621902	—	4	1	no	—
65	C₂×D₁₇₂	—	2	1	no	—
66	C₂×D₁₇₂	—	2	1	no	—
67	C₂×D₁₇₂	—	2	1	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	D₄	1	1	yes	characteristic
70	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
71	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	—	2	1	no	—
74	C₂×C₄×D₄₃	—	2	1	no	—
75	C₂×C₄×D₄₃	—	2	1	no	—
76	C₂×C₄×D₄₃	—	2	1	no	—
77	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
78	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
79	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
80	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	—	2	1	no	—
82	C₄⋊C₄×C₄₃	—	2	1	no	—
83	C₄⋊C₄×C₄₃	—	2	1	no	—
84	C₄⋊C₄×C₄₃	—	2	1	no	—
85	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
86	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
87	C₄²×C₄₃	C₂³	1	1	yes	characteristic
88	#10621884	—	4	1	no	—
89	#10621884	—	4	1	no	—
90	(C₂×D₄₃)⋊C₄	—	2	1	no	—
91	(C₂×D₄₃)⋊C₄	—	2	1	no	—
92	(C₂×D₄₃)⋊C₄	—	2	1	no	—
93	(C₂×D₄₃)⋊C₄	—	2	1	no	—
94	(C₂×D₄₃)⋊C₄	—	2	1	no	—
95	(C₂×D₄₃)⋊C₄	—	2	1	no	—
96	(C₂×D₄₃)⋊C₄	—	2	1	no	—
97	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
98	(C₂×D₄₃)⋊C₄	D₄	1	1	yes	characteristic
99	C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
100	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
101	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
102	Dic₄₃⋊C₄	—	2	1	no	—
103	Dic₄₃⋊C₄	—	2	1	no	—
104	Dic₄₃⋊C₄	—	2	1	no	—
105	C₄×Dic₄₃	—	2	1	no	—
106	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
107	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
108	D₃₄₄	—	2	2	no	—
109	#10621872	—	4	1	no	—
110	C₈×D₄₃	—	4	1	no	—
111	C₂²×D₄₃	—	2	1	no	—
112	C₂²×D₄₃	—	2	1	no	—
113	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
115	Q₈×C₄₃	—	2	1	no	—
116	Q₈×C₄₃	—	2	1	no	—
117	Q₈×C₄₃	—	2	1	no	—
118	Q₈×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	—	2	1	no	—
123	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	D₁₇₂	—	2	1	no	—
131	D₁₇₂	—	2	1	no	—
132	D₁₇₂	—	2	1	no	—
133	D₁₇₂	—	4	1	no	—
134	D₁₇₂	—	4	2	no	—
135	D₁₇₂	—	4	2	no	—
136	D₁₇₂	—	4	2	no	—
137	D₁₇₂	—	8	1	no	—
138	D₁₇₂	#41	1	2	yes	—
139	C₄×D₄₃	—	2	1	no	—
140	C₄×D₄₃	—	2	1	no	—
141	C₄×D₄₃	—	2	1	no	—
142	C₄×D₄₃	—	2	1	no	—
143	C₄×D₄₃	—	4	2	no	—
144	C₄×D₄₃	—	4	2	no	—
145	C₄×D₄₃	—	4	2	no	—
146	C₄×D₄₃	—	8	1	no	—
147	C₄×D₄₃	—	8	1	no	—
148	C₈×C₄₃	—	2	1	no	—
149	C₈×C₄₃	—	2	1	no	—
150	C₄₃⋊C₈	—	2	1	no	—
151	C₄₃⋊C₈	—	2	1	no	—
152	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
153	C₂×D₄₃	—	2	2	no	—
154	C₂×D₄₃	—	2	1	no	—
155	C₂×D₄₃	—	2	1	no	—
156	C₂×D₄₃	—	2	1	no	—
157	C₂×D₄₃	—	2	1	no	—
158	C₂×D₄₃	—	2	1	no	—
159	C₂×D₄₃	—	2	1	no	—
160	C₂×D₄₃	—	4	1	no	—
161	C₂×D₄₃	—	4	1	no	—
162	C₂×D₄₃	—	4	1	no	—
163	C₂×D₄₃	—	4	1	no	—
164	C₂×D₄₃	—	4	1	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	2	1	no	—
167	C₄×C₄₃	—	4	1	no	—
168	C₄×C₄₃	—	4	1	no	—
169	C₄×C₄₃	—	4	1	no	—
170	C₄×C₄₃	—	4	1	no	—
171	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
172	C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
173	Dic₄₃	—	2	1	no	—
174	Dic₄₃	—	2	1	no	—
175	Dic₄₃	—	4	1	no	—
176	Dic₄₃	—	4	1	no	—
177	#3307	—	43	1	no	maximal, Sylow
178	C₂×C₄₃	#546	1	1	yes	characteristic
179	C₂×C₄₃	#575	1	1	yes	characteristic
180	C₂×C₄₃	#577	1	1	yes	characteristic
181	D₄₃	—	4	1	no	—
182	D₄₃	—	4	1	no	—
183	D₄₃	—	4	2	no	—
184	D₄₃	—	8	1	no	—
185	#553	—	43	1	no	—
186	#547	—	43	1	no	—
187	#491	D₄₃	1	1	yes	characteristic, complemented
188	#485	—	43	1	no	—
189	#483	—	43	1	no	—
190	#478	—	43	1	no	—
191	#468	—	43	1	no	—
192	#465	—	43	1	no	—
193	#461	—	43	1	no	—
194	#449	—	43	1	no	—
195	#449	—	43	1	no	—
196	#442	—	43	1	no	—
197	#435	—	43	1	no	—
198	#417	—	43	1	no	—
199	#416	—	43	1	no	—
200	C₄₃	#3307	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	43	1	no	—
202	C₂×#41	—	43	1	no	—
203	C₂×SD₁₆	—	43	1	no	—
204	C₂×SD₁₆	—	43	1	no	—
205	C₂×D₈	—	43	1	no	—
206	C₂×M₁₆	—	43	1	no	—
207	C₂²×C₈	—	43	1	no	—
208	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
209	#126	—	86	1	no	—
210	#125	C₂×D₄₃	1	1	yes	characteristic
211	#123	—	86	1	no	—
212	#123	—	86	1	no	—
213	#123	—	86	1	no	—
214	C₂²⋊Q₈	—	43	1	no	—
215	C₂²⋊Q₈	—	43	1	no	—
216	#121	—	43	1	no	—
217	#121	—	43	1	no	—
218	#121	—	86	1	no	—
219	#121	—	86	1	no	—
220	C₄×D₄	—	43	1	no	—
221	C₄×D₄	—	43	1	no	—
222	C₄×D₄	—	43	1	no	—
223	C₄×D₄	—	86	1	no	—
224	#117	—	43	1	no	—
225	C₂×C₄⋊C₄	—	43	1	no	—
226	C₂×C₄⋊C₄	—	86	1	no	—
227	#107	—	43	1	no	—
228	#106	—	43	1	no	—
229	#106	—	43	1	no	—
230	C₄⋊C₈	—	43	1	no	—
231	#103	—	43	1	no	—
232	#103	—	43	1	no	—
233	#103	C₂×D₄₃	1	1	yes	characteristic
234	#103	C₂×D₄₃	1	1	yes	characteristic
235	#103	C₂×D₄₃	1	1	yes	characteristic
236	#103	C₂×D₄₃	1	1	yes	characteristic
237	#102	—	43	1	no	—
238	#102	—	43	1	no	—
239	#102	—	43	1	no	—
240	#102	—	43	1	no	—
241	C₂²⋊C₈	—	43	1	no	—
242	C₂²⋊C₈	—	43	1	no	—
243	#97	C₂×D₄₃	1	1	yes	characteristic
244	#41	—	86	2	no	—
245	#41	—	86	2	no	—
246	#41	—	172	1	no	—
247	#41	—	172	1	no	—
248	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
249	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
250	C₂×D₄	—	43	1	no	—
251	C₂×D₄	—	43	1	no	—
252	C₂×D₄	—	43	1	no	—
253	C₂×D₄	—	86	1	no	—
254	C₂×D₄	—	86	1	no	—
255	C₂×D₄	—	86	1	no	—
256	C₂×D₄	—	86	1	no	—
257	C₂²×C₄	—	43	1	no	—
258	C₂²×C₄	—	43	1	no	—
259	C₂²×C₄	—	86	1	no	—
260	C₂²×C₄	—	86	1	no	—
261	C₂²×C₄	—	86	1	no	—
262	C₂²×C₄	—	86	1	no	—
263	C₂²×C₄	—	86	1	no	—
264	SD₁₆	—	172	1	no	—
265	SD₁₆	—	172	1	no	—
266	D₈	—	86	2	no	—
267	M₁₆	—	172	1	no	—
268	C₂×C₈	—	43	1	no	—
269	C₂×C₈	—	43	1	no	—
270	C₂×C₈	—	172	1	no	—
271	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
272	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	2	1	no	—
277	C₄⋊C₄	—	43	1	no	—
278	C₄⋊C₄	—	43	1	no	—
279	C₄⋊C₄	—	43	1	no	—
280	C₄⋊C₄	—	86	1	no	—
281	C₄⋊C₄	—	86	1	no	—
282	C₄⋊C₄	—	86	1	no	—
283	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
284	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
285	C₂²⋊C₄	—	43	1	no	—
286	C₂²⋊C₄	—	43	1	no	—
287	C₂²⋊C₄	—	86	1	no	—
288	C₂²⋊C₄	—	86	1	no	—
289	C₂²⋊C₄	—	86	1	no	—
290	C₂²⋊C₄	—	86	1	no	—
291	C₂²⋊C₄	—	86	1	no	—
292	C₂²⋊C₄	—	86	1	no	—
293	C₂²⋊C₄	—	86	1	no	—
294	C₄²	—	86	1	no	—
295	C₄²	C₂²×D₄₃	1	1	yes	characteristic
296	C₂³	—	43	1	no	—
297	C₂³	—	43	1	no	—
298	C₂³	—	86	1	no	—
299	C₂³	—	86	1	no	—
300	Q₈	—	2	1	no	—
301	Q₈	—	2	1	no	—
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	D₄	—	43	2	no	—
305	D₄	—	86	1	no	—
306	D₄	—	86	1	no	—
307	D₄	—	86	1	no	—
308	D₄	—	344	1	no	—
309	D₄	—	172	1	no	—
310	D₄	—	172	2	no	—
311	D₄	—	172	2	no	—
312	D₄	—	172	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	43	1	no	—
318	C₂×C₄	—	43	1	no	—
319	C₂×C₄	—	86	1	no	—
320	C₂×C₄	—	86	1	no	—
321	C₂×C₄	—	86	1	no	—
322	C₂×C₄	—	86	1	no	—
323	C₂×C₄	—	86	1	no	—
324	C₂×C₄	—	86	1	no	—
325	C₂×C₄	—	344	1	no	—
326	C₂×C₄	—	344	1	no	—
327	C₂×C₄	—	172	2	no	—
328	C₂×C₄	—	172	2	no	—
329	C₂×C₄	—	172	2	no	—
330	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
331	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
332	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
333	C₈	—	2	1	no	—
334	C₈	—	2	1	no	—
335	C₈	—	86	1	no	—
336	C₈	—	86	1	no	—
337	C₂²	—	86	1	no	—
338	C₂²	—	86	1	no	—
339	C₂²	—	86	1	no	—
340	C₂²	—	86	1	no	—
341	C₂²	—	86	1	no	—
342	C₂²	—	86	1	no	—
343	C₂²	—	86	2	no	—
344	C₂²	—	172	1	no	—
345	C₂²	—	172	1	no	—
346	C₂²	—	172	1	no	—
347	C₂²	—	172	1	no	—
348	C₂²	—	172	1	no	—
349	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
350	C₄	—	2	1	no	—
351	C₄	—	2	1	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	86	1	no	—
357	C₄	—	86	1	no	—
358	C₄	—	172	1	no	—
359	C₄	—	172	1	no	—
360	C₄	#41×D₄₃	1	1	yes	characteristic
361	C₄	#13063371	1	1	yes	characteristic
362	C₂	—	344	1	no	—
363	C₂	—	172	1	no	—
364	C₂	—	172	1	no	—
365	C₂	—	172	2	no	—
366	C₂	#14583542	1	1	yes	characteristic
367	C₂	#14583698	1	1	yes	characteristic
368	C₂	#14583708	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#491	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#491	D₄₃
hypocenter	C₄₃	#3307
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #491

Complements

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

N	H ≅ G/N	type
1. #14583602	C₂	semidirect
2. #14583552	C₂	semidirect
3. #14583238	C₂	semidirect
4. #14583228	C₂	semidirect
5. #14583196	C₂	semidirect
6. #14583148	C₂	semidirect
7. #14583131	C₂	semidirect
8. #14583101	C₂	semidirect
9. #14583047	C₂	semidirect
10. #14583047	C₂	semidirect
11. #14583012	C₂	semidirect
12. #14582972	C₂	semidirect
13. #14582899	C₂	semidirect
14. #14582894	C₂	semidirect
15. C₄₃×#491	C₂	semidirect
187. #491	D₄₃	semidirect
200. C₄₃	#3307	semidirect