Subgroups — #19144624

Order 5504.

428 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#14583446	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583219	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583216	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583183	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583182	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583098	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583097	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583039	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583036	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×#10621873	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#10621884	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×SD₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621898	—	2	1	no	—
17	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
18	C₂×#10621873	C₂²	1	1	yes	characteristic
19	C₂×#10621883	C₂²	1	1	yes	characteristic
20	C₂×#10621884	C₂²	1	1	yes	characteristic
21	#13063324	C₂²	1	1	yes	characteristic
22	#13063321	C₂²	1	1	yes	characteristic
23	#13063311	—	2	1	no	—
24	#13063300	C₂²	1	1	yes	characteristic
25	#13063298	—	2	1	no	—
26	#13063296	C₂²	1	1	yes	characteristic
27	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
28	C₄×Dic₈₆	—	2	1	no	—
29	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
30	#13063290	—	2	1	no	—
31	#13063290	C₂²	1	1	yes	characteristic
32	C₄×D₁₇₂	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₄₃	—	2	1	no	—
34	C₄⋊C₄×D₄₃	—	2	1	no	—
35	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
36	C₄²×D₄₃	—	2	1	no	—
37	C₄²×D₄₃	C₂²	1	1	yes	characteristic
38	#13063257	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	#13063249	C₂²	1	1	yes	characteristic
45	#13063248	C₂²	1	1	yes	characteristic
46	#13063247	C₂²	1	1	yes	characteristic
47	#13063246	C₂²	1	1	yes	characteristic
48	#13063245	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×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
55	#106×C₄₃	C₂²	1	1	yes	characteristic
56	#103×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×Q₈×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂×C₂²⋊D₄₃	—	2	1	no	—
63	C₂×C₂²⋊D₄₃	—	2	1	no	—
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	Q₈×D₄₃	—	2	2	no	—
66	Q₈×D₄₃	—	2	2	no	—
67	#10621898	—	4	4	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
70	C₂×C₄×D₄₃	—	2	1	no	—
71	C₂×C₄×D₄₃	—	2	1	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
74	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
75	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
76	C₂×Dic₈₆	—	2	1	no	—
77	C₂×Dic₈₆	—	2	1	no	—
78	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
79	SD₁₆×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
81	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
82	C₄⋊C₄×C₄₃	—	2	1	no	—
83	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
85	C₂²⋊C₄×C₄₃	—	2	1	no	—
86	C₄²×C₄₃	—	2	1	no	—
87	C₄²×C₄₃	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₄₃	—	2	1	no	—
89	#10621884	—	2	2	no	—
90	#10621883	—	2	2	no	—
91	(C₂×D₄₃)⋊C₄	—	2	1	no	—
92	(C₂×D₄₃)⋊C₄	—	2	1	no	—
93	C₄⋊Dic₄₃	—	2	1	no	—
94	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
95	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
96	Dic₄₃⋊C₄	—	2	1	no	—
97	Dic₄₃⋊C₄	—	2	1	no	—
98	Dic₄₃⋊C₄	—	2	1	no	—
99	Dic₄₃⋊C₄	—	2	1	no	—
100	C₄×Dic₄₃	—	2	1	no	—
101	C₄×Dic₄₃	—	2	1	no	—
102	C₄×Dic₄₃	—	2	1	no	—
103	C₄×Dic₄₃	D₄	1	1	yes	characteristic
104	C₄×Dic₄₃	D₄	1	1	yes	characteristic
105	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
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	C₂²×D₄₃	—	2	1	no	—
111	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
112	Q₈×C₄₃	—	2	1	no	—
113	Q₈×C₄₃	#41	1	2	yes	complemented
114	D₄×C₄₃	—	2	1	no	—
115	D₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	4	2	no	—
120	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₄₃	—	4	2	no	—
124	C₂²⋊D₄₃	—	4	2	no	—
125	C₂²⋊D₄₃	—	4	2	no	—
126	C₂×Dic₄₃	—	2	1	no	—
127	C₂×Dic₄₃	—	2	1	no	—
128	C₂×Dic₄₃	—	2	1	no	—
129	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
132	D₁₇₂	—	2	1	no	—
133	D₁₇₂	—	2	1	no	—
134	D₁₇₂	—	4	2	no	—
135	C₄×D₄₃	—	2	4	no	—
136	C₄×D₄₃	—	2	1	no	—
137	C₄×D₄₃	—	2	1	no	—
138	C₄×D₄₃	—	4	1	no	—
139	C₄×D₄₃	—	4	2	no	—
140	C₄×D₄₃	—	4	2	no	—
141	C₄×D₄₃	SD₁₆	1	4	yes	complemented
142	Dic₈₆	—	2	4	no	—
143	Dic₈₆	—	2	1	no	—
144	Dic₈₆	—	4	2	no	—
145	Dic₈₆	#41	1	2	yes	complemented
146	C₈×C₄₃	—	2	1	no	—
147	C₈×C₄₃	—	2	1	no	—
148	C₄₃⋊C₈	—	2	1	no	—
149	C₄₃⋊C₈	—	2	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	—	4	1	no	—
152	C₂²×C₄₃	—	4	1	no	—
153	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
154	C₂×D₄₃	—	2	1	no	—
155	C₂×D₄₃	—	2	1	no	—
156	C₂×D₄₃	—	4	1	no	—
157	C₂×D₄₃	—	4	1	no	—
158	C₂×D₄₃	—	4	1	no	—
159	C₂×D₄₃	C₂×SD₁₆	1	2	yes	—
160	C₄×C₄₃	—	2	2	no	—
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	—	4	1	no	—
163	C₄×C₄₃	—	4	1	no	—
164	C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
165	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
166	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
167	Dic₄₃	—	2	1	no	—
168	Dic₄₃	—	2	2	no	—
169	Dic₄₃	—	2	1	no	—
170	Dic₄₃	—	4	1	no	—
171	Dic₄₃	—	4	1	no	—
172	Dic₄₃	C₂×SD₁₆	1	2	yes	—
173	#3103	—	43	1	no	maximal, Sylow
174	C₂×C₄₃	—	8	1	no	—
175	C₂×C₄₃	#525	1	1	yes	characteristic
176	C₂×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
177	C₂×C₄₃	#575	1	1	yes	characteristic
178	D₄₃	—	2	2	no	—
179	D₄₃	—	8	1	no	—
180	#532	—	43	1	no	—
181	C₂×C₄×Q₈	—	43	1	no	—
182	#481	—	43	1	no	—
183	#481	—	43	1	no	—
184	#475	—	43	1	no	—
185	#475	—	43	1	no	—
186	#460	—	43	1	no	—
187	#460	—	43	1	no	—
188	#448	—	43	1	no	—
189	#448	—	43	1	no	—
190	C₄×SD₁₆	—	43	1	no	—
191	C₄×SD₁₆	—	43	1	no	—
192	C₄×SD₁₆	—	43	1	no	—
193	C₄×SD₁₆	D₄₃	1	1	yes	characteristic, complemented
194	#431	—	43	1	no	—
195	C₄₃	#3103	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	—	86	1	no	—
197	C₂²×Q₈	—	43	1	no	—
198	C₂×SD₁₆	—	43	1	no	—
199	C₂×SD₁₆	—	43	1	no	—
200	C₂×SD₁₆	—	43	1	no	—
201	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
202	C₄⋊Q₈	—	43	1	no	—
203	#127	—	43	1	no	—
204	#124	—	86	1	no	—
205	#121	—	43	1	no	—
206	#121	—	43	1	no	—
207	#121	—	86	1	no	—
208	C₄×Q₈	—	43	1	no	—
209	C₄×Q₈	—	43	1	no	—
210	C₄×Q₈	—	43	1	no	—
211	C₄×Q₈	—	86	1	no	—
212	C₄×Q₈	—	86	1	no	—
213	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
214	C₄×D₄	—	43	1	no	—
215	C₄×D₄	—	86	1	no	—
216	C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
217	C₂×C₄⋊C₄	—	43	1	no	—
218	C₂×C₄⋊C₄	—	86	1	no	—
219	C₂×C₄²	—	43	1	no	—
220	C₂×C₄²	—	86	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	C₄⋊C₈	—	43	1	no	—
226	C₄⋊C₈	—	43	1	no	—
227	#103	—	43	1	no	—
228	#103	—	43	1	no	—
229	#103	—	43	1	no	—
230	#103	C₂×D₄₃	1	1	yes	characteristic
231	#102	—	43	1	no	—
232	#102	—	43	1	no	—
233	#102	—	43	1	no	—
234	#102	C₂×D₄₃	1	1	yes	characteristic
235	C₂²⋊C₈	—	43	1	no	—
236	C₂²⋊C₈	—	43	1	no	—
237	C₄×C₈	—	43	1	no	—
238	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
239	#41	—	172	4	no	—
240	C₂×Q₈	—	43	1	no	—
241	C₂×Q₈	—	86	1	no	—
242	C₂×Q₈	—	86	1	no	—
243	C₂×Q₈	—	86	2	no	—
244	C₂×Q₈	—	86	2	no	—
245	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
246	C₂×D₄	—	43	1	no	—
247	C₂×D₄	—	86	1	no	—
248	C₂×D₄	—	86	1	no	—
249	C₂×D₄	—	86	1	no	—
250	C₂×D₄	—	86	1	no	—
251	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	—	2	1	no	—
253	C₂²×C₄	—	43	1	no	—
254	C₂²×C₄	—	43	1	no	—
255	C₂²×C₄	—	43	1	no	—
256	C₂²×C₄	—	86	1	no	—
257	C₂²×C₄	—	86	1	no	—
258	C₂²×C₄	—	86	1	no	—
259	SD₁₆	—	2	2	no	—
260	SD₁₆	—	86	2	no	—
261	SD₁₆	—	86	2	no	—
262	SD₁₆	—	86	2	no	—
263	C₂×C₈	—	43	1	no	—
264	C₂×C₈	—	43	1	no	—
265	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
266	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	43	1	no	—
269	C₄⋊C₄	—	43	1	no	—
270	C₄⋊C₄	—	86	1	no	—
271	C₄⋊C₄	—	86	1	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₄⋊C₄	—	86	1	no	—
274	C₄⋊C₄	—	86	1	no	—
275	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
276	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
277	C₂²⋊C₄	—	2	1	no	—
278	C₂²⋊C₄	—	86	1	no	—
279	C₂²⋊C₄	—	86	1	no	—
280	C₂²⋊C₄	—	86	1	no	—
281	C₄²	—	2	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	C₄²	—	86	1	no	—
288	C₄²	C₂²×D₄₃	1	1	yes	characteristic
289	C₂³	—	2	1	no	—
290	C₂³	—	43	1	no	—
291	C₂³	—	86	1	no	—
292	Q₈	—	2	1	no	—
293	Q₈	—	43	2	no	—
294	Q₈	—	86	4	no	—
295	Q₈	—	86	1	no	—
296	Q₈	—	172	2	no	—
297	Q₈	#10621898	1	2	yes	complemented
298	D₄	—	2	1	no	—
299	D₄	—	2	1	no	—
300	D₄	—	86	1	no	—
301	D₄	—	86	1	no	—
302	D₄	—	172	2	no	—
303	D₄	—	172	2	no	—
304	D₄	—	172	2	no	—
305	D₄	—	172	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	4	2	no	—
310	C₂×C₄	—	43	1	no	—
311	C₂×C₄	—	43	4	no	—
312	C₂×C₄	—	43	1	no	—
313	C₂×C₄	—	43	1	no	—
314	C₂×C₄	—	86	4	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	86	1	no	—
317	C₂×C₄	—	86	1	no	—
318	C₂×C₄	—	86	1	no	—
319	C₂×C₄	—	86	1	no	—
320	C₂×C₄	—	172	1	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	—	172	2	no	—
323	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
324	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
325	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
326	C₈	—	2	1	no	—
327	C₈	—	2	1	no	—
328	C₈	—	86	1	no	—
329	C₈	—	86	1	no	—
330	C₂²	—	4	1	no	—
331	C₂²	—	4	1	no	—
332	C₂²	—	4	1	no	—
333	C₂²	—	43	2	no	—
334	C₂²	—	86	1	no	—
335	C₂²	—	86	1	no	—
336	C₂²	—	172	1	no	—
337	C₂²	—	172	1	no	—
338	C₂²	—	172	1	no	—
339	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	1	no	—
342	C₄	—	4	1	no	—
343	C₄	—	4	1	no	—
344	C₄	—	43	2	no	—
345	C₄	—	86	1	no	—
346	C₄	—	86	2	no	—
347	C₄	—	86	1	no	—
348	C₄	—	172	1	no	—
349	C₄	—	172	1	no	—
350	C₄	SD₁₆×D₄₃	1	2	yes	—
351	C₄	#41×D₄₃	1	1	yes	characteristic
352	C₄	C₂×#10621898	1	1	yes	characteristic
353	C₂	—	8	1	no	—
354	C₂	—	86	2	no	—
355	C₂	—	344	1	no	—
356	C₂	#14583408	1	1	yes	characteristic
357	C₂	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
358	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×SD₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	C₄×SD₁₆	D₄₃
hypocenter	C₄₃	#3103
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×SD₁₆

Complements

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

N	H ≅ G/N	type
2. C₄×Q₈×D₄₃	C₂	semidirect
3. #14583219	C₂	semidirect
4. #14583216	C₂	semidirect
5. #14583183	C₂	semidirect
6. #14583182	C₂	semidirect
7. #14583098	C₂	semidirect
8. #14583097	C₂	semidirect
9. #14583039	C₂	semidirect
10. #14583036	C₂	semidirect
11. C₄×#10621873	C₂	semidirect
12. C₄×#10621884	C₂	semidirect
13. C₄×#10621883	C₂	semidirect
14. #14582950	C₂	semidirect
15. C₄×SD₁₆×C₄₃	C₂	semidirect
113. Q₈×C₄₃	#41	semidirect
141. C₄×D₄₃	SD₁₆	semidirect
145. Dic₈₆	#41	semidirect
193. C₄×SD₁₆	D₄₃	semidirect
195. C₄₃	#3103	semidirect
297. Q₈	#10621898	semidirect