Subgroups — #19145064

Order 5504.

564 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063354	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063352	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#13063346	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×SD₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583474	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583462	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583291	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583290	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583286	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583285	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583283	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583280	C₂	1	1	yes	maximal, characteristic, complemented
13	#14583270	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582853	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621902	C₂²	1	1	yes	characteristic
17	C₂×#10621900	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
19	C₂×D₄×D₄₃	C₂²	1	1	yes	characteristic
20	#13063354	—	2	4	no	—
21	#13063352	—	2	4	no	—
22	#13063346	—	2	4	no	—
23	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic, complemented
24	C₂×#10621885	C₂²	1	1	yes	characteristic, complemented
25	C₂×#10621885	C₂²	1	1	yes	characteristic, complemented
26	C₂×#10621873	C₂²	1	1	yes	characteristic
27	C₂×#10621873	C₂²	1	1	yes	characteristic, complemented
28	C₂×#10621883	C₂²	1	1	yes	characteristic
29	C₂×#10621883	C₂²	1	1	yes	characteristic, complemented
30	C₂×#10621884	C₂²	1	1	yes	characteristic
31	C₂×#10621884	C₂²	1	1	yes	characteristic
32	SD₁₆×D₄₃	—	2	4	no	—
33	C₂×D₃₄₄	C₂²	1	1	yes	characteristic
34	C₂×#10621882	C₂²	1	1	yes	characteristic
35	C₂×#10621882	C₂²	1	1	yes	characteristic, complemented
36	C₂×#10621872	C₂²	1	1	yes	characteristic
37	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
38	#13063323	C₂²	1	1	yes	characteristic, complemented
39	#13063321	C₂²	1	1	yes	characteristic
40	#13063314	C₂²	1	1	yes	characteristic, complemented
41	#13063313	C₂²	1	1	yes	characteristic
42	#13063313	C₂²	1	1	yes	characteristic, complemented
43	#13063312	C₂²	1	1	yes	characteristic
44	#13063310	—	2	2	no	—
45	#13063303	—	2	2	no	—
46	#13063297	—	2	2	no	—
47	#13063293	—	2	2	no	—
48	#13063282	C₂²	1	1	yes	characteristic
49	#13063236	C₂²	1	1	yes	characteristic, complemented
50	#13063235	C₂²	1	1	yes	characteristic, complemented
51	C₈×Dic₄₃	C₂²	1	1	yes	characteristic, complemented
52	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
55	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic, complemented
56	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
57	#127×C₄₃	C₂²	1	1	yes	characteristic
58	C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic, complemented
59	C₂³×D₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×D₄×C₄₃	—	2	2	no	—
63	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
64	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₄₃	—	2	1	no	—
66	C₂×C₂²⋊D₄₃	—	2	2	no	—
67	C₂×C₂²⋊D₄₃	—	2	1	no	—
68	C₂²×Dic₄₃	—	2	1	no	—
69	#10621902	—	2	2	no	—
70	#10621902	—	2	2	no	—
71	Q₈×D₄₃	—	2	2	no	—
72	Q₈×D₄₃	—	2	2	no	—
73	#10621900	—	2	2	no	—
74	#10621900	—	2	2	no	—
75	D₄×D₄₃	—	2	2	no	—
76	D₄×D₄₃	—	2	2	no	—
77	C₂×D₁₇₂	—	2	1	no	—
78	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
79	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×C₄×D₄₃	—	2	1	no	—
82	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
83	C₂×Dic₈₆	—	2	1	no	—
84	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
85	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
86	SD₁₆×C₄₃	—	2	2	no	—
87	SD₁₆×C₄₃	—	2	2	no	—
88	SD₁₆×C₄₃	—	2	2	no	—
89	SD₁₆×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
91	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₄⋊C₄×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	C₂³	1	1	yes	characteristic
94	C₂²⋊Dic₄₃	—	2	2	no	—
95	#10621885	—	2	2	no	—
96	#10621885	—	2	2	no	—
97	#10621884	—	2	2	no	—
98	#10621884	—	2	2	no	—
99	#10621883	—	2	2	no	—
100	#10621883	—	2	2	no	—
101	#10621882	—	2	2	no	—
102	#10621882	—	2	2	no	—
103	(C₂×D₄₃)⋊C₄	—	2	2	no	—
104	(C₂×D₄₃)⋊C₄	—	2	2	no	—
105	(C₂×D₄₃)⋊C₄	—	2	2	no	—
106	(C₂×D₄₃)⋊C₄	D₄	1	2	yes	—
107	Dic₄₃⋊C₄	—	2	2	no	—
108	Dic₄₃⋊C₄	D₄	1	2	yes	—
109	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
110	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
111	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
112	Dic₁₇₂	—	2	2	no	—
113	D₃₄₄	—	2	2	no	—
114	#10621873	—	2	2	no	—
115	#10621873	—	2	2	no	—
116	#10621872	D₄	1	4	yes	complemented
117	C₈×D₄₃	D₄	1	4	yes	complemented
118	C₂³×C₄₃	—	2	1	no	—
119	C₂³×C₄₃	—	2	1	no	—
120	C₂²×D₄₃	—	2	1	no	—
121	C₂²×D₄₃	—	2	1	no	—
122	C₂²×D₄₃	—	4	2	no	—
123	C₂²×D₄₃	—	8	2	no	—
124	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
125	Q₈×C₄₃	—	2	1	no	—
126	Q₈×C₄₃	—	2	1	no	—
127	Q₈×C₄₃	—	2	1	no	—
128	Q₈×C₄₃	—	2	1	no	—
129	D₄×C₄₃	—	2	1	no	—
130	D₄×C₄₃	—	2	1	no	—
131	D₄×C₄₃	—	2	1	no	—
132	D₄×C₄₃	—	2	1	no	—
133	D₄×C₄₃	—	8	2	no	—
134	C₂×C₄×C₄₃	—	2	1	no	—
135	C₂×C₄×C₄₃	—	2	1	no	—
136	C₂×C₄×C₄₃	C₂×D₄	1	2	yes	—
137	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
138	C₂²⋊D₄₃	—	4	2	no	—
139	C₂²⋊D₄₃	—	4	2	no	—
140	C₂²⋊D₄₃	—	8	2	no	—
141	C₂×Dic₄₃	—	2	1	no	—
142	C₂×Dic₄₃	—	2	1	no	—
143	C₂×Dic₄₃	—	4	2	no	—
144	C₂×Dic₄₃	C₂×D₄	1	2	yes	—
145	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
146	D₁₇₂	—	2	1	no	—
147	D₁₇₂	—	2	1	no	—
148	D₁₇₂	—	2	1	no	—
149	D₁₇₂	—	2	1	no	—
150	D₁₇₂	—	4	2	no	—
151	C₄×D₄₃	—	4	2	no	—
152	C₄×D₄₃	—	4	2	no	—
153	C₄×D₄₃	C₂×D₄	1	2	yes	—
154	C₄×D₄₃	C₂×D₄	1	2	yes	—
155	Dic₈₆	—	2	1	no	—
156	Dic₈₆	—	2	1	no	—
157	Dic₈₆	—	2	1	no	—
158	Dic₈₆	—	2	1	no	—
159	Dic₈₆	—	4	2	no	—
160	C₈×C₄₃	C₂×D₄	1	2	yes	—
161	C₈×C₄₃	C₂×D₄	1	2	yes	—
162	C₄₃⋊C₈	C₂×D₄	1	2	yes	—
163	C₄₃⋊C₈	C₂×D₄	1	2	yes	—
164	C₂²×C₄₃	—	4	1	no	—
165	C₂²×C₄₃	—	4	1	no	—
166	C₂²×C₄₃	—	4	2	no	—
167	C₂²×C₄₃	—	4	2	no	—
168	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
169	C₂×D₄₃	—	2	2	no	—
170	C₂×D₄₃	—	4	1	no	—
171	C₂×D₄₃	—	4	1	no	—
172	C₂×D₄₃	—	4	2	no	—
173	C₂×D₄₃	—	4	2	no	—
174	C₂×D₄₃	—	8	4	no	—
175	C₂×D₄₃	#127	1	2	yes	—
176	C₄×C₄₃	—	2	2	no	—
177	C₄×C₄₃	—	4	1	no	—
178	C₄×C₄₃	—	4	1	no	—
179	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
180	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
181	Dic₄₃	—	2	2	no	—
182	Dic₄₃	—	4	1	no	—
183	Dic₄₃	—	4	1	no	—
184	Dic₄₃	#127	1	2	yes	—
185	#3154	—	43	1	no	maximal, Sylow
186	C₂×C₄₃	—	8	1	no	—
187	C₂×C₄₃	—	8	1	no	—
188	C₂×C₄₃	C₂×#127	1	1	yes	characteristic
189	C₂×C₄₃	#577	1	2	yes	—
190	D₄₃	—	2	2	no	—
191	D₄₃	—	8	1	no	—
192	D₄₃	—	8	1	no	—
193	C₂×#137	—	43	1	no	—
194	C₂×AGL(1,ℤ/8)	—	43	1	no	—
195	C₂×#135	—	43	1	no	—
196	C₂²×SD₁₆	—	43	1	no	—
197	#536	—	43	1	no	—
198	#535	—	43	1	no	—
199	#497	—	43	1	no	—
200	#497	—	43	1	no	—
201	#496	—	43	1	no	—
202	#496	—	43	1	no	—
203	#495	—	43	1	no	—
204	#495	—	43	1	no	—
205	#492	—	43	1	no	—
206	#492	D₄₃	1	1	yes	characteristic, complemented
207	#405	—	43	1	no	—
208	C₄₃	#3154	1	1	yes	characteristic, complemented, Sylow
209	C₂×#41	—	43	1	no	—
210	C₂×#41	—	43	1	no	—
211	C₂²×Q₈	—	43	1	no	—
212	C₂²×D₄	—	43	1	no	—
213	#137	—	86	4	no	—
214	AGL(1,ℤ/8)	—	86	4	no	—
215	#135	—	86	4	no	—
216	C₂×Q₁₆	—	43	1	no	—
217	C₂×Q₁₆	—	43	1	no	—
218	C₂×Q₁₆	—	43	1	no	—
219	C₂×SD₁₆	—	43	1	no	—
220	C₂×SD₁₆	—	43	1	no	—
221	C₂×SD₁₆	—	43	1	no	—
222	C₂×SD₁₆	—	43	1	no	—
223	C₂×SD₁₆	—	43	1	no	—
224	C₂×SD₁₆	—	43	1	no	—
225	C₂×SD₁₆	—	86	4	no	—
226	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
227	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
228	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
229	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic, complemented
230	C₂×D₈	—	43	1	no	—
231	C₂×D₈	—	43	1	no	—
232	C₂×D₈	—	43	1	no	—
233	C₂×M₁₆	—	43	1	no	—
234	C₂²×C₈	—	43	1	no	—
235	C₄⋊Q₈	—	43	1	no	—
236	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic, complemented
237	#127	—	43	1	no	—
238	#127	C₂×D₄₃	1	1	yes	characteristic
239	#124	—	43	1	no	—
240	#124	—	43	1	no	—
241	#124	—	43	1	no	—
242	#124	—	43	1	no	—
243	#123	—	86	2	no	—
244	C₂²⋊Q₈	—	86	2	no	—
245	#121	—	86	2	no	—
246	#120	—	86	2	no	—
247	#117	—	43	1	no	—
248	#97	—	43	1	no	—
249	#97	—	43	1	no	—
250	C₄×C₈	—	43	1	no	—
251	C₄×C₈	C₂×D₄₃	1	1	yes	characteristic, complemented
252	C₂⁴	—	86	1	no	—
253	#41	—	86	2	no	—
254	#41	—	86	2	no	—
255	#41	—	86	2	no	—
256	#41	—	86	2	no	—
257	C₂×Q₈	—	43	1	no	—
258	C₂×Q₈	—	43	1	no	—
259	C₂×Q₈	—	86	1	no	—
260	C₂×Q₈	—	86	2	no	—
261	C₂×Q₈	—	86	2	no	—
262	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
263	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
264	C₂×D₄	—	2	2	no	—
265	C₂×D₄	—	43	1	no	—
266	C₂×D₄	—	43	1	no	—
267	C₂×D₄	—	86	1	no	—
268	C₂×D₄	—	86	2	no	—
269	C₂×D₄	—	86	2	no	—
270	C₂×D₄	—	86	2	no	—
271	C₂×D₄	—	86	1	no	—
272	C₂×D₄	—	86	1	no	—
273	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
274	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
275	C₂²×C₄	—	43	1	no	—
276	C₂²×C₄	—	86	1	no	—
277	C₂²×C₄	—	86	1	no	—
278	C₂²×C₄	—	86	1	no	—
279	Q₁₆	—	86	2	no	—
280	Q₁₆	—	86	2	no	—
281	Q₁₆	—	86	2	no	—
282	SD₁₆	—	2	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	2	2	no	—
285	SD₁₆	—	2	2	no	—
286	SD₁₆	—	86	2	no	—
287	SD₁₆	—	86	2	no	—
288	SD₁₆	—	86	2	no	—
289	SD₁₆	—	86	2	no	—
290	SD₁₆	—	86	2	no	—
291	SD₁₆	—	86	2	no	—
292	D₈	—	86	2	no	—
293	D₈	—	86	2	no	—
294	D₈	—	86	2	no	—
295	M₁₆	—	43	4	no	—
296	C₂×C₈	—	43	1	no	—
297	C₂×C₈	—	43	1	no	—
298	C₂×C₈	—	43	4	no	—
299	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
300	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
301	C₄⋊C₄	—	2	2	no	—
302	C₄⋊C₄	—	43	2	no	—
303	C₄⋊C₄	—	86	2	no	—
304	C₂²⋊C₄	—	43	2	no	—
305	C₂²⋊C₄	—	86	2	no	—
306	C₂²⋊C₄	—	86	2	no	—
307	C₂²⋊C₄	—	86	2	no	—
308	C₂²⋊C₄	—	86	2	no	—
309	C₄²	—	43	1	no	—
310	C₄²	C₂²×D₄₃	1	1	yes	characteristic
311	C₂³	—	2	1	no	—
312	C₂³	—	2	1	no	—
313	C₂³	—	43	1	no	—
314	C₂³	—	86	1	no	—
315	C₂³	—	86	1	no	—
316	C₂³	—	344	2	no	—
317	C₂³	—	172	2	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	86	1	no	—
323	Q₈	—	86	1	no	—
324	Q₈	—	86	1	no	—
325	Q₈	—	86	1	no	—
326	Q₈	—	172	2	no	—
327	D₄	—	2	1	no	—
328	D₄	—	2	1	no	—
329	D₄	—	2	1	no	—
330	D₄	—	2	1	no	—
331	D₄	—	8	2	no	—
332	D₄	—	86	1	no	—
333	D₄	—	86	1	no	—
334	D₄	—	86	1	no	—
335	D₄	—	86	1	no	—
336	D₄	—	344	2	no	—
337	D₄	—	172	2	no	—
338	D₄	—	172	2	no	—
339	D₄	—	172	2	no	—
340	C₂×C₄	—	2	1	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	43	1	no	—
343	C₂×C₄	—	43	2	no	—
344	C₂×C₄	—	43	2	no	—
345	C₂×C₄	—	43	2	no	—
346	C₂×C₄	—	86	1	no	—
347	C₂×C₄	—	86	1	no	—
348	C₂×C₄	—	172	2	no	—
349	C₂×C₄	—	172	2	no	—
350	C₂×C₄	—	172	2	no	—
351	C₂×C₄	D₄×D₄₃	1	2	yes	—
352	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
353	C₈	—	43	2	no	—
354	C₈	—	43	2	no	—
355	C₈	D₄×D₄₃	1	2	yes	—
356	C₈	D₄×D₄₃	1	2	yes	—
357	C₂²	—	4	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	2	no	—
360	C₂²	—	4	2	no	—
361	C₂²	—	43	2	no	—
362	C₂²	—	86	2	no	—
363	C₂²	—	344	4	no	—
364	C₂²	—	172	1	no	—
365	C₂²	—	172	1	no	—
366	C₂²	—	172	2	no	—
367	C₂²	—	172	2	no	—
368	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
369	C₄	—	2	2	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	43	2	no	—
373	C₄	—	86	2	no	—
374	C₄	—	172	1	no	—
375	C₄	—	172	1	no	—
376	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
377	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
378	C₂	—	8	1	no	—
379	C₂	—	8	1	no	—
380	C₂	—	86	2	no	—
381	C₂	—	344	1	no	—
382	C₂	—	344	1	no	—
383	C₂	#127×D₄₃	1	1	yes	characteristic
384	C₂	#14583705	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#492	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#492	D₄₃
hypocenter	C₄₃	#3154
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

Complements

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

N	H ≅ G/N	type
1. C₂×#13063354	C₂	semidirect
2. C₂×#13063352	C₂	semidirect
3. C₂×#13063346	C₂	semidirect
4. C₂×SD₁₆×D₄₃	C₂	semidirect
5. #14583474	C₂	semidirect
6. #14583462	C₂	semidirect
7. #14583291	C₂	semidirect
8. #14583290	C₂	semidirect
9. #14583286	C₂	semidirect
10. #14583285	C₂	semidirect
11. #14583283	C₂	semidirect
12. #14583280	C₂	semidirect
13. #14583270	C₂	semidirect
14. #14582853	C₂	semidirect
15. C₄₃×#492	C₂	semidirect
23. C₂×Dic₁₇₂	C₂²	semidirect
24. C₂×#10621885	C₂²	semidirect
25. C₂×#10621885	C₂²	semidirect
27. C₂×#10621873	C₂²	semidirect
29. C₂×#10621883	C₂²	semidirect
35. C₂×#10621882	C₂²	semidirect
38. #13063323	C₂²	semidirect
40. #13063314	C₂²	semidirect
42. #13063313	C₂²	semidirect
49. #13063236	C₂²	semidirect
50. #13063235	C₂²	semidirect
51. C₈×Dic₄₃	C₂²	semidirect
54. C₂×SD₁₆×C₄₃	C₂²	semidirect
55. C₂×SD₁₆×C₄₃	C₂²	semidirect
56. C₄⋊Q₈×C₄₃	C₂²	semidirect
58. C₄×C₈×C₄₃	C₂²	semidirect
116. #10621872	D₄	semidirect
117. C₈×D₄₃	D₄	semidirect
206. #492	D₄₃	semidirect
208. C₄₃	#3154	semidirect
228. C₂×SD₁₆	C₂×D₄₃	semidirect
229. C₂×SD₁₆	C₂×D₄₃	semidirect
236. C₄⋊Q₈	C₂×D₄₃	semidirect
251. C₄×C₈	C₂×D₄₃	semidirect