Subgroups — #19144123

Order 5504.

458 conjugacy classes of subgroups.

The subgroup lattice has 372 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
2	C₂×Q₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583469	C₂	1	1	yes	maximal, characteristic
4	#14583375	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583120	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583119	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583118	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583110	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583109	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583107	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582924	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582906	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582892	C₂	1	1	yes	maximal, characteristic
15	C₄₃×#464	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#10621900	C₂²	1	1	yes	characteristic
17	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
18	C₂×Q₈×D₄₃	C₂²	1	1	yes	characteristic
19	#13063354	—	2	4	no	—
20	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₇₂	C₂²	1	1	yes	characteristic
22	C₂×#10621885	C₂²	1	1	yes	characteristic
23	C₂×#10621885	C₂²	1	1	yes	characteristic
24	C₂×#10621885	C₂²	1	1	yes	characteristic
25	Q₁₆×D₄₃	—	2	4	no	—
26	C₂×#10621883	C₂²	1	1	yes	characteristic
27	C₂×#10621872	C₂²	1	1	yes	characteristic
28	C₂×C₈×D₄₃	C₂²	1	1	yes	characteristic
29	#13063325	C₂²	1	1	yes	characteristic
30	#13063323	C₂²	1	1	yes	characteristic
31	#13063312	C₂²	1	1	yes	characteristic
32	#13063307	—	2	1	no	—
33	#13063307	—	2	1	no	—
34	#13063302	—	2	1	no	—
35	#13063302	—	2	1	no	—
36	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
37	C₄×Dic₈₆	—	2	1	no	—
38	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
39	#13063290	—	2	1	no	—
40	#13063290	C₂²	1	1	yes	characteristic
41	#13063283	—	2	1	no	—
42	#13063283	C₂²	1	1	yes	characteristic
43	#13063282	—	2	1	no	—
44	#13063282	C₂²	1	1	yes	characteristic
45	C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
46	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
47	Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
48	#13063250	C₂²	1	1	yes	characteristic
49	#13063250	C₂²	1	1	yes	characteristic
50	#13063249	C₂²	1	1	yes	characteristic
51	#13063248	C₂²	1	1	yes	characteristic
52	#13063248	C₂²	1	1	yes	characteristic
53	#13063246	C₂²	1	1	yes	characteristic
54	C₂×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₄₃	C₂²	1	1	yes	characteristic
56	#124×C₄₃	C₂²	1	1	yes	characteristic
57	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
58	C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
59	#103×C₄₃	C₂²	1	1	yes	characteristic
60	#102×C₄₃	C₂²	1	1	yes	characteristic
61	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
63	C₂×D₄×C₄₃	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	1	no	—
66	Q₈×D₄₃	—	2	2	no	—
67	Q₈×D₄₃	—	2	2	no	—
68	Q₈×D₄₃	—	2	2	no	—
69	Q₈×D₄₃	D₄	1	4	yes	—
70	#10621900	—	2	2	no	—
71	#10621900	—	2	2	no	—
72	C₂×C₄×D₄₃	—	2	1	no	—
73	C₂×C₄×D₄₃	—	2	1	no	—
74	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
75	C₂×Dic₈₆	—	2	1	no	—
76	C₂×Dic₈₆	—	2	1	no	—
77	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
78	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
79	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
80	Q₁₆×C₄₃	—	2	2	no	—
81	SD₁₆×C₄₃	—	2	2	no	—
82	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
83	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
84	C₄⋊C₄×C₄₃	—	2	1	no	—
85	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
86	C₂²⋊C₄×C₄₃	—	2	1	no	—
87	C₂²⋊C₄×C₄₃	—	2	1	no	—
88	C₄²×C₄₃	—	2	1	no	—
89	C₄²×C₄₃	C₂³	1	1	yes	characteristic
90	C₂²⋊Dic₄₃	—	2	1	no	—
91	C₂²⋊Dic₄₃	—	2	1	no	—
92	#10621885	—	2	2	no	—
93	#10621885	—	2	2	no	—
94	#10621885	—	2	2	no	—
95	#10621883	—	2	2	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	—	2	1	no	—
100	C₄⋊Dic₄₃	—	2	1	no	—
101	C₄⋊Dic₄₃	—	2	1	no	—
102	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
103	Dic₄₃⋊C₄	—	2	1	no	—
104	Dic₄₃⋊C₄	—	2	1	no	—
105	Dic₄₃⋊C₄	—	2	1	no	—
106	Dic₄₃⋊C₄	—	2	1	no	—
107	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
108	Dic₄₃⋊C₄	D₄	1	1	yes	characteristic
109	C₄×Dic₄₃	—	2	1	no	—
110	C₄×Dic₄₃	—	2	1	no	—
111	C₄×Dic₄₃	—	2	1	no	—
112	C₄×Dic₄₃	—	2	1	no	—
113	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
114	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
115	C₂×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
116	Dic₁₇₂	—	2	2	no	—
117	Dic₁₇₂	—	2	2	no	—
118	#10621872	—	2	2	no	—
119	C₈×D₄₃	—	2	2	no	—
120	C₂³×C₄₃	—	2	1	no	—
121	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
122	Q₈×C₄₃	—	2	1	no	—
123	Q₈×C₄₃	—	2	1	no	—
124	Q₈×C₄₃	—	2	1	no	—
125	Q₈×C₄₃	C₂×D₄	1	2	yes	—
126	D₄×C₄₃	—	2	1	no	—
127	D₄×C₄₃	—	2	1	no	—
128	C₂×C₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
134	C₂²⋊D₄₃	—	4	2	no	—
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	2	1	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
141	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
142	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
143	C₄×D₄₃	—	2	4	no	—
144	C₄×D₄₃	—	4	2	no	—
145	C₄×D₄₃	C₂×D₄	1	2	yes	—
146	C₄×D₄₃	#41	1	2	yes	—
147	Dic₈₆	—	2	1	no	—
148	Dic₈₆	—	2	4	no	—
149	Dic₈₆	—	2	1	no	—
150	Dic₈₆	—	2	1	no	—
151	Dic₈₆	—	2	1	no	—
152	Dic₈₆	—	2	1	no	—
153	Dic₈₆	—	4	2	no	—
154	Dic₈₆	C₂×D₄	1	2	yes	—
155	C₈×C₄₃	—	2	1	no	—
156	C₈×C₄₃	—	2	1	no	—
157	C₄₃⋊C₈	—	2	1	no	—
158	C₄₃⋊C₈	—	2	1	no	—
159	C₂²×C₄₃	—	4	1	no	—
160	C₂²×C₄₃	—	4	1	no	—
161	C₂²×C₄₃	—	4	1	no	—
162	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
163	C₂×D₄₃	—	2	1	no	—
164	C₂×D₄₃	—	2	1	no	—
165	C₂×D₄₃	#121	1	2	yes	—
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	2	1	no	—
168	C₄×C₄₃	—	2	1	no	—
169	C₄×C₄₃	—	4	1	no	—
170	C₄×C₄₃	—	4	1	no	—
171	C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
172	C₄×C₄₃	C₂×#41	1	1	yes	characteristic
173	Dic₄₃	—	2	2	no	—
174	Dic₄₃	—	2	1	no	—
175	Dic₄₃	—	2	1	no	—
176	Dic₄₃	—	4	1	no	—
177	Dic₄₃	—	4	1	no	—
178	Dic₄₃	—	4	1	no	—
179	Dic₄₃	#121	1	2	yes	—
180	#3044	—	43	1	no	maximal, Sylow
181	C₂×C₄₃	—	8	1	no	—
182	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
183	C₂×C₄₃	#578	1	1	yes	characteristic
184	C₂×C₄₃	#578	1	1	yes	characteristic
185	D₄₃	—	2	2	no	—
186	C₂×#137	—	43	1	no	—
187	C₂²×Q₁₆	—	43	1	no	—
188	#536	—	43	1	no	—
189	#519	—	43	1	no	—
190	#464	—	43	1	no	—
191	#464	—	43	1	no	—
192	#464	—	43	1	no	—
193	#464	D₄₃	1	1	yes	characteristic, complemented
194	#462	—	43	1	no	—
195	#462	—	43	1	no	—
196	#462	—	43	1	no	—
197	#462	—	43	1	no	—
198	#424	—	43	1	no	—
199	#419	—	43	1	no	—
200	#416	—	43	1	no	—
201	C₄₃	#3044	1	1	yes	characteristic, complemented, Sylow
202	C₂×#41	—	43	1	no	—
203	C₂²×Q₈	—	43	1	no	—
204	C₂²×Q₈	—	43	1	no	—
205	#137	—	86	4	no	—
206	C₂×Q₁₆	—	43	1	no	—
207	C₂×Q₁₆	—	43	1	no	—
208	C₂×Q₁₆	—	43	1	no	—
209	C₂×Q₁₆	—	43	1	no	—
210	C₂×Q₁₆	—	43	1	no	—
211	C₂×Q₁₆	—	86	4	no	—
212	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
213	C₂×SD₁₆	—	43	1	no	—
214	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
215	C₂×M₁₆	—	43	1	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₄⋊Q₈	—	43	1	no	—
218	C₄⋊Q₈	—	43	1	no	—
219	#124	—	43	1	no	—
220	#124	C₂×D₄₃	1	1	yes	characteristic
221	#123	—	86	1	no	—
222	#123	—	86	1	no	—
223	C₂²⋊Q₈	—	86	1	no	—
224	C₂²⋊Q₈	—	86	1	no	—
225	C₄×Q₈	—	43	1	no	—
226	C₄×Q₈	—	43	1	no	—
227	C₄×Q₈	—	43	1	no	—
228	C₄×Q₈	—	86	1	no	—
229	C₄×Q₈	—	86	1	no	—
230	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
231	#117	—	43	1	no	—
232	#117	—	43	1	no	—
233	#117	—	86	1	no	—
234	#117	—	86	1	no	—
235	C₄⋊C₈	—	43	1	no	—
236	C₄⋊C₈	—	43	1	no	—
237	C₄⋊C₈	—	43	1	no	—
238	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
239	#103	—	43	1	no	—
240	#103	—	43	1	no	—
241	#103	—	43	1	no	—
242	#103	—	43	1	no	—
243	#103	—	43	1	no	—
244	#103	C₂×D₄₃	1	1	yes	characteristic
245	#102	—	43	1	no	—
246	#102	C₂×D₄₃	1	1	yes	characteristic
247	#41	—	86	2	no	—
248	#41	—	86	2	no	—
249	C₂×Q₈	—	43	4	no	—
250	C₂×Q₈	—	43	1	no	—
251	C₂×Q₈	—	43	1	no	—
252	C₂×Q₈	—	43	1	no	—
253	C₂×Q₈	—	86	1	no	—
254	C₂×Q₈	—	86	1	no	—
255	C₂×Q₈	—	86	2	no	—
256	C₂×Q₈	—	86	2	no	—
257	C₂×Q₈	—	86	2	no	—
258	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
259	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
260	C₂×D₄	—	86	1	no	—
261	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
262	C₂²×C₄	—	43	1	no	—
263	C₂²×C₄	—	86	1	no	—
264	C₂²×C₄	—	86	1	no	—
265	C₂²×C₄	—	86	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	86	2	no	—
268	Q₁₆	—	86	2	no	—
269	Q₁₆	—	86	2	no	—
270	Q₁₆	—	86	2	no	—
271	Q₁₆	—	86	2	no	—
272	SD₁₆	—	2	2	no	—
273	SD₁₆	—	86	2	no	—
274	M₁₆	—	86	2	no	—
275	C₂×C₈	—	43	1	no	—
276	C₂×C₈	—	43	1	no	—
277	C₂×C₈	—	86	2	no	—
278	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
279	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	43	1	no	—
282	C₄⋊C₄	—	43	1	no	—
283	C₄⋊C₄	—	43	1	no	—
284	C₄⋊C₄	—	86	1	no	—
285	C₄⋊C₄	—	86	1	no	—
286	C₄⋊C₄	—	86	1	no	—
287	C₄⋊C₄	—	86	1	no	—
288	C₄⋊C₄	—	86	1	no	—
289	C₄⋊C₄	—	86	1	no	—
290	C₄⋊C₄	—	86	1	no	—
291	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
292	C₂²⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₂²⋊C₄	—	43	1	no	—
295	C₂²⋊C₄	—	43	1	no	—
296	C₂²⋊C₄	—	86	1	no	—
297	C₂²⋊C₄	—	86	1	no	—
298	C₂²⋊C₄	—	86	1	no	—
299	C₄²	—	2	1	no	—
300	C₄²	—	43	1	no	—
301	C₄²	—	86	1	no	—
302	C₄²	—	86	1	no	—
303	C₄²	—	86	1	no	—
304	C₄²	—	86	1	no	—
305	C₄²	C₂²×D₄₃	1	1	yes	characteristic
306	C₂³	—	2	1	no	—
307	C₂³	—	43	1	no	—
308	Q₈	—	2	1	no	—
309	Q₈	—	2	1	no	—
310	Q₈	—	2	1	no	—
311	Q₈	—	43	2	no	—
312	Q₈	—	86	4	no	—
313	Q₈	—	86	1	no	—
314	Q₈	—	86	1	no	—
315	Q₈	—	86	1	no	—
316	Q₈	—	86	1	no	—
317	Q₈	—	86	1	no	—
318	Q₈	—	172	2	no	—
319	Q₈	D₄×D₄₃	1	2	yes	—
320	D₄	—	2	1	no	—
321	D₄	—	2	1	no	—
322	D₄	—	172	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	43	1	no	—
327	C₂×C₄	—	43	2	no	—
328	C₂×C₄	—	43	2	no	—
329	C₂×C₄	—	43	1	no	—
330	C₂×C₄	—	43	1	no	—
331	C₂×C₄	—	86	4	no	—
332	C₂×C₄	—	86	1	no	—
333	C₂×C₄	—	86	1	no	—
334	C₂×C₄	—	86	1	no	—
335	C₂×C₄	—	86	1	no	—
336	C₂×C₄	—	172	2	no	—
337	C₂×C₄	—	172	2	no	—
338	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
339	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
340	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
341	C₈	—	2	1	no	—
342	C₈	—	2	1	no	—
343	C₈	—	86	1	no	—
344	C₈	—	86	1	no	—
345	C₂²	—	4	1	no	—
346	C₂²	—	4	1	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	43	2	no	—
349	C₂²	—	86	1	no	—
350	C₂²	—	86	1	no	—
351	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
352	C₄	—	2	2	no	—
353	C₄	—	2	1	no	—
354	C₄	—	2	1	no	—
355	C₄	—	4	1	no	—
356	C₄	—	4	1	no	—
357	C₄	—	43	2	no	—
358	C₄	—	86	2	no	—
359	C₄	—	86	1	no	—
360	C₄	—	86	1	no	—
361	C₄	—	172	1	no	—
362	C₄	—	172	1	no	—
363	C₄	—	172	1	no	—
364	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
365	C₄	#41×D₄₃	1	1	yes	characteristic
366	C₂	—	8	1	no	—
367	C₂	—	86	2	no	—
368	C₂	#121×D₄₃	1	1	yes	characteristic
369	C₂	#14583713	1	1	yes	characteristic
370	C₂	#14583715	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₃
Fitting	C₄₃×#464	C₂
socle	C₂²×C₄₃	C₂²×D₄
hypercenter	#464	D₄₃
hypocenter	C₄₃	#3044
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. C₂×Q₁₆×D₄₃	C₂	semidirect
4. #14583375	C₂	semidirect
5. #14583120	C₂	semidirect
6. #14583119	C₂	semidirect
7. #14583118	C₂	semidirect
8. #14583110	C₂	semidirect
9. #14583109	C₂	semidirect
10. #14583108	C₂	semidirect
11. #14583107	C₂	semidirect
12. #14582924	C₂	semidirect
13. #14582906	C₂	semidirect
15. C₄₃×#464	C₂	semidirect
193. #464	D₄₃	semidirect
201. C₄₃	#3044	semidirect