Subgroups — #19144065

Order 5504.

484 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13063357	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#13063354	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583448	C₂	1	1	yes	maximal, characteristic
4	C₄×Q₈×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#14583122	C₂	1	1	yes	maximal, characteristic, complemented
6	#14583121	C₂	1	1	yes	maximal, characteristic, complemented
7	#14583118	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583108	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583107	C₂	1	1	yes	maximal, characteristic, complemented
10	#14583098	C₂	1	1	yes	maximal, characteristic, complemented
11	#14583097	C₂	1	1	yes	maximal, characteristic, complemented
12	#14582921	C₂	1	1	yes	maximal, characteristic, complemented
13	#14582899	C₂	1	1	yes	maximal, characteristic, complemented
14	#14582897	C₂	1	1	yes	maximal, characteristic, complemented
15	C₄₃×#464	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	#13063357	—	2	4	no	—
20	#13063354	—	2	4	no	—
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×#10621873	C₂²	1	1	yes	characteristic
25	C₂×#10621883	C₂²	1	1	yes	characteristic
26	C₂×#10621884	C₂²	1	1	yes	characteristic
27	C₂×#10621872	C₂²	1	1	yes	characteristic
28	C₂×#10621872	C₂²	1	1	yes	characteristic
29	#13063323	C₂²	1	1	yes	characteristic
30	#13063321	C₂²	1	1	yes	characteristic
31	#13063313	C₂²	1	1	yes	characteristic
32	#13063298	—	2	1	no	—
33	#13063298	—	2	1	no	—
34	Q₈×Dic₄₃	C₂²	1	1	yes	characteristic
35	C₄×Dic₈₆	—	2	1	no	—
36	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
37	#13063290	—	2	1	no	—
38	#13063290	C₂²	1	1	yes	characteristic
39	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
40	(C₂²⋊D₄₃)⋊C₄	—	2	1	no	—
41	C₄⋊C₄×D₄₃	—	2	1	no	—
42	C₄⋊C₄×D₄₃	C₂²	1	1	yes	characteristic
43	C₄²×D₄₃	—	2	1	no	—
44	C₄²×D₄₃	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	#13063249	C₂²	1	1	yes	characteristic
50	#13063248	C₂²	1	1	yes	characteristic
51	#13063247	C₂²	1	1	yes	characteristic
52	#13063246	C₂²	1	1	yes	characteristic
53	#13063245	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	C₂×C₂²⋊D₄₃	—	2	1	no	—
66	C₂×C₂²⋊D₄₃	—	2	1	no	—
67	C₂²×Dic₄₃	—	2	1	no	—
68	#10621902	—	2	2	no	—
69	#10621902	—	2	2	no	—
70	Q₈×D₄₃	—	2	2	no	—
71	Q₈×D₄₃	D₄	1	4	yes	complemented
72	#10621900	—	2	2	no	—
73	#10621900	—	2	2	no	—
74	C₂×D₁₇₂	—	2	1	no	—
75	C₂×D₁₇₂	C₂³	1	1	yes	characteristic
76	C₂×C₄×D₄₃	—	2	1	no	—
77	C₂×C₄×D₄₃	—	2	1	no	—
78	C₂×C₄×D₄₃	—	2	1	no	—
79	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
80	C₂×C₄×D₄₃	D₄	1	1	yes	characteristic
81	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
82	C₂×Dic₈₆	—	2	1	no	—
83	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
84	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
85	Q₁₆×C₄₃	—	2	2	no	—
86	SD₁₆×C₄₃	—	2	2	no	—
87	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
88	C₂×C₈×C₄₃	C₂³	1	1	yes	characteristic
89	C₄⋊C₄×C₄₃	—	2	1	no	—
90	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
91	C₂²⋊C₄×C₄₃	—	2	1	no	—
92	C₂²⋊C₄×C₄₃	—	2	1	no	—
93	C₄²×C₄₃	—	2	1	no	—
94	C₄²×C₄₃	C₂³	1	1	yes	characteristic
95	#10621885	—	2	2	no	—
96	#10621885	—	2	2	no	—
97	#10621884	—	2	2	no	—
98	#10621883	—	2	2	no	—
99	(C₂×D₄₃)⋊C₄	—	2	1	no	—
100	(C₂×D₄₃)⋊C₄	—	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	—	2	1	no	—
108	C₄×Dic₄₃	—	2	1	no	—
109	C₄×Dic₄₃	—	2	1	no	—
110	C₄×Dic₄₃	—	2	1	no	—
111	C₄×Dic₄₃	D₄	1	1	yes	characteristic
112	C₄×Dic₄₃	D₄	1	1	yes	characteristic
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	#10621873	—	2	2	no	—
118	#10621872	—	2	2	no	—
119	#10621872	—	2	2	no	—
120	C₂³×C₄₃	—	2	1	no	—
121	C₂²×D₄₃	—	2	1	no	—
122	C₂²×D₄₃	C₂×D₄	1	1	yes	characteristic
123	Q₈×C₄₃	—	2	1	no	—
124	Q₈×C₄₃	—	2	1	no	—
125	Q₈×C₄₃	—	2	1	no	—
126	Q₈×C₄₃	C₂×D₄	1	2	yes	—
127	D₄×C₄₃	—	2	1	no	—
128	D₄×C₄₃	—	2	1	no	—
129	C₂×C₄×C₄₃	—	2	1	no	—
130	C₂×C₄×C₄₃	—	2	1	no	—
131	C₂×C₄×C₄₃	—	2	1	no	—
132	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
135	C₂²⋊D₄₃	—	4	2	no	—
136	C₂²⋊D₄₃	—	4	2	no	—
137	C₂²⋊D₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	2	1	no	—
139	C₂×Dic₄₃	—	2	1	no	—
140	C₂×Dic₄₃	—	2	1	no	—
141	C₂×Dic₄₃	—	4	2	no	—
142	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
143	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
144	C₂×Dic₄₃	C₂×D₄	1	1	yes	characteristic
145	D₁₇₂	—	2	1	no	—
146	D₁₇₂	—	2	1	no	—
147	D₁₇₂	—	4	2	no	—
148	C₄×D₄₃	—	2	4	no	—
149	C₄×D₄₃	—	2	2	no	—
150	C₄×D₄₃	—	4	1	no	—
151	C₄×D₄₃	—	4	2	no	—
152	C₄×D₄₃	—	4	2	no	—
153	C₄×D₄₃	C₂×D₄	1	2	yes	—
154	C₄×D₄₃	#41	1	2	yes	—
155	Dic₈₆	—	2	1	no	—
156	Dic₈₆	—	2	4	no	—
157	Dic₈₆	—	2	1	no	—
158	Dic₈₆	—	2	1	no	—
159	Dic₈₆	C₂×D₄	1	2	yes	—
160	C₈×C₄₃	—	2	1	no	—
161	C₈×C₄₃	—	2	1	no	—
162	C₄₃⋊C₈	—	2	1	no	—
163	C₄₃⋊C₈	—	2	1	no	—
164	C₂²×C₄₃	—	4	1	no	—
165	C₂²×C₄₃	—	4	1	no	—
166	C₂²×C₄₃	—	4	1	no	—
167	C₂²×C₄₃	C₂²×D₄	1	1	yes	characteristic
168	C₂×D₄₃	—	2	1	no	—
169	C₂×D₄₃	—	2	1	no	—
170	C₂×D₄₃	—	4	1	no	—
171	C₂×D₄₃	—	4	1	no	—
172	C₂×D₄₃	—	4	1	no	—
173	C₂×D₄₃	#121	1	2	yes	—
174	C₄×C₄₃	—	2	2	no	—
175	C₄×C₄₃	—	2	1	no	—
176	C₄×C₄₃	—	2	1	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	C₂×#41	1	1	yes	characteristic
181	Dic₄₃	—	2	2	no	—
182	Dic₄₃	—	2	1	no	—
183	Dic₄₃	—	4	1	no	—
184	Dic₄₃	—	4	1	no	—
185	Dic₄₃	#121	1	2	yes	—
186	Dic₄₃	#137	1	2	yes	complemented
187	#3037	—	43	1	no	maximal, Sylow
188	C₂×C₄₃	—	8	1	no	—
189	C₂×C₄₃	C₂×#121	1	1	yes	characteristic
190	C₂×C₄₃	C₂×#137	1	1	yes	characteristic
191	C₂×C₄₃	#575	1	1	yes	characteristic
192	D₄₃	—	2	2	no	—
193	D₄₃	—	8	1	no	—
194	C₂×#137	—	43	1	no	—
195	C₂×#137	—	43	1	no	—
196	#532	—	43	1	no	—
197	C₂×C₄×Q₈	—	43	1	no	—
198	#464	—	43	1	no	—
199	#464	—	43	1	no	—
200	#464	—	43	1	no	—
201	#464	D₄₃	1	1	yes	characteristic, complemented
202	#462	—	43	1	no	—
203	#462	—	43	1	no	—
204	#460	—	43	1	no	—
205	#460	—	43	1	no	—
206	#423	—	43	1	no	—
207	#417	—	43	1	no	—
208	#417	—	43	1	no	—
209	C₄₃	#3037	1	1	yes	characteristic, complemented, Sylow
210	C₂×#41	—	43	1	no	—
211	C₂×#41	—	43	1	no	—
212	C₂²×Q₈	—	43	1	no	—
213	#137	—	86	4	no	—
214	#137	—	86	4	no	—
215	C₂×Q₁₆	—	43	1	no	—
216	C₂×Q₁₆	—	43	1	no	—
217	C₂×Q₁₆	—	43	1	no	—
218	C₂×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
219	C₂×SD₁₆	—	43	1	no	—
220	C₂×SD₁₆	—	43	1	no	—
221	C₂×SD₁₆	—	43	1	no	—
222	C₂×SD₁₆	C₂×D₄₃	1	1	yes	characteristic
223	C₂×M₁₆	—	43	1	no	—
224	C₂×M₁₆	—	43	1	no	—
225	C₄⋊Q₈	—	43	1	no	—
226	#127	—	43	1	no	—
227	#124	—	43	1	no	—
228	#124	C₂×D₄₃	1	1	yes	characteristic
229	#121	—	86	1	no	—
230	#121	—	86	1	no	—
231	C₄×Q₈	—	43	1	no	—
232	C₄×Q₈	—	43	1	no	—
233	C₄×Q₈	—	43	1	no	—
234	C₄×Q₈	—	86	1	no	—
235	C₄×Q₈	—	86	1	no	—
236	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
237	C₄×D₄	—	86	1	no	—
238	C₄×D₄	—	86	1	no	—
239	C₂×C₄⋊C₄	—	43	1	no	—
240	C₂×C₄⋊C₄	—	86	1	no	—
241	C₂×C₄²	—	43	1	no	—
242	C₂×C₄²	—	86	1	no	—
243	C₄⋊C₈	—	43	1	no	—
244	C₄⋊C₈	—	43	1	no	—
245	C₄⋊C₈	—	43	1	no	—
246	C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
247	#103	—	43	1	no	—
248	#103	—	43	1	no	—
249	#103	—	43	1	no	—
250	#103	C₂×D₄₃	1	1	yes	characteristic
251	#102	—	43	1	no	—
252	#102	—	43	1	no	—
253	#102	—	43	1	no	—
254	#102	C₂×D₄₃	1	1	yes	characteristic
255	#41	—	86	2	no	—
256	#41	—	86	2	no	—
257	#41	—	86	2	no	—
258	#41	—	86	2	no	—
259	C₂×Q₈	—	43	4	no	—
260	C₂×Q₈	—	43	1	no	—
261	C₂×Q₈	—	43	1	no	—
262	C₂×Q₈	—	86	1	no	—
263	C₂×Q₈	—	86	2	no	—
264	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
265	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
266	C₂×D₄	—	43	1	no	—
267	C₂×D₄	—	86	1	no	—
268	C₂×D₄	—	86	1	no	—
269	C₂×D₄	—	86	1	no	—
270	C₂×D₄	—	86	1	no	—
271	C₂×D₄	C₂²×D₄₃	1	1	yes	characteristic
272	C₂²×C₄	—	43	1	no	—
273	C₂²×C₄	—	43	1	no	—
274	C₂²×C₄	—	43	1	no	—
275	C₂²×C₄	—	86	1	no	—
276	C₂²×C₄	—	86	1	no	—
277	C₂²×C₄	—	86	1	no	—
278	C₂²×C₄	—	86	1	no	—
279	Q₁₆	—	2	2	no	—
280	Q₁₆	—	86	2	no	—
281	Q₁₆	—	86	2	no	—
282	Q₁₆	—	86	2	no	—
283	SD₁₆	—	2	2	no	—
284	SD₁₆	—	86	2	no	—
285	SD₁₆	—	86	2	no	—
286	SD₁₆	—	86	2	no	—
287	M₁₆	—	86	2	no	—
288	M₁₆	—	86	2	no	—
289	C₂×C₈	—	43	1	no	—
290	C₂×C₈	—	43	1	no	—
291	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
292	C₂×C₈	C₂²×D₄₃	1	1	yes	characteristic
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	43	1	no	—
295	C₄⋊C₄	—	86	1	no	—
296	C₄⋊C₄	—	86	1	no	—
297	C₄⋊C₄	—	86	1	no	—
298	C₄⋊C₄	—	86	1	no	—
299	C₄⋊C₄	—	86	1	no	—
300	C₄⋊C₄	—	86	1	no	—
301	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
302	C₂²⋊C₄	—	2	1	no	—
303	C₂²⋊C₄	—	2	1	no	—
304	C₂²⋊C₄	—	86	1	no	—
305	C₂²⋊C₄	—	86	1	no	—
306	C₄²	—	2	1	no	—
307	C₄²	—	43	1	no	—
308	C₄²	—	43	1	no	—
309	C₄²	—	43	1	no	—
310	C₄²	—	86	1	no	—
311	C₄²	—	86	1	no	—
312	C₄²	—	86	1	no	—
313	C₄²	C₂²×D₄₃	1	1	yes	characteristic
314	C₂³	—	2	1	no	—
315	C₂³	—	43	1	no	—
316	C₂³	—	86	1	no	—
317	Q₈	—	2	1	no	—
318	Q₈	—	2	1	no	—
319	Q₈	—	2	1	no	—
320	Q₈	—	43	2	no	—
321	Q₈	—	86	4	no	—
322	Q₈	—	86	1	no	—
323	Q₈	—	86	1	no	—
324	Q₈	—	86	1	no	—
325	Q₈	D₄×D₄₃	1	2	yes	—
326	D₄	—	2	1	no	—
327	D₄	—	2	1	no	—
328	D₄	—	86	1	no	—
329	D₄	—	86	1	no	—
330	D₄	—	172	2	no	—
331	D₄	—	172	2	no	—
332	D₄	—	172	2	no	—
333	D₄	—	172	2	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	43	1	no	—
338	C₂×C₄	—	43	1	no	—
339	C₂×C₄	—	43	2	no	—
340	C₂×C₄	—	43	2	no	—
341	C₂×C₄	—	43	1	no	—
342	C₂×C₄	—	86	2	no	—
343	C₂×C₄	—	86	4	no	—
344	C₂×C₄	—	86	1	no	—
345	C₂×C₄	—	86	1	no	—
346	C₂×C₄	—	86	1	no	—
347	C₂×C₄	—	172	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	1	yes	characteristic
352	C₂×C₄	D₄×D₄₃	1	1	yes	characteristic
353	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
354	C₈	—	2	1	no	—
355	C₈	—	2	1	no	—
356	C₈	—	86	1	no	—
357	C₈	—	86	1	no	—
358	C₂²	—	4	1	no	—
359	C₂²	—	4	1	no	—
360	C₂²	—	4	1	no	—
361	C₂²	—	43	2	no	—
362	C₂²	—	86	1	no	—
363	C₂²	—	86	1	no	—
364	C₂²	—	172	1	no	—
365	C₂²	—	172	1	no	—
366	C₂²	—	172	1	no	—
367	C₂²	C₂×D₄×D₄₃	1	1	yes	characteristic
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	2	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	1	no	—
373	C₄	—	43	2	no	—
374	C₄	—	43	2	no	—
375	C₄	—	86	2	no	—
376	C₄	—	86	1	no	—
377	C₄	—	172	1	no	—
378	C₄	—	172	1	no	—
379	C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
380	C₄	#41×D₄₃	1	1	yes	characteristic
381	C₂	—	8	1	no	—
382	C₂	—	86	2	no	—
383	C₂	—	344	1	no	—
384	C₂	#121×D₄₃	1	1	yes	characteristic
385	C₂	#137×D₄₃	1	1	yes	characteristic
386	C₂	#14583696	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	#3037
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
1. C₂×#13063357	C₂	semidirect
2. C₂×#13063354	C₂	semidirect
4. C₄×Q₈×D₄₃	C₂	semidirect
5. #14583122	C₂	semidirect
6. #14583121	C₂	semidirect
7. #14583118	C₂	semidirect
8. #14583108	C₂	semidirect
9. #14583107	C₂	semidirect
10. #14583098	C₂	semidirect
11. #14583097	C₂	semidirect
12. #14582921	C₂	semidirect
13. #14582899	C₂	semidirect
14. #14582897	C₂	semidirect
15. C₄₃×#464	C₂	semidirect
71. Q₈×D₄₃	D₄	semidirect
186. Dic₄₃	#137	semidirect
201. #464	D₄₃	semidirect
209. C₄₃	#3037	semidirect