Subgroups — #19149087

Order 5504.

890 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄₃⋊#564	C₂	1	4	yes	maximal, complemented
2	#14583628	C₂	1	1	yes	maximal, characteristic, complemented
3	#14583627	C₂	1	1	yes	maximal, characteristic, complemented
4	#14583627	C₂	1	2	yes	maximal, complemented
5	#14583626	C₂	1	2	yes	maximal, complemented
6	#14583625	C₂	1	2	yes	maximal, complemented
7	#14583624	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583624	C₂	1	2	yes	maximal, complemented
9	#14583512	C₂	1	2	yes	maximal
10	#14583510	C₂	1	2	yes	maximal
11	#14583508	C₂	1	4	yes	maximal
12	#14583507	C₂	1	2	yes	maximal
13	#14583507	C₂	1	2	yes	maximal
14	#14583454	C₂	1	1	yes	maximal, characteristic
15	#14583453	C₂	1	1	yes	maximal, characteristic
16	#14583453	C₂	1	1	yes	maximal, characteristic
17	C₄₃×#557	C₂	1	1	yes	maximal, characteristic, complemented
18	#13063325	C₂²	1	1	yes	characteristic
19	#13063325	C₂²	1	2	yes	—
20	#13063324	C₂²	1	1	yes	characteristic
21	#13063324	C₂²	1	4	yes	—
22	#13063324	C₂²	1	1	yes	characteristic
23	#13063323	C₂²	1	1	yes	characteristic
24	#13063323	C₂²	1	2	yes	—
25	#13063323	C₂²	1	1	yes	characteristic
26	#13063323	C₂²	1	2	yes	—
27	#13063320	C₂²	1	4	yes	—
28	#13063320	C₂²	1	4	yes	—
29	#13063320	C₂²	1	4	yes	—
30	#13063319	C₂²	1	4	yes	—
31	#13063317	C₂²	1	2	yes	—
32	#13063317	C₂²	1	4	yes	—
33	#13063317	C₂²	1	4	yes	—
34	#13063316	C₂²	1	2	yes	—
35	#13063316	C₂²	1	4	yes	—
36	#13063316	C₂²	1	2	yes	—
37	#13063316	C₂²	1	4	yes	—
38	#13063316	C₂²	1	4	yes	—
39	#13063316	C₂²	1	2	yes	—
40	#13063315	C₂²	1	4	yes	—
41	#13063315	C₂²	1	4	yes	—
42	#13063315	C₂²	1	4	yes	—
43	#13063315	C₂²	1	4	yes	—
44	#13063315	C₂²	1	4	yes	—
45	#13063315	C₂²	1	4	yes	—
46	#13063315	C₂²	1	4	yes	—
47	#13063315	C₂²	1	2	yes	—
48	#13063315	C₂²	1	2	yes	—
49	#13063305	C₂²	1	1	yes	characteristic
50	#13063305	C₂²	1	1	yes	characteristic
51	#13063305	C₂²	1	2	yes	—
52	#13063303	C₂²	1	2	yes	—
53	#13063303	C₂²	1	2	yes	—
54	#13063301	C₂²	1	1	yes	characteristic
55	#13063301	C₂²	1	1	yes	characteristic
56	#13063301	C₂²	1	2	yes	—
57	Q₈×Dic₄₃	C₂²	1	2	yes	—
58	Q₈×Dic₄₃	C₂²	1	2	yes	—
59	Q₈×Dic₄₃	C₂²	1	2	yes	—
60	C₄×Dic₈₆	C₂²	1	2	yes	—
61	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
62	C₄×Dic₈₆	C₂²	1	2	yes	—
63	C₄×Dic₈₆	C₂²	1	1	yes	characteristic
64	C₄×Dic₈₆	C₂²	1	4	yes	—
65	C₄×Dic₈₆	C₂²	1	2	yes	—
66	C₄×Dic₈₆	C₂²	1	2	yes	—
67	#13063290	C₂²	1	4	yes	—
68	#13063290	C₂²	1	2	yes	—
69	#13063290	C₂²	1	2	yes	—
70	#13063290	C₂²	1	2	yes	—
71	#13063283	C₂²	1	2	yes	—
72	#13063282	C₂²	1	2	yes	—
73	#13063282	C₂²	1	2	yes	—
74	C₄²×D₄₃	C₂²	1	1	yes	characteristic
75	C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
76	C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
77	#125×C₄₃	C₂²	1	2	yes	—
78	#125×C₄₃	C₂²	1	4	yes	—
79	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
80	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
81	C₄×Q₈×C₄₃	C₂²	1	2	yes	—
82	C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
83	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
84	C₂×Q₈×C₄₃	C₂³	1	2	yes	—
85	C₂×Q₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
87	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
88	C₂×C₄×D₄₃	C₂³	1	1	yes	characteristic
89	C₂×Dic₈₆	C₂³	1	2	yes	—
90	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
91	C₂×Dic₈₆	C₂³	1	2	yes	—
92	C₂×Dic₈₆	C₂³	1	2	yes	—
93	C₂×Dic₈₆	C₂³	1	1	yes	characteristic
94	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₄₃	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
97	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
98	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
99	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
100	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
101	C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
102	C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
103	C₄²×C₄₃	C₂³	1	2	yes	—
104	C₄²×C₄₃	C₂³	1	4	yes	—
105	C₄²×C₄₃	C₂³	1	1	yes	characteristic
106	C₄²×C₄₃	C₂³	1	2	yes	—
107	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
108	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
109	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
110	(C₂×D₄₃)⋊C₄	C₂³	1	2	yes	—
111	(C₂×D₄₃)⋊C₄	C₂³	1	4	yes	—
112	C₄⋊Dic₄₃	C₂³	1	2	yes	—
113	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
114	C₄⋊Dic₄₃	C₂³	1	2	yes	—
115	C₄⋊Dic₄₃	C₂³	1	1	yes	characteristic
116	C₄⋊Dic₄₃	C₂³	1	2	yes	—
117	C₄⋊Dic₄₃	C₂³	1	2	yes	—
118	C₄⋊Dic₄₃	C₂³	1	4	yes	—
119	C₄⋊Dic₄₃	C₂³	1	4	yes	—
120	C₄⋊Dic₄₃	C₂³	1	4	yes	—
121	C₄⋊Dic₄₃	C₂³	1	4	yes	—
122	Dic₄₃⋊C₄	C₂³	1	2	yes	—
123	Dic₄₃⋊C₄	C₂³	1	2	yes	—
124	Dic₄₃⋊C₄	C₂³	1	2	yes	—
125	Dic₄₃⋊C₄	C₂³	1	2	yes	—
126	Dic₄₃⋊C₄	C₂³	1	2	yes	—
127	Dic₄₃⋊C₄	C₂³	1	4	yes	—
128	Dic₄₃⋊C₄	C₂³	1	4	yes	—
129	Dic₄₃⋊C₄	C₂³	1	4	yes	—
130	Dic₄₃⋊C₄	C₂³	1	4	yes	—
131	Dic₄₃⋊C₄	C₂³	1	2	yes	—
132	Dic₄₃⋊C₄	C₂³	1	2	yes	—
133	Dic₄₃⋊C₄	C₂³	1	2	yes	—
134	Dic₄₃⋊C₄	C₂³	1	2	yes	—
135	Dic₄₃⋊C₄	C₂³	1	2	yes	—
136	Dic₄₃⋊C₄	C₂³	1	4	yes	—
137	Dic₄₃⋊C₄	C₂³	1	4	yes	—
138	Dic₄₃⋊C₄	C₂³	1	4	yes	—
139	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
140	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
141	C₄×Dic₄₃	C₂³	1	1	yes	characteristic
142	C₄×Dic₄₃	C₂³	1	2	yes	—
143	C₄×Dic₄₃	C₂³	1	2	yes	—
144	C₄×Dic₄₃	C₂³	1	4	yes	—
145	C₄×Dic₄₃	C₂³	1	4	yes	—
146	C₄×Dic₄₃	C₂³	1	4	yes	—
147	C₄×Dic₄₃	C₂³	1	4	yes	—
148	C₂²×D₄₃	C₂⁴	1	1	yes	characteristic
149	Q₈×C₄₃	—	2	4	no	—
150	Q₈×C₄₃	—	2	2	no	—
151	Q₈×C₄₃	—	2	2	no	—
152	C₂×C₄×C₄₃	C₂⁴	1	4	yes	—
153	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
154	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
155	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
156	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
157	C₂×C₄×C₄₃	C₂⁴	1	2	yes	—
158	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
159	C₂×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
160	C₂×Dic₄₃	C₂⁴	1	2	yes	—
161	C₂×Dic₄₃	C₂⁴	1	2	yes	—
162	C₂×Dic₄₃	C₂⁴	1	2	yes	—
163	C₂×Dic₄₃	C₂⁴	1	2	yes	—
164	C₂×Dic₄₃	C₂⁴	1	4	yes	—
165	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
166	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
167	C₂×Dic₄₃	C₂⁴	1	1	yes	characteristic
168	C₄×D₄₃	—	2	2	no	—
169	C₄×D₄₃	—	2	2	no	—
170	C₄×D₄₃	—	2	2	no	—
171	Dic₈₆	—	2	2	no	—
172	Dic₈₆	—	2	2	no	—
173	Dic₈₆	—	2	4	no	—
174	Dic₈₆	—	2	4	no	—
175	Dic₈₆	—	2	4	no	—
176	C₂²×C₄₃	C₂⁵	1	1	yes	characteristic
177	C₂×D₄₃	—	2	1	no	—
178	C₂×D₄₃	—	2	1	no	—
179	C₂×D₄₃	—	2	1	no	—
180	C₄×C₄₃	—	2	2	no	—
181	C₄×C₄₃	—	2	2	no	—
182	C₄×C₄₃	—	2	1	no	—
183	C₄×C₄₃	—	2	4	no	—
184	C₄×C₄₃	—	2	2	no	—
185	C₄×C₄₃	—	2	2	no	—
186	C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
187	C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
188	Dic₄₃	—	2	2	no	—
189	Dic₄₃	—	2	2	no	—
190	Dic₄₃	—	2	2	no	—
191	Dic₄₃	—	2	2	no	—
192	Dic₄₃	—	2	4	no	—
193	Dic₄₃	—	2	1	no	—
194	Dic₄₃	—	2	1	no	—
195	Dic₄₃	2₋¹⁺⁴	1	2	yes	—
196	#3556	—	43	1	no	maximal, Sylow
197	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
198	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×C₄₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
200	D₄₃	—	4	1	no	—
201	#564	—	43	4	no	—
202	#557	—	43	2	no	—
203	#557	—	43	1	no	—
204	#557	—	43	2	no	—
205	#557	—	43	2	no	—
206	#557	—	43	2	no	—
207	#557	—	43	1	no	—
208	#557	—	43	1	no	—
209	#557	D₄₃	1	1	yes	characteristic, complemented
210	#541	—	43	2	no	—
211	#541	—	43	4	no	—
212	#541	—	43	2	no	—
213	#541	—	43	2	no	—
214	#541	—	43	2	no	—
215	#533	—	43	1	no	—
216	#533	—	43	1	no	—
217	#533	—	43	1	no	—
218	C₄₃	#3556	1	1	yes	characteristic, complemented, Sylow
219	C₄⋊Q₈	—	43	1	no	—
220	C₄⋊Q₈	—	43	1	no	—
221	C₄⋊Q₈	—	43	2	no	—
222	C₄⋊Q₈	—	43	2	no	—
223	C₄⋊Q₈	—	43	1	no	—
224	C₄⋊Q₈	—	43	1	no	—
225	C₄⋊Q₈	—	43	4	no	—
226	C₄⋊Q₈	—	43	1	no	—
227	C₄⋊Q₈	—	43	2	no	—
228	C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
229	C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
230	#126	—	43	4	no	—
231	#126	—	43	4	no	—
232	#126	—	43	4	no	—
233	#126	—	43	4	no	—
234	#125	—	43	2	no	—
235	#125	—	43	2	no	—
236	#125	—	43	2	no	—
237	#125	—	43	4	no	—
238	#125	—	43	2	no	—
239	#125	—	43	2	no	—
240	#125	—	43	4	no	—
241	#125	—	43	4	no	—
242	#125	—	43	4	no	—
243	#125	—	43	4	no	—
244	#125	—	43	4	no	—
245	#125	—	43	4	no	—
246	#125	—	43	4	no	—
247	#125	—	43	4	no	—
248	#125	—	43	4	no	—
249	#125	—	43	2	no	—
250	#125	—	43	4	no	—
251	#125	—	43	4	no	—
252	#125	C₂×D₄₃	1	2	yes	—
253	#125	C₂×D₄₃	1	4	yes	—
254	C₂²⋊Q₈	—	43	2	no	—
255	C₂²⋊Q₈	—	43	1	no	—
256	C₂²⋊Q₈	—	43	1	no	—
257	C₂²⋊Q₈	—	43	2	no	—
258	C₂²⋊Q₈	—	43	2	no	—
259	C₂²⋊Q₈	—	43	2	no	—
260	C₂²⋊Q₈	—	43	1	no	—
261	C₂²⋊Q₈	—	43	1	no	—
262	C₄×Q₈	—	43	2	no	—
263	C₄×Q₈	—	43	2	no	—
264	C₄×Q₈	—	43	2	no	—
265	C₄×Q₈	—	43	2	no	—
266	C₄×Q₈	—	43	4	no	—
267	C₄×Q₈	—	43	2	no	—
268	C₄×Q₈	—	43	2	no	—
269	C₄×Q₈	—	43	2	no	—
270	C₄×Q₈	—	43	2	no	—
271	C₄×Q₈	—	43	1	no	—
272	C₄×Q₈	—	43	1	no	—
273	C₄×Q₈	—	43	4	no	—
274	C₄×Q₈	—	43	2	no	—
275	C₄×Q₈	—	43	2	no	—
276	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
277	C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
278	C₄×Q₈	C₂×D₄₃	1	2	yes	—
279	C₄×Q₈	C₂×D₄₃	1	2	yes	—
280	#117	—	43	2	no	—
281	#117	—	43	2	no	—
282	#117	—	43	2	no	—
283	C₂×C₄²	—	43	1	no	—
284	C₂×Q₈	—	43	2	no	—
285	C₂×Q₈	—	43	2	no	—
286	C₂×Q₈	—	43	1	no	—
287	C₂×Q₈	—	43	2	no	—
288	C₂×Q₈	—	43	1	no	—
289	C₂×Q₈	C₂²×D₄₃	1	2	yes	—
290	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
291	C₂×Q₈	C₂²×D₄₃	1	1	yes	characteristic
292	C₂²×C₄	—	43	1	no	—
293	C₂²×C₄	—	43	1	no	—
294	C₂²×C₄	—	43	1	no	—
295	C₄⋊C₄	—	43	2	no	—
296	C₄⋊C₄	—	43	2	no	—
297	C₄⋊C₄	—	43	2	no	—
298	C₄⋊C₄	—	43	2	no	—
299	C₄⋊C₄	—	43	2	no	—
300	C₄⋊C₄	—	43	2	no	—
301	C₄⋊C₄	—	43	2	no	—
302	C₄⋊C₄	—	43	2	no	—
303	C₄⋊C₄	—	43	1	no	—
304	C₄⋊C₄	—	43	1	no	—
305	C₄⋊C₄	—	43	2	no	—
306	C₄⋊C₄	—	43	2	no	—
307	C₄⋊C₄	—	43	4	no	—
308	C₄⋊C₄	—	43	2	no	—
309	C₄⋊C₄	—	43	2	no	—
310	C₄⋊C₄	—	43	4	no	—
311	C₄⋊C₄	—	43	2	no	—
312	C₄⋊C₄	—	43	2	no	—
313	C₄⋊C₄	—	43	4	no	—
314	C₄⋊C₄	—	43	4	no	—
315	C₄⋊C₄	—	43	4	no	—
316	C₄⋊C₄	—	43	4	no	—
317	C₄⋊C₄	—	43	4	no	—
318	C₄⋊C₄	—	43	4	no	—
319	C₄⋊C₄	—	43	4	no	—
320	C₄⋊C₄	—	43	4	no	—
321	C₄⋊C₄	—	43	4	no	—
322	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
323	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
324	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
325	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
326	C₄⋊C₄	C₂²×D₄₃	1	1	yes	characteristic
327	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
328	C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
329	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
330	C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
331	C₂²⋊C₄	—	43	4	no	—
332	C₂²⋊C₄	—	43	2	no	—
333	C₂²⋊C₄	—	43	2	no	—
334	C₂²⋊C₄	—	43	2	no	—
335	C₂²⋊C₄	—	43	2	no	—
336	C₄²	—	43	1	no	—
337	C₄²	—	43	1	no	—
338	C₄²	—	43	1	no	—
339	C₄²	—	43	4	no	—
340	C₄²	—	43	2	no	—
341	C₄²	—	43	2	no	—
342	C₄²	—	43	4	no	—
343	C₄²	—	43	4	no	—
344	C₄²	—	43	4	no	—
345	C₄²	C₂²×D₄₃	1	1	yes	characteristic
346	C₄²	C₂²×D₄₃	1	2	yes	—
347	C₄²	C₂²×D₄₃	1	2	yes	—
348	C₄²	C₂²×D₄₃	1	4	yes	—
349	C₂³	—	43	1	no	—
350	Q₈	—	2	4	no	—
351	Q₈	—	2	2	no	—
352	Q₈	—	2	2	no	—
353	Q₈	—	86	4	no	—
354	Q₈	—	86	4	no	—
355	Q₈	—	86	2	no	—
356	Q₈	—	86	4	no	—
357	Q₈	—	86	2	no	—
358	C₂×C₄	—	43	1	no	—
359	C₂×C₄	—	43	2	no	—
360	C₂×C₄	—	43	2	no	—
361	C₂×C₄	—	43	2	no	—
362	C₂×C₄	—	43	2	no	—
363	C₂×C₄	—	43	4	no	—
364	C₂×C₄	—	43	1	no	—
365	C₂×C₄	—	43	1	no	—
366	C₂×C₄	—	86	2	no	—
367	C₂×C₄	—	86	2	no	—
368	C₂×C₄	—	86	2	no	—
369	C₂×C₄	C₂³×D₄₃	1	4	yes	—
370	C₂×C₄	C₂³×D₄₃	1	2	yes	—
371	C₂×C₄	C₂³×D₄₃	1	2	yes	—
372	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
373	C₂×C₄	C₂³×D₄₃	1	2	yes	—
374	C₂×C₄	C₂³×D₄₃	1	2	yes	—
375	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
376	C₂×C₄	C₂³×D₄₃	1	1	yes	characteristic
377	C₂²	—	86	1	no	—
378	C₂²	—	86	1	no	—
379	C₂²	—	86	1	no	—
380	C₂²	C₂⁴×D₄₃	1	1	yes	characteristic
381	C₄	—	2	1	no	—
382	C₄	—	2	2	no	—
383	C₄	—	2	2	no	—
384	C₄	—	2	4	no	—
385	C₄	—	2	2	no	—
386	C₄	—	2	2	no	—
387	C₄	—	43	2	no	—
388	C₄	—	86	2	no	—
389	C₄	—	86	2	no	—
390	C₄	—	86	1	no	—
391	C₄	—	86	2	no	—
392	C₄	—	86	1	no	—
393	C₄	—	86	2	no	—
394	C₄	—	86	4	no	—
395	C₄	#13063372	1	2	yes	—
396	C₄	#13063373	1	2	yes	—
397	C₂	—	172	1	no	—
398	C₂	2₋¹⁺⁴×D₄₃	1	1	yes	characteristic
399	C₂	C₂×#13063372	1	1	yes	characteristic
400	C₂	C₂×#13063373	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₄₃
commutator, socle	C₂²×C₄₃	C₂⁵
Fitting	C₄₃×#557	C₂
hypercenter	#557	D₄₃
hypocenter	C₄₃	#3556
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ #557

Complements

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

N	H ≅ G/N	type
1. C₄₃⋊#564	C₂	semidirect
2. #14583628	C₂	semidirect
3. #14583627	C₂	semidirect
4. #14583627	C₂	semidirect
5. #14583626	C₂	semidirect
6. #14583625	C₂	semidirect
7. #14583624	C₂	semidirect
8. #14583624	C₂	semidirect
17. C₄₃×#557	C₂	semidirect
209. #557	D₄₃	semidirect
218. C₄₃	#3556	semidirect