Subgroups — #25116404

Order 11008.

770 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	#19137551	C₂	1	2	yes	maximal
3	#19135467	C₂	1	1	yes	maximal, characteristic
4	#19135232	C₂	1	2	yes	maximal
5	#19135226	C₂	1	2	yes	maximal
6	#19134750	C₂	1	1	yes	maximal, characteristic
7	#19134598	C₂	1	2	yes	maximal
8	#19134495	C₂	1	1	yes	maximal, characteristic
9	#19134492	C₂	1	1	yes	maximal, characteristic
10	#19134463	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#1971	C₂	1	1	yes	maximal, characteristic
12	C₂×Q₈×Dic₄₃	C₂²	1	2	yes	—
13	Q₁₆×Dic₄₃	C₄	1	8	yes	complemented
14	C₂×#13063259	C₂²	1	1	yes	characteristic
15	C₂×#13063257	C₂²	1	1	yes	characteristic
16	C₂×#13063256	C₂²	1	2	yes	—
17	C₂×Dic₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×#13063249	C₂²	1	2	yes	—
19	C₂×#13063249	C₂²	1	2	yes	—
20	C₂×#13063235	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
22	#14582806	C₂²	1	2	yes	—
23	#14582784	—	2	2	no	—
24	#14582782	—	2	2	no	—
25	#14582768	C₂²	1	2	yes	—
26	#14582763	—	2	2	no	—
27	C₄⋊C₄×Dic₄₃	C₂²	1	2	yes	—
28	C₄×Dic₄₃⋊C₄	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
30	#14582750	—	2	2	no	—
31	#14582662	C₂²	1	2	yes	—
32	#14582661	C₂²	1	2	yes	—
33	#14582654	C₂²	1	1	yes	characteristic
34	#14582653	C₂²	1	1	yes	characteristic
35	C₂²×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
36	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
37	C₂×#103×C₄₃	C₂²	1	2	yes	—
38	C₂×#97×C₄₃	C₂²	1	1	yes	characteristic
39	C₄₃×#391	C₂²	1	2	yes	—
40	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	Q₈×Dic₄₃	—	2	4	no	—
43	Q₈×Dic₄₃	C₂×C₄	1	8	yes	—
44	C₂×C₄⋊Dic₄₃	—	2	2	no	—
45	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
46	C₂×C₄⋊Dic₄₃	C₂³	1	2	yes	—
47	C₂×Dic₄₃⋊C₄	—	2	2	no	—
48	C₂×Dic₄₃⋊C₄	—	2	2	no	—
49	C₂×Dic₄₃⋊C₄	D₄	1	2	yes	—
50	C₂×C₄×Dic₄₃	—	2	2	no	—
51	C₂×C₄×Dic₄₃	—	2	2	no	—
52	C₂×C₄×Dic₄₃	—	2	2	no	—
53	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
55	#13063259	C₂×C₄	1	4	yes	—
56	#13063257	—	2	2	no	—
57	#13063256	—	4	2	no	—
58	Dic₄₃⋊C₈	—	2	2	no	—
59	#13063249	—	4	2	no	—
60	#13063249	C₂×C₄	1	8	yes	—
61	#13063235	—	2	2	no	—
62	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
63	C₄₃⋊#95	—	2	2	no	—
64	C₄₃⋊#95	—	2	2	no	—
65	C₄₃⋊#95	—	2	2	no	—
66	C₄₃⋊#95	—	2	2	no	—
67	C₄₃⋊#95	D₄	1	2	yes	—
68	C₂²×Q₈×C₄₃	C₂³	1	2	yes	—
69	C₂×Q₁₆×C₄₃	—	2	2	no	—
70	C₂×Q₁₆×C₄₃	—	2	2	no	—
71	C₂×Q₁₆×C₄₃	C₂×C₄	1	4	yes	—
72	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
73	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
75	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
76	#106×C₄₃	—	2	2	no	—
77	#103×C₄₃	—	4	2	no	—
78	#97×C₄₃	—	2	2	no	—
79	#95×C₄₃	—	2	2	no	—
80	#95×C₄₃	—	2	2	no	—
81	C₂×Q₈×C₄₃	—	2	2	no	—
82	C₂×Q₈×C₄₃	—	2	2	no	—
83	C₂×Q₈×C₄₃	—	2	2	no	—
84	C₂×Q₈×C₄₃	—	2	2	no	—
85	C₂×Q₈×C₄₃	—	2	2	no	—
86	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₄₃	—	2	2	no	—
88	C₂²×C₄×C₄₃	—	2	2	no	—
89	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
90	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
91	C₂²×Dic₄₃	—	2	2	no	—
92	C₂²×Dic₄₃	—	2	2	no	—
93	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
94	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
96	Q₁₆×C₄₃	—	2	8	no	—
97	C₂×C₈×C₄₃	—	2	1	no	—
98	C₂×C₈×C₄₃	—	2	1	no	—
99	C₂×C₈×C₄₃	—	2	1	no	—
100	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₄₃	#41	1	2	yes	—
102	C₂×C₈×C₄₃	#41	1	2	yes	—
103	C₄⋊C₄×C₄₃	—	2	2	no	—
104	C₄⋊C₄×C₄₃	—	2	2	no	—
105	C₄²×C₄₃	—	2	1	no	—
106	C₄²×C₄₃	—	2	1	no	—
107	C₄⋊Dic₄₃	—	2	2	no	—
108	C₄⋊Dic₄₃	—	2	8	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
112	Dic₄₃⋊C₄	—	2	4	no	—
113	Dic₄₃⋊C₄	—	4	4	no	—
114	Dic₄₃⋊C₄	—	8	2	no	—
115	C₄×Dic₄₃	—	2	1	no	—
116	C₄×Dic₄₃	—	2	1	no	—
117	C₄×Dic₄₃	—	2	8	no	—
118	C₄×Dic₄₃	—	8	2	no	—
119	C₄×Dic₄₃	—	8	2	no	—
120	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
121	C₄×Dic₄₃	#41	1	2	yes	—
122	C₂×C₄₃⋊C₈	—	2	1	no	—
123	C₂×C₄₃⋊C₈	—	2	1	no	—
124	C₂×C₄₃⋊C₈	—	2	1	no	—
125	C₂×C₄₃⋊C₈	—	2	1	no	—
126	C₂×C₄₃⋊C₈	—	2	1	no	—
127	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
128	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
129	Q₈×C₄₃	—	2	4	no	—
130	Q₈×C₄₃	—	2	4	no	—
131	Q₈×C₄₃	—	4	2	no	—
132	Q₈×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	2	4	no	—
134	C₂×C₄×C₄₃	—	2	2	no	—
135	C₂×C₄×C₄₃	—	2	2	no	—
136	C₂×C₄×C₄₃	—	2	2	no	—
137	C₂×C₄×C₄₃	—	4	2	no	—
138	C₂×C₄×C₄₃	—	4	2	no	—
139	C₂×C₄×C₄₃	—	4	2	no	—
140	C₂×C₄×C₄₃	—	4	2	no	—
141	C₂×C₄×C₄₃	—	4	2	no	—
142	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	2	4	no	—
150	C₂×Dic₄₃	—	2	2	no	—
151	C₂×Dic₄₃	—	2	2	no	—
152	C₂×Dic₄₃	—	2	2	no	—
153	C₂×Dic₄₃	—	4	2	no	—
154	C₂×Dic₄₃	—	4	2	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	C₂×Dic₄₃	—	4	2	no	—
158	C₂×Dic₄₃	C₄×D₄	1	2	yes	—
159	C₂×Dic₄₃	C₄×D₄	1	2	yes	—
160	C₂×Dic₄₃	#137	1	4	yes	—
161	C₈×C₄₃	—	4	1	no	—
162	C₈×C₄₃	#117	1	4	yes	—
163	C₄₃⋊C₈	—	2	2	no	—
164	C₄₃⋊C₈	—	4	1	no	—
165	#22814	—	43	1	no	maximal, Sylow
166	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
167	C₂²×C₄₃	#547	1	1	yes	characteristic
168	C₂²×C₄₃	#550	1	1	yes	characteristic
169	C₂²×C₄₃	C₂×#137	1	2	yes	—
170	C₂²×C₄₃	#577	1	2	yes	—
171	C₄×C₄₃	—	4	2	no	—
172	C₄×C₄₃	—	4	4	no	—
173	C₄×C₄₃	—	8	2	no	—
174	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
175	C₄×C₄₃	#520	1	1	yes	characteristic
176	C₄×C₄₃	#555	1	1	yes	characteristic
177	C₄×C₄₃	#556	1	1	yes	characteristic
178	Dic₄₃	—	2	2	no	—
179	Dic₄₃	—	4	2	no	—
180	Dic₄₃	—	4	4	no	—
181	Dic₄₃	—	8	2	no	—
182	Dic₄₃	#441	1	4	yes	—
183	C₂×C₄×Q₁₆	—	43	1	no	—
184	#2355	—	43	2	no	—
185	#1971	—	43	1	no	—
186	#1971	D₄₃	1	1	yes	characteristic
187	#1927	—	43	2	no	—
188	#1927	—	43	2	no	—
189	#1844	—	43	1	no	—
190	#1803	—	43	2	no	—
191	#1776	—	43	1	no	—
192	#1776	—	43	1	no	—
193	#1770	—	43	1	no	—
194	C₂×C₄₃	#2354	1	1	yes	characteristic
195	C₂×C₄₃	C₂×#441	1	1	yes	characteristic
196	C₂×C₄₃	#2952	1	1	yes	characteristic
197	C₂×C₄₃	#3356	1	2	yes	—
198	C₂×C₄₃	#3400	1	2	yes	—
199	C₂²×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
200	C₂×C₄×Q₈	—	43	2	no	—
201	C₄×Q₁₆	—	43	8	no	—
202	C₂×#107	—	43	1	no	—
203	C₂×#106	—	43	1	no	—
204	C₂×#106	—	43	2	no	—
205	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
206	C₂×C₄⋊C₈	—	43	1	no	—
207	C₂×#103	—	43	2	no	—
208	C₂×#103	—	43	2	no	—
209	C₂×#103	C₂×D₄₃	1	2	yes	—
210	C₂×#97	—	43	1	no	—
211	C₂×#97	C₂×D₄₃	1	1	yes	characteristic
212	C₂×C₄×C₈	—	43	1	no	—
213	#391	—	43	2	no	—
214	#391	C₂×D₄₃	1	2	yes	—
215	#387	—	86	2	no	—
216	#387	—	86	2	no	—
217	#384	—	43	2	no	—
218	#382	—	86	2	no	—
219	C₄×C₄⋊C₄	—	43	1	no	—
220	C₄×C₄⋊C₄	—	43	2	no	—
221	C₄×C₄⋊C₄	—	43	2	no	—
222	C₄×C₄⋊C₄	—	86	2	no	—
223	#340	—	43	2	no	—
224	#340	—	43	2	no	—
225	#336	—	43	1	no	—
226	#336	—	43	1	no	—
227	C₄₃	#22814	1	1	yes	characteristic, complemented, Sylow
228	C₂²×Q₈	C₂²×D₄₃	1	2	yes	—
229	C₂×Q₁₆	—	2	2	no	—
230	C₂×Q₁₆	—	2	2	no	—
231	C₂×Q₁₆	C₄×D₄₃	1	4	yes	—
232	C₂²×C₈	—	43	1	no	—
233	C₂²×C₈	—	43	1	no	—
234	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
235	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
236	C₄×Q₈	—	43	8	no	—
237	C₄×Q₈	—	86	4	no	—
238	C₂×C₄⋊C₄	—	43	2	no	—
239	C₂×C₄⋊C₄	—	43	2	no	—
240	C₂×C₄⋊C₄	—	43	2	no	—
241	C₂×C₄⋊C₄	—	86	2	no	—
242	C₂×C₄⋊C₄	—	86	2	no	—
243	C₂×C₄⋊C₄	—	86	2	no	—
244	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
245	C₂×C₄²	—	43	1	no	—
246	C₂×C₄²	—	43	1	no	—
247	C₂×C₄²	—	86	2	no	—
248	C₂×C₄²	—	86	2	no	—
249	C₂×C₄²	—	86	2	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	#107	—	43	4	no	—
252	#106	—	2	2	no	—
253	#106	—	86	2	no	—
254	#106	—	172	2	no	—
255	C₄⋊C₈	—	86	2	no	—
256	#103	—	4	2	no	—
257	#103	—	43	8	no	—
258	#103	—	172	2	no	—
259	#97	—	2	2	no	—
260	#97	—	86	2	no	—
261	C₄×C₈	—	43	4	no	—
262	#95	—	2	2	no	—
263	#95	—	2	2	no	—
264	#95	—	43	2	no	—
265	#95	—	86	2	no	—
266	#95	—	86	2	no	—
267	#95	—	86	2	no	—
268	#95	—	86	2	no	—
269	C₂×Q₈	—	2	2	no	—
270	C₂×Q₈	—	2	2	no	—
271	C₂×Q₈	—	2	2	no	—
272	C₂×Q₈	—	2	2	no	—
273	C₂×Q₈	—	2	2	no	—
274	C₂×Q₈	C₂×C₄×D₄₃	1	4	yes	—
275	C₂²×C₄	—	2	2	no	—
276	C₂²×C₄	—	2	2	no	—
277	C₂²×C₄	—	43	1	no	—
278	C₂²×C₄	—	43	1	no	—
279	C₂²×C₄	—	43	2	no	—
280	C₂²×C₄	—	86	2	no	—
281	C₂²×C₄	—	86	2	no	—
282	C₂²×C₄	D₄×D₄₃	1	2	yes	—
283	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
284	Q₁₆	—	2	8	no	—
285	C₂×C₈	—	2	1	no	—
286	C₂×C₈	—	2	1	no	—
287	C₂×C₈	—	2	1	no	—
288	C₂×C₈	—	43	2	no	—
289	C₂×C₈	—	86	1	no	—
290	C₂×C₈	—	86	1	no	—
291	C₂×C₈	—	86	1	no	—
292	C₂×C₈	—	86	1	no	—
293	C₂×C₈	—	86	1	no	—
294	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
295	C₂×C₈	#10621900	1	2	yes	—
296	C₂×C₈	#10621900	1	2	yes	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	43	4	no	—
300	C₄⋊C₄	—	86	4	no	—
301	C₄⋊C₄	—	86	8	no	—
302	C₄⋊C₄	—	86	2	no	—
303	C₄⋊C₄	—	86	2	no	—
304	C₄⋊C₄	—	86	2	no	—
305	C₄⋊C₄	—	344	2	no	—
306	C₄⋊C₄	—	172	4	no	—
307	C₄²	—	2	1	no	—
308	C₄²	—	2	1	no	—
309	C₄²	—	43	2	no	—
310	C₄²	—	43	2	no	—
311	C₄²	—	86	8	no	—
312	C₄²	—	86	1	no	—
313	C₄²	—	86	1	no	—
314	C₄²	—	344	2	no	—
315	C₄²	—	344	2	no	—
316	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
317	Q₈	—	2	4	no	—
318	Q₈	—	2	4	no	—
319	Q₈	—	4	2	no	—
320	Q₈	—	4	2	no	—
321	C₂×C₄	—	2	4	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	—	43	2	no	—
331	C₂×C₄	—	43	4	no	—
332	C₂×C₄	—	43	2	no	—
333	C₂×C₄	—	86	2	no	—
334	C₂×C₄	—	86	4	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	86	2	no	—
337	C₂×C₄	—	86	2	no	—
338	C₂×C₄	—	172	2	no	—
339	C₂×C₄	—	172	2	no	—
340	C₂×C₄	—	172	2	no	—
341	C₂×C₄	—	172	2	no	—
342	C₂×C₄	—	172	2	no	—
343	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
344	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
345	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
346	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
347	C₂×C₄	#13063370	1	1	yes	characteristic
348	C₂×C₄	#13063372	1	1	yes	characteristic
349	C₈	—	4	1	no	—
350	C₈	—	86	2	no	—
351	C₈	—	172	1	no	—
352	C₈	#13063281	1	4	yes	—
353	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
354	C₂²	#14583549	1	1	yes	characteristic
355	C₂²	#14583569	1	1	yes	characteristic
356	C₂²	#137×D₄₃	1	2	yes	—
357	C₂²	#14583708	1	2	yes	—
358	C₄	—	4	4	no	—
359	C₄	—	4	2	no	—
360	C₄	—	8	2	no	—
361	C₄	—	43	4	no	—
362	C₄	—	86	2	no	—
363	C₄	—	344	2	no	—
364	C₄	—	172	4	no	—
365	C₄	—	172	2	no	—
366	C₄	C₂×#13063281	1	1	yes	characteristic
367	C₄	#14583378	1	1	yes	characteristic
368	C₄	#14583613	1	1	yes	characteristic
369	C₄	#14583618	1	1	yes	characteristic
370	C₂	#19137543	1	1	yes	characteristic
371	C₂	#441×D₄₃	1	1	yes	characteristic
372	C₂	#19143475	1	1	yes	characteristic
373	C₂	#19147205	1	2	yes	—
374	C₂	#19147645	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1971	C₂
socle, prime radical	C₂³×C₄₃	C₂²×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1971	D₄₃
hypocenter	C₄₃	#22814
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1971

Complements

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

N	H ≅ G/N	type
13. Q₁₆×Dic₄₃	C₄	semidirect
227. C₄₃	#22814	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×Q₁₆×Dic₄₃	2
#19137551	2
#19135467	2
#19135232	2
#19135226	2
#19134750	2
#19134598	2
#19134495	2
#19134492	2
#19134463	2
C₄₃×#1971	2
#22814	43