Subgroups — #25251642

Order 11008.

586 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145172	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136774	C₂	1	2	yes	maximal, complemented
4	#19136748	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136747	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136632	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136510	C₂	1	2	yes	maximal, complemented
8	#19136460	C₂	1	2	yes	maximal, complemented
9	#19136456	C₂	1	2	yes	maximal, complemented
10	#19136397	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#2207	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₄₃	C₂²	1	1	yes	characteristic
13	#14583447	—	2	1	no	—
14	C₄×Q₈×D₄₃	—	2	1	no	—
15	C₂×#13063269	C₂²	1	2	yes	—
16	#14583304	C₂²	1	1	yes	characteristic
17	#14583245	—	2	1	no	—
18	#14583132	—	2	1	no	—
19	#14583132	C₂²	1	2	yes	—
20	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
21	Q₁₆×Dic₄₃	C₂²	1	1	yes	characteristic
22	#14582993	C₂²	1	1	yes	characteristic
23	C₄×#10621885	—	2	1	no	—
24	#14582991	—	2	1	no	—
25	D₈×Dic₄₃	—	2	1	no	—
26	D₈×Dic₄₃	C₂²	1	2	yes	—
27	#107×D₄₃	C₂²	1	1	yes	characteristic
28	#103×D₄₃	—	2	1	no	—
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	#14582725	C₂²	1	2	yes	—
31	#14582724	C₂²	1	2	yes	—
32	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
33	#14582704	C₂²	1	2	yes	—
34	#14582703	C₂²	1	2	yes	—
35	#14582700	C₂²	1	2	yes	—
36	(C₂×D₄₃)⋊C₁₆	C₂²	1	2	yes	—
37	#14582674	C₂²	1	2	yes	—
38	C₂×SD₃₂×C₄₃	C₂²	1	2	yes	—
39	C₄₃×#493	C₂²	1	1	yes	characteristic
40	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₄₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
42	C₄₃×#357	C₂²	1	2	yes	—
43	C₂×#10621900	—	4	1	no	—
44	C₂×Q₈×D₄₃	—	2	1	no	—
45	C₂×Dic₁₇₂	C₂³	1	1	yes	characteristic
46	C₂×#10621885	—	2	1	no	—
47	Q₁₆×D₄₃	—	2	2	no	—
48	Q₁₆×D₄₃	—	2	2	no	—
49	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
50	C₂×C₈×D₄₃	C₂³	1	1	yes	characteristic
51	#13063325	—	2	1	no	—
52	#13063312	—	4	1	no	—
53	#13063296	—	2	2	no	—
54	#13063296	—	4	1	no	—
55	Q₈×Dic₄₃	—	2	1	no	—
56	C₄×Dic₈₆	—	2	1	no	—
57	C₄×Dic₈₆	—	4	1	no	—
58	#13063290	—	2	1	no	—
59	#13063290	—	4	1	no	—
60	D₄×Dic₄₃	—	2	2	no	—
61	D₄×Dic₄₃	—	4	1	no	—
62	C₄⋊C₄×D₄₃	—	2	1	no	—
63	C₄⋊C₄×D₄₃	—	4	1	no	—
64	C₄²×D₄₃	—	4	1	no	—
65	C₄²×D₄₃	D₄	1	1	yes	characteristic
66	#13063269	—	2	4	no	—
67	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
68	#13063259	—	2	1	no	—
69	#13063259	C₂³	1	2	yes	—
70	#13063259	C₂³	1	1	yes	characteristic
71	#13063258	—	2	1	no	—
72	#13063250	—	2	1	no	—
73	#13063249	—	2	1	no	—
74	#13063248	—	2	1	no	—
75	#13063246	—	2	2	no	—
76	#13063246	—	2	2	no	—
77	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
78	C₈×Dic₄₃	D₄	1	1	yes	characteristic
79	C₈×Dic₄₃	C₂³	1	1	yes	characteristic
80	C₂×Q₁₆×C₄₃	C₂³	1	1	yes	characteristic
81	C₂×D₈×C₄₃	—	2	1	no	—
82	C₂×D₈×C₄₃	C₂³	1	2	yes	—
83	#127×C₄₃	—	2	1	no	—
84	C₄×Q₈×C₄₃	—	2	1	no	—
85	SD₃₂×C₄₃	—	2	4	no	—
86	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
87	#107×C₄₃	C₂³	1	1	yes	characteristic
88	#103×C₄₃	—	2	1	no	—
89	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
90	C₂×Q₈×C₄₃	—	2	1	no	—
91	C₂×D₄×C₄₃	—	2	2	no	—
92	C₂×D₄×C₄₃	—	4	1	no	—
93	C₂×D₄×C₄₃	—	4	1	no	—
94	C₂×C₂²⋊D₄₃	—	4	2	no	—
95	C₂²×Dic₄₃	—	4	2	no	—
96	Q₈×D₄₃	—	2	4	no	—
97	Q₈×D₄₃	—	4	2	no	—
98	#10621900	—	8	4	no	—
99	C₂×C₄×D₄₃	—	4	1	no	—
100	C₂×C₄×D₄₃	—	4	1	no	—
101	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
102	C₂×C₄×D₄₃	D₈	1	1	yes	characteristic
103	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₈₆	—	2	1	no	—
105	C₂×Dic₈₆	—	4	1	no	—
106	C₂×Dic₈₆	—	4	1	no	—
107	Q₁₆×C₄₃	—	2	1	no	—
108	Q₁₆×C₄₃	#41	1	2	yes	complemented
109	D₈×C₄₃	—	2	2	no	—
110	D₈×C₄₃	—	2	2	no	—
111	D₈×C₄₃	—	4	2	no	—
112	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
114	C₄⋊C₄×C₄₃	—	2	1	no	—
115	C₄⋊C₄×C₄₃	—	4	1	no	—
116	C₄²×C₄₃	—	4	1	no	—
117	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂²⋊Dic₄₃	—	4	2	no	—
119	C₂²⋊Dic₄₃	—	4	2	no	—
120	#10621885	—	2	4	no	—
121	C₄⋊Dic₄₃	—	2	2	no	—
122	C₄⋊Dic₄₃	—	2	1	no	—
123	C₄⋊Dic₄₃	—	4	1	no	—
124	C₄⋊Dic₄₃	—	4	1	no	—
125	Dic₄₃⋊C₄	—	4	1	no	—
126	Dic₄₃⋊C₄	—	4	1	no	—
127	Dic₄₃⋊C₄	—	4	1	no	—
128	C₄×Dic₄₃	—	4	1	no	—
129	C₄×Dic₄₃	—	4	1	no	—
130	C₄×Dic₄₃	—	4	1	no	—
131	C₄×Dic₄₃	D₈	1	1	yes	characteristic
132	C₄×Dic₄₃	D₈	1	1	yes	characteristic
133	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
134	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
135	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
136	Dic₁₇₂	—	2	1	no	—
137	Dic₁₇₂	#41	1	2	yes	complemented
138	C₈×D₄₃	—	2	1	no	—
139	C₈×D₄₃	—	2	1	no	—
140	C₈×D₄₃	—	4	1	no	—
141	C₁₆×C₄₃	—	2	2	no	—
142	C₄₃⋊C₁₆	—	2	2	no	—
143	C₂³×C₄₃	—	4	2	no	—
144	C₂²×D₄₃	C₂×D₈	1	1	yes	characteristic
145	Q₈×C₄₃	—	2	2	no	—
146	Q₈×C₄₃	—	4	1	no	—
147	D₄×C₄₃	—	4	2	no	—
148	D₄×C₄₃	—	4	2	no	—
149	D₄×C₄₃	—	8	2	no	—
150	D₄×C₄₃	—	8	2	no	—
151	C₂×C₄×C₄₃	—	4	1	no	—
152	C₂×C₄×C₄₃	—	4	1	no	—
153	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
155	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
156	C₂²⋊D₄₃	—	8	4	no	—
157	C₂×Dic₄₃	—	4	1	no	—
158	C₂×Dic₄₃	—	4	1	no	—
159	C₂×Dic₄₃	—	4	2	no	—
160	C₂×Dic₄₃	—	8	4	no	—
161	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
162	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
163	C₂×Dic₄₃	C₂×D₈	1	1	yes	characteristic
164	C₄×D₄₃	—	2	2	no	—
165	C₄×D₄₃	—	2	1	no	—
166	C₄×D₄₃	—	2	1	no	—
167	C₄×D₄₃	—	2	2	no	—
168	C₄×D₄₃	—	4	4	no	—
169	C₄×D₄₃	—	8	2	no	—
170	Dic₈₆	—	2	2	no	—
171	Dic₈₆	—	4	4	no	—
172	Dic₈₆	—	4	1	no	—
173	Dic₈₆	—	8	2	no	—
174	C₈×C₄₃	—	2	1	no	—
175	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
176	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
177	C₄₃⋊C₈	—	2	1	no	—
178	C₄₃⋊C₈	—	2	1	no	—
179	#33672	—	43	1	no	maximal, Sylow
180	C₂²×C₄₃	—	8	2	no	—
181	C₂²×C₄₃	—	8	2	no	—
182	C₂²×C₄₃	—	8	2	no	—
183	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
184	C₂×D₄₃	—	2	1	no	—
185	C₂×D₄₃	—	2	1	no	—
186	C₂×D₄₃	#509	1	2	yes	—
187	C₄×C₄₃	—	2	1	no	—
188	C₄×C₄₃	—	4	2	no	—
189	C₄×C₄₃	—	8	1	no	—
190	C₄×C₄₃	#510	1	2	yes	—
191	C₄×C₄₃	#525	1	1	yes	characteristic
192	C₄×C₄₃	#575	1	1	yes	characteristic
193	Dic₄₃	—	2	1	no	—
194	Dic₄₃	—	2	1	no	—
195	Dic₄₃	—	2	1	no	—
196	Dic₄₃	—	4	2	no	—
197	Dic₄₃	—	8	2	no	—
198	Dic₄₃	—	8	1	no	—
199	#3169	—	43	1	no	—
200	C₂×C₄×Q₁₆	—	43	1	no	—
201	#2232	—	43	2	no	—
202	#2227	—	43	1	no	—
203	#2227	—	43	1	no	—
204	#2207	—	43	1	no	—
205	#2207	D₄₃	1	1	yes	characteristic, complemented
206	#2185	—	43	2	no	—
207	#2175	—	43	2	no	—
208	#2175	—	43	2	no	—
209	#2163	—	43	1	no	—
210	C₂×C₄₃	—	16	2	no	—
211	C₂×C₄₃	#3100	1	1	yes	characteristic
212	C₂×C₄₃	C₂×#509	1	1	yes	characteristic
213	C₂×C₄₃	C₂×#510	1	1	yes	characteristic
214	D₄₃	—	2	2	no	—
215	C₂²×Q₁₆	—	43	1	no	—
216	#532	—	86	1	no	—
217	C₂×C₄×Q₈	—	86	1	no	—
218	C₂×SD₃₂	—	43	2	no	—
219	C₂×SD₃₂	C₂×D₄₃	1	2	yes	—
220	#500	—	43	1	no	—
221	#493	C₂×D₄₃	1	1	yes	characteristic
222	#486	—	86	1	no	—
223	#466	—	43	2	no	—
224	#466	—	86	1	no	—
225	C₄×Q₁₆	—	43	1	no	—
226	C₄×Q₁₆	—	43	1	no	—
227	C₄×Q₁₆	—	43	1	no	—
228	C₄×Q₁₆	—	86	1	no	—
229	C₄×Q₁₆	—	86	1	no	—
230	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
231	C₄×D₈	—	43	2	no	—
232	C₄×D₈	—	86	1	no	—
233	C₂×#107	—	43	1	no	—
234	C₂×#103	—	86	1	no	—
235	C₂×C₄×C₈	—	43	1	no	—
236	#366	—	43	2	no	—
237	#366	—	43	2	no	—
238	C₄⋊C₁₆	—	43	1	no	—
239	C₄⋊C₁₆	C₂×D₄₃	1	1	yes	characteristic
240	#358	—	43	2	no	—
241	#358	—	43	2	no	—
242	#357	—	43	2	no	—
243	#357	C₂×D₄₃	1	2	yes	—
244	C₂²⋊C₁₆	—	43	2	no	—
245	#346	—	43	2	no	—
246	C₄₃	#33672	1	1	yes	characteristic, complemented, Sylow
247	C₂×#41	—	172	1	no	—
248	C₂²×Q₈	—	86	1	no	—
249	C₂×Q₁₆	—	43	1	no	—
250	C₂×Q₁₆	—	86	1	no	—
251	C₂×Q₁₆	—	86	2	no	—
252	C₂×Q₁₆	—	86	2	no	—
253	C₂×Q₁₆	C₂²×D₄₃	1	1	yes	characteristic
254	C₂×D₈	—	2	1	no	—
255	C₂×D₈	C₂²×D₄₃	1	2	yes	—
256	C₂²×C₈	—	43	1	no	—
257	C₂²×C₈	—	43	1	no	—
258	C₄⋊Q₈	—	86	1	no	—
259	#127	—	2	1	no	—
260	#124	—	172	1	no	—
261	#121	—	86	2	no	—
262	#121	—	172	1	no	—
263	C₄×Q₈	—	2	1	no	—
264	C₄×Q₈	—	86	1	no	—
265	C₄×Q₈	—	86	1	no	—
266	C₄×Q₈	—	86	1	no	—
267	C₄×Q₈	—	172	1	no	—
268	C₄×Q₈	—	172	1	no	—
269	C₄×D₄	—	86	2	no	—
270	C₄×D₄	—	172	1	no	—
271	C₂×C₄⋊C₄	—	86	1	no	—
272	C₂×C₄⋊C₄	—	172	1	no	—
273	C₂×C₄²	—	43	1	no	—
274	C₂×C₄²	—	172	1	no	—
275	SD₃₂	—	2	4	no	—
276	SD₃₂	—	86	4	no	—
277	C₂×C₁₆	—	43	2	no	—
278	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
279	#107	—	43	1	no	—
280	#107	—	43	2	no	—
281	#107	—	86	1	no	—
282	#107	—	86	1	no	—
283	#107	C₂²×D₄₃	1	1	yes	characteristic
284	#103	—	2	1	no	—
285	#103	—	86	1	no	—
286	#103	—	86	1	no	—
287	#103	—	86	1	no	—
288	#102	—	86	2	no	—
289	#102	—	86	2	no	—
290	C₄×C₈	—	43	1	no	—
291	C₄×C₈	—	43	1	no	—
292	C₄×C₈	—	43	1	no	—
293	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
294	#41	—	344	4	no	—
295	C₂×Q₈	—	2	1	no	—
296	C₂×Q₈	—	86	4	no	—
297	C₂×Q₈	—	86	1	no	—
298	C₂×Q₈	—	172	1	no	—
299	C₂×Q₈	—	172	2	no	—
300	C₂×Q₈	—	172	1	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂×D₄	—	4	1	no	—
303	C₂×D₄	—	4	1	no	—
304	C₂×D₄	—	172	2	no	—
305	C₂²×C₄	—	43	1	no	—
306	C₂²×C₄	—	43	1	no	—
307	C₂²×C₄	—	43	1	no	—
308	C₂²×C₄	—	172	1	no	—
309	C₂²×C₄	—	172	1	no	—
310	C₂²×C₄	—	172	2	no	—
311	Q₁₆	—	2	1	no	—
312	Q₁₆	—	43	2	no	—
313	Q₁₆	—	86	4	no	—
314	Q₁₆	—	86	1	no	—
315	Q₁₆	#10621900	1	2	yes	complemented
316	D₈	—	2	2	no	—
317	D₈	—	2	2	no	—
318	D₈	—	4	2	no	—
319	C₂×C₈	—	43	1	no	—
320	C₂×C₈	—	43	1	no	—
321	C₂×C₈	—	86	1	no	—
322	C₂×C₈	—	86	1	no	—
323	C₂×C₈	—	172	1	no	—
324	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
325	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
326	C₄⋊C₄	—	2	1	no	—
327	C₄⋊C₄	—	4	1	no	—
328	C₄⋊C₄	—	86	2	no	—
329	C₄⋊C₄	—	86	1	no	—
330	C₄⋊C₄	—	172	1	no	—
331	C₄⋊C₄	—	172	1	no	—
332	C₄⋊C₄	—	172	1	no	—
333	C₄⋊C₄	—	172	1	no	—
334	C₄⋊C₄	—	172	1	no	—
335	C₂²⋊C₄	—	172	2	no	—
336	C₂²⋊C₄	—	172	2	no	—
337	C₄²	—	4	1	no	—
338	C₄²	—	43	1	no	—
339	C₄²	—	43	1	no	—
340	C₄²	—	43	1	no	—
341	C₄²	—	172	1	no	—
342	C₄²	—	172	1	no	—
343	C₄²	—	172	1	no	—
344	C₄²	D₄×D₄₃	1	1	yes	characteristic
345	C₁₆	—	2	2	no	—
346	C₁₆	—	86	2	no	—
347	C₂³	—	4	2	no	—
348	C₂³	—	43	1	no	—
349	Q₈	—	2	2	no	—
350	Q₈	—	4	1	no	—
351	Q₈	—	86	2	no	—
352	Q₈	—	344	2	no	—
353	Q₈	—	172	4	no	—
354	Q₈	—	172	1	no	—
355	D₄	—	4	2	no	—
356	D₄	—	4	2	no	—
357	D₄	—	8	2	no	—
358	D₄	—	8	2	no	—
359	D₄	—	344	4	no	—
360	C₂×C₄	—	4	1	no	—
361	C₂×C₄	—	4	1	no	—
362	C₂×C₄	—	43	1	no	—
363	C₂×C₄	—	43	1	no	—
364	C₂×C₄	—	43	1	no	—
365	C₂×C₄	—	86	2	no	—
366	C₂×C₄	—	86	2	no	—
367	C₂×C₄	—	86	1	no	—
368	C₂×C₄	—	86	1	no	—
369	C₂×C₄	—	344	2	no	—
370	C₂×C₄	—	344	4	no	—
371	C₂×C₄	—	172	4	no	—
372	C₂×C₄	—	172	1	no	—
373	C₂×C₄	—	172	2	no	—
374	C₂×C₄	—	172	1	no	—
375	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
376	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
377	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
378	C₈	—	2	1	no	—
379	C₈	—	86	1	no	—
380	C₈	—	86	1	no	—
381	C₈	#41×D₄₃	1	1	yes	characteristic
382	C₈	C₂×#10621900	1	1	yes	characteristic
383	C₂²	—	8	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	43	2	no	—
387	C₂²	—	86	1	no	—
388	C₂²	—	86	1	no	—
389	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
390	C₄	—	2	1	no	—
391	C₄	—	4	2	no	—
392	C₄	—	8	1	no	—
393	C₄	—	86	1	no	—
394	C₄	—	86	1	no	—
395	C₄	—	86	1	no	—
396	C₄	—	344	2	no	—
397	C₄	—	344	1	no	—
398	C₄	—	172	2	no	—
399	C₄	#14583338	1	2	yes	—
400	C₄	#14583409	1	1	yes	characteristic
401	C₄	#14583695	1	1	yes	characteristic
402	C₂	—	16	2	no	—
403	C₂	—	86	2	no	—
404	C₂	#19144594	1	1	yes	characteristic
405	C₂	#509×D₄₃	1	1	yes	characteristic
406	C₂	C₂×#14583338	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#2207	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19145172	C₂
hypercenter	#2207	D₄₃
hypocenter	C₄₃	#33672
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2207

Complements

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

N	H ≅ G/N	type
2. C₄×Q₁₆×D₄₃	C₂	semidirect
3. #19136774	C₂	semidirect
4. #19136748	C₂	semidirect
5. #19136747	C₂	semidirect
6. #19136632	C₂	semidirect
7. #19136510	C₂	semidirect
8. #19136460	C₂	semidirect
9. #19136456	C₂	semidirect
10. #19136397	C₂	semidirect
11. C₄₃×#2207	C₂	semidirect
108. Q₁₆×C₄₃	#41	semidirect
137. Dic₁₇₂	#41	semidirect
205. #2207	D₄₃	semidirect
246. C₄₃	#33672	semidirect
315. Q₁₆	#10621900	semidirect

Maximal subgroups

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

subgroup	index
#19145172	2
C₄×Q₁₆×D₄₃	2
#19136774	2
#19136748	2
#19136747	2
#19136632	2
#19136510	2
#19136460	2
#19136456	2
#19136397	2
C₄₃×#2207	2
#33672	43