Subgroups — #25067124

Order 11008.

614 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145178	C₂	1	1	yes	maximal, characteristic
2	#19143524	C₂	1	2	yes	maximal
3	#19133492	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133491	C₂	1	1	yes	maximal, characteristic, complemented
5	#19133489	C₂	1	2	yes	maximal, complemented
6	C₈×Dic₁₇₂	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×#10621885	C₂	1	2	yes	maximal, complemented
8	#19133352	C₂	1	1	yes	maximal, characteristic
9	#19133048	C₂	1	2	yes	maximal
10	C₈²×D₄₃	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583453	C₂²	1	2	yes	—
13	#14583303	C₄	1	1	yes	characteristic
14	#14583277	C₄	1	1	yes	characteristic
15	#14583248	C₄	1	2	yes	—
16	#14583141	C₄	1	1	yes	characteristic
17	#14583139	C₄	1	2	yes	—
18	#14583137	C₄	1	1	yes	characteristic
19	#14583030	C₂²	1	2	yes	—
20	#14583028	—	2	2	no	—
21	#14583026	C₂²	1	2	yes	—
22	C₈×Dic₈₆	—	2	2	no	—
23	C₈×Dic₈₆	C₂²	1	2	yes	—
24	C₄×Dic₁₇₂	C₂²	1	1	yes	characteristic
25	C₄×#10621885	C₂²	1	2	yes	—
26	#14582924	—	2	2	no	—
27	#14582921	C₂²	1	2	yes	—
28	#14582853	—	2	2	no	—
29	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
30	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₄₃	C₂²	1	1	yes	characteristic
32	#14582652	C₂²	1	1	yes	characteristic
33	#14582651	C₂²	1	2	yes	—
34	#14582631	C₂²	1	2	yes	—
35	Q₈⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
36	#14582629	C₂²	1	2	yes	—
37	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
38	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
39	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₄₃	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₄₃	C₂²	1	1	yes	characteristic
42	C₄₃×#335	C₂²	1	1	yes	characteristic
43	C₄₃×Q₈⋊C₈	C₂²	1	2	yes	—
44	C₈²×C₄₃	C₂²	1	1	yes	characteristic
45	C₂×Dic₁₇₂	C₂×C₄	1	1	yes	characteristic
46	C₂×#10621885	C₂×C₄	1	2	yes	—
47	C₂×#10621872	—	2	2	no	—
48	C₂×C₈×D₄₃	—	2	1	no	—
49	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₈×D₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
52	C₂×C₈×D₄₃	D₄	1	1	yes	characteristic
53	#13063324	C₂×C₄	1	2	yes	—
54	#13063323	C₂×C₄	1	2	yes	—
55	#13063315	—	2	2	no	—
56	#13063305	C₂×C₄	1	2	yes	—
57	#13063303	—	2	2	no	—
58	#13063301	C₂×C₄	1	2	yes	—
59	C₄×Dic₈₆	—	2	2	no	—
60	C₄×Dic₈₆	C₂³	1	2	yes	—
61	#13063282	—	2	2	no	—
62	C₄²×D₄₃	C₂³	1	1	yes	characteristic
63	#13063259	C₂×C₄	1	1	yes	characteristic
64	#13063258	C₂×C₄	1	2	yes	—
65	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
66	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
67	Dic₄₃⋊C₈	—	2	2	no	—
68	Dic₄₃⋊C₈	—	2	2	no	—
69	Dic₄₃⋊C₈	—	2	2	no	—
70	#13063250	C₂×C₄	1	2	yes	—
71	#13063249	C₂×C₄	1	2	yes	—
72	#13063248	C₂×C₄	1	2	yes	—
73	#13063236	—	2	2	no	—
74	#13063235	—	2	2	no	—
75	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
76	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
77	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
78	C₈×Dic₄₃	—	2	2	no	—
79	C₈×Dic₄₃	—	2	1	no	—
80	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
81	C₈×Dic₄₃	C₂×C₄	1	1	yes	characteristic
82	C₈×Dic₄₃	D₄	1	1	yes	characteristic
83	C₈×Dic₄₃	D₄	1	1	yes	characteristic
84	C₂×Q₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
85	C₄×Q₈×C₄₃	C₂³	1	2	yes	—
86	#107×C₄₃	C₂×C₄	1	1	yes	characteristic
87	C₄⋊C₈×C₄₃	—	2	2	no	—
88	C₄⋊C₈×C₄₃	C₂³	1	2	yes	—
89	#103×C₄₃	C₂×C₄	1	2	yes	—
90	C₄×C₈×C₄₃	—	2	2	no	—
91	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
92	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₄₃	C₂²×C₄	1	2	yes	—
95	C₂×C₄×D₄₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×D₄₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×D₄₃	#41	1	1	yes	characteristic
98	C₂×Dic₈₆	—	2	2	no	—
99	C₂×Dic₈₆	C₂²×C₄	1	2	yes	—
100	Q₁₆×C₄₃	—	2	2	no	—
101	C₂×C₈×C₄₃	—	2	1	no	—
102	C₂×C₈×C₄₃	—	2	2	no	—
103	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄⋊C₄×C₄₃	—	2	2	no	—
108	C₄⋊C₄×C₄₃	C₂²×C₄	1	2	yes	—
109	C₄²×C₄₃	—	2	2	no	—
110	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
111	#10621885	—	2	4	no	—
112	(C₂×D₄₃)⋊C₄	—	2	2	no	—
113	(C₂×D₄₃)⋊C₄	—	2	2	no	—
114	C₄⋊Dic₄₃	—	2	2	no	—
115	C₄⋊Dic₄₃	C₂²×C₄	1	2	yes	—
116	Dic₄₃⋊C₄	—	2	2	no	—
117	Dic₄₃⋊C₄	—	2	2	no	—
118	Dic₄₃⋊C₄	—	2	2	no	—
119	Dic₄₃⋊C₄	—	2	2	no	—
120	Dic₄₃⋊C₄	—	2	2	no	—
121	C₄×Dic₄₃	—	2	2	no	—
122	C₄×Dic₄₃	C₂²×C₄	1	1	yes	characteristic
123	C₄×Dic₄₃	C₂×D₄	1	1	yes	characteristic
124	C₄×Dic₄₃	#41	1	1	yes	characteristic
125	C₂×C₄₃⋊C₈	—	2	1	no	—
126	C₂×C₄₃⋊C₈	—	2	2	no	—
127	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
128	C₂×C₄₃⋊C₈	C₂²×C₄	1	1	yes	characteristic
129	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
130	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
131	Dic₁₇₂	—	2	2	no	—
132	#10621872	—	4	4	no	—
133	C₈×D₄₃	—	2	4	no	—
134	C₈×D₄₃	—	2	2	no	—
135	C₈×D₄₃	—	2	2	no	—
136	C₈×D₄₃	—	2	2	no	—
137	C₈×D₄₃	Q₁₆	1	4	yes	complemented
138	C₂²×D₄₃	C₄×D₄	1	1	yes	characteristic
139	Q₈×C₄₃	—	2	2	no	—
140	Q₈×C₄₃	—	2	2	no	—
141	C₂×C₄×C₄₃	—	2	2	no	—
142	C₂×C₄×C₄₃	—	2	2	no	—
143	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₂×Dic₄₃	—	2	2	no	—
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
149	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
150	C₂×Dic₄₃	C₄×D₄	1	1	yes	characteristic
151	C₄×D₄₃	—	2	1	no	—
152	C₄×D₄₃	—	2	1	no	—
153	C₄×D₄₃	—	2	2	no	—
154	C₄×D₄₃	C₂×Q₁₆	1	2	yes	—
155	C₄×D₄₃	#135	1	2	yes	—
156	Dic₈₆	—	2	2	no	—
157	Dic₈₆	—	2	2	no	—
158	Dic₈₆	—	4	4	no	—
159	C₈×C₄₃	—	2	2	no	—
160	C₈×C₄₃	—	2	1	no	—
161	C₈×C₄₃	—	2	1	no	—
162	C₈×C₄₃	—	4	2	no	—
163	C₈×C₄₃	#131	1	2	yes	—
164	C₈×C₄₃	C₂×Q₁₆	1	2	yes	—
165	C₄₃⋊C₈	—	2	2	no	—
166	C₄₃⋊C₈	—	2	1	no	—
167	C₄₃⋊C₈	—	2	1	no	—
168	C₄₃⋊C₈	—	4	2	no	—
169	C₄₃⋊C₈	#131	1	2	yes	—
170	C₄₃⋊C₈	C₂×Q₁₆	1	2	yes	—
171	#17958	—	43	1	no	maximal, Sylow
172	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
173	C₂×D₄₃	—	2	1	no	—
174	C₂×D₄₃	—	2	1	no	—
175	C₂×D₄₃	C₄×Q₁₆	1	2	yes	—
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	4	2	no	—
178	C₄×C₄₃	—	4	2	no	—
179	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
180	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
181	C₄×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
182	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
183	Dic₄₃	—	2	1	no	—
184	Dic₄₃	—	4	2	no	—
185	Dic₄₃	—	4	2	no	—
186	Dic₄₃	C₄×Q₁₆	1	2	yes	—
187	Dic₄₃	#443	1	2	yes	—
188	#3170	—	43	1	no	—
189	#2960	—	43	2	no	—
190	#1591	—	43	1	no	—
191	#1591	—	43	1	no	—
192	#1591	—	43	2	no	—
193	C₈×Q₁₆	—	43	2	no	—
194	C₈×Q₁₆	—	43	1	no	—
195	C₈×Q₁₆	D₄₃	1	1	yes	characteristic, complemented
196	#1567	—	43	1	no	—
197	#1493	—	43	2	no	—
198	C₂×C₈²	—	43	1	no	—
199	C₂×C₄₃	#2931	1	1	yes	characteristic
200	C₂×C₄₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
201	C₂×C₄₃	C₂×#443	1	1	yes	characteristic
202	D₄₃	—	2	2	no	—
203	#533	—	43	2	no	—
204	#500	—	43	1	no	—
205	#494	—	43	1	no	—
206	#487	—	43	2	no	—
207	#467	—	43	1	no	—
208	#467	—	43	1	no	—
209	#467	—	43	2	no	—
210	#446	—	43	2	no	—
211	#446	—	43	2	no	—
212	#446	—	86	2	no	—
213	C₈×Q₈	—	43	2	no	—
214	C₈×Q₈	—	86	2	no	—
215	C₈×Q₈	C₂×D₄₃	1	2	yes	—
216	C₄×Q₁₆	—	43	2	no	—
217	C₄×Q₁₆	—	43	1	no	—
218	C₄×Q₁₆	C₂×D₄₃	1	1	yes	characteristic
219	#424	—	86	2	no	—
220	#423	—	43	2	no	—
221	#405	—	86	2	no	—
222	C₂×C₄×C₈	—	43	1	no	—
223	C₂×C₄×C₈	—	43	1	no	—
224	C₂×C₄×C₈	—	43	1	no	—
225	#335	—	43	2	no	—
226	#335	—	43	1	no	—
227	#335	C₂×D₄₃	1	1	yes	characteristic
228	Q₈⋊C₈	—	43	2	no	—
229	Q₈⋊C₈	—	43	2	no	—
230	Q₈⋊C₈	—	43	2	no	—
231	Q₈⋊C₈	C₂×D₄₃	1	2	yes	—
232	C₈²	—	43	1	no	—
233	C₈²	—	43	1	no	—
234	C₈²	—	43	1	no	—
235	C₈²	C₂×D₄₃	1	1	yes	characteristic
236	C₄₃	#17958	1	1	yes	characteristic, complemented, Sylow
237	C₂×Q₁₆	—	43	2	no	—
238	C₂×Q₁₆	—	43	1	no	—
239	C₂×Q₁₆	C₄×D₄₃	1	1	yes	characteristic
240	C₂×M₁₆	—	86	2	no	—
241	C₂²×C₈	—	43	1	no	—
242	C₂²×C₈	—	43	1	no	—
243	C₂²×C₈	—	43	1	no	—
244	C₂²×C₈	—	43	1	no	—
245	C₂²×C₈	—	86	1	no	—
246	C₄⋊Q₈	—	43	2	no	—
247	C₄⋊Q₈	—	43	2	no	—
248	#125	—	86	2	no	—
249	C₂²⋊Q₈	—	43	2	no	—
250	C₂²⋊Q₈	—	43	2	no	—
251	C₂²⋊Q₈	—	86	2	no	—
252	C₄×Q₈	—	43	2	no	—
253	C₄×Q₈	—	86	2	no	—
254	C₄×Q₈	C₂²×D₄₃	1	2	yes	—
255	#117	—	86	2	no	—
256	C₂×C₄²	—	43	1	no	—
257	#107	—	43	1	no	—
258	#107	—	43	2	no	—
259	#107	C₄×D₄₃	1	1	yes	characteristic
260	C₄⋊C₈	—	2	2	no	—
261	C₄⋊C₈	—	43	2	no	—
262	C₄⋊C₈	—	86	2	no	—
263	C₄⋊C₈	—	86	2	no	—
264	C₄⋊C₈	—	86	2	no	—
265	C₄⋊C₈	—	86	2	no	—
266	C₄⋊C₈	C₂²×D₄₃	1	2	yes	—
267	#103	—	43	2	no	—
268	#103	—	43	2	no	—
269	#103	—	43	2	no	—
270	#103	C₄×D₄₃	1	2	yes	—
271	#97	—	86	2	no	—
272	#97	—	86	2	no	—
273	C₄×C₈	—	2	2	no	—
274	C₄×C₈	—	43	1	no	—
275	C₄×C₈	—	43	1	no	—
276	C₄×C₈	—	43	1	no	—
277	C₄×C₈	—	43	1	no	—
278	C₄×C₈	—	43	1	no	—
279	C₄×C₈	—	43	1	no	—
280	C₄×C₈	—	43	1	no	—
281	C₄×C₈	—	86	1	no	—
282	C₄×C₈	—	86	2	no	—
283	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
284	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
285	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
286	C₂×Q₈	—	43	2	no	—
287	C₂×Q₈	—	86	2	no	—
288	C₂×Q₈	C₂×C₄×D₄₃	1	2	yes	—
289	C₂²×C₄	—	43	1	no	—
290	C₂²×C₄	—	43	1	no	—
291	C₂²×C₄	—	43	1	no	—
292	Q₁₆	—	2	2	no	—
293	Q₁₆	—	86	4	no	—
294	Q₁₆	—	86	2	no	—
295	M₁₆	—	172	4	no	—
296	C₂×C₈	—	2	1	no	—
297	C₂×C₈	—	2	2	no	—
298	C₂×C₈	—	43	1	no	—
299	C₂×C₈	—	43	1	no	—
300	C₂×C₈	—	43	4	no	—
301	C₂×C₈	—	43	1	no	—
302	C₂×C₈	—	43	1	no	—
303	C₂×C₈	—	86	2	no	—
304	C₂×C₈	—	86	2	no	—
305	C₂×C₈	—	86	4	no	—
306	C₂×C₈	—	86	1	no	—
307	C₂×C₈	—	86	2	no	—
308	C₂×C₈	—	86	2	no	—
309	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
310	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
311	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
312	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
313	C₄⋊C₄	—	2	2	no	—
314	C₄⋊C₄	—	43	2	no	—
315	C₄⋊C₄	—	86	2	no	—
316	C₄⋊C₄	—	86	2	no	—
317	C₄⋊C₄	—	86	2	no	—
318	C₄⋊C₄	—	86	2	no	—
319	C₄⋊C₄	—	86	2	no	—
320	C₄⋊C₄	—	86	2	no	—
321	C₄⋊C₄	C₂×C₄×D₄₃	1	2	yes	—
322	C₂²⋊C₄	—	86	2	no	—
323	C₂²⋊C₄	—	86	2	no	—
324	C₄²	—	2	2	no	—
325	C₄²	—	43	1	no	—
326	C₄²	—	43	1	no	—
327	C₄²	—	43	1	no	—
328	C₄²	—	86	2	no	—
329	C₄²	C₂³×D₄₃	1	1	yes	characteristic
330	C₂³	—	43	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	2	no	—
333	Q₈	—	86	2	no	—
334	Q₈	—	86	2	no	—
335	Q₈	—	172	4	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	43	1	no	—
339	C₂×C₄	—	43	1	no	—
340	C₂×C₄	—	43	2	no	—
341	C₂×C₄	—	43	2	no	—
342	C₂×C₄	—	43	1	no	—
343	C₂×C₄	—	86	2	no	—
344	C₂×C₄	—	86	1	no	—
345	C₂×C₄	—	86	1	no	—
346	C₂×C₄	—	86	2	no	—
347	C₂×C₄	—	86	2	no	—
348	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
349	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
350	C₂×C₄	#41×D₄₃	1	1	yes	characteristic
351	C₈	—	2	2	no	—
352	C₈	—	2	1	no	—
353	C₈	—	2	1	no	—
354	C₈	—	4	2	no	—
355	C₈	—	43	2	no	—
356	C₈	—	43	2	no	—
357	C₈	—	86	2	no	—
358	C₈	—	86	1	no	—
359	C₈	—	86	1	no	—
360	C₈	—	172	2	no	—
361	C₈	#13063331	1	2	yes	—
362	C₈	Q₁₆×D₄₃	1	2	yes	—
363	C₂²	—	43	2	no	—
364	C₂²	—	86	1	no	—
365	C₂²	—	86	1	no	—
366	C₂²	C₄×D₄×D₄₃	1	1	yes	characteristic
367	C₄	—	2	1	no	—
368	C₄	—	4	2	no	—
369	C₄	—	4	2	no	—
370	C₄	—	43	2	no	—
371	C₄	—	43	2	no	—
372	C₄	—	86	1	no	—
373	C₄	—	172	2	no	—
374	C₄	—	172	2	no	—
375	C₄	#131×D₄₃	1	1	yes	characteristic
376	C₄	C₂×#13063331	1	1	yes	characteristic
377	C₄	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
378	C₄	#135×D₄₃	1	1	yes	characteristic
379	C₂	—	86	2	no	—
380	C₂	#19143302	1	1	yes	characteristic
381	C₂	C₄×Q₁₆×D₄₃	1	1	yes	characteristic
382	C₂	#443×D₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₄²	C₂³×D₄₃
Fitting	C₈×Q₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂²×D₄₃	C₄×D₄
cosocle	C₄²×C₄₃	C₂⁴
hypercenter	C₈×Q₁₆	D₄₃
hypocenter	C₄₃	#17958
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #19133492	C₂	semidirect
4. #19133491	C₂	semidirect
5. #19133489	C₂	semidirect
6. C₈×Dic₁₇₂	C₂	semidirect
7. C₈×#10621885	C₂	semidirect
11. C₈×Q₁₆×C₄₃	C₂	semidirect
137. C₈×D₄₃	Q₁₆	semidirect
195. C₈×Q₁₆	D₄₃	semidirect
236. C₄₃	#17958	semidirect

Maximal subgroups

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

subgroup	index
#19145178	2
#19143524	2
#19133492	2
#19133491	2
#19133489	2
C₈×Dic₁₇₂	2
C₈×#10621885	2
#19133352	2
#19133048	2
C₈²×D₄₃	2
C₈×Q₁₆×C₄₃	2
#17958	43