Subgroups — #25087930

Order 11008.

710 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#19146261	C₂	1	2	yes	maximal, complemented
3	#19134184	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134058	C₂	1	2	yes	maximal, complemented
5	#19133989	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133952	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133788	C₂	1	2	yes	maximal, complemented
8	#19133697	C₂	1	2	yes	maximal, complemented
9	#19133480	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10621885	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3391	C₂	1	1	yes	maximal, characteristic
12	C₂×#13063355	—	2	2	no	—
13	D₄×Dic₈₆	C₂²	1	2	yes	—
14	#14583304	C₂²	1	1	yes	characteristic
15	#14583303	—	2	1	no	—
16	#14583271	C₂²	1	2	yes	—
17	#14583245	C₂²	1	2	yes	—
18	#14583179	C₂²	1	2	yes	—
19	#14583177	—	2	2	no	—
20	#14583143	—	2	2	no	—
21	#14583109	C₂²	1	2	yes	—
22	#14583096	C₂²	1	2	yes	—
23	#14583061	—	2	2	no	—
24	#14583037	—	2	2	no	—
25	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
26	C₄×#10621885	C₂²	1	1	yes	characteristic
27	D₈×Dic₄₃	—	2	1	no	—
28	#14582977	C₂²	1	2	yes	—
29	#14582652	C₂²	1	1	yes	characteristic
30	#14582651	C₂²	1	1	yes	characteristic
31	#14582644	C₂²	1	2	yes	—
32	#14582643	C₂²	1	2	yes	—
33	Q₈⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
34	#14582629	C₂²	1	1	yes	characteristic
35	D₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
36	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
37	C₄₃×#550	C₂²	1	2	yes	—
38	C₄₃×#493	—	2	1	no	—
39	C₄₃×#493	C₂²	1	1	yes	characteristic
40	C₄₃×#459	—	2	2	no	—
41	C₄₃×#459	C₂²	1	2	yes	—
42	C₈×Q₈×C₄₃	C₂²	1	1	yes	characteristic
43	C₄×D₈×C₄₃	—	2	1	no	—
44	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
45	C₂²×Dic₈₆	—	2	2	no	—
46	#13063355	—	4	8	no	—
47	C₂×#10621885	—	2	2	no	—
48	C₂×#10621885	D₄	1	1	yes	characteristic
49	C₂×#10621883	—	2	2	no	—
50	C₂×#10621883	—	2	2	no	—
51	C₂×#10621883	—	2	2	no	—
52	C₂×#10621877	—	2	2	no	—
53	#13063325	C₂³	1	2	yes	—
54	#13063324	—	2	2	no	—
55	#13063304	—	2	2	no	—
56	#13063302	—	4	2	no	—
57	C₄×Dic₈₆	C₂³	1	1	yes	characteristic
58	D₄×Dic₄₃	—	2	2	no	—
59	#13063259	—	2	1	no	—
60	#13063259	—	2	1	no	—
61	#13063258	—	2	2	no	—
62	#13063258	D₄	1	1	yes	characteristic
63	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
64	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
65	Dic₄₃⋊C₈	—	2	1	no	—
66	#13063249	D₄	1	1	yes	characteristic
67	#13063248	—	2	2	no	—
68	#13063248	D₄	1	1	yes	characteristic
69	#13063246	—	2	2	no	—
70	#13063246	—	2	2	no	—
71	#13063246	—	4	2	no	—
72	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
73	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
74	C₄×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
75	C₈×Dic₄₃	—	2	1	no	—
76	C₂×#41×C₄₃	—	2	2	no	—
77	C₂×D₈×C₄₃	—	2	2	no	—
78	C₂×D₈×C₄₃	—	2	1	no	—
79	C₂×D₈×C₄₃	—	2	1	no	—
80	C₂×D₈×C₄₃	D₄	1	1	yes	characteristic
81	#127×C₄₃	—	2	2	no	—
82	#127×C₄₃	C₂³	1	2	yes	—
83	#121×C₄₃	—	2	2	no	—
84	#121×C₄₃	—	4	2	no	—
85	C₄×Q₈×C₄₃	C₂³	1	1	yes	characteristic
86	C₄×D₄×C₄₃	—	2	2	no	—
87	C₄×D₄×C₄₃	C₂³	1	2	yes	—
88	#107×C₄₃	D₄	1	1	yes	characteristic
89	C₄⋊C₈×C₄₃	—	2	1	no	—
90	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
91	#102×C₄₃	—	2	2	no	—
92	#102×C₄₃	D₄	1	2	yes	—
93	C₄×C₈×C₄₃	—	2	1	no	—
94	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
95	#41×C₄₃	—	4	4	no	—
96	#41×C₄₃	—	8	2	no	—
97	C₂×Q₈×C₄₃	C₂×D₄	1	1	yes	characteristic
98	C₂×D₄×C₄₃	—	2	2	no	—
99	C₂×D₄×C₄₃	—	2	2	no	—
100	C₂×D₄×C₄₃	—	2	2	no	—
101	C₂×D₄×C₄₃	—	4	2	no	—
102	C₂×D₄×C₄₃	—	4	2	no	—
103	C₂×D₄×C₄₃	C₂×D₄	1	2	yes	—
104	C₂²×C₄×C₄₃	—	2	2	no	—
105	C₂²×C₄×C₄₃	—	4	2	no	—
106	C₂²×Dic₄₃	—	4	2	no	—
107	C₂×Dic₈₆	—	2	2	no	—
108	C₂×Dic₈₆	—	4	4	no	—
109	C₂×Dic₈₆	—	4	2	no	—
110	C₂×Dic₈₆	—	8	2	no	—
111	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
112	D₈×C₄₃	—	2	2	no	—
113	D₈×C₄₃	—	4	2	no	—
114	D₈×C₄₃	—	8	1	no	—
115	D₈×C₄₃	—	8	2	no	—
116	C₂×C₈×C₄₃	—	2	1	no	—
117	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₄₃	—	2	1	no	—
120	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
121	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
122	C₂²⋊C₄×C₄₃	—	2	2	no	—
123	C₂²⋊C₄×C₄₃	—	4	2	no	—
124	C₄²×C₄₃	—	2	1	no	—
125	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
126	C₂²⋊Dic₄₃	—	4	2	no	—
127	#10621885	—	4	4	no	—
128	#10621885	D₈	1	4	yes	complemented
129	#10621883	—	4	4	no	—
130	#10621883	—	4	4	no	—
131	#10621883	—	8	2	no	—
132	C₄⋊Dic₄₃	—	2	2	no	—
133	C₄⋊Dic₄₃	—	2	2	no	—
134	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
135	Dic₄₃⋊C₄	—	2	1	no	—
136	Dic₄₃⋊C₄	—	4	2	no	—
137	C₄×Dic₄₃	—	2	1	no	—
138	#10621877	—	4	4	no	—
139	C₂×C₄₃⋊C₈	—	2	1	no	—
140	C₂×C₄₃⋊C₈	—	2	2	no	—
141	C₂×C₄₃⋊C₈	—	2	1	no	—
142	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
143	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
144	C₂³×C₄₃	—	2	2	no	—
145	C₂³×C₄₃	—	4	2	no	—
146	Q₈×C₄₃	—	2	1	no	—
147	Q₈×C₄₃	C₂×D₈	1	2	yes	—
148	D₄×C₄₃	—	2	2	no	—
149	D₄×C₄₃	—	2	2	no	—
150	D₄×C₄₃	—	4	2	no	—
151	D₄×C₄₃	—	4	2	no	—
152	D₄×C₄₃	—	4	2	no	—
153	D₄×C₄₃	—	4	2	no	—
154	D₄×C₄₃	—	8	4	no	—
155	D₄×C₄₃	—	8	2	no	—
156	D₄×C₄₃	—	16	2	no	—
157	C₂×C₄×C₄₃	—	2	1	no	—
158	C₂×C₄×C₄₃	—	2	2	no	—
159	C₂×C₄×C₄₃	—	2	1	no	—
160	C₂×C₄×C₄₃	—	4	2	no	—
161	C₂×C₄×C₄₃	—	4	2	no	—
162	C₂×C₄×C₄₃	—	8	4	no	—
163	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
164	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
165	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂×Dic₄₃	—	2	1	no	—
167	C₂×Dic₄₃	—	2	1	no	—
168	C₂×Dic₄₃	—	4	2	no	—
169	C₂×Dic₄₃	—	8	4	no	—
170	Dic₈₆	—	2	1	no	—
171	Dic₈₆	—	4	2	no	—
172	Dic₈₆	—	4	2	no	—
173	Dic₈₆	—	8	4	no	—
174	Dic₈₆	C₂×D₈	1	2	yes	—
175	C₈×C₄₃	—	2	1	no	—
176	C₈×C₄₃	—	4	1	no	—
177	C₈×C₄₃	#137	1	2	yes	complemented
178	C₄₃⋊C₈	—	2	1	no	—
179	C₄₃⋊C₈	—	4	1	no	—
180	C₄₃⋊C₈	—	4	2	no	—
181	C₄₃⋊C₈	—	4	1	no	—
182	C₄₃⋊C₈	C₂×D₈	1	2	yes	—
183	#19889	—	43	1	no	maximal, Sylow
184	C₂²×C₄₃	—	4	2	no	—
185	C₂²×C₄₃	—	4	2	no	—
186	C₂²×C₄₃	—	4	2	no	—
187	C₂²×C₄₃	—	8	2	no	—
188	C₂²×C₄₃	—	8	2	no	—
189	C₂²×C₄₃	—	8	2	no	—
190	C₂²×C₄₃	D₄²	1	1	yes	characteristic
191	C₄×C₄₃	—	2	2	no	—
192	C₄×C₄₃	—	2	1	no	—
193	C₄×C₄₃	—	4	2	no	—
194	C₄×C₄₃	—	4	1	no	—
195	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
196	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
197	C₄×C₄₃	#576	1	1	yes	characteristic
198	C₄×C₄₃	#577	1	1	yes	characteristic
199	Dic₄₃	—	2	2	no	—
200	Dic₄₃	—	4	1	no	—
201	Dic₄₃	—	8	2	no	—
202	#3391	D₄₃	1	1	yes	characteristic
203	Q₈×D₈	—	43	1	no	—
204	#3278	—	43	2	no	—
205	#1706	—	43	1	no	—
206	#1685	—	43	2	no	—
207	#1674	—	43	1	no	—
208	#1668	—	43	1	no	—
209	#1641	—	43	2	no	—
210	#1627	—	43	2	no	—
211	#1589	—	43	1	no	—
212	C₈×Q₁₆	—	43	1	no	—
213	C₂×C₄₃	—	8	2	no	—
214	C₂×C₄₃	—	16	2	no	—
215	C₂×C₄₃	#3278	1	1	yes	characteristic
216	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
217	C₂×C₄₃	#3290	1	1	yes	characteristic
218	C₂×#137	—	86	2	no	—
219	#550	C₂×D₄₃	1	2	yes	—
220	D₄×Q₈	—	43	2	no	—
221	#500	—	43	1	no	—
222	#500	—	86	1	no	—
223	#493	—	2	1	no	—
224	#493	C₂×D₄₃	1	1	yes	characteristic
225	#492	—	43	2	no	—
226	#486	—	43	2	no	—
227	#474	—	43	2	no	—
228	#474	—	86	2	no	—
229	#468	—	86	2	no	—
230	#462	—	43	2	no	—
231	#460	—	43	2	no	—
232	#459	—	2	2	no	—
233	#459	C₂×D₄₃	1	2	yes	—
234	#452	—	86	2	no	—
235	#448	—	86	2	no	—
236	C₈×Q₈	—	43	1	no	—
237	C₈×Q₈	C₂×D₄₃	1	1	yes	characteristic
238	C₄×Q₁₆	—	43	1	no	—
239	C₄×D₈	—	2	1	no	—
240	C₄×D₈	—	86	1	no	—
241	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
242	#436	—	43	2	no	—
243	#335	—	43	1	no	—
244	#335	—	43	1	no	—
245	#332	—	43	2	no	—
246	#331	—	43	2	no	—
247	Q₈⋊C₈	—	43	1	no	—
248	Q₈⋊C₈	—	43	1	no	—
249	D₄⋊C₈	—	43	2	no	—
250	C₈²	—	43	1	no	—
251	C₄₃	#19889	1	1	yes	characteristic, complemented, Sylow
252	C₂×#41	—	2	2	no	—
253	C₂²×Q₈	—	86	2	no	—
254	#137	—	172	8	no	—
255	C₂×Q₁₆	—	43	1	no	—
256	C₂×Q₁₆	—	86	2	no	—
257	C₂×SD₁₆	—	86	2	no	—
258	C₂×SD₁₆	—	86	2	no	—
259	C₂×SD₁₆	—	86	2	no	—
260	C₂×D₈	—	2	2	no	—
261	C₂×D₈	—	2	1	no	—
262	C₂×D₈	—	2	1	no	—
263	C₂×D₈	C₂²⋊D₄₃	1	1	yes	characteristic
264	C₂×M₁₆	—	86	2	no	—
265	C₄⋊Q₈	—	43	2	no	—
266	C₄⋊Q₈	—	86	2	no	—
267	#127	—	2	2	no	—
268	#127	C₂²×D₄₃	1	2	yes	—
269	C₂²⋊Q₈	—	86	2	no	—
270	C₂²⋊Q₈	—	172	2	no	—
271	#121	—	2	2	no	—
272	#121	—	4	2	no	—
273	C₄×Q₈	—	43	1	no	—
274	C₄×Q₈	C₂²×D₄₃	1	1	yes	characteristic
275	C₄×D₄	—	2	2	no	—
276	C₄×D₄	—	86	2	no	—
277	C₄×D₄	C₂²×D₄₃	1	2	yes	—
278	#107	—	43	1	no	—
279	#107	—	86	1	no	—
280	#107	—	86	1	no	—
281	#107	—	86	2	no	—
282	#107	C₂²⋊D₄₃	1	1	yes	characteristic
283	C₄⋊C₈	—	2	1	no	—
284	C₄⋊C₈	—	43	2	no	—
285	C₄⋊C₈	—	43	1	no	—
286	C₄⋊C₈	—	86	1	no	—
287	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
288	#103	—	43	1	no	—
289	#103	—	43	1	no	—
290	#103	—	86	2	no	—
291	#102	—	2	2	no	—
292	#102	—	86	2	no	—
293	#102	—	86	2	no	—
294	#102	—	172	2	no	—
295	#102	C₂²⋊D₄₃	1	2	yes	—
296	C₂²⋊C₈	—	86	2	no	—
297	C₄×C₈	—	2	1	no	—
298	C₄×C₈	—	43	1	no	—
299	C₄×C₈	—	43	1	no	—
300	C₄×C₈	—	86	1	no	—
301	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
302	#41	—	4	4	no	—
303	#41	—	8	2	no	—
304	C₂×Q₈	—	43	1	no	—
305	C₂×Q₈	—	86	2	no	—
306	C₂×Q₈	—	344	2	no	—
307	C₂×Q₈	—	172	2	no	—
308	C₂×Q₈	—	172	4	no	—
309	C₂×Q₈	D₄×D₄₃	1	1	yes	characteristic
310	C₂×D₄	—	2	2	no	—
311	C₂×D₄	—	2	2	no	—
312	C₂×D₄	—	2	2	no	—
313	C₂×D₄	—	4	2	no	—
314	C₂×D₄	—	4	2	no	—
315	C₂×D₄	C₂×C₂²⋊D₄₃	1	2	yes	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	4	2	no	—
318	C₂²×C₄	—	172	2	no	—
319	Q₁₆	—	43	4	no	—
320	Q₁₆	—	172	4	no	—
321	SD₁₆	—	344	2	no	—
322	SD₁₆	—	172	4	no	—
323	SD₁₆	—	172	4	no	—
324	D₈	—	2	2	no	—
325	D₈	—	4	2	no	—
326	D₈	—	8	1	no	—
327	D₈	—	8	2	no	—
328	M₁₆	—	172	4	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	43	1	no	—
331	C₂×C₈	—	43	1	no	—
332	C₂×C₈	—	86	2	no	—
333	C₂×C₈	—	86	1	no	—
334	C₂×C₈	—	86	1	no	—
335	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
336	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
337	C₄⋊C₄	—	2	1	no	—
338	C₄⋊C₄	—	43	1	no	—
339	C₄⋊C₄	—	86	1	no	—
340	C₄⋊C₄	—	86	2	no	—
341	C₄⋊C₄	—	86	2	no	—
342	C₄⋊C₄	—	172	2	no	—
343	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
344	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
345	C₂²⋊C₄	—	2	2	no	—
346	C₂²⋊C₄	—	4	2	no	—
347	C₂²⋊C₄	—	172	2	no	—
348	C₄²	—	2	1	no	—
349	C₄²	—	86	1	no	—
350	C₄²	C₂³×D₄₃	1	1	yes	characteristic
351	C₂³	—	2	2	no	—
352	C₂³	—	4	2	no	—
353	Q₈	—	2	1	no	—
354	Q₈	—	43	2	no	—
355	Q₈	—	86	1	no	—
356	Q₈	—	344	4	no	—
357	Q₈	—	172	2	no	—
358	Q₈	—	172	2	no	—
359	Q₈	D₈×D₄₃	1	2	yes	—
360	D₄	—	2	2	no	—
361	D₄	—	2	2	no	—
362	D₄	—	4	2	no	—
363	D₄	—	4	2	no	—
364	D₄	—	4	2	no	—
365	D₄	—	4	2	no	—
366	D₄	—	8	4	no	—
367	D₄	—	8	2	no	—
368	D₄	—	16	2	no	—
369	C₂×C₄	—	2	1	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	8	4	no	—
375	C₂×C₄	—	86	1	no	—
376	C₂×C₄	—	86	1	no	—
377	C₂×C₄	—	344	4	no	—
378	C₂×C₄	—	172	2	no	—
379	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
380	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
381	C₂×C₄	#13063370	1	1	yes	characteristic
382	C₈	—	2	1	no	—
383	C₈	—	4	1	no	—
384	C₈	—	43	2	no	—
385	C₈	—	86	1	no	—
386	C₈	—	172	1	no	—
387	C₈	—	172	2	no	—
388	C₈	—	172	1	no	—
389	C₈	#13063355	1	2	yes	complemented
390	C₂²	—	4	2	no	—
391	C₂²	—	4	2	no	—
392	C₂²	—	4	2	no	—
393	C₂²	—	8	2	no	—
394	C₂²	—	8	2	no	—
395	C₂²	—	8	2	no	—
396	C₂²	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
397	C₄	—	2	2	no	—
398	C₄	—	2	1	no	—
399	C₄	—	4	2	no	—
400	C₄	—	4	1	no	—
401	C₄	—	86	2	no	—
402	C₄	—	344	2	no	—
403	C₄	—	172	1	no	—
404	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
405	C₄	C₂×#13063355	1	1	yes	characteristic
406	C₄	#14583700	1	1	yes	characteristic
407	C₄	#14583710	1	1	yes	characteristic
408	C₂	—	8	2	no	—
409	C₂	—	16	2	no	—
410	C₂	#19146261	1	1	yes	characteristic
411	C₂	D₈×C₂²⋊D₄₃	1	1	yes	characteristic
412	C₂	#19146377	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×C₂²⋊D₄₃
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3391	C₂
socle	C₂²×C₄₃	D₄²
hypercenter	#3391	D₄₃
hypocenter	C₄₃	#19889
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3391

Complements

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

N	H ≅ G/N	type
1. D₈×Dic₈₆	C₂	semidirect
2. #19146261	C₂	semidirect
3. #19134184	C₂	semidirect
4. #19134058	C₂	semidirect
5. #19133989	C₂	semidirect
6. #19133952	C₂	semidirect
7. #19133788	C₂	semidirect
8. #19133697	C₂	semidirect
9. #19133480	C₂	semidirect
10. C₈×#10621885	C₂	semidirect
128. #10621885	D₈	semidirect
177. C₈×C₄₃	#137	semidirect
251. C₄₃	#19889	semidirect
389. C₈	#13063355	semidirect

Maximal subgroups

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

subgroup	index
D₈×Dic₈₆	2
#19146261	2
#19134184	2
#19134058	2
#19133989	2
#19133952	2
#19133788	2
#19133697	2
#19133480	2
C₈×#10621885	2
C₄₃×#3391	2
#19889	43