Subgroups — #25087906

Order 11008.

690 conjugacy classes of subgroups.

The subgroup lattice has 406 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	#19146237	C₂	1	2	yes	maximal, complemented
3	#19134212	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134082	C₂	1	2	yes	maximal, complemented
5	#19133986	C₂	1	1	yes	maximal, characteristic, complemented
6	#19133984	C₂	1	1	yes	maximal, characteristic, complemented
7	#19133843	C₂	1	2	yes	maximal, complemented
8	#19133830	C₂	1	2	yes	maximal, complemented
9	#19133480	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈×#10621883	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄₃×#3334	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	#14583302	C₂²	1	2	yes	—
17	#14583278	C₂²	1	2	yes	—
18	#14583179	C₂²	1	2	yes	—
19	#14583177	—	2	2	no	—
20	#14583155	—	2	2	no	—
21	#14583106	C₂²	1	2	yes	—
22	#14583104	C₂²	1	2	yes	—
23	#14583061	—	2	2	no	—
24	#14583054	—	2	2	no	—
25	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
26	C₄×#10621883	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	#14582649	C₂²	1	1	yes	characteristic
31	#14582645	C₂²	1	2	yes	—
32	#14582644	C₂²	1	2	yes	—
33	#14582629	C₂²	1	1	yes	characteristic
34	D₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
35	D₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
36	C₈×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
37	C₄₃×#547	C₂²	1	2	yes	—
38	C₄₃×#500	C₂²	1	1	yes	characteristic
39	C₄₃×#480	—	2	2	no	—
40	C₄₃×#474	C₂²	1	2	yes	—
41	C₄₃×#466	—	2	1	no	—
42	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
43	C₈×D₄×C₄₃	C₂²	1	1	yes	characteristic
44	C₂×#107×C₄₃	—	2	1	no	—
45	C₂²×Dic₈₆	—	2	2	no	—
46	#13063355	—	4	8	no	—
47	C₂×#10621885	—	2	2	no	—
48	C₂×#10621885	—	2	2	no	—
49	C₂×#10621885	—	2	2	no	—
50	C₂×#10621883	—	2	2	no	—
51	C₂×#10621883	D₄	1	1	yes	characteristic
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	—	4	2	no	—
63	#13063256	D₄	1	1	yes	characteristic
64	C₄⋊(C₄₃⋊C₈)	C₂³	1	2	yes	—
65	C₄⋊(C₄₃⋊C₈)	C₂³	1	1	yes	characteristic
66	Dic₄₃⋊C₈	—	2	1	no	—
67	#13063249	—	2	2	no	—
68	#13063248	—	2	2	no	—
69	#13063248	D₄	1	1	yes	characteristic
70	#13063246	—	2	2	no	—
71	#13063246	D₄	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
78	C₂²×C₈×C₄₃	—	2	1	no	—
79	C₄⋊Q₈×C₄₃	C₂³	1	2	yes	—
80	#123×C₄₃	—	4	2	no	—
81	C₂²⋊Q₈×C₄₃	—	2	2	no	—
82	#121×C₄₃	—	2	2	no	—
83	C₄×D₄×C₄₃	C₂³	1	2	yes	—
84	C₄×D₄×C₄₃	C₂³	1	1	yes	characteristic
85	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
86	#107×C₄₃	—	2	1	no	—
87	#107×C₄₃	—	2	1	no	—
88	#107×C₄₃	—	2	2	no	—
89	#107×C₄₃	D₄	1	1	yes	characteristic
90	C₄⋊C₈×C₄₃	C₂³	1	1	yes	characteristic
91	#102×C₄₃	—	2	2	no	—
92	#102×C₄₃	D₄	1	2	yes	—
93	C₂²⋊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	—	2	2	no	—
98	C₂×D₄×C₄₃	—	4	2	no	—
99	C₂×D₄×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×D₄×C₄₃	C₂×D₄	1	2	yes	—
101	C₂²×C₄×C₄₃	—	2	1	no	—
102	C₂²×C₄×C₄₃	—	2	2	no	—
103	C₂²×C₄×C₄₃	—	4	2	no	—
104	C₂²×Dic₄₃	—	4	2	no	—
105	C₂×Dic₈₆	—	2	2	no	—
106	C₂×Dic₈₆	—	4	4	no	—
107	C₂×Dic₈₆	—	4	2	no	—
108	C₂×Dic₈₆	—	8	2	no	—
109	C₂×Dic₈₆	C₂×D₄	1	1	yes	characteristic
110	D₈×C₄₃	—	2	2	no	—
111	C₂×C₈×C₄₃	—	2	1	no	—
112	C₂×C₈×C₄₃	—	4	2	no	—
113	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
115	C₄⋊C₄×C₄₃	—	2	2	no	—
116	C₄⋊C₄×C₄₃	—	2	2	no	—
117	C₄⋊C₄×C₄₃	—	4	2	no	—
118	C₄⋊C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
119	C₄⋊C₄×C₄₃	C₂×D₄	1	2	yes	—
120	C₂²⋊C₄×C₄₃	—	2	1	no	—
121	C₂²⋊C₄×C₄₃	—	2	2	no	—
122	C₂²⋊C₄×C₄₃	—	4	2	no	—
123	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
124	C₂²⋊Dic₄₃	—	4	2	no	—
125	#10621885	—	4	4	no	—
126	#10621885	—	4	4	no	—
127	#10621885	—	8	2	no	—
128	#10621883	—	4	4	no	—
129	#10621883	D₈	1	4	yes	complemented
130	C₄⋊Dic₄₃	—	2	2	no	—
131	C₄⋊Dic₄₃	—	2	2	no	—
132	C₄⋊Dic₄₃	C₂×D₄	1	1	yes	characteristic
133	Dic₄₃⋊C₄	—	2	1	no	—
134	Dic₄₃⋊C₄	—	4	2	no	—
135	C₄×Dic₄₃	—	2	1	no	—
136	#10621877	—	4	4	no	—
137	C₂×C₄₃⋊C₈	—	2	1	no	—
138	C₂×C₄₃⋊C₈	—	2	1	no	—
139	C₂×C₄₃⋊C₈	—	2	2	no	—
140	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
141	C₂×C₄₃⋊C₈	C₂×D₄	1	1	yes	characteristic
142	C₂³×C₄₃	—	2	1	no	—
143	C₂³×C₄₃	—	2	2	no	—
144	Q₈×C₄₃	—	4	2	no	—
145	Q₈×C₄₃	—	4	2	no	—
146	D₄×C₄₃	—	2	2	no	—
147	D₄×C₄₃	—	2	1	no	—
148	D₄×C₄₃	—	2	2	no	—
149	D₄×C₄₃	—	8	4	no	—
150	D₄×C₄₃	C₂×D₈	1	2	yes	—
151	C₂×C₄×C₄₃	—	2	2	no	—
152	C₂×C₄×C₄₃	—	2	1	no	—
153	C₂×C₄×C₄₃	—	4	2	no	—
154	C₂×C₄×C₄₃	—	4	1	no	—
155	C₂×C₄×C₄₃	—	4	2	no	—
156	C₂×C₄×C₄₃	—	4	1	no	—
157	C₂×C₄×C₄₃	—	4	2	no	—
158	C₂×C₄×C₄₃	—	8	4	no	—
159	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
160	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
162	C₂×Dic₄₃	—	2	1	no	—
163	C₂×Dic₄₃	—	2	1	no	—
164	C₂×Dic₄₃	—	4	2	no	—
165	C₂×Dic₄₃	—	8	4	no	—
166	Dic₈₆	—	2	1	no	—
167	Dic₈₆	—	4	2	no	—
168	Dic₈₆	—	4	2	no	—
169	Dic₈₆	—	8	4	no	—
170	Dic₈₆	C₂×D₈	1	2	yes	—
171	C₈×C₄₃	—	2	1	no	—
172	C₈×C₄₃	—	4	1	no	—
173	C₈×C₄₃	#137	1	2	yes	complemented
174	C₄₃⋊C₈	—	2	2	no	—
175	C₄₃⋊C₈	—	2	1	no	—
176	C₄₃⋊C₈	—	4	2	no	—
177	C₄₃⋊C₈	—	4	1	no	—
178	C₄₃⋊C₈	C₂×D₈	1	2	yes	—
179	#19887	—	43	1	no	maximal, Sylow
180	C₂²×C₄₃	—	2	2	no	—
181	C₂²×C₄₃	—	4	1	no	—
182	C₂²×C₄₃	—	4	1	no	—
183	C₂²×C₄₃	—	4	2	no	—
184	C₂²×C₄₃	—	4	2	no	—
185	C₂²×C₄₃	—	4	2	no	—
186	C₂²×C₄₃	D₄²	1	1	yes	characteristic
187	C₄×C₄₃	—	2	1	no	—
188	C₄×C₄₃	—	4	1	no	—
189	C₄×C₄₃	—	4	2	no	—
190	C₄×C₄₃	—	8	2	no	—
191	C₄×C₄₃	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₄₃	C₂×#137	1	1	yes	characteristic
193	C₄×C₄₃	#576	1	1	yes	characteristic
194	C₄×C₄₃	#578	1	1	yes	characteristic
195	Dic₄₃	—	2	2	no	—
196	Dic₄₃	—	4	1	no	—
197	Dic₄₃	—	8	2	no	—
198	Q₈×D₈	—	43	1	no	—
199	#3334	D₄₃	1	1	yes	characteristic
200	#3276	—	43	2	no	—
201	#1713	—	43	1	no	—
202	#1689	—	43	2	no	—
203	#1674	—	43	1	no	—
204	#1673	—	43	1	no	—
205	#1649	—	43	2	no	—
206	#1647	—	43	2	no	—
207	#1589	—	43	1	no	—
208	C₈×SD₁₆	—	43	1	no	—
209	C₂×C₄₃	—	4	2	no	—
210	C₂×C₄₃	—	8	2	no	—
211	C₂×C₄₃	#3276	1	1	yes	characteristic
212	C₂×C₄₃	D₄×D₈	1	1	yes	characteristic
213	C₂×C₄₃	#3293	1	1	yes	characteristic
214	C₂×#137	—	86	2	no	—
215	D₄×Q₈	—	43	2	no	—
216	#547	C₂×D₄₃	1	2	yes	—
217	#500	—	43	2	no	—
218	#500	—	43	1	no	—
219	#500	—	86	1	no	—
220	#500	C₂×D₄₃	1	1	yes	characteristic
221	#494	—	43	2	no	—
222	#480	—	2	2	no	—
223	#474	—	43	2	no	—
224	#474	—	86	2	no	—
225	#474	C₂×D₄₃	1	2	yes	—
226	#470	—	86	2	no	—
227	#466	—	2	1	no	—
228	#462	—	43	2	no	—
229	#461	—	43	2	no	—
230	#452	—	86	2	no	—
231	#451	—	86	2	no	—
232	C₈×Q₈	—	43	1	no	—
233	C₄×SD₁₆	—	43	1	no	—
234	C₄×D₈	—	86	1	no	—
235	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
236	#436	—	43	2	no	—
237	C₈×D₄	C₂×D₄₃	1	1	yes	characteristic
238	C₂×#107	—	2	1	no	—
239	#335	—	43	1	no	—
240	#334	—	43	1	no	—
241	#332	—	43	2	no	—
242	#332	—	43	2	no	—
243	Q₈⋊C₈	—	43	1	no	—
244	D₄⋊C₈	—	43	1	no	—
245	D₄⋊C₈	—	43	2	no	—
246	C₈²	—	43	1	no	—
247	C₄₃	#19887	1	1	yes	characteristic, complemented, Sylow
248	C₂×#41	—	2	2	no	—
249	C₂²×Q₈	—	86	2	no	—
250	#137	—	172	8	no	—
251	C₂×Q₁₆	—	86	2	no	—
252	C₂×Q₁₆	—	86	2	no	—
253	C₂×Q₁₆	—	86	2	no	—
254	C₂×SD₁₆	—	43	1	no	—
255	C₂×SD₁₆	—	86	2	no	—
256	C₂×D₈	C₂²⋊D₄₃	1	1	yes	characteristic
257	C₂×M₁₆	—	86	2	no	—
258	C₂²×C₈	—	2	1	no	—
259	C₄⋊Q₈	—	43	2	no	—
260	C₄⋊Q₈	—	86	2	no	—
261	C₄⋊Q₈	C₂²×D₄₃	1	2	yes	—
262	#123	—	4	2	no	—
263	C₂²⋊Q₈	—	2	2	no	—
264	C₂²⋊Q₈	—	86	2	no	—
265	C₂²⋊Q₈	—	172	2	no	—
266	#121	—	2	2	no	—
267	C₄×Q₈	—	43	1	no	—
268	C₄×D₄	—	86	2	no	—
269	C₄×D₄	C₂²×D₄₃	1	1	yes	characteristic
270	C₄×D₄	C₂²×D₄₃	1	2	yes	—
271	C₂×C₄⋊C₄	—	2	2	no	—
272	#107	—	2	1	no	—
273	#107	—	2	1	no	—
274	#107	—	2	2	no	—
275	#107	—	86	1	no	—
276	#107	—	86	1	no	—
277	#107	—	86	2	no	—
278	#107	—	172	2	no	—
279	#107	C₂²⋊D₄₃	1	1	yes	characteristic
280	#106	—	43	1	no	—
281	C₄⋊C₈	—	43	1	no	—
282	C₄⋊C₈	—	43	2	no	—
283	C₄⋊C₈	—	86	1	no	—
284	C₄⋊C₈	C₂²×D₄₃	1	1	yes	characteristic
285	#103	—	43	1	no	—
286	#103	—	86	2	no	—
287	#103	—	86	2	no	—
288	#102	—	2	2	no	—
289	#102	—	43	1	no	—
290	#102	—	86	2	no	—
291	#102	C₂²⋊D₄₃	1	2	yes	—
292	C₂²⋊C₈	—	2	1	no	—
293	C₂²⋊C₈	—	86	2	no	—
294	C₄×C₈	—	43	1	no	—
295	C₄×C₈	—	43	1	no	—
296	C₄×C₈	—	86	1	no	—
297	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
298	#41	—	4	4	no	—
299	#41	—	8	2	no	—
300	C₂×Q₈	—	2	2	no	—
301	C₂×Q₈	—	43	1	no	—
302	C₂×Q₈	—	86	2	no	—
303	C₂×Q₈	—	344	2	no	—
304	C₂×Q₈	—	172	2	no	—
305	C₂×Q₈	—	172	4	no	—
306	C₂×D₄	—	4	2	no	—
307	C₂×D₄	D₄×D₄₃	1	1	yes	characteristic
308	C₂×D₄	C₂×C₂²⋊D₄₃	1	2	yes	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	4	2	no	—
312	C₂²×C₄	—	172	2	no	—
313	Q₁₆	—	344	2	no	—
314	Q₁₆	—	172	4	no	—
315	Q₁₆	—	172	4	no	—
316	SD₁₆	—	43	4	no	—
317	SD₁₆	—	172	4	no	—
318	D₈	—	2	2	no	—
319	M₁₆	—	172	4	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	4	2	no	—
322	C₂×C₈	—	43	1	no	—
323	C₂×C₈	—	43	1	no	—
324	C₂×C₈	—	86	1	no	—
325	C₂×C₈	—	86	2	no	—
326	C₂×C₈	—	86	1	no	—
327	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
328	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
329	C₄⋊C₄	—	2	2	no	—
330	C₄⋊C₄	—	2	2	no	—
331	C₄⋊C₄	—	4	2	no	—
332	C₄⋊C₄	—	43	1	no	—
333	C₄⋊C₄	—	86	1	no	—
334	C₄⋊C₄	—	86	2	no	—
335	C₄⋊C₄	—	86	2	no	—
336	C₄⋊C₄	—	172	2	no	—
337	C₄⋊C₄	D₄×D₄₃	1	1	yes	characteristic
338	C₄⋊C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
339	C₂²⋊C₄	—	2	2	no	—
340	C₂²⋊C₄	—	2	1	no	—
341	C₂²⋊C₄	—	4	2	no	—
342	C₂²⋊C₄	—	172	2	no	—
343	C₄²	—	86	1	no	—
344	C₄²	C₂³×D₄₃	1	1	yes	characteristic
345	C₂³	—	2	1	no	—
346	C₂³	—	2	2	no	—
347	Q₈	—	4	2	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	43	2	no	—
350	Q₈	—	86	1	no	—
351	Q₈	—	344	4	no	—
352	Q₈	—	172	2	no	—
353	Q₈	—	172	2	no	—
354	D₄	—	2	2	no	—
355	D₄	—	2	1	no	—
356	D₄	—	2	2	no	—
357	D₄	—	8	4	no	—
358	D₄	D₈×D₄₃	1	2	yes	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	4	2	no	—
362	C₂×C₄	—	4	1	no	—
363	C₂×C₄	—	4	2	no	—
364	C₂×C₄	—	4	1	no	—
365	C₂×C₄	—	4	2	no	—
366	C₂×C₄	—	8	4	no	—
367	C₂×C₄	—	86	1	no	—
368	C₂×C₄	—	86	1	no	—
369	C₂×C₄	—	344	4	no	—
370	C₂×C₄	—	172	2	no	—
371	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
372	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
373	C₂×C₄	#13063370	1	1	yes	characteristic
374	C₈	—	2	1	no	—
375	C₈	—	4	1	no	—
376	C₈	—	43	2	no	—
377	C₈	—	86	2	no	—
378	C₈	—	86	1	no	—
379	C₈	—	172	2	no	—
380	C₈	—	172	1	no	—
381	C₈	#13063355	1	2	yes	complemented
382	C₂²	—	2	2	no	—
383	C₂²	—	4	1	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	4	2	no	—
386	C₂²	—	4	2	no	—
387	C₂²	—	4	2	no	—
388	C₂²	D₄×C₂²⋊D₄₃	1	1	yes	characteristic
389	C₄	—	2	1	no	—
390	C₄	—	4	2	no	—
391	C₄	—	4	1	no	—
392	C₄	—	8	2	no	—
393	C₄	—	86	2	no	—
394	C₄	—	344	2	no	—
395	C₄	—	172	1	no	—
396	C₄	C₂×D₈×D₄₃	1	1	yes	characteristic
397	C₄	C₂×#13063355	1	1	yes	characteristic
398	C₄	#14583700	1	1	yes	characteristic
399	C₄	#14583714	1	1	yes	characteristic
400	C₂	—	4	2	no	—
401	C₂	—	8	2	no	—
402	C₂	#19146237	1	1	yes	characteristic
403	C₂	D₈×C₂²⋊D₄₃	1	1	yes	characteristic
404	C₂	#19146397	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×#3334	C₂
socle	C₂²×C₄₃	D₄²
hypercenter	#3334	D₄₃
hypocenter	C₄₃	#19887
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #3334

Complements

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

N	H ≅ G/N	type
1. D₈×Dic₈₆	C₂	semidirect
2. #19146237	C₂	semidirect
3. #19134212	C₂	semidirect
4. #19134082	C₂	semidirect
5. #19133986	C₂	semidirect
6. #19133984	C₂	semidirect
7. #19133843	C₂	semidirect
8. #19133830	C₂	semidirect
9. #19133480	C₂	semidirect
10. C₈×#10621883	C₂	semidirect
129. #10621883	D₈	semidirect
173. C₈×C₄₃	#137	semidirect
247. C₄₃	#19887	semidirect
381. 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
#19146237	2
#19134212	2
#19134082	2
#19133986	2
#19133984	2
#19133843	2
#19133830	2
#19133480	2
C₈×#10621883	2
C₄₃×#3334	2
#19887	43