Subgroups — #25254240

Order 11008.

562 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147569	C₂	1	1	yes	maximal, characteristic, complemented
2	D₈×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136819	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136817	C₂	1	2	yes	maximal, complemented
5	#19136732	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136731	C₂	1	2	yes	maximal, complemented
7	#19136630	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136628	C₂	1	2	yes	maximal, complemented
9	C₁₆×Dic₈₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#19136439	C₂	1	2	yes	maximal, complemented
11	C₄×D₁₆×C₄₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#14583570	—	2	1	no	—
13	D₄×Dic₈₆	—	2	1	no	—
14	C₂×#13063266	C₂²	1	2	yes	—
15	#14583304	C₂²	1	1	yes	characteristic
16	#14583303	C₂²	1	2	yes	—
17	#14583276	C₂²	1	1	yes	characteristic
18	#14583274	C₂²	1	2	yes	—
19	#14583179	—	2	1	no	—
20	#14583177	—	2	2	no	—
21	#14583094	—	2	1	no	—
22	#14583093	—	2	2	no	—
23	C₈×Dic₈₆	C₂²	1	1	yes	characteristic
24	D₈×Dic₄₃	C₂²	1	2	yes	—
25	#14582981	C₂²	1	2	yes	—
26	#14582724	C₂²	1	2	yes	—
27	C₄⋊(C₄₃⋊C₁₆)	C₂²	1	1	yes	characteristic
28	Dic₄₃⋊C₁₆	C₂²	1	2	yes	—
29	#14582701	C₂²	1	2	yes	—
30	#14582701	C₂²	1	2	yes	—
31	#14582700	C₂²	1	2	yes	—
32	#14582699	C₂²	1	2	yes	—
33	C₁₆×Dic₄₃	C₂²	1	2	yes	—
34	C₂×D₁₆×C₄₃	C₂²	1	1	yes	characteristic
35	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
36	C₄×D₈×C₄₃	C₂²	1	1	yes	characteristic
37	C₄₃×#366	C₂²	1	1	yes	characteristic
38	C₄₃×#357	C₂²	1	1	yes	characteristic
39	C₄₃×#357	C₂²	1	1	yes	characteristic
40	C₄×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
41	C₂×#10621898	—	4	1	no	—
42	C₂²×Dic₈₆	—	4	1	no	—
43	C₂×D₃₄₄	—	2	1	no	—
44	C₂×D₃₄₄	C₂³	1	2	yes	—
45	C₂×#10621882	—	2	2	no	—
46	#13063325	—	2	1	no	—
47	#13063324	—	2	2	no	—
48	#13063322	—	2	1	no	—
49	#13063321	—	2	2	no	—
50	#13063304	—	4	1	no	—
51	#13063302	—	4	2	no	—
52	#13063299	—	4	1	no	—
53	#13063298	—	4	2	no	—
54	C₄×Dic₈₆	D₄	1	1	yes	characteristic
55	D₄×Dic₄₃	—	2	2	no	—
56	#13063287	—	2	2	no	—
57	#13063266	—	2	4	no	—
58	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
59	#13063259	—	2	1	no	—
60	#13063259	C₂³	1	2	yes	—
61	#13063258	—	2	2	no	—
62	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
63	Dic₄₃⋊C₈	D₄	1	2	yes	—
64	#13063246	—	2	2	no	—
65	#13063245	—	2	2	no	—
66	C₈×Dic₄₃	C₂³	1	2	yes	—
67	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×D₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄×D₄×C₄₃	—	2	1	no	—
70	C₄×D₄×C₄₃	—	2	1	no	—
71	D₁₆×C₄₃	—	2	2	no	—
72	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
73	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
74	#107×C₄₃	C₂³	1	1	yes	characteristic
75	#107×C₄₃	C₂³	1	1	yes	characteristic
76	#102×C₄₃	—	2	1	no	—
77	#102×C₄₃	—	2	1	no	—
78	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₄₃	—	2	1	no	—
80	C₂×D₄×C₄₃	—	2	1	no	—
81	C₂²×C₄×C₄₃	—	4	1	no	—
82	C₂²×C₄×C₄₃	—	4	1	no	—
83	C₂×C₂²⋊D₄₃	—	4	2	no	—
84	C₂×C₂²⋊D₄₃	—	4	2	no	—
85	C₂²×Dic₄₃	—	4	2	no	—
86	#10621898	—	8	4	no	—
87	C₂×D₁₇₂	—	2	2	no	—
88	C₂×D₁₇₂	—	4	1	no	—
89	C₂×D₁₇₂	—	4	1	no	—
90	C₂×C₄×D₄₃	—	4	2	no	—
91	C₂×Dic₈₆	—	4	1	no	—
92	C₂×Dic₈₆	—	4	2	no	—
93	C₂×Dic₈₆	—	8	4	no	—
94	C₂×Dic₈₆	D₈	1	1	yes	characteristic
95	D₈×C₄₃	—	2	1	no	—
96	D₈×C₄₃	—	2	1	no	—
97	D₈×C₄₃	—	2	1	no	—
98	D₈×C₄₃	#41	1	2	yes	complemented
99	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
101	C₄⋊C₄×C₄₃	—	2	1	no	—
102	C₄⋊C₄×C₄₃	—	2	1	no	—
103	C₂²⋊C₄×C₄₃	—	4	1	no	—
104	C₂²⋊C₄×C₄₃	—	4	1	no	—
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₂²⋊Dic₄₃	—	4	2	no	—
107	#10621882	—	8	2	no	—
108	(C₂×D₄₃)⋊C₄	—	4	2	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	4	1	no	—
111	C₄⋊Dic₄₃	D₈	1	1	yes	characteristic
112	Dic₄₃⋊C₄	—	4	2	no	—
113	Dic₄₃⋊C₄	D₈	1	2	yes	—
114	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
115	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
116	D₃₄₄	—	2	2	no	—
117	D₃₄₄	—	2	2	no	—
118	D₃₄₄	—	8	1	no	—
119	C₁₆×C₄₃	—	2	1	no	—
120	C₁₆×C₄₃	—	2	1	no	—
121	C₄₃⋊C₁₆	—	2	2	no	—
122	C₂³×C₄₃	—	4	1	no	—
123	C₂³×C₄₃	—	4	1	no	—
124	C₂²×D₄₃	—	4	2	no	—
125	D₄×C₄₃	—	2	2	no	—
126	D₄×C₄₃	—	4	1	no	—
127	D₄×C₄₃	—	4	1	no	—
128	D₄×C₄₃	—	4	1	no	—
129	C₂×C₄×C₄₃	—	4	1	no	—
130	C₂×C₄×C₄₃	—	4	1	no	—
131	C₂×C₄×C₄₃	—	8	2	no	—
132	C₂×C₄×C₄₃	—	8	2	no	—
133	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
136	C₂²⋊D₄₃	—	8	4	no	—
137	C₂²⋊D₄₃	—	16	2	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	—	8	4	no	—
140	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
141	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
142	D₁₇₂	—	4	2	no	—
143	D₁₇₂	—	4	2	no	—
144	D₁₇₂	—	8	2	no	—
145	D₁₇₂	—	16	1	no	—
146	C₄×D₄₃	—	8	4	no	—
147	Dic₈₆	—	8	2	no	—
148	Dic₈₆	—	8	4	no	—
149	Dic₈₆	D₁₆	1	4	yes	complemented
150	C₈×C₄₃	—	2	1	no	—
151	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₈×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
153	C₄₃⋊C₈	—	2	2	no	—
154	#33925	—	43	1	no	maximal, Sylow
155	C₂²×C₄₃	—	4	2	no	—
156	C₂²×C₄₃	—	8	1	no	—
157	C₂²×C₄₃	—	8	1	no	—
158	C₂²×C₄₃	—	8	1	no	—
159	C₂²×C₄₃	—	8	1	no	—
160	C₂²×C₄₃	—	8	1	no	—
161	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
162	C₂×D₄₃	—	8	2	no	—
163	C₂×D₄₃	—	8	2	no	—
164	C₂×D₄₃	—	8	2	no	—
165	C₄×C₄₃	—	2	1	no	—
166	C₄×C₄₃	—	8	1	no	—
167	C₄×C₄₃	—	8	1	no	—
168	C₄×C₄₃	C₂×D₁₆	1	2	yes	—
169	C₄×C₄₃	#548	1	1	yes	characteristic
170	C₄×C₄₃	#577	1	1	yes	characteristic
171	Dic₄₃	—	2	2	no	—
172	Dic₄₃	—	8	2	no	—
173	Dic₄₃	C₂×D₁₆	1	4	yes	—
174	#3391	—	43	1	no	—
175	Q₈×D₈	—	43	1	no	—
176	#2240	—	43	2	no	—
177	#2240	—	43	1	no	—
178	#2224	—	43	1	no	—
179	#2224	—	43	2	no	—
180	#2206	—	43	2	no	—
181	#2206	—	43	1	no	—
182	Q₈×C₁₆	—	43	1	no	—
183	C₄×D₁₆	—	43	2	no	—
184	C₄×D₁₆	D₄₃	1	1	yes	characteristic, complemented
185	C₂×C₄₃	—	8	2	no	—
186	C₂×C₄₃	—	16	1	no	—
187	C₂×C₄₃	#3358	1	1	yes	characteristic
188	C₂×C₄₃	C₂²×D₁₆	1	1	yes	characteristic
189	C₂×C₄₃	#3416	1	1	yes	characteristic
190	D₄₃	—	16	2	no	—
191	#550	—	86	1	no	—
192	D₄×Q₈	—	86	1	no	—
193	C₂×D₁₆	—	43	2	no	—
194	C₂×D₁₆	C₂×D₄₃	1	1	yes	characteristic
195	#500	—	43	2	no	—
196	#500	—	43	1	no	—
197	#493	—	43	2	no	—
198	#493	—	43	1	no	—
199	#474	—	86	2	no	—
200	#474	—	86	1	no	—
201	#459	—	86	2	no	—
202	#459	—	86	1	no	—
203	C₈×Q₈	—	43	1	no	—
204	C₄×D₈	—	43	2	no	—
205	C₄×D₈	—	43	2	no	—
206	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
207	C₄×D₈	C₂×D₄₃	1	1	yes	characteristic
208	#366	—	43	2	no	—
209	#366	C₂×D₄₃	1	1	yes	characteristic
210	C₄⋊C₁₆	—	43	2	no	—
211	C₄⋊C₁₆	—	43	1	no	—
212	#357	—	43	2	no	—
213	#357	—	43	2	no	—
214	#357	—	43	2	no	—
215	#357	—	43	2	no	—
216	#357	C₂×D₄₃	1	1	yes	characteristic
217	#357	C₂×D₄₃	1	1	yes	characteristic
218	C₄×C₁₆	—	43	2	no	—
219	C₄×C₁₆	C₂×D₄₃	1	1	yes	characteristic
220	C₄₃	#33925	1	1	yes	characteristic, complemented, Sylow
221	C₂×#41	—	172	1	no	—
222	C₂²×Q₈	—	172	1	no	—
223	C₂×D₈	—	43	2	no	—
224	C₂×D₈	—	86	2	no	—
225	C₂×D₈	—	86	1	no	—
226	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
227	C₂×D₈	C₂²×D₄₃	1	1	yes	characteristic
228	C₄⋊Q₈	—	86	2	no	—
229	C₄⋊Q₈	—	86	1	no	—
230	#127	—	86	2	no	—
231	#127	—	86	1	no	—
232	C₂²⋊Q₈	—	172	2	no	—
233	C₂²⋊Q₈	—	172	1	no	—
234	#121	—	172	2	no	—
235	#121	—	172	1	no	—
236	C₄×Q₈	—	43	1	no	—
237	C₄×D₄	—	2	1	no	—
238	C₄×D₄	—	2	1	no	—
239	C₄×D₄	—	86	2	no	—
240	C₄×D₄	—	86	2	no	—
241	D₁₆	—	2	2	no	—
242	D₁₆	—	86	4	no	—
243	C₂×C₁₆	—	43	2	no	—
244	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
245	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
246	#107	—	43	2	no	—
247	#107	—	86	2	no	—
248	#107	—	86	1	no	—
249	#107	C₂²×D₄₃	1	1	yes	characteristic
250	#107	C₂²×D₄₃	1	1	yes	characteristic
251	C₄⋊C₈	—	43	2	no	—
252	C₄⋊C₈	—	43	1	no	—
253	#102	—	2	1	no	—
254	#102	—	2	1	no	—
255	#102	—	86	2	no	—
256	#102	—	86	2	no	—
257	C₄×C₈	—	43	2	no	—
258	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
259	#41	—	344	4	no	—
260	C₂×Q₈	—	43	1	no	—
261	C₂×Q₈	—	344	4	no	—
262	C₂×Q₈	—	172	2	no	—
263	C₂×Q₈	—	172	1	no	—
264	C₂×D₄	—	2	1	no	—
265	C₂×D₄	—	2	1	no	—
266	C₂×D₄	—	86	2	no	—
267	C₂×D₄	—	172	2	no	—
268	C₂×D₄	—	172	2	no	—
269	C₂×D₄	—	172	1	no	—
270	C₂×D₄	—	172	1	no	—
271	C₂²×C₄	—	4	1	no	—
272	C₂²×C₄	—	4	1	no	—
273	C₂²×C₄	—	172	2	no	—
274	C₂²×C₄	—	172	2	no	—
275	D₈	—	2	1	no	—
276	D₈	—	2	1	no	—
277	D₈	—	2	1	no	—
278	D₈	—	86	2	no	—
279	D₈	—	86	2	no	—
280	D₈	—	344	1	no	—
281	D₈	—	344	2	no	—
282	D₈	#10621898	1	2	yes	complemented
283	C₂×C₈	—	43	2	no	—
284	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
285	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
286	C₄⋊C₄	—	2	1	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	43	2	no	—
289	C₄⋊C₄	—	43	1	no	—
290	C₄⋊C₄	—	86	2	no	—
291	C₄⋊C₄	—	172	2	no	—
292	C₄⋊C₄	—	172	1	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	1	no	—
295	C₂²⋊C₄	—	172	2	no	—
296	C₂²⋊C₄	—	172	2	no	—
297	C₄²	—	43	2	no	—
298	C₄²	D₄×D₄₃	1	1	yes	characteristic
299	C₁₆	—	2	1	no	—
300	C₁₆	—	2	1	no	—
301	C₁₆	—	86	2	no	—
302	C₂³	—	4	1	no	—
303	C₂³	—	4	1	no	—
304	C₂³	—	172	2	no	—
305	Q₈	—	43	4	no	—
306	Q₈	—	344	4	no	—
307	Q₈	—	344	2	no	—
308	D₄	—	2	2	no	—
309	D₄	—	4	1	no	—
310	D₄	—	4	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	344	4	no	—
313	D₄	—	344	2	no	—
314	D₄	—	172	2	no	—
315	D₄	—	172	2	no	—
316	D₄	—	688	1	no	—
317	D₄	—	688	2	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	8	2	no	—
321	C₂×C₄	—	8	2	no	—
322	C₂×C₄	—	43	2	no	—
323	C₂×C₄	—	43	2	no	—
324	C₂×C₄	—	344	4	no	—
325	C₂×C₄	—	344	4	no	—
326	C₂×C₄	—	172	2	no	—
327	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
328	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
329	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
330	C₈	—	2	1	no	—
331	C₈	—	86	2	no	—
332	C₈	C₂×#10621898	1	1	yes	characteristic
333	C₈	#13063372	1	1	yes	characteristic
334	C₂²	—	4	2	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	1	no	—
337	C₂²	—	8	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	344	2	no	—
341	C₂²	—	344	2	no	—
342	C₂²	—	344	2	no	—
343	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	—	43	4	no	—
348	C₄	—	86	2	no	—
349	C₄	—	344	2	no	—
350	C₄	D₁₆×D₄₃	1	2	yes	—
351	C₄	#14583558	1	1	yes	characteristic
352	C₄	#14583706	1	1	yes	characteristic
353	C₂	—	8	2	no	—
354	C₂	—	16	1	no	—
355	C₂	—	688	2	no	—
356	C₂	#19147228	1	1	yes	characteristic
357	C₂	C₂×D₁₆×D₄₃	1	1	yes	characteristic
358	C₂	#19147738	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₄₃
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄×D₁₆×C₄₃	C₂
socle	C₂²×C₄₃	C₂²×D₈
hypercenter	C₄×D₁₆	D₄₃
hypocenter	C₄₃	#33925
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×D₁₆

Complements

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

N	H ≅ G/N	type
1. #19147569	C₂	semidirect
2. D₈×Dic₈₆	C₂	semidirect
3. #19136819	C₂	semidirect
4. #19136817	C₂	semidirect
5. #19136732	C₂	semidirect
6. #19136731	C₂	semidirect
7. #19136630	C₂	semidirect
8. #19136628	C₂	semidirect
9. C₁₆×Dic₈₆	C₂	semidirect
10. #19136439	C₂	semidirect
11. C₄×D₁₆×C₄₃	C₂	semidirect
98. D₈×C₄₃	#41	semidirect
149. Dic₈₆	D₁₆	semidirect
184. C₄×D₁₆	D₄₃	semidirect
220. C₄₃	#33925	semidirect
282. D₈	#10621898	semidirect

Maximal subgroups

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

subgroup	index
#19147569	2
D₈×Dic₈₆	2
#19136819	2
#19136817	2
#19136732	2
#19136731	2
#19136630	2
#19136628	2
C₁₆×Dic₈₆	2
#19136439	2
C₄×D₁₆×C₄₃	2
#33925	43