Subgroups — #25251108

Order 11008.

682 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063270	C₂	1	1	yes	maximal, characteristic, complemented
2	#19144250	C₂	1	1	yes	maximal, characteristic, complemented
3	#19144243	C₂	1	1	yes	maximal, characteristic
4	#19136708	C₂	1	4	yes	maximal, complemented
5	#19136691	C₂	1	4	yes	maximal, complemented
6	#19136379	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136316	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136312	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₄₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
11	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
12	#14583470	—	2	1	no	—
13	#14583463	—	2	1	no	—
14	C₂×#13063270	—	2	4	no	—
15	C₂×#13063270	C₂²	1	2	yes	—
16	C₂×#13063270	C₂²	1	2	yes	—
17	#14583157	—	2	2	no	—
18	#14583152	—	2	2	no	—
19	#14583140	C₂²	1	4	yes	complemented
20	#14583135	C₂²	1	4	yes	—
21	#14582937	C₂²	1	1	yes	characteristic
22	#14582937	C₂²	1	1	yes	characteristic
23	#14582909	—	2	1	no	—
24	#14582905	—	2	1	no	—
25	#14582727	C₂²	1	2	yes	complemented
26	#14582725	C₂²	1	2	yes	complemented
27	#14582704	C₂²	1	4	yes	complemented
28	#14582701	C₂²	1	4	yes	—
29	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#504	—	2	4	no	—
31	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
32	C₂×M₃₂×C₄₃	C₂²	1	2	yes	complemented
33	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
34	C₂²×C₁₆×C₄₃	C₂²	1	2	yes	complemented
35	C₂²×Dic₈₆	—	2	1	no	—
36	C₂²×D₁₇₂	—	2	1	no	—
37	C₂×Dic₁₇₂	—	2	2	no	—
38	C₂×Dic₁₇₂	—	2	2	no	—
39	C₂×Dic₁₇₂	—	2	1	no	—
40	C₂×Dic₁₇₂	C₂³	1	2	yes	—
41	C₂×D₃₄₄	—	2	2	no	—
42	C₂×D₃₄₄	—	2	2	no	—
43	C₂×D₃₄₄	—	2	1	no	—
44	C₂×D₃₄₄	C₂³	1	2	yes	—
45	#13063323	—	4	1	no	—
46	#13063321	—	4	1	no	—
47	#13063314	—	4	1	no	—
48	#13063312	—	4	1	no	—
49	#13063306	—	4	2	no	—
50	#13063304	—	2	4	no	—
51	#13063299	—	2	4	no	—
52	#13063293	—	4	2	no	—
53	#13063284	—	2	1	no	—
54	#13063284	—	2	1	no	—
55	#13063270	—	2	4	no	—
56	#13063270	—	2	4	no	—
57	#13063259	C₂³	1	2	yes	—
58	#13063259	C₂³	1	2	yes	—
59	#13063257	—	2	1	no	—
60	#13063257	—	2	1	no	—
61	#13063250	—	2	4	no	—
62	#13063247	—	2	4	no	—
63	C₂×#41×C₄₃	D₄	1	1	yes	characteristic
64	#131×C₄₃	—	2	2	no	—
65	#131×C₄₃	—	2	2	no	—
66	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
67	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
68	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
69	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
70	M₃₂×C₄₃	—	2	2	no	—
71	M₃₂×C₄₃	—	2	4	no	—
72	C₂×C₁₆×C₄₃	—	2	4	no	—
73	C₂×C₁₆×C₄₃	D₄	1	4	yes	complemented
74	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
75	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
76	C₂×C₁₆×C₄₃	C₂³	1	1	yes	characteristic
77	C₂³×D₄₃	—	4	1	no	—
78	#41×C₄₃	—	2	2	no	—
79	#41×C₄₃	—	2	2	no	—
80	C₂×Q₈×C₄₃	D₈	1	1	yes	characteristic
81	C₂×D₄×C₄₃	D₈	1	2	yes	—
82	C₂×D₄×C₄₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₂²⋊D₄₃	—	4	4	no	—
86	C₂²×Dic₄₃	—	4	1	no	—
87	C₂×D₁₇₂	—	2	2	no	—
88	C₂×D₁₇₂	—	4	1	no	—
89	C₂×D₁₇₂	—	4	2	no	—
90	C₂×D₁₇₂	—	4	2	no	—
91	C₂×Dic₈₆	—	2	2	no	—
92	C₂×Dic₈₆	—	4	2	no	—
93	C₂×Dic₈₆	—	4	2	no	—
94	C₂×Dic₈₆	—	4	1	no	—
95	M₁₆×C₄₃	—	2	2	no	—
96	M₁₆×C₄₃	—	2	4	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₄₃	#41	1	2	yes	—
103	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
104	C₂²⋊Dic₄₃	—	4	1	no	—
105	C₂²⋊Dic₄₃	—	4	1	no	—
106	C₂²⋊Dic₄₃	—	4	4	no	—
107	(C₂×D₄₃)⋊C₄	—	4	4	no	—
108	C₄⋊Dic₄₃	—	2	2	no	—
109	C₄⋊Dic₄₃	—	2	2	no	—
110	Dic₄₃⋊C₄	—	4	4	no	—
111	C₄×Dic₄₃	—	4	1	no	—
112	C₄×Dic₄₃	—	4	1	no	—
113	Dic₁₇₂	—	2	2	no	—
114	Dic₁₇₂	—	2	2	no	—
115	Dic₁₇₂	—	4	2	no	—
116	D₃₄₄	—	2	2	no	—
117	D₃₄₄	—	2	2	no	—
118	D₃₄₄	—	4	2	no	—
119	C₁₆×C₄₃	—	2	2	no	—
120	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
121	C₁₆×C₄₃	C₂×D₄	1	2	yes	—
122	C₂³×C₄₃	C₂×D₈	1	2	yes	—
123	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
124	C₂²×D₄₃	—	4	2	no	—
125	C₂²×D₄₃	—	8	2	no	—
126	C₂²×D₄₃	—	16	2	no	—
127	Q₈×C₄₃	—	2	2	no	—
128	D₄×C₄₃	—	2	4	no	—
129	D₄×C₄₃	—	2	2	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	#121	1	2	yes	—
133	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
135	C₂×C₄×C₄₃	#135	1	2	yes	—
136	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
137	C₂²⋊D₄₃	—	16	4	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	—	4	2	no	—
141	C₂×Dic₄₃	—	8	2	no	—
142	D₁₇₂	—	4	2	no	—
143	D₁₇₂	—	4	2	no	—
144	D₁₇₂	—	8	2	no	—
145	Dic₈₆	—	4	2	no	—
146	Dic₈₆	—	4	2	no	—
147	Dic₈₆	—	8	2	no	—
148	C₈×C₄₃	—	2	2	no	—
149	C₈×C₄₃	#121	1	2	yes	—
150	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
152	#33609	—	43	1	no	maximal, Sylow
153	C₂²×C₄₃	—	2	2	no	—
154	C₂²×C₄₃	—	2	4	no	—
155	C₂²×C₄₃	—	2	2	no	—
156	C₂²×C₄₃	#466	1	2	yes	—
157	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
158	C₂×D₄₃	—	8	4	no	—
159	C₂×D₄₃	—	8	2	no	—
160	C₂×D₄₃	—	16	4	no	—
161	C₄×C₄₃	—	2	2	no	—
162	C₄×C₄₃	#466	1	2	yes	—
163	C₄×C₄₃	C₂×#121	1	1	yes	characteristic
164	C₄×C₄₃	C₂×#135	1	1	yes	characteristic
165	Dic₄₃	—	8	2	no	—
166	Dic₄₃	—	8	2	no	—
167	Dic₄₃	—	8	2	no	—
168	C₂²×SD₃₂	—	43	1	no	—
169	C₂×#504	D₄₃	1	1	yes	characteristic, complemented
170	#3059	—	43	1	no	—
171	#3058	—	43	1	no	—
172	#2220	—	43	4	no	—
173	#2217	—	43	4	no	—
174	#2158	—	43	1	no	—
175	#2141	—	43	1	no	—
176	#2140	—	43	1	no	—
177	C₂×C₄₃	—	2	2	no	—
178	C₂×C₄₃	—	4	2	no	—
179	C₂×C₄₃	C₂×#466	1	1	yes	characteristic
180	C₂×C₄₃	#3416	1	2	yes	—
181	D₄₃	—	16	2	no	—
182	C₂²×Q₁₆	—	43	1	no	—
183	C₂²×D₈	—	43	1	no	—
184	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
185	#536	—	86	1	no	—
186	#535	—	86	1	no	—
187	C₂×SD₃₂	—	43	2	no	—
188	C₂×SD₃₂	—	43	2	no	—
189	C₂×SD₃₂	—	86	4	no	—
190	#504	—	2	4	no	—
191	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
192	C₂×M₃₂	C₂×D₄₃	1	2	yes	complemented
193	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
194	C₂²×C₁₆	C₂×D₄₃	1	2	yes	complemented
195	#470	—	86	2	no	—
196	#469	—	86	2	no	—
197	#467	—	43	4	no	—
198	#466	—	43	4	no	—
199	#427	—	43	1	no	—
200	#427	—	43	1	no	—
201	#419	—	86	1	no	—
202	#418	—	86	1	no	—
203	#367	—	43	2	no	—
204	#366	—	43	2	no	—
205	#358	—	43	4	no	—
206	#357	—	43	4	no	—
207	C₄₃	#33609	1	1	yes	characteristic, complemented, Sylow
208	C₂×#41	D₁₇₂	1	1	yes	characteristic
209	C₂²×Q₈	—	86	1	no	—
210	C₂²×D₄	—	86	1	no	—
211	C₂×Q₁₆	—	43	2	no	—
212	C₂×Q₁₆	—	86	1	no	—
213	C₂×Q₁₆	—	86	2	no	—
214	C₂×Q₁₆	—	86	2	no	—
215	C₂×D₈	—	43	2	no	—
216	C₂×D₈	—	86	1	no	—
217	C₂×D₈	—	86	2	no	—
218	C₂×D₈	—	86	2	no	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
222	C₂×M₁₆	D₁₇₂	1	2	yes	—
223	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
224	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
225	C₄⋊Q₈	—	172	1	no	—
226	#127	—	172	1	no	—
227	#124	—	172	1	no	—
228	#124	—	172	1	no	—
229	#123	—	172	2	no	—
230	C₂²⋊Q₈	—	86	4	no	—
231	#121	—	86	4	no	—
232	#120	—	172	2	no	—
233	#117	—	86	1	no	—
234	#117	—	86	1	no	—
235	SD₃₂	—	86	4	no	—
236	SD₃₂	—	86	4	no	—
237	M₃₂	—	2	2	no	—
238	M₃₂	—	2	4	no	—
239	C₂×C₁₆	—	2	4	no	—
240	C₂×C₁₆	C₂²⋊D₄₃	1	4	yes	complemented
241	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
242	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
243	C₂×C₁₆	C₂²×D₄₃	1	1	yes	characteristic
244	#107	—	43	2	no	—
245	#107	—	43	2	no	—
246	#106	—	86	1	no	—
247	#106	—	86	1	no	—
248	#103	—	86	4	no	—
249	#102	—	86	4	no	—
250	C₂⁴	—	172	1	no	—
251	#41	—	2	2	no	—
252	#41	—	2	2	no	—
253	C₂×Q₈	—	86	2	no	—
254	C₂×Q₈	—	172	2	no	—
255	C₂×Q₈	—	172	2	no	—
256	C₂×Q₈	—	172	1	no	—
257	C₂×Q₈	D₃₄₄	1	1	yes	characteristic
258	C₂×D₄	—	86	2	no	—
259	C₂×D₄	—	172	4	no	—
260	C₂×D₄	—	172	2	no	—
261	C₂×D₄	—	172	1	no	—
262	C₂×D₄	—	172	2	no	—
263	C₂×D₄	D₃₄₄	1	2	yes	—
264	C₂×D₄	D₃₄₄	1	1	yes	characteristic
265	C₂²×C₄	—	172	1	no	—
266	C₂²×C₄	C₂×D₁₇₂	1	2	yes	—
267	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
268	Q₁₆	—	86	2	no	—
269	Q₁₆	—	86	2	no	—
270	Q₁₆	—	172	2	no	—
271	D₈	—	86	2	no	—
272	D₈	—	86	2	no	—
273	D₈	—	172	2	no	—
274	M₁₆	—	2	2	no	—
275	M₁₆	—	2	4	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	—	2	2	no	—
278	C₂×C₈	C₂×D₁₇₂	1	2	yes	—
279	C₂×C₈	C₂×D₁₇₂	1	1	yes	characteristic
280	C₂×C₈	#10621898	1	2	yes	—
281	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
282	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
283	C₄⋊C₄	—	86	2	no	—
284	C₄⋊C₄	—	86	2	no	—
285	C₄⋊C₄	—	172	4	no	—
286	C₂²⋊C₄	—	172	1	no	—
287	C₂²⋊C₄	—	172	1	no	—
288	C₂²⋊C₄	—	172	4	no	—
289	C₂²⋊C₄	—	172	4	no	—
290	C₄²	—	172	1	no	—
291	C₄²	—	172	1	no	—
292	C₁₆	—	2	2	no	—
293	C₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
294	C₁₆	C₂×C₂²⋊D₄₃	1	2	yes	—
295	C₂³	—	344	2	no	—
296	C₂³	—	172	2	no	—
297	C₂³	—	688	2	no	—
298	C₂³	C₂×D₃₄₄	1	2	yes	—
299	C₂³	C₂×D₃₄₄	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	344	2	no	—
302	Q₈	—	172	2	no	—
303	Q₈	—	172	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	344	2	no	—
307	D₄	—	172	2	no	—
308	D₄	—	172	2	no	—
309	D₄	—	688	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	344	2	no	—
313	C₂×C₄	—	172	2	no	—
314	C₂×C₄	—	172	2	no	—
315	C₂×C₄	—	172	2	no	—
316	C₂×C₄	#13063299	1	2	yes	—
317	C₂×C₄	C₂×D₃₄₄	1	2	yes	—
318	C₂×C₄	C₂×D₃₄₄	1	1	yes	characteristic
319	C₂×C₄	#13063347	1	2	yes	—
320	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	#13063299	1	2	yes	—
323	C₈	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
324	C₈	C₂×#10621898	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	4	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	344	4	no	—
329	C₂²	—	344	2	no	—
330	C₂²	—	688	4	no	—
331	C₂²	#14583135	1	2	yes	—
332	C₂²	C₂²×D₃₄₄	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	344	2	no	—
335	C₄	—	344	2	no	—
336	C₄	—	344	2	no	—
337	C₄	#14583135	1	2	yes	—
338	C₄	C₂×#13063299	1	1	yes	characteristic
339	C₄	C₂×#13063347	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	688	2	no	—
343	C₂	C₂×#14583135	1	1	yes	characteristic
344	C₂	#19147743	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₃₄₄
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#504	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	#19144243	C₂
hypercenter	C₂×#504	D₄₃
hypocenter	C₄₃	#33609
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×#41 ◁ C₂×#131 ◁ C₂×#504

Complements

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

N	H ≅ G/N	type
1. C₂²×#13063270	C₂	semidirect
2. #19144250	C₂	semidirect
4. #19136708	C₂	semidirect
5. #19136691	C₂	semidirect
6. #19136379	C₂	semidirect
7. #19136316	C₂	semidirect
8. #19136312	C₂	semidirect
9. C₂×C₄₃×#504	C₂	semidirect
19. #14583140	C₂²	semidirect
25. #14582727	C₂²	semidirect
26. #14582725	C₂²	semidirect
27. #14582704	C₂²	semidirect
32. C₂×M₃₂×C₄₃	C₂²	semidirect
34. C₂²×C₁₆×C₄₃	C₂²	semidirect
73. C₂×C₁₆×C₄₃	D₄	semidirect
169. C₂×#504	D₄₃	semidirect
192. C₂×M₃₂	C₂×D₄₃	semidirect
194. C₂²×C₁₆	C₂×D₄₃	semidirect
207. C₄₃	#33609	semidirect
240. C₂×C₁₆	C₂²⋊D₄₃	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063270	2
#19144250	2
#19144243	2
#19136708	2
#19136691	2
#19136379	2
#19136316	2
#19136312	2
C₂×C₄₃×#504	2
#33609	43