Subgroups — #25251499

Order 11008.

566 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19144239	C₂	1	1	yes	maximal, characteristic
2	#19143203	C₂	1	1	yes	maximal, characteristic, complemented
3	#19136808	C₂	1	8	yes	maximal, complemented
4	#19136369	C₂	1	1	yes	maximal, characteristic, complemented
5	#19136368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14582712	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄₃×#361	C₂	1	1	yes	maximal, characteristic, complemented
9	#14583665	—	2	1	no	—
10	C₂×#13063331	C₂²	1	1	yes	characteristic
11	#14583466	—	2	1	no	—
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	#14583149	C₂²	1	4	yes	—
14	#14583132	—	2	2	no	—
15	#14582947	—	2	1	no	—
16	#14582946	C₂²	1	4	yes	complemented
17	C₂×#13063261	C₂²	1	1	yes	characteristic
18	#108×D₄₃	—	2	2	no	—
19	#14582937	C₂²	1	1	yes	characteristic
20	#14582904	—	2	1	no	—
21	#14582723	C₂²	1	4	yes	complemented
22	#14582722	C₂²	1	4	yes	complemented
23	#14582712	C₂²	1	4	yes	—
24	#14582680	C₂²	1	4	yes	complemented
25	C₂²×D₈×C₄₃	C₂²	1	1	yes	characteristic
26	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#108×C₄₃	C₂²	1	1	yes	characteristic
28	C₄₃×#361	C₂²	1	4	yes	—
29	C₂×#10621898	D₄	1	1	yes	characteristic
30	#13063333	—	4	2	no	—
31	#13063331	—	2	1	no	—
32	#13063331	—	2	1	no	—
33	#13063331	—	2	1	no	—
34	#13063331	—	2	1	no	—
35	#13063331	—	4	2	no	—
36	C₂×#10621877	—	2	1	no	—
37	C₂×#10621877	D₄	1	1	yes	characteristic
38	C₂×#10621872	C₂³	1	2	yes	—
39	M₁₆×D₄₃	—	2	4	no	—
40	C₂×C₈×D₄₃	D₄	1	2	yes	—
41	#13063321	—	4	1	no	—
42	#13063312	—	4	1	no	—
43	#13063296	—	2	4	no	—
44	#13063294	—	4	2	no	—
45	#13063284	—	2	1	no	—
46	#13063265	—	2	1	no	—
47	#13063265	C₂³	1	2	yes	—
48	C₂×C₄₃⋊C₁₆	—	2	1	no	—
49	#13063261	—	2	1	no	—
50	#13063261	C₂³	1	2	yes	—
51	#13063260	—	2	4	no	—
52	#13063259	—	2	1	no	—
53	#13063257	C₂³	1	2	yes	—
54	#13063246	—	2	4	no	—
55	C₂²×D₄×C₄₃	—	2	1	no	—
56	C₂×D₈×C₄₃	—	2	1	no	—
57	C₂×D₈×C₄₃	—	2	1	no	—
58	C₂×D₈×C₄₃	—	2	1	no	—
59	C₂×D₈×C₄₃	—	2	1	no	—
60	C₂×D₈×C₄₃	—	2	1	no	—
61	C₂×D₈×C₄₃	C₂³	1	2	yes	—
62	C₂×M₁₆×C₄₃	—	2	1	no	—
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	M₃₂×C₄₃	—	2	1	no	—
65	M₃₂×C₄₃	C₂³	1	2	yes	—
66	C₂×C₁₆×C₄₃	—	2	1	no	—
67	#108×C₄₃	—	2	1	no	—
68	#108×C₄₃	C₂³	1	2	yes	—
69	C₂⁴×C₄₃	—	4	1	no	—
70	C₂×D₄×C₄₃	—	2	2	no	—
71	C₂×D₄×C₄₃	—	4	1	no	—
72	C₂×D₄×C₄₃	—	4	1	no	—
73	C₂×D₄×C₄₃	—	4	1	no	—
74	C₂×D₄×C₄₃	—	4	1	no	—
75	C₂×D₄×C₄₃	—	4	1	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₂²⋊D₄₃	—	4	4	no	—
78	C₂×C₂²⋊D₄₃	SD₁₆	1	2	yes	—
79	#10621898	—	2	1	no	—
80	#10621898	—	2	1	no	—
81	#10621898	—	2	1	no	—
82	#10621898	—	2	1	no	—
83	C₂×D₁₇₂	SD₁₆	1	1	yes	characteristic
84	C₂×C₄×D₄₃	C₂×D₄	1	2	yes	—
85	C₂×Dic₈₆	SD₁₆	1	1	yes	characteristic
86	D₈×C₄₃	—	2	2	no	—
87	D₈×C₄₃	—	2	2	no	—
88	D₈×C₄₃	—	4	1	no	—
89	D₈×C₄₃	—	4	1	no	—
90	M₁₆×C₄₃	—	2	2	no	—
91	M₁₆×C₄₃	—	4	1	no	—
92	C₂×C₈×C₄₃	—	4	1	no	—
93	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
95	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
96	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
97	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
98	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
99	C₂²⋊Dic₄₃	—	4	4	no	—
100	C₂²⋊Dic₄₃	—	4	1	no	—
101	C₄⋊Dic₄₃	—	2	2	no	—
102	C₄×Dic₄₃	—	4	1	no	—
103	#10621877	—	2	1	no	—
104	#10621877	—	2	1	no	—
105	#10621877	—	2	2	no	—
106	#10621877	—	4	1	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
109	#10621872	—	2	2	no	—
110	#10621872	—	2	2	no	—
111	#10621872	—	4	4	no	—
112	C₈×D₄₃	—	2	2	no	—
113	C₈×D₄₃	—	2	2	no	—
114	C₈×D₄₃	—	4	4	no	—
115	C₁₆×C₄₃	—	2	2	no	—
116	C₄₃⋊C₁₆	—	2	2	no	—
117	C₂³×C₄₃	—	4	2	no	—
118	C₂³×C₄₃	—	8	1	no	—
119	C₂³×C₄₃	—	8	1	no	—
120	C₂³×C₄₃	—	16	2	no	—
121	C₂³×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
122	C₂²×D₄₃	C₂×SD₁₆	1	2	yes	—
123	D₄×C₄₃	—	4	2	no	—
124	D₄×C₄₃	—	4	2	no	—
125	D₄×C₄₃	—	8	1	no	—
126	D₄×C₄₃	—	8	1	no	—
127	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
128	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
133	C₂²⋊D₄₃	—	2	2	no	—
134	C₂²⋊D₄₃	—	2	2	no	—
135	C₂²⋊D₄₃	—	16	4	no	—
136	C₂×Dic₄₃	—	4	2	no	—
137	C₂×Dic₄₃	C₂×SD₁₆	1	2	yes	—
138	D₁₇₂	—	2	1	no	—
139	D₁₇₂	—	2	1	no	—
140	C₄×D₄₃	—	2	2	no	—
141	C₄×D₄₃	—	2	2	no	—
142	Dic₈₆	—	2	1	no	—
143	Dic₈₆	—	2	1	no	—
144	C₈×C₄₃	—	4	2	no	—
145	C₈×C₄₃	#123	1	1	yes	characteristic
146	C₈×C₄₃	#123	1	1	yes	characteristic
147	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
149	C₄₃⋊C₈	—	2	2	no	—
150	C₄₃⋊C₈	—	4	2	no	—
151	#33657	—	43	1	no	maximal, Sylow
152	C₂²×C₄₃	—	4	1	no	—
153	C₂²×C₄₃	—	8	4	no	—
154	C₂²×C₄₃	—	8	2	no	—
155	C₂²×C₄₃	—	16	2	no	—
156	C₂²×C₄₃	—	16	2	no	—
157	C₂²×C₄₃	#481	1	1	yes	characteristic
158	C₂²×C₄₃	#481	1	1	yes	characteristic
159	C₂²×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
160	C₂×D₄₃	—	2	2	no	—
161	C₂×D₄₃	—	4	2	no	—
162	C₄×C₄₃	#481	1	1	yes	characteristic
163	C₄×C₄₃	#481	1	1	yes	characteristic
164	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
165	C₄×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
166	Dic₄₃	—	2	2	no	—
167	Dic₄₃	—	8	2	no	—
168	#3058	—	43	1	no	—
169	#2916	—	43	1	no	—
170	#2238	—	43	8	no	—
171	#2155	—	43	1	no	—
172	#2155	—	43	1	no	—
173	C₂×#361	—	43	1	no	—
174	C₂×#361	D₄₃	1	1	yes	characteristic, complemented
175	#2115	—	43	1	no	—
176	C₂×C₄₃	—	2	1	no	—
177	C₂×C₄₃	—	2	1	no	—
178	C₂×C₄₃	—	2	1	no	—
179	C₂×C₄₃	—	16	2	no	—
180	C₂×C₄₃	C₂×#481	1	1	yes	characteristic
181	D₄₃	—	4	2	no	—
182	C₂²×D₈	C₂×D₄₃	1	1	yes	characteristic
183	#568	—	86	1	no	—
184	C₂×#131	—	43	1	no	—
185	#535	—	86	1	no	—
186	C₂×M₃₂	—	43	1	no	—
187	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
188	#469	—	43	4	no	—
189	#466	—	86	2	no	—
190	#430	—	43	4	no	—
191	#430	—	86	1	no	—
192	C₂×#108	—	43	1	no	—
193	C₂×#108	—	86	2	no	—
194	C₂×#108	C₂×D₄₃	1	1	yes	characteristic
195	#427	—	43	1	no	—
196	#418	—	86	1	no	—
197	#365	—	43	4	no	—
198	#365	—	43	4	no	—
199	#361	—	43	4	no	—
200	#361	C₂×D₄₃	1	4	yes	—
201	#349	—	43	4	no	—
202	C₄₃	#33657	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	—	43	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂×D₈	—	2	1	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	—	2	1	no	—
208	C₂×D₈	—	2	1	no	—
209	C₂×D₈	—	2	1	no	—
210	C₂×D₈	C₂²×D₄₃	1	2	yes	—
211	#131	—	86	1	no	—
212	#131	—	86	1	no	—
213	#131	—	86	1	no	—
214	#131	—	86	1	no	—
215	#131	—	172	2	no	—
216	#131	—	172	2	no	—
217	C₂×M₁₆	—	2	1	no	—
218	C₂×M₁₆	—	43	1	no	—
219	C₂×M₁₆	—	43	2	no	—
220	C₂×M₁₆	—	86	4	no	—
221	C₂×M₁₆	—	86	1	no	—
222	C₂²×C₈	—	43	2	no	—
223	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
224	#127	—	172	1	no	—
225	#124	—	172	1	no	—
226	#121	—	86	4	no	—
227	#120	—	172	2	no	—
228	#117	—	86	1	no	—
229	M₃₂	—	2	1	no	—
230	M₃₂	—	43	2	no	—
231	M₃₂	—	86	1	no	—
232	M₃₂	C₂²×D₄₃	1	2	yes	—
233	C₂×C₁₆	—	2	1	no	—
234	C₂×C₁₆	—	86	1	no	—
235	#108	—	2	1	no	—
236	#108	—	43	2	no	—
237	#108	—	86	4	no	—
238	#108	—	86	1	no	—
239	#108	C₂²×D₄₃	1	2	yes	—
240	#107	—	86	1	no	—
241	#106	—	43	2	no	—
242	#102	—	86	4	no	—
243	C₂⁴	—	4	1	no	—
244	#41	—	86	1	no	—
245	#41	—	86	1	no	—
246	#41	—	86	1	no	—
247	#41	—	86	1	no	—
248	C₂×Q₈	—	43	1	no	—
249	C₂×D₄	—	2	2	no	—
250	C₂×D₄	—	4	1	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂×D₄	—	4	1	no	—
254	C₂×D₄	—	4	1	no	—
255	C₂×D₄	—	43	2	no	—
256	C₂×D₄	—	43	1	no	—
257	C₂×D₄	—	172	4	no	—
258	C₂²×C₄	—	43	2	no	—
259	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
260	D₈	—	2	2	no	—
261	D₈	—	2	2	no	—
262	D₈	—	4	1	no	—
263	D₈	—	4	1	no	—
264	M₁₆	—	2	2	no	—
265	M₁₆	—	4	1	no	—
266	M₁₆	—	86	1	no	—
267	M₁₆	—	86	1	no	—
268	M₁₆	—	86	2	no	—
269	M₁₆	—	86	2	no	—
270	M₁₆	—	86	2	no	—
271	M₁₆	—	172	4	no	—
272	M₁₆	—	172	1	no	—
273	C₂×C₈	—	4	1	no	—
274	C₂×C₈	—	43	2	no	—
275	C₂×C₈	—	86	2	no	—
276	C₂×C₈	—	86	2	no	—
277	C₂×C₈	—	172	4	no	—
278	C₂×C₈	—	172	1	no	—
279	C₂×C₈	#10621898	1	1	yes	characteristic
280	C₂×C₈	#10621898	1	1	yes	characteristic
281	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
282	C₂×C₈	#10621900	1	1	yes	characteristic
283	C₂×C₈	#10621900	1	1	yes	characteristic
284	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
285	C₄⋊C₄	—	86	2	no	—
286	C₂²⋊C₄	—	172	4	no	—
287	C₂²⋊C₄	—	172	1	no	—
288	C₄²	—	172	1	no	—
289	C₁₆	—	2	2	no	—
290	C₁₆	—	86	2	no	—
291	C₂³	—	4	2	no	—
292	C₂³	—	8	1	no	—
293	C₂³	—	8	1	no	—
294	C₂³	—	16	2	no	—
295	C₂³	—	43	2	no	—
296	C₂³	SD₁₆×D₄₃	1	1	yes	characteristic
297	Q₈	—	86	1	no	—
298	Q₈	—	86	1	no	—
299	D₄	—	4	2	no	—
300	D₄	—	4	2	no	—
301	D₄	—	8	1	no	—
302	D₄	—	8	1	no	—
303	D₄	—	86	2	no	—
304	D₄	—	86	2	no	—
305	D₄	—	86	1	no	—
306	D₄	—	86	1	no	—
307	D₄	—	688	4	no	—
308	C₂×C₄	—	43	2	no	—
309	C₂×C₄	—	86	2	no	—
310	C₂×C₄	—	86	2	no	—
311	C₂×C₄	—	172	2	no	—
312	C₂×C₄	#13063307	1	1	yes	characteristic
313	C₂×C₄	#13063307	1	1	yes	characteristic
314	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
315	C₂×C₄	#13063349	1	1	yes	characteristic
316	C₂×C₄	#13063349	1	1	yes	characteristic
317	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
318	C₈	—	4	2	no	—
319	C₈	—	86	2	no	—
320	C₈	—	172	2	no	—
321	C₈	#13063307	1	1	yes	characteristic
322	C₈	#13063307	1	1	yes	characteristic
323	C₈	C₂×#10621900	1	1	yes	characteristic
324	C₈	C₂×#10621898	1	1	yes	characteristic
325	C₂²	—	4	1	no	—
326	C₂²	—	8	4	no	—
327	C₂²	—	8	2	no	—
328	C₂²	—	16	2	no	—
329	C₂²	—	16	2	no	—
330	C₂²	—	86	2	no	—
331	C₂²	—	172	2	no	—
332	C₂²	#14583216	1	1	yes	characteristic
333	C₂²	#14583216	1	1	yes	characteristic
334	C₂²	C₂×SD₁₆×D₄₃	1	1	yes	characteristic
335	C₄	—	86	2	no	—
336	C₄	—	344	2	no	—
337	C₄	#14583216	1	1	yes	characteristic
338	C₄	#14583216	1	1	yes	characteristic
339	C₄	C₂×#13063307	1	1	yes	characteristic
340	C₄	C₂×#13063349	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	2	1	no	—
344	C₂	—	16	2	no	—
345	C₂	—	172	2	no	—
346	C₂	C₂×#14583216	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14583216
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₂×C₄₃×#361	C₂
socle	C₂×C₄₃	C₂×#481
prime radical	#19144239	C₂
hypercenter	C₂×#361	D₄₃
hypocenter	C₄₃	#33657
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#361

Complements

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

N	H ≅ G/N	type
2. #19143203	C₂	semidirect
3. #19136808	C₂	semidirect
4. #19136369	C₂	semidirect
5. #19136368	C₂	semidirect
6. C₂×#14582712	C₂	semidirect
7. #19136215	C₂	semidirect
8. C₂×C₄₃×#361	C₂	semidirect
16. #14582946	C₂²	semidirect
21. #14582723	C₂²	semidirect
22. #14582722	C₂²	semidirect
24. #14582680	C₂²	semidirect
174. C₂×#361	D₄₃	semidirect
202. C₄₃	#33657	semidirect

Maximal subgroups

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

subgroup	index
#19144239	2
#19143203	2
#19136808	2
#19136369	2
#19136368	2
C₂×#14582712	2
#19136215	2
C₂×C₄₃×#361	2
#33657	43