Subgroups — #25251343

Order 11008.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143513	C₂	1	1	yes	maximal, characteristic
2	#19136768	C₂	1	4	yes	maximal, complemented
3	#19136763	C₂	1	4	yes	maximal, complemented
4	#19136367	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14582721	C₂	1	1	yes	maximal, characteristic, complemented
6	#19136361	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136330	C₂	1	1	yes	maximal, characteristic
8	C₂×#14582708	C₂	1	1	yes	maximal, characteristic
9	C₄₃×#3163	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#13063347	C₂²	1	1	yes	characteristic
11	#14583665	—	2	1	no	—
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	#14583022	—	2	2	no	—
15	#14583017	C₂²	1	4	yes	—
16	C₂×#13063261	C₂²	1	1	yes	characteristic
17	#14582917	—	2	1	no	—
18	#14582721	C₂²	1	2	yes	complemented
19	#14582721	C₂²	1	4	yes	complemented
20	#14582721	C₂²	1	2	yes	complemented
21	#14582720	C₂²	1	4	yes	complemented
22	#14582708	C₂²	1	2	yes	—
23	#14582708	C₂²	1	4	yes	—
24	#14582708	C₂²	1	2	yes	—
25	C₄₃×#544	—	2	1	no	—
26	C₄₃×#501	—	2	2	no	—
27	C₄₃×#499	C₂²	1	2	yes	complemented
28	C₄₃×#498	C₂²	1	2	yes	complemented
29	C₄₃×#428	—	2	1	no	—
30	C₄₃×#427	C₂²	1	1	yes	characteristic
31	C₂×#106×C₄₃	C₂²	1	1	yes	characteristic
32	C₄₃×#405	C₂²	1	1	yes	characteristic
33	C₂×#10621898	—	2	1	no	—
34	#13063347	—	2	1	no	—
35	#13063347	—	2	1	no	—
36	#13063347	Q₈	1	2	yes	complemented
37	#13063347	C₂³	1	2	yes	—
38	C₂×Dic₁₇₂	Q₈	1	1	yes	characteristic, complemented
39	C₂×#10621873	—	2	1	no	—
40	C₂×D₃₄₄	Q₈	1	1	yes	characteristic, complemented
41	#13063333	—	4	2	no	—
42	#13063331	—	2	4	no	—
43	C₂×#10621877	—	2	1	no	—
44	M₁₆×D₄₃	—	4	2	no	—
45	#13063265	—	2	1	no	—
46	#13063265	—	2	1	no	—
47	#13063265	C₂³	1	2	yes	—
48	#13063265	C₂³	1	2	yes	—
49	C₂×C₄₃⋊C₁₆	—	2	1	no	—
50	C₂×C₄₃⋊C₁₆	—	2	1	no	—
51	#13063261	—	2	1	no	—
52	#13063261	C₂³	1	2	yes	—
53	#13063253	—	2	4	no	—
54	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
55	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₄₃	—	2	2	no	—
57	C₄⋊Q₈×C₄₃	—	4	1	no	—
58	#125×C₄₃	—	2	2	no	—
59	#125×C₄₃	—	4	1	no	—
60	C₂²⋊Q₈×C₄₃	—	4	2	no	—
61	#117×C₄₃	—	2	1	no	—
62	#117×C₄₃	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
64	#107×C₄₃	—	2	2	no	—
65	#107×C₄₃	—	2	1	no	—
66	#106×C₄₃	—	2	2	no	—
67	#106×C₄₃	—	2	1	no	—
68	#106×C₄₃	C₂³	1	1	yes	characteristic
69	#106×C₄₃	C₂³	1	2	yes	—
70	#106×C₄₃	C₂³	1	1	yes	characteristic
71	#97×C₄₃	D₄	1	2	yes	—
72	C₄×C₈×C₄₃	C₂³	1	2	yes	—
73	C₂×Q₈×C₄₃	—	4	2	no	—
74	C₂²×C₄×C₄₃	—	4	1	no	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₂²⋊D₄₃	—	4	1	no	—
77	#10621898	—	2	2	no	—
78	#10621898	—	2	2	no	—
79	#10621898	—	4	1	no	—
80	#10621898	—	4	1	no	—
81	C₂×D₁₇₂	—	2	1	no	—
82	C₂×C₄×D₄₃	—	4	1	no	—
83	C₂×Dic₈₆	—	2	1	no	—
84	M₁₆×C₄₃	SD₁₆	1	2	yes	—
85	M₁₆×C₄₃	SD₁₆	1	2	yes	—
86	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
89	C₂×C₈×C₄₃	C₂×Q₈	1	1	yes	characteristic
90	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
91	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
93	C₄⋊C₄×C₄₃	—	2	1	no	—
94	C₄⋊C₄×C₄₃	—	2	2	no	—
95	C₄⋊C₄×C₄₃	—	2	1	no	—
96	C₄⋊C₄×C₄₃	—	2	1	no	—
97	C₄⋊C₄×C₄₃	—	4	2	no	—
98	C₄⋊C₄×C₄₃	—	4	2	no	—
99	C₄⋊C₄×C₄₃	—	4	2	no	—
100	C₄⋊C₄×C₄₃	—	4	1	no	—
101	C₂²⋊C₄×C₄₃	—	2	1	no	—
102	C₂²⋊C₄×C₄₃	—	4	1	no	—
103	C₄²×C₄₃	—	4	1	no	—
104	C₄²×C₄₃	C₂×D₄	1	2	yes	—
105	#10621877	—	2	2	no	—
106	#10621877	—	4	1	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	Dic₁₇₂	—	2	1	no	—
109	Dic₁₇₂	C₂×Q₈	1	2	yes	—
110	D₃₄₄	—	2	1	no	—
111	D₃₄₄	C₂×Q₈	1	2	yes	—
112	#10621873	—	2	2	no	—
113	#10621873	—	2	2	no	—
114	#10621872	—	4	4	no	—
115	C₈×D₄₃	—	4	4	no	—
116	C₄₃⋊C₁₆	—	2	2	no	—
117	C₄₃⋊C₁₆	—	2	2	no	—
118	C₂³×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂²×D₄₃	—	4	1	no	—
120	Q₈×C₄₃	—	16	2	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	4	1	no	—
123	C₂×C₄×C₄₃	—	4	1	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	—	16	1	no	—
126	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×SD₁₆	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	C₂²×D₄	1	1	yes	characteristic
132	C₂²⋊D₄₃	—	4	2	no	—
133	C₂²⋊D₄₃	—	4	2	no	—
134	C₂×Dic₄₃	—	4	1	no	—
135	D₁₇₂	—	2	2	no	—
136	D₁₇₂	—	4	1	no	—
137	C₄×D₄₃	—	4	2	no	—
138	C₄×D₄₃	—	4	2	no	—
139	Dic₈₆	—	2	2	no	—
140	Dic₈₆	—	4	1	no	—
141	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
142	C₈×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
143	C₈×C₄₃	C₂×SD₁₆	1	4	yes	—
144	C₈×C₄₃	C₂²×Q₈	1	1	yes	characteristic
145	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₄₃⋊C₈	—	4	2	no	—
147	#33638	—	43	1	no	maximal, Sylow
148	C₂²×C₄₃	—	4	1	no	—
149	C₂²×C₄₃	#476	1	1	yes	characteristic
150	C₂²×C₄₃	#476	1	1	yes	characteristic
151	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
152	C₂×D₄₃	—	4	2	no	—
153	C₂×D₄₃	—	16	1	no	—
154	C₄×C₄₃	—	4	2	no	—
155	C₄×C₄₃	—	8	1	no	—
156	C₄×C₄₃	—	8	2	no	—
157	C₄×C₄₃	—	8	1	no	—
158	C₄×C₄₃	#476	1	1	yes	characteristic
159	C₄×C₄₃	#476	1	1	yes	characteristic
160	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
161	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
162	Dic₄₃	—	4	2	no	—
163	#3163	D₄₃	1	1	yes	characteristic, complemented
164	#2957	—	43	1	no	—
165	#2231	—	43	4	no	—
166	#2230	—	43	4	no	—
167	#2155	—	43	1	no	—
168	C₂×#365	—	43	1	no	—
169	#2153	—	43	1	no	—
170	#2145	—	43	1	no	—
171	C₂×#360	—	43	1	no	—
172	C₂×C₄₃	—	2	1	no	—
173	C₂×C₄₃	—	2	1	no	—
174	C₂×C₄₃	—	2	1	no	—
175	C₂×C₄₃	C₂×#476	1	1	yes	characteristic
176	D₄₃	—	8	2	no	—
177	C₂×#135	—	43	1	no	—
178	#568	—	86	1	no	—
179	#544	—	2	1	no	—
180	C₂×M₃₂	—	43	1	no	—
181	C₂×M₃₂	—	43	1	no	—
182	#501	—	2	2	no	—
183	#499	C₂×D₄₃	1	2	yes	complemented
184	#498	C₂×D₄₃	1	2	yes	complemented
185	#444	—	86	2	no	—
186	#443	—	43	4	no	—
187	C₂×#108	—	43	1	no	—
188	#428	—	2	1	no	—
189	#427	C₂×D₄₃	1	1	yes	characteristic
190	C₂×#106	C₂×D₄₃	1	1	yes	characteristic
191	#421	—	86	1	no	—
192	#405	C₂×D₄₃	1	1	yes	characteristic
193	#365	—	43	4	no	—
194	#365	—	43	2	no	—
195	#365	—	43	2	no	—
196	#364	—	43	4	no	—
197	#360	—	43	4	no	—
198	#360	—	43	2	no	—
199	#360	—	43	2	no	—
200	C₄₃	#33638	1	1	yes	characteristic, complemented, Sylow
201	C₂×#41	—	86	1	no	—
202	#135	—	43	2	no	—
203	#135	—	43	2	no	—
204	#135	—	86	1	no	—
205	#135	—	86	1	no	—
206	C₂×Q₁₆	—	43	1	no	—
207	C₂×SD₁₆	—	86	1	no	—
208	C₂×D₈	—	43	1	no	—
209	#131	—	86	4	no	—
210	#131	—	172	2	no	—
211	C₂×M₁₆	—	86	1	no	—
212	C₂×M₁₆	—	172	2	no	—
213	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
214	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
215	C₄⋊Q₈	—	2	2	no	—
216	C₄⋊Q₈	—	4	1	no	—
217	#125	—	2	2	no	—
218	#125	—	4	1	no	—
219	C₂²⋊Q₈	—	4	2	no	—
220	#117	—	2	1	no	—
221	#117	C₂²⋊D₄₃	1	1	yes	characteristic
222	C₂×C₄⋊C₄	—	2	1	no	—
223	M₃₂	—	43	2	no	—
224	M₃₂	—	43	2	no	—
225	M₃₂	—	86	1	no	—
226	M₃₂	—	86	1	no	—
227	C₂×C₁₆	—	86	1	no	—
228	C₂×C₁₆	—	86	1	no	—
229	#108	—	43	2	no	—
230	#108	—	86	1	no	—
231	#107	—	2	1	no	—
232	#107	—	2	2	no	—
233	#106	—	2	1	no	—
234	#106	—	2	2	no	—
235	#106	C₂²×D₄₃	1	2	yes	—
236	#106	C₂²×D₄₃	1	1	yes	characteristic
237	#106	C₂²×D₄₃	1	1	yes	characteristic
238	#104	—	86	4	no	—
239	#97	C₂²⋊D₄₃	1	2	yes	—
240	C₄×C₈	C₂²×D₄₃	1	2	yes	—
241	#41	—	86	2	no	—
242	#41	—	86	2	no	—
243	#41	—	172	1	no	—
244	#41	—	172	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	86	1	no	—
247	C₂×D₄	—	86	1	no	—
248	C₂×D₄	—	172	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	—	172	1	no	—
251	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
252	Q₁₆	—	43	2	no	—
253	Q₁₆	—	86	1	no	—
254	SD₁₆	—	86	2	no	—
255	SD₁₆	—	86	2	no	—
256	D₈	—	43	2	no	—
257	D₈	—	86	1	no	—
258	M₁₆	—	86	2	no	—
259	M₁₆	—	172	1	no	—
260	M₁₆	—	172	4	no	—
261	M₁₆	#10621884	1	2	yes	—
262	M₁₆	#10621884	1	2	yes	—
263	C₂×C₈	—	172	1	no	—
264	C₂×C₈	—	172	4	no	—
265	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
266	C₂×C₈	Q₈×D₄₃	1	1	yes	characteristic
267	C₂×C₈	#10621902	1	1	yes	characteristic
268	C₂×C₈	#10621902	1	1	yes	characteristic
269	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
270	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
271	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
272	C₄⋊C₄	—	2	1	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	2	1	no	—
281	C₂²⋊C₄	—	4	1	no	—
282	C₄²	—	4	1	no	—
283	C₄²	C₂×C₂²⋊D₄₃	1	2	yes	—
284	C₁₆	—	86	2	no	—
285	C₁₆	—	86	2	no	—
286	C₂³	—	172	1	no	—
287	C₂³	#13063351	1	1	yes	characteristic
288	Q₈	—	16	2	no	—
289	Q₈	—	86	2	no	—
290	Q₈	—	172	1	no	—
291	D₄	—	86	2	no	—
292	D₄	—	172	2	no	—
293	D₄	—	172	2	no	—
294	D₄	—	172	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	16	1	no	—
300	C₂×C₄	—	172	1	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	—	172	2	no	—
303	C₂×C₄	#13063301	1	1	yes	characteristic
304	C₂×C₄	#13063301	1	1	yes	characteristic
305	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
306	C₂×C₄	C₂×#10621884	1	1	yes	characteristic
307	C₂×C₄	#13063351	1	1	yes	characteristic
308	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
309	C₈	—	172	2	no	—
310	C₈	#13063301	1	1	yes	characteristic
311	C₈	#13063301	1	1	yes	characteristic
312	C₈	C₂×#10621884	1	4	yes	—
313	C₈	C₂×Q₈×D₄₃	1	1	yes	characteristic
314	C₈	C₂×#10621902	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	172	2	no	—
317	C₂²	—	688	1	no	—
318	C₂²	#14583186	1	1	yes	characteristic
319	C₂²	#14583186	1	1	yes	characteristic
320	C₂²	C₂×#13063351	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	172	2	no	—
326	C₄	#14583186	1	1	yes	characteristic
327	C₄	#14583186	1	1	yes	characteristic
328	C₄	C₂×#13063301	1	1	yes	characteristic
329	C₄	C₂²×#10621884	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	344	2	no	—
334	C₂	C₂×#14583186	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×#14583186
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3163	C₂
socle	C₂×C₄₃	C₂×#476
prime radical	C₂×#13063347	C₂²
hypercenter	#3163	D₄₃
hypocenter	C₄₃	#33638
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #405 ◁ #3163

Complements

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

N	H ≅ G/N	type
2. #19136768	C₂	semidirect
3. #19136763	C₂	semidirect
4. #19136367	C₂	semidirect
5. C₂×#14582721	C₂	semidirect
6. #19136361	C₂	semidirect
9. C₄₃×#3163	C₂	semidirect
18. #14582721	C₂²	semidirect
19. #14582721	C₂²	semidirect
20. #14582721	C₂²	semidirect
21. #14582720	C₂²	semidirect
27. C₄₃×#499	C₂²	semidirect
28. C₄₃×#498	C₂²	semidirect
36. #13063347	Q₈	semidirect
38. C₂×Dic₁₇₂	Q₈	semidirect
40. C₂×D₃₄₄	Q₈	semidirect
163. #3163	D₄₃	semidirect
183. #499	C₂×D₄₃	semidirect
184. #498	C₂×D₄₃	semidirect
200. C₄₃	#33638	semidirect

Maximal subgroups

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

subgroup	index
#19143513	2
#19136768	2
#19136763	2
#19136367	2
C₂×#14582721	2
#19136361	2
#19136330	2
C₂×#14582708	2
C₄₃×#3163	2
#33638	43