Subgroups — #24995126

Order 11008.

356 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19135514	C₂	1	1	yes	maximal, characteristic
2	#19133245	C₂	1	1	yes	maximal, characteristic
3	#19133045	C₂	1	1	yes	maximal, characteristic
4	#19132882	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132496	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132434	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1950	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583449	C₂²	1	1	yes	characteristic
9	#14582952	C₂²	1	1	yes	characteristic
10	C₂×#13063251	—	2	1	no	—
11	C₂×#13063251	—	2	1	no	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
14	#14582646	C₄	1	1	yes	characteristic, complemented
15	#14582646	C₄	1	1	yes	characteristic, complemented
16	#14582642	C₄	1	1	yes	characteristic, complemented
17	#14582642	C₄	1	1	yes	characteristic, complemented
18	#14582626	—	2	1	no	—
19	#14582626	—	2	1	no	—
20	#14582621	—	2	1	no	—
21	#14582621	—	2	1	no	—
22	C₂×#108×C₄₃	—	2	1	no	—
23	C₄₃×#423	C₂²	1	1	yes	characteristic
24	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
26	C₄₃×#341	—	2	1	no	—
27	C₂×#10621898	—	2	1	no	—
28	C₂²×C₄₃⋊C₈	—	2	1	no	—
29	#13063325	C₂×C₄	1	1	yes	characteristic
30	#13063322	C₂×C₄	1	1	yes	characteristic
31	#13063313	—	2	1	no	—
32	#13063299	—	2	1	no	—
33	#13063299	—	2	1	no	—
34	C₄×D₁₇₂	—	2	1	no	—
35	C₄×D₁₇₂	—	2	1	no	—
36	C₂×C₄⋊Dic₄₃	—	2	1	no	—
37	C₂×C₄×Dic₄₃	—	2	1	no	—
38	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
39	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
40	#13063251	—	4	2	no	—
41	#13063251	—	4	2	no	—
42	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₄×C₄₃⋊C₈	—	2	1	no	—
44	C₄×C₄₃⋊C₈	—	2	1	no	—
45	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
46	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
47	C₄₃⋊#95	—	2	1	no	—
48	C₂×M₁₆×C₄₃	—	2	1	no	—
49	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
50	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₄₃	—	2	1	no	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	#108×C₄₃	—	4	2	no	—
54	C₄⋊C₈×C₄₃	—	2	1	no	—
55	C₄⋊C₈×C₄₃	—	2	1	no	—
56	C₄⋊C₈×C₄₃	—	2	1	no	—
57	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
59	#97×C₄₃	—	2	1	no	—
60	C₄×C₈×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂×C₂²⋊D₄₃	—	4	1	no	—
65	C₂²×Dic₄₃	—	4	1	no	—
66	#10621898	—	4	1	no	—
67	#10621898	—	4	1	no	—
68	#10621898	—	4	1	no	—
69	#10621898	—	4	1	no	—
70	C₂×D₁₇₂	—	2	1	no	—
71	C₂×D₁₇₂	—	2	1	no	—
72	C₂×D₁₇₂	—	2	1	no	—
73	C₂×C₄×D₄₃	—	4	1	no	—
74	C₂×Dic₈₆	—	2	1	no	—
75	M₁₆×C₄₃	—	2	1	no	—
76	M₁₆×C₄₃	—	2	1	no	—
77	M₁₆×C₄₃	—	2	1	no	—
78	M₁₆×C₄₃	—	2	1	no	—
79	M₁₆×C₄₃	—	4	1	no	—
80	M₁₆×C₄₃	—	4	1	no	—
81	C₂×C₈×C₄₃	—	2	1	no	—
82	C₂×C₈×C₄₃	—	2	1	no	—
83	C₂×C₈×C₄₃	—	2	1	no	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	4	1	no	—
86	C₂×C₈×C₄₃	—	4	1	no	—
87	C₂×C₈×C₄₃	—	4	1	no	—
88	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
90	C₄²×C₄₃	#41	1	1	yes	characteristic
91	C₄²×C₄₃	#41	1	1	yes	characteristic
92	(C₂×D₄₃)⋊C₄	—	4	1	no	—
93	(C₂×D₄₃)⋊C₄	—	4	1	no	—
94	C₄⋊Dic₄₃	—	2	2	no	—
95	C₄⋊Dic₄₃	—	2	1	no	—
96	C₄⋊Dic₄₃	—	2	1	no	—
97	C₄⋊Dic₄₃	—	4	1	no	—
98	C₄×Dic₄₃	—	4	1	no	—
99	C₄×Dic₄₃	—	4	1	no	—
100	C₂×C₄₃⋊C₈	—	2	1	no	—
101	C₂×C₄₃⋊C₈	—	2	1	no	—
102	C₂×C₄₃⋊C₈	—	2	1	no	—
103	C₂×C₄₃⋊C₈	—	2	1	no	—
104	C₂×C₄₃⋊C₈	—	4	2	no	—
105	C₂³×C₄₃	#124	1	1	yes	characteristic
106	C₂²×D₄₃	—	4	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	1	no	—
109	C₂×C₄×C₄₃	—	4	1	no	—
110	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
114	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#126	1	1	yes	characteristic
116	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
117	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
118	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
120	C₂²⋊D₄₃	—	8	1	no	—
121	C₂²⋊D₄₃	—	8	1	no	—
122	C₂×Dic₄₃	—	4	2	no	—
123	C₂×Dic₄₃	—	4	1	no	—
124	C₂×Dic₄₃	—	8	1	no	—
125	C₂×Dic₄₃	—	8	1	no	—
126	D₁₇₂	—	4	1	no	—
127	D₁₇₂	—	4	1	no	—
128	D₁₇₂	—	4	1	no	—
129	D₁₇₂	—	4	1	no	—
130	D₁₇₂	—	8	1	no	—
131	C₄×D₄₃	—	8	1	no	—
132	C₄×D₄₃	—	8	1	no	—
133	Dic₈₆	—	4	1	no	—
134	Dic₈₆	—	4	1	no	—
135	C₈×C₄₃	—	4	1	no	—
136	C₈×C₄₃	—	4	1	no	—
137	C₈×C₄₃	—	4	1	no	—
138	C₈×C₄₃	—	4	1	no	—
139	C₈×C₄₃	—	8	1	no	—
140	C₄₃⋊C₈	—	2	2	no	—
141	C₄₃⋊C₈	—	4	1	no	—
142	C₄₃⋊C₈	—	4	1	no	—
143	C₄₃⋊C₈	—	4	1	no	—
144	#9592	—	43	1	no	maximal, Sylow
145	C₂²×C₄₃	—	2	1	no	—
146	C₂²×C₄₃	—	2	1	no	—
147	C₂²×C₄₃	#388	1	1	yes	characteristic
148	C₂²×C₄₃	#489	1	1	yes	characteristic
149	C₂²×C₄₃	#490	1	1	yes	characteristic
150	C₂×D₄₃	—	8	1	no	—
151	C₂×D₄₃	—	8	1	no	—
152	C₂×D₄₃	—	8	1	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	#440	1	1	yes	characteristic
157	C₄×C₄₃	#442	1	1	yes	characteristic
158	C₄×C₄₃	#443	1	1	yes	characteristic
159	C₄×C₄₃	#444	1	1	yes	characteristic
160	C₄×C₄₃	#468	1	1	yes	characteristic
161	C₄×C₄₃	#469	1	1	yes	characteristic
162	Dic₄₃	—	8	2	no	—
163	Dic₄₃	—	8	1	no	—
164	#1980	—	43	1	no	—
165	#1950	D₄₃	1	1	yes	characteristic, complemented
166	#1540	—	43	1	no	—
167	#1492	—	43	1	no	—
168	#1451	—	43	1	no	—
169	#1303	—	43	1	no	—
170	#1279	—	43	1	no	—
171	C₂×C₄₃	—	2	1	no	—
172	C₂×C₄₃	—	2	1	no	—
173	C₂×C₄₃	#1929	1	1	yes	characteristic
174	C₂×C₄₃	#1932	1	1	yes	characteristic
175	C₂×C₄₃	#1936	1	1	yes	characteristic
176	D₄₃	—	16	1	no	—
177	#532	—	43	1	no	—
178	#431	—	43	1	no	—
179	C₂×#108	—	2	1	no	—
180	#423	C₂×D₄₃	1	1	yes	characteristic
181	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
182	C₂×#104	—	86	1	no	—
183	C₂×#104	—	86	1	no	—
184	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
185	C₂×C₄×C₈	—	43	1	no	—
186	C₄×C₄⋊C₄	—	43	1	no	—
187	#341	—	2	1	no	—
188	#332	—	43	1	no	—
189	#332	—	43	1	no	—
190	#331	—	43	1	no	—
191	#331	—	43	1	no	—
192	D₄⋊C₈	—	86	1	no	—
193	D₄⋊C₈	—	86	1	no	—
194	#322	—	86	1	no	—
195	#322	—	86	1	no	—
196	C₄₃	#9592	1	1	yes	characteristic, complemented, Sylow
197	C₂×#41	—	86	1	no	—
198	C₂×M₁₆	—	2	1	no	—
199	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
200	C₂×M₁₆	C₂²⋊D₄₃	1	1	yes	characteristic
201	C₂²×C₈	—	2	1	no	—
202	C₂²×C₈	—	86	1	no	—
203	C₄⋊Q₈	—	43	1	no	—
204	#127	—	43	1	no	—
205	#124	—	86	1	no	—
206	#121	—	86	1	no	—
207	#121	—	86	1	no	—
208	C₄×D₄	—	86	1	no	—
209	C₄×D₄	—	86	1	no	—
210	C₂×C₄⋊C₄	—	86	1	no	—
211	C₂×C₄²	—	86	1	no	—
212	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
213	#108	—	4	2	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	43	1	no	—
218	C₄⋊C₈	—	43	1	no	—
219	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
220	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
221	#104	—	172	2	no	—
222	#104	—	172	2	no	—
223	C₂²⋊C₈	—	86	1	no	—
224	#97	—	2	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	43	1	no	—
227	C₄×C₈	—	43	1	no	—
228	C₄×C₈	—	86	1	no	—
229	C₄×C₈	—	86	1	no	—
230	#95	—	86	1	no	—
231	#41	—	172	1	no	—
232	#41	—	172	1	no	—
233	#41	—	172	1	no	—
234	#41	—	172	1	no	—
235	C₂×Q₈	—	86	1	no	—
236	C₂×D₄	—	86	1	no	—
237	C₂×D₄	—	86	1	no	—
238	C₂×D₄	—	86	1	no	—
239	C₂×D₄	—	172	1	no	—
240	C₂²×C₄	—	172	1	no	—
241	C₂²×C₄	—	172	1	no	—
242	C₂²×C₄	#10621902	1	1	yes	characteristic
243	C₂²×C₄	#10621902	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
245	M₁₆	—	2	1	no	—
246	M₁₆	—	2	1	no	—
247	M₁₆	—	2	1	no	—
248	M₁₆	—	2	1	no	—
249	M₁₆	—	4	1	no	—
250	M₁₆	—	4	1	no	—
251	C₂×C₈	—	2	1	no	—
252	C₂×C₈	—	2	1	no	—
253	C₂×C₈	—	2	1	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	—	4	1	no	—
258	C₂×C₈	—	86	1	no	—
259	C₂×C₈	—	86	1	no	—
260	C₂×C₈	—	86	1	no	—
261	C₂×C₈	—	86	1	no	—
262	C₂×C₈	—	172	2	no	—
263	C₄⋊C₄	—	86	2	no	—
264	C₄⋊C₄	—	86	1	no	—
265	C₄⋊C₄	—	86	1	no	—
266	C₄⋊C₄	—	172	1	no	—
267	C₂²⋊C₄	—	172	1	no	—
268	C₂²⋊C₄	—	172	1	no	—
269	C₄²	—	172	1	no	—
270	C₄²	—	172	1	no	—
271	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
272	C₄²	#10621898	1	1	yes	characteristic
273	C₄²	D₄×D₄₃	1	1	yes	characteristic
274	C₄²	#10621900	1	1	yes	characteristic
275	C₂³	—	172	1	no	—
276	C₂³	#13063314	1	1	yes	characteristic
277	Q₈	—	172	1	no	—
278	Q₈	—	172	1	no	—
279	D₄	—	344	1	no	—
280	D₄	—	344	1	no	—
281	D₄	—	344	1	no	—
282	D₄	—	172	1	no	—
283	D₄	—	172	1	no	—
284	D₄	—	172	1	no	—
285	D₄	—	172	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	344	1	no	—
290	C₂×C₄	—	344	1	no	—
291	C₂×C₄	—	344	1	no	—
292	C₂×C₄	—	344	1	no	—
293	C₂×C₄	—	172	2	no	—
294	C₂×C₄	—	172	1	no	—
295	C₂×C₄	#13063283	1	1	yes	characteristic
296	C₂×C₄	C₄×C₂²⋊D₄₃	1	1	yes	characteristic
297	C₂×C₄	#13063287	1	1	yes	characteristic
298	C₂×C₄	#13063297	1	1	yes	characteristic
299	C₂×C₄	#13063310	1	1	yes	characteristic
300	C₂×C₄	#13063320	1	1	yes	characteristic
301	C₂×C₄	#13063350	1	1	yes	characteristic
302	C₂×C₄	#13063350	1	1	yes	characteristic
303	C₂×C₄	#13063351	1	1	yes	characteristic
304	C₂×C₄	#13063356	1	1	yes	characteristic
305	C₈	—	4	1	no	—
306	C₈	—	4	1	no	—
307	C₈	—	4	1	no	—
308	C₈	—	4	1	no	—
309	C₈	—	8	1	no	—
310	C₈	—	86	2	no	—
311	C₈	—	172	1	no	—
312	C₈	—	172	1	no	—
313	C₈	—	172	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	344	1	no	—
317	C₂²	—	344	1	no	—
318	C₂²	—	344	1	no	—
319	C₂²	#14582789	1	1	yes	characteristic
320	C₂²	#14583261	1	1	yes	characteristic
321	C₂²	#14583265	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	344	2	no	—
326	C₄	—	344	1	no	—
327	C₄	#14582997	1	1	yes	characteristic
328	C₄	#14583009	1	1	yes	characteristic
329	C₄	#14583015	1	1	yes	characteristic
330	C₄	#14583020	1	1	yes	characteristic
331	C₄	#14583147	1	1	yes	characteristic
332	C₄	#14583150	1	1	yes	characteristic
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	688	1	no	—
336	C₂	#19135242	1	1	yes	characteristic
337	C₂	#19135263	1	1	yes	characteristic
338	C₂	#19135291	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063297
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1950	C₂
socle	C₂²×C₄₃	#388
prime radical	#14583449	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1950	D₄₃
hypocenter	C₄₃	#9592
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1950

Complements

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

N	H ≅ G/N	type
4. #19132882	C₂	semidirect
5. #19132496	C₂	semidirect
6. #19132434	C₂	semidirect
7. C₄₃×#1950	C₂	semidirect
14. #14582646	C₄	semidirect
15. #14582646	C₄	semidirect
16. #14582642	C₄	semidirect
17. #14582642	C₄	semidirect
165. #1950	D₄₃	semidirect
196. C₄₃	#9592	semidirect

Maximal subgroups

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

subgroup	index
#19135514	2
#19133245	2
#19133045	2
#19132882	2
#19132496	2
#19132434	2
C₄₃×#1950	2
#9592	43