Subgroups — #24991088

Order 11008.

426 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	#19143341	C₂	1	1	yes	maximal, characteristic
2	#19135370	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133366	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133242	C₂	1	1	yes	maximal, characteristic
5	#19133047	C₂	1	1	yes	maximal, characteristic
6	#19132430	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1278	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#13063333	—	2	1	no	—
9	#14583449	C₂²	1	1	yes	characteristic
10	#14582978	—	2	1	no	—
11	#14582972	—	2	2	no	—
12	C₈×D₁₇₂	—	2	1	no	—
13	C₂×#13063260	—	2	1	no	—
14	#14582923	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
17	#14582867	—	2	1	no	—
18	#14582663	—	2	1	no	—
19	#14582650	—	2	1	no	—
20	#14582650	—	2	1	no	—
21	#14582644	C₄	1	1	yes	characteristic
22	#14582644	C₄	1	1	yes	characteristic
23	#14582642	C₄	1	1	yes	characteristic
24	#14582642	C₄	1	1	yes	characteristic
25	#14582628	—	2	2	no	—
26	C₄₃×#431	C₂²	1	1	yes	characteristic
27	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#10621898	—	2	1	no	—
30	#13063333	—	4	2	no	—
31	#13063333	—	4	2	no	—
32	C₂×#10621877	—	2	1	no	—
33	C₂×#10621872	—	2	2	no	—
34	C₂²×C₄₃⋊C₈	—	2	1	no	—
35	C₂²×C₄₃⋊C₈	—	2	1	no	—
36	C₂×C₈×D₄₃	—	2	2	no	—
37	#13063325	C₂×C₄	1	1	yes	characteristic
38	#13063322	C₂×C₄	1	1	yes	characteristic
39	#13063313	—	2	1	no	—
40	#13063299	—	2	1	no	—
41	#13063299	Q₈	1	2	yes	—
42	C₄×D₁₇₂	—	2	1	no	—
43	C₄×D₁₇₂	D₄	1	2	yes	—
44	#13063260	—	4	2	no	—
45	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
49	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
50	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
51	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
52	C₄⋊(C₄₃⋊C₈)	D₄	1	1	yes	characteristic
53	(C₂×D₄₃)⋊C₈	—	2	2	no	—
54	(C₂×D₄₃)⋊C₈	—	2	2	no	—
55	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
56	C₂×M₁₆×C₄₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₄₃	—	2	1	no	—
58	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
59	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₂²⋊C₈×C₄₃	—	2	1	no	—
62	#97×C₄₃	—	2	1	no	—
63	C₄×C₈×C₄₃	—	2	1	no	—
64	C₄×C₈×C₄₃	—	2	1	no	—
65	C₄×C₈×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	—	2	1	no	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂×C₂²⋊D₄₃	—	2	2	no	—
71	#10621898	—	4	1	no	—
72	#10621898	—	4	1	no	—
73	#10621898	—	4	1	no	—
74	#10621898	—	4	1	no	—
75	C₂×D₁₇₂	—	2	1	no	—
76	C₂×D₁₇₂	—	2	1	no	—
77	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
78	C₂×C₄×D₄₃	—	2	2	no	—
79	C₂×Dic₈₆	—	2	1	no	—
80	M₁₆×C₄₃	—	2	1	no	—
81	M₁₆×C₄₃	—	2	1	no	—
82	M₁₆×C₄₃	—	2	1	no	—
83	M₁₆×C₄₃	SD₁₆	1	2	yes	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₂×C₈×C₄₃	—	2	1	no	—
87	C₂×C₈×C₄₃	—	2	1	no	—
88	C₂×C₈×C₄₃	—	2	1	no	—
89	C₂×C₈×C₄₃	—	4	1	no	—
90	C₂×C₈×C₄₃	—	4	1	no	—
91	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
93	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
94	C₄²×C₄₃	#41	1	1	yes	characteristic
95	(C₂×D₄₃)⋊C₄	—	2	2	no	—
96	(C₂×D₄₃)⋊C₄	—	2	2	no	—
97	C₄⋊Dic₄₃	—	2	1	no	—
98	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
99	#10621877	—	4	1	no	—
100	#10621877	—	4	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	—	2	1	no	—
105	C₂×C₄₃⋊C₈	—	4	1	no	—
106	C₂×C₄₃⋊C₈	—	4	1	no	—
107	C₂×C₄₃⋊C₈	—	4	1	no	—
108	C₂×C₄₃⋊C₈	—	4	1	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	#10621872	—	4	4	no	—
111	C₈×D₄₃	—	4	4	no	—
112	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂²×D₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	4	1	no	—
117	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
123	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
125	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
126	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
127	C₂²⋊D₄₃	—	4	2	no	—
128	C₂²⋊D₄₃	—	4	2	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	D₁₇₂	—	4	2	no	—
131	D₁₇₂	—	4	1	no	—
132	D₁₇₂	—	4	1	no	—
133	D₁₇₂	—	4	1	no	—
134	D₁₇₂	—	4	1	no	—
135	C₄×D₄₃	—	4	2	no	—
136	C₄×D₄₃	—	4	2	no	—
137	Dic₈₆	—	4	1	no	—
138	Dic₈₆	—	4	1	no	—
139	C₈×C₄₃	—	2	2	no	—
140	C₈×C₄₃	—	4	2	no	—
141	C₈×C₄₃	—	4	1	no	—
142	C₈×C₄₃	—	4	1	no	—
143	C₈×C₄₃	—	4	1	no	—
144	C₄₃⋊C₈	—	4	1	no	—
145	C₄₃⋊C₈	—	4	1	no	—
146	C₄₃⋊C₈	—	4	1	no	—
147	C₄₃⋊C₈	—	4	1	no	—
148	C₄₃⋊C₈	—	8	1	no	—
149	#8932	—	43	1	no	maximal, Sylow
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	—	2	1	no	—
152	C₂²×C₄₃	#385	1	1	yes	characteristic
153	C₂²×C₄₃	#476	1	1	yes	characteristic
154	C₂²×C₄₃	#478	1	1	yes	characteristic
155	C₂×D₄₃	—	4	2	no	—
156	C₂×D₄₃	—	4	2	no	—
157	C₂×D₄₃	—	4	2	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	2	1	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
162	C₄×C₄₃	#442	1	1	yes	characteristic
163	C₄×C₄₃	#443	1	1	yes	characteristic
164	C₄×C₄₃	#444	1	1	yes	characteristic
165	C₄×C₄₃	#449	1	1	yes	characteristic
166	C₄×C₄₃	#450	1	1	yes	characteristic
167	Dic₄₃	—	4	2	no	—
168	#2935	—	43	1	no	—
169	#1950	—	43	1	no	—
170	#1570	—	43	1	no	—
171	#1540	—	43	1	no	—
172	#1492	—	43	1	no	—
173	#1278	D₄₃	1	1	yes	characteristic, complemented
174	#1277	—	43	1	no	—
175	C₂×C₄₃	—	2	1	no	—
176	C₂×C₄₃	—	2	1	no	—
177	C₂×C₄₃	#1858	1	1	yes	characteristic
178	C₂×C₄₃	#1864	1	1	yes	characteristic
179	C₂×C₄₃	#1866	1	1	yes	characteristic
180	D₄₃	—	8	2	no	—
181	C₂×#131	—	86	1	no	—
182	#532	—	43	1	no	—
183	#436	—	86	1	no	—
184	#435	—	86	2	no	—
185	C₈×D₄	—	86	1	no	—
186	#431	C₂×D₄₃	1	1	yes	characteristic
187	C₂×#108	—	86	1	no	—
188	#423	—	43	1	no	—
189	C₂×C₄⋊C₈	—	43	1	no	—
190	C₂×C₄⋊C₈	—	43	1	no	—
191	#408	—	86	1	no	—
192	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
193	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
194	#341	—	86	1	no	—
195	#334	—	86	1	no	—
196	#334	—	86	1	no	—
197	#332	—	43	1	no	—
198	#332	—	43	1	no	—
199	#331	—	43	1	no	—
200	#331	—	43	1	no	—
201	D₄⋊C₈	—	86	2	no	—
202	C₄₃	#8932	1	1	yes	characteristic, complemented, Sylow
203	C₂×#41	—	86	1	no	—
204	#131	—	172	2	no	—
205	#131	—	172	2	no	—
206	C₂×M₁₆	—	86	2	no	—
207	C₂×M₁₆	—	86	1	no	—
208	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
209	C₂×M₁₆	D₁₇₂	1	1	yes	characteristic
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	86	2	no	—
212	C₂²×C₈	—	86	1	no	—
213	C₂²×C₈	—	86	1	no	—
214	C₄⋊Q₈	—	43	1	no	—
215	#127	—	43	1	no	—
216	#124	—	86	1	no	—
217	#121	—	43	2	no	—
218	#121	—	86	1	no	—
219	C₄×D₄	—	43	2	no	—
220	C₄×D₄	—	86	1	no	—
221	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
222	#108	—	172	2	no	—
223	C₄⋊C₈	—	43	1	no	—
224	C₄⋊C₈	—	43	1	no	—
225	C₄⋊C₈	—	43	1	no	—
226	C₄⋊C₈	—	43	1	no	—
227	C₄⋊C₈	—	86	1	no	—
228	C₄⋊C₈	—	86	1	no	—
229	C₄⋊C₈	—	86	1	no	—
230	C₄⋊C₈	—	86	1	no	—
231	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
232	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
233	C₂²⋊C₈	—	2	1	no	—
234	C₂²⋊C₈	—	86	2	no	—
235	C₂²⋊C₈	—	86	2	no	—
236	#97	—	2	1	no	—
237	C₄×C₈	—	2	1	no	—
238	C₄×C₈	—	2	1	no	—
239	C₄×C₈	—	2	1	no	—
240	C₄×C₈	—	2	1	no	—
241	#41	—	172	1	no	—
242	#41	—	172	1	no	—
243	#41	—	172	1	no	—
244	#41	—	172	1	no	—
245	C₂×Q₈	—	86	1	no	—
246	C₂×D₄	—	43	2	no	—
247	C₂×D₄	—	86	2	no	—
248	C₂×D₄	—	86	1	no	—
249	C₂×D₄	—	86	1	no	—
250	C₂²×C₄	—	86	2	no	—
251	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
252	C₂²×C₄	#10621900	1	1	yes	characteristic
253	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
254	M₁₆	—	2	1	no	—
255	M₁₆	—	2	1	no	—
256	M₁₆	—	2	1	no	—
257	M₁₆	—	172	4	no	—
258	M₁₆	—	172	1	no	—
259	M₁₆	—	172	1	no	—
260	M₁₆	#10621873	1	2	yes	—
261	C₂×C₈	—	2	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	1	no	—
264	C₂×C₈	—	2	1	no	—
265	C₂×C₈	—	2	1	no	—
266	C₂×C₈	—	4	1	no	—
267	C₂×C₈	—	4	1	no	—
268	C₂×C₈	—	86	1	no	—
269	C₂×C₈	—	86	1	no	—
270	C₂×C₈	—	86	1	no	—
271	C₂×C₈	—	86	1	no	—
272	C₂×C₈	—	172	4	no	—
273	C₂×C₈	—	172	1	no	—
274	C₂×C₈	—	172	1	no	—
275	C₂×C₈	—	172	1	no	—
276	C₂×C₈	—	172	1	no	—
277	C₂×C₈	—	172	1	no	—
278	C₄⋊C₄	—	43	2	no	—
279	C₄⋊C₄	—	86	1	no	—
280	C₂²⋊C₄	—	86	2	no	—
281	C₂²⋊C₄	—	86	2	no	—
282	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
283	C₄²	#10621898	1	1	yes	characteristic
284	C₄²	D₄×D₄₃	1	1	yes	characteristic
285	C₄²	D₄×D₄₃	1	1	yes	characteristic
286	C₂³	—	86	2	no	—
287	C₂³	#13063305	1	1	yes	characteristic
288	Q₈	—	172	1	no	—
289	Q₈	—	172	1	no	—
290	D₄	—	172	2	no	—
291	D₄	—	172	2	no	—
292	D₄	—	172	2	no	—
293	D₄	—	172	1	no	—
294	D₄	—	172	1	no	—
295	D₄	—	172	1	no	—
296	D₄	—	172	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	86	2	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	—	172	2	no	—
303	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
304	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
305	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
306	C₂×C₄	#13063295	1	1	yes	characteristic
307	C₂×C₄	#13063303	1	1	yes	characteristic
308	C₂×C₄	#13063307	1	1	yes	characteristic
309	C₂×C₄	C₂×#10621873	1	1	yes	characteristic
310	C₂×C₄	#13063347	1	1	yes	characteristic
311	C₂×C₄	#13063353	1	1	yes	characteristic
312	C₂×C₄	#13063353	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	4	2	no	—
315	C₈	—	4	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	4	1	no	—
318	C₈	—	344	1	no	—
319	C₈	—	172	1	no	—
320	C₈	—	172	1	no	—
321	C₈	—	172	1	no	—
322	C₈	—	172	1	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	1	no	—
325	C₂²	—	172	2	no	—
326	C₂²	—	172	2	no	—
327	C₂²	—	172	2	no	—
328	C₂²	#14582773	1	1	yes	characteristic
329	C₂²	#14583190	1	1	yes	characteristic
330	C₂²	#14583200	1	1	yes	characteristic
331	C₄	—	2	1	no	—
332	C₄	—	2	1	no	—
333	C₄	—	2	1	no	—
334	C₄	—	172	2	no	—
335	C₄	C₄×#10621873	1	1	yes	characteristic
336	C₄	#14583012	1	1	yes	characteristic
337	C₄	#14583013	1	1	yes	characteristic
338	C₄	#14583018	1	1	yes	characteristic
339	C₄	#14583046	1	1	yes	characteristic
340	C₄	#14583052	1	1	yes	characteristic
341	C₂	—	2	1	no	—
342	C₂	—	2	1	no	—
343	C₂	—	344	2	no	—
344	C₂	#19134816	1	1	yes	characteristic
345	C₂	#19134859	1	1	yes	characteristic
346	C₂	#19134871	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063295
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1278	C₂
socle	C₂²×C₄₃	#385
prime radical	#14583449	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1278	D₄₃
hypocenter	C₄₃	#8932
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1278

Complements

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

N	H ≅ G/N	type
2. #19135370	C₂	semidirect
3. #19133366	C₂	semidirect
6. #19132430	C₂	semidirect
7. C₄₃×#1278	C₂	semidirect
173. #1278	D₄₃	semidirect
202. C₄₃	#8932	semidirect

Maximal subgroups

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

subgroup	index
#19143341	2
#19135370	2
#19133366	2
#19133242	2
#19133047	2
#19132430	2
C₄₃×#1278	2
#8932	43