Subgroups — #24989750

Order 11008.

434 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	C₂×#14582976	C₂	1	1	yes	maximal, characteristic
2	#19134732	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133341	C₂	1	1	yes	maximal, characteristic, complemented
4	#19133021	C₂	1	1	yes	maximal, characteristic
5	#19133019	C₂	1	1	yes	maximal, characteristic
6	#19132492	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1301	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621872	—	2	1	no	—
9	C₂×C₄×D₁₇₂	C₂²	1	1	yes	characteristic
10	#14582976	—	2	1	no	—
11	#14582976	—	2	1	no	—
12	#14582976	C₄	1	2	yes	—
13	#14582976	C₄	1	2	yes	—
14	#14582952	C₂²	1	1	yes	characteristic
15	#14582923	C₂²	1	1	yes	characteristic
16	C₂×(C₂×D₄₃)⋊C₈	—	2	1	no	—
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	#14582663	—	2	1	no	—
19	#14582663	—	2	1	no	—
20	#14582651	—	2	1	no	—
21	#14582649	—	2	1	no	—
22	#14582628	—	2	2	no	—
23	#14582626	—	2	2	no	—
24	C₄₃×#431	C₂²	1	1	yes	characteristic
25	C₄₃×#423	C₂²	1	1	yes	characteristic
26	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×D₁₇₂	Q₈	1	1	yes	characteristic
28	C₂²×C₄×D₄₃	—	2	1	no	—
29	C₂×#10621877	—	2	1	no	—
30	C₂×#10621872	—	2	2	no	—
31	C₂×#10621872	—	2	2	no	—
32	C₂×#10621872	—	4	2	no	—
33	C₂×#10621872	—	4	2	no	—
34	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
35	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
36	C₄×D₁₇₂	—	2	1	no	—
37	C₄×D₁₇₂	—	2	1	no	—
38	C₄×D₁₇₂	C₂×C₄	1	2	yes	—
39	C₄×D₁₇₂	D₄	1	2	yes	—
40	C₂×C₄⋊Dic₄₃	Q₈	1	1	yes	characteristic
41	C₂×(C₂×D₄₃)⋊C₄	—	2	1	no	—
42	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
43	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
44	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
45	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
46	(C₂×D₄₃)⋊C₈	—	2	2	no	—
47	(C₂×D₄₃)⋊C₈	—	2	2	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×M₁₆×C₄₃	—	2	1	no	—
54	C₂²×C₈×C₄₃	—	2	1	no	—
55	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
56	C₄⋊C₈×C₄₃	—	2	1	no	—
57	C₄⋊C₈×C₄₃	—	2	1	no	—
58	C₄⋊C₈×C₄₃	—	2	1	no	—
59	C₄⋊C₈×C₄₃	—	2	1	no	—
60	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
61	C₂²⋊C₈×C₄₃	—	2	1	no	—
62	C₄×C₈×C₄₃	—	2	1	no	—
63	C₂³×D₄₃	—	2	1	no	—
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
67	C₂²×Dic₄₃	—	2	1	no	—
68	C₂×D₁₇₂	—	2	1	no	—
69	C₂×D₁₇₂	—	4	2	no	—
70	C₂×D₁₇₂	C₄⋊C₄	1	2	yes	—
71	C₂×C₄×D₄₃	—	2	2	no	—
72	C₂×C₄×D₄₃	—	2	2	no	—
73	C₂×C₄×D₄₃	—	4	1	no	—
74	C₂×C₄×D₄₃	—	4	1	no	—
75	C₂×C₄×D₄₃	—	4	1	no	—
76	C₂×C₄×D₄₃	—	4	1	no	—
77	M₁₆×C₄₃	—	4	1	no	—
78	M₁₆×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	—	2	2	no	—
80	C₂×C₈×C₄₃	—	4	1	no	—
81	C₂×C₈×C₄₃	—	4	1	no	—
82	C₂×C₈×C₄₃	—	4	1	no	—
83	C₂×C₈×C₄₃	—	4	1	no	—
84	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
85	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₃	#41	1	1	yes	characteristic
88	(C₂×D₄₃)⋊C₄	—	2	2	no	—
89	(C₂×D₄₃)⋊C₄	—	2	2	no	—
90	C₄⋊Dic₄₃	—	2	1	no	—
91	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
92	#10621877	—	4	1	no	—
93	#10621877	—	4	1	no	—
94	C₂×C₄₃⋊C₈	—	2	1	no	—
95	C₂×C₄₃⋊C₈	—	2	1	no	—
96	C₂×C₄₃⋊C₈	—	2	1	no	—
97	C₂×C₄₃⋊C₈	—	2	1	no	—
98	C₂×C₄₃⋊C₈	—	4	1	no	—
99	C₂×C₄₃⋊C₈	—	4	1	no	—
100	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
101	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
103	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
104	#10621872	—	4	4	no	—
105	#10621872	—	4	4	no	—
106	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
107	C₂²×D₄₃	—	2	2	no	—
108	C₂²×D₄₃	—	4	2	no	—
109	C₂²×D₄₃	—	4	2	no	—
110	C₂²×D₄₃	—	4	1	no	—
111	C₂²×D₄₃	—	4	1	no	—
112	C₂×C₄×C₄₃	—	2	2	no	—
113	C₂×C₄×C₄₃	—	4	1	no	—
114	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
117	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
121	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
122	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
123	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
124	C₂×Dic₄₃	—	2	2	no	—
125	C₂×Dic₄₃	—	4	1	no	—
126	C₂×Dic₄₃	—	4	1	no	—
127	D₁₇₂	—	2	4	no	—
128	D₁₇₂	—	4	2	no	—
129	C₄×D₄₃	—	4	2	no	—
130	C₄×D₄₃	—	4	2	no	—
131	C₄×D₄₃	—	4	2	no	—
132	C₄×D₄₃	—	4	2	no	—
133	C₈×C₄₃	—	4	2	no	—
134	C₈×C₄₃	—	8	1	no	—
135	C₈×C₄₃	—	8	1	no	—
136	C₄₃⋊C₈	—	2	1	no	—
137	C₄₃⋊C₈	—	2	1	no	—
138	C₄₃⋊C₈	—	2	1	no	—
139	C₄₃⋊C₈	—	2	1	no	—
140	C₄₃⋊C₈	—	8	1	no	—
141	C₄₃⋊C₈	—	8	1	no	—
142	#8664	—	43	1	no	maximal, Sylow
143	C₂²×C₄₃	—	2	1	no	—
144	C₂²×C₄₃	—	2	1	no	—
145	C₂²×C₄₃	#380	1	1	yes	characteristic
146	C₂²×C₄₃	#478	1	1	yes	characteristic
147	C₂²×C₄₃	#478	1	1	yes	characteristic
148	C₂×D₄₃	—	4	2	no	—
149	C₂×D₄₃	—	4	2	no	—
150	C₂×D₄₃	—	4	2	no	—
151	C₂×D₄₃	—	8	2	no	—
152	C₂×D₄₃	—	8	2	no	—
153	C₄×C₄₃	—	2	1	no	—
154	C₄×C₄₃	—	2	1	no	—
155	C₄×C₄₃	—	2	1	no	—
156	C₄×C₄₃	#416	1	1	yes	characteristic
157	C₄×C₄₃	#418	1	1	yes	characteristic
158	C₄×C₄₃	#430	1	2	yes	—
159	C₄×C₄₃	#463	1	1	yes	characteristic
160	C₄×C₄₃	#463	1	1	yes	characteristic
161	Dic₄₃	—	4	2	no	—
162	C₂×#436	—	43	1	no	—
163	#1837	—	43	1	no	—
164	#1564	—	43	1	no	—
165	#1486	—	43	1	no	—
166	#1486	—	43	1	no	—
167	#1301	—	43	1	no	—
168	#1301	D₄₃	1	1	yes	characteristic, complemented
169	C₂×C₄₃	—	2	1	no	—
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	#1804	1	1	yes	characteristic
172	C₂×C₄₃	#1811	1	1	yes	characteristic
173	C₂×C₄₃	#1814	1	1	yes	characteristic
174	D₄₃	—	8	2	no	—
175	C₂²×M₁₆	—	86	1	no	—
176	C₂×C₄×D₄	—	43	1	no	—
177	#436	—	43	2	no	—
178	#436	—	43	2	no	—
179	#436	—	86	1	no	—
180	#436	—	86	1	no	—
181	#431	—	43	1	no	—
182	#431	C₂×D₄₃	1	1	yes	characteristic
183	#423	—	43	1	no	—
184	#423	C₂×D₄₃	1	1	yes	characteristic
185	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
186	C₂×C₂²⋊C₈	—	86	1	no	—
187	C₂×C₄×C₈	—	43	1	no	—
188	#341	—	86	1	no	—
189	#341	—	86	1	no	—
190	#335	—	86	1	no	—
191	#334	—	86	1	no	—
192	D₄⋊C₈	—	86	2	no	—
193	D₄⋊C₈	—	86	2	no	—
194	C₄₃	#8664	1	1	yes	characteristic, complemented, Sylow
195	C₂²×D₄	—	43	1	no	—
196	C₂³×C₄	—	86	1	no	—
197	C₂×M₁₆	—	2	1	no	—
198	C₂×M₁₆	—	86	2	no	—
199	C₂×M₁₆	—	86	2	no	—
200	C₂×M₁₆	—	86	1	no	—
201	C₂×M₁₆	—	172	2	no	—
202	C₂×M₁₆	—	172	2	no	—
203	C₂²×C₈	—	2	1	no	—
204	C₂²×C₈	—	43	1	no	—
205	C₂²×C₈	—	43	1	no	—
206	C₄×D₄	—	43	2	no	—
207	C₄×D₄	—	43	2	no	—
208	C₄×D₄	—	86	1	no	—
209	C₄×D₄	—	86	1	no	—
210	C₂×C₄⋊C₄	—	43	1	no	—
211	C₂×C₂²⋊C₄	—	86	1	no	—
212	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
213	C₄⋊C₈	—	2	1	no	—
214	C₄⋊C₈	—	2	1	no	—
215	C₄⋊C₈	—	2	1	no	—
216	C₄⋊C₈	—	2	1	no	—
217	C₄⋊C₈	—	86	1	no	—
218	C₄⋊C₈	—	86	1	no	—
219	C₄⋊C₈	—	86	1	no	—
220	C₄⋊C₈	C₄×D₄₃	1	2	yes	—
221	C₂²⋊C₈	—	2	1	no	—
222	C₂²⋊C₈	—	86	2	no	—
223	C₂²⋊C₈	—	86	2	no	—
224	C₂²⋊C₈	—	86	1	no	—
225	C₄×C₈	—	2	1	no	—
226	C₄×C₈	—	43	1	no	—
227	C₄×C₈	—	43	1	no	—
228	C₄×C₈	—	43	1	no	—
229	C₄×C₈	—	43	1	no	—
230	C₄×C₈	—	86	1	no	—
231	C₂⁴	—	86	1	no	—
232	C₂×D₄	—	43	2	no	—
233	C₂×D₄	—	86	1	no	—
234	C₂×D₄	—	172	2	no	—
235	C₂²×C₄	—	86	1	no	—
236	C₂²×C₄	—	86	2	no	—
237	C₂²×C₄	—	86	2	no	—
238	C₂²×C₄	—	172	1	no	—
239	C₂²×C₄	—	172	1	no	—
240	C₂²×C₄	—	172	1	no	—
241	C₂²×C₄	—	172	1	no	—
242	C₂²×C₄	#10621898	1	1	yes	characteristic
243	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
244	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
245	M₁₆	—	4	1	no	—
246	M₁₆	—	4	1	no	—
247	M₁₆	—	172	4	no	—
248	M₁₆	—	172	4	no	—
249	M₁₆	—	172	1	no	—
250	M₁₆	—	172	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	4	1	no	—
253	C₂×C₈	—	4	1	no	—
254	C₂×C₈	—	4	1	no	—
255	C₂×C₈	—	4	1	no	—
256	C₂×C₈	—	43	1	no	—
257	C₂×C₈	—	43	1	no	—
258	C₂×C₈	—	43	1	no	—
259	C₂×C₈	—	43	1	no	—
260	C₂×C₈	—	86	1	no	—
261	C₂×C₈	—	86	1	no	—
262	C₂×C₈	—	86	1	no	—
263	C₂×C₈	—	86	1	no	—
264	C₂×C₈	—	172	1	no	—
265	C₂×C₈	—	172	1	no	—
266	C₄⋊C₄	—	43	2	no	—
267	C₄⋊C₄	—	86	1	no	—
268	C₂²⋊C₄	—	86	2	no	—
269	C₂²⋊C₄	—	86	2	no	—
270	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
271	C₄²	#10621898	1	1	yes	characteristic
272	C₄²	D₄×D₄₃	1	1	yes	characteristic
273	C₄²	D₄×D₄₃	1	1	yes	characteristic
274	C₂³	—	86	2	no	—
275	C₂³	—	172	2	no	—
276	C₂³	—	172	2	no	—
277	C₂³	—	172	1	no	—
278	C₂³	—	172	1	no	—
279	C₂³	#13063303	1	1	yes	characteristic
280	D₄	—	86	4	no	—
281	D₄	—	172	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	86	2	no	—
285	C₂×C₄	—	172	1	no	—
286	C₂×C₄	—	172	1	no	—
287	C₂×C₄	—	172	2	no	—
288	C₂×C₄	—	172	2	no	—
289	C₂×C₄	—	172	2	no	—
290	C₂×C₄	—	172	2	no	—
291	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
292	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
293	C₂×C₄	#13063282	1	1	yes	characteristic
294	C₂×C₄	#13063298	1	1	yes	characteristic
295	C₂×C₄	#13063298	1	1	yes	characteristic
296	C₂×C₄	#13063303	1	1	yes	characteristic
297	C₂×C₄	#13063346	1	1	yes	characteristic
298	C₂×C₄	#13063348	1	1	yes	characteristic
299	C₂×C₄	#13063349	1	1	yes	characteristic
300	C₂×C₄	#13063352	1	1	yes	characteristic
301	C₈	—	4	2	no	—
302	C₈	—	8	1	no	—
303	C₈	—	8	1	no	—
304	C₈	—	86	1	no	—
305	C₈	—	86	1	no	—
306	C₈	—	86	1	no	—
307	C₈	—	86	1	no	—
308	C₈	—	344	1	no	—
309	C₈	—	344	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	344	2	no	—
313	C₂²	—	344	2	no	—
314	C₂²	—	172	2	no	—
315	C₂²	—	172	2	no	—
316	C₂²	—	172	2	no	—
317	C₂²	#14582758	1	1	yes	characteristic
318	C₂²	#14583198	1	1	yes	characteristic
319	C₂²	#14583202	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	1	no	—
322	C₄	—	2	1	no	—
323	C₄	—	172	2	no	—
324	C₄	#14582894	1	1	yes	characteristic
325	C₄	#14582902	1	1	yes	characteristic
326	C₄	#14582946	1	2	yes	—
327	C₄	#14583113	1	1	yes	characteristic
328	C₄	#14583117	1	1	yes	characteristic
329	C₂	—	2	1	no	—
330	C₂	—	2	1	no	—
331	C₂	—	344	2	no	—
332	C₂	#19134604	1	1	yes	characteristic
333	C₂	#19134636	1	1	yes	characteristic
334	C₂	#19134648	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13063298
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1301	C₂
socle	C₂²×C₄₃	#380
prime radical	C₂²×D₁₇₂	Q₈
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1301	D₄₃
hypocenter	C₄₃	#8664
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1301

Complements

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

N	H ≅ G/N	type
2. #19134732	C₂	semidirect
3. #19133341	C₂	semidirect
6. #19132492	C₂	semidirect
7. C₄₃×#1301	C₂	semidirect
168. #1301	D₄₃	semidirect
194. C₄₃	#8664	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582976	2
#19134732	2
#19133341	2
#19133021	2
#19133019	2
#19132492	2
C₄₃×#1301	2
#8664	43