Subgroups — #24987647

Order 11008.

436 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143564	C₂	1	1	yes	maximal, characteristic
2	#19134282	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133184	C₂	1	1	yes	maximal, characteristic
4	#19133079	C₂	1	1	yes	maximal, characteristic
5	#19132882	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1300	C₂	1	1	yes	maximal, characteristic, complemented
8	#14583433	C₂²	1	1	yes	characteristic
9	#14583026	C₄	1	1	yes	characteristic, complemented
10	#14583023	C₄	1	1	yes	characteristic, complemented
11	#14582976	C₄	1	1	yes	characteristic, complemented
12	#14582967	—	2	1	no	—
13	#14582964	C₄	1	1	yes	characteristic, complemented
14	(C₂²⋊D₄₃)⋊C₈	—	2	1	no	—
15	#14582961	—	2	1	no	—
16	#14582952	C₂²	1	1	yes	characteristic
17	#14582856	—	2	1	no	—
18	C₂×#13063236	—	2	1	no	—
19	C₂×#13063236	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₄₃⋊C₈	—	2	1	no	—
21	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
22	#14582640	C₄	1	2	yes	complemented
23	#14582638	C₄	1	2	yes	complemented
24	#14582635	C₄	1	2	yes	—
25	#14582632	C₄	1	2	yes	—
26	#14582621	—	2	1	no	—
27	#14582621	—	2	1	no	—
28	C₄₃×#432	C₂²	1	1	yes	characteristic
29	C₄×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
30	C₄₃×#376	C₂²	1	1	yes	characteristic
31	C₂×#10621872	—	2	1	no	—
32	C₂²×C₄₃⋊C₈	—	2	1	no	—
33	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
34	C₂²×C₄₃⋊C₈	D₄	1	1	yes	characteristic
35	C₂×C₈×D₄₃	—	2	1	no	—
36	#13063319	—	2	1	no	—
37	#13063317	C₂×C₄	1	1	yes	characteristic
38	#13063313	C₂×C₄	1	1	yes	characteristic
39	#13063308	—	2	1	no	—
40	#13063304	C₂×C₄	1	1	yes	characteristic
41	#13063299	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₈₆	C₂×C₄	1	1	yes	characteristic
43	C₄×D₁₇₂	C₂×C₄	1	1	yes	characteristic
44	C₄×C₂²⋊D₄₃	—	2	1	no	—
45	#13063282	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
47	Dic₄₃⋊C₈	—	2	1	no	—
48	Dic₄₃⋊C₈	—	2	1	no	—
49	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
50	(C₂×D₄₃)⋊C₈	—	2	1	no	—
51	(C₂×D₄₃)⋊C₈	—	2	1	no	—
52	#13063236	—	2	4	no	—
53	#13063236	—	2	1	no	—
54	#13063236	C₂×C₄	1	2	yes	—
55	#13063235	—	2	1	no	—
56	C₄×C₄₃⋊C₈	—	2	4	no	—
57	C₄×C₄₃⋊C₈	—	2	1	no	—
58	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
59	C₄×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
60	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
61	C₄×C₄₃⋊C₈	D₄	1	1	yes	characteristic
62	C₈×Dic₄₃	—	2	1	no	—
63	C₂×M₁₆×C₄₃	—	2	1	no	—
64	C₂×C₄²×C₄₃	—	2	1	no	—
65	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
66	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
67	C₄⋊C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊C₈×C₄₃	—	2	1	no	—
69	#97×C₄₃	—	2	1	no	—
70	#97×C₄₃	C₂×C₄	1	2	yes	—
71	C₄×C₈×C₄₃	—	2	1	no	—
72	#95×C₄₃	—	2	1	no	—
73	#95×C₄₃	C₂×C₄	1	2	yes	—
74	C₂²×C₄×C₄₃	—	2	2	no	—
75	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
78	C₂×C₂²⋊D₄₃	—	2	1	no	—
79	C₂×D₁₇₂	C₄²	1	1	yes	characteristic
80	C₂×C₄×D₄₃	—	2	1	no	—
81	C₂×Dic₈₆	C₄²	1	1	yes	characteristic
82	M₁₆×C₄₃	—	4	2	no	—
83	C₂×C₈×C₄₃	—	2	1	no	—
84	C₂×C₈×C₄₃	—	2	1	no	—
85	C₂×C₈×C₄₃	—	2	2	no	—
86	C₄²×C₄₃	—	2	2	no	—
87	C₄²×C₄₃	—	2	2	no	—
88	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
89	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
90	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
91	C₄²×C₄₃	#41	1	1	yes	characteristic
92	C₂²⋊Dic₄₃	—	2	1	no	—
93	(C₂×D₄₃)⋊C₄	—	2	1	no	—
94	(C₂×D₄₃)⋊C₄	—	2	1	no	—
95	(C₂×D₄₃)⋊C₄	—	2	1	no	—
96	(C₂×D₄₃)⋊C₄	—	2	1	no	—
97	C₄⋊Dic₄₃	C₄²	1	1	yes	characteristic
98	C₄⋊Dic₄₃	C₄²	1	1	yes	characteristic
99	Dic₄₃⋊C₄	—	2	1	no	—
100	Dic₄₃⋊C₄	—	2	1	no	—
101	Dic₄₃⋊C₄	—	2	1	no	—
102	Dic₄₃⋊C₄	—	2	1	no	—
103	C₄×Dic₄₃	—	2	1	no	—
104	C₂×C₄₃⋊C₈	—	2	2	no	—
105	C₂×C₄₃⋊C₈	—	2	2	no	—
106	C₂×C₄₃⋊C₈	—	2	2	no	—
107	C₂×C₄₃⋊C₈	—	2	2	no	—
108	C₂×C₄₃⋊C₈	—	2	2	no	—
109	C₂×C₄₃⋊C₈	—	2	2	no	—
110	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
111	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
112	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
113	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
114	#10621872	—	4	2	no	—
115	C₈×D₄₃	—	4	2	no	—
116	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
117	C₂²×D₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	2	no	—
119	C₂×C₄×C₄₃	—	2	1	no	—
120	C₂×C₄×C₄₃	—	2	1	no	—
121	C₂×C₄×C₄₃	—	2	4	no	—
122	C₂×C₄×C₄₃	—	4	1	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	—	4	2	no	—
125	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₂²⋊D₄₃	—	4	2	no	—
132	C₂×Dic₄₃	—	2	1	no	—
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	—	2	1	no	—
135	D₁₇₂	—	4	1	no	—
136	C₄×D₄₃	—	4	1	no	—
137	C₄×D₄₃	—	4	1	no	—
138	Dic₈₆	—	4	1	no	—
139	C₈×C₄₃	—	4	2	no	—
140	C₈×C₄₃	—	4	1	no	—
141	C₈×C₄₃	—	4	1	no	—
142	C₄₃⋊C₈	—	2	4	no	—
143	C₄₃⋊C₈	—	2	1	no	—
144	C₄₃⋊C₈	—	2	1	no	—
145	C₄₃⋊C₈	—	4	2	no	—
146	C₄₃⋊C₈	#104	1	2	yes	—
147	C₄₃⋊C₈	#104	1	2	yes	—
148	#8249	—	43	1	no	maximal, Sylow
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₃	#444	1	2	yes	—
153	C₂×D₄₃	—	4	1	no	—
154	C₂×D₄₃	—	4	1	no	—
155	C₂×D₄₃	—	4	1	no	—
156	C₄×C₄₃	—	2	1	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	2	1	no	—
159	C₄×C₄₃	—	2	1	no	—
160	C₄×C₄₃	—	2	1	no	—
161	C₄×C₄₃	—	4	4	no	—
162	C₄×C₄₃	C₂×#104	1	1	yes	characteristic
163	C₄×C₄₃	C₂×#104	1	1	yes	characteristic
164	Dic₄₃	—	4	1	no	—
165	Dic₄₃	—	4	1	no	—
166	Dic₄₃	—	4	1	no	—
167	#2966	—	43	1	no	—
168	#1726	—	43	1	no	—
169	#1524	—	43	1	no	—
170	#1500	—	43	1	no	—
171	#1451	—	43	1	no	—
172	#1302	—	43	1	no	—
173	#1300	D₄₃	1	1	yes	characteristic, complemented
174	C₂×C₄₃	—	2	2	no	—
175	C₂×C₄₃	#1757	1	1	yes	characteristic
176	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
177	C₂×C₄₃	#1765	1	1	yes	characteristic
178	D₄₃	—	8	1	no	—
179	#529	—	43	1	no	—
180	#446	—	43	1	no	—
181	C₈×Q₈	—	43	1	no	—
182	#436	—	43	1	no	—
183	#435	—	86	1	no	—
184	C₈×D₄	—	43	1	no	—
185	C₈×D₄	—	86	1	no	—
186	#433	—	86	1	no	—
187	#432	C₂×D₄₃	1	1	yes	characteristic
188	#431	—	43	1	no	—
189	#405	—	86	1	no	—
190	C₄×M₁₆	C₂×D₄₃	1	1	yes	characteristic
191	C₂×#97	—	43	1	no	—
192	C₂×#97	—	86	1	no	—
193	C₂×C₄×C₈	—	43	1	no	—
194	C₂×C₄×C₈	—	86	1	no	—
195	#376	C₂×D₄₃	1	1	yes	characteristic
196	#330	—	43	2	no	—
197	#329	—	43	2	no	—
198	#328	—	43	2	no	—
199	#327	—	43	2	no	—
200	#322	—	86	1	no	—
201	#322	—	86	1	no	—
202	C₄₃	#8249	1	1	yes	characteristic, complemented, Sylow
203	C₂×M₁₆	—	2	1	no	—
204	C₂×M₁₆	—	86	1	no	—
205	C₂²×C₈	—	43	1	no	—
206	C₂²×C₈	—	43	1	no	—
207	C₂²×C₈	—	86	1	no	—
208	C₂²×C₈	—	86	1	no	—
209	#126	—	86	1	no	—
210	#125	—	43	1	no	—
211	#124	—	43	1	no	—
212	#123	—	86	1	no	—
213	C₂²⋊Q₈	—	43	1	no	—
214	#121	—	43	1	no	—
215	C₄×Q₈	—	43	1	no	—
216	C₄×D₄	—	43	1	no	—
217	C₄×D₄	—	86	1	no	—
218	#117	—	86	1	no	—
219	C₂×C₄²	—	2	1	no	—
220	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
221	C₄⋊C₈	—	86	1	no	—
222	C₄⋊C₈	—	86	1	no	—
223	C₄⋊C₈	—	86	1	no	—
224	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
225	C₄⋊C₈	C₄×D₄₃	1	1	yes	characteristic
226	C₂²⋊C₈	—	2	1	no	—
227	C₂²⋊C₈	—	43	2	no	—
228	C₂²⋊C₈	—	86	1	no	—
229	C₂²⋊C₈	—	86	1	no	—
230	#97	—	2	1	no	—
231	#97	—	43	2	no	—
232	#97	—	86	4	no	—
233	#97	—	86	1	no	—
234	#97	—	86	1	no	—
235	#97	C₄×D₄₃	1	2	yes	—
236	C₄×C₈	—	2	1	no	—
237	C₄×C₈	—	43	1	no	—
238	C₄×C₈	—	43	1	no	—
239	C₄×C₈	—	43	1	no	—
240	C₄×C₈	—	43	1	no	—
241	C₄×C₈	—	86	4	no	—
242	C₄×C₈	—	86	1	no	—
243	C₄×C₈	—	86	1	no	—
244	#95	—	2	1	no	—
245	#95	C₄×D₄₃	1	2	yes	—
246	C₂×Q₈	—	43	1	no	—
247	C₂×D₄	—	43	1	no	—
248	C₂×D₄	—	86	1	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	86	1	no	—
251	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	#10621900	1	1	yes	characteristic
254	M₁₆	—	4	2	no	—
255	M₁₆	—	172	2	no	—
256	C₂×C₈	—	2	1	no	—
257	C₂×C₈	—	2	1	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	43	1	no	—
260	C₂×C₈	—	43	1	no	—
261	C₂×C₈	—	43	1	no	—
262	C₂×C₈	—	43	1	no	—
263	C₂×C₈	—	86	2	no	—
264	C₂×C₈	—	86	2	no	—
265	C₂×C₈	—	86	2	no	—
266	C₂×C₈	—	86	2	no	—
267	C₂×C₈	—	86	2	no	—
268	C₂×C₈	—	86	2	no	—
269	C₂×C₈	—	172	2	no	—
270	C₄⋊C₄	—	43	1	no	—
271	C₄⋊C₄	—	43	1	no	—
272	C₄⋊C₄	—	86	1	no	—
273	C₄⋊C₄	—	86	1	no	—
274	C₄⋊C₄	—	86	1	no	—
275	C₄⋊C₄	—	86	1	no	—
276	C₂²⋊C₄	—	86	1	no	—
277	C₂²⋊C₄	—	86	1	no	—
278	C₂²⋊C₄	—	86	1	no	—
279	C₂²⋊C₄	—	86	1	no	—
280	C₂²⋊C₄	—	86	1	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	86	1	no	—
284	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
285	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
286	C₄²	D₄×D₄₃	1	1	yes	characteristic
287	C₄²	#10621902	1	1	yes	characteristic
288	C₂³	—	86	1	no	—
289	C₂³	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
290	Q₈	—	172	1	no	—
291	D₄	—	172	1	no	—
292	D₄	—	172	2	no	—
293	C₂×C₄	—	2	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	4	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	86	1	no	—
301	C₂×C₄	—	86	1	no	—
302	C₂×C₄	—	86	1	no	—
303	C₂×C₄	—	172	1	no	—
304	C₂×C₄	—	172	1	no	—
305	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	#13063283	1	1	yes	characteristic
308	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
309	C₂×C₄	#13063287	1	1	yes	characteristic
310	C₂×C₄	#13063287	1	1	yes	characteristic
311	C₈	—	4	2	no	—
312	C₈	—	4	1	no	—
313	C₈	—	4	1	no	—
314	C₈	—	43	2	no	—
315	C₈	—	43	2	no	—
316	C₈	—	86	4	no	—
317	C₈	—	86	1	no	—
318	C₈	—	86	1	no	—
319	C₈	—	172	2	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	172	1	no	—
323	C₂²	—	172	1	no	—
324	C₂²	—	172	1	no	—
325	C₂²	#14582746	1	1	yes	characteristic
326	C₂²	#14583018	1	2	yes	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	4	no	—
333	C₄	—	172	1	no	—
334	C₄	—	172	1	no	—
335	C₄	—	172	1	no	—
336	C₄	#104×D₄₃	1	1	yes	characteristic
337	C₄	#104×D₄₃	1	1	yes	characteristic
338	C₂	—	2	2	no	—
339	C₂	—	344	1	no	—
340	C₂	#19134394	1	1	yes	characteristic
341	C₂	#19134399	1	1	yes	characteristic
342	C₂	#19134436	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1300	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	#13063299	C₂×C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1300	D₄₃
hypocenter	C₄₃	#8249
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1300

Complements

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

N	H ≅ G/N	type
2. #19134282	C₂	semidirect
5. #19132882	C₂	semidirect
6. #19132493	C₂	semidirect
7. C₄₃×#1300	C₂	semidirect
9. #14583026	C₄	semidirect
10. #14583023	C₄	semidirect
11. #14582976	C₄	semidirect
13. #14582964	C₄	semidirect
22. #14582640	C₄	semidirect
23. #14582638	C₄	semidirect
173. #1300	D₄₃	semidirect
202. C₄₃	#8249	semidirect

Maximal subgroups

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

subgroup	index
#19143564	2
#19134282	2
#19133184	2
#19133079	2
#19132882	2
#19132493	2
C₄₃×#1300	2
#8249	43