Subgroups — #25013469

Order 11008.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19147698	C₂	1	1	yes	maximal, characteristic, complemented
2	#19136223	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14582683	C₂	1	1	yes	maximal, characteristic, complemented
4	#19136209	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132684	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132681	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#2955	C₂	1	1	yes	maximal, characteristic
8	#14583316	—	2	2	no	—
9	#14583316	—	2	2	no	—
10	C₂×#13063265	—	2	2	no	—
11	C₂×#13063265	C₂²	1	1	yes	characteristic
12	C₂×#13063265	C₂²	1	1	yes	characteristic
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
15	#14582683	—	2	2	no	—
16	#14582683	—	2	2	no	—
17	C₂²⋊(C₄₃⋊C₁₆)	—	2	2	no	—
18	C₄₃×#575	C₄	1	1	yes	characteristic
19	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#444	—	2	2	no	—
22	C₄₃×#443	—	2	2	no	—
23	C₄₃×#430	C₄	1	1	yes	characteristic
24	C₄₃×#421	C₄	1	1	yes	characteristic
25	C₄₃×#421	C₄	1	1	yes	characteristic
26	C₄₃×#405	C₂²	1	1	yes	characteristic
27	#13063265	—	2	4	no	—
28	#13063265	—	2	1	no	—
29	#13063265	—	2	1	no	—
30	#13063265	—	2	1	no	—
31	#13063265	—	2	1	no	—
32	#13063265	—	2	1	no	—
33	#13063265	D₄	1	2	yes	—
34	C₂×C₄₃⋊C₁₆	—	2	4	no	—
35	C₂×C₄₃⋊C₁₆	—	2	1	no	—
36	C₂×C₄₃⋊C₁₆	—	2	1	no	—
37	C₂×C₄₃⋊C₁₆	—	2	1	no	—
38	C₂×C₄₃⋊C₁₆	—	2	1	no	—
39	C₂×C₄₃⋊C₁₆	—	2	1	no	—
40	C₂×C₄₃⋊C₁₆	D₄	1	2	yes	—
41	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
42	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
43	#137×C₄₃	—	2	1	no	—
44	#137×C₄₃	C₈	1	2	yes	—
45	AGL(1,ℤ/8)×C₄₃	—	2	1	no	—
46	AGL(1,ℤ/8)×C₄₃	C₈	1	2	yes	—
47	#135×C₄₃	—	2	2	no	—
48	#131×C₄₃	—	2	1	no	—
49	#131×C₄₃	—	2	1	no	—
50	#131×C₄₃	—	2	1	no	—
51	#131×C₄₃	—	2	1	no	—
52	#131×C₄₃	—	2	1	no	—
53	#131×C₄₃	—	2	1	no	—
54	#131×C₄₃	D₄	1	2	yes	—
55	#131×C₄₃	D₄	1	2	yes	—
56	C₂×M₁₆×C₄₃	—	2	1	no	—
57	C₂×M₁₆×C₄₃	—	2	1	no	—
58	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
59	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
60	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
61	C₂²×C₈×C₄₃	—	2	1	no	—
62	C₂²×C₈×C₄₃	—	2	1	no	—
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	#117×C₄₃	C₂×C₄	1	1	yes	characteristic
65	#108×C₄₃	—	2	2	no	—
66	#104×C₄₃	—	2	2	no	—
67	#104×C₄₃	—	2	2	no	—
68	#97×C₄₃	—	2	1	no	—
69	C₄×C₈×C₄₃	—	2	1	no	—
70	#41×C₄₃	—	2	1	no	—
71	#41×C₄₃	—	2	1	no	—
72	#41×C₄₃	—	2	1	no	—
73	#41×C₄₃	—	2	1	no	—
74	#41×C₄₃	—	2	1	no	—
75	#41×C₄₃	—	2	1	no	—
76	#41×C₄₃	C₂²⋊C₄	1	2	yes	—
77	#41×C₄₃	C₂×C₈	1	2	yes	—
78	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
79	C₂×Q₈×C₄₃	M₁₆	1	1	yes	characteristic
80	C₂×D₄×C₄₃	—	2	1	no	—
81	C₂×D₄×C₄₃	—	2	1	no	—
82	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
83	C₂×D₄×C₄₃	M₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	—	2	1	no	—
85	C₂²×C₄×C₄₃	—	2	1	no	—
86	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
87	Q₁₆×C₄₃	—	2	2	no	—
88	SD₁₆×C₄₃	—	2	2	no	—
89	SD₁₆×C₄₃	—	2	2	no	—
90	D₈×C₄₃	—	2	2	no	—
91	M₁₆×C₄₃	—	2	1	no	—
92	M₁₆×C₄₃	—	2	1	no	—
93	M₁₆×C₄₃	—	2	2	no	—
94	M₁₆×C₄₃	—	2	2	no	—
95	M₁₆×C₄₃	—	2	1	no	—
96	M₁₆×C₄₃	—	2	1	no	—
97	M₁₆×C₄₃	—	4	1	no	—
98	M₁₆×C₄₃	—	4	1	no	—
99	M₁₆×C₄₃	C₂²⋊C₄	1	2	yes	—
100	M₁₆×C₄₃	C₂×C₈	1	2	yes	—
101	C₂×C₈×C₄₃	—	2	1	no	—
102	C₂×C₈×C₄₃	—	2	1	no	—
103	C₂×C₈×C₄₃	—	2	2	no	—
104	C₂×C₈×C₄₃	—	2	2	no	—
105	C₂×C₈×C₄₃	—	2	1	no	—
106	C₂×C₈×C₄₃	—	4	1	no	—
107	C₂×C₈×C₄₃	—	4	1	no	—
108	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
113	C₂×C₈×C₄₃	#41	1	1	yes	characteristic
114	C₄⋊C₄×C₄₃	—	2	1	no	—
115	C₂²⋊C₄×C₄₃	—	2	1	no	—
116	C₄²×C₄₃	—	2	1	no	—
117	C₄₃⋊C₁₆	—	2	2	no	—
118	C₄₃⋊C₁₆	—	2	2	no	—
119	C₄₃⋊C₁₆	—	4	1	no	—
120	C₄₃⋊C₁₆	—	4	1	no	—
121	C₂³×C₄₃	—	2	1	no	—
122	C₂³×C₄₃	—	2	1	no	—
123	C₂³×C₄₃	#131	1	1	yes	characteristic
124	Q₈×C₄₃	—	2	1	no	—
125	Q₈×C₄₃	—	2	1	no	—
126	Q₈×C₄₃	—	2	1	no	—
127	Q₈×C₄₃	C₂²⋊C₈	1	2	yes	—
128	D₄×C₄₃	—	2	2	no	—
129	D₄×C₄₃	—	2	2	no	—
130	D₄×C₄₃	—	2	1	no	—
131	D₄×C₄₃	—	2	1	no	—
132	D₄×C₄₃	—	2	1	no	—
133	D₄×C₄₃	—	4	1	no	—
134	D₄×C₄₃	—	4	1	no	—
135	D₄×C₄₃	C₂²⋊C₈	1	2	yes	—
136	C₂×C₄×C₄₃	—	2	1	no	—
137	C₂×C₄×C₄₃	—	2	2	no	—
138	C₂×C₄×C₄₃	—	2	2	no	—
139	C₂×C₄×C₄₃	—	2	1	no	—
140	C₂×C₄×C₄₃	—	4	1	no	—
141	C₂×C₄×C₄₃	—	4	1	no	—
142	C₂×C₄×C₄₃	—	4	1	no	—
143	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
147	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
148	C₂×C₄×C₄₃	#131	1	1	yes	characteristic
149	C₈×C₄₃	—	2	2	no	—
150	C₈×C₄₃	—	2	2	no	—
151	C₈×C₄₃	—	4	1	no	—
152	C₈×C₄₃	#120	1	1	yes	characteristic
153	C₈×C₄₃	#121	1	1	yes	characteristic
154	C₈×C₄₃	#121	1	1	yes	characteristic
155	C₈×C₄₃	#123	1	1	yes	characteristic
156	#12979	—	43	1	no	maximal, Sylow
157	C₂²×C₄₃	—	2	2	no	—
158	C₂²×C₄₃	—	4	1	no	—
159	C₂²×C₄₃	—	4	2	no	—
160	C₂²×C₄₃	—	4	1	no	—
161	C₂²×C₄₃	—	8	1	no	—
162	C₂²×C₄₃	#408	1	1	yes	characteristic
163	C₂²×C₄₃	C₈×D₄	1	1	yes	characteristic
164	C₂²×C₄₃	#435	1	1	yes	characteristic
165	C₄×C₄₃	—	2	2	no	—
166	C₄×C₄₃	—	4	1	no	—
167	C₄×C₄₃	—	8	1	no	—
168	C₄×C₄₃	#386	1	1	yes	characteristic
169	C₄×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
170	C₄×C₄₃	C₈×D₄	1	1	yes	characteristic
171	C₄×C₄₃	#435	1	1	yes	characteristic
172	#3407	—	43	1	no	—
173	#2955	D₄₃	1	1	yes	characteristic
174	#2117	—	43	1	no	—
175	C₂×#350	—	43	1	no	—
176	#2113	—	43	1	no	—
177	#1379	—	43	1	no	—
178	#1378	—	43	1	no	—
179	C₂×C₄₃	—	2	1	no	—
180	C₂×C₄₃	—	2	1	no	—
181	C₂×C₄₃	—	2	1	no	—
182	C₂×C₄₃	—	4	2	no	—
183	C₂×C₄₃	—	8	1	no	—
184	C₂×C₄₃	#1869	1	1	yes	characteristic
185	#575	Dic₄₃	1	1	yes	characteristic
186	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
187	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
188	#504	—	86	2	no	—
189	#504	—	86	2	no	—
190	C₂×M₃₂	—	43	1	no	—
191	C₂×M₃₂	—	43	1	no	—
192	C₂×M₃₂	—	43	1	no	—
193	C₂×M₃₂	—	86	2	no	—
194	C₂²×C₁₆	—	43	1	no	—
195	#444	—	2	2	no	—
196	#443	—	2	2	no	—
197	#430	Dic₄₃	1	1	yes	characteristic
198	#421	Dic₄₃	1	1	yes	characteristic
199	#421	Dic₄₃	1	1	yes	characteristic
200	#405	C₂×D₄₃	1	1	yes	characteristic
201	#350	—	86	2	no	—
202	#350	—	86	2	no	—
203	C₂²⋊C₁₆	—	86	2	no	—
204	C₄₃	#12979	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	C₂×Dic₄₃	1	1	yes	characteristic
206	C₂×#41	C₂×Dic₄₃	1	1	yes	characteristic
207	#137	—	2	1	no	—
208	#137	C₄₃⋊C₈	1	2	yes	—
209	AGL(1,ℤ/8)	—	2	1	no	—
210	AGL(1,ℤ/8)	C₄₃⋊C₈	1	2	yes	—
211	#135	—	2	2	no	—
212	#131	—	2	1	no	—
213	#131	—	2	1	no	—
214	#131	—	2	1	no	—
215	#131	—	2	1	no	—
216	#131	—	2	1	no	—
217	#131	—	2	1	no	—
218	#131	C₂²⋊D₄₃	1	2	yes	—
219	#131	C₂²⋊D₄₃	1	2	yes	—
220	C₂×M₁₆	—	2	1	no	—
221	C₂×M₁₆	—	2	1	no	—
222	C₂×M₁₆	C₂×Dic₄₃	1	1	yes	characteristic
223	C₂×M₁₆	C₂×Dic₄₃	1	1	yes	characteristic
224	C₂×M₁₆	C₂×Dic₄₃	1	1	yes	characteristic
225	C₂²×C₈	—	2	1	no	—
226	C₂²×C₈	—	2	1	no	—
227	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
228	#117	C₂×Dic₄₃	1	1	yes	characteristic
229	M₃₂	—	43	2	no	—
230	M₃₂	—	86	4	no	—
231	M₃₂	—	86	1	no	—
232	M₃₂	—	86	1	no	—
233	M₃₂	—	86	1	no	—
234	M₃₂	—	86	1	no	—
235	M₃₂	—	86	1	no	—
236	C₂×C₁₆	—	43	2	no	—
237	C₂×C₁₆	—	86	4	no	—
238	C₂×C₁₆	—	86	1	no	—
239	C₂×C₁₆	—	86	1	no	—
240	C₂×C₁₆	—	86	1	no	—
241	C₂×C₁₆	—	86	1	no	—
242	C₂×C₁₆	—	86	1	no	—
243	#108	—	2	2	no	—
244	#104	—	2	2	no	—
245	#104	—	2	2	no	—
246	#97	—	2	1	no	—
247	C₄×C₈	—	2	1	no	—
248	#41	—	2	1	no	—
249	#41	—	2	1	no	—
250	#41	—	2	1	no	—
251	#41	—	2	1	no	—
252	#41	—	2	1	no	—
253	#41	—	2	1	no	—
254	#41	C₂×C₄₃⋊C₈	1	2	yes	—
255	#41	C₂²⋊Dic₄₃	1	2	yes	—
256	C₂×Q₈	C₂×C₄₃⋊C₈	1	1	yes	characteristic
257	C₂×Q₈	#10621877	1	1	yes	characteristic
258	C₂×D₄	—	2	1	no	—
259	C₂×D₄	—	2	1	no	—
260	C₂×D₄	C₂×C₄₃⋊C₈	1	1	yes	characteristic
261	C₂×D₄	#10621877	1	1	yes	characteristic
262	C₂²×C₄	—	2	1	no	—
263	C₂²×C₄	—	2	1	no	—
264	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
265	Q₁₆	—	2	2	no	—
266	SD₁₆	—	2	2	no	—
267	SD₁₆	—	2	2	no	—
268	D₈	—	2	2	no	—
269	M₁₆	—	2	1	no	—
270	M₁₆	—	2	1	no	—
271	M₁₆	—	2	2	no	—
272	M₁₆	—	2	2	no	—
273	M₁₆	—	2	1	no	—
274	M₁₆	—	2	1	no	—
275	M₁₆	—	4	1	no	—
276	M₁₆	—	4	1	no	—
277	M₁₆	C₂×C₄₃⋊C₈	1	2	yes	—
278	M₁₆	C₂²⋊Dic₄₃	1	2	yes	—
279	C₂×C₈	—	2	1	no	—
280	C₂×C₈	—	2	1	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	1	no	—
284	C₂×C₈	—	4	1	no	—
285	C₂×C₈	—	4	1	no	—
286	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
287	C₂×C₈	D₄×D₄₃	1	1	yes	characteristic
288	C₂×C₈	#10621900	1	1	yes	characteristic
289	C₂×C₈	#10621900	1	1	yes	characteristic
290	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
291	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
292	C₄⋊C₄	—	2	1	no	—
293	C₂²⋊C₄	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₁₆	—	86	2	no	—
296	C₁₆	—	86	2	no	—
297	C₁₆	—	172	1	no	—
298	C₁₆	—	172	1	no	—
299	C₂³	—	2	1	no	—
300	C₂³	—	2	1	no	—
301	C₂³	#13063332	1	1	yes	characteristic
302	Q₈	—	2	1	no	—
303	Q₈	—	2	1	no	—
304	Q₈	—	2	1	no	—
305	Q₈	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
306	D₄	—	2	2	no	—
307	D₄	—	2	2	no	—
308	D₄	—	2	1	no	—
309	D₄	—	2	1	no	—
310	D₄	—	2	1	no	—
311	D₄	—	4	1	no	—
312	D₄	—	4	1	no	—
313	D₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
322	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
323	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
324	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
325	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
326	C₂×C₄	#13063332	1	1	yes	characteristic
327	C₈	—	2	2	no	—
328	C₈	—	2	2	no	—
329	C₈	—	4	1	no	—
330	C₈	#13063293	1	1	yes	characteristic
331	C₈	#13063297	1	1	yes	characteristic
332	C₈	#13063297	1	1	yes	characteristic
333	C₈	#13063306	1	1	yes	characteristic
334	C₂²	—	2	2	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	2	no	—
337	C₂²	—	4	1	no	—
338	C₂²	—	8	1	no	—
339	C₂²	#14582866	1	1	yes	characteristic
340	C₂²	D₄×C₄₃⋊C₈	1	1	yes	characteristic
341	C₂²	#14582971	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	4	1	no	—
344	C₄	—	8	1	no	—
345	C₄	#14582779	1	1	yes	characteristic
346	C₄	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
347	C₄	D₄×C₄₃⋊C₈	1	1	yes	characteristic
348	C₄	#14582971	1	1	yes	characteristic
349	C₂	—	2	1	no	—
350	C₂	—	2	1	no	—
351	C₂	—	2	1	no	—
352	C₂	—	4	2	no	—
353	C₂	—	8	1	no	—
354	C₂	#19134888	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#13063293
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₄₃
Fitting	C₄₃×#2955	C₂
socle	C₂×C₄₃	#1869
prime radical	C₄₃×#575	C₄
cosocle	C₂²×C₈×C₄₃	C₂³
hypercenter	#2955	D₄₃
hypocenter	C₄₃	#12979
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #2955

Complements

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

N	H ≅ G/N	type
1. #19147698	C₂	semidirect
2. #19136223	C₂	semidirect
3. C₂×#14582683	C₂	semidirect
4. #19136209	C₂	semidirect
5. #19132684	C₂	semidirect
6. #19132681	C₂	semidirect
204. C₄₃	#12979	semidirect

Maximal subgroups

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

subgroup	index
#19147698	2
#19136223	2
C₂×#14582683	2
#19136209	2
#19132684	2
#19132681	2
C₄₃×#2955	2
#12979	43