Subgroups — #24993298

Order 11008.

594 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137321	C₂	1	1	yes	maximal, characteristic
2	#19135577	C₂	1	1	yes	maximal, characteristic, complemented
3	#19133263	C₂	1	1	yes	maximal, characteristic
4	#19133159	C₂	1	1	yes	maximal, characteristic, complemented
5	#19132454	C₂	1	1	yes	maximal, characteristic, complemented
6	#19132423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#3208	C₂	1	1	yes	maximal, characteristic
8	C₂×D₄×Dic₄₃	—	2	1	no	—
9	#14582952	C₂²	1	1	yes	characteristic
10	#14582923	C₂²	1	1	yes	characteristic
11	#14582802	—	2	1	no	—
12	#14582779	—	2	2	no	—
13	#14582779	—	2	2	no	—
14	#14582765	C₂²	1	1	yes	characteristic
15	#14582745	C₂²	1	1	yes	characteristic
16	#14582658	—	2	1	no	—
17	#14582658	—	2	1	no	—
18	#14582643	—	2	2	no	—
19	#14582636	—	2	2	no	—
20	C₄₃×#533	C₂²	1	1	yes	characteristic
21	C₂×#127×C₄₃	C₂²	1	1	yes	characteristic
22	C₄₃×#492	C₄	1	2	yes	complemented
23	C₄₃×#486	C₄	1	2	yes	complemented
24	C₄₃×#460	—	2	2	no	—
25	C₄₃×#450	—	2	2	no	—
26	C₄₃×#431	C₂²	1	1	yes	characteristic
27	C₂³×Dic₄₃	—	2	2	no	—
28	C₂×#10621877	—	2	1	no	—
29	D₄×Dic₄₃	—	4	2	no	—
30	D₄×Dic₄₃	—	4	2	no	—
31	C₂×C₄⋊Dic₄₃	—	2	1	no	—
32	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
33	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
34	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
36	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
37	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
38	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
39	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
40	C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
41	C₄×C₄₃⋊C₈	—	2	1	no	—
42	C₄₃⋊#95	—	2	1	no	—
43	C₄₃⋊#95	—	2	1	no	—
44	C₄₃⋊#95	—	2	1	no	—
45	C₄₃⋊#95	D₄	1	2	yes	—
46	C₂²×D₄×C₄₃	—	2	1	no	—
47	C₂²×D₄×C₄₃	—	2	1	no	—
48	C₂²×D₄×C₄₃	D₄	1	2	yes	—
49	C₂×SD₁₆×C₄₃	—	2	4	no	—
50	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic, complemented
51	C₄⋊Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic, complemented
52	#127×C₄₃	—	2	1	no	—
53	#127×C₄₃	—	2	1	no	—
54	#127×C₄₃	C₂×C₄	1	2	yes	—
55	#127×C₄₃	D₄	1	2	yes	—
56	#125×C₄₃	—	2	1	no	—
57	C₂²⋊Q₈×C₄₃	—	2	1	no	—
58	C₂²⋊Q₈×C₄₃	—	2	1	no	—
59	C₄×Q₈×C₄₃	—	2	1	no	—
60	C₄×Q₈×C₄₃	—	2	1	no	—
61	#117×C₄₃	—	2	1	no	—
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	C₄⋊C₈×C₄₃	—	2	1	no	—
64	#102×C₄₃	—	2	4	no	—
65	C₂²⋊C₈×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	complemented
67	C₂⁴×C₄₃	—	2	2	no	—
68	C₂×Q₈×C₄₃	—	2	1	no	—
69	C₂×Q₈×C₄₃	—	2	1	no	—
70	C₂×D₄×C₄₃	—	2	2	no	—
71	C₂×D₄×C₄₃	—	2	2	no	—
72	C₂×D₄×C₄₃	—	2	2	no	—
73	C₂×D₄×C₄₃	—	2	2	no	—
74	C₂×D₄×C₄₃	—	2	2	no	—
75	C₂×D₄×C₄₃	—	4	1	no	—
76	C₂×D₄×C₄₃	—	4	1	no	—
77	C₂×D₄×C₄₃	—	4	1	no	—
78	C₂×D₄×C₄₃	—	4	1	no	—
79	C₂×D₄×C₄₃	—	8	2	no	—
80	C₂×D₄×C₄₃	—	8	2	no	—
81	C₂×D₄×C₄₃	C₂²⋊C₄	1	4	yes	—
82	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×Dic₄₃	—	2	1	no	—
86	C₂²×Dic₄₃	—	2	1	no	—
87	C₂²×Dic₄₃	—	2	2	no	—
88	C₂²×Dic₄₃	—	4	4	no	—
89	C₂²×Dic₄₃	—	8	2	no	—
90	C₂²×Dic₄₃	—	8	2	no	—
91	SD₁₆×C₄₃	—	8	4	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₄⋊C₄×C₄₃	—	2	1	no	—
94	C₄⋊C₄×C₄₃	—	2	1	no	—
95	C₄⋊C₄×C₄₃	—	2	1	no	—
96	C₄⋊C₄×C₄₃	—	2	1	no	—
97	C₄⋊C₄×C₄₃	—	4	1	no	—
98	C₄⋊C₄×C₄₃	—	4	1	no	—
99	C₂²⋊C₄×C₄₃	—	4	1	no	—
100	C₄²×C₄₃	—	4	1	no	—
101	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
102	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
103	C₄²×C₄₃	#41	1	1	yes	characteristic
104	C₄²×C₄₃	#41	1	1	yes	characteristic
105	C₂²⋊Dic₄₃	—	4	4	no	—
106	C₂²⋊Dic₄₃	—	8	2	no	—
107	C₂²⋊Dic₄₃	—	8	2	no	—
108	C₄⋊Dic₄₃	—	4	1	no	—
109	C₄⋊Dic₄₃	—	4	1	no	—
110	C₄×Dic₄₃	—	2	1	no	—
111	C₄×Dic₄₃	—	2	1	no	—
112	C₄×Dic₄₃	—	2	1	no	—
113	C₄×Dic₄₃	SD₁₆	1	2	yes	complemented
114	#10621877	—	4	1	no	—
115	#10621877	—	4	1	no	—
116	C₂×C₄₃⋊C₈	—	4	1	no	—
117	C₂×C₄₃⋊C₈	—	4	1	no	—
118	C₂³×C₄₃	—	2	4	no	—
119	C₂³×C₄₃	—	4	2	no	—
120	C₂³×C₄₃	—	4	2	no	—
121	C₂³×C₄₃	—	4	2	no	—
122	C₂³×C₄₃	—	4	2	no	—
123	C₂³×C₄₃	—	4	2	no	—
124	C₂³×C₄₃	—	4	2	no	—
125	C₂³×C₄₃	#120	1	1	yes	characteristic
126	Q₈×C₄₃	—	4	2	no	—
127	Q₈×C₄₃	—	8	1	no	—
128	D₄×C₄₃	—	4	4	no	—
129	D₄×C₄₃	—	4	2	no	—
130	D₄×C₄₃	—	4	2	no	—
131	D₄×C₄₃	—	4	2	no	—
132	D₄×C₄₃	—	4	2	no	—
133	D₄×C₄₃	—	8	2	no	—
134	D₄×C₄₃	—	8	2	no	—
135	D₄×C₄₃	—	8	2	no	—
136	C₂×C₄×C₄₃	—	4	1	no	—
137	C₂×C₄×C₄₃	—	4	1	no	—
138	C₂×C₄×C₄₃	—	4	1	no	—
139	C₂×C₄×C₄₃	—	4	1	no	—
140	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
141	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
147	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
148	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	1	yes	characteristic
149	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
150	C₂×Dic₄₃	—	2	2	no	—
151	C₂×Dic₄₃	—	4	1	no	—
152	C₂×Dic₄₃	—	4	1	no	—
153	C₂×Dic₄₃	—	4	1	no	—
154	C₂×Dic₄₃	—	4	1	no	—
155	C₂×Dic₄₃	—	4	2	no	—
156	C₂×Dic₄₃	—	4	2	no	—
157	C₂×Dic₄₃	—	4	2	no	—
158	C₂×Dic₄₃	—	4	1	no	—
159	C₂×Dic₄₃	—	8	8	no	—
160	C₈×C₄₃	—	4	2	no	—
161	C₄₃⋊C₈	—	8	1	no	—
162	C₄₃⋊C₈	—	8	1	no	—
163	#9295	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	1	no	—
165	C₂²×C₄₃	—	2	1	no	—
166	C₂²×C₄₃	—	4	4	no	—
167	C₂²×C₄₃	—	4	4	no	—
168	C₂²×C₄₃	—	4	4	no	—
169	C₂²×C₄₃	—	8	2	no	—
170	C₂²×C₄₃	—	8	2	no	—
171	C₂²×C₄₃	—	8	2	no	—
172	C₂²×C₄₃	—	8	2	no	—
173	C₂²×C₄₃	#386	1	1	yes	characteristic
174	C₂²×C₄₃	#450	1	1	yes	characteristic
175	C₂²×C₄₃	#452	1	1	yes	characteristic
176	C₄×C₄₃	—	2	1	no	—
177	C₄×C₄₃	—	2	1	no	—
178	C₄×C₄₃	—	2	1	no	—
179	C₄×C₄₃	—	2	1	no	—
180	C₄×C₄₃	—	8	1	no	—
181	C₄×C₄₃	—	8	1	no	—
182	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
183	C₄×C₄₃	#440	1	1	yes	characteristic
184	C₄×C₄₃	#461	1	1	yes	characteristic
185	C₄×C₄₃	#463	1	1	yes	characteristic
186	Dic₄₃	—	4	2	no	—
187	Dic₄₃	—	8	1	no	—
188	Dic₄₃	—	8	2	no	—
189	#3208	D₄₃	1	1	yes	characteristic
190	#2328	—	43	1	no	—
191	#1995	—	43	1	no	—
192	#1545	—	43	1	no	—
193	#1518	—	43	1	no	—
194	#1286	—	43	1	no	—
195	#1274	—	43	1	no	—
196	C₂×C₄₃	—	2	1	no	—
197	C₂×C₄₃	—	2	1	no	—
198	C₂×C₄₃	—	8	4	no	—
199	C₂×C₄₃	#1877	1	1	yes	characteristic
200	C₂×C₄₃	#1904	1	1	yes	characteristic
201	C₂×C₄₃	#1911	1	1	yes	characteristic
202	#533	C₂×D₄₃	1	1	yes	characteristic
203	C₂×#127	C₂×D₄₃	1	1	yes	characteristic
204	C₂×C₄×D₄	—	86	1	no	—
205	#492	Dic₄₃	1	2	yes	complemented
206	#486	Dic₄₃	1	2	yes	complemented
207	#460	—	2	2	no	—
208	#450	—	2	2	no	—
209	#431	—	43	1	no	—
210	#431	C₂×D₄₃	1	1	yes	characteristic
211	#423	—	43	1	no	—
212	#390	—	86	1	no	—
213	#386	—	86	2	no	—
214	#386	—	86	2	no	—
215	#382	—	43	1	no	—
216	#376	—	43	1	no	—
217	#339	—	86	1	no	—
218	#339	—	86	1	no	—
219	#331	—	86	2	no	—
220	#328	—	86	2	no	—
221	C₄₃	#9295	1	1	yes	characteristic, complemented, Sylow
222	C₂²×D₄	—	2	1	no	—
223	C₂²×D₄	—	2	1	no	—
224	C₂²×D₄	C₂²⋊D₄₃	1	2	yes	—
225	C₂³×C₄	—	86	2	no	—
226	C₂×SD₁₆	—	2	4	no	—
227	C₂×M₁₆	—	86	1	no	—
228	C₄⋊Q₈	C₂×Dic₄₃	1	1	yes	characteristic, complemented
229	C₄⋊Q₈	C₂×Dic₄₃	1	1	yes	characteristic, complemented
230	#127	—	2	1	no	—
231	#127	—	2	1	no	—
232	#127	C₂×Dic₄₃	1	2	yes	—
233	#127	C₂²⋊D₄₃	1	2	yes	—
234	#125	—	2	1	no	—
235	C₂²⋊Q₈	—	2	1	no	—
236	C₂²⋊Q₈	—	2	1	no	—
237	C₄×Q₈	—	2	1	no	—
238	C₄×Q₈	—	2	1	no	—
239	C₄×D₄	—	172	2	no	—
240	C₄×D₄	—	172	2	no	—
241	#117	—	2	1	no	—
242	C₂×C₄⋊C₄	—	86	1	no	—
243	C₂×C₂²⋊C₄	—	86	2	no	—
244	C₂×C₂²⋊C₄	—	86	2	no	—
245	C₂×C₂²⋊C₄	—	86	2	no	—
246	C₂×C₄²	—	43	1	no	—
247	C₂×C₄²	—	43	1	no	—
248	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
249	C₄⋊C₈	—	2	1	no	—
250	C₄⋊C₈	—	86	1	no	—
251	C₄⋊C₈	—	86	1	no	—
252	C₄⋊C₈	—	86	1	no	—
253	#102	—	2	4	no	—
254	C₂²⋊C₈	—	2	1	no	—
255	C₂²⋊C₈	—	86	1	no	—
256	C₄×C₈	—	86	1	no	—
257	C₄×C₈	C₂×Dic₄₃	1	2	yes	complemented
258	#95	—	43	2	no	—
259	#95	—	86	1	no	—
260	#95	—	86	1	no	—
261	#95	—	86	1	no	—
262	C₂⁴	—	2	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	2	2	no	—
268	C₂×D₄	—	2	2	no	—
269	C₂×D₄	—	2	2	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	—	4	1	no	—
272	C₂×D₄	—	4	1	no	—
273	C₂×D₄	—	4	1	no	—
274	C₂×D₄	—	8	2	no	—
275	C₂×D₄	—	8	2	no	—
276	C₂×D₄	C₂²⋊Dic₄₃	1	4	yes	—
277	C₂²×C₄	—	86	1	no	—
278	C₂²×C₄	—	86	2	no	—
279	C₂²×C₄	—	86	1	no	—
280	C₂²×C₄	—	344	2	no	—
281	C₂²×C₄	—	344	2	no	—
282	C₂²×C₄	—	172	4	no	—
283	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
284	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
285	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
286	SD₁₆	—	8	4	no	—
287	M₁₆	—	172	1	no	—
288	M₁₆	—	172	1	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	172	1	no	—
291	C₂×C₈	—	172	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	—	2	1	no	—
294	C₄⋊C₄	—	2	1	no	—
295	C₄⋊C₄	—	2	1	no	—
296	C₄⋊C₄	—	4	1	no	—
297	C₄⋊C₄	—	4	1	no	—
298	C₄⋊C₄	—	172	1	no	—
299	C₄⋊C₄	—	172	1	no	—
300	C₂²⋊C₄	—	4	1	no	—
301	C₂²⋊C₄	—	344	2	no	—
302	C₂²⋊C₄	—	344	2	no	—
303	C₂²⋊C₄	—	172	4	no	—
304	C₄²	—	4	1	no	—
305	C₄²	—	43	2	no	—
306	C₄²	—	86	1	no	—
307	C₄²	—	86	1	no	—
308	C₄²	—	86	1	no	—
309	C₄²	#10621900	1	1	yes	characteristic
310	C₄²	#10621900	1	1	yes	characteristic
311	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
312	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
313	C₂³	—	2	4	no	—
314	C₂³	—	4	2	no	—
315	C₂³	—	4	2	no	—
316	C₂³	—	4	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	—	4	2	no	—
319	C₂³	—	4	2	no	—
320	C₂³	#13063293	1	1	yes	characteristic
321	Q₈	—	4	2	no	—
322	Q₈	—	8	1	no	—
323	D₄	—	4	4	no	—
324	D₄	—	4	2	no	—
325	D₄	—	4	2	no	—
326	D₄	—	4	2	no	—
327	D₄	—	4	2	no	—
328	D₄	—	8	2	no	—
329	D₄	—	8	2	no	—
330	D₄	—	8	2	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	—	4	1	no	—
333	C₂×C₄	—	4	1	no	—
334	C₂×C₄	—	4	1	no	—
335	C₂×C₄	—	86	2	no	—
336	C₂×C₄	—	344	8	no	—
337	C₂×C₄	—	172	1	no	—
338	C₂×C₄	—	172	1	no	—
339	C₂×C₄	—	172	2	no	—
340	C₂×C₄	—	172	2	no	—
341	C₂×C₄	—	172	2	no	—
342	C₂×C₄	—	172	1	no	—
343	C₂×C₄	—	172	1	no	—
344	C₂×C₄	—	172	1	no	—
345	C₂×C₄	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
346	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
347	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
348	C₂×C₄	#13063297	1	1	yes	characteristic
349	C₂×C₄	#13063297	1	1	yes	characteristic
350	C₂×C₄	#13063306	1	1	yes	characteristic
351	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
352	C₂×C₄	#13063346	1	1	yes	characteristic
353	C₂×C₄	#13063352	1	1	yes	characteristic
354	C₂×C₄	#13063354	1	1	yes	characteristic
355	C₈	—	4	2	no	—
356	C₈	—	344	1	no	—
357	C₈	—	344	1	no	—
358	C₂²	—	2	1	no	—
359	C₂²	—	2	1	no	—
360	C₂²	—	4	4	no	—
361	C₂²	—	4	4	no	—
362	C₂²	—	4	4	no	—
363	C₂²	—	8	2	no	—
364	C₂²	—	8	2	no	—
365	C₂²	—	8	2	no	—
366	C₂²	—	8	2	no	—
367	C₂²	#14582779	1	1	yes	characteristic
368	C₂²	#14583048	1	1	yes	characteristic
369	C₂²	#14583058	1	1	yes	characteristic
370	C₄	—	2	1	no	—
371	C₄	—	2	1	no	—
372	C₄	—	2	1	no	—
373	C₄	—	2	1	no	—
374	C₄	—	8	1	no	—
375	C₄	—	8	1	no	—
376	C₄	—	344	2	no	—
377	C₄	—	344	1	no	—
378	C₄	—	172	2	no	—
379	C₄	SD₁₆×Dic₄₃	1	1	yes	characteristic
380	C₄	#14583000	1	1	yes	characteristic
381	C₄	#14583102	1	1	yes	characteristic
382	C₄	#14583116	1	1	yes	characteristic
383	C₂	—	2	1	no	—
384	C₂	—	2	1	no	—
385	C₂	—	8	4	no	—
386	C₂	#19134931	1	1	yes	characteristic
387	C₂	#19135084	1	1	yes	characteristic
388	C₂	#19135128	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	#14582779
commutator	C₄²×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#3208	C₂
socle	C₂²×C₄₃	#386
prime radical	C₂×#127×C₄₃	C₂²
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#3208	D₄₃
hypocenter	C₄₃	#9295
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3208

Complements

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

N	H ≅ G/N	type
2. #19135577	C₂	semidirect
4. #19133159	C₂	semidirect
5. #19132454	C₂	semidirect
6. #19132423	C₂	semidirect
22. C₄₃×#492	C₄	semidirect
23. C₄₃×#486	C₄	semidirect
50. C₄⋊Q₈×C₄₃	C₂×C₄	semidirect
51. C₄⋊Q₈×C₄₃	C₂×C₄	semidirect
66. C₄×C₈×C₄₃	C₂×C₄	semidirect
113. C₄×Dic₄₃	SD₁₆	semidirect
205. #492	Dic₄₃	semidirect
206. #486	Dic₄₃	semidirect
221. C₄₃	#9295	semidirect
228. C₄⋊Q₈	C₂×Dic₄₃	semidirect
229. C₄⋊Q₈	C₂×Dic₄₃	semidirect
257. C₄×C₈	C₂×Dic₄₃	semidirect

Maximal subgroups

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

subgroup	index
#19137321	2
#19135577	2
#19133263	2
#19133159	2
#19132454	2
#19132423	2
C₄₃×#3208	2
#9295	43