Subgroups — #24999431

Order 11008.

734 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583295	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#14583273	C₂	1	1	yes	maximal, characteristic
3	C₂×#14583272	C₂	1	1	yes	maximal, characteristic
4	#19135555	C₂	1	1	yes	maximal, characteristic, complemented
5	#19135400	C₂	1	1	yes	maximal, characteristic
6	C₄₃⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621873	—	2	2	no	—
9	C₂²×#10621884	—	2	2	no	—
10	C₂×#13063325	C₂²	1	1	yes	characteristic
11	C₂×#13063322	C₂²	1	1	yes	characteristic
12	#14583295	—	2	2	no	—
13	#14583295	—	2	2	no	—
14	#14583273	—	2	2	no	—
15	#14583273	—	2	2	no	—
16	#14583272	—	2	2	no	—
17	#14583272	—	2	2	no	—
18	C₂×#13063256	—	2	2	no	—
19	C₂×#13063245	—	2	2	no	—
20	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
22	#14582661	—	2	2	no	—
23	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
25	C₄₃×#384	C₂²	1	1	yes	characteristic
26	C₄₃×#340	—	2	2	no	—
27	C₂²×Dic₈₆	—	2	1	no	—
28	C₂²×D₁₇₂	—	2	1	no	—
29	C₂²×D₁₇₂	—	2	1	no	—
30	C₂²×D₁₇₂	—	2	1	no	—
31	C₂×#10621873	—	2	8	no	—
32	C₂×#10621873	—	4	4	no	—
33	C₂×#10621873	—	4	4	no	—
34	C₂×#10621884	—	2	8	no	—
35	C₂×#10621884	—	4	4	no	—
36	C₂×#10621884	—	4	4	no	—
37	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
38	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
39	#13063325	—	2	1	no	—
40	#13063325	—	2	1	no	—
41	#13063325	—	2	1	no	—
42	#13063325	D₄	1	2	yes	—
43	#13063322	—	2	1	no	—
44	#13063322	—	2	1	no	—
45	#13063322	—	2	1	no	—
46	#13063322	D₄	1	2	yes	—
47	C₂×C₄⋊Dic₄₃	—	2	1	no	—
48	C₂×C₄⋊Dic₄₃	—	2	1	no	—
49	#13063256	—	2	8	no	—
50	#13063245	—	4	4	no	—
51	C₄×C₄₃⋊C₈	—	2	1	no	—
52	C₄×C₄₃⋊C₈	—	2	1	no	—
53	C₄×C₄₃⋊C₈	—	2	1	no	—
54	C₄×C₄₃⋊C₈	—	2	1	no	—
55	C₂²×Q₈×C₄₃	—	2	1	no	—
56	C₂²×C₈×C₄₃	D₄	1	2	yes	—
57	C₄⋊Q₈×C₄₃	—	2	1	no	—
58	C₄⋊Q₈×C₄₃	—	2	1	no	—
59	C₄⋊Q₈×C₄₃	—	2	1	no	—
60	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
63	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
64	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
65	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
66	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₃	—	2	1	no	—
68	C₄×C₈×C₄₃	—	2	1	no	—
69	C₂³×D₄₃	—	4	1	no	—
70	C₂×Q₈×C₄₃	—	2	2	no	—
71	C₂×Q₈×C₄₃	—	4	1	no	—
72	C₂×Q₈×C₄₃	—	4	2	no	—
73	C₂×Q₈×C₄₃	—	4	2	no	—
74	C₂²×C₄×C₄₃	—	4	1	no	—
75	C₂²×C₄×C₄₃	—	4	1	no	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×Dic₄₃	—	4	1	no	—
80	C₂×D₁₇₂	—	2	2	no	—
81	C₂×D₁₇₂	—	2	2	no	—
82	C₂×D₁₇₂	—	2	2	no	—
83	C₂×D₁₇₂	—	4	1	no	—
84	C₂×D₁₇₂	—	4	1	no	—
85	C₂×D₁₇₂	—	4	1	no	—
86	C₂×D₁₇₂	—	4	2	no	—
87	C₂×D₁₇₂	—	4	2	no	—
88	C₂×D₁₇₂	—	4	2	no	—
89	C₂×D₁₇₂	—	4	2	no	—
90	C₂×D₁₇₂	—	4	2	no	—
91	C₂×D₁₇₂	—	4	2	no	—
92	C₂×Dic₈₆	—	2	2	no	—
93	C₂×Dic₈₆	—	4	2	no	—
94	C₂×Dic₈₆	—	4	2	no	—
95	C₂×Dic₈₆	—	4	1	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	SD₁₆	1	4	yes	complemented
99	C₄⋊C₄×C₄₃	—	2	2	no	—
100	C₄⋊C₄×C₄₃	—	2	2	no	—
101	C₄⋊C₄×C₄₃	—	4	1	no	—
102	C₄⋊C₄×C₄₃	—	4	1	no	—
103	C₄⋊C₄×C₄₃	—	4	2	no	—
104	C₄⋊C₄×C₄₃	—	4	2	no	—
105	C₄⋊C₄×C₄₃	—	4	2	no	—
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
109	C₄²×C₄₃	#41	1	1	yes	characteristic
110	#10621884	—	4	16	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	—	2	2	no	—
113	C₄⋊Dic₄₃	—	4	1	no	—
114	C₄⋊Dic₄₃	—	4	1	no	—
115	C₂×C₄₃⋊C₈	—	2	2	no	—
116	C₂×C₄₃⋊C₈	—	2	2	no	—
117	C₂×C₄₃⋊C₈	—	2	2	no	—
118	C₂×C₄₃⋊C₈	—	2	2	no	—
119	C₂×C₄₃⋊C₈	SD₁₆	1	4	yes	complemented
120	C₂×C₄₃⋊C₈	SD₁₆	1	4	yes	complemented
121	#10621873	—	4	16	no	—
122	C₂³×C₄₃	#127	1	1	yes	characteristic
123	C₂²×D₄₃	—	4	2	no	—
124	C₂²×D₄₃	—	8	1	no	—
125	C₂²×D₄₃	—	8	1	no	—
126	C₂²×D₄₃	—	8	1	no	—
127	C₂²×D₄₃	—	8	1	no	—
128	C₂²×D₄₃	—	8	1	no	—
129	C₂²×D₄₃	—	8	1	no	—
130	Q₈×C₄₃	—	4	4	no	—
131	Q₈×C₄₃	—	8	2	no	—
132	C₂×C₄×C₄₃	—	4	2	no	—
133	C₂×C₄×C₄₃	—	8	1	no	—
134	C₂×C₄×C₄₃	—	8	1	no	—
135	C₂×C₄×C₄₃	—	8	1	no	—
136	C₂×C₄×C₄₃	—	8	1	no	—
137	C₂×C₄×C₄₃	—	8	1	no	—
138	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
141	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
145	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
146	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
147	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
149	C₂×C₄×C₄₃	#137	1	2	yes	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₂×Dic₄₃	—	8	1	no	—
152	C₂×Dic₄₃	—	8	1	no	—
153	D₁₇₂	—	4	4	no	—
154	D₁₇₂	—	4	4	no	—
155	D₁₇₂	—	4	4	no	—
156	D₁₇₂	—	8	2	no	—
157	D₁₇₂	—	8	2	no	—
158	D₁₇₂	—	8	2	no	—
159	Dic₈₆	—	4	4	no	—
160	Dic₈₆	—	8	2	no	—
161	C₈×C₄₃	—	2	4	no	—
162	C₄₃⋊C₈	—	2	4	no	—
163	C₄₃⋊C₈	—	2	4	no	—
164	#10315	—	43	1	no	maximal, Sylow
165	C₂²×C₄₃	#392	1	1	yes	characteristic
166	C₂²×C₄₃	#492	1	1	yes	characteristic
167	C₂²×C₄₃	#492	1	1	yes	characteristic
168	C₂²×C₄₃	#492	1	1	yes	characteristic
169	C₂²×C₄₃	#496	1	1	yes	characteristic
170	C₂²×C₄₃	#496	1	1	yes	characteristic
171	C₂²×C₄₃	#497	1	1	yes	characteristic
172	C₂×D₄₃	—	8	2	no	—
173	C₂×D₄₃	—	8	2	no	—
174	C₂×D₄₃	—	8	2	no	—
175	C₂×D₄₃	—	16	1	no	—
176	C₂×D₄₃	—	16	1	no	—
177	C₂×D₄₃	—	16	1	no	—
178	C₂×D₄₃	—	16	1	no	—
179	C₄×C₄₃	—	8	2	no	—
180	C₄×C₄₃	—	16	1	no	—
181	C₄×C₄₃	#448	1	2	yes	—
182	C₄×C₄₃	#450	1	2	yes	—
183	C₄×C₄₃	#450	1	2	yes	—
184	C₄×C₄₃	#460	1	2	yes	—
185	C₄×C₄₃	#460	1	2	yes	—
186	C₄×C₄₃	#461	1	2	yes	—
187	Dic₄₃	—	8	2	no	—
188	C₂×#498	—	43	1	no	—
189	C₂×#492	—	43	1	no	—
190	C₂×#492	—	43	1	no	—
191	#1990	—	43	1	no	—
192	#1990	D₄₃	1	1	yes	characteristic, complemented
193	#1957	—	43	1	no	—
194	#1273	—	43	1	no	—
195	C₂×C₄₃	#1705	1	1	yes	characteristic
196	C₂×C₄₃	#1714	1	1	yes	characteristic
197	C₂×C₄₃	#1714	1	1	yes	characteristic
198	C₂×C₄₃	#1715	1	1	yes	characteristic
199	C₂×C₄₃	#2013	1	1	yes	characteristic
200	C₂×C₄₃	#2014	1	1	yes	characteristic
201	C₂×C₄₃	#2014	1	1	yes	characteristic
202	D₄₃	—	16	2	no	—
203	C₂²×SD₁₆	—	86	2	no	—
204	C₂²×SD₁₆	—	86	2	no	—
205	C₂×C₄⋊Q₈	—	43	1	no	—
206	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
207	C₂×#127	—	43	1	no	—
208	#498	—	86	2	no	—
209	#498	—	86	2	no	—
210	#492	—	86	2	no	—
211	#492	—	86	2	no	—
212	#492	—	86	2	no	—
213	#492	—	86	2	no	—
214	C₂×#106	—	86	2	no	—
215	C₂×#102	—	86	2	no	—
216	C₂×C₄×C₈	—	43	1	no	—
217	C₂×C₄×C₈	—	43	1	no	—
218	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
219	#384	C₂×D₄₃	1	1	yes	characteristic
220	#340	—	2	2	no	—
221	#340	—	86	2	no	—
222	C₄₃	#10315	1	1	yes	characteristic, complemented, Sylow
223	C₂²×Q₈	—	2	1	no	—
224	C₂²×Q₈	—	86	1	no	—
225	C₂²×D₄	—	86	1	no	—
226	C₂²×D₄	—	86	1	no	—
227	C₂²×D₄	—	86	1	no	—
228	C₂×SD₁₆	—	86	8	no	—
229	C₂×SD₁₆	—	86	8	no	—
230	C₂×SD₁₆	—	172	4	no	—
231	C₂×SD₁₆	—	172	4	no	—
232	C₂×SD₁₆	—	172	4	no	—
233	C₂×SD₁₆	—	172	4	no	—
234	C₂²×C₈	—	43	2	no	—
235	C₂²×C₈	—	43	2	no	—
236	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
237	C₄⋊Q₈	—	2	1	no	—
238	C₄⋊Q₈	—	2	1	no	—
239	C₄⋊Q₈	—	2	1	no	—
240	C₄⋊Q₈	—	43	2	no	—
241	C₄⋊Q₈	—	86	1	no	—
242	C₄⋊Q₈	—	86	1	no	—
243	C₄⋊Q₈	—	86	1	no	—
244	C₄⋊Q₈	D₁₇₂	1	2	yes	—
245	#127	—	43	2	no	—
246	#127	—	86	1	no	—
247	#127	—	86	1	no	—
248	#127	—	86	1	no	—
249	C₂×C₄⋊C₄	—	2	1	no	—
250	C₂×C₄⋊C₄	—	2	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	2	1	no	—
253	C₂×C₄⋊C₄	—	2	1	no	—
254	C₂×C₄⋊C₄	—	86	1	no	—
255	C₂×C₄⋊C₄	—	86	1	no	—
256	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
257	#106	—	86	8	no	—
258	#102	—	172	4	no	—
259	C₄×C₈	—	2	1	no	—
260	C₄×C₈	—	2	1	no	—
261	C₄×C₈	—	86	1	no	—
262	C₄×C₈	—	86	1	no	—
263	C₄×C₈	—	86	1	no	—
264	C₄×C₈	—	86	1	no	—
265	C₂⁴	—	172	1	no	—
266	C₂×Q₈	—	2	2	no	—
267	C₂×Q₈	—	4	1	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×Q₈	—	4	2	no	—
270	C₂×Q₈	—	86	2	no	—
271	C₂×Q₈	—	172	2	no	—
272	C₂×Q₈	—	172	2	no	—
273	C₂×Q₈	—	172	1	no	—
274	C₂×D₄	—	86	2	no	—
275	C₂×D₄	—	86	2	no	—
276	C₂×D₄	—	86	2	no	—
277	C₂×D₄	—	172	1	no	—
278	C₂×D₄	—	172	1	no	—
279	C₂×D₄	—	172	1	no	—
280	C₂×D₄	—	172	2	no	—
281	C₂×D₄	—	172	2	no	—
282	C₂×D₄	—	172	2	no	—
283	C₂×D₄	—	172	2	no	—
284	C₂×D₄	—	172	2	no	—
285	C₂×D₄	—	172	2	no	—
286	C₂²×C₄	—	4	1	no	—
287	C₂²×C₄	—	4	1	no	—
288	C₂²×C₄	—	172	1	no	—
289	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
290	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
291	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
292	SD₁₆	—	172	16	no	—
293	SD₁₆	—	172	16	no	—
294	C₂×C₈	—	2	2	no	—
295	C₂×C₈	—	2	2	no	—
296	C₂×C₈	—	43	4	no	—
297	C₂×C₈	—	43	4	no	—
298	C₂×C₈	—	86	2	no	—
299	C₂×C₈	—	86	2	no	—
300	C₂×C₈	—	86	2	no	—
301	C₂×C₈	—	86	2	no	—
302	C₂×C₈	#10621884	1	4	yes	complemented
303	C₄⋊C₄	—	2	2	no	—
304	C₄⋊C₄	—	2	2	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	1	no	—
307	C₄⋊C₄	—	4	2	no	—
308	C₄⋊C₄	—	4	2	no	—
309	C₄⋊C₄	—	4	2	no	—
310	C₄⋊C₄	—	86	2	no	—
311	C₄⋊C₄	—	86	2	no	—
312	C₄⋊C₄	—	172	1	no	—
313	C₄⋊C₄	—	172	1	no	—
314	C₄²	C₂×D₁₇₂	1	1	yes	characteristic
315	C₄²	#10621898	1	1	yes	characteristic
316	C₄²	D₄×D₄₃	1	1	yes	characteristic
317	C₄²	D₄×D₄₃	1	1	yes	characteristic
318	C₂³	—	344	1	no	—
319	C₂³	—	344	1	no	—
320	C₂³	—	344	1	no	—
321	C₂³	—	344	1	no	—
322	C₂³	—	344	1	no	—
323	C₂³	—	344	1	no	—
324	C₂³	—	172	2	no	—
325	C₂³	#13063321	1	1	yes	characteristic
326	Q₈	—	4	4	no	—
327	Q₈	—	8	2	no	—
328	Q₈	—	344	2	no	—
329	Q₈	—	172	4	no	—
330	D₄	—	344	2	no	—
331	D₄	—	344	2	no	—
332	D₄	—	344	2	no	—
333	D₄	—	172	4	no	—
334	D₄	—	172	4	no	—
335	D₄	—	172	4	no	—
336	C₂×C₄	—	4	2	no	—
337	C₂×C₄	—	8	1	no	—
338	C₂×C₄	—	8	1	no	—
339	C₂×C₄	—	8	1	no	—
340	C₂×C₄	—	8	1	no	—
341	C₂×C₄	—	8	1	no	—
342	C₂×C₄	—	344	1	no	—
343	C₂×C₄	—	344	1	no	—
344	C₂×C₄	—	172	2	no	—
345	C₂×C₄	#13063293	1	1	yes	characteristic
346	C₂×C₄	#13063295	1	1	yes	characteristic
347	C₂×C₄	#13063295	1	1	yes	characteristic
348	C₂×C₄	#13063298	1	1	yes	characteristic
349	C₂×C₄	#13063298	1	1	yes	characteristic
350	C₂×C₄	#13063299	1	1	yes	characteristic
351	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
352	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
353	C₂×C₄	C₂×#10621884	1	2	yes	—
354	C₂×C₄	#13063349	1	2	yes	—
355	C₂×C₄	#13063350	1	2	yes	—
356	C₂×C₄	#13063357	1	2	yes	—
357	C₈	—	2	4	no	—
358	C₈	—	86	4	no	—
359	C₈	—	86	4	no	—
360	C₂²	—	344	2	no	—
361	C₂²	—	344	2	no	—
362	C₂²	—	344	2	no	—
363	C₂²	—	688	1	no	—
364	C₂²	—	688	1	no	—
365	C₂²	—	688	1	no	—
366	C₂²	—	688	1	no	—
367	C₂²	#14582808	1	1	yes	characteristic
368	C₂²	#14583270	1	1	yes	characteristic
369	C₂²	#14583270	1	1	yes	characteristic
370	C₂²	#14583273	1	1	yes	characteristic
371	C₂²	#14583287	1	1	yes	characteristic
372	C₂²	#14583289	1	1	yes	characteristic
373	C₂²	#14583292	1	1	yes	characteristic
374	C₄	—	8	2	no	—
375	C₄	—	16	1	no	—
376	C₄	—	344	2	no	—
377	C₄	#14583039	1	2	yes	—
378	C₄	#14583049	1	2	yes	—
379	C₄	#14583051	1	2	yes	—
380	C₄	#14583098	1	2	yes	—
381	C₄	#14583099	1	2	yes	—
382	C₄	#14583103	1	2	yes	—
383	C₂	—	688	2	no	—
384	C₂	#19134183	1	1	yes	characteristic
385	C₂	#19134218	1	1	yes	characteristic
386	C₂	#19134219	1	1	yes	characteristic
387	C₂	#19134225	1	1	yes	characteristic
388	C₂	#19135679	1	1	yes	characteristic
389	C₂	#19135682	1	1	yes	characteristic
390	C₂	#19135684	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063321
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1990	C₂
socle	C₂³×C₄₃	#127
prime radical	C₂×#13063322	C₂²
hypercenter	#1990	D₄₃
hypocenter	C₄₃	#10315
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1990

Complements

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

N	H ≅ G/N	type
1. C₂×#14583295	C₂	semidirect
4. #19135555	C₂	semidirect
6. C₄₃⋊#1273	C₂	semidirect
7. C₄₃×#1990	C₂	semidirect
98. C₂×C₈×C₄₃	SD₁₆	semidirect
119. C₂×C₄₃⋊C₈	SD₁₆	semidirect
120. C₂×C₄₃⋊C₈	SD₁₆	semidirect
192. #1990	D₄₃	semidirect
222. C₄₃	#10315	semidirect
302. C₂×C₈	#10621884	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583295	2
C₂×#14583273	2
C₂×#14583272	2
#19135555	2
#19135400	2
C₄₃⋊#1273	2
C₄₃×#1990	2
#10315	43