Subgroups — #24999396

Order 11008.

670 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583273	C₂	1	1	yes	maximal, characteristic
2	C₂×#14583247	C₂	1	1	yes	maximal, characteristic
3	#19135557	C₂	1	1	yes	maximal, characteristic, complemented
4	#19135557	C₂	1	1	yes	maximal, characteristic, complemented
5	#19135401	C₂	1	1	yes	maximal, characteristic
6	C₄₃⋊#1273	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#10621884	—	2	2	no	—
9	C₂×#13063322	C₂²	1	1	yes	characteristic
10	#14583273	—	2	2	no	—
11	#14583273	—	2	2	no	—
12	#14583247	—	2	2	no	—
13	#14583247	—	2	2	no	—
14	C₂×#13063247	—	2	2	no	—
15	C₂×#13063245	—	2	2	no	—
16	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
18	#14582769	C₂²	1	1	yes	characteristic
19	#14582661	—	2	2	no	—
20	#14582661	—	2	2	no	—
21	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₄₃×#487	—	2	2	no	—
24	C₄₃×#487	—	2	2	no	—
25	C₂×#103×C₄₃	—	2	2	no	—
26	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
27	C₂²×D₁₇₂	—	2	1	no	—
28	C₂²×D₁₇₂	—	2	1	no	—
29	C₂²×D₁₇₂	—	2	1	no	—
30	C₂×#10621884	—	2	8	no	—
31	C₂×#10621884	—	4	4	no	—
32	C₂×#10621884	—	4	4	no	—
33	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
34	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
35	#13063322	—	2	1	no	—
36	#13063322	—	2	1	no	—
37	#13063322	—	2	1	no	—
38	#13063322	D₄	1	2	yes	—
39	C₂×C₄⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄⋊Dic₄₃	—	2	1	no	—
41	C₂×C₄⋊Dic₄₃	—	2	1	no	—
42	#13063247	—	4	4	no	—
43	#13063245	—	2	8	no	—
44	C₄×C₄₃⋊C₈	—	2	1	no	—
45	C₄×C₄₃⋊C₈	—	2	1	no	—
46	C₄×C₄₃⋊C₈	—	2	1	no	—
47	C₄×C₄₃⋊C₈	—	2	1	no	—
48	C₂²×Q₈×C₄₃	—	2	1	no	—
49	C₂²×Q₈×C₄₃	—	2	1	no	—
50	C₂²×C₈×C₄₃	D₄	1	2	yes	—
51	C₄⋊Q₈×C₄₃	—	2	1	no	—
52	C₄⋊Q₈×C₄₃	—	2	1	no	—
53	C₄⋊Q₈×C₄₃	—	2	1	no	—
54	C₄⋊Q₈×C₄₃	—	2	1	no	—
55	C₄⋊Q₈×C₄₃	—	2	1	no	—
56	C₄⋊Q₈×C₄₃	—	2	1	no	—
57	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
58	C₄⋊Q₈×C₄₃	D₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
60	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
61	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
62	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
63	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
64	#103×C₄₃	—	2	8	no	—
65	C₄×C₈×C₄₃	—	2	1	no	—
66	C₄×C₈×C₄₃	—	2	1	no	—
67	C₂³×D₄₃	—	4	1	no	—
68	C₂×Q₈×C₄₃	—	2	2	no	—
69	C₂×Q₈×C₄₃	—	2	2	no	—
70	C₂×Q₈×C₄₃	—	4	1	no	—
71	C₂×Q₈×C₄₃	—	4	1	no	—
72	C₂×Q₈×C₄₃	—	4	2	no	—
73	C₂×Q₈×C₄₃	—	4	2	no	—
74	C₂×Q₈×C₄₃	—	4	2	no	—
75	C₂×Q₈×C₄₃	—	4	2	no	—
76	C₂²×C₄×C₄₃	—	4	1	no	—
77	C₂²×C₄×C₄₃	—	4	1	no	—
78	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×Dic₄₃	—	4	1	no	—
82	C₂×D₁₇₂	—	2	2	no	—
83	C₂×D₁₇₂	—	2	2	no	—
84	C₂×D₁₇₂	—	2	2	no	—
85	C₂×D₁₇₂	—	4	1	no	—
86	C₂×D₁₇₂	—	4	1	no	—
87	C₂×D₁₇₂	—	4	1	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×D₁₇₂	—	4	2	no	—
93	C₂×D₁₇₂	—	4	2	no	—
94	C₂×C₈×C₄₃	—	2	2	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₄⋊C₄×C₄₃	—	2	2	no	—
98	C₄⋊C₄×C₄₃	—	2	2	no	—
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	1	no	—
104	C₄⋊C₄×C₄₃	—	4	1	no	—
105	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
106	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
107	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₄₃	#41	1	1	yes	characteristic
109	#10621884	—	4	16	no	—
110	C₄⋊Dic₄₃	—	4	2	no	—
111	C₄⋊Dic₄₃	—	4	2	no	—
112	C₄⋊Dic₄₃	—	4	2	no	—
113	C₂×C₄₃⋊C₈	—	2	2	no	—
114	C₂×C₄₃⋊C₈	—	2	2	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	SD₁₆	1	4	yes	—
119	C₂³×C₄₃	#127	1	1	yes	characteristic
120	C₂²×D₄₃	—	4	2	no	—
121	C₂²×D₄₃	—	8	1	no	—
122	C₂²×D₄₃	—	8	1	no	—
123	C₂²×D₄₃	—	8	1	no	—
124	C₂²×D₄₃	—	8	1	no	—
125	C₂²×D₄₃	—	8	1	no	—
126	C₂²×D₄₃	—	8	1	no	—
127	Q₈×C₄₃	—	4	4	no	—
128	Q₈×C₄₃	—	4	4	no	—
129	Q₈×C₄₃	—	8	2	no	—
130	Q₈×C₄₃	—	8	2	no	—
131	C₂×C₄×C₄₃	—	4	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	#120	1	1	yes	characteristic
138	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
139	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
140	C₂×C₄×C₄₃	#121	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	C₂×SD₁₆	1	2	yes	—
144	C₂×C₄×C₄₃	#135	1	2	yes	—
145	C₂×C₄×C₄₃	#135	1	2	yes	—
146	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
147	C₂×C₄×C₄₃	AGL(1,ℤ/8)	1	2	yes	—
148	C₂×C₄×C₄₃	#137	1	2	yes	—
149	C₂×Dic₄₃	—	8	1	no	—
150	C₂×Dic₄₃	—	8	1	no	—
151	C₂×Dic₄₃	—	8	1	no	—
152	D₁₇₂	—	4	4	no	—
153	D₁₇₂	—	4	4	no	—
154	D₁₇₂	—	4	4	no	—
155	D₁₇₂	—	8	2	no	—
156	D₁₇₂	—	8	2	no	—
157	D₁₇₂	—	8	2	no	—
158	C₈×C₄₃	—	4	2	no	—
159	C₄₃⋊C₈	—	2	4	no	—
160	C₄₃⋊C₈	—	4	2	no	—
161	#10310	—	43	1	no	maximal, Sylow
162	C₂²×C₄₃	#392	1	1	yes	characteristic
163	C₂²×C₄₃	#492	1	1	yes	characteristic
164	C₂²×C₄₃	#495	1	1	yes	characteristic
165	C₂²×C₄₃	#495	1	1	yes	characteristic
166	C₂²×C₄₃	#496	1	1	yes	characteristic
167	C₂²×C₄₃	#496	1	1	yes	characteristic
168	C₂²×C₄₃	#497	1	1	yes	characteristic
169	C₂×D₄₃	—	8	2	no	—
170	C₂×D₄₃	—	8	2	no	—
171	C₂×D₄₃	—	8	2	no	—
172	C₂×D₄₃	—	16	1	no	—
173	C₂×D₄₃	—	16	1	no	—
174	C₂×D₄₃	—	16	1	no	—
175	C₂×D₄₃	—	16	1	no	—
176	C₄×C₄₃	—	8	2	no	—
177	C₄×C₄₃	—	8	2	no	—
178	C₄×C₄₃	#449	1	2	yes	—
179	C₄×C₄₃	#450	1	2	yes	—
180	C₄×C₄₃	#452	1	2	yes	—
181	C₄×C₄₃	#460	1	2	yes	—
182	C₄×C₄₃	#463	1	2	yes	—
183	C₄×C₄₃	#464	1	2	yes	—
184	Dic₄₃	—	16	1	no	—
185	C₂×#492	—	43	1	no	—
186	C₂×#487	D₄₃	1	1	yes	characteristic, complemented
187	C₂×#486	—	43	1	no	—
188	#1990	—	43	1	no	—
189	#1990	—	43	1	no	—
190	#1957	—	43	1	no	—
191	#1273	—	43	1	no	—
192	C₂×C₄₃	#1708	1	1	yes	characteristic
193	C₂×C₄₃	#1714	1	1	yes	characteristic
194	C₂×C₄₃	#1717	1	1	yes	characteristic
195	C₂×C₄₃	#1719	1	1	yes	characteristic
196	C₂×C₄₃	#2012	1	1	yes	characteristic
197	C₂×C₄₃	#2014	1	1	yes	characteristic
198	C₂×C₄₃	#2015	1	1	yes	characteristic
199	D₄₃	—	16	2	no	—
200	C₂²×SD₁₆	—	86	2	no	—
201	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
202	C₂×C₄⋊Q₈	C₂×D₄₃	1	1	yes	characteristic
203	C₂×#127	—	43	1	no	—
204	#492	—	86	2	no	—
205	#492	—	86	2	no	—
206	#487	—	2	2	no	—
207	#487	—	2	2	no	—
208	#486	—	86	2	no	—
209	#486	—	86	2	no	—
210	C₂×#103	—	2	2	no	—
211	C₂×#102	—	86	2	no	—
212	C₂×#102	—	86	2	no	—
213	C₂×C₄×C₈	—	43	1	no	—
214	C₂×C₄×C₈	—	43	1	no	—
215	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
216	#384	—	43	1	no	—
217	#340	—	86	2	no	—
218	#340	—	86	2	no	—
219	C₄₃	#10310	1	1	yes	characteristic, complemented, Sylow
220	C₂²×Q₈	—	2	1	no	—
221	C₂²×Q₈	—	2	1	no	—
222	C₂²×D₄	—	86	1	no	—
223	C₂²×D₄	—	86	1	no	—
224	C₂²×D₄	—	86	1	no	—
225	C₂×SD₁₆	—	86	8	no	—
226	C₂×SD₁₆	—	172	4	no	—
227	C₂×SD₁₆	—	172	4	no	—
228	C₂²×C₈	—	43	2	no	—
229	C₂²×C₈	—	43	2	no	—
230	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
231	C₄⋊Q₈	—	2	1	no	—
232	C₄⋊Q₈	—	2	1	no	—
233	C₄⋊Q₈	—	2	1	no	—
234	C₄⋊Q₈	—	2	1	no	—
235	C₄⋊Q₈	—	2	1	no	—
236	C₄⋊Q₈	—	2	1	no	—
237	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
238	C₄⋊Q₈	C₂²⋊D₄₃	1	2	yes	—
239	#127	—	43	2	no	—
240	#127	—	86	1	no	—
241	#127	—	86	1	no	—
242	#127	—	86	1	no	—
243	C₂×C₄⋊C₄	—	2	1	no	—
244	C₂×C₄⋊C₄	—	2	1	no	—
245	C₂×C₄⋊C₄	—	2	1	no	—
246	C₂×C₄⋊C₄	—	2	1	no	—
247	C₂×C₄⋊C₄	—	86	1	no	—
248	C₂×C₄⋊C₄	—	86	1	no	—
249	C₂×C₄⋊C₄	—	86	1	no	—
250	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
251	#103	—	2	8	no	—
252	#102	—	86	8	no	—
253	#102	—	172	4	no	—
254	C₄×C₈	—	2	1	no	—
255	C₄×C₈	—	2	1	no	—
256	C₄×C₈	—	86	1	no	—
257	C₄×C₈	—	86	1	no	—
258	C₄×C₈	—	86	1	no	—
259	C₄×C₈	—	86	1	no	—
260	C₂⁴	—	172	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	2	2	no	—
263	C₂×Q₈	—	4	1	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	4	2	no	—
266	C₂×Q₈	—	4	2	no	—
267	C₂×Q₈	—	4	2	no	—
268	C₂×Q₈	—	4	2	no	—
269	C₂×D₄	—	86	2	no	—
270	C₂×D₄	—	86	2	no	—
271	C₂×D₄	—	86	2	no	—
272	C₂×D₄	—	172	1	no	—
273	C₂×D₄	—	172	1	no	—
274	C₂×D₄	—	172	1	no	—
275	C₂×D₄	—	172	2	no	—
276	C₂×D₄	—	172	2	no	—
277	C₂×D₄	—	172	2	no	—
278	C₂×D₄	—	172	2	no	—
279	C₂×D₄	—	172	2	no	—
280	C₂×D₄	—	172	2	no	—
281	C₂²×C₄	—	4	1	no	—
282	C₂²×C₄	—	4	1	no	—
283	C₂²×C₄	—	172	1	no	—
284	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
285	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
286	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
287	SD₁₆	—	172	16	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	2	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	43	4	no	—
292	C₂×C₈	—	86	2	no	—
293	C₂×C₈	—	86	2	no	—
294	C₂×C₈	—	86	2	no	—
295	C₂×C₈	—	86	2	no	—
296	C₂×C₈	—	86	2	no	—
297	C₄⋊C₄	—	2	2	no	—
298	C₄⋊C₄	—	2	2	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	2	2	no	—
301	C₄⋊C₄	—	4	1	no	—
302	C₄⋊C₄	—	4	1	no	—
303	C₄⋊C₄	—	4	1	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	172	2	no	—
306	C₄⋊C₄	—	172	2	no	—
307	C₄⋊C₄	—	172	2	no	—
308	C₄²	D₄×D₄₃	1	1	yes	characteristic
309	C₄²	#10621900	1	1	yes	characteristic
310	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
311	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
312	C₂³	—	344	1	no	—
313	C₂³	—	344	1	no	—
314	C₂³	—	344	1	no	—
315	C₂³	—	344	1	no	—
316	C₂³	—	344	1	no	—
317	C₂³	—	344	1	no	—
318	C₂³	—	172	2	no	—
319	C₂³	#13063321	1	1	yes	characteristic
320	Q₈	—	4	4	no	—
321	Q₈	—	4	4	no	—
322	Q₈	—	8	2	no	—
323	Q₈	—	8	2	no	—
324	D₄	—	344	2	no	—
325	D₄	—	344	2	no	—
326	D₄	—	344	2	no	—
327	D₄	—	172	4	no	—
328	D₄	—	172	4	no	—
329	D₄	—	172	4	no	—
330	C₂×C₄	—	4	2	no	—
331	C₂×C₄	—	4	2	no	—
332	C₂×C₄	—	8	1	no	—
333	C₂×C₄	—	8	1	no	—
334	C₂×C₄	—	8	1	no	—
335	C₂×C₄	—	8	1	no	—
336	C₂×C₄	—	344	1	no	—
337	C₂×C₄	—	344	1	no	—
338	C₂×C₄	—	344	1	no	—
339	C₂×C₄	#13063293	1	1	yes	characteristic
340	C₂×C₄	#13063293	1	1	yes	characteristic
341	C₂×C₄	#13063294	1	1	yes	characteristic
342	C₂×C₄	#13063296	1	1	yes	characteristic
343	C₂×C₄	#13063297	1	1	yes	characteristic
344	C₂×C₄	#13063297	1	1	yes	characteristic
345	C₂×C₄	SD₁₆×D₄₃	1	2	yes	—
346	C₂×C₄	#13063345	1	2	yes	—
347	C₂×C₄	#13063348	1	2	yes	—
348	C₂×C₄	#13063350	1	2	yes	—
349	C₂×C₄	#13063352	1	2	yes	—
350	C₂×C₄	#13063357	1	2	yes	—
351	C₈	—	4	2	no	—
352	C₈	—	86	4	no	—
353	C₈	—	172	2	no	—
354	C₂²	—	344	2	no	—
355	C₂²	—	344	2	no	—
356	C₂²	—	344	2	no	—
357	C₂²	—	688	1	no	—
358	C₂²	—	688	1	no	—
359	C₂²	—	688	1	no	—
360	C₂²	—	688	1	no	—
361	C₂²	#14582807	1	1	yes	characteristic
362	C₂²	#14583270	1	1	yes	characteristic
363	C₂²	#14583281	1	1	yes	characteristic
364	C₂²	#14583284	1	1	yes	characteristic
365	C₂²	#14583285	1	1	yes	characteristic
366	C₂²	#14583289	1	1	yes	characteristic
367	C₂²	#14583292	1	1	yes	characteristic
368	C₄	—	8	2	no	—
369	C₄	—	8	2	no	—
370	C₄	—	688	1	no	—
371	C₄	#14583044	1	2	yes	—
372	C₄	#14583048	1	2	yes	—
373	C₄	#14583060	1	2	yes	—
374	C₄	#14583097	1	2	yes	—
375	C₄	#14583111	1	2	yes	—
376	C₄	#14583123	1	2	yes	—
377	C₂	—	688	2	no	—
378	C₂	#19134198	1	1	yes	characteristic
379	C₂	#19134219	1	1	yes	characteristic
380	C₂	#19134237	1	1	yes	characteristic
381	C₂	#19134249	1	1	yes	characteristic
382	C₂	#19135672	1	1	yes	characteristic
383	C₂	#19135681	1	1	yes	characteristic
384	C₂	#19135686	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#13063321
commutator, cosocle	C₂×C₄²×C₄₃	C₂³
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×#487	C₂
socle	C₂³×C₄₃	#127
prime radical	C₂×#13063322	C₂²
hypercenter	C₂×#487	D₄₃
hypocenter	C₄₃	#10310
LCS	G ▷ C₂×C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#487

Complements

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

N	H ≅ G/N	type
3. #19135557	C₂	semidirect
4. #19135557	C₂	semidirect
6. C₄₃⋊#1273	C₂	semidirect
7. C₂×C₄₃×#487	C₂	semidirect
186. C₂×#487	D₄₃	semidirect
219. C₄₃	#10310	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583273	2
C₂×#14583247	2
#19135557	2
#19135401	2
C₄₃⋊#1273	2
C₂×C₄₃×#487	2
#10310	43