Subgroups — #25098866

Order 11008.

790 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582937	C₂	1	1	yes	maximal, characteristic
2	#19143158	C₂	1	1	yes	maximal, characteristic
3	#19143013	C₂	1	1	yes	maximal, characteristic
4	#19135195	C₂	1	8	yes	maximal, complemented
5	C₂×#14582664	C₂	1	1	yes	maximal, characteristic
6	#19134305	C₂	1	1	yes	maximal, characteristic
7	#19134305	C₂	1	1	yes	maximal, characteristic
8	C₄₃×#1729	C₂	1	1	yes	maximal, characteristic
9	C₂²×#10621877	C₂²	1	1	yes	characteristic
10	C₂²×#10621877	C₂²	1	1	yes	characteristic
11	C₂×#13063284	C₂²	1	1	yes	characteristic
12	C₂×#13063284	C₂²	1	1	yes	characteristic
13	#14582937	C₄	1	4	yes	complemented
14	#14582937	C₂²	1	4	yes	—
15	C₂×#13063259	C₄	1	2	yes	complemented
16	C₂×#13063257	C₄	1	2	yes	complemented
17	#14582927	—	2	2	no	—
18	#14582925	C₂²	1	4	yes	—
19	#14582922	—	2	2	no	—
20	#14582857	—	2	2	no	—
21	#14582854	C₂²	1	4	yes	—
22	M₁₆×Dic₄₃	—	2	2	no	—
23	#14582664	C₂²	1	4	yes	—
24	#14582659	C₂²	1	4	yes	—
25	#14582659	C₂²	1	4	yes	—
26	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
27	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
29	C₄₃×#336	C₂²	1	4	yes	—
30	C₂×#10621877	—	2	1	no	—
31	C₂×#10621877	—	2	2	no	—
32	C₂×#10621877	—	2	1	no	—
33	C₂×#10621877	—	2	2	no	—
34	C₂×#10621877	—	2	2	no	—
35	C₂×#10621877	Q₈	1	4	yes	—
36	C₂×#10621877	C₂³	1	2	yes	—
37	C₂×#10621877	C₂³	1	2	yes	—
38	C₂²×C₄₃⋊C₈	—	2	1	no	—
39	C₂²×C₄₃⋊C₈	—	2	1	no	—
40	#13063284	—	2	1	no	—
41	#13063284	—	2	1	no	—
42	#13063284	—	2	1	no	—
43	#13063284	—	2	1	no	—
44	#13063284	C₂×C₄	1	2	yes	—
45	#13063284	C₂×C₄	1	2	yes	—
46	#13063284	C₂³	1	2	yes	—
47	#13063284	C₂³	1	2	yes	—
48	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
50	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
51	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
52	C₂×C₄×Dic₄₃	—	2	1	no	—
53	C₂×C₄×Dic₄₃	—	2	1	no	—
54	#13063259	C₂×C₄	1	8	yes	—
55	#13063257	C₂×C₄	1	8	yes	—
56	Dic₄₃⋊C₈	—	2	8	no	—
57	#13063235	—	2	4	no	—
58	C₈×Dic₄₃	—	2	4	no	—
59	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂×M₁₆×C₄₃	—	2	2	no	—
61	C₂×M₁₆×C₄₃	—	2	2	no	—
62	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
63	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
64	C₂²×C₈×C₄₃	—	2	2	no	—
65	C₂²×C₈×C₄₃	—	2	2	no	—
66	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
69	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
70	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
71	#117×C₄₃	—	2	2	no	—
72	#117×C₄₃	—	2	2	no	—
73	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
74	C₂×C₂²⋊C₄×C₄₃	—	2	1	no	—
75	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
76	C₂⁴×C₄₃	#41	1	1	yes	characteristic
77	C₂²×C₄×C₄₃	—	2	2	no	—
78	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
79	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
83	C₂²×C₄×C₄₃	#41	1	2	yes	—
84	C₂²×C₄×C₄₃	#41	1	2	yes	—
85	C₂²×C₄×C₄₃	#41	1	2	yes	—
86	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
87	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²×Dic₄₃	—	2	2	no	—
89	C₂²×Dic₄₃	—	2	2	no	—
90	M₁₆×C₄₃	—	2	4	no	—
91	M₁₆×C₄₃	—	2	4	no	—
92	C₂×C₈×C₄₃	—	2	2	no	—
93	C₂×C₈×C₄₃	—	2	2	no	—
94	C₂×C₈×C₄₃	—	2	4	no	—
95	C₂×C₈×C₄₃	—	2	2	no	—
96	C₂×C₈×C₄₃	—	2	2	no	—
97	C₂×C₈×C₄₃	—	2	2	no	—
98	C₂×C₈×C₄₃	—	2	2	no	—
99	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
100	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
101	C₂×C₈×C₄₃	C₂×D₄	1	4	yes	—
102	C₄⋊C₄×C₄₃	—	4	2	no	—
103	C₂²⋊C₄×C₄₃	—	4	2	no	—
104	C₄²×C₄₃	—	2	2	no	—
105	C₄²×C₄₃	—	2	2	no	—
106	C₂²⋊Dic₄₃	—	2	2	no	—
107	C₂²⋊Dic₄₃	—	2	2	no	—
108	C₂²⋊Dic₄₃	—	2	2	no	—
109	C₂²⋊Dic₄₃	—	2	2	no	—
110	C₄⋊Dic₄₃	—	2	2	no	—
111	C₄⋊Dic₄₃	—	2	2	no	—
112	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
113	C₄⋊Dic₄₃	C₂²×C₄	1	4	yes	—
114	C₄×Dic₄₃	—	2	2	no	—
115	C₄×Dic₄₃	—	2	2	no	—
116	C₄×Dic₄₃	—	2	2	no	—
117	C₄×Dic₄₃	—	2	2	no	—
118	#10621877	—	2	4	no	—
119	#10621877	—	2	2	no	—
120	#10621877	—	2	2	no	—
121	#10621877	—	2	2	no	—
122	#10621877	—	4	2	no	—
123	#10621877	C₂×Q₈	1	4	yes	—
124	C₂×C₄₃⋊C₈	—	2	4	no	—
125	C₂×C₄₃⋊C₈	—	2	2	no	—
126	C₂×C₄₃⋊C₈	—	2	2	no	—
127	C₂×C₄₃⋊C₈	—	4	2	no	—
128	C₂³×C₄₃	—	4	2	no	—
129	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
130	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
131	C₂³×C₄₃	#125	1	2	yes	—
132	C₂³×C₄₃	#126	1	2	yes	—
133	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
134	C₂×C₄×C₄₃	—	4	4	no	—
135	C₂×C₄×C₄₃	—	4	2	no	—
136	C₂×C₄×C₄₃	#117	1	2	yes	—
137	C₂×C₄×C₄₃	#117	1	2	yes	—
138	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
141	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
142	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
143	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
144	C₂×C₄×C₄₃	#123	1	2	yes	—
145	C₂×C₄×C₄₃	#123	1	2	yes	—
146	C₂×C₄×C₄₃	#125	1	2	yes	—
147	C₂×C₄×C₄₃	#126	1	2	yes	—
148	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
149	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
150	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
151	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
152	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
153	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
154	C₂×Dic₄₃	—	2	4	no	—
155	C₂×Dic₄₃	—	2	4	no	—
156	C₂×Dic₄₃	—	8	2	no	—
157	C₂×Dic₄₃	—	8	2	no	—
158	C₈×C₄₃	—	2	2	no	—
159	C₈×C₄₃	—	2	2	no	—
160	C₈×C₄₃	—	2	4	no	—
161	C₄₃⋊C₈	—	2	4	no	—
162	C₄₃⋊C₈	—	4	2	no	—
163	#21043	—	43	1	no	maximal, Sylow
164	C₂²×C₄₃	—	2	2	no	—
165	C₂²×C₄₃	—	2	2	no	—
166	C₂²×C₄₃	—	2	2	no	—
167	C₂²×C₄₃	—	4	2	no	—
168	C₂²×C₄₃	—	4	2	no	—
169	C₂²×C₄₃	#387	1	2	yes	—
170	C₂²×C₄₃	#387	1	2	yes	—
171	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
172	C₂²×C₄₃	C₂×#125	1	1	yes	characteristic
173	C₂²×C₄₃	C₂×#126	1	1	yes	characteristic
174	C₄×C₄₃	—	8	2	no	—
175	C₄×C₄₃	#387	1	2	yes	—
176	C₄×C₄₃	#387	1	2	yes	—
177	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
178	C₄×C₄₃	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₄×C₄₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
180	C₄×C₄₃	C₂×#123	1	1	yes	characteristic
181	Dic₄₃	—	4	4	no	—
182	Dic₄₃	—	4	4	no	—
183	C₂×#427	—	43	1	no	—
184	#2905	—	43	1	no	—
185	#2873	—	43	1	no	—
186	#1922	—	43	8	no	—
187	C₂×#341	—	43	1	no	—
188	#1733	—	43	1	no	—
189	#1733	—	43	1	no	—
190	#1729	D₄₃	1	1	yes	characteristic
191	C₂×C₄₃	—	2	1	no	—
192	C₂×C₄₃	—	2	1	no	—
193	C₂×C₄₃	—	2	2	no	—
194	C₂×C₄₃	—	2	2	no	—
195	C₂×C₄₃	C₂×#387	1	1	yes	characteristic
196	C₂×C₄₃	#2957	1	2	yes	—
197	C₂²×M₁₆	—	43	1	no	—
198	C₂²×M₁₆	—	43	1	no	—
199	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
200	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
201	C₂×#117	—	43	1	no	—
202	C₂×#117	—	43	1	no	—
203	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
204	#427	—	43	4	no	—
205	#427	—	43	4	no	—
206	C₂×#107	—	43	2	no	—
207	C₂×#106	—	43	2	no	—
208	#424	—	43	4	no	—
209	#424	—	86	2	no	—
210	#423	—	86	2	no	—
211	#405	—	43	4	no	—
212	#405	—	86	2	no	—
213	C₄×M₁₆	—	86	2	no	—
214	#341	—	43	4	no	—
215	#339	—	43	4	no	—
216	#339	—	43	4	no	—
217	#336	C₂×D₄₃	1	4	yes	—
218	C₄₃	#21043	1	1	yes	characteristic, complemented, Sylow
219	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
220	C₂×M₁₆	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	—	43	4	no	—
223	C₂×M₁₆	—	43	2	no	—
224	C₂×M₁₆	—	43	2	no	—
225	C₂×M₁₆	—	86	1	no	—
226	C₂×M₁₆	—	86	2	no	—
227	C₂×M₁₆	—	86	1	no	—
228	C₂×M₁₆	—	86	2	no	—
229	C₂×M₁₆	—	86	2	no	—
230	C₂×M₁₆	D₁₇₂	1	2	yes	—
231	C₂×M₁₆	D₁₇₂	1	2	yes	—
232	C₂²×C₈	—	2	2	no	—
233	C₂²×C₈	—	2	2	no	—
234	C₂²×C₈	—	86	1	no	—
235	C₂²×C₈	—	86	1	no	—
236	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
237	C₂²×C₈	C₄×D₄₃	1	1	yes	characteristic
238	C₂²×C₈	C₄×D₄₃	1	2	yes	—
239	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
240	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
241	#117	—	2	2	no	—
242	#117	—	2	2	no	—
243	#117	—	43	2	no	—
244	#117	—	43	2	no	—
245	#117	—	43	2	no	—
246	#117	—	43	2	no	—
247	#117	—	86	1	no	—
248	#117	—	86	1	no	—
249	#117	—	86	1	no	—
250	#117	—	86	1	no	—
251	C₂×C₄⋊C₄	—	2	1	no	—
252	C₂×C₄⋊C₄	—	43	2	no	—
253	C₂×C₄⋊C₄	—	43	2	no	—
254	C₂×C₂²⋊C₄	—	2	1	no	—
255	C₂×C₂²⋊C₄	—	86	1	no	—
256	C₂×C₂²⋊C₄	—	86	1	no	—
257	C₂×C₄²	—	86	1	no	—
258	C₂×C₄²	—	86	1	no	—
259	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
260	#107	—	43	8	no	—
261	#106	—	43	8	no	—
262	C₄⋊C₈	—	86	8	no	—
263	#97	—	86	4	no	—
264	C₄×C₈	—	86	4	no	—
265	C₂⁴	#10621898	1	1	yes	characteristic
266	C₂²×C₄	—	2	2	no	—
267	C₂²×C₄	—	86	2	no	—
268	C₂²×C₄	—	86	2	no	—
269	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
270	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
271	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
273	C₂²×C₄	#10621898	1	1	yes	characteristic
274	C₂²×C₄	#10621900	1	2	yes	—
275	C₂²×C₄	#10621900	1	2	yes	—
276	C₂²×C₄	#10621900	1	2	yes	—
277	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
278	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
279	M₁₆	—	2	4	no	—
280	M₁₆	—	2	4	no	—
281	M₁₆	—	43	4	no	—
282	M₁₆	—	86	4	no	—
283	M₁₆	—	86	2	no	—
284	M₁₆	—	86	2	no	—
285	M₁₆	—	86	2	no	—
286	M₁₆	—	172	2	no	—
287	C₂×C₈	—	2	2	no	—
288	C₂×C₈	—	2	2	no	—
289	C₂×C₈	—	2	4	no	—
290	C₂×C₈	—	2	2	no	—
291	C₂×C₈	—	2	2	no	—
292	C₂×C₈	—	2	2	no	—
293	C₂×C₈	—	2	2	no	—
294	C₂×C₈	—	86	4	no	—
295	C₂×C₈	—	86	2	no	—
296	C₂×C₈	—	86	2	no	—
297	C₂×C₈	—	172	2	no	—
298	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
299	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
300	C₂×C₈	C₂×D₁₇₂	1	4	yes	—
301	C₄⋊C₄	—	4	2	no	—
302	C₄⋊C₄	—	43	4	no	—
303	C₄⋊C₄	—	43	4	no	—
304	C₄⋊C₄	—	86	2	no	—
305	C₄⋊C₄	—	86	2	no	—
306	C₂²⋊C₄	—	4	2	no	—
307	C₂²⋊C₄	—	86	2	no	—
308	C₂²⋊C₄	—	86	2	no	—
309	C₂²⋊C₄	—	86	2	no	—
310	C₂²⋊C₄	—	86	2	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	2	no	—
313	C₄²	—	86	2	no	—
314	C₄²	—	86	2	no	—
315	C₄²	—	86	2	no	—
316	C₄²	—	86	2	no	—
317	C₂³	—	4	2	no	—
318	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
319	C₂³	C₄×D₁₇₂	1	1	yes	characteristic
320	C₂³	#13063315	1	2	yes	—
321	C₂³	#13063318	1	2	yes	—
322	C₂³	C₂×#10621898	1	1	yes	characteristic
323	C₂×C₄	—	4	4	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	86	4	no	—
326	C₂×C₄	—	86	4	no	—
327	C₂×C₄	—	344	2	no	—
328	C₂×C₄	—	344	2	no	—
329	C₂×C₄	#13063281	1	2	yes	—
330	C₂×C₄	#13063281	1	2	yes	—
331	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
332	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
333	C₂×C₄	#13063290	1	2	yes	—
334	C₂×C₄	#13063290	1	2	yes	—
335	C₂×C₄	#13063305	1	2	yes	—
336	C₂×C₄	#13063305	1	2	yes	—
337	C₂×C₄	#13063309	1	2	yes	—
338	C₂×C₄	#13063309	1	2	yes	—
339	C₂×C₄	#13063315	1	2	yes	—
340	C₂×C₄	#13063318	1	2	yes	—
341	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
342	C₂×C₄	C₂²×D₁₇₂	1	1	yes	characteristic
343	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
344	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
345	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
346	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
347	C₈	—	2	2	no	—
348	C₈	—	2	2	no	—
349	C₈	—	2	4	no	—
350	C₈	—	86	4	no	—
351	C₈	—	172	2	no	—
352	C₂²	—	2	2	no	—
353	C₂²	—	2	2	no	—
354	C₂²	—	2	2	no	—
355	C₂²	—	4	2	no	—
356	C₂²	—	4	2	no	—
357	C₂²	#14582784	1	2	yes	—
358	C₂²	#14582784	1	2	yes	—
359	C₂²	C₂×C₄×D₁₇₂	1	1	yes	characteristic
360	C₂²	C₂×#13063315	1	1	yes	characteristic
361	C₂²	C₂×#13063318	1	1	yes	characteristic
362	C₄	—	8	2	no	—
363	C₄	—	172	4	no	—
364	C₄	—	172	4	no	—
365	C₄	#14582784	1	2	yes	—
366	C₄	#14582784	1	2	yes	—
367	C₄	C₂×#13063281	1	1	yes	characteristic
368	C₄	C₂×#13063290	1	1	yes	characteristic
369	C₄	C₂×#13063305	1	1	yes	characteristic
370	C₄	C₂×#13063309	1	1	yes	characteristic
371	C₂	—	2	1	no	—
372	C₂	—	2	2	no	—
373	C₂	—	2	2	no	—
374	C₂	—	2	1	no	—
375	C₂	C₂×#14582784	1	1	yes	characteristic
376	C₂	#19143513	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²×D₁₇₂
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1729	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1729	D₄₃
hypocenter	C₄₃	#21043
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ #1729

Complements

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

N	H ≅ G/N	type
4. #19135195	C₂	semidirect
13. #14582937	C₄	semidirect
15. C₂×#13063259	C₄	semidirect
16. C₂×#13063257	C₄	semidirect
218. C₄₃	#21043	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582937	2
#19143158	2
#19143013	2
#19135195	2
C₂×#14582664	2
#19134305	2
C₄₃×#1729	2
#21043	43