Subgroups — #25002355

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582854	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582668	C₂	1	2	yes	maximal
3	C₂×#14582664	C₂	1	1	yes	maximal, characteristic
4	#19134314	C₂	1	2	yes	maximal
5	C₂×C₄₃×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂²×#10621877	C₂²	1	1	yes	characteristic
8	C₂×#13063284	C₂²	1	1	yes	characteristic
9	#14582854	—	2	2	no	—
10	#14582854	C₄	1	4	yes	complemented
11	C₂×#13063235	—	2	1	no	—
12	C₂×C₈×Dic₄₃	—	2	1	no	—
13	#14582668	C₄	1	8	yes	complemented
14	#14582664	—	2	2	no	—
15	#14582661	—	2	2	no	—
16	#14582661	—	2	2	no	—
17	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
19	C₄₃×#340	—	2	1	no	—
20	C₄₃×#340	—	2	1	no	—
21	C₂×#10621877	—	2	1	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	—	2	2	no	—
26	C₂×#10621877	—	2	2	no	—
27	C₂×#10621877	—	2	2	no	—
28	C₂×#10621877	C₂×C₄	1	4	yes	—
29	C₂×#10621877	C₂×C₄	1	2	yes	—
30	C₂×#10621877	C₂×C₄	1	4	yes	—
31	C₂²×C₄₃⋊C₈	—	2	1	no	—
32	C₂²×C₄₃⋊C₈	—	2	2	no	—
33	#13063284	—	2	1	no	—
34	#13063284	—	2	1	no	—
35	#13063284	C₂×C₄	1	2	yes	—
36	#13063284	Q₈	1	2	yes	—
37	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
38	C₂×C₄⋊Dic₄₃	D₄	1	1	yes	characteristic
39	C₂×C₂²⋊Dic₄₃	—	2	1	no	—
40	C₂×C₄×Dic₄₃	—	2	1	no	—
41	#13063235	—	2	4	no	—
42	C₈×Dic₄₃	—	2	4	no	—
43	C₂³×C₄×C₄₃	—	2	2	no	—
44	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₄₃	—	2	2	no	—
46	C₂²×C₈×C₄₃	—	2	2	no	—
47	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
48	C₂²×C₈×C₄₃	D₄	1	2	yes	—
49	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
52	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
53	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
54	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
55	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
57	C₂⁴×C₄₃	#41	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	—	2	4	no	—
59	C₂²×C₄×C₄₃	—	4	2	no	—
60	C₂²×C₄×C₄₃	—	4	2	no	—
61	C₂²×C₄×C₄₃	—	8	2	no	—
62	C₂²×C₄×C₄₃	—	8	2	no	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	#41	1	2	yes	—
70	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	#41	1	2	yes	—
72	C₂²×C₄×C₄₃	#41	1	2	yes	—
73	C₂²×Dic₄₃	—	2	1	no	—
74	C₂²×Dic₄₃	—	2	1	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₂×C₈×C₄₃	—	2	2	no	—
79	C₂×C₈×C₄₃	—	2	4	no	—
80	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
81	C₂×C₈×C₄₃	C₂²⋊C₄	1	2	yes	—
82	C₄⋊C₄×C₄₃	—	2	4	no	—
83	C₄⋊C₄×C₄₃	—	2	4	no	—
84	C₄⋊C₄×C₄₃	—	4	2	no	—
85	C₄⋊C₄×C₄₃	—	4	2	no	—
86	C₂²⋊Dic₄₃	—	2	2	no	—
87	C₂²⋊Dic₄₃	—	2	2	no	—
88	C₄⋊Dic₄₃	—	2	1	no	—
89	C₄⋊Dic₄₃	—	2	1	no	—
90	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
91	C₄⋊Dic₄₃	C₄⋊C₄	1	2	yes	—
92	C₄×Dic₄₃	—	2	2	no	—
93	C₄×Dic₄₃	—	2	2	no	—
94	#10621877	—	2	2	no	—
95	#10621877	—	2	4	no	—
96	#10621877	—	2	4	no	—
97	#10621877	—	2	4	no	—
98	#10621877	—	4	2	no	—
99	#10621877	—	4	2	no	—
100	#10621877	C₄²	1	4	yes	—
101	C₂×C₄₃⋊C₈	—	2	2	no	—
102	C₂×C₄₃⋊C₈	—	2	4	no	—
103	C₂×C₄₃⋊C₈	—	2	4	no	—
104	C₂×C₄₃⋊C₈	—	4	2	no	—
105	C₂×C₄₃⋊C₈	—	4	2	no	—
106	C₂³×C₄₃	—	2	2	no	—
107	C₂³×C₄₃	—	2	1	no	—
108	C₂³×C₄₃	—	2	1	no	—
109	C₂³×C₄₃	#117	1	2	yes	—
110	C₂³×C₄₃	#117	1	1	yes	characteristic
111	C₂³×C₄₃	#121	1	1	yes	characteristic
112	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
113	C₂³×C₄₃	#126	1	2	yes	—
114	C₂×C₄×C₄₃	—	4	4	no	—
115	C₂×C₄×C₄₃	—	8	4	no	—
116	C₂×C₄×C₄₃	—	8	2	no	—
117	C₂×C₄×C₄₃	—	8	2	no	—
118	C₂×C₄×C₄₃	—	8	4	no	—
119	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
125	C₂×C₄×C₄₃	#117	1	2	yes	—
126	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#123	1	2	yes	—
133	C₂×C₄×C₄₃	#123	1	2	yes	—
134	C₂×C₄×C₄₃	#124	1	2	yes	—
135	C₂×C₄×C₄₃	#125	1	2	yes	—
136	C₂×C₄×C₄₃	#126	1	2	yes	—
137	C₂×Dic₄₃	—	2	2	no	—
138	C₂×Dic₄₃	—	2	2	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	—	4	1	no	—
141	C₂×Dic₄₃	—	4	1	no	—
142	C₈×C₄₃	—	2	2	no	—
143	C₈×C₄₃	—	2	2	no	—
144	C₄₃⋊C₈	—	2	4	no	—
145	C₄₃⋊C₈	—	4	4	no	—
146	#10827	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	—	2	1	no	—
148	C₂²×C₄₃	—	2	1	no	—
149	C₂²×C₄₃	—	2	1	no	—
150	C₂²×C₄₃	—	2	1	no	—
151	C₂²×C₄₃	—	2	2	no	—
152	C₂²×C₄₃	—	2	2	no	—
153	C₂²×C₄₃	—	2	2	no	—
154	C₂²×C₄₃	—	2	2	no	—
155	C₂²×C₄₃	#376	1	1	yes	characteristic
156	C₂²×C₄₃	#380	1	1	yes	characteristic
157	C₂²×C₄₃	#383	1	1	yes	characteristic
158	C₂²×C₄₃	#387	1	2	yes	—
159	C₂²×C₄₃	#388	1	2	yes	—
160	C₂²×C₄₃	#471	1	2	yes	—
161	C₂²×C₄₃	#472	1	1	yes	characteristic
162	C₂²×C₄₃	#473	1	1	yes	characteristic
163	C₄×C₄₃	—	8	4	no	—
164	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
165	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
166	C₄×C₄₃	#381	1	1	yes	characteristic
167	C₄×C₄₃	#382	1	1	yes	characteristic
168	C₄×C₄₃	#387	1	2	yes	—
169	C₄×C₄₃	#388	1	2	yes	—
170	Dic₄₃	—	4	2	no	—
171	Dic₄₃	—	4	2	no	—
172	C₂×#405	—	43	1	no	—
173	C₂×#343	—	43	2	no	—
174	C₂×#341	—	43	1	no	—
175	#1736	—	43	2	no	—
176	C₂×#340	D₄₃	1	1	yes	characteristic
177	C₂×C₄₃	—	2	1	no	—
178	C₂×C₄₃	—	2	1	no	—
179	C₂×C₄₃	—	2	2	no	—
180	C₂×C₄₃	#1434	1	1	yes	characteristic
181	C₂×C₄₃	#1815	1	1	yes	characteristic
182	C₂×C₄₃	#1816	1	1	yes	characteristic
183	C₂×C₄₃	#1930	1	2	yes	—
184	C₂×C₄₃	#1931	1	2	yes	—
185	C₂²×M₁₆	—	43	1	no	—
186	C₂²×M₁₆	—	43	2	no	—
187	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
188	C₂×#117	—	43	1	no	—
189	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
190	#405	—	43	4	no	—
191	#405	—	86	2	no	—
192	C₂×#97	—	86	1	no	—
193	C₂×C₄×C₈	—	86	1	no	—
194	#343	—	43	8	no	—
195	#341	—	86	2	no	—
196	#340	—	2	1	no	—
197	#340	—	2	1	no	—
198	#340	—	86	2	no	—
199	#340	—	86	2	no	—
200	C₄₃	#10827	1	1	yes	characteristic, complemented, Sylow
201	C₂³×C₄	—	2	2	no	—
202	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
203	C₂×M₁₆	—	43	4	no	—
204	C₂×M₁₆	—	43	2	no	—
205	C₂×M₁₆	—	43	4	no	—
206	C₂×M₁₆	—	86	1	no	—
207	C₂×M₁₆	—	86	2	no	—
208	C₂×M₁₆	—	86	2	no	—
209	C₂×M₁₆	—	86	2	no	—
210	C₂×M₁₆	—	86	2	no	—
211	C₂×M₁₆	—	86	2	no	—
212	C₂×M₁₆	—	86	2	no	—
213	C₂²×C₈	—	2	2	no	—
214	C₂²×C₈	—	2	2	no	—
215	C₂²×C₈	—	86	1	no	—
216	C₂²×C₈	—	86	2	no	—
217	C₂²×C₈	C₄×D₄₃	1	2	yes	—
218	C₂²×C₈	D₁₇₂	1	2	yes	—
219	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
220	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
221	#117	—	43	2	no	—
222	#117	—	43	2	no	—
223	#117	—	86	1	no	—
224	#117	—	86	1	no	—
225	C₂×C₄⋊C₄	—	2	2	no	—
226	C₂×C₄⋊C₄	—	2	2	no	—
227	C₂×C₄⋊C₄	—	2	2	no	—
228	C₂×C₄⋊C₄	—	2	2	no	—
229	C₂×C₄⋊C₄	—	2	2	no	—
230	C₂×C₄⋊C₄	—	43	1	no	—
231	C₂×C₄⋊C₄	—	43	1	no	—
232	C₂×C₄⋊C₄	C₄×D₄₃	1	4	yes	—
233	C₂×C₂²⋊C₄	—	86	1	no	—
234	C₂×C₄²	—	86	1	no	—
235	#97	—	86	4	no	—
236	C₄×C₈	—	86	4	no	—
237	C₂⁴	#10621900	1	1	yes	characteristic
238	C₂²×C₄	—	2	4	no	—
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	—	8	2	no	—
242	C₂²×C₄	—	8	2	no	—
243	C₂²×C₄	—	86	1	no	—
244	C₂²×C₄	—	86	1	no	—
245	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
246	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
247	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
248	C₂²×C₄	#10621898	1	2	yes	—
249	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
250	C₂²×C₄	#10621900	1	2	yes	—
251	C₂²×C₄	#10621900	1	2	yes	—
252	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
253	C₂²×C₄	#10621902	1	1	yes	characteristic
254	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
255	M₁₆	—	43	4	no	—
256	M₁₆	—	86	2	no	—
257	M₁₆	—	86	4	no	—
258	M₁₆	—	86	4	no	—
259	M₁₆	—	86	4	no	—
260	M₁₆	—	172	2	no	—
261	M₁₆	—	172	2	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	4	no	—
267	C₂×C₈	—	86	2	no	—
268	C₂×C₈	—	86	4	no	—
269	C₂×C₈	—	86	4	no	—
270	C₂×C₈	—	172	2	no	—
271	C₂×C₈	—	172	2	no	—
272	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
273	C₂×C₈	(C₂×D₄₃)⋊C₄	1	2	yes	—
274	C₄⋊C₄	—	2	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	43	2	no	—
279	C₄⋊C₄	—	43	2	no	—
280	C₄⋊C₄	—	86	1	no	—
281	C₄⋊C₄	—	86	1	no	—
282	C₂²⋊C₄	—	86	2	no	—
283	C₂²⋊C₄	—	86	2	no	—
284	C₄²	—	86	2	no	—
285	C₄²	—	86	2	no	—
286	C₂³	—	2	2	no	—
287	C₂³	—	2	1	no	—
288	C₂³	—	2	1	no	—
289	C₂³	#13063281	1	2	yes	—
290	C₂³	#13063283	1	1	yes	characteristic
291	C₂³	#13063296	1	1	yes	characteristic
292	C₂³	#13063305	1	1	yes	characteristic
293	C₂³	#13063318	1	2	yes	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	—	8	2	no	—
297	C₂×C₄	—	8	2	no	—
298	C₂×C₄	—	8	4	no	—
299	C₂×C₄	—	86	2	no	—
300	C₂×C₄	—	86	2	no	—
301	C₂×C₄	—	172	2	no	—
302	C₂×C₄	—	172	1	no	—
303	C₂×C₄	—	172	1	no	—
304	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
305	C₂×C₄	C₂×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
306	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	#13063281	1	1	yes	characteristic
308	C₂×C₄	#13063281	1	1	yes	characteristic
309	C₂×C₄	#13063282	1	1	yes	characteristic
310	C₂×C₄	#13063283	1	2	yes	—
311	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
312	C₂×C₄	C₄×C₂²⋊D₄₃	1	2	yes	—
313	C₂×C₄	C₄×D₁₇₂	1	2	yes	—
314	C₂×C₄	#13063290	1	2	yes	—
315	C₂×C₄	#13063300	1	1	yes	characteristic
316	C₂×C₄	#13063301	1	1	yes	characteristic
317	C₂×C₄	#13063306	1	2	yes	—
318	C₂×C₄	#13063309	1	2	yes	—
319	C₂×C₄	#13063311	1	2	yes	—
320	C₂×C₄	#13063315	1	2	yes	—
321	C₂×C₄	#13063320	1	2	yes	—
322	C₈	—	2	2	no	—
323	C₈	—	2	2	no	—
324	C₈	—	86	4	no	—
325	C₈	—	172	4	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	2	no	—
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	2	1	no	—
334	C₂²	#14582744	1	1	yes	characteristic
335	C₂²	#14582756	1	1	yes	characteristic
336	C₂²	#14582766	1	1	yes	characteristic
337	C₂²	#14582784	1	2	yes	—
338	C₂²	#14582787	1	2	yes	—
339	C₂²	#14583159	1	2	yes	—
340	C₂²	#14583166	1	1	yes	characteristic
341	C₂²	#14583171	1	1	yes	characteristic
342	C₄	—	8	4	no	—
343	C₄	—	172	2	no	—
344	C₄	—	172	2	no	—
345	C₄	C₄×(C₂×D₄₃)⋊C₄	1	1	yes	characteristic
346	C₄	#14582750	1	1	yes	characteristic
347	C₄	#14582760	1	1	yes	characteristic
348	C₄	#14582763	1	1	yes	characteristic
349	C₄	#14582780	1	2	yes	—
350	C₄	#14582791	1	2	yes	—
351	C₂	—	2	2	no	—
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	#19132819	1	1	yes	characteristic
355	C₂	#19134655	1	1	yes	characteristic
356	C₂	#19134660	1	1	yes	characteristic
357	C₂	#19135247	1	2	yes	—
358	C₂	#19135254	1	2	yes	—

	subgroup	G/H
center	C₂³	#13063296
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×#340	C₂
socle	C₂³×C₄₃	#121
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×#340	D₄₃
hypocenter	C₄₃	#10827
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#340

Complements

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

N	H ≅ G/N	type
10. #14582854	C₄	semidirect
13. #14582668	C₄	semidirect
200. C₄₃	#10827	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582854	2
C₂×#14582668	2
C₂×#14582664	2
#19134314	2
C₂×C₄₃×#340	2
#10827	43