Subgroups — #25105228

Order 11008.

998 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583021	C₂	1	2	yes	maximal, complemented
2	C₂×#14583016	C₂	1	2	yes	maximal, complemented
3	#19143313	C₂	1	2	yes	maximal, complemented
4	#19135489	C₂	1	2	yes	maximal, complemented
5	#19135484	C₂	1	2	yes	maximal, complemented
6	#19135428	C₂	1	2	yes	maximal, complemented
7	#19134767	C₂	1	1	yes	maximal, characteristic, complemented
8	#19134281	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#3147	C₂	1	1	yes	maximal, characteristic
10	C₂×#13063332	C₂²	1	4	yes	—
11	#14583021	—	2	8	no	—
12	#14583016	—	2	8	no	—
13	#14582971	—	2	8	no	—
14	C₂×#13063261	C₂²	1	2	yes	—
15	C₂×#13063261	C₂²	1	2	yes	—
16	#14582923	C₂²	1	2	yes	—
17	C₂×#13063252	C₂²	1	4	yes	—
18	C₂×#13063252	C₂²	1	4	yes	—
19	#14582866	C₂²	1	4	yes	—
20	C₂×#13063236	C₂²	1	1	yes	characteristic
21	C₂×#13063235	C₂²	1	1	yes	characteristic
22	C₂×#13063235	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
24	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
25	#14582745	C₂²	1	1	yes	characteristic
26	C₂×#135×C₄₃	C₂²	1	2	yes	—
27	C₂×#135×C₄₃	C₂²	1	2	yes	—
28	C₄₃×#532	C₂²	1	2	yes	—
29	C₄₃×#495	C₄	1	2	yes	complemented
30	C₄₃×#495	C₄	1	2	yes	complemented
31	C₄₃×#494	C₄	1	1	yes	characteristic, complemented
32	C₄₃×#493	C₄	1	1	yes	characteristic, complemented
33	C₄₃×#492	C₄	1	2	yes	complemented
34	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
35	#13063332	—	2	8	no	—
36	#13063332	—	2	8	no	—
37	C₂×#10621877	—	2	4	no	—
38	C₂×#10621877	C₂³	1	4	yes	—
39	C₂²×C₄₃⋊C₈	—	2	4	no	—
40	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
43	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
44	#13063261	—	2	4	no	—
45	#13063261	—	2	4	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
47	#13063252	—	2	8	no	—
48	#13063252	—	2	8	no	—
49	C₂²⋊(C₄₃⋊C₈)	—	2	8	no	—
50	#13063236	D₄	1	4	yes	—
51	#13063235	—	2	2	no	—
52	#13063235	D₄	1	4	yes	complemented
53	C₈×Dic₄₃	—	2	2	no	—
54	C₈×Dic₄₃	D₄	1	4	yes	complemented
55	C₄₃⋊#95	—	2	2	no	—
56	C₂×#41×C₄₃	C₂³	1	4	yes	—
57	#135×C₄₃	—	2	4	no	—
58	#135×C₄₃	—	2	4	no	—
59	#135×C₄₃	—	2	4	no	—
60	#135×C₄₃	—	2	4	no	—
61	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	—
62	C₂×Q₁₆×C₄₃	C₂×C₄	1	2	yes	—
63	C₂×SD₁₆×C₄₃	C₂×C₄	1	4	yes	—
64	C₂×SD₁₆×C₄₃	C₂×C₄	1	4	yes	—
65	C₂×D₈×C₄₃	C₂×C₄	1	2	yes	—
66	C₂×D₈×C₄₃	C₂×C₄	1	2	yes	—
67	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
69	C₄⋊Q₈×C₄₃	C₂×C₄	1	2	yes	—
70	#127×C₄₃	C₂×C₄	1	2	yes	—
71	#124×C₄₃	C₂×C₄	1	4	yes	—
72	#121×C₄₃	—	2	4	no	—
73	C₄×D₄×C₄₃	—	2	4	no	—
74	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
76	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
77	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
78	C₄×C₈×C₄₃	C₂×C₄	1	1	yes	characteristic
79	#41×C₄₃	—	2	4	no	—
80	#41×C₄₃	—	2	4	no	—
81	#41×C₄₃	—	2	4	no	—
82	#41×C₄₃	—	2	4	no	—
83	C₂×Q₈×C₄₃	C₂²×C₄	1	4	yes	—
84	C₂×D₄×C₄₃	—	2	4	no	—
85	C₂×D₄×C₄₃	—	2	4	no	—
86	C₂×D₄×C₄₃	C₂²×C₄	1	4	yes	—
87	C₂²×C₄×C₄₃	—	2	4	no	—
88	C₂²×C₄×C₄₃	#41	1	2	yes	—
89	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
90	C₂²×Dic₄₃	—	2	1	no	—
91	C₂²×Dic₄₃	—	2	1	no	—
92	Q₁₆×C₄₃	—	2	4	no	—
93	Q₁₆×C₄₃	—	2	4	no	—
94	SD₁₆×C₄₃	—	2	8	no	—
95	SD₁₆×C₄₃	—	2	8	no	—
96	D₈×C₄₃	—	2	4	no	—
97	D₈×C₄₃	—	2	4	no	—
98	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₄₃	#41	1	2	yes	—
105	C₂×C₈×C₄₃	#41	1	2	yes	—
106	C₄⋊C₄×C₄₃	—	2	4	no	—
107	C₂²⋊C₄×C₄₃	—	2	8	no	—
108	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
109	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
110	C₄²×C₄₃	C₂×D₄	1	2	yes	—
111	C₄×Dic₄₃	—	2	1	no	—
112	C₄×Dic₄₃	—	2	1	no	—
113	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
114	C₄×Dic₄₃	C₂×D₄	1	2	yes	—
115	#10621877	—	2	4	no	—
116	#10621877	—	2	4	no	—
117	#10621877	—	4	8	no	—
118	C₂×C₄₃⋊C₈	—	2	1	no	—
119	C₂×C₄₃⋊C₈	—	2	1	no	—
120	C₂×C₄₃⋊C₈	—	2	4	no	—
121	C₂×C₄₃⋊C₈	—	4	8	no	—
122	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
123	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
124	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
125	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
126	C₂³×C₄₃	—	2	4	no	—
127	C₂³×C₄₃	C₂×#41	1	1	yes	characteristic
128	Q₈×C₄₃	—	2	4	no	—
129	Q₈×C₄₃	—	2	4	no	—
130	D₄×C₄₃	—	2	4	no	—
131	D₄×C₄₃	—	2	4	no	—
132	D₄×C₄₃	—	4	4	no	—
133	D₄×C₄₃	—	4	4	no	—
134	D₄×C₄₃	—	8	4	no	—
135	C₂×C₄×C₄₃	—	2	4	no	—
136	C₂×C₄×C₄₃	—	4	2	no	—
137	C₂×C₄×C₄₃	—	4	4	no	—
138	C₂×C₄×C₄₃	—	4	4	no	—
139	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
140	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
141	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
145	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
147	C₂×Dic₄₃	—	2	2	no	—
148	C₂×Dic₄₃	—	2	2	no	—
149	C₂×Dic₄₃	—	4	2	no	—
150	C₂×Dic₄₃	—	4	2	no	—
151	C₈×C₄₃	C₄×D₄	1	2	yes	—
152	C₈×C₄₃	C₄×D₄	1	2	yes	—
153	C₈×C₄₃	C₄×D₄	1	2	yes	—
154	C₈×C₄₃	C₄×D₄	1	2	yes	—
155	C₄₃⋊C₈	—	2	2	no	—
156	C₄₃⋊C₈	—	4	4	no	—
157	C₄₃⋊C₈	#127	1	4	yes	—
158	#21707	—	43	1	no	maximal, Sylow
159	C₂²×C₄₃	—	2	2	no	—
160	C₂²×C₄₃	—	4	4	no	—
161	C₂²×C₄₃	—	4	8	no	—
162	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
163	C₂²×C₄₃	#532	1	1	yes	characteristic
164	C₂²×C₄₃	#532	1	1	yes	characteristic
165	C₄×C₄₃	—	2	2	no	—
166	C₄×C₄₃	—	2	2	no	—
167	C₄×C₄₃	—	4	4	no	—
168	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
169	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₄×C₄₃	C₂×#127	1	1	yes	characteristic
171	C₄×C₄₃	#532	1	1	yes	characteristic
172	Dic₄₃	—	4	2	no	—
173	Dic₄₃	—	4	2	no	—
174	#3147	D₄₃	1	1	yes	characteristic
175	C₂×#444	—	43	2	no	—
176	C₂×#443	—	43	2	no	—
177	#2932	—	43	2	no	—
178	#1975	—	43	2	no	—
179	#1974	—	43	2	no	—
180	#1963	—	43	2	no	—
181	#1848	—	43	1	no	—
182	#1726	—	43	1	no	—
183	C₂×C₄₃	—	2	1	no	—
184	C₂×C₄₃	—	2	1	no	—
185	C₂×C₄₃	—	8	4	no	—
186	C₂×C₄₃	C₄×#127	1	1	yes	characteristic
187	C₂×C₄₃	#2957	1	2	yes	—
188	C₂×#135	C₂×D₄₃	1	2	yes	—
189	C₂×#135	C₂×D₄₃	1	2	yes	—
190	C₂×#131	—	43	4	no	—
191	#532	C₂×D₄₃	1	2	yes	—
192	#495	Dic₄₃	1	2	yes	complemented
193	#495	Dic₄₃	1	2	yes	complemented
194	#494	Dic₄₃	1	1	yes	characteristic, complemented
195	#493	Dic₄₃	1	1	yes	characteristic, complemented
196	#492	Dic₄₃	1	2	yes	complemented
197	#444	—	86	8	no	—
198	#443	—	86	8	no	—
199	#435	—	86	8	no	—
200	C₂×#108	—	43	2	no	—
201	C₂×#108	—	43	2	no	—
202	#423	—	43	2	no	—
203	C₂×#104	—	43	4	no	—
204	C₂×#104	—	43	4	no	—
205	#408	—	43	4	no	—
206	C₂×#97	—	43	1	no	—
207	C₂×#97	—	43	1	no	—
208	C₂×#97	—	43	1	no	—
209	C₂×C₄×C₈	—	43	1	no	—
210	C₂×C₄×C₈	—	43	1	no	—
211	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
212	#376	—	43	1	no	—
213	C₄₃	#21707	1	1	yes	characteristic, complemented, Sylow
214	C₂×#41	C₂²×D₄₃	1	4	yes	—
215	#135	—	2	4	no	—
216	#135	—	2	4	no	—
217	#135	—	2	4	no	—
218	#135	—	2	4	no	—
219	C₂×Q₁₆	C₂×Dic₄₃	1	2	yes	—
220	C₂×Q₁₆	C₂×Dic₄₃	1	2	yes	—
221	C₂×SD₁₆	C₂×Dic₄₃	1	4	yes	—
222	C₂×SD₁₆	C₂×Dic₄₃	1	4	yes	—
223	C₂×D₈	C₂×Dic₄₃	1	2	yes	—
224	C₂×D₈	C₂×Dic₄₃	1	2	yes	—
225	#131	—	86	8	no	—
226	#131	—	86	8	no	—
227	C₂×M₁₆	—	43	4	no	—
228	C₂×M₁₆	—	86	4	no	—
229	C₂²×C₈	—	43	1	no	—
230	C₂²×C₈	—	43	1	no	—
231	C₂²×C₈	—	86	4	no	—
232	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
233	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
234	C₄⋊Q₈	C₂×Dic₄₃	1	2	yes	—
235	#127	C₂×Dic₄₃	1	2	yes	—
236	#124	C₂×Dic₄₃	1	4	yes	—
237	#121	—	2	4	no	—
238	C₄×D₄	—	2	4	no	—
239	C₂×C₄²	—	43	1	no	—
240	C₂×C₄²	—	43	1	no	—
241	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
242	#108	—	86	4	no	—
243	#108	—	86	4	no	—
244	C₄⋊C₈	—	86	4	no	—
245	#104	—	86	8	no	—
246	#104	—	86	8	no	—
247	C₂²⋊C₈	—	86	8	no	—
248	#97	—	43	4	no	—
249	#97	—	43	4	no	—
250	#97	—	86	2	no	—
251	C₄×C₈	—	43	4	no	—
252	C₄×C₈	—	86	2	no	—
253	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
254	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
255	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
256	C₄×C₈	C₂×Dic₄₃	1	1	yes	characteristic
257	#95	—	86	2	no	—
258	#41	—	2	4	no	—
259	#41	—	2	4	no	—
260	#41	—	2	4	no	—
261	#41	—	2	4	no	—
262	C₂×Q₈	C₂²×Dic₄₃	1	4	yes	—
263	C₂×D₄	—	2	4	no	—
264	C₂×D₄	—	2	4	no	—
265	C₂×D₄	C₂²×Dic₄₃	1	4	yes	—
266	C₂²×C₄	—	2	4	no	—
267	C₂²×C₄	—	86	1	no	—
268	C₂²×C₄	—	86	1	no	—
269	C₂²×C₄	#10621900	1	2	yes	—
270	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
271	Q₁₆	—	2	4	no	—
272	Q₁₆	—	2	4	no	—
273	SD₁₆	—	2	8	no	—
274	SD₁₆	—	2	8	no	—
275	D₈	—	2	4	no	—
276	D₈	—	2	4	no	—
277	M₁₆	—	86	4	no	—
278	M₁₆	—	86	4	no	—
279	M₁₆	—	172	8	no	—
280	C₂×C₈	—	43	2	no	—
281	C₂×C₈	—	43	2	no	—
282	C₂×C₈	—	43	2	no	—
283	C₂×C₈	—	43	2	no	—
284	C₂×C₈	—	86	1	no	—
285	C₂×C₈	—	86	1	no	—
286	C₂×C₈	—	86	4	no	—
287	C₂×C₈	—	172	8	no	—
288	C₂×C₈	D₄×D₄₃	1	2	yes	—
289	C₂×C₈	D₄×D₄₃	1	2	yes	—
290	C₂×C₈	#10621900	1	2	yes	—
291	C₂×C₈	#10621900	1	2	yes	—
292	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
293	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
294	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
295	C₂×C₈	C₂²×Dic₄₃	1	1	yes	characteristic
296	C₄⋊C₄	—	2	4	no	—
297	C₂²⋊C₄	—	2	8	no	—
298	C₄²	—	43	2	no	—
299	C₄²	—	43	2	no	—
300	C₄²	—	86	1	no	—
301	C₄²	—	86	1	no	—
302	C₄²	D₄×D₄₃	1	2	yes	—
303	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
304	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
305	C₂³	—	2	4	no	—
306	C₂³	C₂×#10621900	1	1	yes	characteristic
307	Q₈	—	2	4	no	—
308	Q₈	—	2	4	no	—
309	D₄	—	2	4	no	—
310	D₄	—	2	4	no	—
311	D₄	—	4	4	no	—
312	D₄	—	4	4	no	—
313	D₄	—	8	4	no	—
314	C₂×C₄	—	2	4	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	4	no	—
317	C₂×C₄	—	4	4	no	—
318	C₂×C₄	—	86	2	no	—
319	C₂×C₄	—	86	2	no	—
320	C₂×C₄	—	172	2	no	—
321	C₂×C₄	—	172	2	no	—
322	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
323	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
324	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
325	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
326	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
327	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
328	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
329	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
330	C₈	—	43	4	no	—
331	C₈	—	86	2	no	—
332	C₈	—	172	4	no	—
333	C₈	D₄×Dic₄₃	1	2	yes	—
334	C₈	D₄×Dic₄₃	1	2	yes	—
335	C₈	D₄×Dic₄₃	1	2	yes	—
336	C₈	D₄×Dic₄₃	1	2	yes	—
337	C₂²	—	2	2	no	—
338	C₂²	—	4	8	no	—
339	C₂²	—	4	4	no	—
340	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
341	C₂²	#14583447	1	1	yes	characteristic
342	C₂²	#14583447	1	1	yes	characteristic
343	C₄	—	2	2	no	—
344	C₄	—	2	2	no	—
345	C₄	—	4	4	no	—
346	C₄	—	172	2	no	—
347	C₄	—	172	2	no	—
348	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
349	C₄	C₂×D₄×Dic₄₃	1	1	yes	characteristic
350	C₄	#127×D₄₃	1	1	yes	characteristic
351	C₄	#14583447	1	1	yes	characteristic
352	C₂	—	2	1	no	—
353	C₂	—	2	1	no	—
354	C₂	—	8	4	no	—
355	C₂	#127×Dic₄₃	1	1	yes	characteristic
356	C₂	#19143512	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting, prime radical	C₄₃×#3147	C₂
socle	C₂²×C₄₃	C₂×C₄×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#3147	D₄₃
hypocenter	C₄₃	#21707
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #3147

Complements

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

N	H ≅ G/N	type
1. C₂×#14583021	C₂	semidirect
2. C₂×#14583016	C₂	semidirect
3. #19143313	C₂	semidirect
4. #19135489	C₂	semidirect
5. #19135484	C₂	semidirect
6. #19135428	C₂	semidirect
7. #19134767	C₂	semidirect
8. #19134281	C₂	semidirect
29. C₄₃×#495	C₄	semidirect
30. C₄₃×#495	C₄	semidirect
31. C₄₃×#494	C₄	semidirect
32. C₄₃×#493	C₄	semidirect
33. C₄₃×#492	C₄	semidirect
52. #13063235	D₄	semidirect
54. C₈×Dic₄₃	D₄	semidirect
192. #495	Dic₄₃	semidirect
193. #495	Dic₄₃	semidirect
194. #494	Dic₄₃	semidirect
195. #493	Dic₄₃	semidirect
196. #492	Dic₄₃	semidirect
213. C₄₃	#21707	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583021	2
C₂×#14583016	2
#19143313	2
#19135489	2
#19135484	2
#19135428	2
#19134767	2
#19134281	2
C₄₃×#3147	2
#21707	43