Subgroups — #25001812

Order 11008.

842 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582857	C₂	1	2	yes	maximal
2	C₂×#14582855	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582671	C₂	1	2	yes	maximal
4	#19134328	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1742	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	1	yes	characteristic
7	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×#13063284	C₂²	1	2	yes	—
9	#14582857	—	2	4	no	—
10	#14582857	C₄	1	8	yes	complemented
11	#14582855	—	2	2	no	—
12	#14582855	—	2	2	no	—
13	C₂×#13063236	C₄	1	2	yes	complemented
14	C₂×#13063235	—	2	2	no	—
15	C₂×C₄×C₄₃⋊C₈	C₄	1	2	yes	complemented
16	C₂×C₈×Dic₄₃	—	2	2	no	—
17	#14582671	C₄	1	8	yes	complemented
18	#14582666	C₄	1	4	yes	—
19	C₂²×M₁₆×C₄₃	C₂²	1	2	yes	—
20	C₂×#117×C₄₃	C₂²	1	1	yes	characteristic
21	C₄₃×#343	C₄	1	4	yes	complemented
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	1	no	—
24	C₂×#10621877	—	2	1	no	—
25	C₂×#10621877	C₂×C₄	1	4	yes	—
26	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
30	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
31	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
32	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
33	C₂²×C₄₃⋊C₈	Q₈	1	2	yes	—
34	#13063284	—	2	2	no	—
35	#13063284	—	2	2	no	—
36	#13063284	C₂×C₄	1	4	yes	—
37	#13063284	C₂×C₄	1	4	yes	—
38	C₂×C₄⋊Dic₄₃	—	2	2	no	—
39	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
40	C₂×C₄×Dic₄₃	—	2	2	no	—
41	#13063236	—	2	4	no	—
42	#13063235	—	2	8	no	—
43	C₄×C₄₃⋊C₈	—	2	2	no	—
44	C₄×C₄₃⋊C₈	—	2	2	no	—
45	C₈×Dic₄₃	—	2	8	no	—
46	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
47	C₂×M₁₆×C₄₃	—	2	2	no	—
48	C₂×M₁₆×C₄₃	—	2	2	no	—
49	C₂×M₁₆×C₄₃	—	2	2	no	—
50	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
51	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
52	C₂×M₁₆×C₄₃	C₂×C₄	1	4	yes	—
53	C₂²×C₈×C₄₃	—	2	2	no	—
54	#117×C₄₃	—	2	2	no	—
55	#117×C₄₃	—	2	2	no	—
56	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
57	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
59	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
61	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	#41	1	2	yes	—
69	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
71	C₂²×Dic₄₃	—	2	4	no	—
72	M₁₆×C₄₃	—	2	4	no	—
73	M₁₆×C₄₃	—	2	4	no	—
74	M₁₆×C₄₃	—	2	4	no	—
75	M₁₆×C₄₃	C₄²	1	8	yes	—
76	C₂×C₈×C₄₃	—	2	4	no	—
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	—	2	4	no	—
79	C₄⋊C₄×C₄₃	—	2	4	no	—
80	C₂²⋊C₄×C₄₃	—	2	4	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₂²⋊Dic₄₃	—	2	4	no	—
84	C₂²⋊Dic₄₃	C₄²	1	8	yes	—
85	C₄⋊Dic₄₃	—	2	4	no	—
86	C₄⋊Dic₄₃	—	2	4	no	—
87	C₄×Dic₄₃	—	2	4	no	—
88	C₄×Dic₄₃	—	2	4	no	—
89	#10621877	—	2	4	no	—
90	#10621877	—	2	4	no	—
91	C₂×C₄₃⋊C₈	—	2	2	no	—
92	C₂×C₄₃⋊C₈	—	2	2	no	—
93	C₂×C₄₃⋊C₈	C₄²	1	4	yes	—
94	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
95	C₂×C₄₃⋊C₈	C₄²	1	2	yes	—
96	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	4	yes	—
97	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	4	yes	—
100	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
101	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
102	C₂×C₄₃⋊C₈	C₄⋊C₄	1	2	yes	—
103	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
104	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
105	C₂³×C₄₃	—	4	2	no	—
106	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
107	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
108	C₂³×C₄₃	C₄×D₄	1	2	yes	—
109	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
110	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	4	4	no	—
113	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
116	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	2	yes	—
117	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
120	C₂×C₄×C₄₃	#117	1	2	yes	—
121	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#117	1	2	yes	—
124	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
126	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
129	C₂×C₄×C₄₃	C₄×Q₈	1	2	yes	—
130	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
131	C₂×Dic₄₃	—	2	8	no	—
132	C₂×Dic₄₃	—	4	8	no	—
133	C₈×C₄₃	—	2	8	no	—
134	C₄₃⋊C₈	—	2	4	no	—
135	C₄₃⋊C₈	#95	1	2	yes	—
136	C₄₃⋊C₈	#95	1	4	yes	—
137	C₄₃⋊C₈	#95	1	2	yes	—
138	#10711	—	43	1	no	maximal, Sylow
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	2	4	no	—
141	C₂²×C₄₃	—	4	2	no	—
142	C₂²×C₄₃	—	4	1	no	—
143	C₂²×C₄₃	—	4	1	no	—
144	C₂²×C₄₃	C₄³	1	1	yes	characteristic
145	C₂²×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
146	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
147	C₂²×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
148	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₄₃	—	4	4	no	—
150	C₄×C₄₃	C₂×#95	1	1	yes	characteristic
151	C₄×C₄₃	#376	1	1	yes	characteristic
152	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
153	C₄×C₄₃	C₄×C₂²⋊C₄	1	2	yes	—
154	C₄×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
155	C₄×C₄₃	C₄×C₄⋊C₄	1	2	yes	—
156	Dic₄₃	—	4	8	no	—
157	C₂×#405	—	43	2	no	—
158	C₂×#405	—	43	1	no	—
159	C₂×#344	—	43	2	no	—
160	#1742	D₄₃	1	1	yes	characteristic
161	#1740	—	43	1	no	—
162	C₂×C₄₃	—	2	2	no	—
163	C₂×C₄₃	—	2	1	no	—
164	C₂×C₄₃	—	2	2	no	—
165	C₂×C₄₃	—	2	1	no	—
166	C₂×C₄₃	C₄×#95	1	1	yes	characteristic
167	C₂×C₄₃	#1757	1	2	yes	—
168	C₂²×M₁₆	—	43	1	no	—
169	C₂²×M₁₆	C₂×D₄₃	1	2	yes	—
170	C₂³×C₈	—	43	1	no	—
171	C₂×#117	—	43	2	no	—
172	C₂×#117	C₂×D₄₃	1	1	yes	characteristic
173	#405	—	43	8	no	—
174	#405	—	86	2	no	—
175	#405	—	86	2	no	—
176	#405	—	86	4	no	—
177	C₂×#97	—	43	2	no	—
178	C₂×#97	—	86	2	no	—
179	C₂×C₄×C₈	—	43	2	no	—
180	C₂×C₄×C₈	—	86	2	no	—
181	#344	—	43	8	no	—
182	#343	Dic₄₃	1	4	yes	complemented
183	#342	—	43	4	no	—
184	C₄₃	#10711	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	43	4	no	—
190	C₂×M₁₆	—	86	2	no	—
191	C₂×M₁₆	—	86	1	no	—
192	C₂×M₁₆	—	86	1	no	—
193	C₂×M₁₆	C₄×D₄₃	1	4	yes	—
194	C₂×M₁₆	C₄×D₄₃	1	4	yes	—
195	C₂×M₁₆	C₂×Dic₄₃	1	4	yes	—
196	C₂²×C₈	—	2	2	no	—
197	C₂²×C₈	—	43	1	no	—
198	C₂²×C₈	—	43	1	no	—
199	C₂²×C₈	—	43	4	no	—
200	C₂²×C₈	—	43	2	no	—
201	C₂²×C₈	—	43	2	no	—
202	C₂²×C₈	—	43	2	no	—
203	C₂²×C₈	—	43	2	no	—
204	C₂²×C₈	—	43	2	no	—
205	#117	—	2	2	no	—
206	#117	—	2	2	no	—
207	#117	—	43	4	no	—
208	#117	—	43	4	no	—
209	#117	—	86	2	no	—
210	#117	—	86	2	no	—
211	C₂×C₄⋊C₄	—	86	2	no	—
212	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
213	C₂×C₂²⋊C₄	—	43	4	no	—
214	C₂×C₂²⋊C₄	C₄×D₄₃	1	2	yes	—
215	C₂×C₄²	—	86	2	no	—
216	C₂×C₄²	C₄×D₄₃	1	2	yes	—
217	#97	—	86	4	no	—
218	#97	—	86	8	no	—
219	C₄×C₈	—	86	2	no	—
220	C₄×C₈	—	86	2	no	—
221	C₄×C₈	—	86	8	no	—
222	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
223	C₂²×C₄	—	86	4	no	—
224	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
225	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
227	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
228	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
229	C₂²×C₄	D₄×D₄₃	1	2	yes	—
230	C₂²×C₄	#10621900	1	1	yes	characteristic
231	C₂²×C₄	#10621900	1	2	yes	—
232	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
233	C₂²×C₄	#10621902	1	1	yes	characteristic
234	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	86	4	no	—
239	M₁₆	—	86	4	no	—
240	M₁₆	C₄×Dic₄₃	1	8	yes	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₂×C₈	—	43	2	no	—
245	C₂×C₈	—	43	2	no	—
246	C₂×C₈	—	43	4	no	—
247	C₂×C₈	—	43	4	no	—
248	C₂×C₈	—	43	1	no	—
249	C₂×C₈	—	43	1	no	—
250	C₂×C₈	—	43	2	no	—
251	C₂×C₈	—	43	2	no	—
252	C₂×C₈	—	43	2	no	—
253	C₂×C₈	—	43	4	no	—
254	C₂×C₈	—	43	4	no	—
255	C₂×C₈	—	43	4	no	—
256	C₂×C₈	—	86	2	no	—
257	C₂×C₈	—	86	2	no	—
258	C₄⋊C₄	—	2	4	no	—
259	C₄⋊C₄	—	86	4	no	—
260	C₄⋊C₄	—	86	4	no	—
261	C₂²⋊C₄	—	2	4	no	—
262	C₂²⋊C₄	—	43	8	no	—
263	C₂²⋊C₄	—	86	4	no	—
264	C₄²	—	2	2	no	—
265	C₄²	—	2	2	no	—
266	C₄²	—	86	4	no	—
267	C₄²	—	86	4	no	—
268	C₂³	—	4	2	no	—
269	C₂³	C₄²×D₄₃	1	2	yes	—
270	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
271	C₂³	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
272	C₂³	#13063287	1	1	yes	characteristic
273	C₂³	#13063290	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	4	4	no	—
276	C₂×C₄	—	86	8	no	—
277	C₂×C₄	—	172	8	no	—
278	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
279	C₂×C₄	C₄²×D₄₃	1	1	yes	characteristic
280	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
281	C₂×C₄	C₂²⋊C₄×D₄₃	1	2	yes	—
282	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
283	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
284	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
285	C₂×C₄	#13063281	1	2	yes	—
286	C₂×C₄	#13063281	1	1	yes	characteristic
287	C₂×C₄	#13063283	1	1	yes	characteristic
288	C₂×C₄	#13063283	1	2	yes	—
289	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
290	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
291	C₂×C₄	#13063287	1	2	yes	—
292	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
293	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
294	C₂×C₄	#13063290	1	2	yes	—
295	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
296	C₈	—	2	8	no	—
297	C₈	—	43	2	no	—
298	C₈	—	43	4	no	—
299	C₈	—	43	2	no	—
300	C₈	—	86	4	no	—
301	C₂²	—	2	2	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	4	2	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	C₄²×Dic₄₃	1	1	yes	characteristic
307	C₂²	#14582746	1	2	yes	—
308	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
309	C₂²	#14582750	1	2	yes	—
310	C₂²	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
311	C₄	—	4	4	no	—
312	C₄	—	172	8	no	—
313	C₄	#95×D₄₃	1	1	yes	characteristic
314	C₄	#14582744	1	1	yes	characteristic
315	C₄	#14582746	1	1	yes	characteristic
316	C₄	C₂²⋊C₄×Dic₄₃	1	2	yes	—
317	C₄	#14582750	1	1	yes	characteristic
318	C₄	C₄⋊C₄×Dic₄₃	1	2	yes	—
319	C₂	—	2	2	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	—	2	1	no	—
323	C₂	#95×Dic₄₃	1	1	yes	characteristic
324	C₂	#19134398	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₄₃
commutator	C₂²×C₄₃	C₄³
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1742	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1742	D₄₃
hypocenter	C₄₃	#10711
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1742

Complements

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

N	H ≅ G/N	type
10. #14582857	C₄	semidirect
13. C₂×#13063236	C₄	semidirect
15. C₂×C₄×C₄₃⋊C₈	C₄	semidirect
17. #14582671	C₄	semidirect
21. C₄₃×#343	C₄	semidirect
182. #343	Dic₄₃	semidirect
184. C₄₃	#10711	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582857	2
C₂×#14582855	2
C₂×#14582671	2
#19134328	2
C₄₃×#1742	2
#10711	43