Subgroups — #25097006

Order 11008.

1126 conjugacy classes of subgroups.

The subgroup lattice has 338 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	C₂×#14582935	C₂	1	2	yes	maximal
3	C₂²×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	#19136998	C₂	1	2	yes	maximal
5	#19134758	C₂	1	2	yes	maximal, complemented
6	#19134756	C₂	1	2	yes	maximal, complemented
7	#19134753	C₂	1	2	yes	maximal, complemented
8	#19134750	C₂	1	2	yes	maximal, complemented
9	C₂×C₄₃×#427	C₂	1	1	yes	maximal, characteristic
10	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#13063284	C₂²	1	2	yes	—
12	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
13	#14582937	—	2	4	no	—
14	#14582935	—	2	8	no	—
15	C₂×#13063259	C₂²	1	2	yes	—
16	C₂×#13063258	C₂²	1	4	yes	—
17	C₂×#13063257	C₂²	1	2	yes	—
18	C₂×#13063256	C₂²	1	4	yes	—
19	C₂×C₈×Dic₄₃	C₄	1	8	yes	complemented
20	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
21	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₈×Dic₄₃	C₂²	1	1	yes	characteristic
24	#14582768	C₂²	1	4	yes	—
25	#14582764	C₂²	1	4	yes	—
26	#14582757	—	2	4	no	—
27	C₄×Dic₄₃⋊C₄	—	2	4	no	—
28	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂×#117×C₄₃	C₂²	1	2	yes	—
30	C₄₃×#427	—	2	4	no	—
31	C₂×#107×C₄₃	C₂²	1	2	yes	—
32	C₂×#106×C₄₃	C₂²	1	2	yes	—
33	C₂³×Dic₄₃	D₄	1	2	yes	—
34	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
35	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
36	C₂²×C₄₃⋊C₈	Q₈	1	4	yes	—
37	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
38	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
39	#13063284	—	2	4	no	—
40	#13063284	—	2	4	no	—
41	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
42	C₂×Dic₄₃⋊C₄	—	2	8	no	—
43	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
44	C₂×C₄×Dic₄₃	—	2	2	no	—
45	C₂×C₄×Dic₄₃	C₂×C₄	1	4	yes	—
46	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
47	C₂×C₄×Dic₄₃	Q₈	1	4	yes	—
48	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
49	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
50	#13063259	—	2	4	no	—
51	#13063258	—	2	8	no	—
52	#13063257	—	2	4	no	—
53	#13063256	—	2	8	no	—
54	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
55	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
56	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
57	C₈×Dic₄₃	C₂×C₄	1	4	yes	—
58	C₄₃⋊#95	—	2	8	no	—
59	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
61	C₂²×C₈×C₄₃	D₄	1	4	yes	—
62	C₂²×C₈×C₄₃	Q₈	1	4	yes	—
63	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
64	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
65	#117×C₄₃	—	2	4	no	—
66	#117×C₄₃	—	2	4	no	—
67	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
68	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
69	C₂×C₄²×C₄₃	—	2	2	no	—
70	#107×C₄₃	—	2	4	no	—
71	#106×C₄₃	—	2	4	no	—
72	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	—	2	4	no	—
74	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
76	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
82	C₂²×Dic₄₃	—	2	8	no	—
83	C₂²×Dic₄₃	—	2	4	no	—
84	C₂²×Dic₄₃	C₄⋊C₄	1	8	yes	—
85	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
86	C₂²×Dic₄₃	C₂×D₄	1	2	yes	—
87	C₂×C₈×C₄₃	C₄⋊C₄	1	16	yes	complemented
88	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
89	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
90	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
91	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
93	C₂×C₈×C₄₃	C₂×Q₈	1	2	yes	—
94	C₄⋊C₄×C₄₃	—	2	4	no	—
95	C₄⋊C₄×C₄₃	—	2	4	no	—
96	C₂²⋊C₄×C₄₃	—	4	4	no	—
97	C₄²×C₄₃	—	4	4	no	—
98	C₂²⋊Dic₄₃	—	4	4	no	—
99	C₄⋊Dic₄₃	—	2	4	no	—
100	C₄⋊Dic₄₃	—	2	4	no	—
101	Dic₄₃⋊C₄	—	4	16	no	—
102	C₄×Dic₄₃	—	4	4	no	—
103	C₄×Dic₄₃	C₄⋊C₄	1	8	yes	—
104	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
105	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
106	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
107	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
108	C₂×C₄₃⋊C₈	C₄⋊C₄	1	16	yes	complemented
109	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
110	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
111	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
112	C₂×C₄₃⋊C₈	C₂×D₄	1	2	yes	—
113	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
114	C₂×C₄₃⋊C₈	C₂×Q₈	1	2	yes	—
115	C₂³×C₄₃	—	2	4	no	—
116	C₂³×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂³×C₄₃	#127	1	2	yes	—
118	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
119	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄×C₄₃	—	4	4	no	—
121	C₂×C₄×C₄₃	—	4	8	no	—
122	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
123	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
125	C₂×C₄×C₄₃	#127	1	2	yes	—
126	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
127	C₂×C₄×C₄₃	C₄⋊Q₈	1	4	yes	—
128	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
129	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
134	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
135	C₂×Dic₄₃	—	2	4	no	—
136	C₂×Dic₄₃	—	2	16	no	—
137	C₂×Dic₄₃	—	4	4	no	—
138	C₂×Dic₄₃	—	4	8	no	—
139	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×Dic₄₃	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×Dic₄₃	#135	1	8	yes	—
142	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
143	C₈×C₄₃	C₂×C₄⋊C₄	1	4	yes	—
144	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
145	C₄₃⋊C₈	C₂×C₄⋊C₄	1	4	yes	—
146	#20797	—	43	1	no	maximal, Sylow
147	C₂²×C₄₃	—	2	4	no	—
148	C₂²×C₄₃	—	2	4	no	—
149	C₂²×C₄₃	—	2	4	no	—
150	C₂²×C₄₃	#384	1	4	yes	—
151	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
153	C₂²×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
154	C₂²×C₄₃	C₂×#135	1	4	yes	—
155	C₄×C₄₃	—	8	4	no	—
156	C₄×C₄₃	#384	1	4	yes	—
157	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
158	C₄×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
159	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
160	C₄×C₄₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
161	Dic₄₃	—	2	4	no	—
162	Dic₄₃	—	8	4	no	—
163	Dic₄₃	#427	1	8	yes	complemented
164	C₂×#427	—	43	1	no	—
165	C₂×#427	—	43	2	no	—
166	C₂×#427	D₄₃	1	1	yes	characteristic
167	C₂²×C₄×C₈	—	43	1	no	—
168	#2285	—	43	2	no	—
169	#1846	—	43	2	no	—
170	#1846	—	43	2	no	—
171	#1845	—	43	2	no	—
172	#1844	—	43	2	no	—
173	C₂×C₄₃	—	2	4	no	—
174	C₂×C₄₃	C₂×#384	1	1	yes	characteristic
175	C₂×C₄₃	C₂×#427	1	2	yes	—
176	C₂×C₄₃	#3147	1	2	yes	—
177	C₂×C₄₃	#3160	1	2	yes	—
178	C₂³×C₈	—	43	1	no	—
179	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
180	C₂×#117	—	43	2	no	—
181	C₂×#117	C₂×D₄₃	1	2	yes	—
182	C₂²×C₄²	—	43	1	no	—
183	#427	—	2	4	no	—
184	#427	—	86	8	no	—
185	#427	—	86	4	no	—
186	C₂×#107	—	43	2	no	—
187	C₂×#107	—	43	4	no	—
188	C₂×#107	C₂×D₄₃	1	2	yes	—
189	C₂×#106	—	43	2	no	—
190	C₂×#106	—	43	4	no	—
191	C₂×#106	C₂×D₄₃	1	2	yes	—
192	C₂×C₄×C₈	—	43	1	no	—
193	C₂×C₄×C₈	—	43	1	no	—
194	C₂×C₄×C₈	—	43	1	no	—
195	C₂×C₄×C₈	—	43	1	no	—
196	C₂×C₄×C₈	—	43	8	no	—
197	#384	—	43	4	no	—
198	#382	—	43	4	no	—
199	#380	—	86	4	no	—
200	C₄×C₄⋊C₄	—	86	4	no	—
201	C₄₃	#20797	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₄	—	43	2	no	—
203	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
204	C₂²×C₈	—	43	4	no	—
205	C₂²×C₈	—	43	4	no	—
206	C₂²×C₈	—	43	4	no	—
207	C₂²×C₈	—	43	1	no	—
208	C₂²×C₈	—	43	1	no	—
209	C₂²×C₈	Dic₈₆	1	4	yes	—
210	C₂²×C₈	C₄×D₄₃	1	4	yes	—
211	C₂²×C₈	C₂²⋊D₄₃	1	4	yes	—
212	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
213	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
214	#117	—	2	4	no	—
215	#117	—	2	4	no	—
216	#117	—	86	4	no	—
217	#117	—	86	4	no	—
218	C₂×C₄⋊C₄	—	43	4	no	—
219	C₂×C₄⋊C₄	—	86	8	no	—
220	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	—	86	2	no	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	—	43	4	no	—
225	C₂×C₄²	—	43	2	no	—
226	C₂×C₄²	—	43	4	no	—
227	C₂×C₄²	—	43	1	no	—
228	C₂×C₄²	—	43	1	no	—
229	C₂×C₄²	—	86	2	no	—
230	#107	—	2	4	no	—
231	#107	—	86	8	no	—
232	#107	—	86	4	no	—
233	#106	—	2	4	no	—
234	#106	—	86	8	no	—
235	#106	—	86	4	no	—
236	C₄×C₈	—	43	4	no	—
237	C₄×C₈	—	43	4	no	—
238	C₄×C₈	—	43	4	no	—
239	C₄×C₈	—	43	4	no	—
240	#95	—	86	8	no	—
241	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	—	2	4	no	—
243	C₂²×C₄	—	43	2	no	—
244	C₂²×C₄	—	43	8	no	—
245	C₂²×C₄	—	43	2	no	—
246	C₂²×C₄	—	86	8	no	—
247	C₂²×C₄	—	86	4	no	—
248	C₂²×C₄	C₂×Dic₈₆	1	2	yes	—
249	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
250	C₂²×C₄	D₄×D₄₃	1	2	yes	—
251	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
252	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
253	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
254	C₂²×C₄	C₂×C₂²⋊D₄₃	1	2	yes	—
255	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
256	C₂×C₈	—	43	16	no	—
257	C₂×C₈	—	43	2	no	—
258	C₂×C₈	—	43	2	no	—
259	C₂×C₈	—	43	2	no	—
260	C₂×C₈	—	43	2	no	—
261	C₂×C₈	—	43	2	no	—
262	C₂×C₈	—	43	2	no	—
263	C₂×C₈	Dic₄₃⋊C₄	1	16	yes	complemented
264	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
265	C₂×C₈	C₂×Dic₈₆	1	2	yes	—
266	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
267	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
268	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
269	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
270	C₄⋊C₄	—	2	4	no	—
271	C₄⋊C₄	—	2	4	no	—
272	C₄⋊C₄	—	86	4	no	—
273	C₄⋊C₄	—	86	4	no	—
274	C₄⋊C₄	—	172	16	no	—
275	C₂²⋊C₄	—	4	4	no	—
276	C₂²⋊C₄	—	172	4	no	—
277	C₄²	—	4	4	no	—
278	C₄²	—	43	8	no	—
279	C₄²	—	43	2	no	—
280	C₄²	—	43	2	no	—
281	C₄²	—	43	2	no	—
282	C₄²	—	43	2	no	—
283	C₄²	—	172	4	no	—
284	C₂³	—	2	4	no	—
285	C₂³	C₄⋊C₄×D₄₃	1	2	yes	—
286	C₂³	#13063321	1	2	yes	—
287	C₂³	#13063324	1	2	yes	—
288	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	4	4	no	—
291	C₂×C₄	—	43	4	no	—
292	C₂×C₄	—	43	8	no	—
293	C₂×C₄	—	43	4	no	—
294	C₂×C₄	—	86	16	no	—
295	C₂×C₄	—	86	4	no	—
296	C₂×C₄	—	172	8	no	—
297	C₂×C₄	—	172	4	no	—
298	C₂×C₄	C₄⋊C₄×D₄₃	1	4	yes	—
299	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
300	C₂×C₄	C₂×Dic₄₃⋊C₄	1	4	yes	—
301	C₂×C₄	#13063321	1	2	yes	—
302	C₂×C₄	#13063323	1	4	yes	—
303	C₂×C₄	#13063324	1	4	yes	—
304	C₂×C₄	#13063324	1	2	yes	—
305	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
306	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
307	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
308	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
309	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
310	C₂×C₄	C₂²×Dic₈₆	1	1	yes	characteristic
311	C₈	—	43	4	no	—
312	C₈	—	43	4	no	—
313	C₈	C₂×Dic₄₃⋊C₄	1	4	yes	—
314	C₈	C₂×Dic₄₃⋊C₄	1	4	yes	—
315	C₂²	—	2	4	no	—
316	C₂²	—	2	4	no	—
317	C₂²	—	2	4	no	—
318	C₂²	#14582768	1	4	yes	—
319	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
320	C₂²	C₂×#13063321	1	1	yes	characteristic
321	C₂²	C₂×#13063324	1	1	yes	characteristic
322	C₂²	#135×D₄₃	1	4	yes	—
323	C₄	—	8	4	no	—
324	C₄	—	43	8	no	—
325	C₄	—	86	4	no	—
326	C₄	—	344	4	no	—
327	C₄	#14582768	1	4	yes	—
328	C₄	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
329	C₄	C₂²×Dic₄₃⋊C₄	1	1	yes	characteristic
330	C₄	C₂×#13063323	1	1	yes	characteristic
331	C₄	C₂×#13063324	1	1	yes	characteristic
332	C₂	—	2	4	no	—
333	C₂	C₂×#14582768	1	1	yes	characteristic
334	C₂	#427×D₄₃	1	2	yes	—
335	C₂	#19145019	1	2	yes	—
336	C₂	#19145107	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂×C₄₃×#427	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#427	D₄₃
hypocenter	C₄₃	#20797
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

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

N	H ≅ G/N	type
5. #19134758	C₂	semidirect
6. #19134756	C₂	semidirect
7. #19134753	C₂	semidirect
8. #19134750	C₂	semidirect
19. C₂×C₈×Dic₄₃	C₄	semidirect
87. C₂×C₈×C₄₃	C₄⋊C₄	semidirect
108. C₂×C₄₃⋊C₈	C₄⋊C₄	semidirect
163. Dic₄₃	#427	semidirect
201. C₄₃	#20797	semidirect
263. C₂×C₈	Dic₄₃⋊C₄	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
C₂×#14582935	2
C₂²×C₈×Dic₄₃	2
#19136998	2
#19134758	2
#19134756	2
#19134753	2
#19134750	2
C₂×C₄₃×#427	2
#20797	43