Subgroups — #25095678

Order 11008.

806 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583030	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#19134470	C₂	1	2	yes	maximal
4	#19134464	C₂	1	2	yes	maximal
5	#19134464	C₂	1	1	yes	maximal, characteristic
6	C₄⋊C₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
7	#19134452	C₂	1	2	yes	maximal
8	#19134448	C₂	1	1	yes	maximal, characteristic
9	#19134347	C₂	1	2	yes	maximal
10	#19134341	C₂	1	1	yes	maximal, characteristic
11	C₄₃×#2295	C₂	1	1	yes	maximal, characteristic
12	#14583030	C₄	1	8	yes	complemented
13	Q₈×C₄₃⋊C₈	C₄	1	8	yes	complemented
14	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×#13063236	C₂²	1	2	yes	—
19	C₂×#13063236	C₂²	1	2	yes	—
20	C₂×#13063236	C₂²	1	2	yes	—
21	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
22	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
23	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
25	#14582654	C₂²	1	1	yes	characteristic
26	#14582654	C₂²	1	2	yes	—
27	#14582654	C₂²	1	2	yes	—
28	#14582654	C₂²	1	1	yes	characteristic
29	#14582654	C₂²	1	1	yes	characteristic
30	#14582654	C₂²	1	1	yes	characteristic
31	#14582654	C₂²	1	2	yes	—
32	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₄₃×#391	C₄	1	2	yes	—
34	C₄₃×#389	C₄	1	2	yes	—
35	C₄₃×#387	C₄	1	4	yes	—
36	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
37	C₄×C₄⋊C₄×C₄₃	C₂²	1	1	yes	characteristic
38	C₄₃×#376	C₂²	1	2	yes	—
39	C₄³×C₄₃	C₂²	1	1	yes	characteristic
40	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
42	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
43	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
44	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
45	C₂²×C₄₃⋊C₈	C₂³	1	1	yes	characteristic
46	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
47	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
48	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
49	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
50	#13063236	—	4	2	no	—
51	#13063236	—	4	2	no	—
52	#13063236	C₂×C₄	1	8	yes	—
53	C₄×C₄₃⋊C₈	—	4	2	no	—
54	C₄×C₄₃⋊C₈	C₂×C₄	1	8	yes	—
55	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
56	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
57	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
58	C₄×Q₈×C₄₃	C₂×C₄	1	4	yes	—
59	C₄×Q₈×C₄₃	C₂×C₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
62	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
65	C₂×C₄²×C₄₃	C₂×C₄	1	2	yes	—
66	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
67	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
70	C₂×C₄²×C₄₃	C₂³	1	2	yes	—
71	#95×C₄₃	C₂×C₄	1	4	yes	—
72	#95×C₄₃	C₂×C₄	1	4	yes	—
73	#95×C₄₃	C₂×C₄	1	2	yes	—
74	#95×C₄₃	C₂×C₄	1	4	yes	—
75	C₂×Q₈×C₄₃	—	2	4	no	—
76	C₂×Q₈×C₄₃	C₄²	1	4	yes	—
77	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
78	C₂²×C₄×C₄₃	C₂²×C₄	1	4	yes	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
85	C₄⋊C₄×C₄₃	—	2	4	no	—
86	C₄⋊C₄×C₄₃	—	4	4	no	—
87	C₄⋊C₄×C₄₃	C₄²	1	8	yes	—
88	C₄⋊C₄×C₄₃	C₄²	1	4	yes	—
89	C₄²×C₄₃	—	2	4	no	—
90	C₄²×C₄₃	—	2	2	no	—
91	C₄²×C₄₃	—	2	1	no	—
92	C₄²×C₄₃	—	2	2	no	—
93	C₄²×C₄₃	—	2	1	no	—
94	C₄²×C₄₃	—	2	1	no	—
95	C₄²×C₄₃	—	2	1	no	—
96	C₄²×C₄₃	—	4	2	no	—
97	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
98	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
99	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₄₃	C₂²×C₄	1	4	yes	—
101	C₂×C₄₃⋊C₈	—	2	1	no	—
102	C₂×C₄₃⋊C₈	—	2	2	no	—
103	C₂×C₄₃⋊C₈	—	2	2	no	—
104	C₂×C₄₃⋊C₈	—	2	1	no	—
105	C₂×C₄₃⋊C₈	—	2	1	no	—
106	C₂×C₄₃⋊C₈	—	2	1	no	—
107	C₂×C₄₃⋊C₈	—	2	1	no	—
108	C₂×C₄₃⋊C₈	—	2	1	no	—
109	C₂×C₄₃⋊C₈	—	2	1	no	—
110	C₂×C₄₃⋊C₈	—	2	2	no	—
111	C₂×C₄₃⋊C₈	—	2	2	no	—
112	C₂×C₄₃⋊C₈	—	2	1	no	—
113	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
114	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
115	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
116	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
117	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
118	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
119	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
120	Q₈×C₄₃	—	2	8	no	—
121	C₂×C₄×C₄₃	—	2	4	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	—	2	4	no	—
124	C₂×C₄×C₄₃	—	2	2	no	—
125	C₂×C₄×C₄₃	—	2	4	no	—
126	C₂×C₄×C₄₃	—	2	2	no	—
127	C₂×C₄×C₄₃	—	2	2	no	—
128	C₂×C₄×C₄₃	—	2	2	no	—
129	C₂×C₄×C₄₃	—	2	4	no	—
130	C₂×C₄×C₄₃	—	2	2	no	—
131	C₂×C₄×C₄₃	—	2	2	no	—
132	C₂×C₄×C₄₃	—	2	4	no	—
133	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
134	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
135	C₂×C₄×C₄₃	C₂×C₄²	1	4	yes	—
136	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
137	C₂×C₄×C₄₃	#131	1	4	yes	—
138	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
140	C₂×C₄×C₄₃	H(5,𝔽₂)	1	1	yes	characteristic
141	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
142	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
143	C₂×C₄×C₄₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₄₃⋊C₈	—	2	2	no	—
145	C₄₃⋊C₈	—	2	2	no	—
146	C₄₃⋊C₈	—	2	2	no	—
147	C₄₃⋊C₈	—	2	4	no	—
148	C₄₃⋊C₈	—	2	4	no	—
149	C₄₃⋊C₈	—	2	2	no	—
150	#20574	—	43	1	no	maximal, Sylow
151	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
152	C₂²×C₄₃	#519	1	1	yes	characteristic
153	C₂²×C₄₃	#520	1	1	yes	characteristic
154	C₂²×C₄₃	C₂×#131	1	2	yes	—
155	C₂²×C₄₃	#568	1	2	yes	—
156	C₄×C₄₃	—	2	4	no	—
157	C₄×C₄₃	—	2	2	no	—
158	C₄×C₄₃	—	2	4	no	—
159	C₄×C₄₃	—	4	2	no	—
160	C₄×C₄₃	—	4	2	no	—
161	C₄×C₄₃	—	4	4	no	—
162	C₄×C₄₃	#405	1	4	yes	—
163	C₄×C₄₃	#519	1	1	yes	characteristic
164	C₄×C₄₃	#519	1	1	yes	characteristic
165	C₄×C₄₃	#520	1	1	yes	characteristic
166	C₄×C₄₃	#520	1	1	yes	characteristic
167	C₂×#446	—	43	1	no	—
168	C₂×C₈×Q₈	—	43	1	no	—
169	#2295	D₄₃	1	1	yes	characteristic
170	#1771	—	43	2	no	—
171	#1770	—	43	2	no	—
172	#1770	—	43	1	no	—
173	C₈×C₄⋊C₄	—	43	1	no	—
174	#1768	—	43	2	no	—
175	#1767	—	43	1	no	—
176	#1746	—	43	2	no	—
177	#1744	—	43	1	no	—
178	C₂×C₄₃	#2276	1	1	yes	characteristic
179	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
180	C₂×C₄₃	#2875	1	1	yes	characteristic
181	C₂×C₄₃	#2985	1	2	yes	—
182	C₂×C₄₃	#2993	1	2	yes	—
183	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
184	#446	—	43	8	no	—
185	C₈×Q₈	—	43	8	no	—
186	C₂×C₄⋊C₈	—	43	1	no	—
187	C₂×C₄⋊C₈	—	43	2	no	—
188	C₂×C₄⋊C₈	—	43	1	no	—
189	C₂×C₄⋊C₈	—	43	2	no	—
190	C₂×#97	—	43	2	no	—
191	C₂×#97	—	43	2	no	—
192	C₂×#97	—	43	2	no	—
193	C₂×C₄×C₈	—	43	1	no	—
194	C₂×C₄×C₈	—	43	2	no	—
195	C₂×C₄×C₈	—	43	1	no	—
196	C₂×C₄×C₈	—	43	2	no	—
197	#391	Dic₄₃	1	2	yes	—
198	#389	Dic₄₃	1	2	yes	—
199	#387	Dic₄₃	1	4	yes	—
200	C₄×C₄⋊C₄	C₂×D₄₃	1	1	yes	characteristic
201	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
202	#376	C₂×D₄₃	1	2	yes	—
203	C₄³	C₂×D₄₃	1	1	yes	characteristic
204	#336	—	43	1	no	—
205	#336	—	43	1	no	—
206	#336	—	43	1	no	—
207	#336	—	43	1	no	—
208	#336	—	43	2	no	—
209	#336	—	43	2	no	—
210	#336	—	43	2	no	—
211	C₄₃	#20574	1	1	yes	characteristic, complemented, Sylow
212	C₂²×Q₈	C₂×Dic₄₃	1	1	yes	characteristic
213	C₂²×C₈	—	43	1	no	—
214	C₂²×C₈	—	43	1	no	—
215	C₂²×C₈	—	43	1	no	—
216	C₂²×C₈	—	43	1	no	—
217	C₂²×C₈	—	43	2	no	—
218	C₂²×C₈	—	43	2	no	—
219	C₄×Q₈	C₄×D₄₃	1	4	yes	—
220	C₄×Q₈	C₄×D₄₃	1	4	yes	—
221	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
222	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
223	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
224	C₂×C₄⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
225	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
226	C₂×C₄²	C₂×Dic₄₃	1	2	yes	—
227	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
229	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
230	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
231	C₂×C₄²	C₂²×D₄₃	1	2	yes	—
232	C₄⋊C₈	—	43	4	no	—
233	C₄⋊C₈	—	43	8	no	—
234	C₄⋊C₈	—	43	4	no	—
235	C₄⋊C₈	—	43	8	no	—
236	#97	—	43	8	no	—
237	#97	—	172	2	no	—
238	#97	—	172	2	no	—
239	C₄×C₈	—	43	4	no	—
240	C₄×C₈	—	43	8	no	—
241	C₄×C₈	—	43	4	no	—
242	C₄×C₈	—	172	2	no	—
243	#95	C₂×Dic₄₃	1	4	yes	—
244	#95	C₂×Dic₄₃	1	4	yes	—
245	#95	C₂×Dic₄₃	1	2	yes	—
246	#95	C₂×Dic₄₃	1	4	yes	—
247	C₂×Q₈	—	2	4	no	—
248	C₂×Q₈	C₄×Dic₄₃	1	4	yes	—
249	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
250	C₂²×C₄	C₂²×Dic₄₃	1	4	yes	—
251	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
252	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
253	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
254	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
255	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
256	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
257	C₂×C₈	—	43	2	no	—
258	C₂×C₈	—	43	2	no	—
259	C₂×C₈	—	43	2	no	—
260	C₂×C₈	—	43	4	no	—
261	C₂×C₈	—	43	2	no	—
262	C₂×C₈	—	43	4	no	—
263	C₂×C₈	—	86	1	no	—
264	C₂×C₈	—	86	2	no	—
265	C₂×C₈	—	86	1	no	—
266	C₂×C₈	—	86	2	no	—
267	C₂×C₈	—	86	1	no	—
268	C₂×C₈	—	86	1	no	—
269	C₂×C₈	—	86	1	no	—
270	C₂×C₈	—	86	1	no	—
271	C₂×C₈	—	86	1	no	—
272	C₂×C₈	—	86	1	no	—
273	C₂×C₈	—	86	2	no	—
274	C₂×C₈	—	86	2	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	4	4	no	—
277	C₄⋊C₄	C₄×Dic₄₃	1	8	yes	—
278	C₄⋊C₄	C₄×Dic₄₃	1	4	yes	—
279	C₄²	—	2	4	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	1	no	—
282	C₄²	—	2	1	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	1	no	—
285	C₄²	—	2	1	no	—
286	C₄²	—	4	2	no	—
287	C₄²	C₂×C₄×D₄₃	1	4	yes	—
288	C₄²	C₂×C₄×D₄₃	1	2	yes	—
289	C₄²	C₂×C₄×D₄₃	1	2	yes	—
290	C₄²	C₂×C₄×D₄₃	1	4	yes	—
291	C₂³	C₂³×Dic₄₃	1	1	yes	characteristic
292	Q₈	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	4	no	—
305	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
306	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
307	C₂×C₄	C₂×C₄×Dic₄₃	1	4	yes	—
308	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
309	C₂×C₄	#13063332	1	4	yes	—
310	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
312	C₂×C₄	#13063371	1	1	yes	characteristic
313	C₂×C₄	#13063372	1	1	yes	characteristic
314	C₂×C₄	#13063373	1	1	yes	characteristic
315	C₂×C₄	#13063373	1	1	yes	characteristic
316	C₈	—	86	2	no	—
317	C₈	—	86	2	no	—
318	C₈	—	86	2	no	—
319	C₈	—	86	4	no	—
320	C₈	—	86	2	no	—
321	C₈	—	86	4	no	—
322	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
323	C₂²	#14583377	1	1	yes	characteristic
324	C₂²	#14583382	1	1	yes	characteristic
325	C₂²	C₂×#13063332	1	2	yes	—
326	C₂²	#14583666	1	2	yes	—
327	C₄	—	2	4	no	—
328	C₄	—	2	2	no	—
329	C₄	—	2	4	no	—
330	C₄	—	4	2	no	—
331	C₄	—	4	2	no	—
332	C₄	—	4	4	no	—
333	C₄	#14582855	1	4	yes	—
334	C₄	#14583375	1	1	yes	characteristic
335	C₄	#14583375	1	1	yes	characteristic
336	C₄	#14583381	1	1	yes	characteristic
337	C₄	#14583381	1	1	yes	characteristic
338	C₂	#19136960	1	1	yes	characteristic
339	C₂	C₂×#14582855	1	1	yes	characteristic
340	C₂	#19143023	1	1	yes	characteristic
341	C₂	#19143690	1	2	yes	—
342	C₂	#19143744	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₄₃×#2295	C₂
socle, prime radical	C₂³×C₄₃	C₂³×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#2295	D₄₃
hypocenter	C₄₃	#20574
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
12. #14583030	C₄	semidirect
13. Q₈×C₄₃⋊C₈	C₄	semidirect
211. C₄₃	#20574	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583030	2
C₂×Q₈×C₄₃⋊C₈	2
#19134470	2
#19134464	2
C₄⋊C₄×C₄₃⋊C₈	2
#19134452	2
#19134448	2
#19134347	2
#19134341	2
C₄₃×#2295	2
#20574	43