Subgroups — #24989944

Order 11008.

680 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19134575	C₂	1	1	yes	maximal, characteristic
2	C₄×C₂²⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#19132436	C₂	1	4	yes	maximal, complemented
4	C₂×C₄×C₄⋊C₄×C₄₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
6	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
7	C₂×C₂²⋊(C₄₃⋊C₈)	—	2	1	no	—
8	C₂×C₄×C₄₃⋊C₈	C₂²	1	2	yes	—
9	#14582654	—	2	1	no	—
10	#14582654	—	2	1	no	—
11	C₂²×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₄₃	C₄	1	1	yes	characteristic
13	C₂²×C₄²×C₄₃	—	2	1	no	—
14	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
15	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
16	C₄×C₄⋊C₄×C₄₃	—	2	1	no	—
17	C₄×C₄⋊C₄×C₄₃	C₄	1	2	yes	—
18	C₄×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
19	C₂×#95×C₄₃	—	2	1	no	—
20	C₂²×C₄₃⋊C₈	—	2	2	no	—
21	C₂²×C₄₃⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₃⋊C₈)	—	2	4	no	—
23	C₂²⋊(C₄₃⋊C₈)	—	2	4	no	—
24	C₂²⋊(C₄₃⋊C₈)	—	4	2	no	—
25	C₄×C₄₃⋊C₈	—	2	4	no	—
26	C₂³×C₄×C₄₃	—	2	1	no	—
27	C₂³×C₄×C₄₃	—	2	1	no	—
28	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
29	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
30	C₂³×C₄×C₄₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
35	C₂×C₄⋊C₄×C₄₃	C₈	1	4	yes	complemented
36	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
38	C₂×C₄²×C₄₃	—	2	4	no	—
39	C₂×C₄²×C₄₃	—	2	2	no	—
40	C₂×C₄²×C₄₃	—	2	2	no	—
41	C₂×C₄²×C₄₃	—	4	2	no	—
42	C₂×C₄²×C₄₃	—	4	2	no	—
43	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
45	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
46	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
47	#95×C₄₃	—	2	2	no	—
48	#95×C₄₃	—	2	2	no	—
49	C₂⁴×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	—	2	2	no	—
51	C₂²×C₄×C₄₃	—	2	2	no	—
52	C₂²×C₄×C₄₃	—	2	2	no	—
53	C₂²×C₄×C₄₃	—	2	2	no	—
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	—	2	2	no	—
56	C₂²×C₄×C₄₃	—	2	4	no	—
57	C₂²×C₄×C₄₃	—	2	2	no	—
58	C₂²×C₄×C₄₃	—	2	2	no	—
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	4	1	no	—
61	C₂²×C₄×C₄₃	—	4	2	no	—
62	C₂²×C₄×C₄₃	—	4	2	no	—
63	C₂²×C₄×C₄₃	—	4	2	no	—
64	C₂²×C₄×C₄₃	—	4	2	no	—
65	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂×C₈	1	2	yes	—
69	C₂²×C₄×C₄₃	M₁₆	1	2	yes	—
70	C₂²×C₄×C₄₃	D₈	1	2	yes	—
71	C₂²×C₄×C₄₃	SD₁₆	1	2	yes	—
72	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₄⋊C₄×C₄₃	—	2	4	no	—
76	C₄⋊C₄×C₄₃	—	2	4	no	—
77	C₄⋊C₄×C₄₃	—	2	2	no	—
78	C₄⋊C₄×C₄₃	—	2	2	no	—
79	C₄⋊C₄×C₄₃	C₂×C₈	1	4	yes	—
80	C₄⋊C₄×C₄₃	M₁₆	1	4	yes	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	C₄²×C₄₃	—	2	2	no	—
84	C₄²×C₄₃	—	4	4	no	—
85	C₄²×C₄₃	—	4	2	no	—
86	C₄²×C₄₃	—	4	2	no	—
87	C₄²×C₄₃	#41	1	4	yes	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₂×C₄₃⋊C₈	—	4	2	no	—
92	C₂×C₄₃⋊C₈	—	4	2	no	—
93	C₂×C₄₃⋊C₈	—	4	2	no	—
94	C₂³×C₄₃	—	2	1	no	—
95	C₂³×C₄₃	—	2	1	no	—
96	C₂³×C₄₃	—	2	1	no	—
97	C₂³×C₄₃	—	2	1	no	—
98	C₂³×C₄₃	C₂²⋊C₈	1	2	yes	—
99	C₂³×C₄₃	#102	1	2	yes	—
100	C₂³×C₄₃	#104	1	2	yes	—
101	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	1	no	—
104	C₂×C₄×C₄₃	—	2	1	no	—
105	C₂×C₄×C₄₃	—	2	1	no	—
106	C₂×C₄×C₄₃	—	2	1	no	—
107	C₂×C₄×C₄₃	—	2	1	no	—
108	C₂×C₄×C₄₃	—	2	4	no	—
109	C₂×C₄×C₄₃	—	2	4	no	—
110	C₂×C₄×C₄₃	—	2	4	no	—
111	C₂×C₄×C₄₃	—	2	4	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	4	no	—
114	C₂×C₄×C₄₃	—	2	4	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	4	4	no	—
118	C₂×C₄×C₄₃	—	4	8	no	—
119	C₂×C₄×C₄₃	—	4	8	no	—
120	C₂×C₄×C₄₃	—	4	2	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	4	2	no	—
124	C₂×C₄×C₄₃	C₂²⋊C₈	1	2	yes	—
125	C₂×C₄×C₄₃	#102	1	2	yes	—
126	C₂×C₄×C₄₃	#117	1	2	yes	—
127	C₂×C₄×C₄₃	#123	1	2	yes	—
128	C₂×C₄×C₄₃	#123	1	2	yes	—
129	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×M₁₆	1	2	yes	—
131	C₂×C₄×C₄₃	C₂×M₁₆	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
133	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
134	C₂×C₄×C₄₃	#137	1	2	yes	—
135	C₄₃⋊C₈	—	4	4	no	—
136	C₄₃⋊C₈	—	8	2	no	—
137	#8706	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	—	2	2	no	—
140	C₂²×C₄₃	—	2	2	no	—
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	2	2	no	—
144	C₂²×C₄₃	D₄⋊C₈	1	4	yes	—
145	C₂²×C₄₃	#381	1	1	yes	characteristic
146	C₂²×C₄₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
147	C₂²×C₄₃	#407	1	1	yes	characteristic
148	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
149	C₂²×C₄₃	#419	1	1	yes	characteristic
150	C₂²×C₄₃	C₂×#104	1	1	yes	characteristic
151	C₂²×C₄₃	#421	1	1	yes	characteristic
152	C₄×C₄₃	—	2	2	no	—
153	C₄×C₄₃	—	2	2	no	—
154	C₄×C₄₃	—	2	2	no	—
155	C₄×C₄₃	—	4	1	no	—
156	C₄×C₄₃	—	4	4	no	—
157	C₄×C₄₃	—	4	4	no	—
158	C₄×C₄₃	#431	1	2	yes	—
159	C₄×C₄₃	#432	1	2	yes	—
160	C₄×C₄₃	#480	1	2	yes	—
161	C₄×C₄₃	#483	1	2	yes	—
162	C₂×C₄×C₄⋊C₄	D₄₃	1	1	yes	characteristic
163	#1797	—	43	1	no	—
164	C₄×C₂²⋊C₈	—	43	1	no	—
165	#1279	—	43	4	no	—
166	C₂×C₄₃	—	2	4	no	—
167	C₂×C₄₃	C₂×D₄⋊C₈	1	1	yes	characteristic
168	C₂×C₄₃	#1479	1	1	yes	characteristic
169	C₂×C₄₃	#1486	1	1	yes	characteristic
170	C₂×C₄₃	#1491	1	1	yes	characteristic
171	C₂×C₄₃	#1817	1	1	yes	characteristic
172	C₂×C₄₃	#1823	1	1	yes	characteristic
173	C₂×C₄₃	#1826	1	1	yes	characteristic
174	C₂²×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
175	C₂²×C₄⋊C₄	Dic₄₃	1	1	yes	characteristic
176	C₂²×C₄²	—	2	1	no	—
177	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
178	C₂×C₄⋊C₈	—	43	2	no	—
179	C₂×C₂²⋊C₈	—	86	1	no	—
180	C₂×C₂²⋊C₈	—	86	1	no	—
181	C₂×C₄×C₈	—	43	2	no	—
182	C₄×C₄⋊C₄	—	2	1	no	—
183	C₄×C₄⋊C₄	—	2	1	no	—
184	C₄×C₄⋊C₄	Dic₄₃	1	2	yes	—
185	C₄×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
186	C₂×#95	—	2	1	no	—
187	#336	—	86	1	no	—
188	#336	—	86	1	no	—
189	C₄₃	#8706	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	—	2	1	no	—
191	C₂³×C₄	—	2	1	no	—
192	C₂³×C₄	C₂×Dic₄₃	1	1	yes	characteristic
193	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
194	C₂³×C₄	C₂²⋊D₄₃	1	1	yes	characteristic
195	C₂²×C₈	—	86	2	no	—
196	C₂²×C₈	—	86	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	1	no	—
199	C₂×C₄⋊C₄	—	2	1	no	—
200	C₂×C₄⋊C₄	—	2	4	no	—
201	C₂×C₄⋊C₄	C₄₃⋊C₈	1	4	yes	complemented
202	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
203	C₂×C₄⋊C₄	C₂×Dic₄₃	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	—	2	2	no	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	—	4	2	no	—
208	C₂×C₄²	—	4	2	no	—
209	C₂×C₄²	C₂×Dic₄₃	1	1	yes	characteristic
210	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
212	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
213	C₄⋊C₈	—	86	4	no	—
214	C₂²⋊C₈	—	86	4	no	—
215	C₂²⋊C₈	—	172	2	no	—
216	C₄×C₈	—	86	4	no	—
217	#95	—	2	2	no	—
218	#95	—	2	2	no	—
219	C₂⁴	C₂²⋊Dic₄₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	4	no	—
225	C₂²×C₄	—	2	4	no	—
226	C₂²×C₄	—	2	2	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	2	2	no	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	4	1	no	—
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	—	4	2	no	—
235	C₂²×C₄	C₂×C₄₃⋊C₈	1	2	yes	—
236	C₂²×C₄	#10621877	1	2	yes	—
237	C₂²×C₄	#10621882	1	2	yes	—
238	C₂²×C₄	#10621883	1	2	yes	—
239	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂²⋊Dic₄₃	1	1	yes	characteristic
242	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
245	C₂×C₈	—	86	4	no	—
246	C₂×C₈	—	172	2	no	—
247	C₂×C₈	—	172	2	no	—
248	C₂×C₈	—	172	2	no	—
249	C₂×C₈	—	172	2	no	—
250	C₂×C₈	—	172	2	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	4	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	C₂×C₄₃⋊C₈	1	4	yes	—
256	C₄⋊C₄	#10621877	1	4	yes	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	4	no	—
261	C₄²	—	4	2	no	—
262	C₄²	—	4	2	no	—
263	C₄²	#10621898	1	4	yes	—
264	C₂³	—	2	1	no	—
265	C₂³	—	2	1	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
269	C₂³	#13063246	1	2	yes	—
270	C₂³	#13063252	1	2	yes	—
271	C₂³	C₂×C₂²⋊Dic₄₃	1	1	yes	characteristic
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	4	no	—
279	C₂×C₄	—	2	4	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	4	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	4	4	no	—
288	C₂×C₄	—	4	8	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	C₂²⋊(C₄₃⋊C₈)	1	2	yes	—
295	C₂×C₄	#13063246	1	2	yes	—
296	C₂×C₄	#13063284	1	2	yes	—
297	C₂×C₄	#13063308	1	2	yes	—
298	C₂×C₄	#13063308	1	2	yes	—
299	C₂×C₄	C₂²×C₄₃⋊C₈	1	1	yes	characteristic
300	C₂×C₄	C₂×#10621877	1	2	yes	—
301	C₂×C₄	C₂×#10621877	1	1	yes	characteristic
302	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
303	C₂×C₄	C₂×#10621883	1	1	yes	characteristic
304	C₂×C₄	#13063356	1	2	yes	—
305	C₈	—	344	2	no	—
306	C₈	—	172	4	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	2	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	D₄⋊(C₄₃⋊C₈)	1	4	yes	—
314	C₂²	#14582762	1	1	yes	characteristic
315	C₂²	C₂×C₂²⋊(C₄₃⋊C₈)	1	1	yes	characteristic
316	C₂²	#14582863	1	1	yes	characteristic
317	C₂²	C₂×#13063246	1	1	yes	characteristic
318	C₂²	#14582908	1	1	yes	characteristic
319	C₂²	C₂×#13063252	1	1	yes	characteristic
320	C₂²	#14582916	1	1	yes	characteristic
321	C₄	—	2	2	no	—
322	C₄	—	2	2	no	—
323	C₄	—	2	2	no	—
324	C₄	—	4	4	no	—
325	C₄	—	4	4	no	—
326	C₄	—	4	1	no	—
327	C₄	#14582952	1	2	yes	—
328	C₄	#14582956	1	2	yes	—
329	C₄	#14583212	1	2	yes	—
330	C₄	#14583229	1	2	yes	—
331	C₂	—	2	4	no	—
332	C₂	C₂×D₄⋊(C₄₃⋊C₈)	1	1	yes	characteristic
333	C₂	#19132990	1	1	yes	characteristic
334	C₂	#19133020	1	1	yes	characteristic
335	C₂	#19133042	1	1	yes	characteristic
336	C₂	#19134663	1	1	yes	characteristic
337	C₂	#19134681	1	1	yes	characteristic
338	C₂	#19134694	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₂×C₄×C₄₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄×C₄⋊C₄×C₄₃	C₂
socle	C₂³×C₄₃	C₂×C₂²⋊C₄
prime radical	C₂⁴×C₄₃	C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×C₄×C₄⋊C₄	D₄₃
hypocenter	C₄₃	#8706
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ C₂×C₄×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
3. #19132436	C₂	semidirect
35. C₂×C₄⋊C₄×C₄₃	C₈	semidirect
189. C₄₃	#8706	semidirect
201. C₂×C₄⋊C₄	C₄₃⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#19134575	2
C₄×C₂²⋊(C₄₃⋊C₈)	2
#19132436	2
C₂×C₄×C₄⋊C₄×C₄₃	2
#8706	43