Subgroups — #99×C₄₃⋊C₈ #24987140

Order 11008.

614 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	C₂×D₄×C₄₃⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂²⋊C₄×C₄₃⋊C₈	C₂	1	2	yes	maximal, complemented
3	#19132913	C₂	1	1	yes	maximal, characteristic
4	#19132443	C₂	1	2	yes	maximal, complemented
5	C₄×#99×C₄₃	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₃⋊C₈	C₄	1	1	yes	characteristic, complemented
7	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	D₄×C₄₃⋊C₈	—	2	2	no	—
9	D₄×C₄₃⋊C₈	—	2	2	no	—
10	C₂×C₄⋊(C₄₃⋊C₈)	C₄	1	1	yes	characteristic, complemented
11	C₂×C₂²⋊(C₄₃⋊C₈)	C₄	1	1	yes	characteristic, complemented
12	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄×C₄₃⋊C₈	—	2	2	no	—
15	C₂×C₄×C₄₃⋊C₈	C₄	1	1	yes	characteristic, complemented
16	#14582654	—	2	2	no	—
17	#14582625	C₄	1	2	yes	complemented
18	#14582623	C₄	1	2	yes	complemented
19	C₂×C₄×D₄×C₄₃	C₂²	1	1	yes	characteristic
20	C₂×#99×C₄₃	C₄	1	2	yes	—
21	C₄×C₂²⋊C₄×C₄₃	C₂²	1	2	yes	—
22	C₄₃×#342	C₄	1	2	yes	—
23	C₂²×C₄₃⋊C₈	—	2	2	no	—
24	C₂²×C₄₃⋊C₈	—	2	1	no	—
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	—	4	2	no	—
27	C₂²×C₄₃⋊C₈	—	4	2	no	—
28	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
33	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
34	C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
35	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
36	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
37	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
38	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
39	C₄×C₄₃⋊C₈	—	2	1	no	—
40	C₄×C₄₃⋊C₈	—	2	1	no	—
41	C₄×C₄₃⋊C₈	—	4	4	no	—
42	C₂²×D₄×C₄₃	C₂×C₄	1	1	yes	characteristic
43	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
45	C₄×D₄×C₄₃	—	2	2	no	—
46	C₄×D₄×C₄₃	—	2	2	no	—
47	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₂²⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₃	—	2	2	no	—
52	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
53	#99×C₄₃	C₈	1	8	yes	complemented
54	#95×C₄₃	—	2	2	no	—
55	C₂⁴×C₄₃	C₄²	1	1	yes	characteristic
56	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
57	C₂×D₄×C₄₃	—	2	2	no	—
58	C₂×D₄×C₄₃	—	2	2	no	—
59	C₂×D₄×C₄₃	C₂×C₈	1	4	yes	—
60	C₂²×C₄×C₄₃	—	2	1	no	—
61	C₂²×C₄×C₄₃	—	2	1	no	—
62	C₂²×C₄×C₄₃	—	2	4	no	—
63	C₂²×C₄×C₄₃	—	4	1	no	—
64	C₂²×C₄×C₄₃	—	4	1	no	—
65	C₂²×C₄×C₄₃	—	4	1	no	—
66	C₂²×C₄×C₄₃	—	4	1	no	—
67	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₄²	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₄₃	#41	1	2	yes	—
74	C₄⋊C₄×C₄₃	—	2	2	no	—
75	C₂²⋊C₄×C₄₃	—	2	2	no	—
76	C₂²⋊C₄×C₄₃	—	2	2	no	—
77	C₂²⋊C₄×C₄₃	—	2	4	no	—
78	C₂²⋊C₄×C₄₃	C₂×C₈	1	8	yes	complemented
79	C₄²×C₄₃	—	2	1	no	—
80	C₄²×C₄₃	—	2	1	no	—
81	C₄²×C₄₃	—	4	2	no	—
82	C₄²×C₄₃	—	4	2	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	2	no	—
85	C₂×C₄₃⋊C₈	—	2	1	no	—
86	C₂×C₄₃⋊C₈	—	2	1	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	4	1	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	C₂×C₄₃⋊C₈	—	4	1	no	—
95	C₂×C₄₃⋊C₈	—	4	1	no	—
96	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	2	yes	—
98	C₂×C₄₃⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
99	C₂³×C₄₃	—	2	1	no	—
100	C₂³×C₄₃	—	2	1	no	—
101	C₂³×C₄₃	—	4	1	no	—
102	C₂³×C₄₃	—	4	1	no	—
103	C₂³×C₄₃	—	4	1	no	—
104	C₂³×C₄₃	—	4	1	no	—
105	C₂³×C₄₃	C₄×C₈	1	2	yes	complemented
106	C₂³×C₄₃	C₂×C₄²	1	1	yes	characteristic
107	C₂³×C₄₃	C₄×D₄	1	2	yes	—
108	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
109	C₂³×C₄₃	#131	1	2	yes	—
110	D₄×C₄₃	—	2	8	no	—
111	C₂×C₄×C₄₃	—	2	1	no	—
112	C₂×C₄×C₄₃	—	2	1	no	—
113	C₂×C₄×C₄₃	—	2	2	no	—
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	2	no	—
116	C₂×C₄×C₄₃	—	2	1	no	—
117	C₂×C₄×C₄₃	—	2	1	no	—
118	C₂×C₄×C₄₃	—	2	1	no	—
119	C₂×C₄×C₄₃	—	2	8	no	—
120	C₂×C₄×C₄₃	—	4	1	no	—
121	C₂×C₄×C₄₃	—	4	2	no	—
122	C₂×C₄×C₄₃	—	4	2	no	—
123	C₂×C₄×C₄₃	—	4	4	no	—
124	C₂×C₄×C₄₃	—	4	4	no	—
125	C₂×C₄×C₄₃	—	4	1	no	—
126	C₂×C₄×C₄₃	—	4	1	no	—
127	C₂×C₄×C₄₃	—	4	1	no	—
128	C₂×C₄×C₄₃	—	4	1	no	—
129	C₂×C₄×C₄₃	C₄×C₈	1	2	yes	complemented
130	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
132	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
133	C₄₃⋊C₈	—	2	1	no	—
134	C₄₃⋊C₈	—	2	2	no	—
135	C₄₃⋊C₈	—	4	4	no	—
136	C₄₃⋊C₈	—	4	1	no	—
137	C₄₃⋊C₈	—	4	1	no	—
138	C₄₃⋊C₈	#99	1	2	yes	direct product factor
139	C₈×#99	—	43	1	no	maximal, Sylow
140	C₂²×C₄₃	—	2	1	no	—
141	C₂²×C₄₃	—	2	2	no	—
142	C₂²×C₄₃	—	2	1	no	—
143	C₂²×C₄₃	—	2	2	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	—	4	2	no	—
146	C₂²×C₄₃	—	4	2	no	—
147	C₂²×C₄₃	—	4	2	no	—
148	C₂²×C₄₃	—	4	2	no	—
149	C₂²×C₄₃	—	4	1	no	—
150	C₂²×C₄₃	—	4	1	no	—
151	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
152	C₂²×C₄₃	C₂×C₄×C₈	1	1	yes	characteristic
153	C₂²×C₄₃	#405	1	1	yes	characteristic
154	C₂²×C₄₃	C₈×D₄	1	4	yes	—
155	C₄×C₄₃	—	2	2	no	—
156	C₄×C₄₃	—	2	2	no	—
157	C₄×C₄₃	—	2	1	no	—
158	C₄×C₄₃	—	4	8	no	—
159	C₄×C₄₃	—	4	1	no	—
160	C₄×C₄₃	—	4	1	no	—
161	C₄×C₄₃	C₂×#99	1	1	yes	characteristic
162	C₄×C₄₃	#410	1	1	yes	characteristic
163	C₂×C₈×D₄	—	43	1	no	—
164	C₄×#99	D₄₃	1	1	yes	characteristic
165	C₈×C₂²⋊C₄	—	43	2	no	—
166	#1459	—	43	1	no	—
167	#1282	—	43	2	no	—
168	C₂×C₄₃	—	2	4	no	—
169	C₂×C₄₃	—	2	1	no	—
170	C₂×C₄₃	—	2	1	no	—
171	C₂×C₄₃	—	4	2	no	—
172	C₂×C₄₃	C₈×C₂²⋊C₄	1	1	yes	characteristic
173	C₂×C₄₃	C₄×#99	1	1	yes	characteristic
174	C₂×C₄₃	#1757	1	1	yes	characteristic
175	C₂³×C₈	—	43	1	no	—
176	C₂³×C₈	—	43	1	no	—
177	C₂×C₄×D₄	C₂×D₄₃	1	1	yes	characteristic
178	C₈×D₄	—	86	2	no	—
179	C₈×D₄	—	86	2	no	—
180	C₂×C₄⋊C₈	—	43	1	no	—
181	C₂×#99	Dic₄₃	1	2	yes	—
182	C₂×C₂²⋊C₈	—	43	2	no	—
183	C₂×C₂²⋊C₈	—	43	1	no	—
184	C₂×C₂²⋊C₈	—	43	1	no	—
185	C₂×C₄×C₈	—	43	1	no	—
186	C₂×C₄×C₈	—	86	2	no	—
187	C₄×C₂²⋊C₄	C₂×D₄₃	1	2	yes	—
188	#342	Dic₄₃	1	2	yes	—
189	#336	—	86	2	no	—
190	#324	—	43	2	no	—
191	#323	—	43	2	no	—
192	C₄₃	C₈×#99	1	1	yes	characteristic, complemented, Sylow
193	C₂²×D₄	C₂×Dic₄₃	1	1	yes	characteristic
194	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
195	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
196	C₂²×C₈	—	43	1	no	—
197	C₂²×C₈	—	43	2	no	—
198	C₂²×C₈	—	43	1	no	—
199	C₂²×C₈	—	43	2	no	—
200	C₂²×C₈	—	43	1	no	—
201	C₂²×C₈	—	43	1	no	—
202	C₂²×C₈	—	86	1	no	—
203	C₂²×C₈	—	86	1	no	—
204	C₂²×C₈	—	86	2	no	—
205	C₂²×C₈	—	172	2	no	—
206	C₂²×C₈	—	172	2	no	—
207	C₄×D₄	—	2	2	no	—
208	C₄×D₄	—	2	2	no	—
209	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
210	C₂×C₂²⋊C₄	C₄×D₄₃	1	1	yes	characteristic
211	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	1	yes	characteristic
212	C₂×C₂²⋊C₄	C₂×Dic₄₃	1	4	yes	—
213	C₂×C₄²	—	2	2	no	—
214	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
215	C₄⋊C₈	—	86	2	no	—
216	#99	C₄₃⋊C₈	1	8	yes	direct product factor
217	C₂²⋊C₈	—	43	4	no	—
218	C₂²⋊C₈	—	86	2	no	—
219	C₂²⋊C₈	—	86	2	no	—
220	C₂²⋊C₈	—	86	2	no	—
221	C₄×C₈	—	86	1	no	—
222	C₄×C₈	—	86	1	no	—
223	C₄×C₈	—	172	4	no	—
224	#95	—	2	2	no	—
225	C₂⁴	C₄×Dic₄₃	1	1	yes	characteristic
226	C₂⁴	C₂²×Dic₄₃	1	1	yes	characteristic
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	—	2	2	no	—
229	C₂×D₄	C₂×C₄₃⋊C₈	1	4	yes	—
230	C₂²×C₄	—	2	4	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	4	1	no	—
234	C₂²×C₄	—	4	1	no	—
235	C₂²×C₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
238	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₄×Dic₄₃	1	1	yes	characteristic
240	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
241	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
242	C₂²×C₄	D₄×D₄₃	1	2	yes	—
243	C₂²×C₄	#10621900	1	2	yes	—
244	C₂×C₈	—	43	1	no	—
245	C₂×C₈	—	43	1	no	—
246	C₂×C₈	—	43	2	no	—
247	C₂×C₈	—	86	4	no	—
248	C₂×C₈	—	86	2	no	—
249	C₂×C₈	—	86	1	no	—
250	C₂×C₈	—	86	1	no	—
251	C₂×C₈	—	86	4	no	—
252	C₂×C₈	—	172	2	no	—
253	C₂×C₈	—	172	2	no	—
254	C₂×C₈	—	172	1	no	—
255	C₂×C₈	—	172	2	no	—
256	C₂×C₈	—	172	2	no	—
257	C₂×C₈	—	172	2	no	—
258	C₂×C₈	—	172	1	no	—
259	C₂×C₈	—	172	1	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₂²⋊C₄	—	2	2	no	—
262	C₂²⋊C₄	—	2	4	no	—
263	C₂²⋊C₄	—	2	2	no	—
264	C₂²⋊C₄	C₂×C₄₃⋊C₈	1	8	yes	complemented
265	C₄²	—	2	1	no	—
266	C₄²	—	2	1	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	2	no	—
269	C₂³	—	2	1	no	—
270	C₂³	—	2	1	no	—
271	C₂³	—	4	1	no	—
272	C₂³	—	4	1	no	—
273	C₂³	—	4	1	no	—
274	C₂³	—	4	1	no	—
275	C₂³	C₄×C₄₃⋊C₈	1	2	yes	complemented
276	C₂³	C₂×C₄×Dic₄₃	1	1	yes	characteristic
277	C₂³	D₄×Dic₄₃	1	2	yes	—
278	C₂³	C₂²×C₄₃⋊C₈	1	2	yes	—
279	C₂³	#13063332	1	2	yes	—
280	D₄	—	2	8	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	8	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	4	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	4	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	C₄×C₄₃⋊C₈	1	2	yes	complemented
300	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
301	C₂×C₄	#13063281	1	1	yes	characteristic
302	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
303	C₈	—	43	2	no	—
304	C₈	—	86	1	no	—
305	C₈	—	86	2	no	—
306	C₈	—	172	4	no	—
307	C₈	—	172	1	no	—
308	C₈	—	172	1	no	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	2	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	4	2	no	—
315	C₂²	—	4	2	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	4	2	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
321	C₂²	C₂×C₄×C₄₃⋊C₈	1	1	yes	characteristic
322	C₂²	#14582855	1	1	yes	characteristic
323	C₂²	D₄×C₄₃⋊C₈	1	4	yes	—
324	C₄	—	2	2	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	4	8	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	#99×D₄₃	1	1	yes	characteristic
331	C₄	#14582871	1	1	yes	characteristic
332	C₂	—	2	4	no	—
333	C₂	—	2	1	no	—
334	C₂	—	2	1	no	—
335	C₂	—	4	2	no	—
336	C₂	C₂²⋊C₄×C₄₃⋊C₈	1	1	yes	characteristic
337	C₂	#99×Dic₄₃	1	1	yes	characteristic
338	C₂	#19134397	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊C₄×D₄₃
commutator	C₂²×C₄₃	C₂×C₄×C₈
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄×#99×C₄₃	C₂
socle	C₂²×C₄₃	C₄×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₃	C₂×C₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₄×#99	D₄₃
hypocenter	C₄₃	C₈×#99
LCS	G ▷ C₂²×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₄×#99

Complements

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

N	H ≅ G/N	type
2. C₂²⋊C₄×C₄₃⋊C₈	C₂	semidirect
4. #19132443	C₂	semidirect
6. C₂³×C₄₃⋊C₈	C₄	semidirect
10. C₂×C₄⋊(C₄₃⋊C₈)	C₄	semidirect
11. C₂×C₂²⋊(C₄₃⋊C₈)	C₄	semidirect
15. C₂×C₄×C₄₃⋊C₈	C₄	semidirect
17. #14582625	C₄	semidirect
18. #14582623	C₄	semidirect
53. #99×C₄₃	C₈	semidirect
78. C₂²⋊C₄×C₄₃	C₂×C₈	semidirect
105. C₂³×C₄₃	C₄×C₈	semidirect
129. C₂×C₄×C₄₃	C₄×C₈	semidirect
138. C₄₃⋊C₈	#99	direct (N × H)
192. C₄₃	C₈×#99	semidirect
216. #99	C₄₃⋊C₈	direct (N × H)
264. C₂²⋊C₄	C₂×C₄₃⋊C₈	semidirect
275. C₂³	C₄×C₄₃⋊C₈	semidirect
299. 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
C₂×D₄×C₄₃⋊C₈	2
C₂²⋊C₄×C₄₃⋊C₈	2
#19132913	2
#19132443	2
C₄×#99×C₄₃	2
C₈×#99	43

Subgroups — #99×C43⋊C8 #24987140

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #99×C₄₃⋊C₈ #24987140