Subgroups — #24996720

Order 11008.

538 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19142992	C₂	1	1	yes	maximal, characteristic
2	#19134746	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	#19132479	C₂	1	1	yes	maximal, characteristic
5	#19132479	C₂	1	1	yes	maximal, characteristic
6	#19132478	C₂	1	1	yes	maximal, characteristic
7	C₄₃×#1296	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063324	—	2	1	no	—
9	C₂×#13063324	—	2	1	no	—
10	C₂×#13063324	C₄	1	2	yes	—
11	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
13	C₂×Dic₄₃⋊C₈	—	2	1	no	—
14	C₂×Dic₄₃⋊C₈	—	2	1	no	—
15	C₂×C₈×Dic₄₃	—	2	1	no	—
16	#14582769	C₂²	1	1	yes	characteristic
17	#14582768	—	2	1	no	—
18	#14582768	—	2	1	no	—
19	#14582768	C₄	1	2	yes	—
20	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
21	#14582653	—	2	1	no	—
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#384	C₂²	1	1	yes	characteristic
25	C₂²×Dic₈₆	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	C₂²×C₄₃⋊C₈	—	2	1	no	—
28	#13063324	—	4	4	no	—
29	#13063324	—	4	2	no	—
30	#13063324	—	4	2	no	—
31	#13063324	—	4	2	no	—
32	#13063324	—	4	2	no	—
33	C₂×C₄⋊Dic₄₃	—	2	1	no	—
34	C₂×C₄⋊Dic₄₃	—	2	1	no	—
35	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
36	C₂×Dic₄₃⋊C₄	—	2	2	no	—
37	C₂×Dic₄₃⋊C₄	—	2	2	no	—
38	C₂×Dic₄₃⋊C₄	—	2	2	no	—
39	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
42	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
43	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
44	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
45	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
46	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
47	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
48	C₄⋊(C₄₃⋊C₈)	Q₈	1	2	yes	complemented
49	Dic₄₃⋊C₈	—	4	2	no	—
50	Dic₄₃⋊C₈	—	4	2	no	—
51	C₈×Dic₄₃	—	4	2	no	—
52	C₂²×C₈×C₄₃	—	2	1	no	—
53	C₂²×C₈×C₄₃	—	2	1	no	—
54	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
55	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
56	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
58	C₄⋊C₈×C₄₃	—	2	1	no	—
59	C₄⋊C₈×C₄₃	—	2	1	no	—
60	C₄⋊C₈×C₄₃	—	2	1	no	—
61	C₄⋊C₈×C₄₃	Q₈	1	2	yes	complemented
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₄₃	—	2	2	no	—
67	C₂²×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
70	C₂²×Dic₄₃	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×Dic₈₆	—	4	4	no	—
72	C₂×Dic₈₆	—	4	4	no	—
73	C₂×Dic₈₆	—	4	4	no	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₂×C₈×C₄₃	—	4	1	no	—
77	C₂×C₈×C₄₃	—	4	1	no	—
78	C₂×C₈×C₄₃	—	4	1	no	—
79	C₂×C₈×C₄₃	—	4	1	no	—
80	C₄⋊C₄×C₄₃	—	4	2	no	—
81	C₄⋊C₄×C₄₃	—	4	1	no	—
82	C₄⋊C₄×C₄₃	—	4	1	no	—
83	C₄⋊C₄×C₄₃	—	4	1	no	—
84	C₄⋊C₄×C₄₃	—	4	1	no	—
85	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
86	C₄²×C₄₃	C₂×Q₈	1	1	yes	characteristic
87	C₄²×C₄₃	#41	1	1	yes	characteristic
88	C₄²×C₄₃	#41	1	1	yes	characteristic
89	C₄⋊Dic₄₃	—	4	1	no	—
90	C₄⋊Dic₄₃	—	4	1	no	—
91	C₄⋊Dic₄₃	—	4	1	no	—
92	C₄⋊Dic₄₃	—	4	1	no	—
93	C₄⋊Dic₄₃	—	4	2	no	—
94	Dic₄₃⋊C₄	—	4	4	no	—
95	Dic₄₃⋊C₄	—	4	4	no	—
96	Dic₄₃⋊C₄	—	4	4	no	—
97	C₄×Dic₄₃	—	2	1	no	—
98	C₄×Dic₄₃	—	2	1	no	—
99	C₄×Dic₄₃	—	2	2	no	—
100	C₄×Dic₄₃	—	2	2	no	—
101	C₄×Dic₄₃	D₈	1	2	yes	—
102	C₄×Dic₄₃	D₈	1	2	yes	—
103	C₄×Dic₄₃	SD₁₆	1	2	yes	—
104	C₄×Dic₄₃	SD₁₆	1	2	yes	—
105	C₄×Dic₄₃	SD₁₆	1	2	yes	—
106	C₄×Dic₄₃	Q₁₆	1	2	yes	—
107	C₂×C₄₃⋊C₈	—	2	2	no	—
108	C₂×C₄₃⋊C₈	—	4	1	no	—
109	C₂×C₄₃⋊C₈	—	4	1	no	—
110	C₂×C₄₃⋊C₈	—	4	1	no	—
111	C₂×C₄₃⋊C₈	—	4	1	no	—
112	C₂×C₄₃⋊C₈	—	4	1	no	—
113	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	—	2	1	no	—
115	C₂×C₄×C₄₃	—	2	1	no	—
116	C₂×C₄×C₄₃	—	4	2	no	—
117	C₂×C₄×C₄₃	—	4	2	no	—
118	C₂×C₄×C₄₃	—	4	2	no	—
119	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
124	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
125	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
129	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
130	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
131	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
132	C₂×C₄×C₄₃	#137	1	1	yes	characteristic
133	C₂×Dic₄₃	—	2	1	no	—
134	C₂×Dic₄₃	—	2	1	no	—
135	C₂×Dic₄₃	—	2	1	no	—
136	C₂×Dic₄₃	—	2	1	no	—
137	C₂×Dic₄₃	—	4	2	no	—
138	C₂×Dic₄₃	—	4	2	no	—
139	C₂×Dic₄₃	—	4	2	no	—
140	C₂×Dic₄₃	#100	1	2	yes	—
141	C₂×Dic₄₃	#101	1	2	yes	—
142	C₂×Dic₄₃	#102	1	2	yes	—
143	C₂×Dic₄₃	#102	1	2	yes	—
144	C₂×Dic₄₃	#102	1	2	yes	—
145	C₂×Dic₄₃	#102	1	2	yes	—
146	C₂×Dic₄₃	#103	1	2	yes	—
147	C₂×Dic₄₃	#103	1	2	yes	—
148	Dic₈₆	—	8	8	no	—
149	C₈×C₄₃	—	4	2	no	—
150	C₈×C₄₃	—	8	1	no	—
151	C₄₃⋊C₈	—	4	2	no	—
152	C₄₃⋊C₈	—	8	1	no	—
153	#9898	—	43	1	no	maximal, Sylow
154	C₂²×C₄₃	#391	1	1	yes	characteristic
155	C₂²×C₄₃	C₂×#100	1	1	yes	characteristic
156	C₂²×C₄₃	C₂×#101	1	1	yes	characteristic
157	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
158	C₂²×C₄₃	C₂×#102	1	1	yes	characteristic
159	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
160	C₂²×C₄₃	#419	1	1	yes	characteristic
161	C₄×C₄₃	—	2	1	no	—
162	C₄×C₄₃	—	2	1	no	—
163	C₄×C₄₃	—	8	2	no	—
164	C₄×C₄₃	#474	1	1	yes	characteristic
165	C₄×C₄₃	#474	1	1	yes	characteristic
166	C₄×C₄₃	#475	1	1	yes	characteristic
167	C₄×C₄₃	#475	1	1	yes	characteristic
168	C₄×C₄₃	#486	1	1	yes	characteristic
169	C₄×C₄₃	#487	1	1	yes	characteristic
170	C₄×C₄₃	#498	1	1	yes	characteristic
171	C₄×C₄₃	#500	1	1	yes	characteristic
172	Dic₄₃	—	2	2	no	—
173	Dic₄₃	—	2	2	no	—
174	Dic₄₃	—	8	2	no	—
175	Dic₄₃	#331	1	4	yes	—
176	Dic₄₃	#332	1	4	yes	—
177	#2868	—	43	1	no	—
178	#1842	—	43	1	no	—
179	C₄×C₄⋊C₈	—	43	1	no	—
180	#1296	—	43	1	no	—
181	#1296	—	43	1	no	—
182	#1296	—	43	1	no	—
183	#1296	D₄₃	1	1	yes	characteristic
184	C₂×C₄₃	C₂×#331	1	1	yes	characteristic
185	C₂×C₄₃	C₂×#332	1	1	yes	characteristic
186	C₂×C₄₃	#1546	1	1	yes	characteristic
187	C₂×C₄₃	#1548	1	1	yes	characteristic
188	C₂×C₄₃	#1988	1	1	yes	characteristic
189	C₂×C₄₃	#1990	1	1	yes	characteristic
190	C₂×C₄₃	#1997	1	1	yes	characteristic
191	C₂×C₄⋊Q₈	—	43	2	no	—
192	C₂×C₄⋊Q₈	—	86	1	no	—
193	C₂×C₄⋊Q₈	—	86	1	no	—
194	C₂×C₄⋊C₈	—	43	1	no	—
195	C₂×C₄⋊C₈	—	43	1	no	—
196	C₂×C₄⋊C₈	—	86	1	no	—
197	C₂×C₄⋊C₈	—	86	1	no	—
198	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
199	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
200	C₂×C₄×C₈	—	86	1	no	—
201	#384	—	43	2	no	—
202	#384	—	43	1	no	—
203	#384	—	86	1	no	—
204	#384	—	86	1	no	—
205	#384	C₂×D₄₃	1	1	yes	characteristic
206	C₄³	—	43	1	no	—
207	#336	—	86	1	no	—
208	C₄₃	#9898	1	1	yes	characteristic, complemented, Sylow
209	C₂²×Q₈	—	86	2	no	—
210	C₂²×C₈	—	2	1	no	—
211	C₂²×C₈	—	2	1	no	—
212	C₂²×C₈	—	86	1	no	—
213	C₂²×C₈	—	86	1	no	—
214	C₄⋊Q₈	—	172	2	no	—
215	C₄⋊Q₈	—	172	2	no	—
216	C₄⋊Q₈	—	172	2	no	—
217	C₄⋊Q₈	—	172	2	no	—
218	C₄⋊Q₈	—	172	4	no	—
219	C₂×C₄⋊C₄	—	2	1	no	—
220	C₂×C₄⋊C₄	—	2	1	no	—
221	C₂×C₄⋊C₄	—	43	2	no	—
222	C₂×C₄⋊C₄	—	86	2	no	—
223	C₂×C₄⋊C₄	—	86	2	no	—
224	C₂×C₄⋊C₄	—	86	2	no	—
225	C₂×C₄⋊C₄	—	86	1	no	—
226	C₂×C₄⋊C₄	—	86	1	no	—
227	C₂×C₄⋊C₄	C₄×D₄₃	1	2	yes	—
228	C₂×C₄²	—	43	1	no	—
229	C₂×C₄²	—	43	1	no	—
230	C₂×C₄²	—	43	1	no	—
231	C₂×C₄²	—	43	1	no	—
232	C₂×C₄²	—	43	1	no	—
233	C₂×C₄²	—	43	1	no	—
234	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
235	C₄⋊C₈	—	2	1	no	—
236	C₄⋊C₈	—	2	1	no	—
237	C₄⋊C₈	—	2	1	no	—
238	C₄⋊C₈	—	43	2	no	—
239	C₄⋊C₈	—	86	1	no	—
240	C₄⋊C₈	—	86	1	no	—
241	C₄⋊C₈	—	86	1	no	—
242	C₄⋊C₈	—	172	2	no	—
243	C₄⋊C₈	—	172	2	no	—
244	C₄⋊C₈	Dic₈₆	1	2	yes	complemented
245	C₄×C₈	—	172	2	no	—
246	C₂×Q₈	—	172	4	no	—
247	C₂×Q₈	—	172	4	no	—
248	C₂×Q₈	—	172	4	no	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	43	1	no	—
251	C₂²×C₄	—	43	1	no	—
252	C₂²×C₄	—	43	1	no	—
253	C₂²×C₄	—	43	1	no	—
254	C₂²×C₄	—	86	2	no	—
255	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
256	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
257	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	—	4	1	no	—
260	C₂×C₈	—	4	1	no	—
261	C₂×C₈	—	4	1	no	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	1	no	—
264	C₂×C₈	—	86	2	no	—
265	C₂×C₈	—	172	1	no	—
266	C₂×C₈	—	172	1	no	—
267	C₂×C₈	—	172	1	no	—
268	C₂×C₈	—	172	1	no	—
269	C₂×C₈	—	172	1	no	—
270	C₄⋊C₄	—	4	2	no	—
271	C₄⋊C₄	—	4	1	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	—	4	1	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	172	2	no	—
276	C₄⋊C₄	—	172	4	no	—
277	C₄⋊C₄	—	172	4	no	—
278	C₄⋊C₄	—	172	4	no	—
279	C₄⋊C₄	—	172	1	no	—
280	C₄⋊C₄	—	172	1	no	—
281	C₄⋊C₄	—	172	1	no	—
282	C₄⋊C₄	—	172	1	no	—
283	C₄²	—	43	2	no	—
284	C₄²	—	43	2	no	—
285	C₄²	—	43	2	no	—
286	C₄²	—	43	2	no	—
287	C₄²	—	43	2	no	—
288	C₄²	—	43	2	no	—
289	C₄²	—	86	2	no	—
290	C₄²	—	86	2	no	—
291	C₄²	—	86	1	no	—
292	C₄²	—	86	1	no	—
293	C₄²	C₂×Dic₈₆	1	1	yes	characteristic
294	C₄²	#10621898	1	1	yes	characteristic
295	C₄²	#10621900	1	1	yes	characteristic
296	C₄²	Q₈×D₄₃	1	1	yes	characteristic
297	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
298	Q₈	—	344	8	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	43	2	no	—
305	C₂×C₄	—	43	2	no	—
306	C₂×C₄	—	43	2	no	—
307	C₂×C₄	—	43	2	no	—
308	C₂×C₄	—	43	2	no	—
309	C₂×C₄	—	43	2	no	—
310	C₂×C₄	—	43	2	no	—
311	C₂×C₄	—	43	2	no	—
312	C₂×C₄	—	86	1	no	—
313	C₂×C₄	—	86	1	no	—
314	C₂×C₄	—	86	1	no	—
315	C₂×C₄	—	86	1	no	—
316	C₂×C₄	—	172	2	no	—
317	C₂×C₄	—	172	2	no	—
318	C₂×C₄	—	172	2	no	—
319	C₂×C₄	#13063290	1	1	yes	characteristic
320	C₂×C₄	C₄×Dic₈₆	1	1	yes	characteristic
321	C₂×C₄	#13063302	1	1	yes	characteristic
322	C₂×C₄	#13063303	1	1	yes	characteristic
323	C₂×C₄	#13063311	1	1	yes	characteristic
324	C₂×C₄	#13063324	1	1	yes	characteristic
325	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
326	C₂×C₄	D₈×D₄₃	1	1	yes	characteristic
327	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
328	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
329	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
330	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
331	C₂×C₄	#13063354	1	1	yes	characteristic
332	C₂×C₄	#13063357	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	8	1	no	—
335	C₈	—	344	1	no	—
336	C₈	—	172	2	no	—
337	C₂²	#14582803	1	1	yes	characteristic
338	C₂²	#100×D₄₃	1	1	yes	characteristic
339	C₂²	#101×D₄₃	1	1	yes	characteristic
340	C₂²	#102×D₄₃	1	1	yes	characteristic
341	C₂²	#102×D₄₃	1	1	yes	characteristic
342	C₂²	#103×D₄₃	1	1	yes	characteristic
343	C₂²	#14582906	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	8	2	no	—
347	C₄	—	43	4	no	—
348	C₄	—	43	4	no	—
349	C₄	—	86	2	no	—
350	C₄	—	86	2	no	—
351	C₄	—	344	2	no	—
352	C₄	#14583177	1	1	yes	characteristic
353	C₄	#14583178	1	1	yes	characteristic
354	C₄	#14583182	1	1	yes	characteristic
355	C₄	#14583183	1	1	yes	characteristic
356	C₄	#14583244	1	1	yes	characteristic
357	C₄	#14583248	1	1	yes	characteristic
358	C₄	#14583296	1	1	yes	characteristic
359	C₄	#14583303	1	1	yes	characteristic
360	C₂	#331×D₄₃	1	1	yes	characteristic
361	C₂	#332×D₄₃	1	1	yes	characteristic
362	C₂	#19133265	1	1	yes	characteristic
363	C₂	#19133274	1	1	yes	characteristic
364	C₂	#19135545	1	1	yes	characteristic
365	C₂	#19135554	1	1	yes	characteristic
366	C₂	#19135582	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊C₄×D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₄₃×#1296	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	#1296	D₄₃
hypocenter	C₄₃	#9898
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1296

Complements

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

N	H ≅ G/N	type
48. C₄⋊(C₄₃⋊C₈)	Q₈	semidirect
61. C₄⋊C₈×C₄₃	Q₈	semidirect
208. C₄₃	#9898	semidirect
244. C₄⋊C₈	Dic₈₆	semidirect

Maximal subgroups

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

subgroup	index
#19142992	2
#19134746	2
C₄⋊C₈×Dic₄₃	2
#19132479	2
#19132478	2
C₄₃×#1296	2
#9898	43