Subgroups — #24987231

Order 11008.

542 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137073	C₂	1	1	yes	maximal, characteristic
2	#19134450	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₄₃	C₂	1	1	yes	maximal, characteristic
4	C₂×Q₈⋊(C₄₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#19132435	C₂	1	1	yes	maximal, characteristic
6	#19132434	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#13063235	—	2	1	no	—
10	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×C₈×Dic₄₃	—	2	2	no	—
12	#14582806	C₄	1	2	yes	—
13	#14582798	C₄	1	2	yes	—
14	#14582785	—	2	2	no	—
15	C₄×C₄⋊Dic₄₃	—	2	1	no	—
16	C₄×C₄⋊Dic₄₃	C₂²	1	1	yes	characteristic
17	#14582745	—	2	1	no	—
18	C₄²×Dic₄₃	C₂²	1	1	yes	characteristic
19	#14582653	—	2	1	no	—
20	Q₈⋊(C₄₃⋊C₈)	C₄	1	4	yes	complemented
21	C₂×C₄×Q₈×C₄₃	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊C₈×C₄₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×Q₈⋊C₈	C₄	1	4	yes	complemented
25	C₂²×C₄₃⋊C₈	—	2	1	no	—
26	C₂²×C₄₃⋊C₈	—	2	1	no	—
27	C₂×C₄⋊Dic₄₃	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	C₂×C₄	1	2	yes	—
29	C₂×C₄×Dic₄₃	—	2	1	no	—
30	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₄₃	D₄	1	2	yes	—
32	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
33	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
34	C₄⋊(C₄₃⋊C₈)	—	2	1	no	—
35	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	2	yes	—
36	#13063235	—	4	2	no	—
37	C₄×C₄₃⋊C₈	—	2	1	no	—
38	C₄×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
39	C₈×Dic₄₃	—	2	8	no	—
40	C₄₃⋊#95	—	2	1	no	—
41	C₄₃⋊#95	—	2	2	no	—
42	C₄₃⋊#95	—	2	2	no	—
43	C₄₃⋊#95	—	2	2	no	—
44	C₂²×Q₈×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₄₃	—	2	1	no	—
46	C₂²×C₈×C₄₃	—	2	1	no	—
47	C₄×Q₈×C₄₃	—	2	1	no	—
48	C₄×Q₈×C₄₃	—	2	1	no	—
49	C₄×Q₈×C₄₃	C₂×C₄	1	2	yes	—
50	C₄×Q₈×C₄₃	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₄₃	—	2	1	no	—
52	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	—	2	1	no	—
54	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
55	C₄⋊C₈×C₄₃	—	2	1	no	—
56	C₄⋊C₈×C₄₃	C₂×C₄	1	2	yes	—
57	C₄×C₈×C₄₃	—	2	1	no	—
58	C₄×C₈×C₄₃	C₂×C₄	1	2	yes	—
59	C₂×Q₈×C₄₃	—	2	1	no	—
60	C₂×Q₈×C₄₃	—	2	2	no	—
61	C₂×Q₈×C₄₃	—	2	2	no	—
62	C₂×Q₈×C₄₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₄₃	—	2	1	no	—
64	C₂²×C₄×C₄₃	—	2	1	no	—
65	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×Dic₄₃	—	2	2	no	—
69	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
70	C₂²×Dic₄₃	C₂²⋊C₄	1	2	yes	—
71	C₂×C₈×C₄₃	—	2	4	no	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	2	no	—
74	C₂×C₈×C₄₃	—	4	1	no	—
75	C₂×C₈×C₄₃	—	4	1	no	—
76	C₄⋊C₄×C₄₃	—	2	1	no	—
77	C₄⋊C₄×C₄₃	—	2	2	no	—
78	C₄⋊C₄×C₄₃	—	2	2	no	—
79	C₄⋊C₄×C₄₃	C₄²	1	2	yes	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
83	C₄²×C₄₃	C₂²×C₄	1	1	yes	characteristic
84	C₄²×C₄₃	#41	1	1	yes	characteristic
85	C₄²×C₄₃	#41	1	1	yes	characteristic
86	C₄⋊Dic₄₃	—	2	2	no	—
87	C₄⋊Dic₄₃	—	2	2	no	—
88	C₄⋊Dic₄₃	—	4	4	no	—
89	C₄×Dic₄₃	—	2	2	no	—
90	C₄×Dic₄₃	—	2	2	no	—
91	C₄×Dic₄₃	—	2	1	no	—
92	C₄×Dic₄₃	—	2	1	no	—
93	C₄×Dic₄₃	—	2	4	no	—
94	C₄×Dic₄₃	—	4	1	no	—
95	C₄×Dic₄₃	—	4	1	no	—
96	C₄×Dic₄₃	SD₁₆	1	2	yes	—
97	C₄×Dic₄₃	Q₁₆	1	2	yes	—
98	C₂×C₄₃⋊C₈	—	2	4	no	—
99	C₂×C₄₃⋊C₈	—	2	2	no	—
100	C₂×C₄₃⋊C₈	—	2	2	no	—
101	C₂×C₄₃⋊C₈	—	4	1	no	—
102	C₂×C₄₃⋊C₈	—	4	1	no	—
103	C₂³×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	Q₈×C₄₃	—	2	2	no	—
105	Q₈×C₄₃	—	2	2	no	—
106	Q₈×C₄₃	—	4	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	2	2	no	—
110	C₂×C₄×C₄₃	—	4	2	no	—
111	C₂×C₄×C₄₃	—	4	2	no	—
112	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
113	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#131	1	2	yes	—
119	C₂×C₄×C₄₃	#131	1	2	yes	—
120	C₂×C₄×C₄₃	C₂×SD₁₆	1	1	yes	characteristic
121	C₂×C₄×C₄₃	C₂×Q₁₆	1	1	yes	characteristic
122	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
123	C₂×C₄×C₄₃	#135	1	1	yes	characteristic
124	C₂×Dic₄₃	—	2	2	no	—
125	C₂×Dic₄₃	—	2	2	no	—
126	C₂×Dic₄₃	—	2	2	no	—
127	C₂×Dic₄₃	—	2	2	no	—
128	C₂×Dic₄₃	—	4	2	no	—
129	C₂×Dic₄₃	—	4	2	no	—
130	C₂×Dic₄₃	—	4	2	no	—
131	C₂×Dic₄₃	#103	1	4	yes	—
132	C₂×Dic₄₃	#104	1	4	yes	—
133	C₈×C₄₃	—	2	4	no	—
134	C₈×C₄₃	—	4	2	no	—
135	C₄₃⋊C₈	—	2	4	no	—
136	C₄₃⋊C₈	—	4	2	no	—
137	#8176	—	43	1	no	maximal, Sylow
138	C₂²×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
139	C₂²×C₄₃	#408	1	2	yes	—
140	C₂²×C₄₃	C₂×#103	1	1	yes	characteristic
141	C₂²×C₄₃	#416	1	1	yes	characteristic
142	C₂²×C₄₃	C₂×#104	1	2	yes	—
143	C₄×C₄₃	—	2	1	no	—
144	C₄×C₄₃	—	2	1	no	—
145	C₄×C₄₃	—	4	2	no	—
146	C₄×C₄₃	—	4	2	no	—
147	C₄×C₄₃	#405	1	1	yes	characteristic
148	C₄×C₄₃	#405	1	1	yes	characteristic
149	C₄×C₄₃	#433	1	2	yes	—
150	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
151	C₄×C₄₃	C₄×SD₁₆	1	1	yes	characteristic
152	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
153	C₄×C₄₃	C₄×Q₁₆	1	1	yes	characteristic
154	Dic₄₃	—	2	4	no	—
155	Dic₄₃	—	2	4	no	—
156	Dic₄₃	—	8	2	no	—
157	#2295	—	43	1	no	—
158	#1767	—	43	1	no	—
159	C₄²×C₈	—	43	1	no	—
160	C₂×Q₈⋊C₈	—	43	1	no	—
161	C₂×Q₈⋊C₈	D₄₃	1	1	yes	characteristic
162	#1279	—	43	1	no	—
163	#1279	—	43	1	no	—
164	C₂×C₄₃	#1493	1	2	yes	—
165	C₂×C₄₃	#1494	1	2	yes	—
166	C₂×C₄₃	#1750	1	1	yes	characteristic
167	C₂×C₄₃	C₄×#104	1	1	yes	characteristic
168	C₂×C₄₃	C₄×#103	1	1	yes	characteristic
169	C₂×C₄×Q₈	C₂×D₄₃	1	1	yes	characteristic
170	C₂×C₄⋊C₈	—	43	1	no	—
171	C₂×C₄⋊C₈	C₂×D₄₃	1	1	yes	characteristic
172	C₂×#97	—	86	1	no	—
173	C₂×C₄×C₈	—	43	1	no	—
174	C₂×C₄×C₈	—	86	2	no	—
175	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
176	#391	—	43	2	no	—
177	#389	—	43	2	no	—
178	#387	—	86	2	no	—
179	C₄×C₄⋊C₄	—	43	1	no	—
180	C₄×C₄⋊C₄	—	86	1	no	—
181	#376	—	86	1	no	—
182	C₄³	—	43	1	no	—
183	#336	—	86	1	no	—
184	Q₈⋊C₈	—	43	4	no	—
185	Q₈⋊C₈	Dic₄₃	1	4	yes	complemented
186	C₄₃	#8176	1	1	yes	characteristic, complemented, Sylow
187	C₂²×Q₈	C₄×D₄₃	1	1	yes	characteristic
188	C₂²×C₈	—	2	1	no	—
189	C₂²×C₈	—	2	1	no	—
190	C₂²×C₈	—	86	1	no	—
191	C₂²×C₈	—	86	1	no	—
192	C₄×Q₈	—	2	1	no	—
193	C₄×Q₈	—	2	1	no	—
194	C₄×Q₈	C₄×D₄₃	1	2	yes	—
195	C₄×Q₈	C₂×Dic₄₃	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	1	no	—
197	C₂×C₄⋊C₄	—	43	2	no	—
198	C₂×C₄⋊C₄	—	86	2	no	—
199	C₂×C₄⋊C₄	C₄×D₄₃	1	1	yes	characteristic
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	43	1	no	—
202	C₂×C₄²	—	43	2	no	—
203	C₂×C₄²	—	43	2	no	—
204	C₂×C₄²	—	43	1	no	—
205	C₂×C₄²	—	86	1	no	—
206	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
207	C₄⋊C₈	—	2	1	no	—
208	C₄⋊C₈	—	43	2	no	—
209	C₄⋊C₈	—	86	1	no	—
210	C₄⋊C₈	C₂×Dic₄₃	1	2	yes	—
211	#97	—	172	2	no	—
212	C₄×C₈	—	2	1	no	—
213	C₄×C₈	—	43	2	no	—
214	C₄×C₈	—	86	8	no	—
215	C₄×C₈	—	86	1	no	—
216	C₄×C₈	C₂×Dic₄₃	1	2	yes	—
217	#95	—	86	2	no	—
218	#95	—	86	2	no	—
219	#95	—	86	2	no	—
220	#95	—	86	1	no	—
221	C₂×Q₈	—	2	1	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	C₄×Dic₄₃	1	2	yes	—
225	C₂²×C₄	—	2	1	no	—
226	C₂²×C₄	—	2	1	no	—
227	C₂²×C₄	—	43	2	no	—
228	C₂²×C₄	—	43	2	no	—
229	C₂²×C₄	—	86	2	no	—
230	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
231	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
232	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	4	1	no	—
237	C₂×C₈	—	4	1	no	—
238	C₂×C₈	—	86	4	no	—
239	C₂×C₈	—	86	2	no	—
240	C₂×C₈	—	86	2	no	—
241	C₂×C₈	—	172	1	no	—
242	C₂×C₈	—	172	1	no	—
243	C₄⋊C₄	—	2	1	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	86	2	no	—
247	C₄⋊C₄	—	86	2	no	—
248	C₄⋊C₄	—	172	4	no	—
249	C₄⋊C₄	C₄×Dic₄₃	1	2	yes	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	43	2	no	—
253	C₄²	—	43	2	no	—
254	C₄²	—	86	4	no	—
255	C₄²	—	86	2	no	—
256	C₄²	—	86	2	no	—
257	C₄²	—	86	1	no	—
258	C₄²	—	86	1	no	—
259	C₄²	—	172	1	no	—
260	C₄²	—	172	1	no	—
261	C₄²	C₂×C₄×D₄₃	1	1	yes	characteristic
262	C₄²	#10621900	1	1	yes	characteristic
263	C₄²	#10621900	1	1	yes	characteristic
264	C₄²	C₂²×Dic₄₃	1	1	yes	characteristic
265	C₂³	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
266	Q₈	—	2	2	no	—
267	Q₈	—	2	2	no	—
268	Q₈	—	4	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	43	4	no	—
275	C₂×C₄	—	43	4	no	—
276	C₂×C₄	—	86	2	no	—
277	C₂×C₄	—	86	2	no	—
278	C₂×C₄	—	86	2	no	—
279	C₂×C₄	—	86	2	no	—
280	C₂×C₄	—	172	2	no	—
281	C₂×C₄	—	172	2	no	—
282	C₂×C₄	—	172	2	no	—
283	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
284	C₂×C₄	#13063281	1	1	yes	characteristic
285	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
286	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
287	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
288	C₂×C₄	D₄×Dic₄₃	1	1	yes	characteristic
289	C₂×C₄	#13063331	1	2	yes	—
290	C₂×C₄	#13063333	1	2	yes	—
291	C₂×C₄	SD₁₆×D₄₃	1	1	yes	characteristic
292	C₂×C₄	Q₁₆×D₄₃	1	1	yes	characteristic
293	C₂×C₄	#13063346	1	1	yes	characteristic
294	C₂×C₄	#13063348	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	4	2	no	—
297	C₈	—	86	4	no	—
298	C₈	—	172	2	no	—
299	C₂²	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
300	C₂²	#14582864	1	2	yes	—
301	C₂²	#103×D₄₃	1	1	yes	characteristic
302	C₂²	#14582894	1	1	yes	characteristic
303	C₂²	#104×D₄₃	1	2	yes	—
304	C₄	—	2	1	no	—
305	C₄	—	2	1	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	2	no	—
308	C₄	—	86	4	no	—
309	C₄	—	86	4	no	—
310	C₄	—	344	2	no	—
311	C₄	#14582854	1	1	yes	characteristic
312	C₄	#14582857	1	1	yes	characteristic
313	C₄	#14582957	1	2	yes	—
314	C₄	#14582984	1	1	yes	characteristic
315	C₄	SD₁₆×Dic₄₃	1	1	yes	characteristic
316	C₄	#14582991	1	1	yes	characteristic
317	C₄	Q₁₆×Dic₄₃	1	1	yes	characteristic
318	C₂	#19133048	1	2	yes	—
319	C₂	#19133054	1	2	yes	—
320	C₂	#19134363	1	1	yes	characteristic
321	C₂	#104×Dic₄₃	1	1	yes	characteristic
322	C₂	#103×Dic₄₃	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₃
Fitting	C₂×C₄₃×Q₈⋊C₈	C₂
socle, prime radical	C₂³×C₄₃	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₄₃	C₂³
hypercenter	C₂×Q₈⋊C₈	D₄₃
hypocenter	C₄₃	#8176
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×Q₈⋊C₈

Complements

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

N	H ≅ G/N	type
20. Q₈⋊(C₄₃⋊C₈)	C₄	semidirect
24. C₄₃×Q₈⋊C₈	C₄	semidirect
185. Q₈⋊C₈	Dic₄₃	semidirect
186. C₄₃	#8176	semidirect

Maximal subgroups

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

subgroup	index
#19137073	2
#19134450	2
C₄×C₈×Dic₄₃	2
C₂×Q₈⋊(C₄₃⋊C₈)	2
#19132435	2
#19132434	2
C₂×C₄₃×Q₈⋊C₈	2
#8176	43