Subgroups — #25097134

Order 11008.

982 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈×D₄₃	C₂	1	1	yes	maximal, characteristic
2	#19137000	C₂	1	2	yes	maximal
3	#19134755	C₂	1	1	yes	maximal, characteristic, complemented
4	#19134722	C₂	1	2	yes	maximal, complemented
5	#19134621	C₂	1	4	yes	maximal
6	#107×Dic₄₃	C₂	1	4	yes	maximal, complemented
7	C₄₃×#1845	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₈×D₄₃	C₂²	1	2	yes	—
9	C₂×#13063283	—	2	2	no	—
10	C₂×C₄²×D₄₃	C₂²	1	1	yes	characteristic
11	C₂×#13063259	C₂²	1	4	yes	—
12	C₂×C₄×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₄₃	C₂²	1	2	yes	—
14	C₄×C₈×D₄₃	C₄	1	8	yes	complemented
15	#14582769	C₂²	1	2	yes	—
16	#14582764	—	2	2	no	—
17	#14582756	—	2	2	no	—
18	#14582756	C₂²	1	4	yes	—
19	C₄⋊C₄×Dic₄₃	—	2	2	no	—
20	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
21	#14582661	C₂²	1	8	yes	—
22	C₂×#107×C₄₃	C₂²	1	4	yes	—
23	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#384	C₂²	1	2	yes	—
25	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
26	C₂²×C₄×D₄₃	D₄	1	1	yes	characteristic
27	C₂²×C₄×D₄₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
29	C₂×C₈×D₄₃	—	2	4	no	—
30	C₂×C₈×D₄₃	—	2	4	no	—
31	C₂×C₈×D₄₃	C₂×C₄	1	8	yes	—
32	#13063283	—	4	8	no	—
33	C₂×C₄⋊Dic₄₃	—	2	2	no	—
34	C₂×C₄⋊Dic₄₃	—	2	2	no	—
35	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
36	C₂×(C₂×D₄₃)⋊C₄	—	2	4	no	—
37	C₂×C₄×Dic₄₃	—	2	4	no	—
38	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
39	C₂×C₄×Dic₄₃	D₄	1	1	yes	characteristic
40	C₂×C₄×Dic₄₃	C₂³	1	1	yes	characteristic
41	C₄²×D₄₃	C₂×C₄	1	4	yes	—
42	C₄²×D₄₃	Q₈	1	4	yes	—
43	#13063259	—	2	8	no	—
44	C₄×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
45	C₈×Dic₄₃	C₂×C₄	1	8	yes	—
46	C₄₃⋊#95	—	2	4	no	—
47	C₄₃⋊#95	—	2	4	no	—
48	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
49	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
50	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₃	C₂³	1	4	yes	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	#107×C₄₃	—	2	8	no	—
54	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
55	C₂³×D₄₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	—	2	4	no	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
60	C₂²×Dic₄₃	—	2	4	no	—
61	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×Dic₄₃	C₂×D₄	1	1	yes	characteristic
64	C₂×C₄×D₄₃	—	2	2	no	—
65	C₂×C₄×D₄₃	—	2	2	no	—
66	C₂×C₄×D₄₃	—	2	2	no	—
67	C₂×C₄×D₄₃	—	2	2	no	—
68	C₂×C₄×D₄₃	C₄⋊C₄	1	4	yes	—
69	C₂×C₄×D₄₃	C₄⋊C₄	1	4	yes	—
70	C₂×C₄×D₄₃	D₈	1	4	yes	—
71	C₂×C₄×D₄₃	Q₁₆	1	4	yes	—
72	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
73	C₂×C₄×D₄₃	C₂×Q₈	1	2	yes	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
75	C₂×C₈×C₄₃	#41	1	4	yes	—
76	C₂×C₈×C₄₃	#41	1	4	yes	—
77	C₄⋊C₄×C₄₃	—	2	8	no	—
78	C₄⋊C₄×C₄₃	—	4	4	no	—
79	C₄⋊C₄×C₄₃	—	4	4	no	—
80	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
81	C₄²×C₄₃	C₂×Q₈	1	2	yes	—
82	(C₂×D₄₃)⋊C₄	—	4	8	no	—
83	C₄⋊Dic₄₃	—	2	8	no	—
84	C₄⋊Dic₄₃	—	4	4	no	—
85	C₄⋊Dic₄₃	—	4	4	no	—
86	C₄×Dic₄₃	—	4	8	no	—
87	C₄×Dic₄₃	C₄⋊C₄	1	4	yes	—
88	C₄×Dic₄₃	C₄⋊C₄	1	4	yes	—
89	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
90	C₄×Dic₄₃	C₂×Q₈	1	2	yes	—
91	C₂×C₄₃⋊C₈	—	2	2	no	—
92	C₂×C₄₃⋊C₈	—	2	2	no	—
93	C₂×C₄₃⋊C₈	C₂²×C₄	1	4	yes	—
94	C₈×D₄₃	—	2	16	no	—
95	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
96	C₂²×D₄₃	—	2	1	no	—
97	C₂²×D₄₃	—	2	1	no	—
98	C₂²×D₄₃	—	2	1	no	—
99	C₂²×D₄₃	—	2	1	no	—
100	C₂²×D₄₃	C₂×C₄⋊C₄	1	2	yes	—
101	C₂²×D₄₃	C₂×D₈	1	2	yes	—
102	C₂²×D₄₃	C₂×Q₁₆	1	2	yes	—
103	C₂×C₄×C₄₃	—	4	4	no	—
104	C₂×C₄×C₄₃	—	4	4	no	—
105	C₂×C₄×C₄₃	—	4	4	no	—
106	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
109	C₂×C₄×C₄₃	C₂×Q₁₆	1	2	yes	—
110	C₂×C₄×C₄₃	#135	1	2	yes	—
111	C₂×C₄×C₄₃	#135	1	2	yes	—
112	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
115	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
116	C₂×Dic₄₃	—	2	1	no	—
117	C₂×Dic₄₃	—	2	1	no	—
118	C₂×Dic₄₃	—	2	1	no	—
119	C₂×Dic₄₃	—	2	1	no	—
120	C₂×Dic₄₃	—	4	4	no	—
121	C₂×Dic₄₃	—	4	4	no	—
122	C₂×Dic₄₃	—	4	4	no	—
123	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
124	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×Dic₄₃	C₂×C₄⋊C₄	1	2	yes	—
126	C₂×Dic₄₃	C₂×D₈	1	2	yes	—
127	C₂×Dic₄₃	C₂×Q₁₆	1	2	yes	—
128	C₄×D₄₃	—	2	4	no	—
129	C₄×D₄₃	—	2	4	no	—
130	C₄×D₄₃	—	2	8	no	—
131	C₄×D₄₃	#107	1	16	yes	complemented
132	C₈×C₄₃	#117	1	8	yes	complemented
133	C₄₃⋊C₈	—	2	4	no	—
134	#20817	—	43	1	no	maximal, Sylow
135	C₂²×C₄₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₃	#532	1	1	yes	characteristic
137	C₂²×C₄₃	#532	1	1	yes	characteristic
138	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
139	C₂²×C₄₃	C₂²×Q₁₆	1	1	yes	characteristic
140	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
141	C₂²×C₄₃	C₂×#135	1	1	yes	characteristic
142	C₂×D₄₃	—	2	2	no	—
143	C₂×D₄₃	—	2	2	no	—
144	C₂×D₄₃	—	2	2	no	—
145	C₂×D₄₃	—	2	2	no	—
146	C₂×D₄₃	—	2	2	no	—
147	C₂×D₄₃	—	2	2	no	—
148	C₂×D₄₃	C₂×#107	1	4	yes	—
149	C₄×C₄₃	—	8	4	no	—
150	C₄×C₄₃	C₂×#107	1	4	yes	—
151	C₄×C₄₃	#427	1	4	yes	—
152	C₄×C₄₃	C₂×#117	1	2	yes	—
153	C₄×C₄₃	#533	1	2	yes	—
154	Dic₄₃	—	2	2	no	—
155	Dic₄₃	—	2	2	no	—
156	Dic₄₃	—	8	4	no	—
157	Dic₄₃	C₂×#107	1	4	yes	—
158	C₂²×C₄×C₈	—	43	1	no	—
159	#2285	—	43	2	no	—
160	#1845	—	43	1	no	—
161	#1845	D₄₃	1	1	yes	characteristic, complemented
162	#1834	—	43	2	no	—
163	#1808	—	43	4	no	—
164	C₄×#107	—	43	4	no	—
165	C₂×C₄₃	#2285	1	1	yes	characteristic
166	C₂×C₄₃	C₂²×#107	1	1	yes	characteristic
167	C₂×C₄₃	C₂×#427	1	1	yes	characteristic
168	C₂×C₄₃	#3169	1	1	yes	characteristic
169	C₂×C₄₃	#3169	1	1	yes	characteristic
170	C₂×C₄₃	#3170	1	1	yes	characteristic
171	C₂×C₄₃	#3170	1	1	yes	characteristic
172	D₄₃	—	2	4	no	—
173	C₂³×C₈	—	43	2	no	—
174	C₂×#117	—	86	2	no	—
175	C₂²×C₄²	—	43	1	no	—
176	C₂×#107	—	43	4	no	—
177	C₂×#107	C₂×D₄₃	1	4	yes	—
178	C₂×C₄×C₈	—	43	2	no	—
179	C₂×C₄×C₈	—	43	1	no	—
180	C₂×C₄×C₈	—	43	8	no	—
181	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
182	#384	—	43	2	no	—
183	#384	C₂×D₄₃	1	2	yes	—
184	#382	—	86	2	no	—
185	#380	—	43	4	no	—
186	#380	—	86	2	no	—
187	C₄×C₄⋊C₄	—	43	4	no	—
188	C₄×C₄⋊C₄	—	86	2	no	—
189	#340	—	43	8	no	—
190	C₄₃	#20817	1	1	yes	characteristic, complemented, Sylow
191	C₂³×C₄	—	43	1	no	—
192	C₂³×C₄	—	43	1	no	—
193	C₂³×C₄	—	43	1	no	—
194	C₂²×C₈	—	43	2	no	—
195	C₂²×C₈	—	43	8	no	—
196	C₂²×C₈	—	86	4	no	—
197	C₂²×C₈	—	86	4	no	—
198	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
199	#117	—	172	8	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	43	4	no	—
203	C₂×C₄⋊C₄	—	86	2	no	—
204	C₂×C₄⋊C₄	—	86	2	no	—
205	C₂×C₄⋊C₄	C₂²×D₄₃	1	4	yes	—
206	C₂×C₂²⋊C₄	—	86	4	no	—
207	C₂×C₄²	—	43	1	no	—
208	C₂×C₄²	—	43	1	no	—
209	C₂×C₄²	—	43	1	no	—
210	C₂×C₄²	—	43	4	no	—
211	C₂×C₄²	—	43	4	no	—
212	C₂×C₄²	—	86	4	no	—
213	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
214	#107	—	2	8	no	—
215	#107	—	86	8	no	—
216	C₄×C₈	—	43	8	no	—
217	C₄×C₈	—	43	4	no	—
218	C₄×C₈	C₄×D₄₃	1	4	yes	—
219	#95	—	86	4	no	—
220	#95	—	86	4	no	—
221	C₂⁴	—	43	1	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	43	1	no	—
224	C₂²×C₄	—	43	4	no	—
225	C₂²×C₄	—	43	4	no	—
226	C₂²×C₄	—	43	4	no	—
227	C₂²×C₄	—	43	1	no	—
228	C₂²×C₄	—	43	1	no	—
229	C₂²×C₄	—	43	4	no	—
230	C₂²×C₄	—	43	2	no	—
231	C₂²×C₄	—	43	2	no	—
232	C₂²×C₄	—	86	2	no	—
233	C₂²×C₄	—	86	2	no	—
234	C₂²×C₄	—	86	2	no	—
235	C₂²×C₄	—	86	2	no	—
236	C₂²×C₄	—	86	4	no	—
237	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
238	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
239	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
240	C₂×C₈	—	43	4	no	—
241	C₂×C₈	—	86	2	no	—
242	C₂×C₈	—	86	2	no	—
243	C₂×C₈	—	86	16	no	—
244	C₂×C₈	C₂×C₄×D₄₃	1	4	yes	—
245	C₂×C₈	#10621900	1	4	yes	—
246	C₂×C₈	#10621902	1	4	yes	—
247	C₄⋊C₄	—	2	8	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	86	8	no	—
251	C₄⋊C₄	—	172	4	no	—
252	C₄⋊C₄	—	172	4	no	—
253	C₂²⋊C₄	—	172	8	no	—
254	C₄²	—	43	4	no	—
255	C₄²	—	43	4	no	—
256	C₄²	—	43	2	no	—
257	C₄²	—	43	2	no	—
258	C₄²	—	172	8	no	—
259	C₄²	C₂×C₄×D₄₃	1	2	yes	—
260	C₄²	Q₈×D₄₃	1	2	yes	—
261	C₂³	—	43	2	no	—
262	C₂³	—	43	2	no	—
263	C₂³	—	43	2	no	—
264	C₂³	—	86	1	no	—
265	C₂³	—	86	1	no	—
266	C₂³	—	86	1	no	—
267	C₂³	—	86	1	no	—
268	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
269	C₂×C₄	—	4	4	no	—
270	C₂×C₄	—	4	4	no	—
271	C₂×C₄	—	4	4	no	—
272	C₂×C₄	—	43	2	no	—
273	C₂×C₄	—	43	2	no	—
274	C₂×C₄	—	43	2	no	—
275	C₂×C₄	—	43	16	no	—
276	C₂×C₄	—	43	2	no	—
277	C₂×C₄	—	43	2	no	—
278	C₂×C₄	—	86	1	no	—
279	C₂×C₄	—	86	1	no	—
280	C₂×C₄	—	86	1	no	—
281	C₂×C₄	—	86	1	no	—
282	C₂×C₄	—	86	4	no	—
283	C₂×C₄	—	86	8	no	—
284	C₂×C₄	—	86	4	no	—
285	C₂×C₄	—	172	4	no	—
286	C₂×C₄	—	172	4	no	—
287	C₂×C₄	—	172	4	no	—
288	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
289	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
290	C₂×C₄	D₈×D₄₃	1	2	yes	—
291	C₂×C₄	Q₁₆×D₄₃	1	2	yes	—
292	C₂×C₄	#13063344	1	2	yes	—
293	C₂×C₄	#13063348	1	2	yes	—
294	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
295	C₂×C₄	C₂×Q₈×D₄₃	1	1	yes	characteristic
296	C₂×C₄	C₂×#10621900	1	2	yes	—
297	C₂×C₄	C₂×#10621902	1	2	yes	—
298	C₈	—	86	4	no	—
299	C₈	#13063283	1	8	yes	complemented
300	C₂²	—	43	4	no	—
301	C₂²	—	86	2	no	—
302	C₂²	—	86	2	no	—
303	C₂²	—	86	2	no	—
304	C₂²	—	86	2	no	—
305	C₂²	—	86	2	no	—
306	C₂²	—	86	2	no	—
307	C₂²	C₂×C₄⋊C₄×D₄₃	1	1	yes	characteristic
308	C₂²	#14583447	1	1	yes	characteristic
309	C₂²	#14583450	1	1	yes	characteristic
310	C₂²	C₂×D₈×D₄₃	1	1	yes	characteristic
311	C₂²	C₂×Q₁₆×D₄₃	1	1	yes	characteristic
312	C₂²	C₂×#13063344	1	1	yes	characteristic
313	C₂²	C₂×#13063348	1	1	yes	characteristic
314	C₄	—	8	4	no	—
315	C₄	—	43	4	no	—
316	C₄	—	86	2	no	—
317	C₄	—	86	2	no	—
318	C₄	—	344	4	no	—
319	C₄	#107×D₄₃	1	4	yes	—
320	C₄	#14582936	1	4	yes	—
321	C₄	C₂×#13063283	1	2	yes	—
322	C₄	#14583454	1	2	yes	—
323	C₂	—	86	4	no	—
324	C₂	#19137000	1	1	yes	characteristic
325	C₂	C₂×#107×D₄₃	1	1	yes	characteristic
326	C₂	C₂×#14582936	1	1	yes	characteristic
327	C₂	#19145172	1	1	yes	characteristic
328	C₂	#19145175	1	1	yes	characteristic
329	C₂	#19145179	1	1	yes	characteristic
330	C₂	#19145182	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₄₃×#1845	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂³×D₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	#1845	D₄₃
hypocenter	C₄₃	#20817
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1845

Complements

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

N	H ≅ G/N	type
3. #19134755	C₂	semidirect
4. #19134722	C₂	semidirect
6. #107×Dic₄₃	C₂	semidirect
7. C₄₃×#1845	C₂	semidirect
14. C₄×C₈×D₄₃	C₄	semidirect
131. C₄×D₄₃	#107	semidirect
132. C₈×C₄₃	#117	semidirect
161. #1845	D₄₃	semidirect
190. C₄₃	#20817	semidirect
299. C₈	#13063283	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈×D₄₃	2
#19137000	2
#19134755	2
#19134722	2
#19134621	2
#107×Dic₄₃	2
C₄₃×#1845	2
#20817	43