Subgroups — #25249771

Order 11008.

710 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	C₂×#14583316	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₆×D₄₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#19143287	C₂	1	1	yes	maximal, characteristic
4	#19136423	C₂	1	4	yes	maximal, complemented
5	(C₂²⋊D₄₃)⋊C₁₆	C₂	1	4	yes	maximal, complemented
6	#19136353	C₂	1	1	yes	maximal, characteristic, complemented
7	#19136205	C₂	1	1	yes	maximal, characteristic, complemented
8	#19136194	C₂	1	1	yes	maximal, characteristic, complemented
9	C₄₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₈×D₄₃	C₂²	1	1	yes	characteristic
11	#14583406	C₄	1	1	yes	characteristic
12	#14583316	—	2	4	no	—
13	C₂×#13063265	C₂²	1	1	yes	characteristic
14	C₂×#13063265	C₂²	1	2	yes	complemented
15	C₂²×C₄₃⋊C₁₆	C₂²	1	1	yes	characteristic
16	C₂²×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
17	C₂×C₁₆×D₄₃	—	2	4	no	—
18	C₂×C₁₆×D₄₃	C₂²	1	2	yes	—
19	C₂×C₁₆×D₄₃	C₂²	1	2	yes	—
20	#14582968	C₄	1	4	yes	—
21	C₈×C₂²⋊D₄₃	C₂²	1	4	yes	—
22	#14582927	C₄	1	1	yes	characteristic
23	#14582866	C₄	1	1	yes	characteristic
24	#14582862	C₄	1	1	yes	characteristic
25	#14582854	C₂²	1	1	yes	characteristic
26	Dic₄₃⋊C₁₆	C₂²	1	4	yes	complemented
27	(C₂×D₄₃)⋊C₁₆	C₂²	1	4	yes	—
28	#14582674	C₂²	1	2	yes	complemented
29	C₁₆×Dic₄₃	C₂²	1	2	yes	complemented
30	C₂×#131×C₄₃	C₂²	1	1	yes	characteristic
31	C₂×M₃₂×C₄₃	C₂²	1	1	yes	characteristic
32	C₂²×C₁₆×C₄₃	C₂²	1	1	yes	characteristic
33	C₄₃×C₂²⋊C₁₆	C₂²	1	4	yes	complemented
34	C₂²×C₄×D₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×#10621877	C₂×C₄	1	1	yes	characteristic
36	C₂²×C₄₃⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₈×D₄₃	—	2	2	no	—
38	C₂×C₈×D₄₃	—	2	2	no	—
39	C₂×C₈×D₄₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₄₃	C₂³	1	2	yes	—
41	#13063306	C₈	1	2	yes	—
42	#13063301	C₈	1	2	yes	—
43	#13063296	C₈	1	2	yes	—
44	#13063293	C₈	1	2	yes	—
45	C₄×C₂²⋊D₄₃	C₂×C₄	1	4	yes	—
46	#13063284	C₂×C₄	1	1	yes	characteristic
47	#13063265	—	2	2	no	—
48	#13063265	—	2	4	no	—
49	C₂×C₄₃⋊C₁₆	—	2	4	no	—
50	C₂×C₄₃⋊C₁₆	D₄	1	4	yes	complemented
51	C₂×C₄₃⋊C₁₆	C₂³	1	2	yes	—
52	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
53	C₂×C₄₃⋊C₁₆	C₂³	1	1	yes	characteristic
54	C₁₆×D₄₃	—	2	4	no	—
55	C₁₆×D₄₃	—	2	4	no	—
56	Dic₄₃⋊C₈	C₂×C₄	1	4	yes	—
57	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	4	yes	—
58	(C₂×D₄₃)⋊C₈	C₂×C₄	1	4	yes	—
59	#13063235	C₂×C₄	1	2	yes	—
60	C₈×Dic₄₃	C₂³	1	2	yes	—
61	C₂×#41×C₄₃	C₂×C₄	1	1	yes	characteristic
62	#131×C₄₃	—	2	2	no	—
63	#131×C₄₃	—	2	2	no	—
64	C₂×M₁₆×C₄₃	C₂×C₄	1	1	yes	characteristic
65	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
66	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
67	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	M₃₂×C₄₃	—	2	2	no	—
69	C₂×C₁₆×C₄₃	—	2	2	no	—
70	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
71	C₂×C₁₆×C₄₃	C₂³	1	2	yes	—
72	C₂³×D₄₃	C₂×C₈	1	1	yes	characteristic
73	#41×C₄₃	—	2	2	no	—
74	#41×C₄₃	—	2	2	no	—
75	C₂×Q₈×C₄₃	C₂×C₈	1	1	yes	characteristic
76	C₂×D₄×C₄₃	C₂×C₈	1	2	yes	—
77	C₂×D₄×C₄₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
79	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₂²⋊D₄₃	C₂×C₈	1	4	yes	—
81	C₂²×Dic₄₃	C₂×C₈	1	1	yes	characteristic
82	C₂×C₄×D₄₃	—	2	2	no	—
83	C₂×C₄×D₄₃	—	2	2	no	—
84	C₂×C₄×D₄₃	C₂×C₈	1	2	yes	—
85	C₂×C₄×D₄₃	C₂²×C₄	1	2	yes	—
86	M₁₆×C₄₃	—	2	2	no	—
87	M₁₆×C₄₃	—	2	4	no	—
88	C₂×C₈×C₄₃	—	2	2	no	—
89	C₂×C₈×C₄₃	—	2	2	no	—
90	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
91	C₂×C₈×C₄₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₄₃	#41	1	2	yes	—
94	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
95	C₂²⋊Dic₄₃	C₂×C₈	1	2	yes	—
96	C₂²⋊Dic₄₃	C₂×C₈	1	4	yes	—
97	(C₂×D₄₃)⋊C₄	C₂×C₈	1	4	yes	—
98	C₄⋊Dic₄₃	C₂×C₈	1	2	yes	—
99	Dic₄₃⋊C₄	C₂×C₈	1	4	yes	—
100	C₄×Dic₄₃	C₂²×C₄	1	2	yes	—
101	#10621877	—	2	2	no	—
102	C₂×C₄₃⋊C₈	—	2	2	no	—
103	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
104	C₂×C₄₃⋊C₈	C₂²×C₄	1	2	yes	—
105	C₈×D₄₃	—	2	4	no	—
106	C₈×D₄₃	—	2	2	no	—
107	C₈×D₄₃	—	2	2	no	—
108	C₁₆×C₄₃	—	2	2	no	—
109	C₁₆×C₄₃	—	2	2	no	—
110	C₄₃⋊C₁₆	—	2	2	no	—
111	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
112	C₄₃⋊C₁₆	C₂×D₄	1	2	yes	—
113	C₂³×C₄₃	C₂²×C₈	1	2	yes	—
114	C₂³×C₄₃	C₂²×C₈	1	1	yes	characteristic
115	C₂²×D₄₃	—	2	2	no	—
116	C₂²×D₄₃	—	2	4	no	—
117	C₂²×D₄₃	C₂²×C₈	1	2	yes	—
118	Q₈×C₄₃	—	2	2	no	—
119	D₄×C₄₃	—	2	4	no	—
120	D₄×C₄₃	—	2	2	no	—
121	C₂×C₄×C₄₃	—	2	2	no	—
122	C₂×C₄×C₄₃	—	2	2	no	—
123	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₄₃	C₂²×C₈	1	2	yes	—
125	C₂×C₄×C₄₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₄₃	#131	1	2	yes	—
127	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
128	C₂²⋊D₄₃	—	2	8	no	—
129	C₂×Dic₄₃	—	2	2	no	—
130	C₂×Dic₄₃	C₂²×C₈	1	4	yes	—
131	C₂×Dic₄₃	C₂²×C₈	1	2	yes	—
132	C₄×D₄₃	—	2	2	no	—
133	C₄×D₄₃	—	2	2	no	—
134	C₄×D₄₃	—	2	4	no	—
135	C₈×C₄₃	—	2	2	no	—
136	C₈×C₄₃	C₄×D₄	1	2	yes	—
137	C₈×C₄₃	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₄₃	C₂×#41	1	1	yes	characteristic
139	C₄₃⋊C₈	—	2	2	no	—
140	C₄₃⋊C₈	—	2	2	no	—
141	#33432	—	43	1	no	maximal, Sylow
142	C₂²×C₄₃	—	2	2	no	—
143	C₂²×C₄₃	—	2	4	no	—
144	C₂²×C₄₃	—	2	2	no	—
145	C₂²×C₄₃	C₈×D₄	1	2	yes	—
146	C₂²×C₄₃	C₂³×C₈	1	1	yes	characteristic
147	C₂×D₄₃	—	2	8	no	—
148	C₂×D₄₃	—	2	2	no	—
149	C₂×D₄₃	—	2	2	no	—
150	C₂×D₄₃	#504	1	4	yes	complemented
151	C₄×C₄₃	—	2	2	no	—
152	C₄×C₄₃	C₈×D₄	1	2	yes	—
153	C₄×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
154	C₄×C₄₃	C₂×#131	1	1	yes	characteristic
155	Dic₄₃	—	2	4	no	—
156	Dic₄₃	—	2	2	no	—
157	C₂×#504	—	43	1	no	—
158	C₂³×C₁₆	—	43	1	no	—
159	#2929	—	43	1	no	—
160	#2168	—	43	4	no	—
161	D₄×C₁₆	—	43	4	no	—
162	#2151	—	43	1	no	—
163	#2113	D₄₃	1	1	yes	characteristic, complemented
164	#2112	—	43	1	no	—
165	#2108	—	43	1	no	—
166	C₂×C₄₃	—	2	2	no	—
167	C₂×C₄₃	—	4	2	no	—
168	C₂×C₄₃	C₂×C₈×D₄	1	1	yes	characteristic
169	C₂×C₄₃	C₂×#504	1	2	yes	—
170	D₄₃	—	2	4	no	—
171	C₂×#131	C₂×D₄₃	1	1	yes	characteristic
172	C₂³×C₈	—	43	1	no	—
173	#525	—	43	1	no	—
174	#504	—	86	4	no	—
175	C₂×M₃₂	—	43	1	no	—
176	C₂×M₃₂	—	43	2	no	—
177	C₂×M₃₂	C₂×D₄₃	1	1	yes	characteristic
178	C₂²×C₁₆	—	43	1	no	—
179	C₂²×C₁₆	—	43	2	no	—
180	C₂²×C₁₆	—	43	2	no	—
181	C₂²×C₁₆	—	43	2	no	—
182	C₂²×C₁₆	—	86	4	no	—
183	C₂²×C₁₆	C₂×D₄₃	1	1	yes	characteristic
184	#435	—	43	4	no	—
185	C₈×D₄	—	43	4	no	—
186	#424	—	43	1	no	—
187	#408	—	43	1	no	—
188	#407	—	43	1	no	—
189	#405	—	43	1	no	—
190	C₄⋊C₁₆	—	43	4	no	—
191	C₂²⋊C₁₆	—	43	4	no	—
192	C₂²⋊C₁₆	C₂×D₄₃	1	4	yes	complemented
193	#346	—	43	2	no	—
194	C₄×C₁₆	—	43	2	no	—
195	C₄₃	#33432	1	1	yes	characteristic, complemented, Sylow
196	C₂×#41	C₄×D₄₃	1	1	yes	characteristic
197	C₂³×C₄	—	43	1	no	—
198	#131	—	2	2	no	—
199	#131	—	2	2	no	—
200	C₂×M₁₆	—	43	1	no	—
201	C₂×M₁₆	C₄×D₄₃	1	1	yes	characteristic
202	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
203	C₂²×C₈	—	43	1	no	—
204	C₂²×C₈	—	43	2	no	—
205	C₂²×C₈	—	43	2	no	—
206	C₂²×C₈	—	86	2	no	—
207	C₂²×C₈	—	86	2	no	—
208	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
209	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
210	#123	—	43	2	no	—
211	C₂²⋊Q₈	—	43	2	no	—
212	#121	—	43	2	no	—
213	#120	—	43	2	no	—
214	C₄×D₄	—	43	4	no	—
215	#117	—	43	1	no	—
216	M₃₂	—	2	2	no	—
217	M₃₂	—	86	4	no	—
218	M₃₂	—	86	2	no	—
219	C₂×C₁₆	—	2	2	no	—
220	C₂×C₁₆	—	43	4	no	—
221	C₂×C₁₆	—	43	1	no	—
222	C₂×C₁₆	—	43	1	no	—
223	C₂×C₁₆	—	43	2	no	—
224	C₂×C₁₆	—	86	4	no	—
225	C₂×C₁₆	—	86	4	no	—
226	C₂×C₁₆	—	86	4	no	—
227	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
228	C₂×C₁₆	C₂²×D₄₃	1	2	yes	—
229	C₄⋊C₈	—	43	4	no	—
230	C₂²⋊C₈	—	43	4	no	—
231	C₂²⋊C₈	—	43	4	no	—
232	#97	—	43	2	no	—
233	C₄×C₈	—	43	2	no	—
234	C₂⁴	—	43	1	no	—
235	#41	—	2	2	no	—
236	#41	—	2	2	no	—
237	C₂×Q₈	C₈×D₄₃	1	1	yes	characteristic
238	C₂×D₄	—	43	4	no	—
239	C₂×D₄	C₈×D₄₃	1	2	yes	—
240	C₂×D₄	C₈×D₄₃	1	1	yes	characteristic
241	C₂²×C₄	—	43	1	no	—
242	C₂²×C₄	—	43	2	no	—
243	C₂²×C₄	—	43	2	no	—
244	C₂²×C₄	—	86	2	no	—
245	C₂²×C₄	—	86	2	no	—
246	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
247	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
248	M₁₆	—	2	2	no	—
249	M₁₆	—	2	4	no	—
250	M₁₆	—	86	2	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	43	2	no	—
254	C₂×C₈	—	43	2	no	—
255	C₂×C₈	—	86	2	no	—
256	C₂×C₈	—	86	4	no	—
257	C₂×C₈	—	86	2	no	—
258	C₂×C₈	—	86	2	no	—
259	C₂×C₈	C₂×C₄×D₄₃	1	2	yes	—
260	C₂×C₈	C₂×C₄×D₄₃	1	1	yes	characteristic
261	C₂×C₈	D₄×D₄₃	1	2	yes	—
262	C₂×C₈	#10621900	1	2	yes	—
263	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
264	C₄⋊C₄	—	43	2	no	—
265	C₄⋊C₄	—	43	4	no	—
266	C₂²⋊C₄	—	43	2	no	—
267	C₂²⋊C₄	—	43	4	no	—
268	C₂²⋊C₄	—	43	4	no	—
269	C₄²	—	43	2	no	—
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	2	no	—
272	C₁₆	—	43	2	no	—
273	C₁₆	—	43	2	no	—
274	C₁₆	—	86	2	no	—
275	C₂³	—	43	2	no	—
276	C₂³	—	86	4	no	—
277	C₂³	—	86	2	no	—
278	C₂³	C₂×C₈×D₄₃	1	2	yes	—
279	C₂³	C₂×C₈×D₄₃	1	1	yes	characteristic
280	Q₈	—	2	2	no	—
281	D₄	—	2	4	no	—
282	D₄	—	2	2	no	—
283	D₄	—	86	8	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	43	4	no	—
287	C₂×C₄	—	43	2	no	—
288	C₂×C₄	—	86	2	no	—
289	C₂×C₄	—	86	2	no	—
290	C₂×C₄	—	86	2	no	—
291	C₂×C₄	—	86	4	no	—
292	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
293	C₂×C₄	C₂×C₈×D₄₃	1	2	yes	—
294	C₂×C₄	C₂×C₈×D₄₃	1	1	yes	characteristic
295	C₂×C₄	#13063333	1	2	yes	—
296	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
297	C₈	—	2	2	no	—
298	C₈	—	86	2	no	—
299	C₈	—	86	2	no	—
300	C₈	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
301	C₈	C₂×D₄×D₄₃	1	1	yes	characteristic
302	C₈	C₂×#10621900	1	1	yes	characteristic
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	4	no	—
306	C₂²	—	43	4	no	—
307	C₂²	—	86	8	no	—
308	C₂²	—	86	2	no	—
309	C₂²	—	86	2	no	—
310	C₂²	(C₂²⋊D₄₃)⋊C₈	1	2	yes	—
311	C₂²	C₂²×C₈×D₄₃	1	1	yes	characteristic
312	C₄	—	2	2	no	—
313	C₄	—	86	2	no	—
314	C₄	—	86	4	no	—
315	C₄	(C₂²⋊D₄₃)⋊C₈	1	2	yes	—
316	C₄	C₂×(C₂²⋊D₄₃)⋊C₄	1	1	yes	characteristic
317	C₄	C₂×#13063333	1	1	yes	characteristic
318	C₂	—	2	2	no	—
319	C₂	—	4	2	no	—
320	C₂	—	86	4	no	—
321	C₂	C₂×(C₂²⋊D₄₃)⋊C₈	1	1	yes	characteristic
322	C₂	#504×D₄₃	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂³×D₄₃
commutator, socle	C₂²×C₄₃	C₂³×C₈
Fitting	C₄₃×#2113	C₂
prime radical	#14583406	C₄
cosocle	C₂×C₈×C₄₃	C₂⁴
hypercenter	#2113	D₄₃
hypocenter	C₄₃	#33432
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₈ ◁ #2113

Complements

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

N	H ≅ G/N	type
1. C₂×#14583316	C₂	semidirect
2. C₂²×C₁₆×D₄₃	C₂	semidirect
4. #19136423	C₂	semidirect
5. (C₂²⋊D₄₃)⋊C₁₆	C₂	semidirect
6. #19136353	C₂	semidirect
7. #19136205	C₂	semidirect
8. #19136194	C₂	semidirect
9. C₄₃×#2113	C₂	semidirect
14. C₂×#13063265	C₂²	semidirect
16. C₂²×C₄₃⋊C₁₆	C₂²	semidirect
26. Dic₄₃⋊C₁₆	C₂²	semidirect
28. #14582674	C₂²	semidirect
29. C₁₆×Dic₄₃	C₂²	semidirect
33. C₄₃×C₂²⋊C₁₆	C₂²	semidirect
50. C₂×C₄₃⋊C₁₆	D₄	semidirect
150. C₂×D₄₃	#504	semidirect
163. #2113	D₄₃	semidirect
192. C₂²⋊C₁₆	C₂×D₄₃	semidirect
195. C₄₃	#33432	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583316	2
C₂²×C₁₆×D₄₃	2
#19143287	2
#19136423	2
(C₂²⋊D₄₃)⋊C₁₆	2
#19136353	2
#19136205	2
#19136194	2
C₄₃×#2113	2
#33432	43