Subgroups — #25005223

Order 11008.

730 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063260	C₂	1	2	yes	maximal
2	C₂×#14582923	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582663	C₂	1	2	yes	maximal
4	C₂×#14582654	C₂	1	1	yes	maximal, characteristic
5	C₄₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂³×C₄₃⋊C₈	C₂²	1	2	yes	—
8	C₂×#13063260	—	2	4	no	—
9	C₂×#13063260	—	2	4	no	—
10	C₂×#13063260	—	2	4	no	—
11	#14582923	—	2	4	no	—
12	C₂×C₄⋊(C₄₃⋊C₈)	—	2	2	no	—
13	#14582663	—	2	4	no	—
14	#14582654	C₄	1	2	yes	—
15	#14582654	C₄	1	2	yes	—
16	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
18	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
19	C₄₃×#336	—	2	2	no	—
20	C₂×#10621877	—	2	2	no	—
21	C₂×#10621877	—	2	2	no	—
22	C₂×#10621877	—	2	2	no	—
23	C₂×#10621877	—	2	2	no	—
24	C₂×#10621877	—	2	2	no	—
25	C₂×#10621877	D₄	1	4	yes	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	C₂²×C₄₃⋊C₈	—	2	2	no	—
28	C₂²×C₄₃⋊C₈	—	2	2	no	—
29	C₂²×C₄₃⋊C₈	—	2	2	no	—
30	C₂²×C₄₃⋊C₈	—	2	2	no	—
31	C₂²×C₄₃⋊C₈	C₂×C₄	1	4	yes	—
32	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
33	C₂²×C₄₃⋊C₈	Q₈	1	4	yes	—
34	#13063260	—	2	16	no	—
35	C₄⋊(C₄₃⋊C₈)	—	2	8	no	—
36	C₂³×C₄×C₄₃	Q₈	1	2	yes	—
37	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
38	C₂×M₁₆×C₄₃	—	2	1	no	—
39	C₂×M₁₆×C₄₃	—	2	1	no	—
40	C₂×M₁₆×C₄₃	—	2	2	no	—
41	C₂×M₁₆×C₄₃	—	2	2	no	—
42	C₂×M₁₆×C₄₃	—	2	2	no	—
43	C₂×M₁₆×C₄₃	—	2	2	no	—
44	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
45	C₂×M₁₆×C₄₃	Q₈	1	2	yes	—
46	C₂²×C₈×C₄₃	—	2	1	no	—
47	C₂²×C₈×C₄₃	—	2	1	no	—
48	C₂×C₄²×C₄₃	—	2	2	no	—
49	C₂×C₄²×C₄₃	—	2	2	no	—
50	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₄₃	D₄	1	2	yes	—
53	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	—	2	4	no	—
55	C₂²×C₄×C₄₃	—	2	4	no	—
56	C₂²×C₄×C₄₃	—	4	2	no	—
57	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₄⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
62	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
64	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
66	C₂²×C₄×C₄₃	#41	1	2	yes	—
67	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
69	M₁₆×C₄₃	—	2	4	no	—
70	M₁₆×C₄₃	—	2	4	no	—
71	M₁₆×C₄₃	—	4	2	no	—
72	M₁₆×C₄₃	—	4	2	no	—
73	C₂×C₈×C₄₃	—	2	2	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	C₄²×C₄₃	—	2	2	no	—
80	C₄²×C₄₃	—	2	2	no	—
81	C₄²×C₄₃	—	2	2	no	—
82	C₄²×C₄₃	—	2	2	no	—
83	#10621877	—	2	4	no	—
84	#10621877	—	2	4	no	—
85	#10621877	—	4	2	no	—
86	#10621877	—	4	2	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	2	4	no	—
90	C₂×C₄₃⋊C₈	—	2	4	no	—
91	C₂×C₄₃⋊C₈	—	2	4	no	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	—	2	4	no	—
94	C₂×C₄₃⋊C₈	—	4	2	no	—
95	C₂×C₄₃⋊C₈	—	4	2	no	—
96	C₂×C₄₃⋊C₈	C₄⋊C₄	1	8	yes	—
97	C₂³×C₄₃	—	2	2	no	—
98	C₂³×C₄₃	—	2	2	no	—
99	C₂³×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
101	C₂³×C₄₃	#125	1	1	yes	characteristic
102	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
103	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	2	no	—
107	C₂×C₄×C₄₃	—	2	2	no	—
108	C₂×C₄×C₄₃	—	2	2	no	—
109	C₂×C₄×C₄₃	—	4	8	no	—
110	C₂×C₄×C₄₃	#108	1	4	yes	complemented
111	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
112	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
113	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
116	C₂×C₄×C₄₃	#120	1	2	yes	—
117	C₂×C₄×C₄₃	#121	1	2	yes	—
118	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
120	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
121	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
122	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
123	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
124	C₂×C₄×C₄₃	#123	1	2	yes	—
125	C₂×C₄×C₄₃	#125	1	2	yes	—
126	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
127	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
129	C₈×C₄₃	—	4	2	no	—
130	C₈×C₄₃	—	4	2	no	—
131	C₄₃⋊C₈	—	2	8	no	—
132	C₄₃⋊C₈	—	4	4	no	—
133	#11326	—	43	1	no	maximal, Sylow
134	C₂²×C₄₃	—	2	2	no	—
135	C₂²×C₄₃	—	2	2	no	—
136	C₂²×C₄₃	—	2	4	no	—
137	C₂²×C₄₃	—	2	2	no	—
138	C₂²×C₄₃	—	2	2	no	—
139	C₂²×C₄₃	#384	1	1	yes	characteristic
140	C₂²×C₄₃	#385	1	1	yes	characteristic
141	C₂²×C₄₃	#389	1	1	yes	characteristic
142	C₂²×C₄₃	#395	1	2	yes	—
143	C₂²×C₄₃	#398	1	2	yes	—
144	C₂²×C₄₃	C₂×#108	1	2	yes	—
145	C₂²×C₄₃	#430	1	2	yes	—
146	C₄×C₄₃	—	2	4	no	—
147	C₄×C₄₃	—	4	2	no	—
148	C₄×C₄₃	#385	1	2	yes	—
149	C₄×C₄₃	#389	1	2	yes	—
150	C₄×C₄₃	#392	1	1	yes	characteristic
151	C₄×C₄₃	#395	1	1	yes	characteristic
152	C₄×C₄₃	#398	1	1	yes	characteristic
153	C₄×C₄₃	#399	1	1	yes	characteristic
154	C₂²×#108	—	43	2	no	—
155	C₂×#423	—	43	1	no	—
156	C₂×#341	—	43	2	no	—
157	#1728	D₄₃	1	1	yes	characteristic
158	C₂×#336	—	43	1	no	—
159	C₂×C₄₃	—	2	2	no	—
160	C₂×C₄₃	—	2	2	no	—
161	C₂×C₄₃	#1444	1	1	yes	characteristic
162	C₂×C₄₃	#1848	1	1	yes	characteristic
163	C₂×C₄₃	#1849	1	1	yes	characteristic
164	C₂×C₄₃	#1855	1	2	yes	—
165	C₂×C₄₃	#1950	1	2	yes	—
166	C₂²×M₁₆	—	43	2	no	—
167	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
168	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
169	C₂³×C₈	—	43	2	no	—
170	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
171	C₂×#108	—	86	4	no	—
172	C₂×#108	—	86	4	no	—
173	C₂×#108	—	86	4	no	—
174	#423	—	86	4	no	—
175	C₂×C₄⋊C₈	—	86	2	no	—
176	#341	—	86	4	no	—
177	#336	—	2	2	no	—
178	#336	—	43	2	no	—
179	#336	—	43	2	no	—
180	C₄₃	#11326	1	1	yes	characteristic, complemented, Sylow
181	C₂³×C₄	Dic₈₆	1	2	yes	—
182	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
183	C₂×M₁₆	—	2	1	no	—
184	C₂×M₁₆	—	2	1	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	43	4	no	—
190	C₂×M₁₆	—	86	2	no	—
191	C₂×M₁₆	—	86	2	no	—
192	C₂×M₁₆	—	86	2	no	—
193	C₂×M₁₆	—	86	2	no	—
194	C₂×M₁₆	—	86	2	no	—
195	C₂×M₁₆	Dic₈₆	1	2	yes	—
196	C₂×M₁₆	D₁₇₂	1	2	yes	—
197	C₂²×C₈	—	2	1	no	—
198	C₂²×C₈	—	2	1	no	—
199	C₂²×C₈	—	43	4	no	—
200	C₂²×C₈	—	43	4	no	—
201	C₂²×C₈	—	43	4	no	—
202	C₂²×C₈	—	86	2	no	—
203	C₂²×C₈	—	86	2	no	—
204	C₂²×C₈	—	86	2	no	—
205	C₂²×C₈	—	86	2	no	—
206	C₂²×C₈	—	86	2	no	—
207	C₂×C₄²	—	2	2	no	—
208	C₂×C₄²	—	2	2	no	—
209	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
210	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
211	C₂×C₄²	C₂²⋊D₄₃	1	2	yes	—
212	#108	—	86	16	no	—
213	C₄⋊C₈	—	86	8	no	—
214	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
215	C₂²×C₄	—	2	4	no	—
216	C₂²×C₄	—	2	4	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
219	C₂²×C₄	Dic₄₃⋊C₄	1	2	yes	—
220	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
223	C₂²×C₄	#10621898	1	1	yes	characteristic
224	C₂²×C₄	#10621898	1	1	yes	characteristic
225	C₂²×C₄	D₄×D₄₃	1	2	yes	—
226	C₂²×C₄	D₄×D₄₃	1	2	yes	—
227	C₂²×C₄	#10621900	1	2	yes	—
228	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
230	M₁₆	—	2	4	no	—
231	M₁₆	—	2	4	no	—
232	M₁₆	—	4	2	no	—
233	M₁₆	—	4	2	no	—
234	M₁₆	—	86	4	no	—
235	M₁₆	—	86	4	no	—
236	M₁₆	—	172	2	no	—
237	M₁₆	—	172	2	no	—
238	C₂×C₈	—	2	2	no	—
239	C₂×C₈	—	2	2	no	—
240	C₂×C₈	—	4	1	no	—
241	C₂×C₈	—	4	1	no	—
242	C₂×C₈	—	4	1	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	—	43	8	no	—
245	C₂×C₈	—	86	4	no	—
246	C₂×C₈	—	86	4	no	—
247	C₂×C₈	—	86	4	no	—
248	C₂×C₈	—	86	4	no	—
249	C₂×C₈	—	86	4	no	—
250	C₂×C₈	—	86	4	no	—
251	C₂×C₈	—	86	4	no	—
252	C₂×C₈	—	172	2	no	—
253	C₂×C₈	—	172	2	no	—
254	C₄²	—	2	2	no	—
255	C₄²	—	2	2	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₂³	—	2	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
261	C₂³	#13063302	1	2	yes	—
262	C₂³	#13063317	1	1	yes	characteristic
263	C₂³	#13063324	1	2	yes	—
264	C₂³	#13063325	1	1	yes	characteristic
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	4	8	no	—
271	C₂×C₄	#13063260	1	4	yes	complemented
272	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
273	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
274	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
275	C₂×C₄	C₄×Dic₈₆	1	1	yes	characteristic
276	C₂×C₄	#13063293	1	1	yes	characteristic
277	C₂×C₄	#13063295	1	2	yes	—
278	C₂×C₄	#13063298	1	2	yes	—
279	C₂×C₄	#13063299	1	1	yes	characteristic
280	C₂×C₄	#13063302	1	2	yes	—
281	C₂×C₄	#13063303	1	2	yes	—
282	C₂×C₄	#13063304	1	1	yes	characteristic
283	C₂×C₄	#13063305	1	2	yes	—
284	C₂×C₄	#13063306	1	1	yes	characteristic
285	C₂×C₄	#13063307	1	2	yes	—
286	C₂×C₄	#13063315	1	2	yes	—
287	C₂×C₄	#13063321	1	1	yes	characteristic
288	C₂×C₄	#13063323	1	1	yes	characteristic
289	C₂×C₄	#13063324	1	2	yes	—
290	C₈	—	4	2	no	—
291	C₈	—	4	2	no	—
292	C₈	—	86	8	no	—
293	C₈	—	172	4	no	—
294	C₂²	—	2	2	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	2	2	no	—
299	C₂²	#14582768	1	1	yes	characteristic
300	C₂²	#14582771	1	1	yes	characteristic
301	C₂²	#14582797	1	1	yes	characteristic
302	C₂²	#14582820	1	2	yes	—
303	C₂²	#14582833	1	2	yes	—
304	C₂²	C₂×#13063260	1	2	yes	—
305	C₂²	#14582947	1	2	yes	—
306	C₄	—	2	4	no	—
307	C₄	—	4	2	no	—
308	C₄	#14582773	1	2	yes	—
309	C₄	#14582796	1	2	yes	—
310	C₄	#14582808	1	1	yes	characteristic
311	C₄	#14582819	1	1	yes	characteristic
312	C₄	#14582829	1	1	yes	characteristic
313	C₄	#14582835	1	1	yes	characteristic
314	C₂	—	2	2	no	—
315	C₂	—	2	2	no	—
316	C₂	#19132859	1	1	yes	characteristic
317	C₂	#19134766	1	1	yes	characteristic
318	C₂	#19134769	1	1	yes	characteristic
319	C₂	#19134799	1	2	yes	—
320	C₂	#19135370	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₃
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1728	C₂
socle	C₂³×C₄₃	C₂×C₄⋊C₄
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1728	D₄₃
hypocenter	C₄₃	#11326
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ #1728

Complements

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

N	H ≅ G/N	type
110. C₂×C₄×C₄₃	#108	semidirect
180. C₄₃	#11326	semidirect
271. C₂×C₄	#13063260	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063260	2
C₂×#14582923	2
C₂×#14582663	2
C₂×#14582654	2
C₄₃×#1728	2
#11326	43