Subgroups — #25005282

Order 11008.

702 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582948	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582947	C₂	1	2	yes	maximal
3	C₂×C₄₃⋊#338	C₂	1	1	yes	maximal, characteristic
4	#19134289	C₂	1	2	yes	maximal
5	C₄₃×#1728	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂²×#10621877	C₂²	1	2	yes	—
8	#14582948	—	2	4	no	—
9	#14582947	—	2	4	no	—
10	#14582947	—	2	4	no	—
11	C₂×#13063261	—	2	2	no	—
12	C₂×#13063260	—	2	4	no	—
13	C₄₃⋊#338	C₄	1	4	yes	complemented
14	#14582654	—	2	4	no	—
15	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
17	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
18	C₄₃×#336	—	2	2	no	—
19	C₂×#10621877	—	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	—	2	2	no	—
26	C₂×#10621877	—	2	2	no	—
27	C₂×#10621877	—	2	2	no	—
28	C₂×#10621877	C₂×C₄	1	4	yes	—
29	C₂×#10621877	D₄	1	4	yes	—
30	C₂×#10621877	Q₈	1	4	yes	—
31	C₂²×C₄₃⋊C₈	—	2	2	no	—
32	C₂²×C₄₃⋊C₈	D₄	1	4	yes	—
33	#13063261	—	2	8	no	—
34	#13063260	—	2	16	no	—
35	C₂³×C₄×C₄₃	Q₈	1	2	yes	—
36	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
37	C₂×M₁₆×C₄₃	—	2	2	no	—
38	C₂×M₁₆×C₄₃	—	2	2	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	C₂×C₄	1	2	yes	—
43	C₂×M₁₆×C₄₃	D₄	1	2	yes	—
44	C₂×M₁₆×C₄₃	Q₈	1	2	yes	—
45	C₂²×C₈×C₄₃	—	2	1	no	—
46	C₂²×C₈×C₄₃	D₄	1	2	yes	—
47	C₂×C₄²×C₄₃	—	2	1	no	—
48	C₂×C₄²×C₄₃	—	2	2	no	—
49	C₂×C₄²×C₄₃	—	2	1	no	—
50	C₂×C₄²×C₄₃	—	2	2	no	—
51	C₂⁴×C₄₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	2	4	no	—
53	C₂²×C₄×C₄₃	—	2	2	no	—
54	C₂²×C₄×C₄₃	—	4	4	no	—
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
57	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
60	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
61	C₂²×C₄×C₄₃	C₂×Q₈	1	2	yes	—
62	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
63	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₄₃	#41	1	2	yes	—
65	M₁₆×C₄₃	—	2	4	no	—
66	M₁₆×C₄₃	—	2	4	no	—
67	M₁₆×C₄₃	—	2	4	no	—
68	M₁₆×C₄₃	—	4	2	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	—	2	2	no	—
72	C₂×C₈×C₄₃	—	4	1	no	—
73	C₂×C₈×C₄₃	—	4	1	no	—
74	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
75	C₄²×C₄₃	—	4	1	no	—
76	C₄²×C₄₃	—	4	2	no	—
77	C₄²×C₄₃	—	4	1	no	—
78	#10621877	—	2	4	no	—
79	#10621877	—	2	4	no	—
80	#10621877	—	2	4	no	—
81	#10621877	—	2	4	no	—
82	#10621877	—	2	4	no	—
83	#10621877	—	2	4	no	—
84	#10621877	—	4	2	no	—
85	#10621877	—	4	2	no	—
86	C₂×C₄₃⋊C₈	—	2	4	no	—
87	C₂×C₄₃⋊C₈	—	2	4	no	—
88	C₂×C₄₃⋊C₈	—	2	4	no	—
89	C₂×C₄₃⋊C₈	—	4	2	no	—
90	C₂×C₄₃⋊C₈	—	4	2	no	—
91	C₂×C₄₃⋊C₈	C₄⋊C₄	1	8	yes	—
92	C₂³×C₄₃	—	4	2	no	—
93	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
94	C₂³×C₄₃	C₂²⋊Q₈	1	2	yes	—
95	C₂³×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
96	C₂³×C₄₃	C₄⋊Q₈	1	2	yes	—
97	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
98	C₂×C₄×C₄₃	—	2	8	no	—
99	C₂×C₄×C₄₃	—	2	4	no	—
100	C₂×C₄×C₄₃	—	4	4	no	—
101	C₂×C₄×C₄₃	—	4	2	no	—
102	C₂×C₄×C₄₃	—	4	2	no	—
103	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	2	yes	—
104	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#120	1	2	yes	—
109	C₂×C₄×C₄₃	#121	1	2	yes	—
110	C₂×C₄×C₄₃	#121	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
112	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
113	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
114	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#123	1	2	yes	—
116	C₂×C₄×C₄₃	#125	1	2	yes	—
117	C₂×C₄×C₄₃	#125	1	1	yes	characteristic
118	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
119	C₂×C₄×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
120	C₂×C₄×C₄₃	C₄⋊Q₈	1	2	yes	—
121	C₈×C₄₃	—	2	4	no	—
122	C₈×C₄₃	—	4	2	no	—
123	C₄₃⋊C₈	—	2	8	no	—
124	C₄₃⋊C₈	—	4	4	no	—
125	#11338	—	43	1	no	maximal, Sylow
126	C₂²×C₄₃	—	2	4	no	—
127	C₂²×C₄₃	—	2	2	no	—
128	C₂²×C₄₃	—	4	1	no	—
129	C₂²×C₄₃	—	4	2	no	—
130	C₂²×C₄₃	—	4	1	no	—
131	C₂²×C₄₃	#389	1	2	yes	—
132	C₂²×C₄₃	#389	1	1	yes	characteristic
133	C₂²×C₄₃	#395	1	1	yes	characteristic
134	C₂²×C₄₃	#395	1	2	yes	—
135	C₂²×C₄₃	#399	1	1	yes	characteristic
136	C₄×C₄₃	—	4	4	no	—
137	C₄×C₄₃	#384	1	1	yes	characteristic
138	C₄×C₄₃	#385	1	1	yes	characteristic
139	C₄×C₄₃	#385	1	2	yes	—
140	C₄×C₄₃	#392	1	1	yes	characteristic
141	C₄×C₄₃	#398	1	1	yes	characteristic
142	C₄×C₄₃	#398	1	2	yes	—
143	C₂×#430	—	43	2	no	—
144	C₂×#430	—	43	1	no	—
145	C₂×#338	—	43	1	no	—
146	#1728	—	43	2	no	—
147	#1728	D₄₃	1	1	yes	characteristic
148	C₂×C₄₃	—	2	2	no	—
149	C₂×C₄₃	—	2	1	no	—
150	C₂×C₄₃	—	2	1	no	—
151	C₂×C₄₃	—	2	2	no	—
152	C₂×C₄₃	#1444	1	1	yes	characteristic
153	C₂×C₄₃	#2091	1	2	yes	—
154	C₂²×M₁₆	—	43	2	no	—
155	C₂²×M₁₆	—	43	2	no	—
156	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
157	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
158	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
159	#430	—	86	4	no	—
160	#430	—	86	4	no	—
161	#430	—	86	4	no	—
162	C₂×#108	—	86	4	no	—
163	C₂×#108	—	86	2	no	—
164	#338	—	43	4	no	—
165	#336	—	2	2	no	—
166	#336	—	86	4	no	—
167	C₄₃	#11338	1	1	yes	characteristic, complemented, Sylow
168	C₂³×C₄	Dic₈₆	1	2	yes	—
169	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	2	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	1	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	43	4	no	—
176	C₂×M₁₆	—	43	4	no	—
177	C₂×M₁₆	—	43	4	no	—
178	C₂×M₁₆	—	86	2	no	—
179	C₂×M₁₆	—	86	2	no	—
180	C₂×M₁₆	—	86	2	no	—
181	C₂×M₁₆	—	86	2	no	—
182	C₂×M₁₆	—	86	2	no	—
183	C₂×M₁₆	—	86	2	no	—
184	C₂×M₁₆	—	86	2	no	—
185	C₂×M₁₆	—	86	2	no	—
186	C₂×M₁₆	—	86	2	no	—
187	C₂×M₁₆	Dic₈₆	1	2	yes	—
188	C₂×M₁₆	C₄×D₄₃	1	2	yes	—
189	C₂×M₁₆	D₁₇₂	1	2	yes	—
190	C₂²×C₈	—	2	1	no	—
191	C₂²×C₈	—	43	4	no	—
192	C₂²×C₈	—	86	2	no	—
193	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
194	C₂×C₄²	—	2	1	no	—
195	C₂×C₄²	—	2	2	no	—
196	C₂×C₄²	—	2	2	no	—
197	C₂×C₄²	—	2	1	no	—
198	#108	—	86	8	no	—
199	#108	—	86	16	no	—
200	C₂⁴	C₂×Dic₈₆	1	1	yes	characteristic
201	C₂²×C₄	—	2	4	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	—	4	4	no	—
204	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
205	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
207	C₂²×C₄	#10621898	1	1	yes	characteristic
208	C₂²×C₄	#10621898	1	1	yes	characteristic
209	C₂²×C₄	D₄×D₄₃	1	2	yes	—
210	C₂²×C₄	D₄×D₄₃	1	2	yes	—
211	C₂²×C₄	#10621900	1	2	yes	—
212	C₂²×C₄	Q₈×D₄₃	1	2	yes	—
213	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
214	M₁₆	—	2	4	no	—
215	M₁₆	—	2	4	no	—
216	M₁₆	—	2	4	no	—
217	M₁₆	—	4	2	no	—
218	M₁₆	—	86	4	no	—
219	M₁₆	—	86	4	no	—
220	M₁₆	—	86	4	no	—
221	M₁₆	—	86	4	no	—
222	M₁₆	—	86	4	no	—
223	M₁₆	—	86	4	no	—
224	M₁₆	—	172	2	no	—
225	M₁₆	—	172	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	4	1	no	—
230	C₂×C₈	—	4	1	no	—
231	C₂×C₈	—	43	8	no	—
232	C₂×C₈	—	86	4	no	—
233	C₂×C₈	—	86	4	no	—
234	C₂×C₈	—	86	4	no	—
235	C₂×C₈	—	172	2	no	—
236	C₂×C₈	—	172	2	no	—
237	C₂×C₈	Dic₄₃⋊C₄	1	4	yes	—
238	C₄²	—	4	1	no	—
239	C₄²	—	4	2	no	—
240	C₄²	—	4	1	no	—
241	C₂³	—	4	2	no	—
242	C₂³	C₄×Dic₈₆	1	1	yes	characteristic
243	C₂³	#13063302	1	2	yes	—
244	C₂³	#13063304	1	1	yes	characteristic
245	C₂³	#13063324	1	2	yes	—
246	C₂³	#13063325	1	1	yes	characteristic
247	C₂×C₄	—	2	8	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	4	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	C₄⋊C₄×D₄₃	1	2	yes	—
253	C₂×C₄	C₂×Dic₄₃⋊C₄	1	1	yes	characteristic
254	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
255	C₂×C₄	C₄×D₁₇₂	1	1	yes	characteristic
256	C₂×C₄	#13063293	1	1	yes	characteristic
257	C₂×C₄	#13063295	1	2	yes	—
258	C₂×C₄	#13063298	1	2	yes	—
259	C₂×C₄	#13063299	1	1	yes	characteristic
260	C₂×C₄	#13063302	1	2	yes	—
261	C₂×C₄	#13063303	1	2	yes	—
262	C₂×C₄	#13063305	1	2	yes	—
263	C₂×C₄	#13063306	1	1	yes	characteristic
264	C₂×C₄	#13063307	1	2	yes	—
265	C₂×C₄	#13063315	1	2	yes	—
266	C₂×C₄	#13063317	1	1	yes	characteristic
267	C₂×C₄	#13063321	1	1	yes	characteristic
268	C₂×C₄	#13063323	1	1	yes	characteristic
269	C₂×C₄	#13063324	1	2	yes	—
270	C₈	—	2	4	no	—
271	C₈	—	4	2	no	—
272	C₈	—	86	8	no	—
273	C₈	—	172	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	1	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	1	no	—
279	C₂²	#14582796	1	2	yes	—
280	C₂²	#14582797	1	1	yes	characteristic
281	C₂²	#14582819	1	1	yes	characteristic
282	C₂²	#14582820	1	2	yes	—
283	C₂²	#14582835	1	1	yes	characteristic
284	C₄	—	4	4	no	—
285	C₄	#14582768	1	1	yes	characteristic
286	C₄	#14582771	1	1	yes	characteristic
287	C₄	#14582773	1	2	yes	—
288	C₄	#14582808	1	1	yes	characteristic
289	C₄	#14582829	1	1	yes	characteristic
290	C₄	#14582833	1	2	yes	—
291	C₂	—	2	2	no	—
292	C₂	—	2	1	no	—
293	C₂	—	2	1	no	—
294	C₂	—	2	2	no	—
295	C₂	#19132859	1	1	yes	characteristic
296	C₂	#19136111	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063321
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1728	C₂
socle	C₂²×C₄₃	#399
prime radical	C₂⁴×C₄₃	C₂×Q₈
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1728	D₄₃
hypocenter	C₄₃	#11338
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #1728

Complements

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

N	H ≅ G/N	type
13. C₄₃⋊#338	C₄	semidirect
167. C₄₃	#11338	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582948	2
C₂×#14582947	2
C₂×C₄₃⋊#338	2
#19134289	2
C₄₃×#1728	2
#11338	43