Subgroups — #25002318

Order 11008.

694 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13063235	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582664	C₂	1	1	yes	maximal, characteristic
3	C₂×#14582658	C₂	1	2	yes	maximal
4	#19134292	C₂	1	2	yes	maximal
5	C₄₃×#1735	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
9	C₂×#13063235	—	2	4	no	—
10	C₂×#13063235	C₄	1	4	yes	complemented
11	#14582664	C₄	1	4	yes	complemented
12	#14582658	—	2	4	no	—
13	#14582654	—	2	4	no	—
14	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₃	C₂²	1	2	yes	—
16	C₄₃×#340	C₄	1	4	yes	complemented
17	C₂³×Dic₄₃	—	2	1	no	—
18	C₂×#10621877	—	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	C₂×C₄	1	4	yes	—
24	C₂²×C₄₃⋊C₈	—	2	2	no	—
25	C₂²×C₄₃⋊C₈	—	2	2	no	—
26	C₂²×C₄₃⋊C₈	—	2	2	no	—
27	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×C₄₃⋊C₈	D₄	1	2	yes	—
29	C₂²×C₄₃⋊C₈	Q₈	1	2	yes	—
30	C₂×C₄×Dic₄₃	—	2	2	no	—
31	C₂×C₄×Dic₄₃	—	2	1	no	—
32	C₂×C₄×Dic₄₃	C₂×C₄	1	2	yes	—
33	C₂×C₄×Dic₄₃	D₄	1	4	yes	—
34	#13063235	—	2	8	no	—
35	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₄₃	—	2	2	no	—
37	C₂²×C₈×C₄₃	—	2	2	no	—
38	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
39	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
40	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
41	#117×C₄₃	—	2	2	no	—
42	#117×C₄₃	—	2	2	no	—
43	#117×C₄₃	—	2	2	no	—
44	#117×C₄₃	—	2	2	no	—
45	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	4	yes	—
46	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
47	C₂×C₄²×C₄₃	—	2	2	no	—
48	C₂⁴×C₄₃	#41	1	1	yes	characteristic
49	C₂²×C₄×C₄₃	—	2	4	no	—
50	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
55	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	#41	1	2	yes	—
58	C₂²×C₄×C₄₃	#41	1	2	yes	—
59	C₂²×C₄×C₄₃	#41	1	2	yes	—
60	C₂²×Dic₄₃	—	2	2	no	—
61	C₂²×Dic₄₃	—	2	4	no	—
62	C₂²×Dic₄₃	—	4	4	no	—
63	C₂×C₈×C₄₃	—	2	2	no	—
64	C₂×C₈×C₄₃	—	2	2	no	—
65	C₂×C₈×C₄₃	—	2	4	no	—
66	C₂×C₈×C₄₃	—	2	4	no	—
67	C₂×C₈×C₄₃	C₄²	1	4	yes	—
68	C₄⋊C₄×C₄₃	—	2	4	no	—
69	C₄⋊C₄×C₄₃	—	2	4	no	—
70	C₂²⋊C₄×C₄₃	—	4	2	no	—
71	C₂²⋊C₄×C₄₃	—	4	2	no	—
72	C₄²×C₄₃	—	4	2	no	—
73	C₄²×C₄₃	—	4	2	no	—
74	C₄×Dic₄₃	—	2	2	no	—
75	C₄×Dic₄₃	—	2	4	no	—
76	C₄×Dic₄₃	C₂²⋊C₄	1	4	yes	—
77	#10621877	—	2	4	no	—
78	#10621877	—	2	4	no	—
79	#10621877	—	4	2	no	—
80	#10621877	—	4	2	no	—
81	C₂×C₄₃⋊C₈	—	2	2	no	—
82	C₂×C₄₃⋊C₈	—	2	2	no	—
83	C₂×C₄₃⋊C₈	—	2	4	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	4	no	—
86	C₂×C₄₃⋊C₈	—	4	2	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
89	C₂³×C₄₃	—	2	4	no	—
90	C₂³×C₄₃	#117	1	1	yes	characteristic
91	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
92	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
93	C₂³×C₄₃	#123	1	2	yes	—
94	C₂³×C₄₃	#126	1	2	yes	—
95	C₂×C₄×C₄₃	—	4	4	no	—
96	C₂×C₄×C₄₃	—	4	8	no	—
97	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
98	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
99	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
102	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
103	C₂×C₄×C₄₃	#117	1	2	yes	—
104	C₂×C₄×C₄₃	#117	1	2	yes	—
105	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
106	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
108	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#121	1	2	yes	—
110	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
111	C₂×C₄×C₄₃	#123	1	2	yes	—
112	C₂×C₄×C₄₃	#124	1	2	yes	—
113	C₂×C₄×C₄₃	#125	1	2	yes	—
114	C₂×C₄×C₄₃	#126	1	2	yes	—
115	C₂×Dic₄₃	—	2	4	no	—
116	C₂×Dic₄₃	—	4	8	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₈×C₄₃	—	2	4	no	—
119	C₄₃⋊C₈	—	2	4	no	—
120	C₄₃⋊C₈	—	4	4	no	—
121	#10819	—	43	1	no	maximal, Sylow
122	C₂²×C₄₃	—	2	4	no	—
123	C₂²×C₄₃	—	2	4	no	—
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
127	C₂²×C₄₃	#381	1	1	yes	characteristic
128	C₂²×C₄₃	#383	1	1	yes	characteristic
129	C₂²×C₄₃	#387	1	2	yes	—
130	C₂²×C₄₃	#388	1	2	yes	—
131	C₂²×C₄₃	#444	1	4	yes	—
132	C₄×C₄₃	—	8	4	no	—
133	C₄×C₄₃	#376	1	1	yes	characteristic
134	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
135	C₄×C₄₃	#380	1	1	yes	characteristic
136	C₄×C₄₃	#382	1	1	yes	characteristic
137	C₄×C₄₃	#387	1	2	yes	—
138	C₄×C₄₃	#388	1	2	yes	—
139	Dic₄₃	—	4	4	no	—
140	C₂²×#97	—	43	1	no	—
141	C₂×#341	—	43	1	no	—
142	#1735	D₄₃	1	1	yes	characteristic
143	C₂×#339	—	43	2	no	—
144	#1729	—	43	2	no	—
145	C₂×C₄₃	—	2	1	no	—
146	C₂×C₄₃	—	2	1	no	—
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	#1434	1	1	yes	characteristic
149	C₂×C₄₃	#1773	1	2	yes	—
150	C₂×C₄₃	#1938	1	4	yes	—
151	C₂²×M₁₆	—	43	2	no	—
152	C₂³×C₈	—	43	1	no	—
153	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
154	C₂×#117	C₂×D₄₃	1	2	yes	—
155	C₂²×C₄²	—	43	1	no	—
156	C₂×#97	—	43	4	no	—
157	C₂×#97	—	86	4	no	—
158	#341	—	43	4	no	—
159	#340	Dic₄₃	1	4	yes	complemented
160	#339	—	86	4	no	—
161	#336	—	86	4	no	—
162	C₄₃	#10819	1	1	yes	characteristic, complemented, Sylow
163	C₂³×C₄	—	86	1	no	—
164	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
165	C₂×M₁₆	—	43	4	no	—
166	C₂×M₁₆	—	86	2	no	—
167	C₂×M₁₆	—	86	2	no	—
168	C₂×M₁₆	—	86	2	no	—
169	C₂×M₁₆	—	86	2	no	—
170	C₂×M₁₆	—	86	2	no	—
171	C₂²×C₈	—	2	2	no	—
172	C₂²×C₈	—	2	2	no	—
173	C₂²×C₈	—	43	2	no	—
174	C₂²×C₈	—	43	2	no	—
175	C₂²×C₈	—	43	2	no	—
176	C₂²×C₈	—	86	2	no	—
177	C₂²×C₈	—	86	2	no	—
178	C₂²×C₈	—	86	2	no	—
179	C₂²×C₈	C₄×D₄₃	1	2	yes	—
180	C₂²×C₈	C₄×D₄₃	1	2	yes	—
181	C₂²×C₈	C₂×Dic₄₃	1	2	yes	—
182	#117	—	2	2	no	—
183	#117	—	2	2	no	—
184	#117	—	2	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₄⋊C₄	C₂×Dic₄₃	1	4	yes	—
187	C₂×C₂²⋊C₄	—	2	2	no	—
188	C₂×C₄²	—	2	2	no	—
189	C₂×C₄²	—	43	2	no	—
190	C₂×C₄²	—	43	4	no	—
191	C₂×C₄²	—	86	2	no	—
192	C₂×C₄²	—	86	1	no	—
193	#97	—	86	8	no	—
194	C₂⁴	#10621900	1	1	yes	characteristic
195	C₂²×C₄	—	2	4	no	—
196	C₂²×C₄	—	86	2	no	—
197	C₂²×C₄	—	86	4	no	—
198	C₂²×C₄	—	172	4	no	—
199	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
200	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
201	C₂²×C₄	#10621898	1	2	yes	—
202	C₂²×C₄	#10621898	1	2	yes	—
203	C₂²×C₄	D₄×D₄₃	1	2	yes	—
204	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
205	C₂²×C₄	#10621900	1	2	yes	—
206	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
207	C₂²×C₄	#10621902	1	1	yes	characteristic
208	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
209	M₁₆	—	86	4	no	—
210	M₁₆	—	86	4	no	—
211	M₁₆	—	172	2	no	—
212	M₁₆	—	172	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	2	2	no	—
215	C₂×C₈	—	2	4	no	—
216	C₂×C₈	—	2	4	no	—
217	C₂×C₈	—	43	4	no	—
218	C₂×C₈	—	86	2	no	—
219	C₂×C₈	—	86	2	no	—
220	C₂×C₈	—	86	4	no	—
221	C₂×C₈	—	86	4	no	—
222	C₂×C₈	—	86	4	no	—
223	C₂×C₈	—	172	2	no	—
224	C₂×C₈	—	172	2	no	—
225	C₂×C₈	C₄×Dic₄₃	1	4	yes	—
226	C₄⋊C₄	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₄²	—	4	2	no	—
231	C₄²	—	4	2	no	—
232	C₄²	—	43	4	no	—
233	C₄²	—	86	2	no	—
234	C₄²	—	86	4	no	—
235	C₂³	—	2	4	no	—
236	C₂³	#13063283	1	1	yes	characteristic
237	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
238	C₂³	#13063290	1	1	yes	characteristic
239	C₂³	#13063307	1	2	yes	—
240	C₂³	#13063318	1	2	yes	—
241	C₂×C₄	—	4	4	no	—
242	C₂×C₄	—	4	8	no	—
243	C₂×C₄	—	86	4	no	—
244	C₂×C₄	—	172	4	no	—
245	C₂×C₄	—	172	8	no	—
246	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
247	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
248	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
249	C₂×C₄	#13063281	1	1	yes	characteristic
250	C₂×C₄	#13063282	1	2	yes	—
251	C₂×C₄	#13063282	1	1	yes	characteristic
252	C₂×C₄	#13063282	1	1	yes	characteristic
253	C₂×C₄	#13063284	1	2	yes	—
254	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
255	C₂×C₄	#13063287	1	1	yes	characteristic
256	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
257	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
258	C₂×C₄	#13063298	1	2	yes	—
259	C₂×C₄	#13063303	1	2	yes	—
260	C₂×C₄	#13063310	1	2	yes	—
261	C₂×C₄	#13063311	1	2	yes	—
262	C₂×C₄	#13063315	1	2	yes	—
263	C₂×C₄	#13063320	1	2	yes	—
264	C₈	—	2	4	no	—
265	C₈	—	86	4	no	—
266	C₈	—	172	4	no	—
267	C₂²	—	2	2	no	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
272	C₂²	#14582761	1	1	yes	characteristic
273	C₂²	#14582766	1	1	yes	characteristic
274	C₂²	#14582781	1	2	yes	—
275	C₂²	#14582792	1	2	yes	—
276	C₂²	#14583021	1	4	yes	—
277	C₄	—	8	4	no	—
278	C₄	—	172	4	no	—
279	C₄	#14582745	1	1	yes	characteristic
280	C₄	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
281	C₄	#14582758	1	1	yes	characteristic
282	C₄	#14582763	1	1	yes	characteristic
283	C₄	#14582785	1	2	yes	—
284	C₄	#14582788	1	2	yes	—
285	C₂	—	2	1	no	—
286	C₂	—	2	1	no	—
287	C₂	—	2	2	no	—
288	C₂	#19132820	1	1	yes	characteristic
289	C₂	#19134481	1	2	yes	—
290	C₂	#19135308	1	4	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₄₃×#1735	C₂
socle	C₂³×C₄₃	C₄×D₄
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	#1735	D₄₃
hypocenter	C₄₃	#10819
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #1735

Complements

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

N	H ≅ G/N	type
10. C₂×#13063235	C₄	semidirect
11. #14582664	C₄	semidirect
16. C₄₃×#340	C₄	semidirect
159. #340	Dic₄₃	semidirect
162. C₄₃	#10819	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#13063235	2
C₂×#14582664	2
C₂×#14582658	2
#19134292	2
C₄₃×#1735	2
#10819	43