Subgroups — #25002290

Order 11008.

674 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×M₁₆×Dic₄₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#14582669	C₂	1	2	yes	maximal
3	C₂×#14582658	C₂	1	2	yes	maximal
4	C₂×C₄₃⋊#338	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂²×#10621877	C₂²	1	2	yes	—
7	C₂²×#10621877	C₂²	1	1	yes	characteristic
8	C₂²×C₄×Dic₄₃	C₂²	1	1	yes	characteristic
9	M₁₆×Dic₄₃	—	2	4	no	—
10	C₂×#13063235	C₄	1	2	yes	complemented
11	C₂×C₈×Dic₄₃	C₄	1	2	yes	complemented
12	#14582669	—	2	4	no	—
13	#14582658	—	2	4	no	—
14	C₄₃⋊#338	C₄	1	4	yes	complemented
15	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
16	C₂×#117×C₄₃	C₂²	1	2	yes	—
17	C₄₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
18	C₂³×Dic₄₃	—	2	1	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	C₂×C₄	1	4	yes	—
27	C₂×#10621877	D₄	1	2	yes	—
28	C₂×#10621877	Q₈	1	2	yes	—
29	C₂²×C₄₃⋊C₈	—	2	2	no	—
30	C₂²×C₄₃⋊C₈	C₂×C₄	1	2	yes	—
31	C₂×C₄×Dic₄₃	—	2	2	no	—
32	C₂×C₄×Dic₄₃	—	2	1	no	—
33	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×Dic₄₃	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄×Dic₄₃	D₄	1	4	yes	—
36	#13063235	—	2	4	no	—
37	C₈×Dic₄₃	—	2	4	no	—
38	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
39	C₂×M₁₆×C₄₃	—	2	2	no	—
40	C₂×M₁₆×C₄₃	—	2	2	no	—
41	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₄₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₄₃	C₂×C₄	1	2	yes	—
44	#117×C₄₃	—	2	2	no	—
45	#117×C₄₃	—	2	2	no	—
46	#117×C₄₃	—	2	2	no	—
47	#117×C₄₃	C₂×C₄	1	4	yes	—
48	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
49	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
50	C₂×C₄²×C₄₃	—	2	2	no	—
51	C₂⁴×C₄₃	#41	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	—	4	2	no	—
53	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
56	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
58	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	#41	1	1	yes	characteristic
60	C₂²×C₄×C₄₃	#41	1	2	yes	—
61	C₂²×C₄×C₄₃	#41	1	2	yes	—
62	C₂²×C₄×C₄₃	#41	1	2	yes	—
63	C₂²×Dic₄₃	—	2	1	no	—
64	C₂²×Dic₄₃	—	2	1	no	—
65	C₂²×Dic₄₃	—	2	4	no	—
66	C₂²×Dic₄₃	—	4	4	no	—
67	M₁₆×C₄₃	—	2	4	no	—
68	M₁₆×C₄₃	—	2	4	no	—
69	C₂×C₈×C₄₃	—	2	2	no	—
70	C₂×C₈×C₄₃	—	2	2	no	—
71	C₂×C₈×C₄₃	C₄²	1	4	yes	—
72	C₄⋊C₄×C₄₃	—	2	4	no	—
73	C₄⋊C₄×C₄₃	—	4	2	no	—
74	C₂²⋊C₄×C₄₃	—	2	4	no	—
75	C₂²⋊C₄×C₄₃	—	4	2	no	—
76	C₄²×C₄₃	—	2	4	no	—
77	C₄²×C₄₃	—	4	2	no	—
78	C₄×Dic₄₃	—	2	1	no	—
79	C₄×Dic₄₃	—	2	1	no	—
80	C₄×Dic₄₃	—	2	4	no	—
81	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
82	C₄×Dic₄₃	C₂²⋊C₄	1	2	yes	—
83	#10621877	—	2	4	no	—
84	#10621877	—	2	4	no	—
85	#10621877	—	2	4	no	—
86	#10621877	—	2	4	no	—
87	#10621877	—	4	2	no	—
88	#10621877	—	4	2	no	—
89	C₂×C₄₃⋊C₈	—	2	2	no	—
90	C₂×C₄₃⋊C₈	—	2	2	no	—
91	C₂×C₄₃⋊C₈	—	2	4	no	—
92	C₂×C₄₃⋊C₈	—	4	2	no	—
93	C₂×C₄₃⋊C₈	—	4	2	no	—
94	C₂×C₄₃⋊C₈	C₄⋊C₄	1	4	yes	—
95	C₂³×C₄₃	—	4	2	no	—
96	C₂³×C₄₃	#117	1	1	yes	characteristic
97	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
98	C₂³×C₄₃	C₄×Q₈	1	1	yes	characteristic
99	C₂³×C₄₃	#123	1	2	yes	—
100	C₂³×C₄₃	#126	1	2	yes	—
101	C₂×C₄×C₄₃	—	4	4	no	—
102	C₂×C₄×C₄₃	—	8	4	no	—
103	C₂×C₄×C₄₃	C₂×C₄²	1	1	yes	characteristic
104	C₂×C₄×C₄₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
105	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
107	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#117	1	2	yes	—
109	C₂×C₄×C₄₃	#117	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#117	1	2	yes	—
111	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
112	C₂×C₄×C₄₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
114	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
115	C₂×C₄×C₄₃	#121	1	2	yes	—
116	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
117	C₂×C₄×C₄₃	#123	1	2	yes	—
118	C₂×C₄×C₄₃	#124	1	2	yes	—
119	C₂×C₄×C₄₃	#125	1	2	yes	—
120	C₂×C₄×C₄₃	#126	1	2	yes	—
121	C₂×Dic₄₃	—	2	4	no	—
122	C₂×Dic₄₃	—	2	2	no	—
123	C₂×Dic₄₃	—	2	2	no	—
124	C₂×Dic₄₃	—	4	8	no	—
125	C₂×Dic₄₃	—	4	2	no	—
126	C₈×C₄₃	—	2	4	no	—
127	C₄₃⋊C₈	—	2	4	no	—
128	C₄₃⋊C₈	—	4	4	no	—
129	#10813	—	43	1	no	maximal, Sylow
130	C₂²×C₄₃	—	2	2	no	—
131	C₂²×C₄₃	—	2	4	no	—
132	C₂²×C₄₃	—	4	1	no	—
133	C₂²×C₄₃	—	4	1	no	—
134	C₂²×C₄₃	—	4	2	no	—
135	C₂²×C₄₃	C₄×C₄⋊C₄	1	1	yes	characteristic
136	C₂²×C₄₃	#381	1	1	yes	characteristic
137	C₂²×C₄₃	#383	1	1	yes	characteristic
138	C₂²×C₄₃	#387	1	2	yes	—
139	C₂²×C₄₃	#388	1	2	yes	—
140	C₄×C₄₃	—	8	4	no	—
141	C₄×C₄₃	#376	1	1	yes	characteristic
142	C₄×C₄₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
143	C₄×C₄₃	#380	1	1	yes	characteristic
144	C₄×C₄₃	#382	1	1	yes	characteristic
145	C₄×C₄₃	#387	1	2	yes	—
146	C₄×C₄₃	#388	1	2	yes	—
147	Dic₄₃	—	2	4	no	—
148	Dic₄₃	—	4	2	no	—
149	C₂×C₄×M₁₆	—	43	1	no	—
150	C₂×#344	—	43	2	no	—
151	C₂×#339	—	43	2	no	—
152	C₂×#338	—	43	1	no	—
153	C₂×H(3,ℤ/4)	D₄₃	1	1	yes	characteristic
154	C₂×C₄₃	—	2	1	no	—
155	C₂×C₄₃	—	2	2	no	—
156	C₂×C₄₃	—	2	1	no	—
157	C₂×C₄₃	—	2	2	no	—
158	C₂×C₄₃	#1434	1	1	yes	characteristic
159	C₂×C₄₃	#1841	1	2	yes	—
160	C₂²×M₁₆	—	43	1	no	—
161	C₂²×M₁₆	—	43	2	no	—
162	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
163	C₂×#117	C₂×D₄₃	1	2	yes	—
164	C₂²×C₄²	—	43	1	no	—
165	C₄×M₁₆	—	86	4	no	—
166	C₂×#97	—	43	2	no	—
167	C₂×C₄×C₈	—	43	2	no	—
168	#344	—	86	4	no	—
169	#339	—	86	4	no	—
170	#338	—	43	4	no	—
171	H(3,ℤ/4)	Dic₄₃	1	4	yes	complemented
172	C₄₃	#10813	1	1	yes	characteristic, complemented, Sylow
173	C₂³×C₄	—	86	1	no	—
174	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	—	43	2	no	—
178	C₂×M₁₆	—	43	2	no	—
179	C₂×M₁₆	—	43	4	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	C₄×D₄₃	1	2	yes	—
188	C₂×M₁₆	C₂×Dic₄₃	1	2	yes	—
189	C₂²×C₈	—	43	2	no	—
190	C₂²×C₈	—	86	2	no	—
191	C₂²×C₈	C₄×D₄₃	1	2	yes	—
192	#117	—	2	2	no	—
193	#117	—	2	2	no	—
194	#117	—	2	2	no	—
195	#117	C₂×Dic₄₃	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₄²	—	2	2	no	—
199	C₂×C₄²	—	43	1	no	—
200	C₂×C₄²	—	43	1	no	—
201	C₂×C₄²	—	43	4	no	—
202	C₂×C₄²	—	86	2	no	—
203	C₂×C₄²	—	86	1	no	—
204	#97	—	86	4	no	—
205	C₄×C₈	—	86	4	no	—
206	C₂⁴	#10621900	1	1	yes	characteristic
207	C₂²×C₄	—	4	2	no	—
208	C₂²×C₄	—	86	1	no	—
209	C₂²×C₄	—	86	1	no	—
210	C₂²×C₄	—	86	4	no	—
211	C₂²×C₄	—	172	4	no	—
212	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
213	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	#10621898	1	2	yes	—
215	C₂²×C₄	#10621898	1	2	yes	—
216	C₂²×C₄	D₄×D₄₃	1	2	yes	—
217	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
218	C₂²×C₄	#10621900	1	2	yes	—
219	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
220	C₂²×C₄	#10621902	1	1	yes	characteristic
221	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
222	M₁₆	—	2	4	no	—
223	M₁₆	—	2	4	no	—
224	M₁₆	—	86	4	no	—
225	M₁₆	—	86	4	no	—
226	M₁₆	—	86	4	no	—
227	M₁₆	—	86	4	no	—
228	M₁₆	—	172	2	no	—
229	M₁₆	—	172	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	43	4	no	—
233	C₂×C₈	—	86	2	no	—
234	C₂×C₈	—	86	2	no	—
235	C₂×C₈	—	86	4	no	—
236	C₂×C₈	—	172	2	no	—
237	C₂×C₈	—	172	2	no	—
238	C₂×C₈	C₄×Dic₄₃	1	4	yes	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₂²⋊C₄	—	2	4	no	—
242	C₂²⋊C₄	—	4	2	no	—
243	C₄²	—	2	4	no	—
244	C₄²	—	4	2	no	—
245	C₄²	—	43	2	no	—
246	C₄²	—	43	2	no	—
247	C₄²	—	86	1	no	—
248	C₄²	—	86	1	no	—
249	C₄²	—	86	4	no	—
250	C₂³	—	4	2	no	—
251	C₂³	#13063283	1	1	yes	characteristic
252	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
253	C₂³	#13063290	1	1	yes	characteristic
254	C₂³	#13063307	1	2	yes	—
255	C₂³	#13063318	1	2	yes	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	8	4	no	—
258	C₂×C₄	—	86	4	no	—
259	C₂×C₄	—	86	2	no	—
260	C₂×C₄	—	86	2	no	—
261	C₂×C₄	—	172	8	no	—
262	C₂×C₄	—	172	2	no	—
263	C₂×C₄	C₂×C₄×Dic₄₃	1	1	yes	characteristic
264	C₂×C₄	C₂²⋊C₄×D₄₃	1	1	yes	characteristic
265	C₂×C₄	C₄⋊C₄×D₄₃	1	1	yes	characteristic
266	C₂×C₄	#13063281	1	1	yes	characteristic
267	C₂×C₄	#13063282	1	2	yes	—
268	C₂×C₄	#13063282	1	1	yes	characteristic
269	C₂×C₄	#13063282	1	1	yes	characteristic
270	C₂×C₄	#13063284	1	2	yes	—
271	C₂×C₄	(C₂²⋊D₄₃)⋊C₄	1	2	yes	—
272	C₂×C₄	#13063287	1	1	yes	characteristic
273	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
274	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
275	C₂×C₄	#13063298	1	2	yes	—
276	C₂×C₄	#13063303	1	2	yes	—
277	C₂×C₄	#13063310	1	2	yes	—
278	C₂×C₄	#13063311	1	2	yes	—
279	C₂×C₄	#13063315	1	2	yes	—
280	C₂×C₄	#13063320	1	2	yes	—
281	C₈	—	2	4	no	—
282	C₈	—	86	4	no	—
283	C₈	—	172	4	no	—
284	C₂²	—	2	4	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	1	no	—
288	C₂²	—	4	1	no	—
289	C₂²	C₄⋊C₄×Dic₄₃	1	1	yes	characteristic
290	C₂²	#14582761	1	1	yes	characteristic
291	C₂²	#14582766	1	1	yes	characteristic
292	C₂²	#14582781	1	2	yes	—
293	C₂²	#14582792	1	2	yes	—
294	C₄	—	8	4	no	—
295	C₄	—	86	4	no	—
296	C₄	—	172	2	no	—
297	C₄	#14582745	1	1	yes	characteristic
298	C₄	C₂²⋊C₄×Dic₄₃	1	1	yes	characteristic
299	C₄	#14582758	1	1	yes	characteristic
300	C₄	#14582763	1	1	yes	characteristic
301	C₄	#14582785	1	2	yes	—
302	C₄	#14582788	1	2	yes	—
303	C₂	—	2	2	no	—
304	C₂	—	2	1	no	—
305	C₂	—	2	1	no	—
306	C₂	—	2	2	no	—
307	C₂	#19132820	1	1	yes	characteristic
308	C₂	#19134742	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063282
commutator	C₂×C₄×C₄₃	C₂×C₄²
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×H(3,ℤ/4)	C₂
socle	C₂²×C₄₃	#383
prime radical	C₂⁴×C₄₃	#41
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₄₃
hypocenter	C₄₃	#10813
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
10. C₂×#13063235	C₄	semidirect
11. C₂×C₈×Dic₄₃	C₄	semidirect
14. C₄₃⋊#338	C₄	semidirect
17. C₄₃×H(3,ℤ/4)	C₄	semidirect
171. H(3,ℤ/4)	Dic₄₃	semidirect
172. C₄₃	#10813	semidirect

Maximal subgroups

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

subgroup	index
C₂×M₁₆×Dic₄₃	2
C₂×#14582669	2
C₂×#14582658	2
C₂×C₄₃⋊#338	2
C₂×C₄₃×H(3,ℤ/4)	2
#10813	43