Subgroups — #25004970

Order 11008.

606 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19143185	C₂	1	1	yes	maximal, characteristic
2	#19134327	C₂	1	2	yes	maximal
3	#19134311	C₂	1	2	yes	maximal
4	#19134293	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic
6	C₂³×C₄₃⋊C₈	C₂²	1	1	yes	characteristic
7	C₂×#13063284	C₂²	1	2	yes	—
8	C₂×#13063284	C₂²	1	1	yes	characteristic
9	#14582941	—	2	4	no	—
10	#14582937	—	2	2	no	—
11	#14582665	—	2	4	no	—
12	#14582661	—	2	4	no	—
13	#14582653	—	2	2	no	—
14	C₂²×M₁₆×C₄₃	C₂²	1	1	yes	characteristic
15	C₂×#117×C₄₃	C₂²	1	2	yes	—
16	C₄₃×H(3,ℤ/4)	C₄	1	4	yes	complemented
17	C₂²×C₄₃⋊C₈	—	2	1	no	—
18	C₂²×C₄₃⋊C₈	—	2	2	no	—
19	C₂²×C₄₃⋊C₈	—	2	2	no	—
20	C₂²×C₄₃⋊C₈	—	2	1	no	—
21	C₂²×C₄₃⋊C₈	Q₈	1	2	yes	—
22	#13063284	—	2	2	no	—
23	#13063284	—	2	2	no	—
24	#13063284	—	2	2	no	—
25	#13063284	—	2	2	no	—
26	#13063284	—	2	2	no	—
27	#13063284	—	2	2	no	—
28	C₂×C₄⋊Dic₄₃	—	2	1	no	—
29	C₂×C₄⋊Dic₄₃	D₄	1	4	yes	—
30	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
31	C₂×C₂²⋊Dic₄₃	D₄	1	2	yes	—
32	C₂×C₄×Dic₄₃	—	2	1	no	—
33	C₂×C₄×Dic₄₃	—	2	2	no	—
34	#13063259	—	2	4	no	—
35	#13063257	—	2	4	no	—
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	C₂×C₄	1	2	yes	—
40	C₂×M₁₆×C₄₃	Q₈	1	2	yes	—
41	C₂²×C₈×C₄₃	D₄	1	2	yes	—
42	#117×C₄₃	—	2	2	no	—
43	#117×C₄₃	—	2	2	no	—
44	#117×C₄₃	—	2	2	no	—
45	#117×C₄₃	C₂×C₄	1	4	yes	—
46	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
47	C₂×C₂²⋊C₄×C₄₃	—	2	2	no	—
48	C₂×C₄²×C₄₃	—	2	2	no	—
49	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
50	C₂²×C₄×C₄₃	—	4	2	no	—
51	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₄₃	C₂×D₄	1	2	yes	—
54	C₂²×C₄×C₄₃	C₂×Q₈	1	1	yes	characteristic
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	1	yes	characteristic
59	C₂²×C₄×C₄₃	#41	1	2	yes	—
60	C₂²×C₄×C₄₃	#41	1	2	yes	—
61	C₂²×Dic₄₃	—	2	2	no	—
62	C₂²×Dic₄₃	—	2	4	no	—
63	M₁₆×C₄₃	—	2	4	no	—
64	M₁₆×C₄₃	—	2	4	no	—
65	C₂×C₈×C₄₃	—	2	2	no	—
66	C₂×C₈×C₄₃	—	2	2	no	—
67	C₂×C₈×C₄₃	C₄⋊C₄	1	4	yes	—
68	C₄⋊C₄×C₄₃	—	2	4	no	—
69	C₄⋊C₄×C₄₃	—	4	2	no	—
70	C₂²⋊C₄×C₄₃	—	2	4	no	—
71	C₂²⋊C₄×C₄₃	—	4	2	no	—
72	C₄²×C₄₃	—	2	4	no	—
73	C₄²×C₄₃	—	4	2	no	—
74	C₂²⋊Dic₄₃	—	2	4	no	—
75	C₂²⋊Dic₄₃	—	4	2	no	—
76	C₂²⋊Dic₄₃	—	4	2	no	—
77	C₄⋊Dic₄₃	—	2	4	no	—
78	C₄⋊Dic₄₃	—	2	4	no	—
79	C₄⋊Dic₄₃	—	4	2	no	—
80	C₄×Dic₄₃	—	4	1	no	—
81	C₄×Dic₄₃	—	4	2	no	—
82	C₄×Dic₄₃	—	4	1	no	—
83	C₄×Dic₄₃	—	4	2	no	—
84	C₂×C₄₃⋊C₈	—	2	4	no	—
85	C₂×C₄₃⋊C₈	—	2	2	no	—
86	C₂×C₄₃⋊C₈	—	4	1	no	—
87	C₂×C₄₃⋊C₈	—	4	2	no	—
88	C₂×C₄₃⋊C₈	—	4	1	no	—
89	C₂³×C₄₃	—	4	2	no	—
90	C₂³×C₄₃	C₄×D₄	1	1	yes	characteristic
91	C₂³×C₄₃	#121	1	2	yes	—
92	C₂³×C₄₃	#121	1	1	yes	characteristic
93	C₂³×C₄₃	#123	1	2	yes	—
94	C₂³×C₄₃	C₄⋊Q₈	1	1	yes	characteristic
95	C₂×C₄×C₄₃	—	4	4	no	—
96	C₂×C₄×C₄₃	—	8	4	no	—
97	C₂×C₄×C₄₃	C₂×C₄⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	#117	1	2	yes	—
99	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
100	C₂×C₄×C₄₃	C₄×Q₈	1	1	yes	characteristic
101	C₂×C₄×C₄₃	#120	1	1	yes	characteristic
102	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
103	C₂×C₄×C₄₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₄₃	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄×C₄₃	#123	1	1	yes	characteristic
106	C₂×C₄×C₄₃	#123	1	2	yes	—
107	C₂×C₄×C₄₃	#124	1	2	yes	—
108	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#124	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#125	1	2	yes	—
111	C₂×C₄×C₄₃	#125	1	1	yes	characteristic
112	C₂×C₄×C₄₃	#125	1	2	yes	—
113	C₂×C₄×C₄₃	#126	1	2	yes	—
114	C₂×C₄×C₄₃	#126	1	2	yes	—
115	C₂×Dic₄₃	—	4	4	no	—
116	C₂×Dic₄₃	—	4	4	no	—
117	C₂×Dic₄₃	—	4	2	no	—
118	C₂×Dic₄₃	—	8	4	no	—
119	C₈×C₄₃	—	2	4	no	—
120	C₄₃⋊C₈	—	4	2	no	—
121	#11285	—	43	1	no	maximal, Sylow
122	C₂²×C₄₃	—	2	2	no	—
123	C₂²×C₄₃	—	2	4	no	—
124	C₂²×C₄₃	—	4	1	no	—
125	C₂²×C₄₃	—	4	2	no	—
126	C₂²×C₄₃	—	4	1	no	—
127	C₂²×C₄₃	#388	1	2	yes	—
128	C₂²×C₄₃	#389	1	1	yes	characteristic
129	C₂²×C₄₃	#394	1	1	yes	characteristic
130	C₂²×C₄₃	#397	1	1	yes	characteristic
131	C₂²×C₄₃	#398	1	2	yes	—
132	C₄×C₄₃	—	8	4	no	—
133	C₄×C₄₃	#382	1	1	yes	characteristic
134	C₄×C₄₃	#385	1	1	yes	characteristic
135	C₄×C₄₃	#387	1	2	yes	—
136	C₄×C₄₃	#393	1	1	yes	characteristic
137	C₄×C₄₃	#400	1	1	yes	characteristic
138	C₄×C₄₃	#400	1	2	yes	—
139	Dic₄₃	—	8	2	no	—
140	Dic₄₃	—	8	4	no	—
141	#2911	—	43	1	no	—
142	#1740	—	43	2	no	—
143	#1735	—	43	2	no	—
144	C₂×H(3,ℤ/4)	D₄₃	1	1	yes	characteristic
145	#1729	—	43	1	no	—
146	C₂×C₄₃	—	2	1	no	—
147	C₂×C₄₃	—	2	2	no	—
148	C₂×C₄₃	—	2	1	no	—
149	C₂×C₄₃	—	2	2	no	—
150	C₂×C₄₃	#1442	1	1	yes	characteristic
151	C₂×C₄₃	#2053	1	2	yes	—
152	C₂²×M₁₆	C₂×D₄₃	1	1	yes	characteristic
153	C₂³×C₈	—	43	1	no	—
154	C₂×#117	—	43	1	no	—
155	C₂×#117	—	43	2	no	—
156	C₂×#117	C₂×D₄₃	1	2	yes	—
157	#428	—	86	4	no	—
158	#427	—	86	2	no	—
159	#342	—	86	4	no	—
160	#340	—	86	4	no	—
161	H(3,ℤ/4)	Dic₄₃	1	4	yes	complemented
162	#336	—	86	2	no	—
163	C₄₃	#11285	1	1	yes	characteristic, complemented, Sylow
164	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
165	C₂×M₁₆	—	2	2	no	—
166	C₂×M₁₆	—	2	2	no	—
167	C₂×M₁₆	Dic₈₆	1	2	yes	—
168	C₂×M₁₆	C₂×Dic₄₃	1	2	yes	—
169	C₂²×C₈	—	43	2	no	—
170	C₂²×C₈	—	86	1	no	—
171	C₂²×C₈	—	86	2	no	—
172	C₂²×C₈	—	86	2	no	—
173	C₂²×C₈	—	86	1	no	—
174	C₂²×C₈	D₁₇₂	1	2	yes	—
175	#117	—	2	2	no	—
176	#117	—	2	2	no	—
177	#117	—	2	2	no	—
178	#117	—	86	2	no	—
179	#117	—	86	2	no	—
180	#117	—	86	2	no	—
181	#117	—	86	2	no	—
182	#117	—	86	2	no	—
183	#117	—	86	2	no	—
184	#117	C₂×Dic₄₃	1	4	yes	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	43	4	no	—
187	C₂×C₄⋊C₄	—	86	1	no	—
188	C₂×C₂²⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	43	2	no	—
190	C₂×C₂²⋊C₄	—	86	2	no	—
191	C₂×C₄²	—	2	2	no	—
192	C₂×C₄²	—	86	1	no	—
193	C₂×C₄²	—	86	2	no	—
194	#107	—	86	4	no	—
195	#106	—	86	4	no	—
196	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	86	4	no	—
199	C₂²×C₄	—	86	2	no	—
200	C₂²×C₄	C₂×Dic₈₆	1	1	yes	characteristic
201	C₂²×C₄	C₂×D₁₇₂	1	1	yes	characteristic
202	C₂²×C₄	#10621898	1	2	yes	—
203	C₂²×C₄	#10621898	1	2	yes	—
204	C₂²×C₄	D₄×D₄₃	1	2	yes	—
205	C₂²×C₄	#10621900	1	2	yes	—
206	C₂²×C₄	#10621900	1	1	yes	characteristic
207	C₂²×C₄	Q₈×D₄₃	1	1	yes	characteristic
208	C₂²×C₄	#10621902	1	1	yes	characteristic
209	C₂²×C₄	C₂²×Dic₄₃	1	1	yes	characteristic
210	M₁₆	—	2	4	no	—
211	M₁₆	—	2	4	no	—
212	C₂×C₈	—	2	2	no	—
213	C₂×C₈	—	2	2	no	—
214	C₂×C₈	—	86	4	no	—
215	C₂×C₈	—	86	2	no	—
216	C₂×C₈	—	172	1	no	—
217	C₂×C₈	—	172	2	no	—
218	C₂×C₈	—	172	1	no	—
219	C₂×C₈	C₄⋊Dic₄₃	1	4	yes	—
220	C₄⋊C₄	—	2	4	no	—
221	C₄⋊C₄	—	4	2	no	—
222	C₄⋊C₄	—	86	4	no	—
223	C₄⋊C₄	—	86	4	no	—
224	C₄⋊C₄	—	172	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	4	2	no	—
227	C₂²⋊C₄	—	86	4	no	—
228	C₂²⋊C₄	—	172	2	no	—
229	C₂²⋊C₄	—	172	2	no	—
230	C₄²	—	2	4	no	—
231	C₄²	—	4	2	no	—
232	C₄²	—	172	1	no	—
233	C₄²	—	172	2	no	—
234	C₄²	—	172	1	no	—
235	C₄²	—	172	2	no	—
236	C₂³	—	4	2	no	—
237	C₂³	D₄×Dic₄₃	1	1	yes	characteristic
238	C₂³	#13063298	1	2	yes	—
239	C₂³	#13063300	1	1	yes	characteristic
240	C₂³	#13063307	1	2	yes	—
241	C₂³	#13063324	1	1	yes	characteristic
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	8	4	no	—
244	C₂×C₄	—	344	4	no	—
245	C₂×C₄	—	172	4	no	—
246	C₂×C₄	—	172	4	no	—
247	C₂×C₄	—	172	2	no	—
248	C₂×C₄	C₂×C₄⋊Dic₄₃	1	1	yes	characteristic
249	C₂×C₄	#13063284	1	2	yes	—
250	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
251	C₂×C₄	Q₈×Dic₄₃	1	1	yes	characteristic
252	C₂×C₄	#13063295	1	1	yes	characteristic
253	C₂×C₄	#13063302	1	2	yes	—
254	C₂×C₄	#13063303	1	2	yes	—
255	C₂×C₄	#13063305	1	1	yes	characteristic
256	C₂×C₄	#13063309	1	1	yes	characteristic
257	C₂×C₄	#13063310	1	2	yes	—
258	C₂×C₄	#13063311	1	2	yes	—
259	C₂×C₄	#13063313	1	1	yes	characteristic
260	C₂×C₄	#13063313	1	1	yes	characteristic
261	C₂×C₄	#13063315	1	2	yes	—
262	C₂×C₄	#13063316	1	1	yes	characteristic
263	C₂×C₄	#13063317	1	2	yes	—
264	C₂×C₄	#13063318	1	2	yes	—
265	C₂×C₄	#13063320	1	2	yes	—
266	C₈	—	2	4	no	—
267	C₈	—	172	2	no	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	1	no	—
273	C₂²	#14582792	1	2	yes	—
274	C₂²	#14582798	1	1	yes	characteristic
275	C₂²	#14582817	1	1	yes	characteristic
276	C₂²	#14582826	1	1	yes	characteristic
277	C₂²	#14582833	1	2	yes	—
278	C₄	—	8	4	no	—
279	C₄	—	344	2	no	—
280	C₄	—	344	4	no	—
281	C₄	#14582765	1	1	yes	characteristic
282	C₄	#14582774	1	1	yes	characteristic
283	C₄	#14582785	1	2	yes	—
284	C₄	#14582812	1	1	yes	characteristic
285	C₄	#14582839	1	1	yes	characteristic
286	C₄	#14582841	1	2	yes	—
287	C₂	—	2	2	no	—
288	C₂	—	2	1	no	—
289	C₂	—	2	1	no	—
290	C₂	—	2	2	no	—
291	C₂	#19132852	1	1	yes	characteristic
292	C₂	#19135902	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#13063313
commutator	C₂²×C₄×C₄₃	C₂²×C₄
Frattini	C₂³×C₄	C₂²×D₄₃
Fitting	C₂×C₄₃×H(3,ℤ/4)	C₂
socle	C₂²×C₄₃	#394
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂³×C₄×C₄₃	C₂³
hypercenter	C₂×H(3,ℤ/4)	D₄₃
hypocenter	C₄₃	#11285
LCS	G ▷ C₂²×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×H(3,ℤ/4)

Complements

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

N	H ≅ G/N	type
16. C₄₃×H(3,ℤ/4)	C₄	semidirect
161. H(3,ℤ/4)	Dic₄₃	semidirect
163. C₄₃	#11285	semidirect

Maximal subgroups

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

subgroup	index
#19143185	2
#19134327	2
#19134311	2
#19134293	2
C₂×C₄₃×H(3,ℤ/4)	2
#11285	43