Subgroups — #25095349

Order 11008.

894 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14582956	C₂	1	1	yes	maximal, characteristic
2	#19137002	C₂	1	1	yes	maximal, characteristic
3	#19134472	C₂	1	4	yes	maximal, complemented
4	#19134462	C₂	1	2	yes	maximal, complemented
5	C₈×C₄⋊Dic₄₃	C₂	1	2	yes	maximal, complemented
6	#19134363	C₂	1	4	yes	maximal
7	C₂×C₄₃×#431	C₂	1	1	yes	maximal, characteristic
8	C₂×#13063284	C₂²	1	2	yes	—
9	#14582956	C₄	1	8	yes	complemented
10	C₂×C₄⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	2	yes	—
12	C₂×#13063236	C₂²	1	2	yes	—
13	C₂×#13063235	C₂²	1	4	yes	—
14	C₂×C₈×Dic₄₃	C₂²	1	4	yes	—
15	#14582769	C₄	1	2	yes	—
16	#14582765	C₄	1	2	yes	—
17	#14582759	C₄	1	2	yes	—
18	#14582759	C₄	1	2	yes	—
19	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
20	C₄×C₄⋊Dic₄₃	C₂²	1	2	yes	—
21	#14582653	C₂²	1	4	yes	—
22	#14582653	C₂²	1	4	yes	—
23	C₂²×C₄²×C₄₃	C₂²	1	1	yes	characteristic
24	C₄₃×#431	C₄	1	8	yes	complemented
25	C₂×C₄⋊C₈×C₄₃	C₂²	1	2	yes	—
26	C₂×C₂²⋊C₈×C₄₃	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×C₄₃⋊C₈	C₂³	1	4	yes	—
30	#13063284	—	4	4	no	—
31	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
32	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
33	C₂×C₄⋊Dic₄₃	C₂×C₄	1	4	yes	—
34	C₂×C₂²⋊Dic₄₃	C₂×C₄	1	4	yes	—
35	C₂×C₄×Dic₄₃	C₂³	1	4	yes	—
36	C₄⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
37	C₂²⋊(C₄₃⋊C₈)	C₂×C₄	1	8	yes	—
38	#13063236	C₂×C₄	1	8	yes	—
39	#13063235	—	4	4	no	—
40	C₈×Dic₄₃	—	4	4	no	—
41	C₄₃⋊#95	C₂×C₄	1	4	yes	—
42	C₄₃⋊#95	C₂×C₄	1	4	yes	—
43	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
44	C₂³×C₄×C₄₃	C₂×C₄	1	1	yes	characteristic
45	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
47	C₂²×C₈×C₄₃	C₂³	1	2	yes	—
48	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄²×C₄₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
51	C₂×C₄²×C₄₃	C₂×C₄	1	4	yes	—
52	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
53	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
54	C₄⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
55	C₂²⋊C₈×C₄₃	C₂×C₄	1	8	yes	—
56	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
57	C₄×C₈×C₄₃	C₂×C₄	1	4	yes	—
58	C₂⁴×C₄₃	C₂²×C₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	—	2	4	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	2	2	no	—
62	C₂²×C₄×C₄₃	—	4	2	no	—
63	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
64	C₂²×C₄×C₄₃	C₄²	1	4	yes	—
65	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
70	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄×C₄₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
73	C₂²×Dic₄₃	C₂²×C₄	1	8	yes	—
74	C₂×C₈×C₄₃	—	2	2	no	—
75	C₂×C₈×C₄₃	—	2	2	no	—
76	C₂×C₈×C₄₃	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	2	no	—
78	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
79	C₂×C₈×C₄₃	C₂²×C₄	1	4	yes	—
80	C₄²×C₄₃	C₄²	1	4	yes	—
81	C₄²×C₄₃	C₄²	1	4	yes	—
82	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
83	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
84	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
85	C₄²×C₄₃	C₂²×C₄	1	2	yes	—
86	C₂²⋊Dic₄₃	—	4	4	no	—
87	C₄⋊Dic₄₃	—	2	8	no	—
88	C₄⋊Dic₄₃	—	4	4	no	—
89	C₄⋊Dic₄₃	—	4	4	no	—
90	C₄×Dic₄₃	—	2	4	no	—
91	C₄×Dic₄₃	—	2	4	no	—
92	C₂×C₄₃⋊C₈	—	2	4	no	—
93	C₂×C₄₃⋊C₈	—	2	4	no	—
94	C₂×C₄₃⋊C₈	C₂²×C₄	1	8	yes	—
95	C₂³×C₄₃	—	2	1	no	—
96	C₂³×C₄₃	—	2	1	no	—
97	C₂³×C₄₃	—	2	2	no	—
98	C₂³×C₄₃	—	2	2	no	—
99	C₂³×C₄₃	C₂×C₄²	1	2	yes	—
100	C₂³×C₄₃	C₂³×C₄	1	1	yes	characteristic
101	C₂×C₄×C₄₃	—	2	1	no	—
102	C₂×C₄×C₄₃	—	2	1	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	2	2	no	—
106	C₂×C₄×C₄₃	—	2	4	no	—
107	C₂×C₄×C₄₃	—	2	4	no	—
108	C₂×C₄×C₄₃	—	2	8	no	—
109	C₂×C₄×C₄₃	—	4	4	no	—
110	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
111	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
112	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
113	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₄₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₄₃	#131	1	4	yes	—
116	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
118	C₂×C₄×C₄₃	H(5,𝔽₂)	1	2	yes	—
119	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
120	C₂×Dic₄₃	—	2	8	no	—
121	C₂×Dic₄₃	—	2	8	no	—
122	C₂×Dic₄₃	—	2	8	no	—
123	C₈×C₄₃	—	2	4	no	—
124	C₈×C₄₃	—	2	4	no	—
125	C₄₃⋊C₈	—	2	8	no	—
126	#20537	—	43	1	no	maximal, Sylow
127	C₂²×C₄₃	—	2	2	no	—
128	C₂²×C₄₃	—	2	2	no	—
129	C₂²×C₄₃	—	2	2	no	—
130	C₂²×C₄₃	—	4	2	no	—
131	C₂²×C₄₃	—	4	2	no	—
132	C₂²×C₄₃	C₂²×C₄²	1	1	yes	characteristic
133	C₂²×C₄₃	#518	1	1	yes	characteristic
134	C₂²×C₄₃	#519	1	1	yes	characteristic
135	C₂²×C₄₃	C₂×#131	1	2	yes	—
136	C₂²×C₄₃	#568	1	2	yes	—
137	C₄×C₄₃	—	2	2	no	—
138	C₄×C₄₃	—	2	2	no	—
139	C₄×C₄₃	—	2	2	no	—
140	C₄×C₄₃	—	2	2	no	—
141	C₄×C₄₃	#405	1	4	yes	—
142	C₄×C₄₃	#520	1	2	yes	—
143	C₄×C₄₃	#520	1	2	yes	—
144	Dic₄₃	—	4	8	no	—
145	C₂×#432	—	43	1	no	—
146	C₂×#431	D₄₃	1	1	yes	characteristic
147	#2285	—	43	1	no	—
148	#1771	—	43	4	no	—
149	#1770	—	43	2	no	—
150	C₈×C₄⋊C₄	—	43	2	no	—
151	#1750	—	43	4	no	—
152	C₂×C₄₃	—	4	2	no	—
153	C₂×C₄₃	#2274	1	1	yes	characteristic
154	C₂×C₄₃	C₂×#405	1	1	yes	characteristic
155	C₂×C₄₃	#2873	1	1	yes	characteristic
156	C₂×C₄₃	#2981	1	2	yes	—
157	C₂×C₄₃	#2985	1	2	yes	—
158	C₂×#117	—	43	2	no	—
159	C₂²×C₄²	C₂×D₄₃	1	1	yes	characteristic
160	#432	—	43	8	no	—
161	#431	Dic₄₃	1	8	yes	complemented
162	C₂×C₄⋊C₈	—	43	2	no	—
163	C₂×C₄⋊C₈	C₂×D₄₃	1	2	yes	—
164	C₂×C₂²⋊C₈	—	43	2	no	—
165	C₂×C₂²⋊C₈	C₂×D₄₃	1	2	yes	—
166	C₂×#97	—	43	2	no	—
167	C₂×#97	—	43	4	no	—
168	C₂×C₄×C₈	—	43	4	no	—
169	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
170	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
171	#384	—	43	2	no	—
172	#382	—	43	2	no	—
173	#380	—	43	2	no	—
174	#380	—	43	2	no	—
175	C₄×C₄⋊C₄	—	43	2	no	—
176	C₄×C₄⋊C₄	—	43	2	no	—
177	#336	—	43	4	no	—
178	#336	—	43	4	no	—
179	C₄₃	#20537	1	1	yes	characteristic, complemented, Sylow
180	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
181	C₂³×C₄	C₄×D₄₃	1	1	yes	characteristic
182	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
183	C₂²×C₈	—	43	4	no	—
184	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
185	C₂²×C₈	C₂²×D₄₃	1	2	yes	—
186	#117	—	172	4	no	—
187	C₂×C₄⋊C₄	—	43	4	no	—
188	C₂×C₄⋊C₄	—	43	4	no	—
189	C₂×C₄⋊C₄	—	43	4	no	—
190	C₂×C₂²⋊C₄	—	43	4	no	—
191	C₂×C₄²	—	43	4	no	—
192	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
193	C₂×C₄²	C₄×D₄₃	1	1	yes	characteristic
194	C₂×C₄²	C₄×D₄₃	1	4	yes	—
195	C₂×C₄²	C₂×Dic₄₃	1	4	yes	—
196	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
197	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
198	C₄⋊C₈	—	43	8	no	—
199	C₄⋊C₈	C₂×Dic₄₃	1	8	yes	—
200	C₂²⋊C₈	—	43	8	no	—
201	C₂²⋊C₈	C₂×Dic₄₃	1	8	yes	—
202	#97	—	43	8	no	—
203	#97	—	172	4	no	—
204	C₄×C₈	—	172	4	no	—
205	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
206	C₄×C₈	C₂×Dic₄₃	1	4	yes	—
207	#95	—	43	4	no	—
208	#95	—	43	4	no	—
209	C₂⁴	C₂×C₄×D₄₃	1	1	yes	characteristic
210	C₂²×C₄	—	2	4	no	—
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	2	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	—	43	8	no	—
215	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
216	C₂²×C₄	C₄×Dic₄₃	1	4	yes	—
217	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
219	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
220	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
221	C₂²×C₄	C₂×C₄×D₄₃	1	2	yes	—
222	C₂²×C₄	C₂×C₄×D₄₃	1	1	yes	characteristic
223	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
224	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	—	2	2	no	—
227	C₂×C₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	43	8	no	—
230	C₂×C₈	—	86	4	no	—
231	C₂×C₈	—	86	4	no	—
232	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
233	C₂×C₈	C₂²×Dic₄₃	1	4	yes	—
234	C₄⋊C₄	—	86	8	no	—
235	C₄⋊C₄	—	172	4	no	—
236	C₄⋊C₄	—	172	4	no	—
237	C₂²⋊C₄	—	172	4	no	—
238	C₄²	—	86	4	no	—
239	C₄²	—	86	4	no	—
240	C₄²	C₄×Dic₄₃	1	4	yes	—
241	C₄²	C₄×Dic₄₃	1	4	yes	—
242	C₄²	C₂×C₄×D₄₃	1	2	yes	—
243	C₄²	C₂×C₄×D₄₃	1	2	yes	—
244	C₄²	C₂²×Dic₄₃	1	2	yes	—
245	C₄²	C₂²×Dic₄₃	1	2	yes	—
246	C₂³	—	2	1	no	—
247	C₂³	—	2	1	no	—
248	C₂³	—	2	2	no	—
249	C₂³	—	2	2	no	—
250	C₂³	C₂×C₄×Dic₄₃	1	2	yes	—
251	C₂³	C₂²×C₄×D₄₃	1	1	yes	characteristic
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	2	1	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	8	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	86	8	no	—
262	C₂×C₄	—	86	8	no	—
263	C₂×C₄	—	86	8	no	—
264	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
265	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
266	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
267	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
268	C₂×C₄	C₂×C₄×Dic₄₃	1	2	yes	—
269	C₂×C₄	#13063331	1	4	yes	—
270	C₂×C₄	C₂²×C₄×D₄₃	1	1	yes	characteristic
271	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
272	C₂×C₄	#13063371	1	2	yes	—
273	C₂×C₄	#13063372	1	2	yes	—
274	C₈	—	2	4	no	—
275	C₈	—	2	4	no	—
276	C₈	—	86	8	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	2	no	—
282	C₂²	C₂²×C₄×Dic₄₃	1	1	yes	characteristic
283	C₂²	#14583373	1	1	yes	characteristic
284	C₂²	#14583376	1	1	yes	characteristic
285	C₂²	C₂×#13063331	1	2	yes	—
286	C₂²	#14583665	1	2	yes	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	172	8	no	—
292	C₄	#14582854	1	4	yes	—
293	C₄	#14583379	1	2	yes	—
294	C₄	#14583382	1	2	yes	—
295	C₂	—	4	2	no	—
296	C₂	#19136951	1	1	yes	characteristic
297	C₂	C₂×#14582854	1	1	yes	characteristic
298	C₂	#19143013	1	1	yes	characteristic
299	C₂	#19143670	1	2	yes	—
300	C₂	#19143689	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₄₃
commutator	C₂²×C₄₃	C₂²×C₄²
Fitting	C₂×C₄₃×#431	C₂
socle	C₂³×C₄₃	C₂³×C₄
prime radical	C₂⁴×C₄₃	C₂²×C₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂×#431	D₄₃
hypocenter	C₄₃	#20537
LCS	G ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431

Complements

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

N	H ≅ G/N	type
3. #19134472	C₂	semidirect
4. #19134462	C₂	semidirect
5. C₈×C₄⋊Dic₄₃	C₂	semidirect
9. #14582956	C₄	semidirect
24. C₄₃×#431	C₄	semidirect
161. #431	Dic₄₃	semidirect
179. C₄₃	#20537	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14582956	2
#19137002	2
#19134472	2
#19134462	2
C₈×C₄⋊Dic₄₃	2
#19134363	2
C₂×C₄₃×#431	2
#20537	43