Subgroups — #25114295

Order 11008.

866 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19137464	C₂	1	2	yes	maximal
2	#19135334	C₂	1	2	yes	maximal, complemented
3	#19135189	C₂	1	4	yes	maximal, complemented
4	#19134681	C₂	1	2	yes	maximal
5	#19134620	C₂	1	1	yes	maximal, characteristic
6	#19134494	C₂	1	2	yes	maximal, complemented
7	#19134371	C₂	1	1	yes	maximal, characteristic
8	C₂²×#107×C₄₃	C₂	1	1	yes	maximal, characteristic
9	C₂×#13063284	—	2	2	no	—
10	C₂×#13063257	C₂²	1	2	yes	—
11	C₂×C₂²⋊(C₄₃⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×#13063235	C₂²	1	2	yes	—
13	#14582798	C₂²	1	4	yes	—
14	#14582785	—	2	4	no	—
15	#14582762	—	2	2	no	—
16	#14582759	C₂²	1	2	yes	—
17	C₄⋊C₄×Dic₄₃	C₂²	1	4	yes	—
18	C₄×C₂²⋊Dic₄₃	C₂²	1	1	yes	characteristic
19	#14582662	C₂²	1	2	yes	—
20	#14582661	C₂²	1	4	yes	—
21	#14582661	C₂²	1	4	yes	—
22	#14582653	C₂²	1	2	yes	—
23	C₂³×C₈×C₄₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄⋊C₄×C₄₃	C₂²	1	2	yes	—
25	C₂×#107×C₄₃	C₄	1	8	yes	complemented
26	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
27	C₂×#107×C₄₃	C₂²	1	1	yes	characteristic
28	C₂×#107×C₄₃	C₂²	1	2	yes	—
29	C₂²×C₄₃⋊C₈	C₂³	1	2	yes	—
30	#13063284	—	4	8	no	—
31	C₂×C₄⋊Dic₄₃	—	2	4	no	—
32	C₂×C₄⋊Dic₄₃	C₂³	1	4	yes	—
33	C₂×C₂²⋊Dic₄₃	—	2	2	no	—
34	C₂×C₂²⋊Dic₄₃	D₄	1	2	yes	—
35	C₂×C₄×Dic₄₃	—	2	4	no	—
36	C₂×C₄×Dic₄₃	C₂³	1	2	yes	—
37	#13063257	—	2	4	no	—
38	C₂²⋊(C₄₃⋊C₈)	—	2	2	no	—
39	#13063235	—	2	4	no	—
40	C₄₃⋊#95	—	2	2	no	—
41	C₄₃⋊#95	—	2	4	no	—
42	C₄₃⋊#95	—	2	4	no	—
43	C₄₃⋊#95	D₄	1	2	yes	—
44	C₂³×C₄×C₄₃	—	2	2	no	—
45	C₂³×C₄×C₄₃	C₂³	1	1	yes	characteristic
46	C₂²×C₈×C₄₃	—	2	4	no	—
47	C₂²×C₈×C₄₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
51	C₂×C₄⋊C₄×C₄₃	—	2	4	no	—
52	C₂×C₄⋊C₄×C₄₃	C₂×C₄	1	8	yes	—
53	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
54	C₂×C₄⋊C₄×C₄₃	C₂³	1	2	yes	—
55	#107×C₄₃	C₂×C₄	1	8	yes	—
56	#107×C₄₃	C₂×C₄	1	4	yes	—
57	#107×C₄₃	C₂×C₄	1	4	yes	—
58	C₂⁴×C₄₃	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₄₃	—	2	2	no	—
60	C₂²×C₄×C₄₃	—	2	2	no	—
61	C₂²×C₄×C₄₃	—	2	8	no	—
62	C₂²×C₄×C₄₃	—	2	2	no	—
63	C₂²×C₄×C₄₃	—	2	2	no	—
64	C₂²×C₄×C₄₃	—	4	4	no	—
65	C₂²×C₄×C₄₃	—	4	4	no	—
66	C₂²×C₄×C₄₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₄₃	#41	1	2	yes	—
69	C₂²×C₄×C₄₃	C₂⁴	1	1	yes	characteristic
70	C₂²×Dic₄₃	—	2	4	no	—
71	C₂²×Dic₄₃	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₄₃	—	2	2	no	—
73	C₂×C₈×C₄₃	—	2	8	no	—
74	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₄₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₄₃	#41	1	4	yes	—
77	C₄⋊C₄×C₄₃	—	2	8	no	—
78	C₄⋊C₄×C₄₃	—	2	2	no	—
79	C₄⋊C₄×C₄₃	—	2	2	no	—
80	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
81	C₄⋊C₄×C₄₃	C₂²×C₄	1	4	yes	—
82	C₂²⋊Dic₄₃	—	2	4	no	—
83	C₂²⋊Dic₄₃	—	4	4	no	—
84	C₄⋊Dic₄₃	—	2	4	no	—
85	C₄⋊Dic₄₃	—	2	4	no	—
86	C₄⋊Dic₄₃	—	4	8	no	—
87	C₄×Dic₄₃	—	2	2	no	—
88	C₄×Dic₄₃	—	2	2	no	—
89	C₄×Dic₄₃	—	4	8	no	—
90	C₂×C₄₃⋊C₈	—	2	2	no	—
91	C₂×C₄₃⋊C₈	—	2	2	no	—
92	C₂×C₄₃⋊C₈	—	2	2	no	—
93	C₂³×C₄₃	—	2	2	no	—
94	C₂³×C₄₃	—	2	2	no	—
95	C₂³×C₄₃	C₄×D₄	1	2	yes	—
96	C₂³×C₄₃	AGL(1,ℤ/8)	1	4	yes	—
97	C₂³×C₄₃	C₂²×D₄	1	1	yes	characteristic
98	C₂×C₄×C₄₃	—	2	2	no	—
99	C₂×C₄×C₄₃	—	2	2	no	—
100	C₂×C₄×C₄₃	—	2	2	no	—
101	C₂×C₄×C₄₃	—	2	4	no	—
102	C₂×C₄×C₄₃	—	2	4	no	—
103	C₂×C₄×C₄₃	—	2	4	no	—
104	C₂×C₄×C₄₃	—	4	2	no	—
105	C₂×C₄×C₄₃	—	4	2	no	—
106	C₂×C₄×C₄₃	—	4	2	no	—
107	C₂×C₄×C₄₃	—	4	2	no	—
108	C₂×C₄×C₄₃	—	4	16	no	—
109	C₂×C₄×C₄₃	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₃	C₂³×C₄	1	1	yes	characteristic
111	C₂×C₄×C₄₃	C₂×#41	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂×#41	1	2	yes	—
113	C₂×C₄×C₄₃	2₋¹⁺⁴	1	2	yes	—
114	C₂×Dic₄₃	—	2	4	no	—
115	C₂×Dic₄₃	—	2	4	no	—
116	C₂×Dic₄₃	—	2	4	no	—
117	C₂×Dic₄₃	—	4	4	no	—
118	C₂×Dic₄₃	—	4	4	no	—
119	C₂×Dic₄₃	—	4	4	no	—
120	C₈×C₄₃	—	2	2	no	—
121	C₈×C₄₃	#117	1	4	yes	complemented
122	C₄₃⋊C₈	—	4	2	no	—
123	#22617	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	8	no	—
125	C₂²×C₄₃	—	2	2	no	—
126	C₂²×C₄₃	—	2	2	no	—
127	C₂²×C₄₃	#442	1	4	yes	—
128	C₂²×C₄₃	C₂×C₄×D₄	1	1	yes	characteristic
129	C₂²×C₄₃	#547	1	2	yes	—
130	C₂²×C₄₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
131	C₂²×C₄₃	#578	1	2	yes	—
132	C₄×C₄₃	—	2	2	no	—
133	C₄×C₄₃	—	4	4	no	—
134	C₄×C₄₃	—	4	4	no	—
135	C₄×C₄₃	C₂×#117	1	1	yes	characteristic
136	C₄×C₄₃	#519	1	1	yes	characteristic
137	C₄×C₄₃	#552	1	2	yes	—
138	Dic₄₃	—	4	4	no	—
139	Dic₄₃	—	8	4	no	—
140	C₂²×#107	D₄₃	1	1	yes	characteristic
141	#2344	—	43	2	no	—
142	#1943	—	43	2	no	—
143	#1921	—	43	4	no	—
144	#1823	—	43	2	no	—
145	#1807	—	43	1	no	—
146	#1776	—	43	2	no	—
147	#1752	—	43	1	no	—
148	C₂×C₄₃	—	2	4	no	—
149	C₂×C₄₃	#2344	1	1	yes	characteristic
150	C₂×C₄₃	C₂×#442	1	1	yes	characteristic
151	C₂×C₄₃	#2950	1	1	yes	characteristic
152	C₂×C₄₃	#3343	1	4	yes	—
153	C₂³×C₈	C₂×D₄₃	1	1	yes	characteristic
154	C₂×#117	—	86	2	no	—
155	C₂²×C₄⋊C₄	C₂×D₄₃	1	2	yes	—
156	C₂×#107	Dic₄₃	1	8	yes	complemented
157	C₂×#107	C₂×D₄₃	1	2	yes	—
158	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
159	C₂×#107	C₂×D₄₃	1	1	yes	characteristic
160	C₂×#106	—	43	2	no	—
161	C₂×C₂²⋊C₈	—	43	1	no	—
162	C₂×#97	—	43	2	no	—
163	#389	—	43	4	no	—
164	#387	—	86	4	no	—
165	#381	—	86	2	no	—
166	#380	—	43	2	no	—
167	C₄×C₄⋊C₄	—	43	4	no	—
168	C₄×C₂²⋊C₄	—	43	1	no	—
169	#340	—	43	2	no	—
170	#340	—	43	4	no	—
171	#340	—	43	4	no	—
172	#336	—	43	2	no	—
173	C₄₃	#22617	1	1	yes	characteristic, complemented, Sylow
174	C₂³×C₄	—	2	2	no	—
175	C₂³×C₄	C₂²×D₄₃	1	1	yes	characteristic
176	C₂²×C₈	—	2	4	no	—
177	C₂²×C₈	—	43	2	no	—
178	C₂²×C₈	C₂×Dic₄₃	1	4	yes	—
179	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
180	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
181	#117	—	172	8	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	4	no	—
184	C₂×C₄⋊C₄	—	43	4	no	—
185	C₂×C₄⋊C₄	—	86	4	no	—
186	C₂×C₄⋊C₄	C₂×Dic₄₃	1	8	yes	—
187	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
188	C₂×C₄⋊C₄	C₂²×D₄₃	1	2	yes	—
189	C₂×C₂²⋊C₄	—	43	2	no	—
190	C₂×C₂²⋊C₄	—	86	2	no	—
191	C₂×C₄²	—	43	2	no	—
192	C₂×C₄²	—	86	4	no	—
193	#107	C₂×Dic₄₃	1	8	yes	—
194	#107	C₂×Dic₄₃	1	4	yes	—
195	#107	C₂×Dic₄₃	1	4	yes	—
196	#106	—	86	4	no	—
197	C₂²⋊C₈	—	86	2	no	—
198	#97	—	86	4	no	—
199	#95	—	43	2	no	—
200	#95	—	86	4	no	—
201	#95	—	86	4	no	—
202	#95	—	86	2	no	—
203	C₂⁴	D₄×D₄₃	1	1	yes	characteristic
204	C₂²×C₄	—	2	2	no	—
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	8	no	—
207	C₂²×C₄	—	2	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	4	4	no	—
210	C₂²×C₄	—	4	4	no	—
211	C₂²×C₄	—	43	4	no	—
212	C₂²×C₄	—	86	4	no	—
213	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
214	C₂²×C₄	#10621900	1	2	yes	—
215	C₂²×C₄	C₂²×Dic₄₃	1	2	yes	—
216	C₂²×C₄	C₂³×D₄₃	1	1	yes	characteristic
217	C₂×C₈	—	2	2	no	—
218	C₂×C₈	—	2	8	no	—
219	C₂×C₈	—	86	2	no	—
220	C₂×C₈	—	86	2	no	—
221	C₂×C₈	—	86	2	no	—
222	C₂×C₈	#10621898	1	4	yes	—
223	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
224	C₂×C₈	C₂²×Dic₄₃	1	2	yes	—
225	C₄⋊C₄	—	2	8	no	—
226	C₄⋊C₄	—	2	2	no	—
227	C₄⋊C₄	—	2	2	no	—
228	C₄⋊C₄	—	86	4	no	—
229	C₄⋊C₄	—	86	4	no	—
230	C₄⋊C₄	—	172	8	no	—
231	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
232	C₄⋊C₄	C₂²×Dic₄₃	1	4	yes	—
233	C₂²⋊C₄	—	86	4	no	—
234	C₂²⋊C₄	—	172	4	no	—
235	C₄²	—	86	2	no	—
236	C₄²	—	86	2	no	—
237	C₄²	—	172	8	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	D₄×Dic₄₃	1	2	yes	—
241	C₂³	#13063349	1	4	yes	—
242	C₂³	C₂×D₄×D₄₃	1	1	yes	characteristic
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	2	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	2	4	no	—
248	C₂×C₄	—	2	4	no	—
249	C₂×C₄	—	4	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	2	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	4	16	no	—
254	C₂×C₄	—	86	4	no	—
255	C₂×C₄	—	86	4	no	—
256	C₂×C₄	—	86	4	no	—
257	C₂×C₄	—	172	4	no	—
258	C₂×C₄	—	172	4	no	—
259	C₂×C₄	—	172	4	no	—
260	C₂×C₄	D₄×Dic₄₃	1	2	yes	—
261	C₂×C₄	C₂³×Dic₄₃	1	1	yes	characteristic
262	C₂×C₄	C₂×#10621900	1	1	yes	characteristic
263	C₂×C₄	C₂×#10621898	1	2	yes	—
264	C₂×C₄	#13063373	1	2	yes	—
265	C₈	—	2	2	no	—
266	C₈	—	172	2	no	—
267	C₈	#13063284	1	4	yes	complemented
268	C₂²	—	2	8	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	2	no	—
271	C₂²	#14583011	1	4	yes	—
272	C₂²	C₂×D₄×Dic₄₃	1	1	yes	characteristic
273	C₂²	#14583550	1	2	yes	—
274	C₂²	C₂×#13063349	1	2	yes	—
275	C₂²	#14583712	1	2	yes	—
276	C₄	—	2	2	no	—
277	C₄	—	4	4	no	—
278	C₄	—	4	4	no	—
279	C₄	—	344	4	no	—
280	C₄	—	172	4	no	—
281	C₄	C₂×#13063284	1	1	yes	characteristic
282	C₄	#14583377	1	1	yes	characteristic
283	C₄	#14583584	1	2	yes	—
284	C₂	—	2	4	no	—
285	C₂	#19137464	1	1	yes	characteristic
286	C₂	C₂×#14583011	1	1	yes	characteristic
287	C₂	#19143463	1	1	yes	characteristic
288	C₂	#19147031	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₄×D₄₃
commutator	C₂×C₄×C₄₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₄₃
Fitting	C₂²×#107×C₄₃	C₂
socle	C₂³×C₄₃	C₂²×D₄
prime radical	C₂⁴×C₄₃	C₂×D₄
cosocle	C₂²×C₄×C₄₃	C₂⁴
hypercenter	C₂²×#107	D₄₃
hypocenter	C₄₃	#22617
LCS	G ▷ C₂×C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×#107

Complements

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

N	H ≅ G/N	type
2. #19135334	C₂	semidirect
3. #19135189	C₂	semidirect
6. #19134494	C₂	semidirect
25. C₂×#107×C₄₃	C₄	semidirect
121. C₈×C₄₃	#117	semidirect
156. C₂×#107	Dic₄₃	semidirect
173. C₄₃	#22617	semidirect
267. C₈	#13063284	semidirect

Maximal subgroups

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

subgroup	index
#19137464	2
#19135334	2
#19135189	2
#19134681	2
#19134620	2
#19134494	2
#19134371	2
C₂²×#107×C₄₃	2
#22617	43