Subgroups — #25084586

Order 11008.

686 conjugacy classes of subgroups.

The subgroup lattice has 280 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19145739	C₂	1	2	yes	maximal
2	#19145181	C₂	1	1	yes	maximal, characteristic
3	#19144975	C₂	1	2	yes	maximal
4	#19134240	C₂	1	4	yes	maximal, complemented
5	#19134204	C₂	1	2	yes	maximal, complemented
6	#19134193	C₂	1	2	yes	maximal, complemented
7	C₈×#10621877	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄₃×#500	C₂	1	1	yes	maximal, characteristic
9	#14583456	C₂²	1	2	yes	—
10	#14583304	C₂²	1	1	yes	characteristic
11	#14583295	C₂²	1	4	yes	—
12	#14583279	C₂²	1	1	yes	characteristic
13	#14583267	—	2	4	no	—
14	#14583251	C₂²	1	4	yes	—
15	#14583250	C₂²	1	2	yes	—
16	#14583249	C₂²	1	4	yes	—
17	#14583249	C₂²	1	4	yes	—
18	#14583229	—	2	4	no	—
19	#14583181	—	2	4	no	—
20	#14583140	—	2	2	no	—
21	C₄×Dic₁₇₂	—	2	2	no	—
22	#14582952	C₂²	1	1	yes	characteristic
23	#14582908	—	2	4	no	—
24	C₄×#10621877	C₂²	1	1	yes	characteristic
25	#14582621	C₂²	1	2	yes	—
26	C₈×C₄₃⋊C₈	C₂²	1	2	yes	—
27	C₂×C₄⋊Q₈×C₄₃	C₂²	1	2	yes	—
28	C₄₃×#500	—	2	2	no	—
29	C₄₃×#500	C₂²	1	2	yes	—
30	C₄₃×#500	C₂²	1	2	yes	—
31	C₂×#107×C₄₃	—	2	2	no	—
32	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
33	C₂×Dic₁₇₂	—	2	2	no	—
34	C₂×#10621877	D₄	1	2	yes	—
35	#13063325	C₂³	1	2	yes	—
36	#13063325	C₂³	1	2	yes	—
37	#13063317	—	2	2	no	—
38	#13063304	—	2	2	no	—
39	#13063304	—	2	2	no	—
40	C₄×Dic₈₆	—	2	2	no	—
41	C₄×Dic₈₆	—	2	2	no	—
42	#13063284	—	2	2	no	—
43	#13063259	—	2	2	no	—
44	#13063256	—	2	8	no	—
45	C₄⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
46	#13063250	—	2	4	no	—
47	#13063249	—	2	8	no	—
48	#13063249	—	2	8	no	—
49	#13063248	—	2	8	no	—
50	C₂²⋊(C₄₃⋊C₈)	D₄	1	2	yes	—
51	#13063236	D₄	1	2	yes	—
52	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
53	C₄×C₄₃⋊C₈	C₂³	1	2	yes	—
54	C₂²×Q₈×C₄₃	—	2	2	no	—
55	C₂²×C₈×C₄₃	D₄	1	2	yes	—
56	C₄⋊Q₈×C₄₃	—	2	2	no	—
57	C₄⋊Q₈×C₄₃	—	2	2	no	—
58	C₄⋊Q₈×C₄₃	—	2	2	no	—
59	C₄⋊Q₈×C₄₃	C₂³	1	4	yes	—
60	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
61	C₂×C₄⋊C₄×C₄₃	—	2	2	no	—
62	C₂×C₄²×C₄₃	C₂³	1	1	yes	characteristic
63	#107×C₄₃	—	2	4	no	—
64	#107×C₄₃	—	2	4	no	—
65	C₄×C₈×C₄₃	D₄	1	2	yes	—
66	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
67	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
68	C₂×Q₈×C₄₃	—	2	4	no	—
69	C₂×Q₈×C₄₃	—	4	2	no	—
70	C₂×Q₈×C₄₃	—	4	8	no	—
71	C₂²×C₄×C₄₃	—	4	2	no	—
72	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
75	C₂×Dic₈₆	—	2	2	no	—
76	C₂×Dic₈₆	—	2	2	no	—
77	C₂×C₈×C₄₃	—	2	4	no	—
78	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
79	C₂×C₈×C₄₃	C₂×D₄	1	2	yes	—
80	C₄⋊C₄×C₄₃	—	2	4	no	—
81	C₄⋊C₄×C₄₃	—	2	4	no	—
82	C₄⋊C₄×C₄₃	—	4	2	no	—
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
85	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
86	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
87	C₄²×C₄₃	C₂⁴	1	1	yes	characteristic
88	C₂²⋊Dic₄₃	—	4	2	no	—
89	C₄⋊Dic₄₃	—	2	2	no	—
90	C₄⋊Dic₄₃	—	2	2	no	—
91	C₄⋊Dic₄₃	—	2	2	no	—
92	C₄⋊Dic₄₃	—	2	2	no	—
93	Dic₄₃⋊C₄	—	4	2	no	—
94	Dic₄₃⋊C₄	—	4	2	no	—
95	C₄×Dic₄₃	—	4	2	no	—
96	#10621877	—	2	4	no	—
97	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
98	C₂×C₄₃⋊C₈	C₂×D₄	1	4	yes	—
99	Dic₁₇₂	—	4	4	no	—
100	C₂³×C₄₃	#127	1	1	yes	characteristic
101	Q₈×C₄₃	—	4	8	no	—
102	Q₈×C₄₃	—	8	4	no	—
103	C₂×C₄×C₄₃	—	2	2	no	—
104	C₂×C₄×C₄₃	—	2	2	no	—
105	C₂×C₄×C₄₃	—	4	4	no	—
106	C₂×C₄×C₄₃	—	8	4	no	—
107	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
108	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
109	C₂×C₄×C₄₃	#127	1	1	yes	characteristic
110	C₂×C₄×C₄₃	#135	1	4	yes	—
111	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
112	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×Dic₄₃	—	4	2	no	—
115	C₂×Dic₄₃	—	4	2	no	—
116	Dic₈₆	—	4	4	no	—
117	Dic₈₆	—	4	4	no	—
118	C₈×C₄₃	—	2	2	no	—
119	C₈×C₄₃	#137	1	4	yes	—
120	C₄₃⋊C₈	—	2	4	no	—
121	C₄₃⋊C₈	—	2	4	no	—
122	#19636	—	43	1	no	maximal, Sylow
123	C₂²×C₄₃	—	2	1	no	—
124	C₂²×C₄₃	—	2	1	no	—
125	C₂²×C₄₃	#495	1	2	yes	—
126	C₂²×C₄₃	C₂×#127	1	1	yes	characteristic
127	C₄×C₄₃	—	2	1	no	—
128	C₄×C₄₃	—	2	1	no	—
129	C₄×C₄₃	—	8	4	no	—
130	C₄×C₄₃	#495	1	2	yes	—
131	C₄×C₄₃	C₂×#135	1	2	yes	—
132	C₄×C₄₃	C₂×#137	1	2	yes	—
133	C₄×C₄₃	#578	1	2	yes	—
134	Dic₄₃	—	8	2	no	—
135	Dic₄₃	—	8	2	no	—
136	#3229	—	43	2	no	—
137	#3170	—	43	1	no	—
138	C₂×#500	D₄₃	1	1	yes	characteristic
139	#3139	—	43	2	no	—
140	#1718	—	43	4	no	—
141	#1710	—	43	2	no	—
142	#1707	—	43	2	no	—
143	C₈×M₁₆	—	43	1	no	—
144	C₂×C₄₃	—	2	2	no	—
145	C₂×C₄₃	C₂×#495	1	1	yes	characteristic
146	C₂×C₄₃	#3152	1	1	yes	characteristic
147	C₂×C₄₃	#3156	1	1	yes	characteristic
148	#533	—	43	2	no	—
149	C₂×C₄⋊Q₈	C₂×D₄₃	1	2	yes	—
150	#500	—	2	2	no	—
151	#500	—	43	1	no	—
152	#500	C₂×D₄₃	1	2	yes	—
153	#500	C₂×D₄₃	1	2	yes	—
154	#498	—	43	4	no	—
155	#494	—	43	1	no	—
156	#491	—	86	4	no	—
157	#487	—	43	4	no	—
158	#487	—	43	4	no	—
159	#487	—	43	4	no	—
160	#487	—	43	2	no	—
161	#483	—	86	4	no	—
162	#475	—	86	4	no	—
163	#467	—	86	2	no	—
164	C₄×Q₁₆	—	86	2	no	—
165	#431	—	43	1	no	—
166	C₂×#107	—	2	2	no	—
167	#419	—	86	4	no	—
168	C₄×M₁₆	—	43	1	no	—
169	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
170	#322	—	43	2	no	—
171	C₈²	—	43	2	no	—
172	C₄₃	#19636	1	1	yes	characteristic, complemented, Sylow
173	C₂²×Q₈	—	2	2	no	—
174	C₂×Q₁₆	—	86	2	no	—
175	C₂×M₁₆	—	43	2	no	—
176	C₂²×C₈	C₂²⋊D₄₃	1	2	yes	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	2	no	—
179	C₄⋊Q₈	—	2	2	no	—
180	C₄⋊Q₈	—	43	2	no	—
181	C₄⋊Q₈	—	43	2	no	—
182	C₄⋊Q₈	C₂²×D₄₃	1	4	yes	—
183	#125	—	86	2	no	—
184	C₂²⋊Q₈	—	86	2	no	—
185	C₂²⋊Q₈	—	86	2	no	—
186	C₄×Q₈	—	86	2	no	—
187	C₄×Q₈	—	86	2	no	—
188	#117	—	86	2	no	—
189	C₂×C₄⋊C₄	—	2	2	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄²	C₂²×D₄₃	1	1	yes	characteristic
192	#107	—	2	4	no	—
193	#107	—	2	4	no	—
194	#107	—	86	2	no	—
195	#106	—	86	8	no	—
196	C₄⋊C₈	—	43	2	no	—
197	#103	—	86	8	no	—
198	#103	—	86	8	no	—
199	#103	—	86	8	no	—
200	#103	—	86	4	no	—
201	C₂²⋊C₈	—	43	2	no	—
202	#97	—	43	2	no	—
203	C₄×C₈	—	43	2	no	—
204	C₄×C₈	—	43	2	no	—
205	C₄×C₈	C₂²⋊D₄₃	1	2	yes	—
206	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
207	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	4	2	no	—
210	C₂×Q₈	—	4	8	no	—
211	C₂×Q₈	—	86	2	no	—
212	C₂×Q₈	—	86	2	no	—
213	C₂²×C₄	—	4	2	no	—
214	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
215	C₂²×C₄	D₄×D₄₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
217	Q₁₆	—	172	4	no	—
218	M₁₆	—	86	4	no	—
219	C₂×C₈	—	2	4	no	—
220	C₂×C₈	—	43	4	no	—
221	C₂×C₈	—	43	4	no	—
222	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₄₃	1	2	yes	—
224	C₄⋊C₄	—	2	4	no	—
225	C₄⋊C₄	—	2	4	no	—
226	C₄⋊C₄	—	4	2	no	—
227	C₄⋊C₄	—	4	2	no	—
228	C₄⋊C₄	—	86	2	no	—
229	C₄⋊C₄	—	86	2	no	—
230	C₄⋊C₄	—	86	2	no	—
231	C₄⋊C₄	—	86	2	no	—
232	C₄⋊C₄	—	172	2	no	—
233	C₄⋊C₄	—	172	2	no	—
234	C₂²⋊C₄	—	172	2	no	—
235	C₄²	—	172	2	no	—
236	C₄²	D₄×D₄₃	1	1	yes	characteristic
237	C₄²	D₄×D₄₃	1	1	yes	characteristic
238	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
239	C₄²	C₂³×D₄₃	1	1	yes	characteristic
240	C₂³	#13063321	1	1	yes	characteristic
241	Q₈	—	4	8	no	—
242	Q₈	—	8	4	no	—
243	Q₈	—	172	4	no	—
244	Q₈	—	172	4	no	—
245	C₂×C₄	—	2	2	no	—
246	C₂×C₄	—	2	2	no	—
247	C₂×C₄	—	4	4	no	—
248	C₂×C₄	—	8	4	no	—
249	C₂×C₄	—	172	2	no	—
250	C₂×C₄	—	172	2	no	—
251	C₂×C₄	#13063321	1	1	yes	characteristic
252	C₂×C₄	#13063321	1	1	yes	characteristic
253	C₂×C₄	#13063321	1	1	yes	characteristic
254	C₂×C₄	#13063344	1	4	yes	—
255	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×D₄₃	1	1	yes	characteristic
257	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
258	C₈	—	2	2	no	—
259	C₈	—	86	4	no	—
260	C₈	—	86	4	no	—
261	C₈	#13063356	1	4	yes	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	#14583282	1	2	yes	—
265	C₂²	C₂×#13063321	1	1	yes	characteristic
266	C₄	—	2	1	no	—
267	C₄	—	2	1	no	—
268	C₄	—	8	4	no	—
269	C₄	—	344	2	no	—
270	C₄	—	344	2	no	—
271	C₄	#14583282	1	2	yes	—
272	C₄	C₂×#13063344	1	2	yes	—
273	C₄	C₂×#13063356	1	2	yes	—
274	C₄	#14583712	1	2	yes	—
275	C₂	—	2	2	no	—
276	C₂	C₂×#14583282	1	1	yes	characteristic
277	C₂	#19145052	1	1	yes	characteristic
278	C₂	#19145084	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#13063321
commutator, cosocle	C₄²×C₄₃	C₂⁴
Frattini	C₄²	C₂³×D₄₃
Fitting	C₂×C₄₃×#500	C₂
socle	C₂²×C₄₃	C₂×#127
prime radical	C₂³×C₄₃	#127
hypercenter	C₂×#500	D₄₃
hypocenter	C₄₃	#19636
LCS	G ▷ C₄²×C₄₃ ▷ C₂²×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
4. #19134240	C₂	semidirect
5. #19134204	C₂	semidirect
6. #19134193	C₂	semidirect
172. C₄₃	#19636	semidirect

Maximal subgroups

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

subgroup	index
#19145739	2
#19145181	2
#19144975	2
#19134240	2
#19134204	2
#19134193	2
C₈×#10621877	2
C₂×C₄₃×#500	2
#19636	43