Subgroups — #25251964

Order 11008.

598 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14583283	C₂	1	1	yes	maximal, characteristic
2	#19136834	C₂	1	4	yes	maximal, complemented
3	#19136829	C₂	1	4	yes	maximal, complemented
4	#19136314	C₂	1	2	yes	maximal, complemented
5	#19136310	C₂	1	2	yes	maximal
6	C₄×#13063265	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₁₇₂	C₂²	1	1	yes	characteristic
9	C₂²×#10621873	—	2	1	no	—
10	C₂²×D₃₄₄	C₂²	1	1	yes	characteristic
11	C₂×#13063313	—	2	1	no	—
12	C₂×#13063265	C₂²	1	2	yes	—
13	#14583283	—	2	1	no	—
14	#14583283	—	2	1	no	—
15	#14583283	C₂²	1	2	yes	—
16	#14583283	C₂²	1	2	yes	—
17	#14582702	C₂²	1	8	yes	complemented
18	#14582699	C₂²	1	8	yes	—
19	#14582675	C₂²	1	2	yes	complemented
20	C₄×C₄₃⋊C₁₆	C₂²	1	2	yes	complemented
21	C₄₃×#533	—	2	1	no	—
22	C₄₃×#500	C₂²	1	2	yes	complemented
23	C₄₃×#499	—	2	1	no	—
24	C₄₃×#499	C₂²	1	2	yes	complemented
25	C₄₃×#498	—	2	1	no	—
26	C₄₃×#427	—	2	1	no	—
27	C₄₃×#427	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₄₃	C₂²	1	1	yes	characteristic
29	C₂²×Dic₈₆	—	2	1	no	—
30	C₂²×D₁₇₂	—	2	1	no	—
31	C₂×Dic₁₇₂	—	2	4	no	—
32	C₂×Dic₁₇₂	—	2	1	no	—
33	C₂×Dic₁₇₂	C₂³	1	2	yes	—
34	C₂×#10621873	—	2	2	no	—
35	C₂×#10621873	—	2	2	no	—
36	C₂×#10621873	—	4	4	no	—
37	C₂×D₃₄₄	—	2	4	no	—
38	C₂×D₃₄₄	—	2	1	no	—
39	C₂×D₃₄₄	C₂³	1	2	yes	—
40	#13063313	—	2	2	no	—
41	#13063313	—	2	2	no	—
42	#13063313	—	4	2	no	—
43	C₂×(C₂×D₄₃)⋊C₄	—	4	2	no	—
44	#13063265	—	2	4	no	—
45	C₂×C₄₃⋊C₁₆	C₂³	1	4	yes	—
46	C₂²×C₈×C₄₃	D₄	1	1	yes	characteristic
47	C₂²×C₈×C₄₃	C₂³	1	1	yes	characteristic
48	C₄⋊Q₈×C₄₃	—	2	2	no	—
49	#125×C₄₃	—	2	2	no	—
50	C₂²⋊Q₈×C₄₃	—	4	2	no	—
51	C₄×Q₈×C₄₃	—	4	2	no	—
52	#117×C₄₃	—	2	2	no	—
53	C₂×C₄²×C₄₃	D₄	1	1	yes	characteristic
54	#107×C₄₃	—	2	2	no	—
55	#107×C₄₃	C₂³	1	4	yes	—
56	#106×C₄₃	—	2	2	no	—
57	#106×C₄₃	—	2	2	no	—
58	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₄₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₄₃	C₂³	1	1	yes	characteristic
62	C₂³×D₄₃	—	4	1	no	—
63	C₂×Q₈×C₄₃	—	4	2	no	—
64	C₂²×C₄×C₄₃	D₈	1	2	yes	—
65	C₂²×C₄×C₄₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×Dic₄₃	—	4	1	no	—
67	C₂×D₁₇₂	—	2	2	no	—
68	C₂×D₁₇₂	—	4	1	no	—
69	C₂×D₁₇₂	—	4	2	no	—
70	C₂×D₁₇₂	—	4	2	no	—
71	C₂×Dic₈₆	—	2	2	no	—
72	C₂×Dic₈₆	—	4	2	no	—
73	C₂×Dic₈₆	—	4	2	no	—
74	C₂×Dic₈₆	—	4	1	no	—
75	C₂×C₈×C₄₃	SD₁₆	1	4	yes	—
76	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
77	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₄₃	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₄₃	#41	1	4	yes	—
80	C₂×C₈×C₄₃	C₂⁴	1	1	yes	characteristic
81	C₄⋊C₄×C₄₃	—	2	4	no	—
82	C₄⋊C₄×C₄₃	—	4	4	no	—
83	C₄⋊C₄×C₄₃	—	4	2	no	—
84	C₂²⋊C₄×C₄₃	—	4	2	no	—
85	C₄²×C₄₃	—	4	2	no	—
86	C₄²×C₄₃	D₈	1	2	yes	—
87	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
88	C₄²×C₄₃	C₂×D₄	1	1	yes	characteristic
89	(C₂×D₄₃)⋊C₄	—	4	4	no	—
90	(C₂×D₄₃)⋊C₄	—	4	4	no	—
91	Dic₁₇₂	—	2	4	no	—
92	Dic₁₇₂	—	4	2	no	—
93	D₃₄₄	—	2	4	no	—
94	D₃₄₄	—	4	2	no	—
95	#10621873	—	4	4	no	—
96	#10621873	—	4	4	no	—
97	C₄₃⋊C₁₆	—	2	4	no	—
98	C₂³×C₄₃	C₂×D₈	1	1	yes	characteristic
99	C₂²×D₄₃	—	4	2	no	—
100	C₂²×D₄₃	—	8	4	no	—
101	C₂²×D₄₃	—	8	2	no	—
102	Q₈×C₄₃	—	8	4	no	—
103	C₂×C₄×C₄₃	—	4	2	no	—
104	C₂×C₄×C₄₃	—	4	4	no	—
105	C₂×C₄×C₄₃	#124	1	2	yes	—
106	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₄₃	C₂×D₈	1	2	yes	—
108	C₂×C₄×C₄₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₄₃	C₂×SD₁₆	1	2	yes	—
110	C₂×C₄×C₄₃	C₂²×D₄	1	1	yes	characteristic
111	C₂×Dic₄₃	—	4	2	no	—
112	C₂×Dic₄₃	—	8	2	no	—
113	D₁₇₂	—	4	2	no	—
114	D₁₇₂	—	4	2	no	—
115	D₁₇₂	—	8	2	no	—
116	Dic₈₆	—	4	2	no	—
117	Dic₈₆	—	4	2	no	—
118	Dic₈₆	—	8	2	no	—
119	C₈×C₄₃	#124	1	2	yes	—
120	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
121	C₈×C₄₃	C₂×SD₁₆	1	2	yes	—
122	C₈×C₄₃	C₂×#41	1	2	yes	—
123	#33706	—	43	1	no	maximal, Sylow
124	C₂²×C₄₃	—	2	2	no	—
125	C₂²×C₄₃	#486	1	2	yes	—
126	C₂²×C₄₃	C₂²×D₈	1	1	yes	characteristic
127	C₂×D₄₃	—	8	2	no	—
128	C₂×D₄₃	—	8	4	no	—
129	C₂×D₄₃	—	16	4	no	—
130	C₄×C₄₃	—	2	2	no	—
131	C₄×C₄₃	—	2	2	no	—
132	C₄×C₄₃	—	8	4	no	—
133	C₄×C₄₃	#486	1	2	yes	—
134	C₄×C₄₃	C₂×#124	1	1	yes	characteristic
135	C₄×C₄₃	C₂²×SD₁₆	1	1	yes	characteristic
136	Dic₄₃	—	8	2	no	—
137	#3160	D₄₃	1	1	yes	characteristic, complemented
138	C₂×#495	—	43	1	no	—
139	#2243	—	43	4	no	—
140	#2242	—	43	4	no	—
141	#2141	—	43	2	no	—
142	#2140	—	43	2	no	—
143	C₄×M₃₂	—	43	1	no	—
144	C₂×C₄₃	—	2	2	no	—
145	C₂×C₄₃	C₂×#486	1	1	yes	characteristic
146	C₂×C₄₃	#3414	1	2	yes	—
147	D₄₃	—	16	2	no	—
148	C₂²×Q₁₆	—	43	1	no	—
149	C₂²×SD₁₆	—	86	1	no	—
150	C₂²×D₈	—	43	1	no	—
151	#533	—	2	1	no	—
152	C₂×#124	—	86	1	no	—
153	C₂×M₃₂	—	43	2	no	—
154	#500	C₂×D₄₃	1	2	yes	complemented
155	#499	—	2	1	no	—
156	#499	C₂×D₄₃	1	2	yes	complemented
157	#498	—	2	1	no	—
158	#495	—	43	2	no	—
159	#495	—	43	2	no	—
160	#495	—	86	1	no	—
161	#495	—	86	1	no	—
162	#427	—	2	1	no	—
163	#427	C₂×D₄₃	1	2	yes	—
164	C₂×C₄×C₈	C₂×D₄₃	1	1	yes	characteristic
165	#358	—	43	8	no	—
166	#357	—	43	8	no	—
167	#346	—	43	2	no	—
168	C₄×C₁₆	—	43	2	no	—
169	C₄₃	#33706	1	1	yes	characteristic, complemented, Sylow
170	C₂²×Q₈	—	86	1	no	—
171	C₂²×D₄	—	86	1	no	—
172	C₂×Q₁₆	—	43	2	no	—
173	C₂×Q₁₆	—	86	4	no	—
174	C₂×Q₁₆	—	86	1	no	—
175	C₂×SD₁₆	—	86	2	no	—
176	C₂×SD₁₆	—	86	2	no	—
177	C₂×SD₁₆	—	172	4	no	—
178	C₂×D₈	—	43	2	no	—
179	C₂×D₈	—	86	4	no	—
180	C₂×D₈	—	86	1	no	—
181	C₂²×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
182	C₂²×C₈	C₂²×D₄₃	1	1	yes	characteristic
183	C₄⋊Q₈	—	2	2	no	—
184	#125	—	2	2	no	—
185	#124	—	86	2	no	—
186	#124	—	86	2	no	—
187	#124	—	172	2	no	—
188	C₂²⋊Q₈	—	4	2	no	—
189	C₄×Q₈	—	4	2	no	—
190	#117	—	2	2	no	—
191	C₂×C₂²⋊C₄	—	172	2	no	—
192	C₂×C₄²	C₂²⋊D₄₃	1	1	yes	characteristic
193	M₃₂	—	86	4	no	—
194	C₂×C₁₆	—	43	4	no	—
195	#107	—	2	2	no	—
196	#107	C₂²×D₄₃	1	4	yes	—
197	#106	—	2	2	no	—
198	#106	—	2	2	no	—
199	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
200	C₄×C₈	C₂²⋊D₄₃	1	1	yes	characteristic
201	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
202	C₄×C₈	C₂²×D₄₃	1	1	yes	characteristic
203	C₂⁴	—	172	1	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	86	2	no	—
206	C₂×Q₈	—	172	2	no	—
207	C₂×Q₈	—	172	2	no	—
208	C₂×Q₈	—	172	1	no	—
209	C₂×D₄	—	86	2	no	—
210	C₂×D₄	—	172	2	no	—
211	C₂×D₄	—	172	1	no	—
212	C₂×D₄	—	172	2	no	—
213	C₂²×C₄	—	172	1	no	—
214	C₂²×C₄	#10621882	1	2	yes	—
215	C₂²×C₄	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
216	Q₁₆	—	86	4	no	—
217	Q₁₆	—	172	2	no	—
218	SD₁₆	—	172	4	no	—
219	SD₁₆	—	172	4	no	—
220	D₈	—	86	4	no	—
221	D₈	—	172	2	no	—
222	C₂×C₈	#10621884	1	4	yes	—
223	C₂×C₈	#10621902	1	4	yes	—
224	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
225	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
226	C₂×C₈	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
227	C₂×C₈	C₂³×D₄₃	1	1	yes	characteristic
228	C₄⋊C₄	—	2	4	no	—
229	C₄⋊C₄	—	4	4	no	—
230	C₄⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	2	no	—
232	C₂²⋊C₄	—	172	4	no	—
233	C₂²⋊C₄	—	172	4	no	—
234	C₄²	—	4	2	no	—
235	C₄²	#10621882	1	2	yes	—
236	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
237	C₄²	C₂×C₂²⋊D₄₃	1	1	yes	characteristic
238	C₁₆	—	86	4	no	—
239	C₂³	—	344	4	no	—
240	C₂³	—	344	2	no	—
241	C₂³	—	172	2	no	—
242	C₂³	C₂×#10621882	1	1	yes	characteristic
243	Q₈	—	8	4	no	—
244	Q₈	—	344	2	no	—
245	Q₈	—	172	2	no	—
246	Q₈	—	172	2	no	—
247	D₄	—	344	2	no	—
248	D₄	—	172	2	no	—
249	D₄	—	172	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	344	2	no	—
253	C₂×C₄	—	172	2	no	—
254	C₂×C₄	#13063314	1	2	yes	—
255	C₂×C₄	C₂×#10621882	1	2	yes	—
256	C₂×C₄	C₂×#10621882	1	2	yes	—
257	C₂×C₄	C₂×#10621882	1	1	yes	characteristic
258	C₂×C₄	C₂×#10621884	1	2	yes	—
259	C₂×C₄	C₂²×C₂²⋊D₄₃	1	1	yes	characteristic
260	C₈	#13063314	1	2	yes	—
261	C₈	C₂×#10621884	1	2	yes	—
262	C₈	C₂×#10621884	1	2	yes	—
263	C₈	C₂×#10621902	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	344	2	no	—
266	C₂²	—	344	4	no	—
267	C₂²	—	688	4	no	—
268	C₂²	#14583247	1	2	yes	—
269	C₂²	C₂²×#10621882	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	8	4	no	—
273	C₄	—	344	2	no	—
274	C₄	#14583247	1	2	yes	—
275	C₄	C₂×#13063314	1	1	yes	characteristic
276	C₄	C₂²×#10621884	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	688	2	no	—
279	C₂	C₂×#14583247	1	1	yes	characteristic
280	C₂	#19147729	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#10621882
commutator, cosocle	C₂×C₈×C₄₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₄₃
Fitting	C₄₃×#3160	C₂
socle	C₂²×C₄₃	C₂²×D₈
prime radical	C₂²×D₃₄₄	C₂²
hypercenter	#3160	D₄₃
hypocenter	C₄₃	#33706
LCS	G ▷ C₂×C₈×C₄₃ ▷ C₄×C₄₃ ▷ C₂×C₄₃ ▷ C₄₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
2. #19136834	C₂	semidirect
3. #19136829	C₂	semidirect
4. #19136314	C₂	semidirect
6. C₄×#13063265	C₂	semidirect
7. C₄₃×#3160	C₂	semidirect
17. #14582702	C₂²	semidirect
19. #14582675	C₂²	semidirect
20. C₄×C₄₃⋊C₁₆	C₂²	semidirect
22. C₄₃×#500	C₂²	semidirect
24. C₄₃×#499	C₂²	semidirect
137. #3160	D₄₃	semidirect
154. #500	C₂×D₄₃	semidirect
156. #499	C₂×D₄₃	semidirect
169. C₄₃	#33706	semidirect

Maximal subgroups

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

subgroup	index
C₂×#14583283	2
#19136834	2
#19136829	2
#19136314	2
#19136310	2
C₄×#13063265	2
C₄₃×#3160	2
#33706	43