Subgroups — #44559270

Order 32128.

458 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#30632484	C₂	1	1	yes	maximal, characteristic
2	#30632452	C₂	1	2	yes	maximal, complemented
3	#30631809	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631808	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631807	C₂	1	2	yes	maximal, complemented
6	#30631805	C₂	1	1	yes	maximal, characteristic, complemented
7	#30631801	C₂	1	2	yes	maximal, complemented
8	#30631734	C₂	1	1	yes	maximal, characteristic, complemented
9	#104×D₂₅₁	C₂	1	2	yes	maximal, complemented
10	C₄×#16860793	C₂	1	1	yes	maximal, characteristic, complemented
11	#443×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#41×D₂₅₁	C₂²	1	2	yes	—
13	#22548747	C₄	1	2	yes	complemented
14	#22548744	C₄	1	2	yes	complemented
15	#22548742	C₄	1	2	yes	complemented
16	#22548739	C₄	1	2	yes	complemented
17	#22548737	C₂²	1	1	yes	characteristic
18	#22548735	C₂²	1	2	yes	—
19	#22548723	—	2	2	no	—
20	#22548723	C₂²	1	2	yes	complemented
21	#22548722	C₂²	1	2	yes	complemented
22	#22548721	—	2	2	no	—
23	#22548721	C₂²	1	2	yes	complemented
24	C₂×#16860793	—	2	2	no	—
25	C₂×#16860793	C₂²	1	1	yes	characteristic
26	C₂×#16860793	C₂²	1	1	yes	characteristic
27	M₁₆×D₂₅₁	—	2	2	no	—
28	M₁₆×D₂₅₁	C₂²	1	2	yes	—
29	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
30	#22548651	C₂²	1	1	yes	characteristic, complemented
31	#22548650	C₂²	1	2	yes	complemented
32	#22548643	C₂²	1	2	yes	complemented
33	#22548642	C₂²	1	2	yes	complemented
34	#22548641	C₂²	1	2	yes	complemented
35	#22548625	C₂²	1	1	yes	characteristic, complemented
36	#22548625	C₂²	1	1	yes	characteristic, complemented
37	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic, complemented
38	#135×C₂₅₁	C₂²	1	1	yes	characteristic
39	#131×C₂₅₁	C₂²	1	2	yes	complemented
40	#108×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
41	#104×C₂₅₁	C₂²	1	2	yes	complemented
42	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic, complemented
43	#16860811	C₂×C₄	1	2	yes	—
44	#16860810	—	2	2	no	—
45	#16860810	C₂³	1	2	yes	—
46	#16860809	—	2	2	no	—
47	#16860809	C₂×C₄	1	2	yes	—
48	Q₈×D₂₅₁	C₂×C₄	1	2	yes	—
49	D₄×D₂₅₁	—	2	2	no	—
50	D₄×D₂₅₁	C₂×C₄	1	2	yes	—
51	C₂×C₄×D₂₅₁	—	2	2	no	—
52	C₂×C₄×D₂₅₁	—	2	1	no	—
53	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
54	Dic₁₀₀₄	C₂×C₄	1	1	yes	characteristic, complemented
55	#16860800	C₂×C₄	1	2	yes	complemented
56	#16860799	C₂×C₄	1	2	yes	complemented
57	#16860798	C₂×C₄	1	2	yes	complemented
58	#16860797	C₂×C₄	1	2	yes	complemented
59	D₂₀₀₈	C₂×C₄	1	1	yes	characteristic, complemented
60	#16860795	C₂×C₄	1	2	yes	complemented
61	#16860794	—	2	2	no	—
62	#16860794	C₂³	1	2	yes	—
63	#16860793	—	2	2	no	—
64	#16860793	—	2	2	no	—
65	#16860793	—	2	2	no	—
66	#16860793	C₂×C₄	1	2	yes	—
67	#16860793	C₂×C₄	1	2	yes	—
68	#16860793	D₄	1	2	yes	—
69	#16860793	D₄	1	2	yes	—
70	C₂×C₂₅₁⋊C₈	—	2	2	no	—
71	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₂₅₁	—	2	2	no	—
74	C₈×D₂₅₁	—	2	2	no	—
75	C₈×D₂₅₁	—	2	2	no	—
76	C₄×Dic₂₅₁	—	2	1	no	—
77	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
78	#41×C₂₅₁	C₂³	1	2	yes	—
79	Q₁₆×C₂₅₁	C₂×C₄	1	1	yes	characteristic, complemented
80	SD₁₆×C₂₅₁	C₂×C₄	1	2	yes	complemented
81	D₈×C₂₅₁	C₂×C₄	1	1	yes	characteristic, complemented
82	M₁₆×C₂₅₁	—	2	2	no	—
83	M₁₆×C₂₅₁	C₂³	1	2	yes	—
84	C₂×C₈×C₂₅₁	—	2	2	no	—
85	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
86	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
87	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
88	C₂²×D₂₅₁	—	2	2	no	—
89	C₂²×D₂₅₁	#41	1	1	yes	characteristic
90	Dic₅₀₂	—	2	2	no	—
91	Dic₅₀₂	C₂²×C₄	1	2	yes	—
92	D₁₀₀₄	—	2	2	no	—
93	D₁₀₀₄	C₂²×C₄	1	2	yes	—
94	C₂²⋊D₂₅₁	—	2	2	no	—
95	C₂²⋊D₂₅₁	—	2	2	no	—
96	C₂²⋊D₂₅₁	—	2	2	no	—
97	C₂×Dic₂₅₁	—	2	2	no	—
98	C₂×Dic₂₅₁	—	2	1	no	—
99	C₂×Dic₂₅₁	#41	1	1	yes	characteristic
100	C₄×D₂₅₁	—	2	2	no	—
101	C₄×D₂₅₁	—	2	2	no	—
102	C₄×D₂₅₁	—	2	2	no	—
103	C₄×D₂₅₁	—	4	2	no	—
104	C₄×D₂₅₁	C₂²×C₄	1	2	yes	—
105	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
106	C₂₅₁⋊C₈	—	2	2	no	—
107	C₂₅₁⋊C₈	C₂²×C₄	1	2	yes	—
108	C₂₅₁⋊C₈	C₂×D₄	1	2	yes	—
109	Q₈×C₂₅₁	C₂²×C₄	1	2	yes	—
110	D₄×C₂₅₁	—	2	2	no	—
111	D₄×C₂₅₁	C₂²×C₄	1	2	yes	—
112	C₂×C₄×C₂₅₁	—	2	1	no	—
113	C₂×C₄×C₂₅₁	—	2	2	no	—
114	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
115	C₈×C₂₅₁	—	2	2	no	—
116	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
117	C₈×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
118	C₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
119	C₈×C₂₅₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₂₅₁	—	2	2	no	—
121	C₂²×C₂₅₁	C₂×#41	1	1	yes	characteristic
122	C₂×D₂₅₁	—	2	2	no	—
123	C₂×D₂₅₁	—	4	1	no	—
124	C₂×D₂₅₁	—	8	2	no	—
125	C₂×D₂₅₁	C₄×D₄	1	2	yes	—
126	C₄×C₂₅₁	—	2	2	no	—
127	C₄×C₂₅₁	—	2	2	no	—
128	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
129	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
130	Dic₂₅₁	—	2	2	no	—
131	Dic₂₅₁	—	4	1	no	—
132	Dic₂₅₁	C₄×D₄	1	2	yes	—
133	C₂×C₂₅₁	—	2	1	no	—
134	C₂×C₂₅₁	—	4	2	no	—
135	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
136	D₂₅₁	—	2	2	no	—
137	D₂₅₁	—	4	2	no	—
138	C₂₅₁	#2956	1	1	yes	characteristic, complemented, Sylow
139	#2956	—	251	1	no	maximal, Sylow
140	#575	—	251	1	no	—
141	#568	—	251	2	no	—
142	#444	—	251	1	no	—
143	#444	—	251	2	no	—
144	#444	—	251	1	no	—
145	#443	—	251	2	no	—
146	#443	—	251	1	no	—
147	#443	D₂₅₁	1	1	yes	characteristic, complemented
148	#430	—	251	1	no	—
149	C₂×#104	—	251	2	no	—
150	C₄×M₁₆	—	251	1	no	—
151	C₂×#41	—	251	2	no	—
152	#137	—	251	2	no	—
153	#137	—	251	2	no	—
154	AGL(1,ℤ/8)	—	251	2	no	—
155	AGL(1,ℤ/8)	—	251	2	no	—
156	#135	—	251	2	no	—
157	#135	—	251	1	no	—
158	#135	C₂×D₂₅₁	1	1	yes	characteristic
159	#131	—	502	2	no	—
160	#131	—	502	2	no	—
161	#131	—	251	2	no	—
162	#131	—	251	2	no	—
163	#131	—	251	2	no	—
164	#131	C₂×D₂₅₁	1	2	yes	complemented
165	C₂×M₁₆	—	502	2	no	—
166	C₂×M₁₆	—	502	2	no	—
167	C₂×M₁₆	—	251	2	no	—
168	C₂×M₁₆	—	251	1	no	—
169	C₂×M₁₆	—	251	1	no	—
170	C₂×C₄²	—	251	1	no	—
171	#108	—	251	2	no	—
172	#108	—	251	1	no	—
173	#108	C₂×D₂₅₁	1	1	yes	characteristic, complemented
174	#104	—	251	2	no	—
175	#104	—	251	2	no	—
176	#104	—	251	2	no	—
177	#104	C₂×D₂₅₁	1	2	yes	complemented
178	#97	—	251	1	no	—
179	#97	—	251	1	no	—
180	C₄×C₈	—	251	1	no	—
181	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic, complemented
182	#41	—	502	2	no	—
183	#41	—	502	2	no	—
184	#41	—	251	2	no	—
185	#41	—	251	2	no	—
186	#41	—	251	2	no	—
187	#41	C₂²×D₂₅₁	1	2	yes	—
188	C₂×Q₈	—	251	2	no	—
189	C₂×D₄	—	502	2	no	—
190	C₂×D₄	—	251	2	no	—
191	C₂²×C₄	—	502	1	no	—
192	C₂²×C₄	—	502	2	no	—
193	C₂²×C₄	—	251	1	no	—
194	Q₁₆	—	251	2	no	—
195	Q₁₆	—	251	1	no	—
196	Q₁₆	C₄×D₂₅₁	1	1	yes	characteristic, complemented
197	SD₁₆	—	251	2	no	—
198	SD₁₆	—	251	2	no	—
199	SD₁₆	—	251	2	no	—
200	SD₁₆	C₄×D₂₅₁	1	2	yes	complemented
201	D₈	—	251	2	no	—
202	D₈	—	251	1	no	—
203	D₈	C₄×D₂₅₁	1	1	yes	characteristic, complemented
204	M₁₆	—	2	2	no	—
205	M₁₆	—	502	2	no	—
206	M₁₆	—	502	2	no	—
207	M₁₆	—	502	2	no	—
208	M₁₆	—	502	2	no	—
209	M₁₆	—	251	2	no	—
210	M₁₆	—	251	2	no	—
211	M₁₆	—	251	2	no	—
212	M₁₆	—	251	2	no	—
213	M₁₆	—	251	2	no	—
214	M₁₆	C₂²×D₂₅₁	1	2	yes	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	502	2	no	—
217	C₂×C₈	—	502	2	no	—
218	C₂×C₈	—	502	2	no	—
219	C₂×C₈	—	502	2	no	—
220	C₂×C₈	—	251	1	no	—
221	C₂×C₈	—	251	1	no	—
222	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
223	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
224	C₄²	—	502	1	no	—
225	C₄²	—	251	1	no	—
226	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
227	C₂³	—	502	2	no	—
228	C₂³	—	251	1	no	—
229	Q₈	—	502	2	no	—
230	Q₈	—	251	2	no	—
231	Q₈	C₂×C₄×D₂₅₁	1	2	yes	—
232	D₄	—	2	2	no	—
233	D₄	—	502	2	no	—
234	D₄	—	502	2	no	—
235	D₄	—	502	2	no	—
236	D₄	—	502	2	no	—
237	D₄	—	251	2	no	—
238	D₄	C₂×C₄×D₂₅₁	1	2	yes	—
239	C₂×C₄	—	2	1	no	—
240	C₂×C₄	—	2	2	no	—
241	C₂×C₄	—	1004	2	no	—
242	C₂×C₄	—	502	2	no	—
243	C₂×C₄	—	502	2	no	—
244	C₂×C₄	—	502	1	no	—
245	C₂×C₄	—	502	2	no	—
246	C₂×C₄	—	502	2	no	—
247	C₂×C₄	—	251	1	no	—
248	C₂×C₄	—	251	2	no	—
249	C₂×C₄	—	251	2	no	—
250	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
251	C₈	—	2	2	no	—
252	C₈	—	502	2	no	—
253	C₈	—	251	2	no	—
254	C₈	—	251	2	no	—
255	C₈	C₂×C₄×D₂₅₁	1	1	yes	characteristic
256	C₈	C₂×C₄×D₂₅₁	1	1	yes	characteristic
257	C₈	D₄×D₂₅₁	1	1	yes	characteristic
258	C₈	D₄×D₂₅₁	1	1	yes	characteristic
259	C₂²	—	2	2	no	—
260	C₂²	—	2008	2	no	—
261	C₂²	—	1004	1	no	—
262	C₂²	—	502	2	no	—
263	C₂²	—	251	2	no	—
264	C₂²	#41×D₂₅₁	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	2	no	—
267	C₄	—	1004	1	no	—
268	C₄	—	502	2	no	—
269	C₄	—	251	2	no	—
270	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
271	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
272	C₂	—	2	1	no	—
273	C₂	—	4	2	no	—
274	C₂	—	1004	2	no	—
275	C₂	—	502	2	no	—
276	C₂	C₄×D₄×D₂₅₁	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#443×C₂₅₁	C₂
socle	C₂×C₂₅₁	C₂×C₄×D₄
prime radical	#30632484	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#443	D₂₅₁
hypocenter	C₂₅₁	#2956
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #443

Complements

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

N	H ≅ G/N	type
2. #30632452	C₂	semidirect
3. #30631809	C₂	semidirect
4. #30631808	C₂	semidirect
5. #30631807	C₂	semidirect
6. #30631805	C₂	semidirect
7. #30631801	C₂	semidirect
8. #30631734	C₂	semidirect
9. #104×D₂₅₁	C₂	semidirect
10. C₄×#16860793	C₂	semidirect
11. #443×C₂₅₁	C₂	semidirect
13. #22548747	C₄	semidirect
14. #22548744	C₄	semidirect
15. #22548742	C₄	semidirect
16. #22548739	C₄	semidirect
20. #22548723	C₂²	semidirect
21. #22548722	C₂²	semidirect
23. #22548721	C₂²	semidirect
30. #22548651	C₂²	semidirect
31. #22548650	C₂²	semidirect
32. #22548643	C₂²	semidirect
33. #22548642	C₂²	semidirect
34. #22548641	C₂²	semidirect
35. #22548625	C₂²	semidirect
36. #22548625	C₂²	semidirect
37. C₄×C₂₅₁⋊C₈	C₂²	semidirect
39. #131×C₂₅₁	C₂²	semidirect
40. #108×C₂₅₁	C₂²	semidirect
41. #104×C₂₅₁	C₂²	semidirect
42. C₄×C₈×C₂₅₁	C₂²	semidirect
54. Dic₁₀₀₄	C₂×C₄	semidirect
55. #16860800	C₂×C₄	semidirect
56. #16860799	C₂×C₄	semidirect
57. #16860798	C₂×C₄	semidirect
58. #16860797	C₂×C₄	semidirect
59. D₂₀₀₈	C₂×C₄	semidirect
60. #16860795	C₂×C₄	semidirect
79. Q₁₆×C₂₅₁	C₂×C₄	semidirect
80. SD₁₆×C₂₅₁	C₂×C₄	semidirect
81. D₈×C₂₅₁	C₂×C₄	semidirect
138. C₂₅₁	#2956	semidirect
147. #443	D₂₅₁	semidirect
164. #131	C₂×D₂₅₁	semidirect
173. #108	C₂×D₂₅₁	semidirect
177. #104	C₂×D₂₅₁	semidirect
181. C₄×C₈	C₂×D₂₅₁	semidirect
196. Q₁₆	C₄×D₂₅₁	semidirect
200. SD₁₆	C₄×D₂₅₁	semidirect
203. D₈	C₄×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632484	2
#30632452	2
#30631809	2
#30631808	2
#30631807	2
#30631805	2
#30631801	2
#30631734	2
#104×D₂₅₁	2
C₄×#16860793	2
#443×C₂₅₁	2
#2956	251