Subgroups — #44559199

Order 32128.

506 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	C₂×#22548743	C₂	1	1	yes	maximal, characteristic
2	#30632167	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632161	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631800	C₂	1	2	yes	maximal, complemented
5	#30631789	C₂	1	2	yes	maximal, complemented
6	C₄×#16860799	C₂	1	2	yes	maximal, complemented
7	C₄×D₂₀₀₈	C₂	1	2	yes	maximal, complemented
8	#30631729	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631684	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#22548637	C₂	1	1	yes	maximal, characteristic, complemented
11	#405×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	1	yes	characteristic
13	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic
14	#22548743	C₄	1	8	yes	complemented
15	C₂×#16860799	C₂²	1	2	yes	—
16	C₂×D₂₀₀₈	C₂²	1	2	yes	—
17	C₄×Dic₅₀₂	C₂²	1	2	yes	complemented
18	C₄×D₁₀₀₄	C₂²	1	2	yes	—
19	C₄×D₁₀₀₄	C₂²	1	2	yes	—
20	C₄×D₁₀₀₄	C₂²	1	2	yes	complemented
21	#22548677	—	2	1	no	—
22	#22548677	—	2	1	no	—
23	(C₂²⋊D₂₅₁)⋊C₄	—	2	2	no	—
24	#22548674	C₂²	1	1	yes	characteristic
25	#22548673	—	2	1	no	—
26	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
27	C₄⋊C₄×D₂₅₁	—	2	1	no	—
28	C₂²⋊C₄×D₂₅₁	—	2	2	no	—
29	#22548649	C₂²	1	2	yes	complemented
30	#22548647	C₂²	1	2	yes	complemented
31	#22548640	C₂²	1	1	yes	characteristic, complemented
32	#22548640	C₂²	1	1	yes	characteristic, complemented
33	#22548637	C₂²	1	2	yes	—
34	#22548637	C₂²	1	2	yes	—
35	#22548637	C₂²	1	1	yes	characteristic, complemented
36	#22548637	C₂²	1	1	yes	characteristic, complemented
37	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
39	#117×C₂₅₁	C₂²	1	1	yes	characteristic
40	#97×C₂₅₁	C₂²	1	2	yes	complemented
41	C₄×C₈×C₂₅₁	C₂²	1	2	yes	complemented
42	C₂³×D₂₅₁	—	2	1	no	—
43	#16860810	—	2	2	no	—
44	#16860810	C₂×C₄	1	4	yes	complemented
45	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
46	C₂×D₁₀₀₄	—	2	1	no	—
47	C₂×D₁₀₀₄	—	2	2	no	—
48	C₂×D₁₀₀₄	C₂×C₄	1	4	yes	complemented
49	C₂×D₁₀₀₄	C₂³	1	2	yes	—
50	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
51	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
52	C₂²×Dic₂₅₁	—	2	1	no	—
53	C₂×C₄×D₂₅₁	—	2	1	no	—
54	C₂×C₄×D₂₅₁	—	2	2	no	—
55	C₂×C₄×D₂₅₁	—	2	2	no	—
56	C₂×C₄×D₂₅₁	—	2	2	no	—
57	#16860799	C₂×C₄	1	8	yes	complemented
58	D₂₀₀₈	C₂×C₄	1	8	yes	complemented
59	C₄⋊Dic₂₅₁	—	2	1	no	—
60	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
61	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
62	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
63	Dic₂₅₁⋊C₄	—	2	2	no	—
64	C₂²⋊Dic₂₅₁	—	2	1	no	—
65	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
66	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
67	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
68	C₄×Dic₂₅₁	—	2	2	no	—
69	C₄×Dic₂₅₁	—	2	1	no	—
70	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
71	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	complemented
72	C₂×C₈×C₂₅₁	—	2	2	no	—
73	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
74	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
76	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
77	C₄⋊C₄×C₂₅₁	D₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₅₁	D₄	1	2	yes	—
79	C₄²×C₂₅₁	C₂³	1	2	yes	—
80	C₂²×D₂₅₁	—	2	1	no	—
81	C₂²×D₂₅₁	—	2	2	no	—
82	C₂²×D₂₅₁	—	2	4	no	—
83	C₂²×D₂₅₁	—	8	1	no	—
84	C₂²×D₂₅₁	—	8	1	no	—
85	Dic₅₀₂	—	2	1	no	—
86	Dic₅₀₂	C₂²×C₄	1	2	yes	—
87	D₁₀₀₄	—	2	4	no	—
88	D₁₀₀₄	—	2	2	no	—
89	D₁₀₀₄	—	2	1	no	—
90	D₁₀₀₄	C₂²×C₄	1	2	yes	—
91	D₁₀₀₄	C₂²×C₄	1	4	yes	—
92	C₂²⋊D₂₅₁	—	2	4	no	—
93	C₂×Dic₂₅₁	—	2	1	no	—
94	C₂×Dic₂₅₁	—	2	1	no	—
95	C₂×Dic₂₅₁	—	2	1	no	—
96	C₂×Dic₂₅₁	—	2	2	no	—
97	C₂×Dic₂₅₁	—	8	1	no	—
98	C₄×D₂₅₁	—	2	4	no	—
99	C₄×D₂₅₁	—	8	2	no	—
100	C₄×D₂₅₁	—	8	2	no	—
101	C₄×D₂₅₁	—	8	2	no	—
102	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
104	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
105	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₂₅₁	#41	1	2	yes	—
108	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
109	C₈×C₂₅₁	—	2	1	no	—
110	C₈×C₂₅₁	—	2	1	no	—
111	C₈×C₂₅₁	C₂²×C₄	1	4	yes	—
112	C₂²×C₂₅₁	—	2	1	no	—
113	C₂²×C₂₅₁	—	2	1	no	—
114	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
115	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
116	C₂×D₂₅₁	—	2	2	no	—
117	C₂×D₂₅₁	—	2	4	no	—
118	C₂×D₂₅₁	—	4	1	no	—
119	C₂×D₂₅₁	—	4	2	no	—
120	C₂×D₂₅₁	—	4	2	no	—
121	C₂×D₂₅₁	—	4	1	no	—
122	C₂×D₂₅₁	—	8	4	no	—
123	C₄×C₂₅₁	—	2	2	no	—
124	C₄×C₂₅₁	—	2	2	no	—
125	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
126	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
127	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
128	Dic₂₅₁	—	2	2	no	—
129	Dic₂₅₁	—	4	1	no	—
130	Dic₂₅₁	—	4	2	no	—
131	Dic₂₅₁	—	4	1	no	—
132	C₂×C₂₅₁	—	2	2	no	—
133	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
134	C₂×C₂₅₁	#576	1	1	yes	characteristic
135	C₂×C₂₅₁	#577	1	1	yes	characteristic
136	D₂₅₁	—	4	2	no	—
137	D₂₅₁	—	4	4	no	—
138	C₂₅₁	#2946	1	1	yes	characteristic, complemented, Sylow
139	#2946	—	251	1	no	maximal, Sylow
140	C₂×AGL(1,ℤ/8)	—	251	1	no	—
141	#520	—	251	1	no	—
142	#518	—	251	1	no	—
143	#442	—	251	2	no	—
144	#440	—	251	2	no	—
145	C₄×SD₁₆	—	251	2	no	—
146	C₄×D₈	—	251	2	no	—
147	#428	—	251	1	no	—
148	#416	—	251	1	no	—
149	C₂×#102	—	251	1	no	—
150	#405	D₂₅₁	1	1	yes	characteristic, complemented
151	C₂×#41	—	251	1	no	—
152	C₂²×D₄	—	251	1	no	—
153	AGL(1,ℤ/8)	—	251	8	no	—
154	C₂×SD₁₆	—	251	2	no	—
155	C₂×D₈	—	251	2	no	—
156	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
157	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
158	C₄×Q₈	—	251	2	no	—
159	C₄×D₄	—	502	1	no	—
160	C₄×D₄	—	502	1	no	—
161	C₄×D₄	—	502	2	no	—
162	C₄×D₄	—	251	2	no	—
163	C₄×D₄	—	251	2	no	—
164	C₄×D₄	—	251	2	no	—
165	#117	—	502	1	no	—
166	#117	—	251	1	no	—
167	#117	C₂×D₂₅₁	1	1	yes	characteristic
168	C₂×C₄⋊C₄	—	502	1	no	—
169	C₂×C₄⋊C₄	—	251	1	no	—
170	C₂×C₂²⋊C₄	—	502	2	no	—
171	#107	—	251	2	no	—
172	#106	—	251	2	no	—
173	#103	—	251	1	no	—
174	#103	—	251	1	no	—
175	#102	—	251	2	no	—
176	#102	—	251	2	no	—
177	#102	—	251	1	no	—
178	#102	—	251	1	no	—
179	#97	C₂×D₂₅₁	1	2	yes	complemented
180	C₄×C₈	C₂×D₂₅₁	1	2	yes	complemented
181	C₂⁴	—	502	1	no	—
182	#41	—	502	2	no	—
183	#41	—	251	4	no	—
184	C₂×Q₈	—	251	1	no	—
185	C₂×D₄	—	502	1	no	—
186	C₂×D₄	—	502	1	no	—
187	C₂×D₄	—	502	2	no	—
188	C₂×D₄	—	251	4	no	—
189	C₂×D₄	—	251	1	no	—
190	C₂×D₄	—	251	2	no	—
191	C₂²×C₄	—	502	1	no	—
192	C₂²×C₄	—	502	2	no	—
193	C₂²×C₄	—	502	2	no	—
194	C₂²×C₄	—	502	2	no	—
195	C₂²×C₄	—	502	1	no	—
196	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
197	SD₁₆	—	251	8	no	—
198	D₈	—	251	8	no	—
199	M₁₆	C₄×D₂₅₁	1	4	yes	complemented
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
202	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
204	C₄⋊C₄	—	502	1	no	—
205	C₄⋊C₄	—	502	2	no	—
206	C₄⋊C₄	—	251	2	no	—
207	C₄⋊C₄	—	251	1	no	—
208	C₄⋊C₄	—	251	1	no	—
209	C₄⋊C₄	D₁₀₀₄	1	1	yes	characteristic
210	C₄⋊C₄	D₁₀₀₄	1	1	yes	characteristic
211	C₂²⋊C₄	—	502	1	no	—
212	C₂²⋊C₄	—	502	2	no	—
213	C₂²⋊C₄	—	502	2	no	—
214	C₂²⋊C₄	—	502	2	no	—
215	C₂²⋊C₄	D₁₀₀₄	1	2	yes	—
216	C₄²	—	502	2	no	—
217	C₄²	—	502	1	no	—
218	C₄²	C₂²×D₂₅₁	1	2	yes	—
219	C₂³	—	2008	1	no	—
220	C₂³	—	2008	1	no	—
221	C₂³	—	502	4	no	—
222	C₂³	—	502	1	no	—
223	C₂³	—	502	2	no	—
224	C₂³	C₂×D₁₀₀₄	1	1	yes	characteristic
225	Q₈	—	502	1	no	—
226	Q₈	—	251	2	no	—
227	D₄	—	502	4	no	—
228	D₄	—	502	2	no	—
229	D₄	—	502	1	no	—
230	D₄	—	502	4	no	—
231	D₄	—	251	2	no	—
232	D₄	—	251	4	no	—
233	C₂×C₄	—	2008	2	no	—
234	C₂×C₄	—	2008	2	no	—
235	C₂×C₄	—	2008	2	no	—
236	C₂×C₄	—	2008	1	no	—
237	C₂×C₄	—	502	1	no	—
238	C₂×C₄	—	502	1	no	—
239	C₂×C₄	—	502	2	no	—
240	C₂×C₄	—	502	1	no	—
241	C₂×C₄	—	502	4	no	—
242	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
243	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
244	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
245	C₂×C₄	C₂×D₁₀₀₄	1	1	yes	characteristic
246	C₂×C₄	#16860810	1	2	yes	—
247	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
248	C₈	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	C₂×C₄×D₂₅₁	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2008	4	no	—
254	C₂²	—	1004	1	no	—
255	C₂²	—	1004	2	no	—
256	C₂²	—	1004	2	no	—
257	C₂²	—	1004	1	no	—
258	C₂²	—	502	2	no	—
259	C₂²	—	502	4	no	—
260	C₂²	C₄×D₁₀₀₄	1	2	yes	—
261	C₂²	C₂²×D₁₀₀₄	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	2	2	no	—
264	C₄	—	1004	1	no	—
265	C₄	—	1004	2	no	—
266	C₄	—	1004	1	no	—
267	C₄	—	502	2	no	—
268	C₄	C₄×D₁₀₀₄	1	2	yes	—
269	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
270	C₄	C₂×#16860810	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	1004	4	no	—
273	C₂	—	1004	2	no	—
274	C₂	C₂×C₄×D₁₀₀₄	1	1	yes	characteristic
275	C₂	#30632492	1	1	yes	characteristic
276	C₂	#30632499	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₁₀₀₄
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#405×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548743	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#405	D₂₅₁
hypocenter	C₂₅₁	#2946
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405

Complements

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

N	H ≅ G/N	type
2. #30632167	C₂	semidirect
3. #30632161	C₂	semidirect
4. #30631800	C₂	semidirect
5. #30631789	C₂	semidirect
6. C₄×#16860799	C₂	semidirect
7. C₄×D₂₀₀₈	C₂	semidirect
8. #30631729	C₂	semidirect
9. #30631684	C₂	semidirect
10. C₂×#22548637	C₂	semidirect
11. #405×C₂₅₁	C₂	semidirect
14. #22548743	C₄	semidirect
17. C₄×Dic₅₀₂	C₂²	semidirect
20. C₄×D₁₀₀₄	C₂²	semidirect
29. #22548649	C₂²	semidirect
30. #22548647	C₂²	semidirect
31. #22548640	C₂²	semidirect
32. #22548640	C₂²	semidirect
35. #22548637	C₂²	semidirect
36. #22548637	C₂²	semidirect
40. #97×C₂₅₁	C₂²	semidirect
41. C₄×C₈×C₂₅₁	C₂²	semidirect
44. #16860810	C₂×C₄	semidirect
48. C₂×D₁₀₀₄	C₂×C₄	semidirect
57. #16860799	C₂×C₄	semidirect
58. D₂₀₀₈	C₂×C₄	semidirect
71. M₁₆×C₂₅₁	C₂×C₄	semidirect
138. C₂₅₁	#2946	semidirect
150. #405	D₂₅₁	semidirect
179. #97	C₂×D₂₅₁	semidirect
180. C₄×C₈	C₂×D₂₅₁	semidirect
199. M₁₆	C₄×D₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548743	2
#30632167	2
#30632161	2
#30631800	2
#30631789	2
C₄×#16860799	2
C₄×D₂₀₀₈	2
#30631729	2
#30631684	2
C₂×#22548637	2
#405×C₂₅₁	2
#2946	251