Subgroups — #44559178

Order 32128.

548 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548740	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#22548677	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631796	C₂	1	2	yes	maximal, complemented
5	#30631790	C₂	1	2	yes	maximal, complemented
6	#30631772	C₂	1	2	yes	maximal, complemented
7	#30631767	C₂	1	2	yes	maximal, complemented
8	#30631727	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631693	C₂	1	1	yes	maximal, characteristic, complemented
10	M₁₆×Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₁₀₀₄	C₂²	1	1	yes	characteristic
13	C₂²×C₄×D₂₅₁	—	2	1	no	—
14	#22548740	C₄	1	8	yes	complemented
15	C₂×#16860797	C₂²	1	2	yes	—
16	C₂×#16860795	C₂²	1	2	yes	—
17	C₂×#16860794	C₂²	1	1	yes	characteristic
18	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
19	Q₈×Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
20	D₄×Dic₂₅₁	—	2	1	no	—
21	D₄×Dic₂₅₁	—	2	1	no	—
22	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
23	D₄×Dic₂₅₁	C₂²	1	1	yes	characteristic, complemented
24	#22548677	—	2	1	no	—
25	#22548677	—	2	1	no	—
26	#22548677	C₂²	1	2	yes	—
27	#22548677	C₂²	1	2	yes	—
28	#22548674	—	2	1	no	—
29	#22548674	C₂²	1	1	yes	characteristic
30	C₂×(C₂×D₂₅₁)⋊C₄	—	2	1	no	—
31	C₂×C₄×Dic₂₅₁	—	2	1	no	—
32	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
33	#22548648	C₂²	1	2	yes	complemented
34	#22548646	C₂²	1	2	yes	complemented
35	#22548637	C₂²	1	2	yes	—
36	#22548637	C₂²	1	2	yes	—
37	#22548625	C₂²	1	2	yes	complemented
38	C₈×Dic₂₅₁	C₂²	1	2	yes	complemented
39	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
40	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
41	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
42	#103×C₂₅₁	C₂²	1	2	yes	complemented
43	#102×C₂₅₁	C₂²	1	2	yes	complemented
44	C₂³×D₂₅₁	—	2	1	no	—
45	C₂×D₁₀₀₄	—	2	1	no	—
46	C₂×D₁₀₀₄	—	2	2	no	—
47	C₂×D₁₀₀₄	C₂×C₄	1	4	yes	complemented
48	C₂×D₁₀₀₄	C₂³	1	2	yes	—
49	C₂²×Dic₂₅₁	—	2	1	no	—
50	C₂²×Dic₂₅₁	—	2	1	no	—
51	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
52	C₂²×Dic₂₅₁	D₄	1	1	yes	characteristic
53	C₂×C₄×D₂₅₁	—	2	2	no	—
54	C₂×C₄×D₂₅₁	—	2	2	no	—
55	C₂×C₄×D₂₅₁	—	4	4	no	—
56	#16860797	C₂×C₄	1	8	yes	complemented
57	#16860795	C₂×C₄	1	8	yes	complemented
58	#16860794	C₂×C₄	1	4	yes	complemented
59	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
60	C₄⋊Dic₂₅₁	—	2	1	no	—
61	C₄⋊Dic₂₅₁	—	2	1	no	—
62	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
63	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
64	C₂²⋊Dic₂₅₁	—	2	1	no	—
65	C₂²⋊Dic₂₅₁	—	2	1	no	—
66	C₂²⋊Dic₂₅₁	—	2	1	no	—
67	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
68	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
69	C₄×Dic₂₅₁	—	2	1	no	—
70	C₄×Dic₂₅₁	—	2	1	no	—
71	C₄×Dic₂₅₁	—	2	1	no	—
72	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
73	C₄×Dic₂₅₁	D₄	1	1	yes	characteristic
74	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
75	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
76	#41×C₂₅₁	—	2	2	no	—
77	#41×C₂₅₁	C₂×C₄	1	4	yes	complemented
78	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
79	C₂×D₄×C₂₅₁	—	2	1	no	—
80	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
81	C₂²×C₄×C₂₅₁	—	2	1	no	—
82	C₂²×C₄×C₂₅₁	—	2	1	no	—
83	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
84	M₁₆×C₂₅₁	—	2	2	no	—
85	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
86	C₄⋊C₄×C₂₅₁	—	2	1	no	—
87	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
88	C₂²×D₂₅₁	—	2	4	no	—
89	C₂²×D₂₅₁	—	2	2	no	—
90	C₂²×D₂₅₁	—	4	2	no	—
91	C₂²×D₂₅₁	—	4	2	no	—
92	D₁₀₀₄	—	2	4	no	—
93	D₁₀₀₄	—	2	2	no	—
94	D₁₀₀₄	C₂²×C₄	1	4	yes	—
95	C₂×Dic₂₅₁	—	2	2	no	—
96	C₂×Dic₂₅₁	—	2	1	no	—
97	C₂×Dic₂₅₁	—	2	1	no	—
98	C₂×Dic₂₅₁	—	4	1	no	—
99	C₂×Dic₂₅₁	—	4	2	no	—
100	C₂×Dic₂₅₁	—	4	2	no	—
101	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
102	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
103	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
104	C₂×Dic₂₅₁	C₂×D₄	1	1	yes	characteristic
105	C₄×D₂₅₁	—	4	4	no	—
106	C₄×D₂₅₁	—	4	4	no	—
107	C₂₅₁⋊C₈	C₂²×C₄	1	4	yes	complemented
108	C₂³×C₂₅₁	—	2	1	no	—
109	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₂₅₁	—	2	1	no	—
111	Q₈×C₂₅₁	C₂²×C₄	1	2	yes	complemented
112	D₄×C₂₅₁	—	2	4	no	—
113	D₄×C₂₅₁	—	2	1	no	—
114	D₄×C₂₅₁	C₂²×C₄	1	2	yes	complemented
115	C₂×C₄×C₂₅₁	—	2	1	no	—
116	C₂×C₄×C₂₅₁	—	2	4	no	—
117	C₂×C₄×C₂₅₁	—	2	2	no	—
118	C₂×C₄×C₂₅₁	—	4	2	no	—
119	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
120	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₂₅₁	#41	1	2	yes	—
122	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
123	C₈×C₂₅₁	—	2	2	no	—
124	C₂²×C₂₅₁	—	2	1	no	—
125	C₂²×C₂₅₁	—	2	1	no	—
126	C₂²×C₂₅₁	—	2	2	no	—
127	C₂²×C₂₅₁	—	4	1	no	—
128	C₂²×C₂₅₁	—	4	1	no	—
129	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
130	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
131	C₂×D₂₅₁	—	2	4	no	—
132	C₂×D₂₅₁	—	4	8	no	—
133	C₂×D₂₅₁	—	4	2	no	—
134	C₂×D₂₅₁	—	4	2	no	—
135	C₄×C₂₅₁	—	2	2	no	—
136	C₄×C₂₅₁	—	4	2	no	—
137	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
138	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
139	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
140	Dic₂₅₁	—	2	1	no	—
141	Dic₂₅₁	—	2	1	no	—
142	Dic₂₅₁	—	2	1	no	—
143	Dic₂₅₁	—	4	1	no	—
144	Dic₂₅₁	—	4	1	no	—
145	Dic₂₅₁	AGL(1,ℤ/8)	1	2	yes	complemented
146	C₂×C₂₅₁	—	2	2	no	—
147	C₂×C₂₅₁	—	4	2	no	—
148	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
149	C₂×C₂₅₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₂₅₁	#575	1	1	yes	characteristic
151	D₂₅₁	—	4	4	no	—
152	C₂₅₁	C₄×AGL(1,ℤ/8)	1	1	yes	characteristic, complemented, Sylow
153	C₄×AGL(1,ℤ/8)	—	251	1	no	maximal, Sylow
154	C₂×AGL(1,ℤ/8)	—	251	1	no	—
155	C₄×#41	—	251	1	no	—
156	C₂×C₄×D₄	—	251	1	no	—
157	#442	—	251	2	no	—
158	#440	—	251	2	no	—
159	C₄×SD₁₆	—	251	2	no	—
160	C₄×D₈	—	251	2	no	—
161	#428	—	251	1	no	—
162	#418	—	251	1	no	—
163	#417	D₂₅₁	1	1	yes	characteristic, complemented
164	C₄×M₁₆	—	251	1	no	—
165	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
166	C₂²×D₄	—	251	1	no	—
167	C₂³×C₄	—	502	1	no	—
168	AGL(1,ℤ/8)	—	251	8	no	—
169	C₂×SD₁₆	—	251	2	no	—
170	C₂×D₈	—	251	2	no	—
171	C₂×M₁₆	—	251	1	no	—
172	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
173	C₄×Q₈	—	251	1	no	—
174	C₄×Q₈	—	251	1	no	—
175	C₄×D₄	—	502	1	no	—
176	C₄×D₄	—	502	1	no	—
177	C₄×D₄	—	502	1	no	—
178	C₄×D₄	—	502	1	no	—
179	C₄×D₄	—	251	1	no	—
180	C₄×D₄	—	251	1	no	—
181	C₄×D₄	—	251	2	no	—
182	C₄×D₄	—	251	2	no	—
183	#117	—	502	1	no	—
184	#117	—	251	1	no	—
185	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
186	C₂×C₂²⋊C₄	—	502	1	no	—
187	C₂×C₄²	—	502	1	no	—
188	C₂×C₄²	—	251	1	no	—
189	#107	—	251	2	no	—
190	#106	—	251	2	no	—
191	#103	C₂×D₂₅₁	1	2	yes	complemented
192	#102	—	251	2	no	—
193	#102	—	251	2	no	—
194	#102	C₂×D₂₅₁	1	2	yes	complemented
195	#97	—	251	2	no	—
196	C₄×C₈	—	251	2	no	—
197	C₂⁴	—	502	1	no	—
198	#41	—	2	2	no	—
199	#41	C₄×D₂₅₁	1	4	yes	complemented
200	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
201	C₂×D₄	—	2	1	no	—
202	C₂×D₄	—	502	1	no	—
203	C₂×D₄	—	502	2	no	—
204	C₂×D₄	—	251	4	no	—
205	C₂×D₄	—	251	2	no	—
206	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
207	C₂²×C₄	—	2	1	no	—
208	C₂²×C₄	—	2	1	no	—
209	C₂²×C₄	—	1004	4	no	—
210	C₂²×C₄	—	502	2	no	—
211	C₂²×C₄	—	502	2	no	—
212	C₂²×C₄	—	502	1	no	—
213	C₂²×C₄	—	502	1	no	—
214	C₂²×C₄	—	251	1	no	—
215	C₂²×C₄	—	251	1	no	—
216	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
217	SD₁₆	—	251	8	no	—
218	D₈	—	251	8	no	—
219	M₁₆	—	2	2	no	—
220	M₁₆	—	251	4	no	—
221	C₂×C₈	—	251	2	no	—
222	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
223	C₄⋊C₄	—	2	1	no	—
224	C₄⋊C₄	—	502	1	no	—
225	C₄⋊C₄	—	502	1	no	—
226	C₄⋊C₄	—	251	1	no	—
227	C₄⋊C₄	—	251	1	no	—
228	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
229	C₂²⋊C₄	—	502	2	no	—
230	C₂²⋊C₄	—	502	2	no	—
231	C₂²⋊C₄	—	502	1	no	—
232	C₂²⋊C₄	—	502	1	no	—
233	C₂²⋊C₄	—	502	1	no	—
234	C₄²	—	502	1	no	—
235	C₄²	—	502	1	no	—
236	C₄²	—	502	1	no	—
237	C₄²	—	251	1	no	—
238	C₄²	—	251	1	no	—
239	C₄²	—	251	1	no	—
240	C₄²	—	251	1	no	—
241	C₂³	—	2	1	no	—
242	C₂³	—	1004	2	no	—
243	C₂³	—	1004	2	no	—
244	C₂³	—	502	2	no	—
245	C₂³	—	502	4	no	—
246	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
247	Q₈	—	2	1	no	—
248	Q₈	C₂×C₄×D₂₅₁	1	2	yes	complemented
249	D₄	—	2	4	no	—
250	D₄	—	2	1	no	—
251	D₄	—	502	2	no	—
252	D₄	—	502	4	no	—
253	D₄	—	251	4	no	—
254	D₄	C₂×C₄×D₂₅₁	1	2	yes	complemented
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	4	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	4	2	no	—
259	C₂×C₄	—	1004	4	no	—
260	C₂×C₄	—	1004	4	no	—
261	C₂×C₄	—	1004	2	no	—
262	C₂×C₄	—	1004	2	no	—
263	C₂×C₄	—	1004	1	no	—
264	C₂×C₄	—	502	1	no	—
265	C₂×C₄	—	502	1	no	—
266	C₂×C₄	—	502	2	no	—
267	C₂×C₄	—	251	1	no	—
268	C₂×C₄	—	251	1	no	—
269	C₂×C₄	—	251	1	no	—
270	C₂×C₄	—	251	1	no	—
271	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
272	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
273	C₂×C₄	#16860809	1	2	yes	—
274	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
275	C₈	—	2	2	no	—
276	C₈	—	251	4	no	—
277	C₂²	—	2	1	no	—
278	C₂²	—	2	1	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	4	1	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	1004	8	no	—
283	C₂²	—	1004	2	no	—
284	C₂²	—	1004	2	no	—
285	C₂²	—	502	4	no	—
286	C₂²	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
287	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
288	C₄	—	2	2	no	—
289	C₄	—	4	2	no	—
290	C₄	—	1004	1	no	—
291	C₄	—	1004	1	no	—
292	C₄	—	502	1	no	—
293	C₄	—	502	1	no	—
294	C₄	—	502	1	no	—
295	C₄	—	251	2	no	—
296	C₄	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
297	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
298	C₄	C₂×#16860809	1	1	yes	characteristic
299	C₂	—	2	2	no	—
300	C₂	—	4	2	no	—
301	C₂	—	1004	4	no	—
302	C₂	C₂×(C₂²⋊D₂₅₁)⋊C₄	1	1	yes	characteristic
303	C₂	AGL(1,ℤ/8)×D₂₅₁	1	1	yes	characteristic
304	C₂	#30632486	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	#417×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548740	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#417	D₂₅₁
hypocenter	C₂₅₁	C₄×AGL(1,ℤ/8)
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #417

Complements

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

N	H ≅ G/N	type
2. #41×Dic₂₅₁	C₂	semidirect
3. C₂×#22548677	C₂	semidirect
4. #30631796	C₂	semidirect
5. #30631790	C₂	semidirect
6. #30631772	C₂	semidirect
7. #30631767	C₂	semidirect
8. #30631727	C₂	semidirect
9. #30631693	C₂	semidirect
10. M₁₆×Dic₂₅₁	C₂	semidirect
11. #417×C₂₅₁	C₂	semidirect
14. #22548740	C₄	semidirect
18. Q₈×Dic₂₅₁	C₂²	semidirect
19. Q₈×Dic₂₅₁	C₂²	semidirect
22. D₄×Dic₂₅₁	C₂²	semidirect
23. D₄×Dic₂₅₁	C₂²	semidirect
33. #22548648	C₂²	semidirect
34. #22548646	C₂²	semidirect
37. #22548625	C₂²	semidirect
38. C₈×Dic₂₅₁	C₂²	semidirect
42. #103×C₂₅₁	C₂²	semidirect
43. #102×C₂₅₁	C₂²	semidirect
47. C₂×D₁₀₀₄	C₂×C₄	semidirect
56. #16860797	C₂×C₄	semidirect
57. #16860795	C₂×C₄	semidirect
58. #16860794	C₂×C₄	semidirect
77. #41×C₂₅₁	C₂×C₄	semidirect
107. C₂₅₁⋊C₈	C₂²×C₄	semidirect
111. Q₈×C₂₅₁	C₂²×C₄	semidirect
114. D₄×C₂₅₁	C₂²×C₄	semidirect
145. Dic₂₅₁	AGL(1,ℤ/8)	semidirect
152. C₂₅₁	C₄×AGL(1,ℤ/8)	semidirect
163. #417	D₂₅₁	semidirect
191. #103	C₂×D₂₅₁	semidirect
194. #102	C₂×D₂₅₁	semidirect
199. #41	C₄×D₂₅₁	semidirect
248. Q₈	C₂×C₄×D₂₅₁	semidirect
254. D₄	C₂×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×#22548740	2
#41×Dic₂₅₁	2
C₂×#22548677	2
#30631796	2
#30631790	2
#30631772	2
#30631767	2
#30631727	2
#30631693	2
M₁₆×Dic₂₅₁	2
#417×C₂₅₁	2
C₄×AGL(1,ℤ/8)	251