Subgroups — #44559198

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	#30632170	C₂	1	1	yes	maximal, characteristic, complemented
2	#30632162	C₂	1	1	yes	maximal, characteristic, complemented
3	#30631799	C₂	1	2	yes	maximal, complemented
4	#30631788	C₂	1	2	yes	maximal, complemented
5	SD₁₆×Dic₂₅₁	C₂	1	2	yes	maximal, complemented
6	D₈×Dic₂₅₁	C₂	1	2	yes	maximal, complemented
7	#30631729	C₂	1	1	yes	maximal, characteristic, complemented
8	#30631683	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#22548636	C₂	1	1	yes	maximal, characteristic, complemented
10	#30631645	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×AGL(1,ℤ/8)×C₂₅₁	C₂	1	1	yes	maximal, characteristic
12	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
13	Q₈×Dic₂₅₁	C₂²	1	2	yes	complemented
14	D₄×Dic₂₅₁	—	2	1	no	—
15	D₄×Dic₂₅₁	—	2	1	no	—
16	D₄×Dic₂₅₁	—	2	2	no	—
17	D₄×Dic₂₅₁	C₂²	1	2	yes	—
18	D₄×Dic₂₅₁	C₂²	1	2	yes	—
19	D₄×Dic₂₅₁	C₂²	1	2	yes	complemented
20	#22548674	—	2	1	no	—
21	#22548674	C₂²	1	1	yes	characteristic
22	#22548674	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Dic₂₅₁	—	2	1	no	—
24	C₂×C₄⋊Dic₂₅₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊Dic₂₅₁	—	2	2	no	—
26	#22548649	C₂²	1	2	yes	complemented
27	#22548647	C₂²	1	2	yes	complemented
28	#22548639	C₂²	1	1	yes	characteristic, complemented
29	#22548639	C₂²	1	1	yes	characteristic, complemented
30	#22548636	C₂²	1	2	yes	—
31	#22548636	C₂²	1	2	yes	—
32	#22548636	C₂²	1	1	yes	characteristic, complemented
33	#22548636	C₂²	1	1	yes	characteristic, complemented
34	#22548625	C₂²	1	2	yes	complemented
35	C₈×Dic₂₅₁	C₂²	1	2	yes	complemented
36	C₂×#41×C₂₅₁	C₂²	1	1	yes	characteristic
37	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
38	AGL(1,ℤ/8)×C₂₅₁	C₄	1	8	yes	complemented
39	C₂×SD₁₆×C₂₅₁	C₂²	1	2	yes	—
40	C₂×D₈×C₂₅₁	C₂²	1	2	yes	—
41	C₂×M₁₆×C₂₅₁	C₂²	1	1	yes	characteristic
42	C₂²×Dic₂₅₁	—	2	2	no	—
43	C₂²×Dic₂₅₁	—	2	2	no	—
44	C₂²×Dic₂₅₁	—	2	2	no	—
45	C₂²×Dic₂₅₁	—	2	1	no	—
46	C₂×C₂₅₁⋊C₈	—	2	2	no	—
47	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
48	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
49	C₄⋊Dic₂₅₁	—	2	2	no	—
50	C₄⋊Dic₂₅₁	—	2	1	no	—
51	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
52	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
53	C₄⋊Dic₂₅₁	C₂³	1	2	yes	—
54	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
55	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
56	C₂²⋊Dic₂₅₁	—	2	2	no	—
57	C₂²⋊Dic₂₅₁	—	2	2	no	—
58	C₂²⋊Dic₂₅₁	—	2	2	no	—
59	C₂²⋊Dic₂₅₁	—	2	1	no	—
60	C₂²⋊Dic₂₅₁	D₄	1	2	yes	—
61	C₄×Dic₂₅₁	—	2	2	no	—
62	C₄×Dic₂₅₁	—	2	1	no	—
63	C₄×Dic₂₅₁	C₂³	1	2	yes	—
64	C₂⁴×C₂₅₁	—	2	1	no	—
65	#41×C₂₅₁	—	2	2	no	—
66	#41×C₂₅₁	C₂×C₄	1	4	yes	complemented
67	C₂×Q₈×C₂₅₁	C₂³	1	1	yes	characteristic
68	C₂×D₄×C₂₅₁	—	2	1	no	—
69	C₂×D₄×C₂₅₁	—	2	1	no	—
70	C₂×D₄×C₂₅₁	—	2	2	no	—
71	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	complemented
72	C₂×D₄×C₂₅₁	C₂³	1	1	yes	characteristic
73	C₂×D₄×C₂₅₁	C₂³	1	2	yes	—
74	C₂²×C₄×C₂₅₁	—	2	1	no	—
75	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
76	SD₁₆×C₂₅₁	C₂×C₄	1	8	yes	complemented
77	D₈×C₂₅₁	C₂×C₄	1	8	yes	complemented
78	M₁₆×C₂₅₁	C₂×C₄	1	4	yes	complemented
79	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
80	C₂×Dic₂₅₁	—	2	1	no	—
81	C₂×Dic₂₅₁	—	2	1	no	—
82	C₂×Dic₂₅₁	—	2	2	no	—
83	C₂×Dic₂₅₁	—	8	2	no	—
84	C₂×Dic₂₅₁	—	8	2	no	—
85	C₂×Dic₂₅₁	—	8	2	no	—
86	C₂×Dic₂₅₁	—	8	1	no	—
87	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
88	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
89	C₂₅₁⋊C₈	—	2	1	no	—
90	C₂₅₁⋊C₈	—	2	1	no	—
91	C₂³×C₂₅₁	—	2	1	no	—
92	C₂³×C₂₅₁	—	2	2	no	—
93	C₂³×C₂₅₁	—	2	4	no	—
94	C₂³×C₂₅₁	—	8	1	no	—
95	C₂³×C₂₅₁	—	8	1	no	—
96	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
97	Q₈×C₂₅₁	—	2	1	no	—
98	Q₈×C₂₅₁	C₂²×C₄	1	2	yes	—
99	D₄×C₂₅₁	—	2	4	no	—
100	D₄×C₂₅₁	—	2	4	no	—
101	D₄×C₂₅₁	—	2	2	no	—
102	D₄×C₂₅₁	—	2	1	no	—
103	D₄×C₂₅₁	C₂²×C₄	1	2	yes	—
104	D₄×C₂₅₁	C₂²×C₄	1	4	yes	—
105	C₂×C₄×C₂₅₁	—	2	1	no	—
106	C₂×C₄×C₂₅₁	—	2	4	no	—
107	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
108	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₂₅₁	#41	1	2	yes	—
110	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
111	C₈×C₂₅₁	C₂²×C₄	1	4	yes	—
112	C₂²×C₂₅₁	—	2	1	no	—
113	C₂²×C₂₅₁	—	2	1	no	—
114	C₂²×C₂₅₁	—	2	2	no	—
115	C₂²×C₂₅₁	—	2	4	no	—
116	C₂²×C₂₅₁	—	4	2	no	—
117	C₂²×C₂₅₁	—	4	1	no	—
118	C₂²×C₂₅₁	—	4	1	no	—
119	C₂²×C₂₅₁	—	4	2	no	—
120	C₂²×C₂₅₁	—	8	4	no	—
121	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
122	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
123	C₄×C₂₅₁	—	2	2	no	—
124	C₄×C₂₅₁	C₄×D₄	1	2	yes	—
125	C₄×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
126	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
127	Dic₂₅₁	—	2	2	no	—
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	—	4	2	no	—
134	C₂×C₂₅₁	—	4	4	no	—
135	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
136	C₂×C₂₅₁	#576	1	1	yes	characteristic
137	C₂×C₂₅₁	#577	1	1	yes	characteristic
138	C₂₅₁	#2946	1	1	yes	characteristic, complemented, Sylow
139	#2946	—	251	1	no	maximal, Sylow
140	C₂×AGL(1,ℤ/8)	D₂₅₁	1	1	yes	characteristic
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	—	251	1	no	—
151	C₂×#41	C₂×D₂₅₁	1	1	yes	characteristic
152	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
153	AGL(1,ℤ/8)	Dic₂₅₁	1	8	yes	complemented
154	C₂×SD₁₆	C₂×D₂₅₁	1	2	yes	—
155	C₂×D₈	C₂×D₂₅₁	1	2	yes	—
156	C₂×M₁₆	C₂×D₂₅₁	1	1	yes	characteristic
157	C₂²×C₈	—	251	1	no	—
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	—	251	1	no	—
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	—	251	2	no	—
180	C₄×C₈	—	251	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	2	2	no	—
183	#41	C₂×Dic₂₅₁	1	4	yes	complemented
184	C₂×Q₈	C₂²×D₂₅₁	1	1	yes	characteristic
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	—	2	2	no	—
188	C₂×D₄	C₂×Dic₂₅₁	1	4	yes	complemented
189	C₂×D₄	C₂²×D₂₅₁	1	1	yes	characteristic
190	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
191	C₂²×C₄	—	2	1	no	—
192	C₂²×C₄	—	502	1	no	—
193	C₂²×C₄	—	502	2	no	—
194	C₂²×C₄	—	502	2	no	—
195	C₂²×C₄	—	502	2	no	—
196	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
197	SD₁₆	C₂×Dic₂₅₁	1	8	yes	complemented
198	D₈	C₂×Dic₂₅₁	1	8	yes	complemented
199	M₁₆	C₂×Dic₂₅₁	1	4	yes	complemented
200	C₂×C₈	—	502	2	no	—
201	C₂×C₈	—	251	1	no	—
202	C₂×C₈	—	251	1	no	—
203	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
204	C₄⋊C₄	—	502	2	no	—
205	C₄⋊C₄	—	502	1	no	—
206	C₄⋊C₄	—	251	2	no	—
207	C₄⋊C₄	—	251	1	no	—
208	C₄⋊C₄	—	251	1	no	—
209	C₄⋊C₄	—	251	1	no	—
210	C₄⋊C₄	—	251	1	no	—
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	—	251	2	no	—
216	C₄²	—	502	2	no	—
217	C₄²	—	502	1	no	—
218	C₄²	—	251	2	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	4	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	8	1	no	—
224	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
225	Q₈	—	2	1	no	—
226	Q₈	C₂²×Dic₂₅₁	1	2	yes	—
227	D₄	—	2	4	no	—
228	D₄	—	2	2	no	—
229	D₄	—	2	1	no	—
230	D₄	—	2	4	no	—
231	D₄	C₂²×Dic₂₅₁	1	2	yes	—
232	D₄	C₂²×Dic₂₅₁	1	4	yes	—
233	C₂×C₄	—	2	1	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2008	1	no	—
236	C₂×C₄	—	2008	2	no	—
237	C₂×C₄	—	2008	2	no	—
238	C₂×C₄	—	2008	2	no	—
239	C₂×C₄	—	502	1	no	—
240	C₂×C₄	—	502	1	no	—
241	C₂×C₄	—	502	2	no	—
242	C₂×C₄	—	251	2	no	—
243	C₂×C₄	—	251	2	no	—
244	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
245	C₂×C₄	D₄×D₂₅₁	1	1	yes	characteristic
246	C₂×C₄	#16860809	1	2	yes	—
247	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
248	C₈	—	502	1	no	—
249	C₈	—	502	1	no	—
250	C₈	C₂²×Dic₂₅₁	1	4	yes	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	2	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	2	no	—
258	C₂²	—	4	1	no	—
259	C₂²	—	8	4	no	—
260	C₂²	D₄×Dic₂₅₁	1	2	yes	—
261	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
262	C₄	—	2	2	no	—
263	C₄	—	1004	2	no	—
264	C₄	—	1004	1	no	—
265	C₄	—	1004	1	no	—
266	C₄	—	502	2	no	—
267	C₄	—	502	2	no	—
268	C₄	D₄×Dic₂₅₁	1	2	yes	—
269	C₄	C₂³×Dic₂₅₁	1	1	yes	characteristic
270	C₄	C₂×#16860809	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	4	2	no	—
274	C₂	C₂×D₄×Dic₂₅₁	1	1	yes	characteristic
275	C₂	#30632491	1	1	yes	characteristic
276	C₂	#30632497	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting, prime radical	C₂×AGL(1,ℤ/8)×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×AGL(1,ℤ/8)	D₂₅₁
hypocenter	C₂₅₁	#2946
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×AGL(1,ℤ/8)

Complements

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

N	H ≅ G/N	type
1. #30632170	C₂	semidirect
2. #30632162	C₂	semidirect
3. #30631799	C₂	semidirect
4. #30631788	C₂	semidirect
5. SD₁₆×Dic₂₅₁	C₂	semidirect
6. D₈×Dic₂₅₁	C₂	semidirect
7. #30631729	C₂	semidirect
8. #30631683	C₂	semidirect
9. C₂×#22548636	C₂	semidirect
10. #30631645	C₂	semidirect
13. Q₈×Dic₂₅₁	C₂²	semidirect
19. D₄×Dic₂₅₁	C₂²	semidirect
26. #22548649	C₂²	semidirect
27. #22548647	C₂²	semidirect
28. #22548639	C₂²	semidirect
29. #22548639	C₂²	semidirect
32. #22548636	C₂²	semidirect
33. #22548636	C₂²	semidirect
34. #22548625	C₂²	semidirect
35. C₈×Dic₂₅₁	C₂²	semidirect
38. AGL(1,ℤ/8)×C₂₅₁	C₄	semidirect
66. #41×C₂₅₁	C₂×C₄	semidirect
71. C₂×D₄×C₂₅₁	C₂×C₄	semidirect
76. SD₁₆×C₂₅₁	C₂×C₄	semidirect
77. D₈×C₂₅₁	C₂×C₄	semidirect
78. M₁₆×C₂₅₁	C₂×C₄	semidirect
138. C₂₅₁	#2946	semidirect
153. AGL(1,ℤ/8)	Dic₂₅₁	semidirect
183. #41	C₂×Dic₂₅₁	semidirect
188. C₂×D₄	C₂×Dic₂₅₁	semidirect
197. SD₁₆	C₂×Dic₂₅₁	semidirect
198. D₈	C₂×Dic₂₅₁	semidirect
199. M₁₆	C₂×Dic₂₅₁	semidirect

Maximal subgroups

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

subgroup	index
#30632170	2
#30632162	2
#30631799	2
#30631788	2
SD₁₆×Dic₂₅₁	2
D₈×Dic₂₅₁	2
#30631729	2
#30631683	2
C₂×#22548636	2
#30631645	2
C₂×AGL(1,ℤ/8)×C₂₅₁	2
#2946	251