Subgroups — #44559191

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×#22548741	C₂	1	1	yes	maximal, characteristic
2	#30632168	C₂	1	1	yes	maximal, characteristic, complemented
3	#30632162	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631796	C₂	1	2	yes	maximal, complemented
5	#30631784	C₂	1	2	yes	maximal, complemented
6	#30631778	C₂	1	2	yes	maximal, complemented
7	#30631767	C₂	1	2	yes	maximal, complemented
8	#30631727	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631684	C₂	1	1	yes	maximal, characteristic, complemented
10	#30631642	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#102×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	1	yes	characteristic
13	#22548741	C₄	1	8	yes	complemented
14	C₂×#16860798	C₂²	1	2	yes	—
15	C₂×#16860795	C₂²	1	2	yes	—
16	C₂×#16860794	C₂²	1	1	yes	characteristic
17	#22548680	C₂²	1	2	yes	complemented
18	D₄×Dic₂₅₁	—	2	1	no	—
19	D₄×Dic₂₅₁	—	2	1	no	—
20	D₄×Dic₂₅₁	C₂²	1	2	yes	—
21	D₄×Dic₂₅₁	C₂²	1	2	yes	—
22	#22548677	C₂²	1	2	yes	complemented
23	C₄×C₂²⋊D₂₅₁	—	2	2	no	—
24	#22548674	C₂²	1	1	yes	characteristic
25	#22548674	C₂²	1	1	yes	characteristic
26	#22548673	—	2	1	no	—
27	C₄⋊C₄×D₂₅₁	—	2	1	no	—
28	C₂×C₂²⋊Dic₂₅₁	—	2	2	no	—
29	#22548648	C₂²	1	2	yes	complemented
30	#22548646	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	1	yes	characteristic, complemented
34	#22548637	C₂²	1	1	yes	characteristic, complemented
35	#22548625	C₂²	1	2	yes	complemented
36	C₈×Dic₂₅₁	C₂²	1	2	yes	complemented
37	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
38	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
39	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
40	#102×C₂₅₁	C₂²	1	2	yes	—
41	#102×C₂₅₁	C₂²	1	2	yes	—
42	#16860810	—	2	2	no	—
43	#16860810	C₂×C₄	1	4	yes	complemented
44	C₂×Dic₅₀₂	C₂³	1	1	yes	characteristic
45	C₂×D₁₀₀₄	C₂³	1	1	yes	characteristic
46	C₂×C₂²⋊D₂₅₁	—	2	1	no	—
47	C₂²×Dic₂₅₁	—	2	2	no	—
48	C₂²×Dic₂₅₁	—	2	2	no	—
49	C₂×C₄×D₂₅₁	—	2	2	no	—
50	C₂×C₄×D₂₅₁	—	2	1	no	—
51	#16860798	C₂×C₄	1	8	yes	complemented
52	#16860795	C₂×C₄	1	8	yes	complemented
53	#16860794	C₂×C₄	1	4	yes	complemented
54	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
55	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
56	C₄⋊Dic₂₅₁	D₄	1	1	yes	characteristic
57	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
58	C₄⋊Dic₂₅₁	C₂³	1	1	yes	characteristic
59	Dic₂₅₁⋊C₄	—	2	2	no	—
60	C₂²⋊Dic₂₅₁	—	2	2	no	—
61	C₂²⋊Dic₂₅₁	—	2	2	no	—
62	C₂²⋊Dic₂₅₁	—	2	1	no	—
63	C₂²⋊Dic₂₅₁	D₄	1	2	yes	—
64	(C₂×D₂₅₁)⋊C₄	—	2	2	no	—
65	C₄×Dic₂₅₁	—	2	2	no	—
66	C₄×Dic₂₅₁	—	2	1	no	—
67	C₄×Dic₂₅₁	C₂³	1	2	yes	—
68	C₂⁴×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	2	yes	—
73	C₂²×C₄×C₂₅₁	—	2	1	no	—
74	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₅₁	—	2	2	no	—
76	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
77	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
78	C₄⋊C₄×C₂₅₁	—	2	1	no	—
79	C₄⋊C₄×C₂₅₁	C₂³	1	2	yes	—
80	C₂²×D₂₅₁	—	2	1	no	—
81	Dic₅₀₂	—	2	1	no	—
82	Dic₅₀₂	C₂²×C₄	1	2	yes	—
83	D₁₀₀₄	—	2	1	no	—
84	D₁₀₀₄	C₂²×C₄	1	2	yes	—
85	C₂²⋊D₂₅₁	—	2	4	no	—
86	C₂×Dic₂₅₁	—	2	1	no	—
87	C₂×Dic₂₅₁	—	2	1	no	—
88	C₂×Dic₂₅₁	—	2	1	no	—
89	C₂×Dic₂₅₁	—	8	2	no	—
90	C₂×Dic₂₅₁	—	8	2	no	—
91	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
92	C₂×Dic₂₅₁	C₂×D₄	1	2	yes	—
93	C₄×D₂₅₁	—	2	4	no	—
94	C₄×D₂₅₁	—	8	2	no	—
95	C₂₅₁⋊C₈	C₂²×C₄	1	4	yes	—
96	C₂³×C₂₅₁	—	2	2	no	—
97	C₂³×C₂₅₁	—	2	4	no	—
98	C₂³×C₂₅₁	—	8	1	no	—
99	C₂³×C₂₅₁	—	8	1	no	—
100	C₂³×C₂₅₁	C₂×D₄	1	1	yes	characteristic
101	D₄×C₂₅₁	—	2	4	no	—
102	D₄×C₂₅₁	—	2	2	no	—
103	D₄×C₂₅₁	C₂²×C₄	1	4	yes	—
104	C₂×C₄×C₂₅₁	—	2	2	no	—
105	C₂×C₄×C₂₅₁	—	8	1	no	—
106	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₂₅₁	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₂₅₁	#41	1	2	yes	—
109	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
110	C₈×C₂₅₁	—	2	1	no	—
111	C₈×C₂₅₁	—	2	1	no	—
112	C₂²×C₂₅₁	—	2	1	no	—
113	C₂²×C₂₅₁	—	2	1	no	—
114	C₂²×C₂₅₁	—	2	4	no	—
115	C₂²×C₂₅₁	—	4	2	no	—
116	C₂²×C₂₅₁	—	4	2	no	—
117	C₂²×C₂₅₁	—	8	4	no	—
118	C₂²×C₂₅₁	C₄×D₄	1	2	yes	—
119	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
120	C₂×D₂₅₁	—	2	2	no	—
121	C₂×D₂₅₁	—	4	1	no	—
122	C₂×D₂₅₁	—	4	1	no	—
123	C₄×C₂₅₁	—	4	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	—	2	2	no	—
130	Dic₂₅₁	—	4	1	no	—
131	Dic₂₅₁	—	4	1	no	—
132	C₂×C₂₅₁	—	2	2	no	—
133	C₂×C₂₅₁	—	4	4	no	—
134	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
135	C₂×C₂₅₁	#576	1	1	yes	characteristic
136	C₂×C₂₅₁	#577	1	1	yes	characteristic
137	D₂₅₁	—	4	2	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	D₂₅₁	1	1	yes	characteristic, complemented
150	#405	—	251	1	no	—
151	C₂×#41	—	251	1	no	—
152	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
153	AGL(1,ℤ/8)	—	251	8	no	—
154	C₂×SD₁₆	—	251	2	no	—
155	C₂×D₈	—	251	2	no	—
156	C₂×M₁₆	—	251	1	no	—
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	—	251	1	no	—
168	C₂×C₄⋊C₄	—	502	1	no	—
169	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
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	1	no	—
176	#102	—	251	1	no	—
177	#102	C₂×D₂₅₁	1	2	yes	—
178	#102	C₂×D₂₅₁	1	2	yes	—
179	#97	—	251	2	no	—
180	C₄×C₈	—	251	2	no	—
181	C₂⁴	—	2	1	no	—
182	#41	—	502	2	no	—
183	#41	—	251	4	no	—
184	C₂×Q₈	—	251	1	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	—	502	1	no	—
188	C₂×D₄	—	251	1	no	—
189	C₂×D₄	C₄×D₂₅₁	1	4	yes	complemented
190	C₂×D₄	C₂²×D₂₅₁	1	2	yes	—
191	C₂²×C₄	—	2	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	—	251	4	no	—
200	C₂×C₈	—	2	2	no	—
201	C₂×C₈	—	251	2	no	—
202	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
203	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
204	C₄⋊C₄	—	2	1	no	—
205	C₄⋊C₄	—	502	2	no	—
206	C₄⋊C₄	—	251	1	no	—
207	C₄⋊C₄	—	251	1	no	—
208	C₄⋊C₄	—	251	1	no	—
209	C₄⋊C₄	—	251	1	no	—
210	C₄⋊C₄	C₂²×D₂₅₁	1	2	yes	—
211	C₂²⋊C₄	—	502	2	no	—
212	C₂²⋊C₄	—	502	1	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	2	no	—
220	C₂³	—	2	4	no	—
221	C₂³	—	8	1	no	—
222	C₂³	—	8	1	no	—
223	C₂³	—	502	1	no	—
224	C₂³	D₄×D₂₅₁	1	1	yes	characteristic
225	Q₈	—	502	1	no	—
226	Q₈	—	251	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	502	4	no	—
230	D₄	—	502	1	no	—
231	D₄	—	251	2	no	—
232	D₄	C₂×C₄×D₂₅₁	1	4	yes	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	8	1	no	—
235	C₂×C₄	—	2008	2	no	—
236	C₂×C₄	—	2008	2	no	—
237	C₂×C₄	—	2008	2	no	—
238	C₂×C₄	—	502	1	no	—
239	C₂×C₄	—	502	1	no	—
240	C₂×C₄	—	502	1	no	—
241	C₂×C₄	—	502	4	no	—
242	C₂×C₄	—	251	2	no	—
243	C₂×C₄	—	251	2	no	—
244	C₂×C₄	C₂×C₄×D₂₅₁	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	—	2	1	no	—
249	C₈	—	2	1	no	—
250	C₈	—	251	4	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	8	4	no	—
257	C₂²	—	1004	1	no	—
258	C₂²	—	1004	1	no	—
259	C₂²	—	502	2	no	—
260	C₂²	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
261	C₂²	C₂×D₄×D₂₅₁	1	1	yes	characteristic
262	C₄	—	4	2	no	—
263	C₄	—	1004	1	no	—
264	C₄	—	1004	1	no	—
265	C₄	—	502	2	no	—
266	C₄	—	502	2	no	—
267	C₄	—	502	2	no	—
268	C₄	(C₂²⋊D₂₅₁)⋊C₄	1	2	yes	—
269	C₄	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
270	C₄	C₂×#16860809	1	1	yes	characteristic
271	C₂	—	2	2	no	—
272	C₂	—	4	4	no	—
273	C₂	—	1004	2	no	—
274	C₂	C₂×(C₂²⋊D₂₅₁)⋊C₄	1	1	yes	characteristic
275	C₂	#30632488	1	1	yes	characteristic
276	C₂	#30632493	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₂₅₁
commutator	C₄×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×#102×C₂₅₁	C₂
socle	C₂²×C₂₅₁	C₂²×D₄
prime radical	C₂×#22548741	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×#102	D₂₅₁
hypocenter	C₂₅₁	#2946
LCS	G ▷ C₄×C₂₅₁ ▷ C₂×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#102

Complements

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

N	H ≅ G/N	type
2. #30632168	C₂	semidirect
3. #30632162	C₂	semidirect
4. #30631796	C₂	semidirect
5. #30631784	C₂	semidirect
6. #30631778	C₂	semidirect
7. #30631767	C₂	semidirect
8. #30631727	C₂	semidirect
9. #30631684	C₂	semidirect
10. #30631642	C₂	semidirect
11. C₂×#102×C₂₅₁	C₂	semidirect
13. #22548741	C₄	semidirect
17. #22548680	C₂²	semidirect
22. #22548677	C₂²	semidirect
29. #22548648	C₂²	semidirect
30. #22548646	C₂²	semidirect
31. #22548640	C₂²	semidirect
32. #22548640	C₂²	semidirect
33. #22548637	C₂²	semidirect
34. #22548637	C₂²	semidirect
35. #22548625	C₂²	semidirect
36. C₈×Dic₂₅₁	C₂²	semidirect
43. #16860810	C₂×C₄	semidirect
51. #16860798	C₂×C₄	semidirect
52. #16860795	C₂×C₄	semidirect
53. #16860794	C₂×C₄	semidirect
71. C₂×D₄×C₂₅₁	C₂×C₄	semidirect
138. C₂₅₁	#2946	semidirect
149. C₂×#102	D₂₅₁	semidirect
189. C₂×D₄	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×#22548741	2
#30632168	2
#30632162	2
#30631796	2
#30631784	2
#30631778	2
#30631767	2
#30631727	2
#30631684	2
#30631642	2
C₂×#102×C₂₅₁	2
#2946	251