Subgroups — #44559072

Order 32128.

500 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548721	C₂	1	2	yes	maximal, complemented
2	#30632234	C₂	1	1	yes	maximal, characteristic
3	#30631765	C₂	1	1	yes	maximal, characteristic, complemented
4	#30631764	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631762	C₂	1	2	yes	maximal, complemented
6	C₈×D₁₀₀₄	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₂₅₁	C₂	1	2	yes	maximal, complemented
8	#30631709	C₂	1	1	yes	maximal, characteristic, complemented
9	#30631652	C₂	1	2	yes	maximal, complemented
10	C₄×C₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#16860810	C₂²	1	2	yes	—
13	#22548721	—	2	8	no	—
14	C₂×#16860794	C₂²	1	2	yes	—
15	C₂×#16860793	C₂²	1	2	yes	—
16	C₂×#16860793	C₂²	1	2	yes	—
17	C₂×C₈×D₂₅₁	C₂²	1	2	yes	—
18	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₂₅₁	C₂²	1	1	yes	characteristic
20	#22548714	C₄	1	1	yes	characteristic
21	#22548711	C₄	1	1	yes	characteristic
22	#22548701	C₄	1	2	yes	—
23	#22548690	C₄	1	1	yes	characteristic
24	#22548688	C₄	1	2	yes	—
25	#22548686	C₄	1	1	yes	characteristic
26	C₄×D₁₀₀₄	C₂²	1	1	yes	characteristic
27	C₄×C₂²⋊D₂₅₁	C₂²	1	2	yes	—
28	C₄²×D₂₅₁	C₂²	1	1	yes	characteristic
29	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
30	Dic₂₅₁⋊C₈	C₂²	1	2	yes	—
31	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
32	(C₂×D₂₅₁)⋊C₈	C₂²	1	2	yes	—
33	(C₂×D₂₅₁)⋊C₈	C₂²	1	2	yes	—
34	C₄×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
35	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
36	C₈×Dic₂₅₁	C₂²	1	1	yes	characteristic
37	C₂²×C₈×C₂₅₁	C₂²	1	2	yes	—
38	C₄×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
39	C₄⋊C₈×C₂₅₁	C₂²	1	1	yes	characteristic
40	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
41	C₄×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
42	#16860810	—	2	4	no	—
43	#16860810	—	2	4	no	—
44	C₂×Dic₅₀₂	C₂×C₄	1	2	yes	—
45	C₂×D₁₀₀₄	C₂×C₄	1	2	yes	—
46	C₂×D₁₀₀₄	C₂×C₄	1	1	yes	characteristic
47	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	2	yes	—
48	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	2	yes	—
49	C₂×C₂²⋊D₂₅₁	C₂×C₄	1	2	yes	—
50	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₂₅₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
53	C₂×C₄×D₂₅₁	C₂³	1	1	yes	characteristic
54	#16860794	—	2	4	no	—
55	#16860793	—	2	4	no	—
56	#16860793	—	2	4	no	—
57	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
58	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
59	C₂×C₂₅₁⋊C₈	C₂³	1	1	yes	characteristic
60	C₈×D₂₅₁	—	2	2	no	—
61	C₈×D₂₅₁	—	2	4	no	—
62	C₈×D₂₅₁	D₄	1	4	yes	complemented
63	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
64	Dic₂₅₁⋊C₄	C₂×C₄	1	2	yes	—
65	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
66	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
67	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	2	yes	—
68	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
69	C₄×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
70	C₄×Dic₂₅₁	C₂³	1	1	yes	characteristic
71	C₂×D₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
73	C₂×C₈×C₂₅₁	—	2	4	no	—
74	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
75	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
76	C₂×C₈×C₂₅₁	C₂³	1	1	yes	characteristic
77	C₄⋊C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
78	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	2	yes	—
79	C₄²×C₂₅₁	C₂³	1	1	yes	characteristic
80	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
81	C₂²×D₂₅₁	C₂²×C₄	1	1	yes	characteristic
82	Dic₅₀₂	—	2	4	no	—
83	D₁₀₀₄	—	2	4	no	—
84	D₁₀₀₄	—	2	2	no	—
85	C₂²⋊D₂₅₁	—	2	4	no	—
86	C₂²⋊D₂₅₁	—	2	4	no	—
87	C₂²⋊D₂₅₁	—	2	4	no	—
88	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
89	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
90	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
91	C₂×Dic₂₅₁	C₂²×C₄	1	1	yes	characteristic
92	C₄×D₂₅₁	—	2	2	no	—
93	C₄×D₂₅₁	—	2	2	no	—
94	C₄×D₂₅₁	—	2	2	no	—
95	C₄×D₂₅₁	—	2	2	no	—
96	C₄×D₂₅₁	C₂×D₄	1	2	yes	—
97	C₄×D₂₅₁	#41	1	2	yes	—
98	C₂₅₁⋊C₈	—	2	2	no	—
99	C₂₅₁⋊C₈	—	2	1	no	—
100	C₂₅₁⋊C₈	C₂×D₄	1	2	yes	—
101	C₂³×C₂₅₁	C₂²×C₄	1	2	yes	—
102	D₄×C₂₅₁	—	2	2	no	—
103	C₂×C₄×C₂₅₁	—	2	2	no	—
104	C₂×C₄×C₂₅₁	—	2	2	no	—
105	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
106	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
107	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
109	C₈×C₂₅₁	—	2	2	no	—
110	C₈×C₂₅₁	—	2	1	no	—
111	C₈×C₂₅₁	C₂×D₄	1	2	yes	—
112	C₂²×C₂₅₁	—	2	2	no	—
113	C₂²×C₂₅₁	—	2	2	no	—
114	C₂²×C₂₅₁	—	2	2	no	—
115	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
116	C₂×D₂₅₁	—	2	1	no	—
117	C₂×D₂₅₁	—	2	1	no	—
118	C₂×D₂₅₁	—	2	2	no	—
119	C₂×D₂₅₁	—	2	2	no	—
120	C₂×D₂₅₁	—	2	2	no	—
121	C₂×D₂₅₁	C₄×D₄	1	2	yes	—
122	C₄×C₂₅₁	—	2	2	no	—
123	C₄×C₂₅₁	—	2	1	no	—
124	C₄×C₂₅₁	#131	1	2	yes	complemented
125	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
126	C₄×C₂₅₁	C₂×#41	1	1	yes	characteristic
127	Dic₂₅₁	—	2	2	no	—
128	Dic₂₅₁	—	2	1	no	—
129	Dic₂₅₁	C₄×D₄	1	2	yes	—
130	Dic₂₅₁	#131	1	2	yes	complemented
131	C₂×C₂₅₁	—	4	2	no	—
132	C₂×C₂₅₁	C₂×C₄×D₄	1	1	yes	characteristic
133	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
134	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
135	D₂₅₁	—	2	2	no	—
136	D₂₅₁	—	4	2	no	—
137	C₂₅₁	#2931	1	1	yes	characteristic, complemented, Sylow
138	#2931	—	251	1	no	maximal, Sylow
139	C₂×#131	—	251	2	no	—
140	#532	—	251	1	no	—
141	#436	—	251	1	no	—
142	#436	—	251	1	no	—
143	#436	—	251	2	no	—
144	C₈×D₄	—	251	2	no	—
145	C₈×D₄	—	251	1	no	—
146	C₈×D₄	D₂₅₁	1	1	yes	characteristic, complemented
147	#423	—	251	1	no	—
148	#408	—	251	2	no	—
149	C₂×C₄×C₈	—	251	1	no	—
150	C₂×#41	—	251	2	no	—
151	#131	—	502	8	no	—
152	C₂×M₁₆	—	251	2	no	—
153	C₂×M₁₆	—	251	2	no	—
154	C₂×M₁₆	—	251	2	no	—
155	C₂²×C₈	—	251	1	no	—
156	C₂²×C₈	—	251	2	no	—
157	C₂²×C₈	—	251	1	no	—
158	C₂²×C₈	C₂×D₂₅₁	1	2	yes	—
159	C₄⋊Q₈	—	251	1	no	—
160	#127	—	251	1	no	—
161	#124	—	251	2	no	—
162	#121	—	251	1	no	—
163	#121	—	251	1	no	—
164	#121	—	251	2	no	—
165	C₄×D₄	—	251	2	no	—
166	C₄×D₄	—	251	1	no	—
167	C₄×D₄	C₂×D₂₅₁	1	1	yes	characteristic
168	C₂×C₄²	—	251	1	no	—
169	C₄⋊C₈	—	251	2	no	—
170	C₄⋊C₈	—	251	1	no	—
171	C₄⋊C₈	C₂×D₂₅₁	1	1	yes	characteristic
172	C₂²⋊C₈	—	251	2	no	—
173	C₂²⋊C₈	—	251	2	no	—
174	C₂²⋊C₈	—	251	2	no	—
175	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
176	C₄×C₈	—	251	1	no	—
177	C₄×C₈	—	251	1	no	—
178	C₄×C₈	—	251	1	no	—
179	C₄×C₈	C₂×D₂₅₁	1	1	yes	characteristic
180	#41	—	502	4	no	—
181	#41	—	502	4	no	—
182	C₂×Q₈	—	251	2	no	—
183	C₂×D₄	—	251	2	no	—
184	C₂×D₄	—	251	2	no	—
185	C₂×D₄	—	251	2	no	—
186	C₂×D₄	—	251	1	no	—
187	C₂×D₄	—	251	2	no	—
188	C₂×D₄	C₄×D₂₅₁	1	1	yes	characteristic
189	C₂²×C₄	—	251	1	no	—
190	C₂²×C₄	—	251	1	no	—
191	C₂²×C₄	—	251	2	no	—
192	C₂²×C₄	—	251	1	no	—
193	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
194	M₁₆	—	502	4	no	—
195	M₁₆	—	502	4	no	—
196	M₁₆	—	502	4	no	—
197	C₂×C₈	—	2	4	no	—
198	C₂×C₈	—	502	2	no	—
199	C₂×C₈	—	502	4	no	—
200	C₂×C₈	—	251	1	no	—
201	C₂×C₈	—	251	2	no	—
202	C₂×C₈	—	251	1	no	—
203	C₂×C₈	—	251	4	no	—
204	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
205	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
206	C₂×C₈	C₂²×D₂₅₁	1	1	yes	characteristic
207	C₄⋊C₄	—	251	2	no	—
208	C₄⋊C₄	—	251	1	no	—
209	C₄⋊C₄	C₄×D₂₅₁	1	1	yes	characteristic
210	C₂²⋊C₄	—	251	2	no	—
211	C₂²⋊C₄	—	251	2	no	—
212	C₂²⋊C₄	—	251	2	no	—
213	C₂²⋊C₄	C₄×D₂₅₁	1	2	yes	—
214	C₄²	—	251	1	no	—
215	C₄²	—	251	1	no	—
216	C₄²	—	251	1	no	—
217	C₄²	C₂²×D₂₅₁	1	1	yes	characteristic
218	C₂³	—	251	2	no	—
219	C₂³	—	251	1	no	—
220	C₂³	C₂×C₄×D₂₅₁	1	2	yes	—
221	Q₈	—	502	4	no	—
222	D₄	—	2	2	no	—
223	D₄	—	502	4	no	—
224	D₄	—	502	4	no	—
225	D₄	—	502	4	no	—
226	D₄	—	502	4	no	—
227	D₄	—	502	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	502	2	no	—
231	C₂×C₄	—	502	2	no	—
232	C₂×C₄	—	502	2	no	—
233	C₂×C₄	—	502	2	no	—
234	C₂×C₄	—	251	1	no	—
235	C₂×C₄	—	251	2	no	—
236	C₂×C₄	—	251	1	no	—
237	C₂×C₄	—	251	1	no	—
238	C₂×C₄	—	251	2	no	—
239	C₂×C₄	—	251	2	no	—
240	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
241	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
242	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
243	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	1	no	—
246	C₈	—	502	2	no	—
247	C₈	—	502	1	no	—
248	C₈	—	251	2	no	—
249	C₈	D₄×D₂₅₁	1	2	yes	—
250	C₂²	—	2	2	no	—
251	C₂²	—	2	2	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	502	2	no	—
254	C₂²	—	502	2	no	—
255	C₂²	—	502	2	no	—
256	C₂²	—	502	1	no	—
257	C₂²	—	502	1	no	—
258	C₂²	—	251	2	no	—
259	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	2	no	—
262	C₄	—	502	1	no	—
263	C₄	—	502	2	no	—
264	C₄	—	251	2	no	—
265	C₄	—	251	2	no	—
266	C₄	#22548721	1	2	yes	complemented
267	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
268	C₄	#41×D₂₅₁	1	1	yes	characteristic
269	C₂	—	4	2	no	—
270	C₂	—	1004	2	no	—
271	C₂	—	502	2	no	—
272	C₂	C₄×D₄×D₂₅₁	1	1	yes	characteristic
273	C₂	#131×D₂₅₁	1	1	yes	characteristic
274	C₂	C₂×#22548721	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂²×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₈×D₄×C₂₅₁	C₂
prime radical	#30632234	C₂
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₈×D₄	D₂₅₁
hypocenter	C₂₅₁	#2931
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₈ ◁ C₈×D₄

Complements

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

N	H ≅ G/N	type
1. C₂×#22548721	C₂	semidirect
3. #30631765	C₂	semidirect
4. #30631764	C₂	semidirect
5. #30631762	C₂	semidirect
6. C₈×D₁₀₀₄	C₂	semidirect
7. C₈×C₂²⋊D₂₅₁	C₂	semidirect
8. #30631709	C₂	semidirect
9. #30631652	C₂	semidirect
10. C₄×C₈×D₂₅₁	C₂	semidirect
11. C₈×D₄×C₂₅₁	C₂	semidirect
62. C₈×D₂₅₁	D₄	semidirect
124. C₄×C₂₅₁	#131	semidirect
130. Dic₂₅₁	#131	semidirect
137. C₂₅₁	#2931	semidirect
146. C₈×D₄	D₂₅₁	semidirect
266. C₄	#22548721	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548721	2
#30632234	2
#30631765	2
#30631764	2
#30631762	2
C₈×D₁₀₀₄	2
C₈×C₂²⋊D₂₅₁	2
#30631709	2
#30631652	2
C₄×C₈×D₂₅₁	2
C₈×D₄×C₂₅₁	2
#2931	251