Subgroups — #44559025

Order 32128.

402 conjugacy classes of subgroups.

The subgroup lattice has 222 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#22548674	C₂	1	1	yes	maximal, characteristic
2	#30631745	C₂	1	4	yes	maximal, complemented
3	#30631741	C₂	1	2	yes	maximal, complemented
4	#30631737	C₂	1	2	yes	maximal, complemented
5	#30631715	C₂	1	2	yes	maximal, complemented
6	C₂×C₂²⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	#30631645	C₂	1	2	yes	maximal, complemented
8	#407×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₂₅₁⋊C₈	C₂²	1	2	yes	—
10	#22548674	C₄	1	2	yes	—
11	#22548674	C₄	1	1	yes	characteristic
12	#22548674	C₄	1	1	yes	characteristic
13	#22548674	C₂²	1	1	yes	characteristic
14	#22548674	C₂²	1	2	yes	—
15	#22548674	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊Dic₂₅₁	C₄	1	1	yes	characteristic
17	C₂×C₄⋊Dic₂₅₁	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊Dic₂₅₁	C₄	1	2	yes	—
19	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
20	Dic₂₅₁⋊C₈	C₂²	1	4	yes	complemented
21	Dic₂₅₁⋊C₈	C₂²	1	4	yes	complemented
22	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
23	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	1	yes	characteristic
24	C₂²⋊(C₂₅₁⋊C₈)	C₂²	1	2	yes	—
25	#22548625	C₂²	1	4	yes	complemented
26	C₈×Dic₂₅₁	C₂²	1	4	yes	complemented
27	C₂³×C₄×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₂×M₁₆×C₂₅₁	C₂²	1	2	yes	—
29	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
30	C₂²⋊C₈×C₂₅₁	C₂²	1	2	yes	—
31	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
32	C₂²×Dic₂₅₁	C₂×C₄	1	2	yes	—
33	C₂×C₂₅₁⋊C₈	—	2	4	no	—
34	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
35	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
36	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
37	C₄⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
41	C₄⋊Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
42	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
43	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
44	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
45	C₂²⋊Dic₂₅₁	C₂×C₄	1	2	yes	—
46	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
47	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
48	C₄×Dic₂₅₁	C₂³	1	2	yes	—
49	C₄×Dic₂₅₁	C₂³	1	2	yes	—
50	C₂⁴×C₂₅₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₅₁	—	2	4	no	—
52	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
56	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
57	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
58	M₁₆×C₂₅₁	—	2	4	no	—
59	C₂×C₈×C₂₅₁	C₂³	1	4	yes	—
60	C₂×Dic₂₅₁	—	2	4	no	—
61	C₂×Dic₂₅₁	—	4	2	no	—
62	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
63	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
64	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
65	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
66	C₂₅₁⋊C₈	—	2	2	no	—
67	C₂₅₁⋊C₈	—	2	2	no	—
68	C₂³×C₂₅₁	—	2	2	no	—
69	C₂³×C₂₅₁	—	2	2	no	—
70	C₂³×C₂₅₁	—	4	1	no	—
71	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
72	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
73	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₄×C₂₅₁	—	2	2	no	—
75	C₂×C₄×C₂₅₁	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	2	2	no	—
77	C₂×C₄×C₂₅₁	—	2	2	no	—
78	C₂×C₄×C₂₅₁	—	2	2	no	—
79	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
81	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
82	C₂×C₄×C₂₅₁	#41	1	4	yes	—
83	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
84	C₈×C₂₅₁	—	2	4	no	—
85	C₂²×C₂₅₁	—	2	1	no	—
86	C₂²×C₂₅₁	—	2	1	no	—
87	C₂²×C₂₅₁	—	2	1	no	—
88	C₂²×C₂₅₁	—	2	1	no	—
89	C₂²×C₂₅₁	—	2	1	no	—
90	C₂²×C₂₅₁	—	2	1	no	—
91	C₂²×C₂₅₁	—	2	1	no	—
92	C₂²×C₂₅₁	—	4	4	no	—
93	C₂²×C₂₅₁	#117	1	2	yes	—
94	C₂²×C₂₅₁	#131	1	2	yes	—
95	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
96	C₄×C₂₅₁	—	2	1	no	—
97	C₄×C₂₅₁	—	2	1	no	—
98	C₄×C₂₅₁	#117	1	2	yes	—
99	C₄×C₂₅₁	C₂×#41	1	2	yes	—
100	Dic₂₅₁	—	2	2	no	—
101	Dic₂₅₁	—	2	2	no	—
102	Dic₂₅₁	—	2	2	no	—
103	Dic₂₅₁	—	2	2	no	—
104	C₂×C₂₅₁	—	2	2	no	—
105	C₂×C₂₅₁	—	2	2	no	—
106	C₂×C₂₅₁	—	4	1	no	—
107	C₂×C₂₅₁	C₂×#117	1	1	yes	characteristic
108	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
109	C₂×C₂₅₁	#568	1	1	yes	characteristic
110	C₂₅₁	#2924	1	1	yes	characteristic, complemented, Sylow
111	#2924	—	251	1	no	maximal, Sylow
112	C₂×#117	—	251	1	no	—
113	#433	—	251	4	no	—
114	#432	—	251	2	no	—
115	#431	—	251	2	no	—
116	#424	—	251	2	no	—
117	#407	D₂₅₁	1	1	yes	characteristic
118	C₂×C₂²⋊C₈	—	251	1	no	—
119	#405	—	251	2	no	—
120	C₂³×C₄	C₂×D₂₅₁	1	1	yes	characteristic
121	C₂×M₁₆	C₂×D₂₅₁	1	2	yes	—
122	C₂²×C₈	—	251	2	no	—
123	#117	—	251	1	no	—
124	#117	—	251	1	no	—
125	#117	—	251	2	no	—
126	#117	—	251	1	no	—
127	#117	—	251	1	no	—
128	#117	—	251	2	no	—
129	C₂×C₄⋊C₄	—	251	1	no	—
130	C₂×C₄⋊C₄	—	251	1	no	—
131	C₂×C₂²⋊C₄	—	251	2	no	—
132	C₂×C₄²	—	251	2	no	—
133	C₄⋊C₈	—	251	4	no	—
134	C₄⋊C₈	—	251	4	no	—
135	C₂²⋊C₈	—	251	1	no	—
136	C₂²⋊C₈	—	251	1	no	—
137	C₂²⋊C₈	—	251	2	no	—
138	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
139	C₂²⋊C₈	C₂×D₂₅₁	1	2	yes	—
140	#97	—	251	4	no	—
141	C₄×C₈	—	251	4	no	—
142	C₂⁴	C₄×D₂₅₁	1	1	yes	characteristic
143	C₂²×C₄	—	2	4	no	—
144	C₂²×C₄	—	251	2	no	—
145	C₂²×C₄	—	251	2	no	—
146	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
147	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
148	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
149	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
150	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
151	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
152	M₁₆	—	2	4	no	—
153	C₂×C₈	—	502	4	no	—
154	C₂×C₈	—	251	2	no	—
155	C₂×C₈	—	251	2	no	—
156	C₂×C₈	C₂²×D₂₅₁	1	4	yes	—
157	C₄⋊C₄	—	251	2	no	—
158	C₄⋊C₄	—	251	2	no	—
159	C₄⋊C₄	—	251	1	no	—
160	C₄⋊C₄	—	251	1	no	—
161	C₄⋊C₄	—	251	1	no	—
162	C₄⋊C₄	—	251	1	no	—
163	C₂²⋊C₄	—	251	2	no	—
164	C₂²⋊C₄	—	251	2	no	—
165	C₂²⋊C₄	—	251	2	no	—
166	C₂²⋊C₄	—	251	2	no	—
167	C₄²	—	251	2	no	—
168	C₄²	—	251	2	no	—
169	C₄²	—	251	2	no	—
170	C₄²	—	251	2	no	—
171	C₂³	—	2	2	no	—
172	C₂³	—	2	2	no	—
173	C₂³	—	4	1	no	—
174	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
175	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
176	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
177	C₂×C₄	—	2	2	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	2	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	1004	2	no	—
183	C₂×C₄	—	502	4	no	—
184	C₂×C₄	—	251	2	no	—
185	C₂×C₄	—	251	2	no	—
186	C₂×C₄	—	251	2	no	—
187	C₂×C₄	—	251	2	no	—
188	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
189	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
190	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
191	C₂×C₄	#16860809	1	4	yes	—
192	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
193	C₈	—	2	4	no	—
194	C₈	—	502	2	no	—
195	C₈	—	502	2	no	—
196	C₂²	—	2	1	no	—
197	C₂²	—	2	1	no	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	—	2	1	no	—
201	C₂²	—	2	1	no	—
202	C₂²	—	2	1	no	—
203	C₂²	—	4	4	no	—
204	C₂²	#22548671	1	2	yes	—
205	C₂²	#22548723	1	2	yes	—
206	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	2	1	no	—
209	C₄	—	502	2	no	—
210	C₄	—	502	2	no	—
211	C₄	—	502	2	no	—
212	C₄	—	502	2	no	—
213	C₄	#22548671	1	2	yes	—
214	C₄	C₂×#16860809	1	2	yes	—
215	C₂	—	2	2	no	—
216	C₂	—	2	2	no	—
217	C₂	—	4	1	no	—
218	C₂	C₂×#22548671	1	1	yes	characteristic
219	C₂	C₂×#22548723	1	1	yes	characteristic
220	C₂	#30632453	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₂₅₁
commutator, socle	C₂²×C₂₅₁	C₂³×C₄
Fitting	#407×C₂₅₁	C₂
prime radical	C₂⁴×C₂₅₁	C₂×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	#407	D₂₅₁
hypocenter	C₂₅₁	#2924
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂×C₄ ◁ #407

Complements

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

N	H ≅ G/N	type
2. #30631745	C₂	semidirect
3. #30631741	C₂	semidirect
4. #30631737	C₂	semidirect
5. #30631715	C₂	semidirect
7. #30631645	C₂	semidirect
20. Dic₂₅₁⋊C₈	C₂²	semidirect
21. Dic₂₅₁⋊C₈	C₂²	semidirect
25. #22548625	C₂²	semidirect
26. C₈×Dic₂₅₁	C₂²	semidirect
110. C₂₅₁	#2924	semidirect

Maximal subgroups

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

subgroup	index
C₂×#22548674	2
#30631745	2
#30631741	2
#30631737	2
#30631715	2
C₂×C₂²⋊(C₂₅₁⋊C₈)	2
#30631645	2
#407×C₂₅₁	2
#2924	251