Subgroups — C₂×#30631712 #44558922

Order 32128.

512 conjugacy classes of subgroups.

The subgroup lattice has 214 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#16860793	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×D₂₅₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#22548672	C₂	1	1	yes	maximal, characteristic
4	#30631712	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₄⋊(C₂₅₁⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×Dic₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×Dic₂₅₁⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄⋊C₈×C₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₄×D₂₅₁	C₂²	1	1	yes	characteristic
10	C₂×#16860793	—	2	4	no	—
11	C₂×#16860793	C₂²	1	4	yes	—
12	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
13	C₂²×C₂₅₁⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₈×D₂₅₁	—	2	4	no	—
15	C₂×C₈×D₂₅₁	C₂²	1	4	yes	—
16	#22548672	C₄	1	4	yes	—
17	#22548672	C₂²	1	4	yes	—
18	C₂×Dic₂₅₁⋊C₄	C₄	1	1	yes	characteristic
19	C₂×Dic₂₅₁⋊C₄	C₄	1	1	yes	characteristic
20	C₂×(C₂×D₂₅₁)⋊C₄	C₄	1	1	yes	characteristic
21	C₂×(C₂×D₂₅₁)⋊C₄	C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₂₅₁	C₂²	1	1	yes	characteristic
23	C₄⋊(C₂₅₁⋊C₈)	C₂²	1	4	yes	complemented
24	Dic₂₅₁⋊C₈	C₂²	1	4	yes	complemented
25	Dic₂₅₁⋊C₈	C₂²	1	4	yes	complemented
26	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
27	C₂²×C₈×C₂₅₁	C₂²	1	1	yes	characteristic
28	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
29	C₄⋊C₈×C₂₅₁	C₂²	1	4	yes	complemented
30	C₂³×D₂₅₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₂₅₁	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄×D₂₅₁	C₂×C₄	1	2	yes	—
35	C₂×C₄×D₂₅₁	D₄	1	4	yes	—
36	C₂×C₄×D₂₅₁	Q₈	1	4	yes	—
37	C₂×C₄×D₂₅₁	C₂³	1	2	yes	—
38	#16860793	—	2	8	no	—
39	C₂×C₂₅₁⋊C₈	—	2	2	no	—
40	C₂×C₂₅₁⋊C₈	—	2	2	no	—
41	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
42	C₂×C₂₅₁⋊C₈	C₂³	1	2	yes	—
43	C₈×D₂₅₁	—	2	8	no	—
44	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
45	Dic₂₅₁⋊C₄	C₂×C₄	1	4	yes	—
46	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	4	yes	—
47	(C₂×D₂₅₁)⋊C₄	C₂×C₄	1	4	yes	—
48	C₄×Dic₂₅₁	C₂×C₄	1	2	yes	—
49	C₄×Dic₂₅₁	C₂³	1	2	yes	—
50	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₄×C₂₅₁	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
53	C₂×C₈×C₂₅₁	—	2	2	no	—
54	C₂×C₈×C₂₅₁	—	2	2	no	—
55	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
56	C₂×C₈×C₂₅₁	C₂³	1	2	yes	—
57	C₄²×C₂₅₁	C₂×C₄	1	2	yes	—
58	C₄²×C₂₅₁	C₂³	1	2	yes	—
59	C₂²×D₂₅₁	—	2	2	no	—
60	C₂²×D₂₅₁	—	2	2	no	—
61	C₂²×D₂₅₁	C₄⋊C₄	1	4	yes	—
62	C₂²×D₂₅₁	C₂²×C₄	1	2	yes	—
63	C₂×Dic₂₅₁	—	2	2	no	—
64	C₂×Dic₂₅₁	—	2	2	no	—
65	C₂×Dic₂₅₁	C₄⋊C₄	1	4	yes	—
66	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
67	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
68	C₂×Dic₂₅₁	C₂²×C₄	1	2	yes	—
69	C₄×D₂₅₁	C₄⋊C₄	1	8	yes	—
70	C₄×D₂₅₁	C₂×D₄	1	4	yes	—
71	C₄×D₂₅₁	C₂×Q₈	1	4	yes	—
72	C₂₅₁⋊C₈	—	2	2	no	—
73	C₂₅₁⋊C₈	—	2	2	no	—
74	C₂³×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₄×C₂₅₁	—	2	2	no	—
76	C₂×C₄×C₂₅₁	—	2	2	no	—
77	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
78	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
79	C₂×C₄×C₂₅₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
81	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
82	C₂×C₄×C₂₅₁	C₂⁴	1	1	yes	characteristic
83	C₈×C₂₅₁	—	2	2	no	—
84	C₈×C₂₅₁	—	2	2	no	—
85	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
86	C₂²×C₂₅₁	#131	1	2	yes	—
87	C₂²×C₂₅₁	#131	1	2	yes	—
88	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×D₂₅₁	—	2	2	no	—
90	C₂×D₂₅₁	—	2	2	no	—
91	C₂×D₂₅₁	—	2	8	no	—
92	C₂×D₂₅₁	C₂×C₄⋊C₄	1	4	yes	—
93	C₄×C₂₅₁	—	2	2	no	—
94	C₄×C₂₅₁	—	2	2	no	—
95	C₄×C₂₅₁	C₂×C₄⋊C₄	1	2	yes	—
96	C₄×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
97	C₄×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
98	Dic₂₅₁	—	2	2	no	—
99	Dic₂₅₁	—	2	2	no	—
100	Dic₂₅₁	C₂×C₄⋊C₄	1	4	yes	—
101	C₂×C₂₅₁	#424	1	4	yes	complemented
102	C₂×C₂₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
104	C₂×C₂₅₁	C₂×#131	1	1	yes	characteristic
105	D₂₅₁	—	2	4	no	—
106	C₂₅₁	C₂×#424	1	1	yes	characteristic, complemented, Sylow
107	C₂×#424	—	251	1	no	maximal, Sylow
108	C₂²×M₁₆	—	251	1	no	—
109	C₂³×C₈	—	251	1	no	—
110	C₂×#117	—	251	1	no	—
111	#424	—	251	8	no	—
112	C₂×C₄⋊C₈	—	251	1	no	—
113	C₂×C₄⋊C₈	—	251	1	no	—
114	C₂×C₄⋊C₈	—	251	1	no	—
115	C₂×C₄⋊C₈	D₂₅₁	1	1	yes	characteristic, complemented
116	C₂³×C₄	—	251	1	no	—
117	C₂×M₁₆	—	502	4	no	—
118	C₂×M₁₆	—	251	4	no	—
119	C₂²×C₈	—	502	4	no	—
120	C₂²×C₈	—	251	1	no	—
121	C₂²×C₈	—	251	1	no	—
122	C₂²×C₈	—	251	4	no	—
123	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
124	C₂²×C₈	C₂×D₂₅₁	1	1	yes	characteristic
125	#117	—	251	4	no	—
126	#117	—	251	4	no	—
127	C₂×C₄⋊C₄	—	251	1	no	—
128	C₂×C₄⋊C₄	—	251	1	no	—
129	C₂×C₂²⋊C₄	—	251	1	no	—
130	C₂×C₂²⋊C₄	—	251	1	no	—
131	C₂×C₄²	—	251	1	no	—
132	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
133	C₄⋊C₈	—	251	4	no	—
134	C₄⋊C₈	—	251	4	no	—
135	C₄⋊C₈	—	251	4	no	—
136	C₄⋊C₈	C₂×D₂₅₁	1	4	yes	complemented
137	C₂⁴	—	251	1	no	—
138	C₂²×C₄	—	251	1	no	—
139	C₂²×C₄	—	251	1	no	—
140	C₂²×C₄	—	251	1	no	—
141	C₂²×C₄	—	251	2	no	—
142	C₂²×C₄	—	251	4	no	—
143	C₂²×C₄	—	251	4	no	—
144	C₂²×C₄	—	251	2	no	—
145	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
146	C₂²×C₄	C₄×D₂₅₁	1	1	yes	characteristic
147	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
148	M₁₆	—	502	8	no	—
149	C₂×C₈	—	2	2	no	—
150	C₂×C₈	—	2	2	no	—
151	C₂×C₈	—	502	8	no	—
152	C₂×C₈	—	502	2	no	—
153	C₂×C₈	—	502	2	no	—
154	C₂×C₈	—	251	2	no	—
155	C₂×C₈	—	251	2	no	—
156	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
157	C₂×C₈	C₂²×D₂₅₁	1	2	yes	—
158	C₄⋊C₄	—	251	4	no	—
159	C₄⋊C₄	—	251	4	no	—
160	C₂²⋊C₄	—	251	4	no	—
161	C₂²⋊C₄	—	251	4	no	—
162	C₄²	—	251	2	no	—
163	C₄²	—	251	2	no	—
164	C₄²	C₄×D₂₅₁	1	2	yes	—
165	C₄²	C₂²×D₂₅₁	1	2	yes	—
166	C₂³	—	502	2	no	—
167	C₂³	—	502	2	no	—
168	C₂³	—	251	2	no	—
169	C₂³	—	251	4	no	—
170	C₂³	C₂×C₄×D₂₅₁	1	1	yes	characteristic
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	—	2	2	no	—
173	C₂×C₄	—	502	2	no	—
174	C₂×C₄	—	502	2	no	—
175	C₂×C₄	—	251	2	no	—
176	C₂×C₄	—	251	2	no	—
177	C₂×C₄	—	251	2	no	—
178	C₂×C₄	—	251	4	no	—
179	C₂×C₄	—	251	4	no	—
180	C₂×C₄	—	251	4	no	—
181	C₂×C₄	—	251	8	no	—
182	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
183	C₂×C₄	C₂×C₄×D₂₅₁	1	2	yes	—
184	C₂×C₄	C₂×C₄×D₂₅₁	1	1	yes	characteristic
185	C₂×C₄	D₄×D₂₅₁	1	2	yes	—
186	C₂×C₄	Q₈×D₂₅₁	1	2	yes	—
187	C₂×C₄	C₂³×D₂₅₁	1	1	yes	characteristic
188	C₈	—	2	2	no	—
189	C₈	—	2	2	no	—
190	C₈	—	502	2	no	—
191	C₈	—	502	2	no	—
192	C₂²	—	502	2	no	—
193	C₂²	—	502	2	no	—
194	C₂²	—	502	8	no	—
195	C₂²	—	251	4	no	—
196	C₂²	C₄⋊C₄×D₂₅₁	1	2	yes	—
197	C₂²	#22548721	1	2	yes	—
198	C₂²	#22548723	1	2	yes	—
199	C₂²	C₂²×C₄×D₂₅₁	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	2	2	no	—
202	C₄	—	502	2	no	—
203	C₄	—	502	2	no	—
204	C₄	—	251	4	no	—
205	C₄	C₄⋊C₄×D₂₅₁	1	2	yes	—
206	C₄	C₂×D₄×D₂₅₁	1	1	yes	characteristic
207	C₄	C₂×Q₈×D₂₅₁	1	1	yes	characteristic
208	C₂	—	502	4	no	—
209	C₂	#30631712	1	4	yes	direct product factor
210	C₂	C₂×C₄⋊C₄×D₂₅₁	1	1	yes	characteristic
211	C₂	C₂×#22548721	1	1	yes	characteristic
212	C₂	C₂×#22548723	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₂₅₁
Fitting	C₂×C₄⋊C₈×C₂₅₁	C₂
socle	C₂³×C₂₅₁	C₂²×C₄
prime radical	C₂³×D₂₅₁	C₂×C₄
cosocle	C₂×C₄×C₂₅₁	C₂⁴
hypercenter	C₂×C₄⋊C₈	D₂₅₁
hypocenter	C₂₅₁	C₂×#424
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

Complements

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

N	H ≅ G/N	type
4. #30631712	C₂	direct (N × H)
5. C₂×C₄⋊(C₂₅₁⋊C₈)	C₂	semidirect
6. C₂×Dic₂₅₁⋊C₈	C₂	semidirect
7. C₂×Dic₂₅₁⋊C₈	C₂	semidirect
8. C₂×C₄⋊C₈×C₂₅₁	C₂	semidirect
23. C₄⋊(C₂₅₁⋊C₈)	C₂²	semidirect
24. Dic₂₅₁⋊C₈	C₂²	semidirect
25. Dic₂₅₁⋊C₈	C₂²	semidirect
29. C₄⋊C₈×C₂₅₁	C₂²	semidirect
101. C₂×C₂₅₁	#424	semidirect
106. C₂₅₁	C₂×#424	semidirect
115. C₂×C₄⋊C₈	D₂₅₁	semidirect
136. C₄⋊C₈	C₂×D₂₅₁	semidirect
209. C₂	#30631712	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#16860793	2
C₂²×C₈×D₂₅₁	2
C₂×#22548672	2
#30631712	2
C₂×C₄⋊(C₂₅₁⋊C₈)	2
C₂×Dic₂₅₁⋊C₈	2
C₂×C₄⋊C₈×C₂₅₁	2
C₂×#424	251

Subgroups — C2×#30631712 #44558922

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#30631712 #44558922